Stochastic Differential Equations: Theory and Applications, a Volume in Honor of Professor Boris L Rozovskii (Interdisciplinary Mathematical Sciences)

Stochastic Differential Equations: Theory and Appllications

INTERDISCIPLINARY MATHEMATICAL SCIENCES Series Editor: Jinqiao Duan (Illinois Inst. of Tech., USA) Editorial Board: Ludwig Arnold, Roberto Camassa, Peter Constantin, Charles Doering, Paul Fischer, Andrei V. Fursikov, Fred R. McMorris, Daniel Schertzer, Bjorn Schmalfuss, Xiangdong Ye, and Jerzy Zabczyk Published Vol. 1: Global Attractors of Nonautonomous Dissipative Dynamical Systems David N. Cheban Vol. 2: Stochastic Differential Equations: Theory and Applications A Volume in Honor of Professor Boris L. Rozovskii eds. Peter H. Baxendale & Sergey V. Lototsky Vol. 3: Amplitude Equations for Stochastic Partial Differential Equations Dirk Blömker Vol. 4: Mathematical Theory of Adaptive Control Vladimir G. Sragovich Vol. 5: The Hilbert–Huang Transform and Its Applications Norden E. Huang & Samuel S. P. Shen Vol. 6: Meshfree Approximation Methods with MATLAB Gregory E. Fasshauer

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Interdiscliplinary Mathematical Sciences - Vol. 2

A Volume in Honor of Professor Boris L. Rozovskii

Stochastic Differential

Equations: Theory and Applications Editors

Peter H. Baxendale

Sergey V. Lototsky University of Southern California, USA

World Scientific NEW JERSEY . LONDON . SINGAPORE . BEIJING . SHANGHAI . HONG KONG . TAIPEI . CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Interdisciplinary Mathematical Sciences — Vol. 2 STOCHASTIC DIFFERENTIAL EQUATIONS: THEORY AND APPLICATIONS A Volume in Honor of Professor Boris L. Rozovskii Copyright © 2007 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN-13 978-981-270-662-1 ISBN-10 981-270-662-3

Printed in Singapore.

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This festschrift volume is dedicated to Boris Rozovskii on his 60th birthday. The first paper in the volume, Stochastic Evolution Equations by N. V. Krylov and B. L. Rozovskii, was originally published in Russian in 1979 (Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Vol, 14, pp. 71–146). The English translation was first published in the Journal of Soviet Mathematics, Vol. c 14, pp. 1233–1277, 1981, Plenum Publishing Co. We are very grateful to the current copyright holder, Springer, for the permission to include the paper in the volume. After more than a quarter-century, this paper remains a standard reference in the field of stochastic partial differential equations (SPDEs) and continues to attract attention of mathematicians of all generations, because, together with a short but thorough introduction to SPDEs, it presents a number of optimal and essentially non-improvable results about solvability for a large class of both linear and non-linear equations. The other papers in this volume were specially written for the occasion. The 14 contributions deal with a wide range of topics in the theory and applications of stochastic differential equations, both ordinary and with partial derivatives. Eight of the contributions are related to stochastic partial differential equations. D. Bl¨ omker and J. Duan investigate behavior of the mean energy and correlation function for the Burgers equation with various types of random perturbation. L. Borcea, G. Papanicolaou, and C. Tsogka study asymptotics of the space-time Wigner transform for the stochastic Schr¨ odinger equation and apply the results to broadband array imaging in random media. A. de Bouard and A. Debussche establish existence and uniqueness of a global square-integrable solution of the stochastic Korteweg-de Vries equation with multiplicative noise. Z. Brze´zniak and L. Debbi establish global existence and uniqueness of a mild solution for a fractional Burgers equation with multiplicative space-time white noise. F. Flandoli and M. Romito study regularity properties of a transition semigroup associated with the NavierStokes equation driven by a non-degenerate additive space-time noise. I. Gy¨ ongy and A. Millet establish the rate of convergence of the implicit Euler scheme for a class of nonlinear SPDEs. N.V. Krylov proves the maximum principle for a large class of linear stochastic parabolic equations and uses the result to study spatial regularity of the solution for equations on the half-line. G. Da Prato investigates

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the Kolmogorov equation associated with a class of non-linear stochastic parabolic equations. Six of the contributions are related to stochastic ordinary differential equations. A. Cadenillas, J. Cvitanic, and F. Zapatero study a stochastic control problem that models the use of stock options for executive compensation. P. Chigansky and R. Liptser establish the large deviations principle for a class of stochastic equations with rapidly growing coefficients and a possibly degenerate diffusion. D. Crisan and S. Ghazali study numerical approximation of the probability distribution of the solution at a fixed time for a class of stochastic equations in the Stratonovich form and apply the results to the nonlinear filtering problem. L. Decreusefond and D. Nualart show that the solutions of stochastic equation driven by a fractional Brownian motion with Hurst parameter bigger than 1/2 generate a flow of homeomorphisms. Yu. A. Kutoyants reviews recent results on statistical inference for a stochastic equation with delay when the delay parameter is unknown. R. Mikulevicius and H. Pragarauskas investigate a linear integro-differential equation connected with jump-diffusion processes. Preparation of this volume was a joint effort by a number of people. We are very grateful to J. Duan, who initiated the project; to Rok Ting Tan and Yubing Zhai, who managed the project at World Scientific; to Inge Weijman and Berendina van Straalen, who processed our copyright clearance at Springer; to Abhinav Guru, who helped to prepare the TEX file of the paper by Krylov and Rozovskii. Our special gratitude goes to all the contributors to this volume and all the referees who carefully reviewed the submitted papers and assured the highest quality of all contributions. P. Baxendale S. Lototsky

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Boris Rozovskii In the life of every mathematician there are usually several key events that facilitate a successful career. For Boris Rozovkii, one such event happened in 1962, when, shortly after entering the Odessa State University, he got a special invitation to transfer to the Faculty of Mechanics and Mathematics (the famous MechMat) at the Moscow State University, and immediately found himself at the center of the Soviet and, to some extent, the world mathematics. The undergraduate program of study at MechMat leads to an analog of Master of Science degree and requires the students to engage in serious research. Boris started to work with I. Girsanov, of the Girsanov Theorem fame. After the tragic death of I. Girsanov in an avalanche, Boris became a student of A. N. Shiryev, and thus an “academic grandson” of A. N. Kolmogorov. While Kolmogorov did not participate directly in the upbringing of his “grandson,” his influence through lectures and seminars was strong and beneficial. From the start, Boris worked at the junction of stochastic analysis and statistics, and with considerable success. His undergraduate work on change detection in a Poisson process was praised by A. N. Kolmogorov, which is probably the highest reward a young mathematician could have received at the time. His Ph.D. dissertation, On stochastic equations arising in filtering of Markov processes, published in 1972, became one of the three works that formed the foundation of the modern theory of stochastic partial differential equations (SPDEs); the other two, by E. Pardoux and M. Viot, appeared in 1975 and were also Ph.D. dissertations. In 1972 Boris became a faculty member at the Moscow Institute of Advanced Studies for Engineers and Managers in Chemical Industry (MIASCME). This institution had a strong statistical group, which Boris joined. Working with chemical engineers was a valuable experience, as it taught him how to effectively communicate mathematics to non-mathematicians and provided ample opportunity to learn about various applications of mathematics. This experience came in very handy later, when Boris became the director of the Center for Applied Mathematical Sciences at the University of Southern California (USC). Meanwhile, Boris continued to work, both independently and with N. V. Krylov, on nonlinear filtering of diffusion processes and the general theory of SPDEs. The result was a complete L2 theory for linear SPDEs, as well as several key results for nonlinear equations. The linear theory was summarized in the book Stochastic Evolution Systems (1983 in Russian, 1990 in English). While the Russian edition looked more like a brochure than a book, the English translation, with extra material on Malliavin Calculus, was in hard cover and had a much more respectable appearance. The inside of the English translation was a different story; because of a technical glitch, the text contained an enormous number of typos, some of them rather serious. In the end, though, this proved to be a blessing in disguise, as reading the book and correcting the typos became a rite of passage for every student interested in SPDEs; correcting all those typos makes the reading much

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more challenging and rewarding. In 1984 Boris was granted the second doctoral degree, Doctor of Science in Physics and Mathematics, by the Vilnius State University. This degree was an important benchmark for a scholar in the former Soviet Union. In 1988 Boris emigrated to the USA. After a short tenure at the University of North Caroline, Charlotte, he moved to Los Angeles, CA, to become director of the Center for Applied Mathematical Sciences (CAMS) at the University of Southern California. Nonlinear filtering and SPDEs are connected with a large number of mathematical problems, both applied and theoretical, and, while at USC, Boris explored these connections to the fullest. As a Director of CAMS, Boris was very successful in attracting extensive external research funds and became involved in numerous projects in computational fluid dynamics, finance, network security, physical oceanography, target recognition and tracking, and many other areas. Applied problem not only help to write a successful grant proposal, but can also lead to new and unexpected theoretical developments. For example, physical oceanography, the subject of the first big grant from the Office of Naval Research Boris secured as the director of CAMS, lead to a new estimation theory for SPDEs, developed in his joint papers with M. Huebner, R. Khasminskii, S. Lototsky, and L. Piterbarg. Similarly, numerical computation of the optimal nonlinear filter, the key component of many successful proposals, lead to a new way of analyzing linear and nonlinear SPDEs using Wiener chaos decomposition. Yet another direction of his research at USC was a comprehensive theory of absolute continuity and singularity of measures generated by solutions of nonlinear SPDEs, developed jointly with R. Mikulevicius. B. L. Rozovskii has published in over 25 different journals with over 40 collaborators. His Erd˝ os number is 4, via several connections, such as A. N. Shiryaev→L. A. Shepp→P. Frankl; R. Z. Khasminskii→O. Zeitouni→P. Diaconis; P. Baxendale→D. Stroock→P. Diaconis. Not long ago Boris became a grandfather (his granddaughter Mia was born in 2005), and shortly after that, an “academic grandfather.” Still, he shows no signs of slowing down, publishing more papers in the past year than in the previous five. In fact, his recent move to the East Coast suggests that he is at the peak of his mathematical career. Main landmarks of B. L. Rozovskii’s scientific career Boris L’vovich Rozovskii was born on June 8, 1945, in Odessa, Ukraine. Currently, he is Professor at the Division of Applied Mathematics, Brown University. 1968 Graduated from the Department of Mechanics and Mathematics, Moscow State University, with Master of Science degree in Probability and Statistics. 1972 Ph.D. in Physical and Mathematical Sciences, Moscow State University. 1984 Doctor of Science in Physics and Mathematics.

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1985 Professor and Head of the Informatics Laboratory, Moscow Institute of Advanced Studies for Engineers and Managers in Chemical Industry. 1991 Professor, Department of Mathematics, USC. 1992 Director, Center for Applied Mathematical Sciences, University of Southern California. 2006 Professor, Division of Applied Mathematics, Brown University. Graduate students of B. L. Rozovskii M. Huebner (USC, 1993) K. Owens (USC, 1994) A. Fung (USC, 1995) S. Lototsky (USC, 1996) C. Rao (USC, 1998) S. Kligys (USC, 1998) A. Petrov (USC, 2000) G. Yaralov (USC, 2000) B. L. Rozovskii: Honors and Awards Fellow of the Institute of Mathematical Statistics (1997) Peter-the-Great Medal (International Academy of Natural and Social Sciences, 1997) Kolmogorov Medal (Kolmogorov Centennial Conference, 2003) Publications of B. L. Rozovskii Books 1. Stochastic evolution systems. Linear theory and applications to the statistics of random processes (in Russian). Moscow: Nauka, 1983. 2. Data analysis in chemical research. Statistical foundations (in Russian). Moscow: Khimija, 1984. 3. Stochastic evolution systems. Linear theory with applications to non-linear filtering. Mathematics and its Applications (Soviet Series), vol. 35. Dordrecht: Kluwer Academic Publishers, 1990. Papers 1. Wiener chaos solutions of linear stochastic evolution equations (with S. Lototsky). Ann. Probab., 34 (2006), no. 2, 638–662. 2. Wiener chaos expansions and numerical solutions of randomly forced equations of fluid mechanics (with T. How et al.), J. Comput. Phys. 216 (2006), no. 2, 687–706. 3. Stochastic differential equations: A Wiener chaos approach (with S. Lototsky). In From Stochastic Calculus to Mathematical Finance, ed. Yu. Kabanov et al.,

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pp. 433–506. Berlin: Springer, 2006. 4. Strong Solutions of Stochastic Generalized Porous Media Equations: Existence, Uniqueness and Ergodicity. (with G. Da Prato et al.) Comm. Partial Dif. Eq., 31 (2006), no. 1-3, 277–291. 5. A novel approach to detection of intrusions in computer networks via adaptive sequential and batch-sequential change-point detection methods (with R. Blazek et al.), IEEE Transactions on Signal Processing, to appear. 6. Detection of intrusions in information systems by sequential change-point methods (with A. Tartakovsky et al.). Stat. Methodol., 3 (2006), no. 3, 252–293. 7. Detection of intrusions in information systems by sequential change-point methods. Authors’ response (with A. Tartakovsky et al.) Stat. Methodol. 3 (2006), no. 3, 329–340. 8. A filtering approach to tracking volatility from prices observed at random times (with J. Cvitanic et al). Ann. Appl. Probab., 16 (2006), no. 3, 1633–1652. 9. Numerical estimation of volatility values from discretely observed diffusion data (with J. Cvitanic et al), Journal of Computational Finance (to appear). 10. A novel approach to detection of intrusions in computer networks via adaptive sequential and batch-sequential change-point detection methods (with R. Blazek et al.) International Journal of Computing and Information Sciences. 11. Global L2 -solutions of stochastic Navier-Stokes equations (with R. Mikulevicius). Ann. Probab., 33 (2005), No. 1, 137–176. 12. Passive Scalar Equation in a Turbulent Incompressible Gaussian Velocity Field (with S. Lototsky), Russian. Math. Surveys, 59 (2004), no.2, 297–312. 13. Stochastic Navier-Stokes equations for turbulent flows (with R. Mikulevicius). SIAM J. Math. Anal. 35 (2004), no. 5, 1250–1310. 14. A diffusion model of roundtrip time (with S. Bohacek). Computational Statistics and Data Analysis, Comput. Statist. Data Anal., 45 (2004) no. 1, 25–50. 15. On martingale problem solutions for stochastic Navier-Stokes equations (with R. Mikulevicius). In Stochastic partial differential equations and applications, ed. G. Da Prato and L. Tubaro. Lecture Notes in Pure and Applied Mathematics Series 227. New York: Marcel Dekker, 2002. 16. A note on Krylov’s Lp -theory for systems of SPDEs (with R. Mikulevicius). Electron. J. Probab. 6 (2001), no. 12, 1–35. 17. On equations of stochastic fluid mechanics (with R. Mikulevicius). In Stochastics in finite and infinite dimensions: in honor of Gopinath Kallianpur, ed. T. Hida et al., 285–302. Trends Math. Boston: Birkhauser, 2001. 18. Stochastic Navier-Stokes equations: propagation of chaos and statistical moments (with R. Mikulevicius). In Optimal control and partial differential equations: in honor of Professor Alain Bensoussan, ed. J. L. Menaldi et al., 258–267. Amsterdam: IOS Press, 2001. 19. Approximation of the Kushner equation of nonlinear filtering (with K. Ito). SIAM J. Control Optim. 38 (2000), no. 3, 893–915.

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20. Parameter estimation for stochastic evolution equations with non-commuting operators (with S. Lototsky). In Skorokhod’s ideas in probability theory, ed. V. Korolyuk et al., pp. 271–280. Kiev: Institute of Mathematics of the National Academy of Sciences of Ukraine, 2000. 21. Fourier-Hermite expansions for nonlinear filtering (with R. Mikulevicius). Teor. Veroyatnost. i Primenen. 44 (1999), no. 3, 675–680. English translation Theory Probab. Appl. 44 (2000), no. 3, 606–612. 22. Spectral asymptotics of some functionals arising in statistical inference for SPDE’s (with S. Lototsky). Stochastic Process. Appl. 79 (1999), no. 1, 69–94. 23. Recursive nonlinear filter for a continuous-discrete time model (with S. Lototsky). IEEE Trans. Automatic Cont. 48 (1998), no. 8. 1154–1158. 24. Martingale problems for stochastic PDE’s (with R. Mikulevicius). In Stochastic partial differential equations: six perspectives, ed. R. Carmona and B. L. Rozovskii, pp. 243–325. Math. Surveys Monogr., vol. 64. Providence, RI: American Mathematical Society, 1998. 25. Normalized stochastic integrals in topological vector spaces (with R. Mikulevicius). In S´eminaire de Probabilit´es XXXII, pp. 137–165. Lecture Notes in Math, vol. 1686. Berlin: Springer, 1998. 26. Linear parabolic stochastic PDE’s and Wiener chaos (with R. Mikulevicius). SIAM J. Math. Anal. 29 (1998), no. 2, 452–480. 27. Weighted stochastic Sobolev spaces and bilinear SPDE’s driven by space-time white noise (with D. Nualart). J. Funct. Anal. 149 (1997), no. 1, 200–225. 28. On asymptotic problems of parameter estimation in stochastic PDE’s: discrete time sampling (with L. Piterbarg). Math. Methods Statist. 6 (1997), no. 2, 200–223. 29. Nonlinear filtering revisited: a spectral approach (with S. Lototsky and R. Mikulevicius). SIAM J. Control Optim. 35 (1997), no. 2, 435–461. 30. On asymptotic properties of an approximate maximum likelihood estimator for stochastic PDEs (with M. Huebner and S. Lototsky). In Statistics and control of stochastic processes, ed. Yu. M. Kabanov et al. pp. 139–155. River Edge, NJ: World Scientific, 1997. 31. Recursive multiple Wiener integral expansion for nonlinear filtering of diffusion processes (with S. Lototsky). In Stochastic processes and functional analysis, ed. J. Goldstein et al., pp. 199–208. Lecture Notes in Pure and App. Math., vol. 186. New York: Marcel Dekker, 1997. 32. Maximum likelihood estimators in the equations of physical oceanography (with L. Piterbarg). In Stochastic modelling in oceanography, ed. R. Adler et al., pp. 397–421. Progr. Probab., vol. 39. Boston: Birkhauser, 1996. 33. On asymptotic properties of maximum likelihood estimators for parabolic stochastic PDE’s (with M. Huebner). Probab. Theory Related Fields 103 (1995), no. 2, 143–163. 34. On stochastic integrals in topological vector spaces (with R. Mikulevicius).

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Stochastic analysis (Ithaca, NY, 1993), pp. 593–602. Proc. Sympos. Pure Math., vol. 57. Providence, RI: American Mathematics Society, 1995. Estimates of turbulent parameters from Lagrangian data using a stochastic particle model (with A. Griffa et al.). Journal of Mar. Res. 53 (1995), no. 3, 371–401. Statistics and physical oceanography (with D. B. Chelton et al.). Stat. Sci. 9 (1994), no. 2, 167–201. Uniqueness and absolute continuity of weak solutions for parabolic SPDE’s (with R. Mikulevicius). Acta Appl. Math. 35 (1994), no. 1–2, 179–192. Soft solutions of linear parabolic SPDE’s and the Wiener chaos expansion (with R. Mikulevicius). In Stochastic analysis on infinite-dimensional spaces, ed. H. Kunita and H.-H. Kuo, pp. 211–220. Pitman Res. Notes Math. Ser., vol. 310. Baton Rouge, LA: Longman Sci. Tech, Harlow, 1994. Kinematic dynamo and intermittence in a turbulent flow. (with P. Baxendale). Geophys. Astrophys. Fluid Dynam. 73 (1993), no. 1-4, 33–60. Two examples of parameter estimation for stochastic partial differential equations (with M. Huebner and R. Khasminskii). In Stochastic processes. A festschrift in honor of Gopinath Kallianpur, pp. 149–160. New York: Springer, 1993. Some results on a diffusion approximation to the induction equation. In Stochastic partial differential equations and applications (Trento, 1990), ed. G. Da Prato and L. Tubaro, pp. 268–281. Pitman Res. Notes in Math. Ser., vol. 268. Baton Rouge, LA: Longman Sci. Tech, Harlow, 1992. A simple proof of uniqueness for Kushner and Zakai equations. In Stochastic analysis, ed. E. Mayer-Wolf, pp. 449–58. Boston: Academic Press, 1991. Measure-valued solutions of second-order stochastic parabolic equations (with O.G. Purtukhiya, in Russian). In Statistics and control of random processes, ed. A. N. Shiryaev, pp. 177–79. Moscow: Nauka, 1989. On the mathematical theory of a hydromagnetic dynamo in a random flow (in Russian). Dokl. Akad. Nauk SSSR 293 (1987), no. 6, 1311–1314. On the statistic estimation of reliability of determining aqueous solution pH by acid-base indicator paper (with V.M. Ostrovskaja et al., in Russian). J. Analit. Chem. USSR Acad. Sci. 42 (1987), no. 9, Part 2, 1369–1371. Nonnegative L1 -solutions of second order stochastic parabolic equations with random coefficients. In Statistics and control of stochastic processes (Moscow, 1984), ed. N. V. Krylov et al. pp. 410–427. Transl. Ser. Math. Engrg., New York: Optimization Software, 1985. Filtering interpolation and extrapolation of degenerate diffusion processes. Backward equations (in Russian). Teor. Veroyatnost. i Primenen. 28 (1983), no. 4, 725–737. Stochastic partial differential equations and diffusion processes (with N. V. Krylov, in Russian). Uspekhi Mat. Nauk 37 (1982), no. 6, 75–95.

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49. Characteristics of second-order degenerate parabolic Itˆ o equations (with N. V. Krylov, in Russian). Trudy Sem. Petrovsk. 8 (1982), 153–168. 50. Smoothness of solutions of stochastic evolution equations and the existence of a filtering transition density (with A. Shimizu). Nagoya Math. J. 84 (1981), 195–208. 51. On the first integrals and Liouville equations for diffusion processes (with N. V. Krylov). In Stochastic differential systems (Visegrad, 1980), pp. 117–125. Lecture Notes in Control and Information Sci., vol. 36. New York: Springer, 1981. 52. On complete integrals of Itˆ o equations (with N.V. Krylov, in Russian). Uspekhi Mat. Nauk, 35 (1980), no. 4, 147. 53. A note on the strong solutions of stochastic differential equations with random coefficients. In Stochastic differential systems. (Proc. IFIP-WG 7/1 Working Conference, Vilnius, 1978), pp. 287–296. Lecture Notes in Control and Information Sci., vol. 25. New York: Springer, 1980. 54. Conditional distributions of degenerate diffusion processes (in Russian). Teor. Veroyatnost. i Primenen. 25 (1980), no. 1, 149–154. 55. Itˆ o equations in Banach spaces and strongly parabolic stochastic partial differential equations (with N. V. Krylov, in Russian). Dokl. Akad. Nauk SSSR 249 (1979), no. 2, 285–289. 56. Stochastic evolution equations (with N. V. Krylov, in Russian), Current Problems in Mathematics, vol. 14, pp. 71–147. Moscow: Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, 1979. English translation J. Soviet Math. 16 (1981), no 4, 1233–1277. 57. Fundamental solutions of stochastic partial differential equations and the filtering of diffusion processes (with L. G. Margulis, in Russian). Uspekhi Mat. Nauk 33, no. 2 (1978), 197. 58. Conditional distributions of diffusion processes (with N. V. Krylov, in Russian). Izv. Akad. Nauk SSR Ser. Mat. 42 (1978), no. 2, 356–378. 59. The Cauchy problem for linear stochastic partial differential equations (with N. V. Krylov, in Russian). Izv. Akad. Nauk SSR Ser. Mat. 41 (1977), no. 6, 1329–1347. 60. Stochastic partial differential equations (in Russian). Mat. Sb. (N.S.) 96 (1975), no. 138, 314–341. 61. Stochastic differential equations in infinite-dimensional spaces and filtering problems (in Russian). In Proceedings of the School and Seminar on the Theory of Random Processes (Druskininkai, 1974), Part II, pp. 147–194. Vilnius: Inst. Fiz. i Mat. Akad. Nauk Litovsk. SSR, 1975. 62. The Itˆ o-Wentzell formula (in Russian). Vestnik Moskov. Univ. Ser. I Mat. Meh. 28 (1973), no. 1, 26–32. 63. On infinite systems of stochastic differential equations that arise in the theory of optimal nonlinear filtering (with A. N. Shiryaev, in Russian). Teor. Verojatnost.

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i Primenen. 17 (1972), 228–237. 64. Stochastic partial differential equations that arise in nonlinear filtering problems (in Russian). Uspekhi Mat. Nauk 27 (1972), no. 3, 213–214. 65. The problem of “disorder” for a Poisson process (with L. I. Galtchuk, in Russian). Teor. Verojatnost. i Primenen. 16 (1971), 729–734. Edited Volumes 1. Applied Mathematics & Optimization. Special issue on Approximation in Stochastic Partial Differential Equations, Guest editor B. Rozovskii, Springer, 2006. 2. Stochastic partial differential equations: six perspectives. Ed. R. Carmona and B. L. Rozovskii. Mathematical Surveys and Monographs, vol. 64. Providence, RI: American Mathematical Society, 1998. 3. Statistics and control of stochastic processes. The Liptser festschrift: papers from the Steklov Seminar (Moscow, 1995/1996). Ed. Yu. M. Kabanov, B. L. Rozovskii, and A. N. Shiryaev. River Edge, NJ: World Scientific, 1997. 4. Stochastic modelling in oceanography. Ed. R. Adler, P. Muller, and B. L. Rozovskii. Progress in Probability 39. Boston: Birkhauser, 1996. 5. Stochastic partial differential equations and their applications. Proceedings of the IFIP WG 7/1 International Conference (Charlotte, NC, 1991). Ed. B. L. Rozovskii and R. B. Sowers. Lecture Notes in Control and Information Sci., vol. 176. Berlin: Springer, 1992. Selected Conference Proceedings 1. Chaos expansions and numerical solutions of randomly forced equations of fluid dynamics (with T. How et al.), Proceedings of the Sixth Helenic-European Conference on Computer Mathematics and its Applications, HERCMA 2003, Vol. 1, ed. E. A. Lipitakis, pp. 12–22. 2. Novel Approach to Detection of “Denial-of-Service” Attacks via Adaptive Sequential and Batch-Sequential Change-Point Detection Methods (with R. Blazek et al.). In Proceedings of the 2nd Annual IEEE Systems, Man, and Cybernetics Information Assurance Workshop (West Point, NY, 2004). New York: Institute of Electrical and Electronics Engineers, 2004. 3. A New Adaptive Batch and Sequential Methods for Rapid Detection of Network Traffic Changes with Emphasis on Detection of “Denial-of-Service” Attacks, (with R. Blazek and H. Kim). In Proceedings of the 53rd Session of the International Statistical Institute (Seoul, 2001). New York: Physica-Verlag, 2001. 4. Tracking Volatility (with J. Cvitanic and R. Liptser). In Proceedings of the 39th IEEE Conference on Decision and Control, IEEE Control Systems Society (Sydney, 2000). New York: Institute of Electrical and Electronics Engineers,

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2000. 5. Interactive Banks of Bayesian Matched Filters (with R. Blazek and A. Petrov). In SPIE Proceedings: Signal and Data Processing of Small Targets (Orlando, FL, 2000), vol. 4048, ed. O. E. Drummond. Bellingham, WA: SPIE (The International Society for Optical Engineering), 2000. 6. Optimal nonlinear filtering for track-before-detect in IR image sequences (with A. Petrov). In SPIE Proceedings: Signal and Data Processing of Small Targets (Denver, CO, 1999), vol. 3809, ed. O. E. Drummond. Bellingham, WA: SPIE (The International Society of Optical Engineering), 1999. 7. An adaptive Bayesian approach to fusion of imaging and kinematic data (with A.Tartakovsky and G.Yaralov). In Proceedings of the 2nd International Conference on Information Fusion (Fusion ’99, Sunnyvale, CA, 1999). Madison, WI: Omnipress, 1999. 8. Matched filters and hidden Markov models with distributed observation (with S. Kligys). In Proceedings of the Fourth Annual U.S. Army Conference on Applied Statistics (Aberdeen Proving Ground, MD, 1998), ed. Barry A. Bodt. ARL-SR-84. Aberdeen, MD: Army Research Laboratory, 1999. 9. State estimation in hidden Markov models with distributed observation (with S. Kligys). In Theory and Practice of Control Systems: Proceedings of the 6th IEEE Mediterranean Conference (Alghero, Sardinia, 1998), ed. A. Tornambe et al. River Edge, NJ: World Scientific, 1998. 10. Splitting-up discretization for Kushner’s equation of nonlinear filtering (with K. Ito). In Proceedings of the 36th IEEE Conference on Decision and Control, IEEE Control Systems Society (San Diego, CA, 1997). New York: Institute of Electrical and Electronics Engineers, 1998. 11. Solving hidden Markov problems by spectral approach (with C.P. Fung). In Proceedings of the 3rd IEEE Mediterranean Symposium, (PLACE, 1995), vol. II. New York: Institute of Electrical and Electronics Engineers, 1995. 12. Separation of observations and parameters in nonlinear filtering (with R. Mikulevicius). In Proceedings of the 32nd IEEE Conference on Decision and Control, (San Antonio, TX, 1993), vol.2. New York: Institute of Electrical and Electronics Engineers, 1993. 13. Statistics and physical oceanography (with A. Griffa et al.). Report of the National Research Council. Washington, D.C.: National Academy Press, 1993. 14. Nonlinear filtering revisited: A spectral approach II (with S. Lototsky and R. Mikulevicius). In Proceedings of the IEEE & SIAM CDC 35th Conference on Decision and Control, (Kobe, Japan, 1996), vol. 4. Madison, WI: Omnipress, 1997. 15. On the kinematic dynamo problem in a random flow. In Probability Theory and Mathematical Statistics: Proceedings of 5th Vilnius Conference on Probabilty Theory and Mathematical Statistics, (Vilnius, 1985), vol. II. Utrecht: VNU Science Press, 1987.

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16. On the kinematic dynamo problem in a random flow. In Probability Theory and Mathematical Statistics: Proceedings of the 4th Vilnius International Conference on Probability Theory and Mathematical Statistics (Vilnius, 1985). Vilnius: Akad. Nauk Litovsk. SSR, Inst. Mat. i Kibernet, 1985. 17. Filtering of degenerate diffusion type processes. Backward equations. In Stochastic Optimization International Conf. Abstracts, (Kiev,1984), Part II. 18. Backward equations of conditional and unconditional diffusion. In Proceedings of the 4th Soviet-Japan Symp. on Probab. Theor. and Math. Stat. Abstracts (Tbilisi, 1982). 19. Backward filtering equations. In 15th All-Union School-Colloquium on Probab. Theor. and Math. Stat. Abstracts (Bakuriani, 1981). Tbilisi: Metsniereba, 1981. 20. Liouville equations for a diffusion Markov process (in Russian). In 14th All Union School on Probab. Theor. and Math. Stat. Abstracts (Bakuriani, 1980). Tbilisi: Metsniereba, 1980. 21. On the first integral and Liouville equations. In Abstracts of 3rd Working Conference on Stochastic Differential Equations (Visegrad, 1980), Budapest: SZAMKI, 1980. 22. On the extrapolation of a signal with a martingale type noise (in Russian). In 5th International Symposium on Inform. Theory. Abstracts (Tbilisi, 1979). 23. Non-linear filtering of diffusion processes: an analytical approach. In International Symposium on Stochastic Differential Equations. Abstracts of Communications (Vilnius, 1978). Vilnius: Inst. Math. and Cybernet. Acad. Sci. Lithuanian SSR, 1978. 24. On Itˆ o equations in Hilbert spaces. In 2nd Vilnius Conference on Probability and Mathematical Statistics. Abstracts of Communications (Vilnius, 1977). Vilnius: Inst. Mat. i Kibernet. Akad. Nauk Litovsk. SSR, 1977. 25. On Cauchy problem for superparabolic stochastic differential equations (with N.V. Krylov). In Proceedings of the Third Japan-USSR Symposium on Probability Theory (Tashkent, 1975). Lecture Notes in Mathematics, vol. 550. New York: Springer, 1976. 26. On stochastic differential equations in partial derivatives. In International Conference on Probability Theory and Mathematical Statistics. Abstracts of Communications (Vilnius, 1973). Vilnius: Akad. Nauk Litovsk. SSR, 1973. 27. Reduced form of non-linear filtering equations (with A. N. Shiryaev). In IFAC Symposium on Stochastic Control. Supplement of abstracts (Budapest, 1974). Selected Technical Reports and Teaching Aids 1. Stochastic Navier-Stokes equations for turbulent flows (with R. Mikulevicius). Warwick Preprint: 21/ 2001. 2. Detection algorithms and track-before-detect architecture based on nonlinear filtering for infrared search and track systems (with S. Kligys and A. Tar-

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3.

4.

5. 6. 7. 8. 9.

10. 11. 12. 13. 14. 15. 16.

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takovsky). Technical Report CAMS-98.9.1, Center for Applied Mathematical Sciences, University of Southern California, 1998. Domain pursuit method for tracking ballistic targets (with R. Rao and A.Tartakovsky). Technical Report CAMS-98.9.2, Center for Applied Mathematical Sciences, University of Southern California, 1998. Parameter estimation for stochastic evolution equations with non-commuting operators (with S. V. Lototsky). IMA Preprint Series #1501, University of Minnesota, 1997. Statistics and physical oceanography (with A. Griffa et al.). Report of the National Research Council, National Academy Press, 1993. Lecture notes on stochastic partial differential equations. University of North Carolina, Charlotte, 1990. Real time statistical quality control (with Yu. P. Adler) (in Russian). Moscow: Znanije Publishing House, 1984. Studies in the theory of stochastic partial differential equations (in Russian). Doctor of Sci. Dissertation (Vilnius, Lithuania, 1984). Statistical methods in chemical industry. Methods and instructions (with B. I. Pashko et al., in Russian). Center for Scientific Organization of Labor in Chemistry, Moscow 1983. Mathematical design of experiments in textile industry (with Yu. P. Adler et al., in Russian). MIASCME, 1984. Mathematics design of experiments. Methods and instructions (with Yu. P. Adler et al., in Russian). MIASCME, 1983. Statistical methods in analytical chemistry. Instructions (with Yu. P. Alder et al., in Russian). MIASCME, 1981. Statistical quality control. Methods and instructions (with Yu. P. Alder et al., in Russian). MIASCME, 1978. Optimal design of experiments. Methods (with Yu. P. Alder et al., in Russian). MIASCME, 1978. Lectures in probability theory (in Russian). MIASCME, 1974. On stochastic equations arising in filtering of Markov processes (in Russian). Ph. D. Dissertation, Moscow State (Lomonosov) University, 1972.

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Contents

Preface

vii

Boris Rozovskii

ix

Publications of B. L. Rozovskii

xi

1. Stochastic Evolution Equations

1

N. V. Krylov and B. L. Rozovskii 2. Predictability of the Burgers Dynamics Under Model Uncertainty

71

D. Bl¨ omker and J. Duan 3. Asymptotics for the Space-Time Wigner Transform with Applications to Imaging

91

L. Borcea, G. Papanicolaou, and C. Tsogka 4. The Korteweg-de Vries Equation with Multiplicative Homogeneous Noise

113

A. de Bouard and A. Debussche 5. On Stochastic Burgers Equation Driven by a Fractional Laplacian and Space-Time White Noise

135

Z. Brze´zniak and L. Debbi 6. Stochastic Control Methods for the Problem of Optimal Compensation of Executives A. Cadenillas, J. Cvitani´c, and F. Zapatero

xxi

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7. The Freidlin-Wentzell LDP with Rapidly Growing Coefficients

197

P. Chigansky and R. Liptser 8. On the Convergence Rates of a General Class of Weak Approximations of SDEs

221

D. Crisan and S. Ghazali 9.

Flow Properties of Differential Equations Driven by Fractional Brownian Motion

249

L. Decreusefond and D. Nualart 10. Regularity of Transition Semigroups Associated to a 3D Stochastic Navier-Stokes Equation

263

F. Flandoli and M. Romito 11. Rate of Convergence of Implicit Approximations for Stochastic Evolution Equations

281

I. Gy¨ ongy and A. Millet 12. Maximum Principle for SPDEs and Its Applications

311

N. V. Krylov 13. On Delay Estimation and Testing for Diffusion Type Processes

339

Yu. A. Kutoyants 14. On Cauchy-Dirichlet Problem for Linear Integro-Differential Equation in Weighted Sobolev Spaces

357

R. Mikulevicius and H. Pragarauskas 15. Strict Solutions of Kolmogorov Equations in Hilbert Spaces and Applications

375

G. Da Prato Author Index

391

Subject Index

393

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Chapter 1 Stochastic Evolution Equations

Nicolai V. Krylov and Boris L. Rozovskii∗ The theory of strong solutions of Itˆ o equations in Banach spaces is expounded. The results of this theory are applied to the investigation of strongly parabolic Itˆ o partial differential equations.

Contents 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Itˆ o Equations in Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Examples of Stochastic Evolution Equations . . . . . . . . . . . . . . . . . . . . 1.3 Stochastic Evolution Equations with Bounded Coefficients and Linear Stochastic Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Nonlinear Stochastic Evolution Equations . . . . . . . . . . . . . . . . . . . . . 1.5 Content and Organization of the Work . . . . . . . . . . . . . . . . . . . . . . . 2 Stochastic Integration in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Stochastic Integrals in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Itˆ o’s Formula for the Square of the Norm . . . . . . . . . . . . . . . . . . . . . . 2.4 Proof of Theorem 2.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Itˆ o Stochastic Equations in Banach Spaces and the Method of Monotonicity . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Assumptions and the Main Results . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Itˆ o Equations in Rd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Uniqueness Theorem: A Priori Estimates and Finite-Dimensional Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Existence of Solution and the Markov Property: Passing to the Limit by the Method of Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Itˆ o Stochastic Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 First Boundary-Value Problem for Nonlinear Stochastic Parabolic Equations . . 4.3 Cauchy Problem for Linear Second-Order Equations . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

∗ This

2 2 2 5 7 8 9 9 10 16 20 27 27 29 34 43 49 54 54 57 61 65

paper was originally published in Russian in 1979 (Itogi Naukt i Tekhniki, Seriya Sovremennye Problemy Matematiki, Vol, 14, pp. 71–146) The English translation was first published c Plenum Publishing Co. in the Journal of Soviet Mathematics, Vol. 14, pp. 1233–1277, 1981, The permission of the current copyright holder, Springer, to include the paper in this volume is gratefully acknowledged. 1

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1. Introduction 1.1. Itˆ o Equations in Banach Spaces The theory of Itˆ o stochastic differential equations is one of the most beautiful and most useful areas of the theory of stochastic processes. However, until recently the range of investigations in this theory had been, in our view, unjustifiably restricted: only equations were studied which could, in analogy with the deterministic case, be called ordinary stochastic equations. The situation has began to change in the last 10–15 years. The necessity of considering equations combining the features of partial differential equations and Itˆ o equations has appeared both in the theory of stochastic processes and in related areas. Such equations have appeared in the statistics of stochastic processes (filtering of diffusion process), statistical hydromechanics, population genetics, Euclidean field theory, classical statistical field theory, and other areas. Concrete examples of equations of this type are presented below. These equations describe the evolution in time of processes with values in function spaces or in other words, random fields in which one coordinate — the “time” — is distinguished. The objective of the present work is to show how to create a unified theory which include both ordinary Itˆ o equations and a rather broad class of stochastic partial differential equations. We realize our program by considering equations of the Itˆ o type in Banach spaces. More precisely we consider the equation du (t, ω) = A (u (t, ω) , t, ω) dt + B (u (t, ω)) dw (t) ,

(1.1)

where A(·, t, ω) and B(·, t, ω) are families of unbound operators in Banach spaces, depending on the elementary outcome ω in a non-anticipating fashion, and w(t) is a process with values in some Hilbert space and with independent (in time) increments. Such equations are usually called stochastic evaluation equations. 1.2. Examples of Stochastic Evolution Equations 1.2.1. Linear Equation of Filtring of Diffusion Processes Filtering is one of the most important problems in statics of random processes. In essence, it consists of the following [1]. Consider a two-component process Z = (x, y), e.g., the (n + m)-dimensional diffusion process dx(t) = a(x(t), y(t), t)dt + b(x(t), y(t), t)dw(t), dy(t) = g(x(t), y(t), t)dt + σ(y(t), t)dw(t), x (0) = x0 , y(0) = y0 , where dim x = n and w(t) is a standard m + n-dimensional Wiener process. It is assumed that the component x of the process z is nonobservable. Given a function

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f = f (x), it is required to find the best mean-square estimate of f (x(t)) on the basis of observations of the trajectory of the observable component y up to time t. In other words, this estimate is to be sought as a functional of the component y. It is well known that such an estimate is the conditional expectation of f (x(t)) given the σ-algebra Fty generated by the values of y up to time t, i.e., E[f (x(t))|Fty ]. The filtering problem consists in computing this conditional expectation. In Ref. [2] we showed that under broad assumptions Z −1 Z E[f (x(t))|Fty ] = f (x)ϕt (x)dx ϕt (x)dx , (1.2) Rn

Rn

where ϕt (x) is the solution of the Cauchy problem 1 dϕt (x) = { tr Dxx (bb∗ ϕt (x)) − Dx (aϕt (x))}dt + [(σσ ∗ )−1/2 gϕt (x) 2 + Dx ((σσ ∗ )−1/2 σb∗ ϕt (x))](σσ ∗ )−1/2 dy(t), ‡ ϕ0 (x) = P (x0 ∈ dx)/dx, Dxx is the matrix of the second derivatives, and Dx is the vector of the first derivatives. This is a linear stochastic differential equation with unbounded operators of “drift” and “diffusion”. 1.2.2. Equations of Population Genetics One of the most important types of models in population genetics are the models with geographic structure. These are models in which the structure of population change not only in time but also in space (geographically). Various probabilistic models of this sort have been proposed by Bailey [3], Crow and Kinmura [4], Malecot [5], and others. All these models are discrete. Dawson [6] and Fleming [7] proposed continuous (in time and space) models which are limits of the corresponding discrete models. These works of Dawson and Fleming continue conceptually the well-known work of Feller [8]. The equation proposed by Dawson for the mass distribution p(t, x) of the population has the form p dp(t, x) = a∆p(t, x)dt + c p(t, x) dw(t, x), (1.3)

while the equations of Fleming has the form r
p(t, x)(1 − p(t, x))+ dp(t, x) = ∆p(t, x) + ap(t, x) − β dt + dw(t, x) 2

(1.4)

In both cases ∆ is the Laplace operator, α, β, c are
constants, (α)+ = max(a, 0), and w (t, x) is a Wiener process with values in L2 Rd , where d is the dimension of ‡

Here ∗ is the symbol for the conjugate; the arguments x, t, and y(t) of the coefficients have been dropped.

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x, and with a nuclear covariance operator. This means that w(t, x) is a stochastic process with values in L2 (Rd ), such that, for every function e ∈ L2 (Rd ), Z we (t) = w(t, x)e(x)dx Rd

is a one dimensional Wiener process and

E(we1 (t) − we1 (s))(we2 (t) − We2 (s)) = (t − s)(e1 , Qe2 )L2 (Rd ) , where Q is a nuclear operator on L2 (Rd ) and (·, ·)L2 (Rd ) is the inner product in L2 (Rd ); for more details, see Ref. [9]). Wiener processes with values in Hilbert spaces are discussed in more detail in Section 2.2 (see, in particular, Definition 2.15). 1.2.3. System of Navier-Stokes Equations with Random External Forces In the physics literature on the theory of turbulence (see, e.g. Novikov [10], Monin and Yagiom [11], Klyatskin [12] and the literature cited there) a model of the motion of an incompressible fluid is considered under the action of random external forces; the model is described by the following system of Navier-Stokes equations: ! 3 X ∂p ∂ui (t, x) − dt + dwi (t, x), dui (t, x) = ν∆ui (t, x) − uk ∂xk ∂xi k=1 (1.5) 3 X ∂uk = 0. ∂xk k=1

Here u = (u1 , u2 , u3 ) is the velocity vector, p is the pressure, ν, the viscosity, and wi (t, x) are independent Wiener processes with values in a function space. For equation (1.5) in a cylinder (0, T ) × G, where G is a domain in R3 with boundary Γ, the first boundary value problem has been considered: u (t, x) |[0,T ]×Γ = 0, u(0, x) = u0 (x) . 1.2.4. Equation of the Free Field
Let L = L Rd+1 be the space of rapidly decreasing function on Rd+1 , and let L0 be the dual space of L, that is, the Schwartz space of slowly increasing generalized functions. We denote by G the σ-algebra in L0 , generated by the cylinder sets. On the measurable space (L0 , G), it is possible to construct a probability measure ν with the characteristic functional (  )  Z √
1 −1 ηω 2 −1 Cν (η) = e ν (dω) = exp − η, −∆t,x + m η , 2 L0 L2 (Rd+1 )

P where η ∈ L, ∆t,x = di=1 functional ω ∈ L0 on η ∈ L.

∂2 ∂x2i

+

∂2 ∂t2 ,

m is a number, and ηω is the value of the

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It is known (see, e.g., the monograph of Simon [13]) that the free field ξ is one of the simplest objects of relativistic quantum mechanics; in the Euclidean model it can be interpreted as a canonical generalized random field on the probability space (L0 , G, ν): ξ (ω, t, x) = ω(t, x) for each ω ∈ L0 . Further, let w˙ be a generalized white noise, i.e., the canonical generalized random field on the probability space (L0 , G, µ), where µ is the Gaussian measure with the characteristic operator   1 Cµ (η) = exp − kηk2L2 (Rd+1 ) ; 2 see e.g. Ref. [9]. Hida and Strett [14] showed that the Euclidean free field ξ(t, x) is a stationary in t solution of the equation p ∂ξ(t, x) = − −∆x + m2 ξ(t, x) + w(t, ˙ x), (1.6) ∂t where ∆x =

d X ∂2 ∂x2i i=1

√ and the operator −∆x + m2 is understood in the sense of the theory of generalized functions. Regarding this equation, see also the survey of Dawson [15]. The work of Albeverio and Hoegh-Krohn [16] is a good example of stochastic evolution equations in Euclidean field theory. The above examples present only a small fraction of stochastic evolution equations considered in recent years. We have selected these examples because they have been studied in detail at a mathematical level of rigor. Modern physics journals are an inexhaustible source of stochastic evolution equations of the most varied type which are studied only at a physical level of rigor. 1.3. Stochastic Evolution Equations with Bounded Coefficients and Linear Stochastic Evolution Equations The impetus for the first mathematical investigations in the area of stochastic evolution equations were not, however, the demands of physics or biology but rather the inner requirements of mathematics, viz., of the theory of differential equations with variational derivatives. In the mid-sixties, Daletskii and Baklan [17–19] studied stochastic evolution equations with the goal of constructing a solution of the Cauchy problem for the Komogorov equation in variational derivatives ∂F (x, t) 1 = tr[B ∗ (x, t)F 00 (x, t)B(x, t)] + A(x, t)F 0 (x, t), t ≤ T, F (x, t) = φ(x). ∂t 2 In these works, a probabilistic method of constructing solutions of the Kolmogorov equation, which is well known in the finite-dimensional case, was used; the method −

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consists in writing the solution in the form F (x, t) = E [Φ(u(T ))|u(t) = x], where u(t) is a solution of the stochastic evolution equation (1.1). To carry out this program, it was necessary to study stochastic equations, and Refs. [17–19] were among the first to address this issue. In these works, B is assumed to be bounded (as an operator) and to satisfy a Lipschitz condition, while A is either the infinitesimal generator of some contractive semigroup or is also bounded and Lipschitz. This direction was further developed by the authors themselves and their students in Refs. [20–24]. Moreover, a number of works [25–32] appeared in which linear stochastic evolution equations were studied for other applications, such as control, filtering, and extrapolation of linear stochastic Itˆ o partial differential equations. From the point of view of proving the existence of a solution, these works are very similar to Ref. [18]: the operator A(s) is always assumed to generate a semigroup Ts,t , and the operator B is bounded and satisfies a Lipschitz condition (we remark that these conditions are often not satisfied for the filtering equations). In the above works, the following equations was considered: Z t u (t) = T0,t u0 + Ts,t B(u(s))dw(s). (1.7) 0

The proof of existence and uniqueness of solution of this equation is accomplished simply by the method of contraction mappings, since Ts,t is a bounded operator. It is then proved, or simply taken for granted, that, under additional conditions the solution of equation (1.7) belongs to the domain of the operator A, and equation (1.1) is equivalent to (1.7). It should be noted that the conditions for the equivalence of equations (1.1) and (1.7) obtained in these works are rather burdensome to be useful in application to stochastic Itˆ o partial differential equations, and the reason is similar to the deterministic case. Indeed, the theorems on the solvability of inhomogeneous parabolic equations obtained e.g., by methods of potential theory (see Chapter 1 in Ref. [33]) are much stronger than their counterparts obtained by the theory of inhomogeneous semigroups for linear operator equations [34]. Methods of potential theory in application to linear equations of type (1.1) were used by Rozovskii and Margulis [35, 36]. In these works it is assumed that the “diffusion” operator has order zero, while the “principal part” of the “drift” operator does not depend on the elementary outcome. In the first work, the unique solvability of the Cauchy problem is proved, while in the second a fundamental solution of the equation is constructed, and precise estimates of it in the H¨ older classes are obtained. By modifying the method of Ref. [35], Shimizu [37] proved the unique solvability of the second boundary-value problem for the same equation as in Ref. [35]. We mention several works of qualitative character. Markus [38] investigated stationary solutions of linear equations, Shimizu [39] proved some comparison theorems, and Mahno [40] studied the averaging principle for equations of type (1.7). Bensoussan [41] used a completely different idea to construct solutions of a linear

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equations of type (1.1) for B ≡ 1. In contrast to all the works cited above, he did not assumed that A generates a semigroup; this assumption was replaced by a coercivity condition. To construct the solution, Bensoussan utilized the method of time discretization. The coercivity condition ensures that the corresponding discrete equation is easily solvable and the solution satisfies the necessary estimates; a weak limiting procedure is then carried out. In the deterministic case, this method was applied earlier by Lions [42].

1.4. Nonlinear Stochastic Evolution Equations The important program started by Bensoussan in Ref. [41] was continued in his joint paper with Temam [43], where the method of discretization was applied to an equation with a nonlinear “drift” operator. It was assumed that B = 1, while the operator A satisfies a monotonicity condition. Theory of monotone operators is one of the most beautiful areas of modern nonlinear analysis. The foundations of this theory were laid in the works of Vainberg, Kachurovskii, Minty, and Browder [44–47]. Further development is reflected in the monographs [42, 48–50]. Application of this theory to nonlinear elliptic and parabolic equations are described in Ref. [51]. The work of Bensoussan and Temam [43] generalizes one of the methods from the monograph of Lions [42]. The method of monotonicity in application to stochastic evolution equations was further developed in the works of Pardoux [52, 53]. In these works a general stochastic evolution equation is considered with unbounded nonlinear operators of “drift” and “diffusion.” As a special case, the result of Pardoux contains the results of [41, 43, 54], as well as the deterministic situation described by Lions [42, Section 2.1]. The results of [25–27, 29–32] are also essentially covered, although here a region where they are formally distinct can be indicated: the situation is analogous to the difference between the theory of differential equations in divergence and nondivergence forms. The solution obtained in Ref. [53] belongs to the domains of the operators A and B and is measurable with respect to the σ-algebra generated by the Wiener process and the initial condition. The solution is constructed on a prescribed probability space; i.e., in correspondence with the terminology of stochastic ordinary differential equations, it is a strong solution. It should be noted that equations (1.3), (1.4), (1.5), and (1.6) do not satisfy the assumptions of Paradox; in fact, solutions of the equations (1.3), (1.4), (1.5) are constructed in the sense of (probabilistic) distributions. The works [55–61] are devoted to an investigation of the Navier-Stokes equation (1.5), while Eqs. (1.3) and (1.4) are investigated in Refs. [15, 61]. In these works, a measure is sought which is supported on the trajectories of solutions and is a solution of either the Kolmogorov or the Hopf equation (formally) associated with the equation in question.

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1.5. Content and Organization of the Work In the present work, we generalize the results of Pardoux. In considering the same situation as in Ref. [53], we admit dependence of the coefficients on the elementary outcome in a nonanticipating way. We show that certain conditions of Pardoux are superfluous; in particular, this is the case of the local Lipschitz condition on the “diffusion coefficient.” The Markov property of the solution (in t) is proved for equations with deterministic coefficients. On the basis of general results regarding the solvability of stochastic evolution equations, we investigate quasilinear stochastic Itˆ o partial differential equations (of any order) which satisfy the so-called condition of strong probability. In the deterministic case (B ≡ 0) this condition coincides with the well-known condition of strong parabolicity of Vishik [62]. In the linear case, we introduced the condition of strong parabolicity [63] and systematically investigated it [64]. An analogous condition in the linear situation was introduced also by Pardoux [53]. We point out that in the finite-dimensional case our results generalize somewhat Itˆ o’s classical theorem on the strong solvability of stochastic equations with random coefficient satisfying Lipschitz conditions [34, 65] (see Example 3.15). The paper is organized as follows. Aside from the introduction, it contains three big sections, each with its own introduction outlining the content and organization. Section 2 is devoted to the theory of stochastic integration in Hilbert spaces. In Section 2.2, the concept of a martingale and of a Wiener process in a Hilbert space is introduced, and stochastic integrals over these processes are described. In Sections 2.3 and 2.4, Itˆ o’s formula for the square of the norm of a semimartingale in a rigged Hilbert space is proved. This result plays an extremely important role in the entire theory. Section 3 is devoted to the proof of the main theorems about solvability and regularity of solutions of stochastic evolution equations. In Section 4, the results of Section 3 are applied to the Itˆ o stochastic partial differential equations. Sections 2.3 and 2.4, where Itˆ o’s formula for the square of the norm of a semimartinagle in a Hilbert space is establishes, and Section 3.4, where the finitedimensional case is considered, may be of independent interest for some readers. The exposition and notations in these sections are independent of the remainder of the work. On the other hand, we go much further here than is required by the rest of the text, so that a reader interested in a rapid acquaintance with the ideas of Itˆ o’s stochastic partial differential equations should concentrate on the statements of the main results in Sections 2.3, 2.4, and 3.4, while skipping most of the proofs.

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2. Stochastic Integration in Hilbert Spaces 2.1. Introduction The theory of stochastic integration in infinite-dimensional spaces is a broard and rapidly developing area of the theory of stochastic processes. A complete survey of this area is beyond the purpose of the present work, and we will discuss only “selected” questions of the theory of stochastic integration in Hilbert spaces which we use directly in the sequel. There are two groups of these “selected” questions: a) construction of a stochastic integral over a square-integrable martingale with values in a Hilbert space; b) derivation of Itˆ o’s formula for the square of the norm of a semimartinagale is a rigged Hilbert space. In Section 2.2 we consider questions of group a) in a very simple situation — the integration with respect to a continuous martingale. This section is entirely of survey character and there are practically no proofs given. It is based on Refs. [32, 66, 67]. The problem formulated in part b) occupies a central spot and is solved in Sections 2.3 and 2.4. It is assumed that a Hilbert space H is rigged by a pair of Banach spaces V and V 0 , i.e. there are the following dense embeddings: V ⊂ H ⊂ V 0 ), and we consider a “semimartingale” of the form Z t v (t) = v 0 (s) ds + m (t) , 0

0

0

where v ∈ V, v ∈ V , and m(t) is a martingale in H. Itˆ o’s formula is derived for kv(t)k2H . The derivation of Itˆ o’s formula for the square of the norm of a semimartingale with all components take value in a single Hilbert space differs little from the finitedimensional case. However, in passing from this situation to the one described above, the jump in complexity is comparable to the jump in complexity in passing from bounded to unbounded operators. This comparison is more apt than is apparent at a first glance. We will see below that evolution equations with unbounded operators cannot be considered in a single space — it is necessary to separate the domain of the operators (the space V ) from the range of the operators (the space V 0 or H). This situation was first considered by Pardoux [53]. Using entirely different methods, we generalize his result to non-reflexive and non-separable spaces and provide a much sorter proof. Itˆ o’s formula for the square of the norm is essentially used in Section 3 to derive a priory estimates and to prove uniqueness and continuity of the solution. To conclude the section we give a brief and, of course, incomplete survey of existing results related to stochastic integrals in Hilbert spaces.

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The first important work in this direction was apparently the work of Daletskii [17], where he constructed a Wiener process with an identity covariance operator in a Hilbert space, or, more precisely, in a certain nuclear extension of this space, and defined a stochastic integral. Later, on the basis of the work of Gross [68, 69], Kuo [70] studied a stochastic integral with respect to an abstract Wiener process in a Banach space. The results of Daletskii were also extended in Refs. [71–73], and others. Kunita [66] initiated the study of the problem of integrability with respect to a square-integrable martingale in a Hilbert space. Metivier and his student subsequently achieved considerable progress in this direction [31, 32, 40, 73–75]. The important work of Meyer [67] should also be mentioned. The interested reader can systematically study the subject using Refs. [40, 67, 70, 74, 76]. 2.2. Stochastic Integrals in Hilbert Spaces We begin this section by recalling and discussing some very basic concepts and results about mappings of measurable spaces into Banach spaces. Let (S, Σ, µ) be a complete measure space, and let (X, X ) be a Banach space with the σ -algebra X of Borel sets relative to the strong topology. We denote by X ∗ the Banach space dual to X and by xx∗ the value of a functional x∗ ∈ X ∗ on x ∈ X. A mapping (function) x : S → X is called measurable if, for each Γ ∈ X , {s : x(s) ∈ Γ} ∈ Σ. A mapping x : S → X is called weakly measurable if for each x∗ ∈ X ∗ , x(s)x∗ is a measurable mapping of S into R. A mapping x : S → X is called strongly measurable if there exists a sequence of measurable simple functions converging to x(s), µ-almost surely. It is clear that the concept of strong measurability coincides with the concept of measurability if X is separable. Theorem 2.1 (Pettis). A mapping x : S → X is strongly measurable if and only if it is weakly measurable and there exists a set B ⊂ Σ such that µ(B) = 0, while the set of values of x(s) on S \ B is separable. In particular, if X is separable, then the concept of weak and strong measurability coincide. Proof.

See Yosida [51].



Let (Ω, F, P ) be a probability space with an expanding system of σ-algebras Ft ⊂ F, t ∈ R+ = [0, ∞). We assume that F0 is complete with respect to the measure P . Definition 2.2. A random variable in X is a measurable mapping of (Ω, F, P ) into X.

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Definition 2.3. We say that x = x(t, ω) is an Ft -consistent stochastic process in X if, for every t ≥ 0, x(t, ·) is a measurable mapping of (Ω, Ft , P ) into X. Definition 2.4. Completely measurable sets are the subsets of [0, ∞) × Ω that are elements of the smallest σ-algebra relative to which all real Ft -consistent right continuous process without discontinuities of the second kind are measurable in (t, ω). A mapping (process) x : [0, ∞) × Ω → X is called completely measurable if for any Borel set Γ ⊂ X the set {(t, ω) : x(t, ω) ∈ Γ} is completely measurable. In this paper we must sometimes consider random variables with values in embedded Banach spaces. Let V and H be two separable Banach spaces, where V is a subset of H and the embedding operator assigning to an element v ∈ V the corresponding element v ∈ H is a continuous operator from V to H. Lemma 2.5. a) If x is a random variable with values in V (relative to the Borel σ-algebra in V ), then x is a random variable with values in H. b) If x is a random variable with values in H, then {ω : x(ω) ∈ V } ∈ F. Proof. Assertion a) of the lemma is a consequence of the measurability of a superposition of measurable mappings. Assertion b) follows from the completeness of F and the fact that the continuous image of a Borel (in V ) set of V is analytic and is hence universally measurable in H; cf. Ref. [77].  We now define a martingale with values in a real Hilbert space H and the stochastic integral over such a martingale. We will consider only martingales m = m(t) that are strongly continuous in t and have a separable range. We temporarily denote by H1 the closed linear hull of the range of m(t, ω), t ≥R0, ω ∈ Ω. t If h(t, ω) ⊥ H1 , then, for all (t, ω), it is natural to set 0 h (s) dm (s) = 0. Therefore, in view of the orthogonal decomposition of H into H1 and H1L , it suffices to study integration of functions with values in H1 . These arguments make the following assumption natural, and we adopt it to the end of the section: H is a separable Hilbert space, identified with its dual in the natural way. For h1, h2 ∈ H we denote by h1 h2 the scalar product of h1 , h2 ; h21 = h1 h1 , |h1 | = 2 1/2 (h1 ) . For random variables in H with finite expectation of the norm, it is possible to define the conditional expectation in complete analogy to the finite-dimensional case. Namely, let G be a sub-σ-algebra of F and let x be a random variable in H with E|x| < ∞. Definition 2.6. The conditional expectation of x given G is the random variable E(x|G) in H such that, for every y ∈ H, yE(x|G) = E(yx|G) (a.s.). It is clear that the random variable so defined in unique (a.s.).

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Definition 2.7. A stochastic process x in H is a martingale relative to the family {Ft } if a) x is Ft -consistent, b) E|x(t)| < ∞ for all t ≥ 0, c) E[x(t)|Fs ] = x(s) (a.s.) for all 0 ≤ s ≤ t. The next theorem follows immediately from Definitions 2.6 and 2.7, and the equivalence of the three concepts of measurability in separable spaces. Theorem 2.8. A stochastic process x(t) in H with E|x(t)| < ∞ for all t ≥ 0 is a martingale relative to the family Ft if and only if, for every y ∈ H, the stochastic process yx(t) is a one-dimensional martingale relative to Ft . Definition 2.9. As in the finite-dimensional case, we say that a stochastic process x(t) in H is a local martingalea and write x ∈ Mloc (R+, H) if there exists a sequence of stopping times τn ↑ ∞ (a.s.) such that the process x(t ∧ τn ) is a martingale for each n. We call the sequence {τn } a localizing sequence. For simplicity we henceforth consider only (strongly) continuous martingales and local martingales. We denote by Mcloc (R+, H) the collection of all continuous local martingales in H issuing from zero. It is easy to see that the following results holds. Theorem 2.10. If x ∈ Mcloc (R+, H), then there exists a sequence {τn0 } of stopping times localizing x for which E supt≥0 |x(t ∧ τn0 )|2 < ∞. If E|x(t)|2 < ∞ for some t ≥ 0, then E sups
This follows from the equality

E |x(t ∧ τn ) − x(s ∧ τn )|2 |Fs = E (|x(t ∧ τn )|2 |Fs − |x(s ∧ τn )|2 ,

which, in turn, follows immediately from the martingale property of the Fourier coefficient of x in a basis in H.  Definition 2.12. An increasing process hxit for x ∈ Mcloc (R+ , H) is the increasing process in the Doob-Meyer decomposition of |x(t)|2 . From the Doob-Meyer theorem it follows that hxit is a.s. uniquely defined and continuous in t. a Here

and below we consider martingales and local martingales only relative to the family {F t }.

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As in the finite-dimensional case, if x, y ∈ Mcloc (R+ , H), then we set  1 hx + yit − hx − yit . hx, yit = 4

It is easy to verify that if the first assertion of Theorem 2.10 holds with the same sequence τn for both x(t) and y(t), then, for each t, s ≥ 0, t ≥ s, and each n, we have
E (x(t ∧ τn ) − x(s ∧ τn ))(y(t ∧ τn ) − y(s ∧ τn ))|Fs
= E hx, yit∧τn − hx, yis∧τn |Fs (a.s.)

Theorem 2.13 (Burkholder’s Inequality). If x ∈ Mcloc (R+ , H) and τ is an a.s. finite stopping time, then E sup |x(t)| ≤ 3Ehxi1/2 τ . t≤τ

A proof of this theorem can be found, e.g., in Ref. [77]. In H we fix an orthogonal basis {hi , i ≥ 1} and set xi (t) = hi x(t). It is known that, almost surely for each i, j, the measure on [0, ∞) generated by the process hxi , xj it is absolutely continuous with respect to the measure generated by hxit . For further exposition of the theory of martingales, we need the following construction. Let E be a separable Hilbert space which is naturally identified with its dual, let {ei i ≥ 1} be an orthonormal basis in E, let L(H, E) be the space of continuous linear operators from H to E, and let L2 (H, E) be subspace of L(H, E) consisting of all Hilbert-Schmidt operators. It is known that L2 (H, E) is a separable Hilbert space with the norm 1/2 !1/2  X X kBk = |Bhi |2 =  (ej , Bhi )2  , i

i,j

and kBk does not depend on the choice of bases in H and E. Given a symmetric nonnegative nuclear operator Q in L(H, H), we denote by LQ (H, E) the set of all linear (bounded or unbounded) operators B defined on Q1/2 H, taking Q1/2 H into E and having the property BQ1/2 ∈ L2 (H, E). For B ∈ LQ (H, E) we define |B|Q = kBQ1/2 k. It is known that if B ∈ L2 (H, E), then |B| ≤ kBk, B ∈ LQ (H, E), and |B|Q ≤ |B|(trQ)1/2 .

We return to x ∈ Mcloc (R+ , H). It can be shown that there exists a completely measurable process Qx = Qx (t, ω) with values in L2 (H, H) such that, for all (t, ω), the operator Qx (t, ω) is symmetric, nonnegative, and nuclear, trQx (t, ω) = 1, and hi Qx (t, ω)hj =

dhxi , xj it (dP × dhxit a.s.) dhxit

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for all i, j and any basis {hi }, where dP × dhxit is the differential of the measure Z ∞ µ(A) = E χA (t, ω)dhxit , 0

defined on the product of F and the Borel σ-algebra on [0, ∞). We call the process Qx the correlation operator of x. If B(t) is a completely measurable process in L2 (H, E) and Z t E kB(s)k2 dhxis < ∞ (a.s.) 0

for any t ≥ 0, then there exists a square-integrable martingale y(t) in E which is strongly continuous in t and such that, for every orthonormal basis {hi } and every v ∈ E, T ≥ 0, 2 n Z t X vB(s)hi d(hi x(s)) = 0. lim E sup v y(t) − n→∞ t≤T 0 i=1

Two processes possessing this property obviously coincide a.s. for all t. It is therefore correct to write Z t y(t) = B(s)dx(s). (2.1) 0

To compute y(t) we fix bases in H, E and set ∞ ∞ Z t X X y(t) = ei y i (t), y i (t) = ei B(s)hj d(hj x(s)), i=1

j=1

0

where the series converges in L2 (Ω, C([0, T ], E)), i.e. in the mean-square, uniformly on each finite time interval. By direct computation, Z t hyit = |B(s)|2Qs (s) dhxis , (2.2) 0

and this together with the inequality |B|Qx ≤ kBk leads to an extension of the stochastic integrals (2.1) to completely measurable functions B(s) such that, for all t ≥ 0, Z t kB(s)k2 dhxis < ∞. 0

This extension is done in the usual way and preserves the property (2.2). The resulting stochastic integral is continuous in t and is a local martingale. The following theorem shows that it is possible to extend the concept of a stochastic integral to an even larger class of processes B(s). 1/2

Theorem 2.14. Let B = B(s, ω) ∈ LQx (s,ω) (H, E) be such that BQx is a completely measurable process in L2 (H, E), and for each t the right-hand side of (2.2)

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is a.s. finite. Then the sequence  −1 Z t 1 1/2 yn (t) = B(s)Q1/2 (s) + Q (s) dx(s) x x n 0 converges in probability, uniformly with respect to t, to a limit denoted by y(t). Moreover, y(t) ∈ Mcloc (R+ , E) and equality (2.2) holds. 1/2

1/2

Proof. Note that Bn (s) = B(s)Qx (s)(1/n + Qx (s))−1 is a completely measurable process with values in L2 (H, E) and kBn (s)k ≤ n|B(s)|Qx (s) ,  X ai (s) 2 1/2 2 − |Bn (s) − Bm (s)|Q = |B(s)Qx (s)ei (s)| 1 n + ai (s) i≥1

1 m

ai(s) + ai (s)

2

,

where ei (s) are the eigenvectors of Qx (s) and a2i (s) are the corresponding eigenvalues. It is evident from this that |Bn (s)−Bm (s)|Q ≤ |B(s)|Q and |Bn (s)−Bm (s)|Q → 0 as n, m → ∞. Hence, Z t hyn − ym it = |Bn (s) − Bm (s)|2Qx (s) dhxis → 0, 0

and the assertions of the theorem are deduced in the well-known way.



Stochastic integrals for functions in LQx (H, E) have previously been defined in Ref. [32], but the construction presented here is somewhat different. By definition, the process y(t) in Theorem 2.14 is taken equal to the right-hand side of (2.1). If X is a separable Hilbert space and A ∈ L(E, X), then for all t (a.s.) Z t Ay(t) = AB(s)dx(s). (2.3) 0

We choose an element e ∈ E and by means of it define an operator eb ∈ L(E, R) by the formula eby = ey, where ey is the scalar product in E. From (2.3) we then have Z t ey(t) = ebB(s)dx(s). 0

1/2

We observe that the operator ebB(s) acts on Qx H by the formula h → eB(s)h, while the latter is equal to (B ∗ e, h) if B ∈ L(H, E). Finally, if h(s) ∈ H is completely measurable, and for all t ≥ 0, Z t Z t 2 2 b |h(s)|Qx (s) dhxis = |Q1/2 x (s)h(s)| dhxis < ∞ (a.s.), 0

then we write

0

Z

t

h(s)dx(s) = 0

Z

t 0

b h(s)dx(s).

We now introduce the concept of a Wiener process in H.

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Definition 2.15. Let Q be a nuclear symmetric nonnegative operator on H with trQ < ∞. A Wiener process relative to {Ft } in H with covariance operator Q is a continuous martingale w(t) with values in H, correlation operator (trQ)−1 Q, and such that w(0) = 0, hwit = trQ · t. It is known that, for every suitable probability space and every nuclear symmetric nonnegative operator Q on H with trQ > 0, there exists a Wiener process with covariance operator Q. Clearly, Ew 2 (t) = trQ · t. Stochastic integrals with respect to a Wiener process possess especially good properties. For example, they are defined not only for completely measurableRB(s) but also for operators measurable in t (s, ω) that are Fs -consistent and satisfy 0 |B(s)|2Q ds < ∞ (a.s.) for all t ≥ 0. We conclude this section with a remark that, in place of an infinite time interval above, we could consider a segment of the form [0, T ]. To have the possibility of doing this, it suffices to extend the processes in question to t ≥ T by setting them equal to the value they assume at t = T . 2.3. Itˆ o’s Formula for the Square of the Norm Let V be a Banach space, let V ∗ be the dual space of V , and let H be a Hilbert space (we assume that all three are real spaces). If v ∈ V (h ∈ H, v ∗ ∈ V ∗ ), then |v| (resp. |h|, |v ∗ |) denotes the norm of v (h, v ∗ ) in V (H, V ∗ ); if h1 , h2 ∈ H, then h1 h2 denotes the scalar product of h1 , h2 ; the result of the action of a functional v ∗ ∈ V ∗ on an element v ∈ V is written as either vv ∗ or v ∗ v. Let Λ be a bounded, linear operator acting from V to H such that ΛV is dense in H. We consider three processes v(t, ω) ∈ V, h(t, ω) ∈ H, v ∗ (t, ω) ∈ V ∗ defined for t ≥ 0 on some complete probability space (Ω, F, P ) and connected with some expanding family of complete σ-algebras Ft ⊂ F, t ≥ 0, in the following way: (1) v(t, ω) is strongly measurable in (t, ω) and is weakly Ft -measurable in ω for almost all t; (2) for every g ∈ V the quantity gv ∗ (t, ω) is Ft - measurable in ω for almost every t and is measurable in (t, ω); (3) h(t, ω) is strongly continuous in t, is strongly measurable in ω relative to Ft for each t, and is a local semimartingale: there exist strongly Ft -measurable continuous processes A(t), m(t) in H such that m(t) is a local martingale, the trajectories A(t, ω) for each ω have a finite variation on bounded time intervals, and h(t) = A(t) + m(t). We fix p ∈ (1, ∞) and set q = p/(p−1). We assume that |v(t)| ∈ Lp ([0, T ]) (a.s.) for all T ≥ 0 and there exists a function f (t, ω) measurable in (t, ω) such that f (t) ∈ Lq ([0, T ]) (a.s.) for all T ≥ 0 and |v ∗ (t)| ≤ f (t) for all (t, ω). Regarding the last condition, it is useful to note that |v ∗ (t)| is, generally speaking, not measurable. This norm is measurable, e.g., if V is separable, and in this case |v ∗ (t)| ∈ Lp ([0, T ])(a.s.) for all T ≥ 0.

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We formulate the main result regarding Itˆ o’s formula. Theorem 2.16. Let τ be a stopping time and suppose that, for every g ∈ V almost everywhere on the set {(t, ω) : t < τ (ω)}, Z t (Λg)(Λv(t)) = gv ∗ (s)ds + (Λg)h(t). (2.4) 0

˜ Then there exist a set Ω ⊂ Ω and a function h(t) with values in H such that 0

˜ a) P (Ω0 = 1), h(t) is strongly Ft - measurable on the set {ω : t < τ (ω)} for any ˜ ˜ t, h(t) is continuous in t on (0, τ (ω)) for every ω, and Λv(t) = h(t) (a.s. on {(τ, ω) : t < τ (ω)}; b) for ω ∈ Ω0 and t < τ (ω) Z t Z t ∗ 2 2 ˜ ˜ v(s)v (s)ds + 2 h(s)dh(s) + hmit ; (2.5) h (t) = h (0) + 2 0

0

0

c) if V is separable, then, for ω ∈ Ω , t < τ (ω), g ∈ V Z t ˜ = (Λg)h(t) gv ∗ (s)ds + (Λg)h(t);

(2.6)

0

d) if V is separable and (2.4) is satisfied for some t ≥ 0 and each g ∈ V (a.s.) ˜ on {ω : t < τ (ω)}, then Λv(t) = h(t) (a.s.) on {ω : t < τ (ω)}. We take up the proof of this theorem after discussing its hypotheses and assertions. ˜ The stochastic integral in (2.5) exists if h(s) is completely measurable and, for t < τ (ω), Z t Z t 2 ˜ ˜ |h(s)|dkAk |h(s)| dhmis < ∞, s+ 0

where

0

   ∞  X k+1 k kAks = lim −A s ∧ n . A s ∧ 2n n→∞ 2 k=0

˜ Both these conditions are satisfied because h(s) is continuous in s and is Fs consistent, while hmit + kAkt < ∞. We point out that by a stochastic integral we always understand a continuous (for all ω) process. Further, since v(s) is strongly measurable, while v ∗ (s) is weakly measurable, v(s)v ∗ (s) is measurable in (s, ω), and by our assumptions it is locally integrable in s (a.s.) All expressions in (2.5) are therefore meaningful. Assertion d) is a simple corollary of the preceding assertions. Indeed, from (2.4) ˜ and (2.6) we have (Λg)h(t) = (Λg)Λv(t) (a.s.) on {ω : t < τ (ω)} for every g ∈ V . ˜ But V is separable, and therefore (Λg)h(t) = (Λg)Λv(t) for all vg ∈ V , a.s. on {ω : τ < τ (ω)}. Since ΛV is dense in H, assertion d) follows.

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We now show that it is enough to prove the theorem for separable H and V Indeed, since process v(t) is strongly measurable in (t, ω), there exists a process v 0 (t) which coincides with v(t) for almost all (t, ω) and has its range in some separable subspace V 0 ⊂ V . It is shown similarly that on a set of full probability Ω00 all the values of h(t, ω), t ≥ 0, ω ∈ Ω00 lie in some separable subspace H 0 ⊂ H. We may assume with no loss of generality that Ω00 = Ω. By hypothesis, ΛV is dense in H. Hence, there exists a separable subspace V 00 ⊂ V such that H 0 ⊂ ΛV 00 . Suppose now that V1 is the closed space spanned by V 0 ∪ V 00 and H1 is the closed space spanned by ΛV1 . The spaces V1 , H1 are separable, and ΛV1 is dense in H1 . Further, v 0 (t) ∈ V1 , h(t) ∈ H1 , while the functionals v ∗ (t) on V are also functionals on V1 . It may therefore be assumed that v ∗ (t) ∈ V1∗ and the norm of v ∗ (t) in V1∗ is no larger than |v ∗ (t)|V ∗ . Relation (2.4) is preserved for every g ∈ V1 (even for g ∈ V ), because v(t) = v 0 (t) a.s. in (t, ω). Hence, if Theorem 2.16 is true for separable V and H, then we obtain the fist assertion in the general case by applying it to V1 , H1 . We may thus assume with no loss of generality until the end of the section that V and H are separable. We will now explain why (2.5) is called Itˆ o’s formula for the square of the norm. For this purpose we place all processes v(t), h(t), v ∗ (t) in a single space. In those cases where the same vector belongs to various spaces we equip its norm with the symbol of the space in which it is considered. Suppose that the space V is a (possibly, non-closed) subspace of H, is dense in H in the norm of H, and |ϕ|H ≤ N |ϕ|V for all ϕ ∈ V , where N does not depend on v. Suppose that H is, in turn, a subspace of some Banach space V 0 and that H is dense in V 0 . Then V ⊂ H ⊂ V 0.

(2.7)

We assume that the scalar product in H possesses the following property: if ϕ ∈ V , ψ ∈ H, then |ϕψ| ≤ |ϕ|V |ψ|V 0 . Since the embeddings in (2.7) are dense, it is possible to uniquely define ϕψ for ϕ ∈ V , ψ ∈ V 0 as limn→∞ ϕψn , where ψn ∈ H and |ψ − ψn |V 0 → 0. Obviously, for a fixed ψ ∈ V 0 , the mapping ϕ 7→ ϕψ is a bounded linear functional on V . We suppose that for ψ ∈ V 0 the equality ϕψ = 0 for all ϕ ∈ V implies that ψ = 0. Then the mapping which, to every ψ ∈ V 0 assigns the corresponding functional ϕ 7→ ϕψ on V , is a one-to-one mapping of V 0 into some subset of the dual space V ∗ of V . We now make an additional assumption that every bounded linear functional on V has the form ϕ 7→ ϕψ for some ψ ∈ V 0 , and then the mapping V 0 → V ∗ mentioned above becomes both one-to-one and onto so that V 0 can be identified with V ∗ if desired. We note that under this identification the norms of ψ as an element of V 0 and as an element of V ∗ are, in general, different, and in this section we will not identify V 0 with V ∗b . Finally, we suppose that that there is a function v 0 (t, ω) with values in V 0 such that, for every b On

the contrary, in the following two sections we will make this identification.

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g ∈ V , gv 0 (t) is measurable in (t, ω) and, for every t ≥ 0, is Ft -measurable in ω, and also |v 0 (t)|V 0 ≤ f (t) for all (t, ω), where the function f is the same as on page 16. Then, for every g ∈ V , t ≥ 0, we have Z t Z t 0 ≤ |g| gv (s)ds f (s)ds. V 0

0

Therefore, from the properties of f (t), it follows that there exists a set Ω0 ⊂ Ω such Rt that P (Ω0 ) = 1 and for ω ∈ Ω0 , t ≥ 0, the mapping g 7→ gv 0 (s)ds is a bounded 0

linear functional on V . Under our assumptions, this functional can be written in the form g 7→ gψ(t), where ψ(t) ∈ V 0 , and, for ω ∈ Ω0 we have ψ(t) =

Zt

v 0 (s)ds.

0

Theorem 2.16 then takes the following form, where, for simplicity of the formulation, we take τ = ∞; generalization to the case of arbitrary τ is obvious. Theorem 2.17. For ω ∈ Ω0 and t ≥ 0 we define Z t ˜ v 0 (s)ds + h(t), h(t) =

(2.8)

0

˜ and suppose that h(t) = v(t) for almost all (t, ω). Then there exists a set Ω00 ⊂ Ω0 00 such that P (Ω ) = 1 and, for ω ∈ Ω00 , the function ˜ h(t) takes values in H, is continuous in H with respect to t, is strongly Ft -measurable with respect to ω for each t (as a function with values in H), and Z t Z t ˜ 2 (t) = h2 (0) + 2 ˜ h v(s)v 0 (s)ds + 2 h(s)dh(s) + hmit . (2.9) 0

0

We note that relation (2.9) is obtained if the rules for computing the stochastic ˜ 2 (t) for the process h(t) ˜ differential dh defined in (2.8) are applied. One of the difficulties in justifying (2.9) is that it is generally not clear why ˜ h(t) ∈ H (in other 2 ˜ ˜ words, why h (t) exists), because equation (2.8) only defines h(t) as a process with ˜ ∈ V for all (t, ω). values in V 0 , and it is generally not true that h(t) Proof. [of Theorem 2.17 using Theorem 2.16] We take for Λ the identify operator and use the fact that the formulation of Theorem 2.17 does not contain the process v ∗ (t). We construct a process v ∗ (t) on the basis of v 0 (t) as the process corresponding to v 0 (t) under the mapping V 0 → V ∗ . Then v ∗ (t) satisfies the assumptions preceding Theorem 2.16, and (2.4) follows from (2.8). On the basis of Theorem 2.16, we ˜1 = h ˜ 1 (t) such that P (Ω00 ) =1, ˜ construct a set Ω00 ⊂ Ω0 and a process h h1 is 1 continuous in H with respect to t, is an Ft -consistent process, and ˜ h (t) = v(t) 00 ˜ ˜ 1 , and for for almost all (t, ω); for ω ∈ Ω formula (2.9) holds if h is replaced by h

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ω ∈ Ω00 , g ∈ V , t ≥ 0, ˜ 1 (t) = gh

Zt

˜ gv 0 (s)ds + gh(t) = g h(t).

0

˜ 1 (t) − h(t)) ˜ From the equality of the extreme terms, it follows that g(h = 0 for 1 00 ˜ ˜ ˜ 1 we now all g ∈ V , and hence h = h for t ≥ 0, ω ∈ Ω ; from the properties of h ˜ The proof of the theorem is complete.  obtain all the required properties of h. We proceed to prepare for the proof of Theorem 2.16. We have already agreed to consider sparable V and H. Further, if in place of τ in Theorem 2.16 we take τ ∧ n, prove the theorem for τ ∧ n and then let n → ∞, then we obtain the proof of the theorem for τ . It may therefore be assumed that τ is a bounded stopping time, and then a non-random change of time reduces everything to the case τ ≤ 1. We further note that both |v(s)| and |v ∗ (s)| are Fs -measurable for almost all s and are measurable in (s, ω). Hence, the process Z t Z t p 2 r(t) = h (0) + kAkt + hmit + |v(s)| ds + |v ∗ (s)|q ds 0

0

is Ft -measurable for each t and is continuous in t (a.s). This implies that for every n ≥ 0 τn = inf {t ≥ 0 : r(t) ≥ n} ∧ τ is a stopping time. Since τn ↑ τ , it suffices to prove Theorem 2.16 with τ in its formulation replaced by τn . Moreover, process r(t) is bounded in (t, ω) on {(t, ω) : t ≤ τn (ω)}, and it may therefore be assumed in the proof of Theorem 2.16 that the process r(t) is bounded on {(t, ω) : t ≤ τ (ω)}. For t > τ , we set v(t) = 0, v ∗ (t) = 0, h(t) = h(τ ), thus ensuring that the process r(t) is bounded on [0, ∞) × Ω. After this, multiplying (2.4) by a suitable constant ε and replacing v, v ∗ , h by εv, εv ∗ , εh, we reduce the matter to the case where r(t) ≤ 1. These arguments show that it suffices to prove Theorem 2.16 in the special case considered below. 2.4. Proof of Theorem 2.16 In this section we prove Theorem 2.16 under the additional assumptions that V and H are separable, τ ≤ 1, and, for all ω, Z1 0

p

|v(s)| ds +

Z1 0

q

|v ∗ (s)| ds + kAk1 + hmi1 + h2 (0) ≤ 1

(2.10)

It was shown above that these additional assumptions do not lead to any loss of generality. Lemma 2.18. There exist (a) a sequence of nested partitions 0 = tn0 < tn1 < . . . < tnk(n)+1 = 1

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of the interval [0, 1] with diameter tending to zero, and (b) a set Ω0 ⊂ Ω with the following properties: 1) P (Ω0 ) = 1 and, if ω ∈ Ω0 and t ∈ I = {tni ; i = 1, . . . , k(n), n ≥ 1} are such that t < τ (ω), then equality (2.4) is satisfied for all g ∈ V and, for every s ∈ I, the quantity v(s) is Fs -measurable; 
(1) 2) Define the processes vn1 (t) and vn2 (t) by vn (t) = v(tni ) for t ∈ tni , tni+1 , n n
(1) (2) n i = 1, . . . , k(n), vn (t) = 0 for t ∈ [0, tn1h); vn (t)  = v(ti+1 ) for t ∈ ti , ti+1 , (2)

i = 1, . . . , k(n) − 1, vn (t) = 0 for t ∈ tnk(n) , 1 . Then, for j = 1, 2 p E sup vn(j) (t) < ∞, t∈[0,1]

lim E

n→∞

Z

1

0

p v(t) − vn(j) (t) dt = 0.

(2.11)

Proof. Let [a] be the integer part of the number a, (for a > 0, [a] is the largest integer smaller than a) and let κ 1 (n, t) = 2−n [2n t], κ 2 (n, t) = 2−n [2n t] + 2−n , v(t) = 0 for t ∈ [0, 1]. Standard arguments of Doob show that there exists a sequence of integers rn → ∞ such that, for j = 1, 2 and almost all s ∈ [0, 1], Z 1 v(t) − v(κ j (rn , t + s) − s) p dt = 0 (2.12) lim E rn →∞

0

Further, it follows from Fubini’s theorem and separability of V that there exists a set T ⊂ [0, 1] of unit Lebesgue measure such that, for all t ∈ T and all g ∈ V , equality (2.4) is satisfied (a.s) on {ω : t < τ (ω)}, and the quantity v(t) is Ft -measurable. It is clear that, for every s ∈ [0, 1], all values of the functions κ j (rn , t+s)−s for t ∈ [0, 1], j = 1, 2, n ≥ 1, lying in [0, 1] also belong to T . We fix a suitable s so that (2.12) is also satisfied; we define {tni } as the set of values of κ 1 (rn , t+s)−s for t ∈ [0, 1] which lie in [0, 1], to which we add the points 0 and 1, and we denote by Ω0 the set of ω for which equation (2.4) is satisfied for all g ∈ V, t = tni < τ (ω), i = 1, . . . , k(n), n ≥ 1. All assertions of the lemma are then valid except possibly for the first inequality in (2.11). We note, however, that by virtue of the second inequality in (2.11), for sufficiently large n, Z 1 (j) p E vn (t) dt < ∞ 0

This inequality is equivalent to the first inequality in (2.11), which is thus valid for large n. For small n it is clearly valid since our partitions are nested. The proof of the lemma is complete.  Lemma 2.19. For ω ∈ Ω0 , t, s ∈ I, s ≤ t ≤ τ (ω) Z t 2 2 2 |Λv(t)| − |Λv(s)| = 2 v(t)v ∗ (u)du + 2Λv(s)(h(t) − h(s)) + |(h(t) − h(s)| s

2

− |Λ(v(t) − v(s)) − (h(t) − h(s))| ,

(2.13)

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2

|Λv(t)| = 2

Z

s

t

2

v(t)v ∗ (u)du + h2 (t) − |Λ(v(t) − (h(t)| .

(2.14)

The proof of this lemma is based on using (2.4) with g = v(t) or g = v(s) and simple algebraic transformations, which we leave to the reader while suggesting that (2.14) be derived first and (2.13) then proved by subtracting the appropriate equalities (2.14). Lemma 2.20. 2

E sup |Λv(t)| < ∞. t∈I, t<τ

Proof.

From (2.14) and (2.11), with t ∈ I, 2

Eχt<τ |Λv(t)| ≤ Eh2 (t) +

2 2 p t E |v(t)| + E p q

Z

t 0

q

|v ∗ (s)| ds < ∞.

(2.15)

Further, by (2.13) and (2.14), with t = tni < τ (ω), ω ∈ Ω0 Z t Z t 1 2 Λvn (t) = h2 (0) + 2 vn2 (s)v ∗ (s)ds + 2 Λvn1 (s)dh(s) + 2h(0)(h(tni ) − h(0)) 0

0

i−1 i−1 X X n
h(t ) − h(tn ) 2 − Λ v(tn ) − v(tn ) − (h(tn ) − h(tn )) 2 , + j+1 j j+1 j j+1 j j=1

j=0

(2.16)

where in the last sum the term corresponding to j = 0 is taken equal to 2 |Λ(v(tn1 ) − h(tn1 )| . To estimate the second therm on the right side of (2.16), we apply (2.10) and the Burkholder inequality. We then obtain Z n Z t t1 1 1 E sup χtn <τ Λvn (u)dh(u) ≤ E sup χu<τ Λvn (u)dh(u) 1 0 i≥1 t≤1 0 ≤ E sup |Λv(tni | χtn <τ kAk1 + 3E i

i≥1

≤ 4E sup |Λv(tni | χtn <τ ≤ i

i≥1

Z

1

0

2 χu<τ Λvn1 (u) dhmiu

1/2

1 2 E sup |Λv(tni | χtn <τ + 16. i 4 i≥1

We note that it follows from (2.15) that the last expression is finite. Moreover, from (2.10) k(n)

E

X h(tn ) − h(tn ) 2 ≤ 4E kAk2 + 2Em2 (1) ≤ 6. j+1 j 1 j=0

From this and from (2.16)

E sup |Λ(v(tni )| χtn <τ ≤ i≥1

i

4 E p

Z

1 0

4 (2) p vn (t) dt + + 100. q

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Letting here n → ∞ and using the fact that by (2.11) the right side is bounded in n, we obtain the assertion of the lemma. The proof of the lemma is complete.  Lemma 2.21. Let

 Ω00 = Ω0 ∩ ω :

sup t∈I, t<τ

 |Λv(t)| < ∞ .

˜ = h(t) ˜ Then there exists a function h with values in H, which, for ω ∈ Ω00 and t ∈ I, t < τ (ω), is weakly continuous in H with respect to t and satisfies (2.6) for ˜ all g ∈ V Moreover, h(t) is Ft -measurable on {ω : t < τ (ω)}, ˜ h(t) = Λv(t) (a.s.) ˜ on {(t, ω) : t < τ (ω)}, h(t) = Λv(t) for ω ∈ Ω00 , t ∈ I, t < τ (ω). Proof. For ω ∈ Ω00 , t ∈ I, t < τ , g ∈ V the function (Λg)Λv(t) coincides with the right side of (2.6), which is continuous in t. Hence, for s < τ there exists lim (Λg)Λv(t) < τ . I 3t→s t<τ

Since |Λv(t)| is bounded on I ∩ {t < τ }, this means that for s < τ , I 3 t → s there exists the weak limit of Λv(t), which we denote by ˜ h(s). It is clear that ˜ h(t) 00 satisfies (2.6) for all ω ∈ Ω , t < τ (ω) g ∈ V . The assertions of the lemma follow from this in an almost obvious way. The proof of the lemma is complete.  ˜ = h(τ ˜ ) for t ≥ τ , h(t) ˜ = 0 for ω ∈ We set h(t) / Ω00 , and we note that for ω ∈ Ω00 ˜ (2.17) sup h(t) = sup |Λv(t)| < ∞. t∈I, t<τ

t≤1

R t ˜ ds is well defined, because h(s) ˜ h(s) is 0 ˜ completely measurable due to separability of H and continuity of π h(s) for ω ∈ Ω00 , Moreover, for t ≤ 1, the integral

where π is the operator of projecting onto some finite-dimensional subspace of H.

˜ n (t) = h(t ˜ n ) for t ∈ [tn , tn ), i = 1, . . . , k(n). Then Lemma 2.22. Define h i i i+1 Z t Z t ˜ ˜ hn (s)dh(s) − h(s)dh(s) = 0 (2.18) lim sup n→∞ t≤1

0

0

in probability.

Proof. Let h1 , . . . , hr , . . . be an orthonormal basis in H, and let πr be the operator ˜ is continuous on projecting H onto the space spanned by h1 , . . . , hr . Since πr h(s) 00 Ω , it follows that (a.s.)   Z t Z t ˜ ˜ ˜ ˜ lim πr hn (s) − πr h(s) d hmis + πr hn (s) − πr h(s) d kAks = 0. n→∞

0

0

It therefore suffices to prove that, for every ε > 0, Z t   ˜ lim sup P sup (1 − πr )hn (s)dh(s) > 2ε = 0 r→∞ n

t≤1

0

(2.19)

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and lim P

r→∞



Z t  ˜ sup (1 − πr )h(s)dh(s) > 2ε = 0. t≤1 0

We shall prove only the first equality. The second is proved similarly. Noting that πr is a self-adjoint operator, for every N , δ > 0 we estimate the probability in (2.19) as follows: Z t Z t   
δ ˜ ˜ P sup hn (s)d (1 − πr )h(s) > 2ε ≤ 2 + P hn (s) d k(1 − πr )Aks > ε ε t≤1 0 0   Z t  2 δ ˜ ˜ +P >N hn (s) d h(1 − πr )mis > δ ≤ 2 + 2P sup h(s) ε s≤1 0 N N + Ek(1 − πr )Ak1 + E |(1 − πr )m(1)|2 . ε δ (2.20) We denote by hi the i-th coordinate of h ∈ H in the basis {hi }, and we set a (t) = dAi (t)/d kAkt . Then as r → ∞ !1/2 Z 1 X
2 k(1 − πr )Ak1 = ai (t) d kAkt → 0, i

0

2

|(1 − πr )m(1)| =

X

i>r

mi (1)

i>r


2

→ 0.

From this and from (2.20) and (2.10) we see that the left side of (2.19) does not exceed   δ ˜ + 2P sup h(s) > N . ε2 s≤1

Since δ, N are arbitrary and (2.17) is satisfied, the last expression can be made arbitrarily small. The proof of the lemma is complete.  We now define the set Ω000 which will play the role of Ω0 in Theorem (2.16). It is possible to find a sequence {n0 } along which the left side of (2.17) tends to zero (a.s.). It may be assumed with no loss of generality that the original sequence has this property. Moreover, we set π0 = 0; then, as is well know, in probability for r > 0, t ∈ [0, 1], X (1 − πr )(h(tn ) − h(tn ) 2 = h(1 − πr )mit . (2.21) lim j+1 j n→∞

tn j+1 ≤t

Therefore, there exists a subsequence along which the last equality, understood in the sense of pointwise convergence, is true for all r > 0, t ∈ I almost surely. To simplify the notation, we assume that this subsequence also coincides with the original sequence. We set Z t   ˜ ˜ Q1 = ω : lim sup hn (s) − h(s))dh(s) = 0 , n→∞ t≤1 0

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Q2 =

 ∞ \ \

r=0 t∈I

  X (1 − πr )(h(tnj+1 ) − h(tnj ) 2 = h(1 − πr )mit , ω : lim  n→∞ n  tj+1 ≤t

Q3 =



ω : lim

n→∞

Z

1

0

n

 p (i) vn (s) − v(s) ds = 0, i = 1, 2 ,

o Q4 = ω : lim h(1 − πr )mii = 0 , r→∞

Q000 = Q00

4 \ \

Qi ,

i=1

˜ ω) = 0 on the complement of Q000 . and set h(t, From what has been said above, P (Qi ) = 1, i = 1, 2. From Lemma 2.18 it follows that it may be assumed that P (Q3 ) = 1. Since the sequence h(1 − πr )mi1 , r ≥ 0, is decreasing (a.s.) and Eh(1 − πr )mi1 = E |(1 − πr )m(1)|2 → 0, it follows that P (Q4 ) = 1. Thus, P (Q000 ) = 1. Lemma 2.23. For ω ∈ Q000 , t, s ∈ I, s < t < τ (ω) Z t 2 ˜ ˜ (v(u) − v(s)) v ∗ (u)du h(t) − h(s) = 2 s Z t  ˜ ˜ h(u) − h(s) dh(u) + hmit − hmis , +2

(2.22)

s

Z t Z t ˜ 2 2 ∗ ˜ h(t) = h (0) + 2 v(u)v (u)du + 2 h(u)dh(u) + hmit . 0

(2.23)

0

Proof. We first prove (2.23). We fix ω ∈ Q000 , t ∈ I, t < τ (ω). From (2.16) and Lemma 2.21 for n such that t is a point of the partition tn0 < tn1 < . . . tnk(n)+1 , we have Z t Z t ˜ 2 (t) = h2 (0) + 2 ˜ n (u)dh(u) h vn(2) (u)v ∗ (u)du + 2 h 0

0

 X X 
2 ˜ n ) − h(t ˜ n ) − h(tn ) − h(tn ) . h(tn ) − h(tn ) 2 − + h(t j j+1 j j+1 j+1 j tn j+1 ≤t

tn j+1 ≤t

Letting n → ∞, we find

˜ 2 (t) = h2 (0) + 2 h where J = lim

n→∞

Z

t ∗

v(u)v (u)du + 2 0

Z

0

t

˜ h(u)dh(u) + hmit − J,

 X 
2 ˜ n ) − h(tn ) − h(tn ) , h(tnj+1 ) − h(t ˜ j j+1 j

tn j+1 ≤t

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and the last limit exists and is finite. For the proof of (2.23) it suffices to show that J = 0. It is convenient to assume that the basis {hi } is formed from elements of the set ΛV . Since the latter is dense in H, this last assumption involves no loss of generality. Moreover, it is obvious that for every r ≥ 1 there exists a function v˜r (t) continuous in the norm of V such that Λ˜ vr (t) = πr h(t) for all (t, ω). On the basis (i) of the function v˜r we construct functions v˜r,n , i = 1, 2 just as in Lemma 2.18 the (i) functions v˜r are constructed on the basis of v. Noting that by (2.4) and Lemma 2.21 for tnj+1 ≤ t and every ϕ ∈ V there is the equality 






˜ n ) − h(t ˜ n ) − h(tn ) − h(tn )) Λϕ = (h(t j+1 j j+1 j

we find easily for any r ≥ 0 J = lim

n→∞

− lim

n→∞

Z

t

(vn(2) (u)

(vn(1) (u))v ∗ (u)du

Z

tn j+1

Z

ϕv ∗ (u)du,

tn j

t

(2) (1) (˜ vr,n (u) − (˜ vr,n (u))v ∗ (u)du  X  ˜ n ) − h(t ˜ n )) − (h(tn ) − h(tn )) (1 − πr )(h(tn ) − h(tn )). (h(t j+1 j j+1 j j+1 j 0

−

− lim

n→∞

0

tn j+1 ≤t

Here the first limit is equal to zero; the second is equal to zero, because ω ∈ Q3 (2) (1) and the continuity of v˜r (u) implies limn→∞ ˜ vr,n (u) − v˜r,n (u) = 0 uniformly in u. Hence,   2  X  ˜ n ) − h(t ˜ n ) − (h(tn ) − h(tn ))  J ≤ lim  (h(t j+1 j j+1 j n→∞



tn j+1 ≤t

1/2 X
2 1/2 (1 − πr ) (h(tnj+1 ) − h(tnj )  × = J 1/2 h(1 − πr )mit . tn j+1 ≤t

As r → ∞ this implies that J = 0. Equality (2.23) is proved. Equality (2.22) is 2 deduced from (2.23) by means of the relations (a − b) = a2 − b2 − 2b(a − b) and ˜ ˜ − h(s)) ˜ −2h(s)( h(t) = −2

Z

s

The proof of the lemma is complete.

t

v(s)v ∗ (u)du − 2

Z

t

˜ h(s)dh(u).

s



We now finish the proof of Theorem 2.16. Because of (2.21) and (2.23) it remains ˜ ω) in t for t < τ (ω), ω ∈ Ω000 . Since for us to prove the strong continuity of h(t, a weakly continuous function with a continuous norm is strongly continuous, it suffices to prove (2.23) for t < τ (ω), ω ∈ Ω000 . For t = 0 (2.23) is obvious. We fix t > 0, t < τ (ω), ω ∈ Ω000 .

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For all sufficiently large n it is possible to define j = j(n) such that 0 < tnj ≤ t < tnj+1 . We set t(n) = tnj(n) and note that t(n) ↑ t, Z t Z 1 lim |v(u) − v(t(n))| · |v ∗ (u)| du ≤ lim v(u) − vn(1) (u) · |v ∗ (u)| du = 0, n→∞

n→∞

t(n)

0

Z Z s s ˜ ˜ ˜ ˜ lim sup h(u) − h(t(n))dh(u) ≤ 2 lim sup h(u) − hn (u))dh(u) = 0, n→∞ s≤t t(n) n→∞ s≤1 0
lim hmit − hmit(n) = 0.

n→∞

Therefore, there exists a subsequence n(k) such that for s(k) = t(n(k)) we have  1/2  s(k+1) Z ∞  X
1/2   |v(u) − v(s(k))| · |v ∗ (u)| du + hmis(k+1) − hmis(k)   k=1   s(k)    s(k+1) Z   1/2    ˜ + h(u) −˜ h(s(k))dh(u)  < ∞.    s(k) From (2.22) we then find that ∞ X ˜ ˜ h(s(k + 1)) − h(s(k)) < ∞. k=1

˜ Therefore, h(s(k)) for k → ∞ has a strong limit. Since s(k) → t, it follows that ˜ ˜ ˜ ˜ h(s(k)) converges weakly to h(t). Thus, h(s(k)) → h(t) strongly in H, and, substituting the numbers s(k) in place of t in (2.23), and passing to the limit k → ∞, we obtain (2.23) for the t chosen. This complete the proof of Theorem 2.16. 3. Itˆ o Stochastic Equations in Banach Spaces and the Method of Monotonicity 3.1. Introduction In this chapter we consider the Itˆ o equations

v(t) = u0 +

Z

t

A(v(s), s)ds + 0

Z

t

B(v(s), s)dw(s)

(3.1)

0

in Banach spaces. The coefficients A(v, s), B(v, s) of “drift” and “diffusion” are generally assumed to be unbounded non-linear operators. They may depend on the elementary outcome in a nonanticipatory fashion. By w we understand a Wiener process with values in some Hilbert space.

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An existence and uniqueness theorem will be proved for the solution of an equation slightly more general than (3.1) and certain qualitative results on the solution will be obtained. A solution is understood to be a trajectory with values in the domain of the operators A(·, t), B(·, t) (the domain of the operators A and B is assumed to be independent of t) that satisfies (3.1) and is consistent with the same system of σalgebras as w(t), A(·, t), and B(·, t). This system is assumed to be given together with the original probability space and the Wiener process. A solution is thus understood in the “strong” sense. The main conditions on A an B are the conditions of monotonicity and coerciveness; see (A2 ), (A3 ) in Section 3.2. The following equations satisfy these assumptions in spaces of Sobolev type: ! m p−2 m m ∂ ∂ m+1 ∂ u(t, x) u(t, x) dt du(t, x) = a(t, ω)(−1) ∂xm ∂xm ∂xm (3.2) p/2 m ∂ + b(t, ω) m u(t, x) dw(t), ∂x p > 1, x ∈ G ⊂ R, −2(p − 1)a + 4−1 p2 b2 ≤ −ε, ε > 0, and w(t) is a Wiener process with values in R;   n X ∂ ∂ du(t, x) = a (t, x, ω) u(t, x) dt ij ∂xi ∂xj i,j=1 (3.3) m X n X ∂ j + bij (t, x, ω) j u(t, x)dw (t), ∂x j=1 i=1 x ∈ Rn , aij , bij are bounded measurable functions such that, for some λ > 0, n X

i,j=1

aij ξ i ξ j −

n m X n X X 1 bi` bj` ξ i ξ j ≥ λ (ξ i )2 2 i=1 i,j=1 `=1

for all t, x, ω and every vector ξ ∈ Rn ; wi (t) are independent Wiener process with values in R, and ω is an elementary outcome. These and other stochastic partial differential equations are considered in detail in Section 4. The results of the present section are a refinement of the results of Pardoux [52, 53] which, in turn, generalize the results of Bensoussan and Temam [43]. As already mentioned above, we have succeeded in showing that certain conditions of Pardoux are superfluous, in particular, the local Lipschitz condition for the operator B. The method of proving the existence theorem (the most difficult and important part of this section) has been borrowed from Pardoux and corresponds to a Galerkin scheme: a finite-dimensional analog of equation 3.1 is considered (Section 3.3), estimates of the solution independent of the dimension are obtained (Section

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3.4), and then (by the method of monotonicity) a passage to the limit is realized (Section 3.5). The basic improvements which make it possible in the final analysis to generalize the results of Pardoux are made at the first step in Section 3.3, where a theorem is obtained generalizing a well-known theorem of Itˆ o on the existence and uniqueness of strong solutions of a stochastic equation with random coefficients satisfying Lipschitz conditions. 3.2. Assumptions and the Main Results Let (Ω, F, P ) be a complete probability space with an expanding system of σalgebras {Ft } (t ∈ [0, T ], T < ∞) imbedded in F. We assume that the family {Ft } is complete with respect to the measure P . Further, let H and E be real separable Hilbert spaces, naturally identified with their duals H ∗ and E ∗ ; let w(t) be a Wiener process in E with nuclear covariance operator Q (see Section 2.2), and let z(t) be a square-integrable martingale in H. We also consider a real, separable, reflexive Banach space V and its dual space V ∗ . As in Section 2, if v is an element of V and v ∗ is an element of V ∗ , then vv ∗ denotes the value of v ∗ on v; |·|X and (·, ·)X denote, respectively, the norm in the space X and the scalar product in the space X if X is a Hilbert space. In Section 3.3, where finite-dimensional spaces are considered, this notation is simplified; special mention is made of this. As before, LQ (E, H) is the space of all linear operators Φ defined on Q1/2 E and taking Q1/2 E into H such that ΦQ1/2 ∈ L2 (E, H) (the space of Hilbert-Schmidt operators from E to H). Recall that LQ (E, H) is a
separable Hilbert space relative ∗ to the scalar product (Φ, Ψ)Q = trΦQ1/2 ΨQ1/2 ; we write |·|Q to denote the norm in this space. The following assumptions are henceforth used: a) b) c) d)

V ⊂ H ≡ H ∗ ⊂ V ∗; V is dense in H (in the norm of H); there exists a constant c such that, for all v ∈ V , |v|H ≤ c |v|V ; vv ∗ = (v, v ∗ )H if v ∗ ∈ H.

An important example of spaces possessing properties a)– d) is the Sobolev space Wpm (G)(= V ) and L2 (G)(= H) where G is a bounded domain in Rd , and dp ≥ 2(d − mp). These spaces are discussed in more detail in Section 4. Some other triples of spaces possessing properties a)–d) are also presented there. We recall also that in Section 2 triples of spaces V , H, V 0 connected by less rigid assumptions have already been considered, and the “implication” of assumptions a) – d) was discussed in some detail; in particular, the possibility of identifying V 0 with V ∗ by means of (·, ·)H was discussed. p We fix numbers p and q, p ∈ (1, ∞), q = . p−1

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Suppose that, for each (v, t, ω) ∈ V × [0, T ] × Ω, A(v, t, ω) ∈ V ∗ , B(v, t, ω) ∈ LQ (E, H). We assume that, for each v ∈ V , the functions A(v, t, ω), B(v, t, ω) are measurable in (t, ω) relative to the measures dt × dP and are Ft -consistent, i.e., for each v ∈ V , t ∈ [0, T ] they are Ft -measurable in ω. We recall that, since V ∗ and LQ (E, H) are separable, the concepts of strong and weak measurability coincide, and we will speak simply of measurability. Suppose further that on Ω there is given an F0 -measurable function u0 with values in H, while on [0, T ] × Ω there is given a nonnegative function f (t, ω) measurable in (t, ω) and Ft -consistent. We assume that, for some constants K, α > 0 and for all v, v1 , v2 ∈ V , (t, ω) ∈ [0, T ] × Ω, the following conditions are satisfied. (A1 ) Semicontinuity of A: the function vA(v1 + λv2 ) is continuous in λ on R. (A2 ) Monotonicity of (A, B): 2

2

2(v1 − v2 )(A(v1 ) − A(v2 )) + |B(v1 ) − B(v2 )|Q ≤ K |v1 − v2 |H . (A3 ) Coercivity of (A, B): 2

p

2

2vA(v) + |B(v)|Q + α |v| ≤ f + K |v|H . (A4 ) Boundedness of the growth of A: |A(v)|V ∗ ≤ f 1/q + K |v|p−1 . V (A5 ) E |u0 |2H < ∞, E

Z

T 0

f (t)dt < ∞.

Under these assumptions we consider on [0, T ]×Ω the stochastic evolution equation Z t Z t v(t, ω) = u0 (ω) + A(v(s, ω), s, ω)ds + B(v(s, ω), s, ω)dw(s, ω) + z(t, ω). (3.4) 0

0

Definition 3.1. A solution (or a V-solution) of equation (3.4) is a function v(t, ω) with values in V defined on [0, T ] × Ω, measurable in (t, ω), Ft -consistent, satisfying the inequality Z T  p 2 E |v(t)|V + |v(t)|H dt < ∞, (3.5) 0

and satisfying equation (3.4) in the sense of equality of elements of V ∗ for almost all (t, ω) ∈ [0, T ] × Ω.

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In this definition it is implicitly assumed that integrals in (3.4) and (3.5) are meaningful. This assumption is, however, superfluous in view of what follows. By Lemma 2.5 the function v is measurable not only as a function with values in V but also as a function with values in H. Therefore, |v(t)|V , |v(t)|H are measurable with respect to (t, ω) and condition (3.5) is meaningful. We show further that the functions A(v(t), t), B(v(t), t) are measurable in (t, ω) and Ft -measurable in ω. We first consider A(v(t), t) and use the same sort of lemma as in the theory of monotone operators (see e.g. Ref. [42]). Lemma 3.2. If a sequence vn converges strongly in V to v, then the sequence A(vn ) converges weakly to A(v) in V ∗ . Proof. Because of assumption (A4 ), for every subsequence {µ} of natural numbers, the sequence A(vµ ) is bounded in V ∗ , and therefore there exists a subsequence {η} of the sequence {µ} along which A(vη ) converges weakly to some A∞ ∈ V ∗ . We will now show that A∞ = A(v). Let u be an arbitrary element of V . By assumption (A2 ) 2

(u − vη )(A(u) − A(vη )) − K |u − vη |H ≤ 0. Passing to the limit η → ∞ in this equality, we obtain 2

(u − v)(A(u) − A∞ ) − K |u − v|H ≤ 0.

(3.6)

We now suppose that u = v + λy where λ ∈ R+ and y is some element of V ; from (3.6) it then follows that 2

y(A(v + λy) − A∞ ) − Kλ |y|H ≤ 0. Passing to the limit λ ↓ 0 in this equality and using the semicontinuity of A (assumption (A1 ),) we obtain y(A(v) − A∞ ) ≤ 0. Since y is arbitrary, it follows that A∞ = A(v) and, since the subsequence {µ} was arbitrary, the proof of the lemma is complete.  It follows from the lemma that the function v → uA(v, t, ω) is continuous for each u ∈ V , (t, ω) ∈ [0, T ] × Ω. As a result, if v = v(t, ω) is measurable in (t, ω) and Ft consistent, then so is A(v(t, ω), t, ω). Moreover, in view of (A4 ), (A5 ), A(v(t, ω), t, ω) is a summable function of t for almost all ω if (3.5) is satisfied. The problem of the measurability of B(v(t, ω), t, ω) is easier, since it is obvious that by assumptions (A2 ) and (A4 ), B(·, t, ω) for all t, ω is a strongly continuous function from V to LQ (E, H). From assumptions (A3 ) and (A4 ) it follows that, for all u, t, ω, 2 2 p
|B(u, t, ω)|Q ≤ c f (t, ω) + |u|H + |u|V . (3.7)

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Therefore, E

Z

T 0

2

|B(v(t), t)|Q dt < ∞,

and the stochastic integral in (3.4) is defined and is a square-integrable martingale in H. Remark 3.3. It is obvious (see the clarifications of Theorem 2.14) that v is a solution of (3.4) if and only if, for some set Y ⊂ V dense in V (in the norm of V) and for every u ∈ Y , the equality Z t (u, v(t))H = (u, u0 )H + uA(v(s), s)ds + u ˜B(v(s), s)dw(s) + (u, z(t))H 0

holds a.e. (t, ω). Definition 3.4. An H-solution of (3.4) is a solution u(t, ω) with values in H defined on [0, T ] × Ω, strongly continuous in H with respect to t, Ft -consistent, and such that (a) u ∈ V a.e. in (t, ω) and Z t  p 2 E |u(t)|V + |u(t)|H dt < ∞. 0

(b) there exists a set Ω0 ⊂ Ω of probability one on which, for all t ∈ [0, T ], Z t Z t u(t) = u0 + A(u(s), s)ds + B(u(s), s)dw(s) + z(t), (3.8) 0

0

where the equality is understood as an equality of elements of V ∗ .

In clarification of (3.8) we note that, by Lemma 2.5, the function χV (u(t)) is measurable in (t, ω), Ft -measurable in ω, equal to one [a.e. in (t, ω)] by condition (a), and the integrals in (3.8) are therefore understood as Z t Z t χV (u(s))A(u(s), s)ds, χV (u(s))B(u(s), s)dw(s). 0

0

Definition 3.5. An H-solution u is called a continuous modification of a solution v in H if u(t, ω) = v(t, ω) a.e. in (t, ω). We now formulate the main results of this Section. All the assumptions above are assumed to be satisfied. Theorem 3.6. A solution v of equation (3.4) exists. From this theorem and Theorem 2.17 we immediately obtain the following result. Corollary 3.7. There exists a continuous modification of the solution v of equation (3.4) in H.

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Theorem 3.4 is proved in Section 3.5. The next theorem contains the uniqueness assertion for the solution of equation (3.4). Theorem 3.8. Let v n (t), n = 0, 1, 2, . . ., be solutions of equation (3.4) with initial 2 2 data u0 = un0 , where E |un0 |H < ∞ and E un0 − u00 H → 0 as n → ∞. Suppose that un (t) are continuous modifications of v n (t) in H. Then, for every ε > 0,     2 lim sup E un (t) − u0 (t) H + P sup un (t) − u0 (t) H ≥ ε = 0. n→∞

t≤T

t≤T

Remark 3.9. We will see from the proof of the theorem that the result is valid if, in the definition of the solution, condition (3.5) is dropped and only condition (A2 ) is required of the coefficients of (3.4). Theorem 3.10. If v is a solution of (3.4) and u is its continuous modification in H, then ! Z Z T

2

E sup |u(t)|H + E t≤T

0

p

T

2

|v(t)|V dt ≤ c E |u0 |H + E

0

2

f (t)dt + E |z(t)|H

,

where c depends only on K, p, T , and α. Theorems 3.8 and 3.10 are proved in Section 3.4. The following theorem on the Markov property of solutions of (3.4) also belongs to the basic results of this section. This theorem is proved at the end of Section 3.5. Theorem 3.11. Suppose that A and B do not depend on ω, z(t) ≡ 0, v = v(t) is a solution of equation (3.4), and u = u(t) is its continuous modification in H. Then u(t) is a Markov random variable. Remark 3.12. With no loss of generality, we can take K = 0 in conditions (A2 ) and (A3 ). Indeed, if v is a solution of (3.4), then v(t)e−Kt is a solution of an equation of the type (3.4) with A replaced by e−Kt (A − KI), where I is the identity operator, and with B replaced by e−Kt B, and these new A and B satisfy conditions (A2 ) and (A3 ) with K = 0. Remark 3.13. Our assumption that the spaces in question are real is not essential and may be relaxed if, in conditions (A2 )–(A3 ), in place of vA(v1 + λv2 ), (v1 − v2 )(A(v1 ) − A(v2 )), vA(v) we write

Re vA(v1 + λv2 ) , Re (v1 − v2 )(A(v1 ) − A(v2 )) , Re vA(v)).

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3.3. Itˆ o Equations in Rd Let Rd be Euclidean space of dimension d with a fixed orthonormal basis, let xi be the i−th coordinate of a point x ∈ Rd , let (Ω, F, P ) be a complete probability space, and let {Ft } be an expanding family of complete σ-algebras Ft ⊂ F. Let m(t) be a d1 -dimensional continuous local martingale relative to {Ft }, with m(0) = 0, and let A(t) be a continuous, real, nondecreasing Ft -consistent process with A(0) = 0. Suppose that, for t ≥ 0, x ∈ Rd , ω ∈ Ω, a d × d1 matrix b(t, x) and a d-dimensional vector a(t, x) are defined. We assume that, for each x ∈ Rd , a(t, x) and b(t, x) are completely measurable relative to {Ft } and are continuous in x for each (t, ω). Let x0 be a d-dimensional F0 -measurable random vector. We consider the following equation: Z t Z t x(t) = x0 + a(s, x(s))dA(s) + b(s, x(s))dm(s). (3.9) 0

0

Equation (3.9) will be considered under certain additional conditions on a, b, A, and m whose formulation requires the following notation. By the Doob-Meyer theorem there exists a continuous increasing process hmit for which (m2 (t)i − hmit ) is a local martingale relative to {Ft } and hmi0 = 0. For i, j = 1, . . . , d1 we further define by means of the Doob-Meyer theorem continuous processes hmi , mj it having locally bounded variation in t for which every process mi (t)mj (t) − hmi , mj i
t is a local martingale and hmi , mj i0 = 0. We recall that the matrix hmi , mj it is nonnegative definite and hmit =

d1 X i=1

hmi , mi it

for all t (a.s.). We fix a continuous real nondecreasing Ft -consistent process Vt such that, for each ω, the measures on the t axis generated by the functions A(t), hmit are absolutely continuous relative to the measure corresponding to Vt (for example, we can take Vt = A(t) + hmit .) We define cij (t) =

dhmi , mj it , dVt

α(t, x) = a(t, x)


C(t) = cij (t) ,

dA(t) 1/2 , β(t, x) = b(t, x)Ct . dVt

We assume that the following conditions are satisfied, in addition to those enumerated above: for each x ∈ Rd , T > 0 Z

T 0

|α(t, x)| dVt =

Z

T 0

|a(t, x)| dA(t) < ∞ (a.s.);

(3.10)

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for every R > 0 there exists a nonnegative completely measurable process K t (R) such that,for all T > 0, Z T K(R)dVt < ∞ (a.s.), 0

d

and, for each z, x, y ∈ R such that |x|, |y| ≤ R,

2

2(x − y)(α(t, x) − α(t, y)) + kβ(t, x) − β(t, y)k ≤ Kt (R)(x − y)2 , 2zα(t, z) + kβ(t, z)k2 ≤ Kt (1 + z 2 )

(3.11) (3.12)

for almost all t relative to the measure dVt , where Kt = Kt (1), kγk for a matrix γ 1/2 means (tr (γγ ∗ )) , and, for ε, δ ∈ Rd , δε denotes the scalar product of δ, ε. Theorem 3.14. There exists a continuous, Ft -consistent process x(t) for which (3.9) is satisfied for all t with probability one. If x(t), y(t) are two continuous Ft -consistent processes satisfying (3.9) for each t (a.s.), then   P sup |x(t) − y(t)| > 0 = 0. t≥0

The present theorem is a generalization of Itˆ o’s classical theorem on the existence of a strong solution of a stochastic equation of type (3.9) with random coefficients. In the present theorem we have avoided the Lipschitz condition and replaced it by condition (3.11) which we shall call the monotonicity condition. Example 3.15. Let w(t) be a one-dimensional Wiener process, let p ∈ (1, 2), and let a(t, ω) and b(t, ω) be completely measurable functions such that Z T a(t, ω)dt < ∞ (a.s.) 0

and

−2(p − 1)a(t, ω) +

p2 2 b (t, ω) ≤ 0 4

(a.s. in (t, ω)).

We consider the equation dx(t) = − |x(t)|

p−1

sgn x(t)a(t)dt + |x(t)|

p/2

b(t)dw(t).

It is clear that for p < 2 the coefficients of this equation do not satisfy Lipschitz conditions, but satisfy the monotonicity condition. Indeed, by the formula of Hadamard, −2a · (|x|

p−1

p−1

p/2

p/2

sgn y)(x − y) + b2 · (|x| − |y| )2 Z 1 p−2 = −2a · (x − y)2 (p − 1) |x + τ (y − x)| dτ

sgn x − |y| Z

0

2 p p 2 2 2 −1 +b · (x − y) |x + τ (y − x)| sgn (x + τ (y − x))dτ 0 2  2 Z 1 p 2 ≤ (x − y)2 b − 2(p − 1)a |x + τ (y − x)|p−2 dτ ≤ 0. 4 0 1

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If we take Vt = t then from this inequality we obtain the monotonicity condition with Kt = 0 and also condition (3.12), since in the present case α(t, 0) = β(t, 0) = 0. Thus, condition (3.11) is weaker than the Lipschitz condition. However, extremely “bad” functions may not satisfy (3.11). We remark without proof that (3.11) implies differentiability of α(t, x) with respect to x for almost all x and that the first generalized derivatives of β(t, x) with respect to x are locally squaresummable. We note that in contrast to Itˆ o’s work we consider the equation over a semimartingale. Generalizations of Itˆ o’s results to stochastic equations over a semimartingale have been obtained by Kazamaki [78], Dol´eans-Dade [79], Protter [80], Gal’chuk [81], Lebedev [54] and others. The results of these works are a special case of our assertion as far as continuous semimartingales are concerned. It seems to us that the extension of Theorem 3.14 to the case of discontinuous semimartingales is an accessible problem, but is beyond our present interest. As always in similar situations, the uniqueness assertion in Theorem 3.14 is very easy to prove. Indeed, let x(t), y(t) be two solutions of (3.9). By the continuity with respect to t of x(t), y(t), and the integrals in (3.9), the processes x(t), y(t) satisfy (3.9) for all t on the same set of probability one. Next, define Z t  Ψt (R) = exp Ks (R)dVs , 0

2

apply Itˆ o’s formula to |x(t) − y(t)| Ψt (R) and use (3.11). We then find that, (a.s.) for all t, |x(t ∧ τ (R)) − y(t ∧ τ (R))|2 Ψt∧τ (R) Z t∧τ (R) Ψs (R) (x(s) − y(s)) (b(s, x(s) − b(s, y(s)))dm(s) ≡ m0t (R), ≤2 0


where τ (R) is the first exit time of max |x(t)| , |y(t)| from [0, R). We see that the local martingale m0t (R) is nonnegative, and hence m0t (R) is a supermartingale. Since m00 (R) = 0, it follows that m0t (R) = 0 (a.s.), so that |x(t ∧ τ (R)) − y(t ∧ τ (R))| = 0 (a.s.) for all t, and, since |x(t) − y(t)| is continuous in t and R is arbitrary, it follows that supt |x(t) − y(t)| = 0 (a.s.) as required. The existence assertion in Theorem 3.14 will be proved after a number of auxiliary propositions. In Theorem 3.14 it is asserted, in particular, that the right side of (3.9) exists for some process x(t). This fact for any Ft -consistent continuous process follows immediately from the following lemma. Lemma 3.16. For every T, R > 0 (a.s.) Z T Z sup |α(t, x)|dV (t) < ∞, 0

|x|≤R

T 0

2

sup kβ(t, x)k dVt < ∞.

x≤R

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Proof. Let {yi } be a countable dense subset of {x : |x| ≤ R + 2}. We define ϕn (x, z) = max((x − yi )z, i = {1, . . . , n}, x, z ∈ Rd . It is clear that each ϕn (x, z) is a continuous function and ϕn (x, z) ↑ xz + |z| (R + 2) as n → ∞. The last function is continuous and greater than two if |x| ≤ R, |z| = 1. By Dini’s theorem there exists an n0 such that ϕn0 (x, z) ≥ 1 for |x| ≤ R, |z| = 1. This implies that ϕn0 (x, z) ≥ |z| for |x| ≤ R and all z ∈ Rd . Substituting the points yi into (3.11) in place of y and computing the upper bounds over i for |x| ≤ R, we find (with α(t, x) in the place of z) that 2 |α(t, x)| ≤ 4(R + 2)2 Kt (R + 2) + 2(R + 2)

n0 X

α(t, yi ).

i=1

Because of (3.10) this proves the first assertion of the lemma. The second assertion follows in an obvious way from the first and from (3.12). The proof of the lemma is complete.  Lemma 3.17. Let f (x) be a real locally bounded function on Rd , let n > 0, and let N = sup {|f (x)| : |x| ≤ n}. Then on Rd there exists a real function g(x) such that g(x) = f (x) for |x| ≤ n and g(x) = 0 for |x| ≥ n + 1, |g(x)| ≤ |f (x)|, and 2

2

|g(x) − g(y)| ≤ |f (x) − f (y)| + N 2 (x − y)

2

(3.13)

for all x, y ∈ Rd . Moreover, if f (x) is continuous in x and depends in a measurable way on several parameters, then g(x) is continuous in x and is measurable with respect to these parameters. Proof.

Definec h(x) = (n + 1)N − N |x| and g(+) (x) = max(min(h(x), f+ (x)), 0), g(−) (x) = max(min(h(x), f− (x)), 0), g(x) = g(+) (x) + g(−) (x).

We will prove that this function g satisfies all the requirements. The last assertion of the lemma is obvious, since N is a measurable function of those parameters on which f depends, while the continuity of g follows from the continuity of f and (3.13). Further, g(+) ≤ f+ , g(−) ≤ f− . Therefore, g± = g(±) , |g| ≤ |f |, and g(x) has the same sign as f (x). Moreover, it is obvious that h(x) ≤ 0 for |x| ≥ n + 1. Therefore, g(x) = 0 for |x| ≥ n + 1. Since h(x) ≥ N ≥ |f (x)| for |x| ≤ n, it follows that g(x) = f (x) for |x| ≤ n. It remains to prove (3.13). We fix points x, y. If f (x) and f (y) have different signs, then the same is true of g(x) and g(y), and |g(x) − g(y)| = |g(x)| + |g(y)| ≤ |f (x)| + |f (y)| = |f (x) − f (y)| .

c This

method of constructing g was suggested to us by A. D. Wentzell.

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We now consider the case where f (x) and f (y) have the same sign. Suppose, to be specific, that f (x) ≥ 0, g(x) ≥ 0; then g(x) = g(+) (x), g(y) = g(+) (y). Since the absolute value of the difference of the upper (lower) bounds does not exceed the upper (lower) bound of the absolute value of the difference, it follows that |g(x) − g(y)| ≤ |min(h(x), f (x)) − min(h(y), f (y))| ≤ max (|h(x) − h(y)| , |f (x) − f (y)|) ,

which implies (3.13). The proof of the lemma is complete.



Lemma 3.18. For every n > 0, there exist processes a ˜(t, x), ˜b(t, x), Nt such that a ˜(t, x) ∈ Rd , ˜b(t, x) is a d × d1 matrix, Nt is a real process, a ˜, ˜b, N are defined d for all x ∈ R , t ≥ 0, ω ∈ Ω, are continuous in x and completely measurable, a(t, x) = a ˜(t, x), b(t, x) = ˜b(t, x) for |x| ≤ n, a ˜(t, x) = 0, ˜b(t, x) = 0 for |x| ≥ n + 3 for all t, Z t Ns dVs < ∞ (a.s.), (3.14) 0

d

and, for all x, y ∈ R ,

2

˜

|˜ a(t, x)| + β(t, x) ≤ Nt ,

˜ ˜ y) + β(t, x) − β(t, where

α(t) ˜ =a ˜(t)

2(x − y)(˜ α(t, x) − (˜ α(t, y))

2

≤ Nt (x − y)2 (a.e. dP × dVt ),

(3.15)

dA(t) ˜ , β(t) = ˜b(t)C 1/2 (t). dVt

Proof. We fix n ≥ 1. Let Tt be an orthogonal d1 × d1 matrix, and let Λt be a diagonal matrix of the same dimension so that Ct = Tt ΛTt∗ . It is well known that such matrices exist, and they can be chosen to be completely measurable. For every element of the matrix b(t, x)Tt , we construct by means of Lemma 3.17 the corresponding truncated elements which we assign to the matrix b0 (t, x), and then define ˜b(t, x) = b0 (t, x)Tt∗ . Consider the function η ∈ C0∞ (Rd ), η(x) = 1 for |x| ≤ n + 2, η(x) = 0 for |x| ≥ n + 3, 0 ≤ η ≤ 1, and define a ˜(t, x) = a(t, x)η(x). We shall prove that there exists a process Nt such that the assertions of the lemma hold for a ˜, ˜b, N . (1) It is not hard to see that by Lemmas 3.16 and 3.17 there exists a process Nt satisfying condition (3.14) and such that

2

2

˜

1/2 ˜ y)

β(t, x) − β(t,

= ˜b(t, x)Tt − ˜b(t, y)Tt Λt

2

(1) 1/2 (3.16) ≤ Nt (x − y)2 + (b(t, x)Tt − b(t, y))Tt Λt (1)

= Nt

(x − y)2 + kβ(t, x) − β(t, y)k .

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Moreover, by Lemma 3.16 there exists a process Nt that

satisfying (3.14) and such

2(x − y)(˜ α(t, x) − (˜ α(t, y)) = 2(x − y)(α(t, x))(η(x) − η(y))

+2(x − y)η(y)(α(t, x) − α(t, y))

≤ 2(η(x) ∧ η(y))(x − y)(α(t, x) − α(t, y)) + (3)

Further, by Lemma 3.16 there exists a process Nt

(2) Nt (x

(3.17)

2

− y) .

satisfying (3.14) and such that

2

˜

(3) 2 |˜ α(t, x)| + β(t, x) ≤ Nt .

(3.18)

Finally, if |x|, |y| ≤ n + 2, then η(y) = 1 and the second inequality in (3.15) is satisfied by (3.16), (3.17), and (3.11), with N = N (2) + N (3) + K(n + 3). If |x|, ˜ ˜ |y| ≥ n + 1, then β(x) = β(y) = 0 and (3.15) is satisfied with N = N (2) + K(n + 3). If one of the values of |x|, |y| is less than n + 1 and the other is greater than n + 2, then |x − y| ≥ 1, and (3.15) is satisfied by (3.17), (3.18), and (3.11), with ˜ x), β(t, ˜ y) is zero. Thus, N = N (2) + N (3) + K(n + 3), since one of the values of β(t, (1) (2) (3) inequalities (3.15) are satisfied with N = N + N + N + K(n + 3) and the proof of the lemma is complete.  Lemma R T 3.19. Suppose there exists a completely measurable process N t ≥ 0 such that 0 Ns dVs < ∞ (a.s.) for all T > 0 and, for all x, y, 2

|α(t, x)| + |β(t, x)| ≤ Nt (a.e. dP × dVt ),

2 |x − y| |α(t, x) − α(t, y)| + kβ(t, x) − β(t, y)k2 ≤ Nt |x − y|2 (a.e. dP × dVt ). Then equation (3.9) has a unique solution. Proof.

We have already proved the uniqueness of the solution of (3.9). Define 

ψ(t) = exp −3

x0t = x0 , xr+1 = x0 +

Z

Z



t

0

Ns dVs − |x0 | ,

t

a(s, xr (s))dA(s) + 0

Using Itˆ o’s formula for xr+1 (t) − xr (t)


2

Z

t

b(s, xr (s))dm(s), r > 0.

(3.19)

0

ψ(t), it is easy to show that, for some

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local martingale mr (t), r ≥ 1, Z tn
2

xr+1 (t) − xr (t) ψ(t) = 2 xr+1 (s) − xr (s) α(s, xr (s)) − α(s, xr−1 (s)) 0

2
2 o r r−1

+ β(s, x (s)) − β(s, x (s)) − 3Ns xr+1 (s) − xr (s) ψs dVs + mr (t) Z tn r+1
x (s) − xr (s) Ns xr (s) − xr−1 (s) + Ns xr (s) − xr−1 (s) 2 ≤ 0
2 o − 3Ns xr+1 (s) − xr (s) ψs dVs + mr (t)  Z t
2 5
2 3 ≤ Ns xr (s) − xr−1 (s) − Ns xr+1 (s) − xr (s) ψs dVs + mr (t) 2 2 0 or Z t
2
2 5 Ns xr+1 (s) − xr (s) ψs dVs xr+1 (t) − xr (t) ψ(t) + 0 2 Z t
2 3 ≤ Ns xr (s) − xr−1 (s) + mr (t). 0 2

In the last inequality,
we put t = τ i ∧ τ , where τ i are stopping times such that τ i ↑ ∞ and mr t ∧ τ i is a martingale, and the stopping time τ will be specified later. Computing the expectations and letting τi ↑ ∞ we conclude that  Z τ 
2
2 E 5 xr+1 (s) − xr (s) Ns ψs dVs + 2 xr+1 (τ ) − xr (τ ) ψτ 0 Z τ
2 ≤ 3E xr (s) − xr−1 (s) Ns ψs dVs , r ≥ 1, 0

r

2

where (x (τ + 1) − xr (τ )) ψτ is taken equal to zero on the set where τ = ∞. Similarly,   Z τ
2
2 E x1 (τ ) ψτ + 2 x1 (s) Ns ψs dVs Z τ0 1 2 −|x0 | ≤ Ex0 e Ns ψs dVs ≤ Ex20 e−|x0 | + < ∞, +E 3 0
2 1 where x (τ ) ψτ is also taken equal to zero on the set where τ = ∞. From these inequalities it follow that, for some constant N 0 and for all r, τ,  r Z τ
2 3 r+1 r 0 E x (s) − x (s) Ns ψs dVs ≤ N , 5 0  r
2 3 r+1 r 0 . E x (τ ) − x (τ ) ψτ ≤ N 5 n o
2 We now take τ = τ (r) = inf t : xr+1 (t) − xr (t) ψt ≥ r−4 and conclude that  r 3 r−4 P {τ (r) < ∞} ≤ N 0 , 5

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P



sup x

r+1

t≥0

r

(t) − x (t)


2

ψt ≥ r

−4



 r 3 ≤N r . 5 0 4

Hence, by the Borel-Cantelli lemma the series ∞ X r+1 x (t) − xr (t)2 ψ 1/2 t

r=1

converges uniformly for almost all ω on each finite time segment to some continuous process x(t). Passing to the limit in (3.19), we complete the proof of the lemma. Lemma 3.20. We choose n > 0, a ˜, ˜b, N from Lemma 3.18. If in equation (3.9) a, b are replaced by a ˜, ˜b, then the equation has a unique solution. Proof. For r = 1, 2, . . . we define functions a ˜r , ˜br by convolution of a ˜, ˜b on x with a δ-type sequence of infinitely differentiable, compactly supported, nonnegative functions of the form ζ(jr x)jrd , where jr ≥ 0, jr → ∞ as r → ∞. Since for any r the first derivatives with respect to x and the functions a ˜ r , ˜br can be bounded in terms of the maxima of |˜ a|, |˜b|, by the first inequality in (3.15) the conditions of Lemma 3.19 are satisfied for a ˜, ˜b. Therefore, for every r > 0, there exists a unique solution of the equation Z t Z t ˜br (s, xr (s))dm(s). (3.20) a ˜r (s, xr (s))dA(s) + xr (t) = x0 + 0

0

Since a ˜(t, x) = 0, ˜b(t, x) = 0 for |x| ≥ n + 3 and ζ is a compactly supported function, it can be assumed with no loss of generality that a ˜ r (t, x) = 0, ˜br (t, x) = 0 for |x| ≥ n + 4. In this case if |x0 | ≤ n + 4, then the process xr (t) never leaves the set {x : |x| ≤ n + 4}. If |x0 | ≥ n + 4, then xr (t) = x0 for all t. This implies that sup sup |xr (t) − x0 | ≤ 2n + 8 (a.s.) t≥0

(3.21)

r

Further, we choose Nt from Lemma 3.18. We note that by (3.15) for α ˜r = r r 1/2 a ˜ (dA/dV ), β˜ = ˜b C , we have r

2

2

2



˜

˜r x) ≤ Nt , |˜ αr (t, xr (t))| ≤ Nt .

β (t, xr (t)) ≤ sup β˜r (t, x) ≤ sup β(t, x

x

 R  t We define ψt = exp − 0 Ns dVs and show next that 2

lim sup E |xr (τ ) − xp (τ )| ψt = 0,

r,p→∞

(3.22)

(3.23)

τ

2

where the supremum is taken over all stopping times τ and (xr (τ ) − xp (τ )) ψt is set equal to zero on when τ = ∞.

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By Itˆ o’s formula, for some local martingale mr,p (t), n d |xr (t) − xp (t)|2 ψt = 2 (xr (t) − xp (t)) (˜ αr (t, xr (t)) − α ˜ r (t, xp (t)))

2 o

2 + β˜r (t, xr (t)) − β˜r (t, xp (t)) − Nt (xr (t) − xp (t)) ψt dVt n + 2 (xr (t) − xp (t)) (˜ αr (t, xp (t)) − α ˜p (t, xp (t)))

2 o

+ β˜r (t, xp (t)) − β˜p (t, xp (t)) ψt dVt h i + 2 β˜r (t, xp (t)) − β˜p (t, xp (t)), β˜r (t, xr (t)) − β˜r (t, xp (t)) ψt dVt + dmr,p (t),

(3.24)

where, for two d × d1 matrices, [σ1, σ2 ] is the sum of all products of the form σ1ij σ2ij . Application of the second inequality in (3.15) and the Cauchy-Schwartz inequality shows that 2 (xr (t) − xp (t)) (˜ αr (t, xr (t)) − α ˜ r (t, xp (t)))

2

+ β˜r (t, xr (t)) − β˜r (t, xp (t)) − Nt (xr (t) − xp (t))2 ≤ 0

(a.e. dP × dVt ). Moreover, the functions

(xr (t) − xp (t)) (˜ αr (t, xp (t)) − α ˜ p (t, xp (t)))ψt by (3.21), (3.22) are bounded in absolute value by 4(2n+8)Ntψt , which is summable with respect to dP × dVt , and for allt, ω. Also, these function tend to zero as r, p → ∞ by continuity in x of the compactly supported α ˜ (t, x). For all t, ω, we have |˜ αr (t, x) − α ˜ (t, x)| as r → ∞ uniformly with respect to x ∈ Rd . Therefore, Z ∞ lim E |(xr (s) − xp (s)) (˜ αr (s, xp (s)) − α ˜ p (s, xp (s)))| ψs dVs = 0. p,r→∞

0

Similarly, lim E

r,p→∞

Z

∞

2

˜r

β (s, xp (s)) − β˜p (s, xp (s)) ψs dVs = 0,

Z0 ∞ ˜r lim E [β (s, xp (s)) − β˜p (s, xp (s)), β˜r (s, xr (s)) − β˜r (s, xp (s))] ψs dVs .

r,p→∞

0

It is now clear that (3.23) follows from (3.24). From (3.23), in turn, we obtain the existence of a subsequence r(i) such that, for all stopping times τ , E xr(i+1) (τ ) − xr(i) (τ ) ψτ ≤ 2−i .

Finally, as in Lemma 3.19, from this we obtain the uniform convergence of xr(i) (t) with respect to t on any finite time interval, and by passing to the limit in (3.19) along this subsequence r(i) we complete the proof of the lemma. 

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Proof of Theorem 3.14. Denote by an , bn the functions a ˜, ˜b constructed in Lemma 3.18 for a given n > 0. By Lemma 3.20, for every n > 0, there exists a solution of the equation Z t Z t an (s, xn (s))dA(s) + bn (s, xn (s))dm(s). xn (t) = x0 + 0

0

n

n

n,m

n

m

Let τ = inf {t : |x (t)| ≥ n}, τ = τ ∧ τ . Since an (t, x) = a(t, x) and b (t, x) = b(t, x) for |x| ≤ n, the processes xn (t ∧ τ n,m ), xm (t ∧ τ n,m ) satisfy the same equation n

dz(t) = χt<τ n,m a(t, z(t))dA(t) + χt<τ n,m b(t, z(t))dm(t). Hence these processes coincide and xn (t) = xm (t) on {t ≤ τ n,m } (a.s.). Therefore, τ n ≤ τ m (a.s.) for n ≤ m and there exists τ = limn→∞ τ n (a.s.). Also, for almost all ω and t < τ , the process x(t) = limn→∞ xn (t) is defined, and for every n and all t (a.s.) Z t∧τ n Z t∧τ n n n n x (t ∧ τ ) = x (t ∧ τ ) = x0 + a(s, x(s))dA(s) + b(s, x(s))dm(s). 0

0

It remains to prove that τ = ∞. Define  Z t  ψt = exp − Ks dVs − |x0 | . 0

As we have done repeatedly, by means of (3.12) and Itˆ o’s formula we find that Z ∞ E(xn (τ n ))2 ψτ n χτ n <∞ ≤ Ex20 ψ0 + E Ktψt dVt < ∞. 0

Therefore n2 Eψτ n χτ n <∞ ≤ N, or Eψτ n χτ n <∞ → 0 and τ n → ∞ (a.s.) as n → ∞. The proof of the theorem is complete.



Remark 3.21. The assumption of complete measurability of a(t, x), b(t, x), Kt (R) ensures that the corresponding stochastic integrals with respect to dm(t) are defined and the integrals with respect to dA(t) and dVt are Ft -consistent. Theorem 3.14 can hold without this assumption. For example, we can instead assume that dA(t)  dt, a(t, x) is measurable in (t, ω) and is Ft -consistent, as are those elements of the matrix b which are multiplied by dmi (t) with dhmi it  dt, while the remaining elements of b are completely measurable, and Kt (R) is progressively measurable. 3.4. Uniqueness Theorem: A Priori Estimates and Finite-Dimensional Approximations In this section we prove Theorems 3.8 and 3.10 and prepare for the proof of Theorem 3.6.

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Proof of Theorem 3.8. By Theorem 2.17, for y n = un − u0 we find for all t (a.s.) Z tn
2 |y n (t)|H = 2(v n (s) − v 0 (s)) A(v n (s), s) − A(v 0 (s), s) 0 2 o 2 + B(v n (s), s) − B(v 0 (s), s) Q ds + mnt + un0 − u00 H ,

where mnt is a local martingale with mn0 = 0. We use this equality and apply Itˆ o’s 2 formula to compute |y n (t)|H e−Kt . After this, we use the monotonicity of (A, B) (condition (A2 )) and the equality y = v n − v 0 (a.e. in (t, ω)). Then, for some local martingales m ˜ nt , we obtain 2 2 |y n (t)| e−Kt ≤ un0 − u00 + m ˜ nt ≡ ξ n (t) (a.s.) H

H

We see that the local martingales ξ n (t) are nonnegative. Hence ξ n (t) is a supermartingale and, by assumption, Eξ n (0) → 0. Therefore, for every ε > 0,    lim sup Eξ n (t) + P sup ξ n (t) ≥ ε = 0. n→∞

t≤τ

Application of the inequality 3.8.

t≤τ

|y n (t)|2H

≤ ξ n (t)eKt completes the proof of Theorem 

Proof of Theorem 3.10. Let v be some solution of equation (3.4) and let u be its continuous modification in H. By Theorem 2.17, for all t ∈ [0, T ] on a single set of full probability, Z t Z t
2 2 |u(t)|H = |u0 |H + 2 v(s)A(v(s))ds + 2 u b(s) B(v(s))dw(s) + dz(s) 0 0 (3.25) Z t  + B(v(s))dw(s) + z . 0

t

Let

n o τn = inf t : |u(t)|2H ≥ n ∧ T.

It is obvious that the stochastic integral on the right side of (3.26) is a local martingale with localizing sequence τn . Therefore, Z t∧τn Z t∧τn 2 2 2 v(s)A(v(s))ds + E |B(v n (s))|Q E |u(t ∧ τn )|H = E |u0 |H + 2E 0 0  Z t∧τn 2 B(v(s))dw(s), z(t ∧ τn ) . + E |z(t ∧ τn |H + 2E 0

Hence, by assumption (A3 ) E |u(t ∧

2 τn )|H

≤

2 E |u0 |H

− αE

Z

t∧τn

p |v(s)|V

H

Z

t∧τn

2

(f (s) + K |u(s)|H ds 0 0 Z t∧τn  2 + E |z(t ∧ τn |H + 2E B(v(s))dw(s), z(t ∧ τn ) . 0

ds +

H

(3.26)

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We now make use of the elementary inequality 1 2ab ≤ εa2 + b2 , ε > 0. ε

(3.27)

Applying this inequality to the last term in (3.26), we obtain Z t∧τn Z t∧τn E |u(t ∧ τn )|2H ≤ E |u0 |2H − αE |v(s)|pV ds + (f (s) + K |u(s)|2H ds 0 0   Z t∧τn 1 EhziT + εE |B(v(s))|2Q ds. + 1+ ε 0 (3.28) On the other hand, in view of inequality (3.7), E

Z

t∧τn 0

|B(v(s))|2Q

ds ≤ cE

Z

t∧τn 0

  f (s) + |v(s)|pV + |u(s)|2H ds.

(3.29)

Combining (3.28) and (3.29), and choosing ε sufficiently small, for some α1 we obtain Z t∧τn p 2 2 |v(s)|V ds E |u(t ∧ τn )|H ≤ E |u0 |H − α1 E 0 (3.30)  Z t  Z t 2 +c E f (s)ds + EhziT + E |u(s ∧ τn )|H ds . 0

0

From this by the Gronwall lemma sup E |u(t ∧ t≤T

2 τn )|H

≤c

2 E |u0 |H

+E

Z

T

f (s)ds + EhziT 0

!

.

(3.31)

Thus, from (3.30) and (3.31) it follows that sup E |u(t ∧ τn )|2H + E

t≤T

Z

t∧τn 0

 Z |v(s)|pV ds ≤ c E |u0 |2H + E

τ

f (s)ds + EhziT 0



.

(3.32) 2 Continuity of |u(t)|H in t implies limn→∞ τn = T . Therefore, passing to the limit as n → ∞ in (3.32) (by Fatou’s lemma and the monotone convergence theorem), we obtain   Z T Z τ 2 p 2 sup E |u(t)|H + E |v(s)|V ds ≤ c E |u0 |H + E f (s)ds + EhziT . (3.33) t≤T

0

0

To complete the proof of Theorem 3.10 it remains to show that an analogous 2 estimate also holds for E supt≤τ |u(t)|H . For this we note that from (3.25) because

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of (A3 ) and (3.27) it follows that E sup |u(t ∧ t≤T

2 τn )|H

≤

2 E |u0 |H

+E

Z

T 0


f (s) + K|u(s)|2H ds

Z t∧τn
2 + 2E sup u b(s) B(v(s))dw(s) + dz(s) t≤T 0 Z 2 T + 2E B(v(s))dw(s) + 2EhziT . 0

(3.34)

H

Define

M (t) =

Z

t

B(v(s))dw(s) + z(t). 0

Using the Burkholder inequality (Theorem 2.13) and (3.27), we obtain Z E sup t≤T

t∧τn 0

Z
2 u b(s) B(v(s))dw(s) + dz(s) ≤ 3E

t∧τn 0

2 |u(s)|H

dhM is

3 3 2 2 ε E sup |u(t ∧ τn )|H + E |M (T )|H 2 2ε t≤T Z τ 3 3 3 2 2 ≤ ε E sup |u(t ∧ τn )|H + E |B(v(s))|Q ds + EhziT . 2 ε ε t≤T 0

≤

1/2

(3.35)

Combining (3.34) and (3.35), choosing ε small, and using (3.29), (3.33), we obtain E sup |u(t ∧ t≤T

2 τn )|H

≤c

2 E |u0 |H

+E

Z

T

f (s)ds + EhziT 0

!

.

Because of the continuity of |u(t)|H , the assertion of Theorem 3.10 follows from this on the basis of Fatou’s lemma. Theorem 3.10 is proved.  We will now prepare for the proof of Theorem 3.6. For this we approximate equation (3.4) by equations in finite-dimensional spaces. Fix orthonormal bases {hi }, {ei } in the spaces H and E, respectively, and assume that hi ∈ V , i = 1, 2, . . ., while ei are the eigenvectors of the covariance operator Q of the Wiener process with the corresponding eigenvalues λi > 0: Qei = λi e1 ; this is possible because Q is a completely continuous (compact) operator. −1

Then, for each i, λi 2 (w(t), ei )E is a one-dimensional standard Wiener process, and for different i these process are independent. For n ≥ 1 we consider the following system of stochastic equations for 1 (un (t), . . . , unn (t)) ∈ Rn

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uin (t) = (hi , u0 )H +

+

n Z t X

Z

0

hi , B

0

j=1

t

  n X hi A  ujn (s)hj , s ds j=1

n X

!

ukn (s)hk , s ej

k=1

!

(3.36)

d(w(s), ej )E H

+ (hi , z(t))H , i = 1, . . . , n, t ∈ [0, T ]. Lemma 3.22. System (3.36) has a unique Ft -measurable solution which is continuous in t. Proof. It suffices to show that system (3.36) can be written in the form (3.9) and to verify assumptions of Theorem 3.14. We note immediately that the measurability conditions are verified by using Remark 3.21 at the end of Section 3.3, and we leave this verification to the reader. We choose d = n, d1 = 2n and we set a(t, x) = (ai (t, x), i = 1, . . . , n), b(t, x) = (bij (t, x), i = 1, . . . , n, j = 1, . . . , 2n), m(t) = (mi (t), i = 1, . . . , 2n), A(t) = t, mi (t) = (w(t), ei )E , mn+i (t) = (z(t), hi )H , 1 ≤ i ≤ n, 

ij

b (t, x) =

(

δ

ai (t, x) = hi A  hi , B n+i,j

,

Pn

k=1

n X j=1



xj hj , t , 1 ≤ i ≤ n,



xk h k , t e j H ,

1 ≤ i, j ≤ n,

1 ≤ i ≤ n, n + 1 ≤ j ≤ 2n,

where δ k,j is the Kronecker symbol. With these notations it is obvious that (3.36) can be written in form (3.9). As in Section 3.3, we introduce the functions cij (t), α(t, x), β(t, x), corresponding to Vt = t + hwit + hzit . It is clear that (3.10) follows from (A4 ) and (A5 ). Pn Pn Further, we choose x, y ∈ Rn and set u = i=1 xi hi , v = i=1 y i hi . It is no hard to see that n  X
2 2 kβ(t, x) − β(t, y)k dVt = hi , B (u, t) − B(v, t) ej λj dt H

i,j=1

≤

∞ X j=1

λj |(B (u, t) −

2 B(v, t)) ej |H

(3.37)

dt = |B (u, t) −

2 B(v, t)|Q

dt.

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Hence, by condition (A2 ) n o 2 2(x − y)(a(t, x) − a(t, y)) + kβ(t, x) − β(t, y)k dVt
2 ≤ 2(u − v)big(A(u, t) − A(v, t) dt + |B (u, t) − B(v, t)|Q dt 2

2

≤ K |u − v|H dt = K (u − v) dt.

Therefore, condition (3.11) is satisfied. Moreover, continuity of a(t, x) in x can be derived from Lemma 3.2, and continuity of b(t, x), from the continuity of B (u, t) which, in turn, follows from (A2 ). It remains to check condition (3.12). Define Z t ξ(t) = b(s, x)dm(s). 0

For every ε > 0 we have by (3.7) in analogy to (3.37)  n  n X 1 X 2 dhmn+i it dhξit ≤ (1 + ε) λj (hi , B (u, t) ej )H dt + 1 + ε i=1 i,j=1  
1 2 dhzit + εc f (t) + |u|2H + |u|pV dt, ≤ |B (u, t)|Q dt + 1 + ε

(3.38)

where c does not depend on ε. We note further that

2(xα(t, x) + kβ(t, x)k2 dVt = 2uA(u, t)dt + dhξit .

(3.39)

Combining (3.38) and (3.39) with condition (A3 ) and then choosing ε sufficiently small, we conclude that  
2xα(t, x) + kβ(t, x)k2 dVt ≤ c x2 + f (t) dt + cdhzit   dt 2 dVt , ≤ c(1 + x ) 1 + f (t) dVt where c is a constant not depending on t, x, ω. Hence, condition (3.12) is satisfied, and the proof of the lemma is complete.  Denote by Πn the projection operator of V ∗ onto the span of {h1 , . . . , hn }, by πn the projection operator of E onto the span of {e1 , . . . , en }, and set un (t) = Pn i i=1 un hi . It is then obvious that (3.36) is equivalent to the equation Z t Z t un (t) = u0 + Πn A(un (s), s)ds + Πn B(un (s), s) πn dw(s) + Πn z(t). (3.40) 0

0

To complete the preparations, we need the following result. Theorem 3.23. 1) There exists a constant C0 such that, for all n at once, ! Z T Z T 2 p 2 E sup |un (t)|H + E |un (t)|V dt ≤ C0 E |u0 |H + E f (t)dt + EhziT ; t≤τ

0

0

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2) With Mn (t) = the function un satisfies 2

Z

t

Πn B(un (s))πn dw(s),

(3.41)

0

2

Ee−ct |un (t)|H = E |Πn u0 |H Z t n o 2 2 +E e−cs 2un (s)A(un (s)) + |Πn B(un (s))πn |H − c |un (s)|H ds 0 Z t Z t + 2E e−cs dhz, Mn is + E e−cs dhΠn zis 0

0

for all t ≤ T and for every constant c.

The proof of this theorem is almost a word-for-word repetition of the proof of Theorem 3.10, and we leave it to the reader. We only point out that un Πn A(un ) = un A(un ), |Πn B(un )πn |Q = |B(un )|Q , and that assertion 2) is proved by applying the usual Itˆ o formula, since un (t) lies in a finite-dimensional subspace of H. 3.5. Existence of Solution and the Markov Property: Passing to the Limit by the Method of Monotonicity In this section we complete the proof of Theorem 3.6 and prove Theorem 3.11. We denote by T the σ-algebra of progressively measurable sets on [0, T ] × Ω and by T˜ , its completion with respect to the measure ` × P , where ` is Lebesgue measure on [0, T ]. We define S = ([0, T ] × Ω, T˜ , ` × P ). From the results of the preceding section (Theorem 3.23) it follows that there exists a subsequence {κ} of the natural numbers along which, for some u, v, u∞ (T ), uκ → u weakly in L2 (S; H),

(3.42)

uκ → v weakly in Lp (S; V ),

(3.43)

uκ (T ) → u∞ (T ) weakly in L2 (Ω, FT , P ; H).

(3.44)

By Theorem 3.23 and conditions (A3 ), (A4 ) it may be assumed that Auκ → A∞ weakly in Lq (S; V ∗ ),

(3.45)

Πκ B(uκ )πκ → A∞ weakly in L2 (S; LQ (E; H)).

(3.46)

Strictly speaking, u, v, u∞ (T ), A∞ , and B∞ are all equivalence classes, but in each class u, v, A∞ , B∞ it is possible to choose a progressive measurable representative, and in the class u∞ (T ), a FT -measurable representative. We henceforth consider only these representatives and preserve for them the notation of the corresponding

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classes. We note also that since the embedding of V in H is dense, it follows that u = v for a.a. (t, ω). Further, it is well known that a strongly continuous linear operator is weakly continuous. Therefore, because of (3.46), Z t Z t Πκ B(uκ (s))πκ dw(s) → B∞ (s)dw(s) (3.47) 0

0

weakly in L2 (S; H) and in L2 (Ω, Ft , P ; H) for each t Let y be a bounded random variable, and let ψ(t) be a bounded function on [0, T ]. It follows from (3.40) that, for each hi and κ ≥ 1 Z T E yψ(t)(uκ (t), hi )H dt 0   Z T Z t Z t =E yψ(t)hi u0 + A(uκ (s))ds + Πκ B(uκ (s))πκ dw(s) + z(t) dt. 0

0

0

Passing to the limit in this equality and using (3.43), (3.45)–(3.47), we obtain Z T E yψ(t)(v(t), hi )H dt 0   Z T Z t Z t =E yψ(t)hi u0 + A∞ (s)ds + B∞ (s)dw(s) + z(t) dt. 0

0

0

From this it follows that for a.a. (t, ω) Z t Z t B∞ (s)dw(s) + z(t). A∞ (s)ds + v(t) = u0 +

(3.48)

0

0

In the same way, using (3.44)–(3.47), we find that (a.s.) Z T Z T u∞ (T ) = u0 + A∞ (s)ds + B∞ (s)dw(s) + z(T ). 0

(3.49)

0

By Theorem 2.17, there exists an Ft -consistent function continuous in t with values in H which coincides with v(t) for a.a. (t, ω) and is equal to the right side of (3.48) for all t ∈ [0, T ] and ω ∈ Ω0 , where P (Ω0 ) = 1. We identify v with u; this is possible, since u = v a.e. in (t, ω). In view of (3.49) we then have u∞ (T ) = u(T ) (a.s.).

(3.50) 0

By the same Theorem 2.17, for all (t, ω) ∈ [0, T ] × Ω , Z t Z t
2 2 |u(t)|H = |u0 |H + 2 v(s)A∞ (s)ds + 2 u b(s) B∞ (s)dw(s) + dz(s) 0

0

+ hM∞ + zit ,

where M∞ (t) =

Z

t 0

B∞ (s)dw(s).

(3.51)

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The remaining argument in the proof of Theorem 3.6 are standard in the theory of monotone operators and do not require special attention of the reader, because, on the first reading, these arguments produce the impression of a collection of trivial computations, mysteriously leading to the required result. Let y(t, ω) be an Ft -consistent function, measurable in (t, ω) with values in V and satisfying E

Z

T 0

  p 2 |(y(t))|V + |(y(t))|H dt < ∞.

(3.52)

Define Z

n 2 e−ct 2(uκ (t) − y(t))(A(uκ (t) − A(y(t)) − c |uv (t) − y(t)|H 0 o 2 + |Πκ B(uκ (t))πκ − Πκ B(y(t))πκ |Q dt.

Oκ = E

T

From (A2 ) it follows that, with a suitable choice of c, Oκ ≤ 0.

(3.53)

Further, we represent Oκ in the form Oκ = Oκ1 + Oκ2 where Oκ1 = E

Z

T 0

n o 2 2 e−ct 2(uκ (t)A(uκ (t)) − c |uκ (t)|H + |Πκ B(uκ (t))πκ |Q dt,

and Oκ2 = Oκ − Oκ1 . By Theorem 3.23, Oκ1

= Ee

−ct

2 |uκ (T )|H

−

2 E |Πκ u0 |H

−2E

Z

T

e 0

−ct

dhMκ , zit − E

Z

T

e−ct dhΠκ zit ,

0

with Mκ defined in (3.41). After integration by parts, E

Z

T

e

−ct

dhMκ , zit = E e

Z

−cT

0

+cE

Z

T

Πκ B(uκ (s))πκ dw(s), z(T ) 0

T

e−ct 0

Z

!

H

t

Πκ B(uκ (s))πκ dw(s), z(t) 0



dt. H

From this and (3.47) it follows that lim supOκ1 κ→∞

= Ee

−cT

−2E 2

Z

0

2 |u(T )|H

T

e

−ct

−

2 E |u0 |H

−E

dhM∞ , zit + δ e 2

Z

−cT

T 0

e−ct dhzit (3.54)

,

where δ = lim sup E |uκ (T )|H −E |u(T )|H ≥ 0 by (3.44). On the other hand, because κ→∞

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of (3.51), Ee−cT |u(T )|2H − E |u0 |2H = E

Z

+ 2E

T 0

Z

0

n o e−ct 2(v(t)A∞ (t) − c |u(t)|2H + |B∞ (t)|2Q dt

T

e−ct dhM∞ , zit + E

Z

T

e−ct dhzit .

0

(3.55)

Comparing (3.54) and (3.55) we see that Z T n o 2 2 lim sup Oκ1 = E e−ct 2(v(t)A∞ (t) − c |u(t)|H + |B∞ (t)|Q dt + δ e−cT . (3.56) κ→∞

0

Further, from (3.42), (3.43), (3.45), and (3.47) it follows that Z T  2 lim Oκ = E e−ct 2y(t)A(y(t)) − 2y(t)A∞ (t) − 2v(t)A(y(t)) κ→∞

0

2

+ 2c(u(t), y(t))H − c |y(t)|H − 2 B∞ (t), B(y(t))




 2 + |B(y(t))| dt. Q Q

Combining (3.56) and (3.57) we find, in view of (3.53), Z T n 2 E e−ct 2(v(t) − y(t))(A∞ (t) − A(y(t))) − c |u(t) − y(t)|H 0 o + |B∞ (t) − B(y(t))|2Q dt + δ e−cT ≤ 0.

(3.57)

(3.58)

Setting y = v in (3.58), we see that B∞ (t) = B(v(t)) (a.e. in (t, ω)), and δ = lim sup E |uκ (T )|2H − E |u(T )|2H = 0.

(3.59)

κ→∞

On the other hand, it follows from (3.58) that Z T n o
2 E e−ct 2(v(t) − y(t)) A∞ (t) − A(y(t)) − c |u(t) − y(t)|H dt ≤ 0.

(3.60)

0

Suppose now that x(t, ω) is a process with values in V which satisfies an inequality analogous to (3.52), and let y = v − λx, λ ∈ R+ . From (3.60) we then obtain the inequality Z T n  o
 2 E e−ct x(t) A∞ (t) − A v(t) + λx(t) − cλ |x(t)|H dt ≤ 0. 0

Letting λ go to zero, we find from this by (A4 ) and the Lebesgue theorem that Z T E x(t)(A∞ (t) − A(v(t)))dt ≤ 0. 0

Since x is arbitrary, the last inequality implies A∞ (t) = A(v(t)) a.e. in (t, ω), which together with (3.48) and an earlier proved equality B∞ (t) = B(v(t)) a.e. in (t, ω) complete the proof of Theorem 3.6. 

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Corollary 3.24. Let v be a solution of (3.4) and u, its continuous modification in H, and let un be solutions of (3.40). Then for any t ≤ T 2

lim E |un (t) − u(t)|H = 0.

n→∞

Proof. Note that we proved equality (3.59) in which u is a continuous modification of v in H. On the other hand, from (3.44) we have E |u(T )|2H ≤ lim E |uκ (T )|2H κ→∞

and so lim E |uκ (T )|2H = E |u(T )|2H .

κ→∞

Returning again to (3.44) and recalling that a weakly convergent sequence converges if the norm of the limit is equal to the limit of the norm, we conclude that 2

lim E |uκ (T ) − u(T )|H = 0.

κ→∞

Here u(T ) does not depend on the sequence {κ} by Theorem 2.3. We have thus proved that any subsequence of the sequence {un (T ), n = 1, 2, 3, . . .} contains a further subsequence converging strongly in L2 (Ω, FT , P ; H) to u(T ). This implies the convergence of the whole sequence: un (T ) → u(T ) as n → ∞. It remains to note that, in place of segment [0, T ], we can consider any smaller segment.  Proof of Theorem 3.11. Together with (3.4), we consider the equation Z t Z t B(v(r), r)dw(r), (3.61) A(v(r), r)dr + v(t) = us + s

s

2 E |u(s)|H

which we solve for t ∈ [s, T ] and < ∞. The definitions of a V -solution and H-solution are made for equation (3.61) in the usual way. All the assertions proved for equation (3.4) carry over to (3.61) in a natural way; in particular, (3.61) has an H-solution u(t) = u(t, s, us ). We now assume that us = x ∈ H is non-random and, for t ∈ [s, T ] and a bounded Borel function f (h) on H, we define Ms,x f (u(t)) = Ef (u(t, s, x)).

(3.62)

To prove the theorem it obviously suffices to show that (3.62) defines a Borel function of x, and for the H-solution u(t) of (3.4), E {f (u(t))|Fs } = Ms,u(s) f (u(t) (a.s.)

(3.63)

The required property of Borel measurability is easily proved by means of a corresponding analog of Theorem 3.8, which shows that for continuous f the function Ms,x f (u(t)) is also continuous in x. It suffices to prove the relation (3.63) only for continuous f . Moreover, for simplicity, we assume that s = 0. This can always be achieved by changing the time variable.

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We note, first of all, that u(t, 0, x) does not depend on F0 ; this follows from Corollary 3.24 and the fact that in Section 3.3 solutions of stochastic equations were constructed in the final analysis by passing to the limit in equations with coefficients that satisfy a Lipschitz condition, and for such equations the independence of a solution with nonrandom initial data from F0 is known [42, 81]. Further, we approximate u0 (ω) by step functions un0 (ω) and let Γn be the set of values of un0 (ω). It is easy to see that u(t, 0, un0 ) and X χ{x} (un0 )u(t, 0, x) x∈Γn

satisfy the same equation and therefore coincide (a.s.) From this and the independence of u(t, 0, x) and F0 , it follows that E {f (u(t, 0, un0 ))|F0 } = M0,un0 f (u(t)) (a.s.)

Letting here n → ∞ and using Theorem 3.8 we obtain (3.63) for continuous f . The proof of Theorem 3.11 is complete.  4. Itˆ o Stochastic Partial Differential Equations 4.1. Introduction This section is devoted to applications of the results of Section 3 to stochastic partial differential equations. We consider the first boundary-value problem for nonlinear equations of second order. The latter merit special attention, since filtering of diffusion process reduces to the investigation of equations of this type [1, 2, 82]. The results pertaining to nonlinear equations are related to Ref. [62], where equations without stochastic terms are considered, while elements of the theory of linear equations are presented following our work [64]. We start by recalling the definitions and some basic facts from the theory of Sobolev spaces. Let Rd be d-dimensional Euclidean space with a fixed basis. Denote by α, β, γ, αi , βi , γi , (i = 1, 2, . . .) the coordinate vectors in this space and also the null vector. If α is the null vector, then D α is the identity operator, while if α is the i − th basis vector, then D α = ∂/∂xi . Suppose further that G is a domain in Rd , Γ is the boundary of G, m ≥ 1 is an integer, p ∈ (1, ∞). Definition 4.1. The Sobolev space Wpm (G) is the space of real functions defined on G with finite norm !1/p X p kukm,p ≡ kDα1 . . . Dαm ukp , α1 ...αm

where Dα1 , . . . , Dαm are generalized derivatives, and Z 1/p p kgkp = |g(x)| dx . G

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∞ Definition 4.2. The Sobolev space W m p (G) is the closure of C0 (G) (the space of infinitely differentiable functions with compact supports in G) in the norm k·km,p . ◦

Theorem 4.3. The spaces W m p (G) are separable reflexive Banach spaces relative to the norm k·km,p . Theorem 4.4 (Sobolev Embedding Theorem). If the domain G is bounded, its boundary Γ is regular, and 2(d − mp) ≤ pd, then Wpm (G) ⊂ L2 (G) and this embedding is dense and continuous. In particular, there exists a constant c such that kuk2 ≤ c kukm,p for all u ∈ Wpm (G). Proofs of these theorems can be found in Refs. [83–85]; the conditions on the boundary we call regular are also presented there. The following assertion is obvious. Theorem 4.5. The Sobolev space W2m (Rd ) is a Hilbert space; it is continuously and densely imbedded in L2 (Rd ). We further need below the so-called Friederichs inequality [83, 85] which is given in the following theorem. Theorem 4.6. Suppose that the domain G is bounded and its boundary is regular. ◦

Then there exists a constant c > 0 such that, for every u ∈ W m p (G),   X α α kD 1 . . . D m ukp  . kukm,p ≤ c  |α1 |+...+|αm |=m

The next fact is also well know; it follows easily by means of the Fourier transform and the Parseval equality. Theorem 4.7. There exists a constant c > 0 such that, for every u ∈ W2m (Rd ),   X kDα1 . . . Dαm uk2 + kuk2  . kukm,2 ≤ c  |α1 |+...+|αm |=m

We will also need the spaces Wq−m (G) where q = p/(p − 1); Refs. [51, 85] present a detailed account of these spaces. It is assumed below that either G is bounded and its boundary is regular, or G = Rd and p = 2. Definition 4.8. The negative norm of an element f ∈ Lq (G) is kf k−m,q = sup(f, u)0 , where (f, u)0 =

Z

G

f (x)u(x)dx

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and the supremum is taken over the set of all functions u ∈ W m p (G) for which kukm,p = 1. Definition 4.9. The space with negative norm Wq−m (G) is the completion of Lq (G) in the norm k·k−m,p . ◦

It is clear that, for f ∈ Lq (G) and u ∈ W m p (G), we have |(f, u)0 | ≤ kf kq · kukp ≤ kf kq · kukm,p . Definition 4.9 is therefore correct. From the above definitions it follows immediately that there are the natural embeddings ◦

−m Wm (G). p (G) ⊂ Lp (G) ⊂ Wp ◦

−m The duality between W m (G) is defined by means of the scalar p (G) and Wq ◦

∗ −m product in L2 (G): if v ∈ W m (G), vn ∈ C0∞ , vn∗ ∈ Lq (G), and p (G), v ∈ Wq kvn − vkm,p → 0, then we define

hhv, v ∗ ii = lim (vn , vn∗ )0 . n→∞

◦

It is known that, for every continuous linear functional on W m p (G), there exists ∗ −m ∗ a v ∈ Wq (G) such that this functional is equal to hhv, v ii. Similarly, every continuous linear functional on Wq−m (G) can be written as hhv, v ∗ ii for some v ∈ ◦

◦

m ∗ −m Wm (G), p (G). Moreover, for v ∈ W p (G) and v ∈ Wp

kvkm,p = sup w∗

hhw, v ∗ ii hhv, v ∗ ii ∗ , kv k = sup . kw∗ k−m,q v ∗ kwkm,q ◦

The relation hh·, ·, ii makes it possible to identify the dual space of W m p (G) with −m Wq (G). We note that under this identification the duality hh·, ·, ii plays a decisive ◦

role. In particular, the space W m 2 (G) is a Hilbert space and, as any other Hilbert space, can be identified in the well-known way with its dual space by means of ◦

−m its own scalar product. At the same time, of course, W m (G). Thus, 2 (G) 6= W2 ◦

different bilinear forms make it possible to construct dual spaces of W m 2 (G) in different ways. We shall use these considerations in the proof of Theorem 4.20 below. ◦ −m We identify the dual space of W m (G), so that the result of p (G) with Wq ◦

applying a functional v ∗ ∈ Wq−m (G) to an element v ∈ W m p (G) is written as hhv, v ∗ ii. This notation differs from the notation vv ∗ used in Section 2 and 3 because both v and v ∗ can now be ordinary functions and then vv ∗ can be misinterpreted as the usual product of two functions.

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−m Theorem 4.10. Let 2(d − mp) ≤ pd. Then W m (G), where p (G) ⊂ L2 (G) ⊂ Wq ◦

each embedding is dense and continuous. If v ∗ ∈ L2 (G) and v ∈ W m p then hhv, v ∗ ii = (v, v ∗ )0 . 4.2. First Boundary-Value Problem for Nonlinear Stochastic Parabolic Equations

Let (Ω, F, P ) be a complete probability space; let {Ft } be an increasing family of complete σ-algebras imbedded in F; let Gbe a bounded domain with a regular boundary or G = Rd and p = 2. We assume that 2(d − mp) ≤ pd. Let z(t) be a continuous square-integrable martingale (relative to Ft ) with values in L2 (G) and let w(t) be a Wiener process with values in some separable Hilbert space E and ◦

with covariance operator Q. By Theorem 4.10 the spaces V = W m p (G), H = L2 (G), V ∗ = Wq−m (G) satisfy assumptions a) – d) of Section 3. In the cylinder [0, T ] × G for fixed T > 0 we consider the problem
du(t, x, ω) = −(−1)|α1 |+...+|αm | Dα1. . . Dαm Aα1 ...αm Dβ1. . . Dβm u(t, x, ω), t, x, ω dt
+ B Dβ1 . . . Dβm u(t, x, ω), t, x, ω dw(t, ω) + dz(t, x, ω), (4.1) u(0, x, ω) = u0 (x, ω), x ∈ G,

(4.2)

Dβ0 . . . Dβm−1 u |s = 0 for all β0 , . . . , βm−1

(4.3)

such that |β0 | + . . . + |βm−1 | ≤ m − 1,

where S is the lateral surface of the cylinder [0, T ] × G; summations over all the values of the repeated indices αi is understood; the functions A, B depend on t, x, ω, and all the derivatives of u with respect to x of order no greater than m; A and u are real functions; B is a function with values in E; in the second term in (4.1) the scalar product in E is understood. β1 βm We assume that for all collections of real numbers ξ = (ξ ... ) and any e ∈ E the functions A(ξ, t, x, ω), B(ξ, t, x, ω)e = (B(ξ, t, x, ω), e)E are (1) measurable in (t, x, ω), (2) measurable in (x, ω) relative to the product of Ft and the Borel σ-algebra on Rd for every fixed t ∈ [0, T ], (3) continuous in ξ for every fixed t, x, ω. Suppose further that there exist a constant K > 0 and a nonnegative function f (t, x, ω) possessing the same measurability properties as A(0, t, x, ω) and such that, for all t, x, ω, ξ, α1 , . . . , αm , X ξ β1 ,...,βm p−1 , |Aα1 ,...,αm (ξ, t, x, ω)| ≤ f 1/q (t, x, ω) + K (4.4) β1 ,...,βm

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|B(ξ, t, x, ω)|2E ≤ f (t, x, ω) + K

X

β1 ,...,βm

2 β ,...,β p m ξ 1 + K ξ 0,...,0 .

(4.5)

Moreover, it is assumed that u0 (x, ω) is measurable relative to the product of F0 and the Borel σ-algebra on Rd , and, for all t, ω, Z T 2 ku0 k2 < ∞, kf (t)k1 < ∞, E ku0 k2 < ∞, kf (t)k1 dt < ∞. 0

These assumptions enable us to give a precise meaning to equation (4.1), to the initial condition (4.2), and to the boundary conditions (4.3) in the following manner.d We note that by (4.4) and the H¨ older inequality  
1/q p−1 Aα1 ...αm (Dβ1 . . . Dβm u), Dα1 . . . Dαm v 0 ≤ c kf (t)k1 + kukm,p kvkm,p . (4.6) Therefore, for t ∈ [0, T ] the left side of (4.6) is a linear functional on Wpm (G) ◦

and, in particular, on V = W m p (G). As we know, there exists a unique element A(u, t, ω) ∈ V ∗ = Wq−m (G) for which for all v ∈ V
hhv, A(u, t, ω)ii = − Aα1 ...αm (Dβ1 . . . Dβm u, t, ω), Dα1 . . . Dαm v 0 .

It is clear that the function A(u, t, ω) satisfies the measurability conditions and conditions (A1 ), (A4 ) of Section 3.2. Further, we note that for g ∈ H by (4.5)

B(Dβ1 . . . Dβm u, t) , g ≤ kf (t)k1/2 · kgk + N kukp/2 · kgk + kuk · kgk . 1 H V H H E 0

This implies that B(u, t, ω) ∈ L(E, H) is defined for t ∈ [0, T ], ω ∈ Ω, u ∈ V according to the formula
(B(u, t, ω)e, g)0 = (B(Dβ1 . . . Dβm u, t, ·, ω), e)E , g 0 ,

where e ∈ E, g ∈ H. Moreover, if ei , hi are orthonormal bases in E, H, respectively, then X 2
2 2 kB(u, t)k = B(u, t)ei , hj ≤ B(Dβ1 . . . Dβm u, t) E 2 i,j (4.7) p 2
≤ kf (t)k1 + N kukV + kukH .

2

A similar inequality obviously holds for B(u, t)Q1/2 . Hence, B(u, t) satisfies inequality (3.7). Moreover, it is clear that if u = u(t, ω) is a function with values in V which is measurable in (t, ω) and Ft -consistent, then B(u(t, ω), t, ω) possesses the same measurability properties as an element of LQ (E, H). Thus, equation (3.4) can be considered for the operators A, B defined above. Definition 4.11. A V -solution (H-solution) of problem (4.1)–(4.3) is a V -solution (H-solution) of equation (3.4). A continuous modification of problem (4.1)–(4.3) in H is defined similarly. d For

notational convenience we will occasionally omit all or some of the arguments in various functions.

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Remark 4.12. A function v(t) ∈ V is a V -solution of problem (4.1)–(4.3) if and only if it is appropriately measurable, inequality (3.5) is satisfied, and, for every η ∈ V and almost all (t, ω), Z t
(v(t), η)0 = (u0 , η)0 − Aα1 ...αm (Dβ1 . . . Dβm v, s), Dα1 . . . Dαm η 0 0 (4.8) Z tZ β1 βm + B(D . . . D v, s, x)η(x)dxdw(s) + (z(t), η)0 . 0

Rd

In others words, a V -solution is a solution of problem (4.1)–(4.3) in the sense of the integral identity. To prove this, it suffices to use Remark 3.3 and the fact that, for η ∈ V , the norm B(Dβ1 . . . Dβm v(t, x), t, x) E multiplied by |η(x)| is integrable over Rd , and hence for any e ∈ E Z
ηbB(v, t)e = (η, B(v, t)e)H = B(Dβ1 . . . Dβm v, t, x), e E η(x)dx d R Z  β1 βm = B(D . . . D v, t, x)η(x)dx, e . Rd

E

A similar remark holds for an H-solution of problem (4.1)–(4.3). Remark 4.13. Relation (4.8) is obtained if we formally multiply (4.1) by η, integrate over t, x, integrate by parts in x, and use the boundary conditions (4.3). In our interpretation of a solution of (4.1)–(4.3) we start from (4.8) in which there are no boundary conditions. They are accounted in (4.8) only through the membership ◦

of v in the space W m p (G). In this connection we note that for p > d by one of the ◦

Sobolev embedding theorems each function of W m p (G) is equal almost everywhere to a function which has derivatives of order less than or equal to m − 1, and these derivatives are continuous in the closure of G and vanish on the boundary of G. Definition 4.14. We say that equation (4.1) satisfies the condition of strong parabolicity if the operators A, B satisfy conditions (A2 ), (A3 ) of Section 3.2. Conditions which are sufficient for strong parabolicity in terms of the original functions Aα1 ...αm (ξ, t, x, ω), B(ξ, t, x, ω) will be given below. The next theorem follows automatically from the results of Section 3.2. Theorem 4.15. If equation (4.1) is strongly parabolic, then the assertions of Theorems 3.6, 3.11 hold. Verification of the condition of strong parabolicity in the general case is a problem of colossal difficulty even for B ≡ 0. However, by generalizing the results of Ref. [62], in certain cases it is possible to give simple sufficient conditions for strong parabolicity of equation (4.1).

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We say that the algebraic condition of strong parabolicity is satisfied if, for every fixed t, x, ω, the functions Aα1 ...αm (ξ β1 ...βm , t, x), and B(ξ β1 ...βm , t, x) are differentiable once with respect to ξ everywhere except for a set having a finite number of points of intersection with each line in the space of coordinates ξ, the derivatives are locally summable on each line in this space, and there exist constants ε > 0 and N ∈ R such that, for all ξ α1 ...αm , η α1 ...αm , t, x, ω,

2 γ1 ...γm m − 2Aβα11...β , t, x)η α1 ...αm η β1 ...βm + B β1 ...βm (ξ γ1 ...γm , t, x)η β1 ...βm Q ...αm (ξ X
2 (4.9) +ε |ξ α1 ...αm |p−2 |η α1 ...αm |2 ≤ N η 0...0 , |α1 |+...+|αm |=m

where m Aβα11...β ...αm =

∂B ∂Aα1 ...αm , B β1 ...βm = β1 ...βm , ∂ξ β1 ...βm ∂ξ

and in the second term of the left side of (4.9) summation over all β1 ..βm is carried out before computing the norm |·|Q . The next theorem justifies the name of condition (4.9). Theorem 4.16. Suppose that the algebraic condition of strong parabolicity is satisfied, together with all other assumptions of this section regarding the functions A(ξ β1 ...βm ), B(ξ β1 ...βm ). Then equation (4.1) is strongly parabolic. Proof.

We use the formula of Hadamard B(ξ) − B(η) =

Z

1 0

B β1 ...βm (ξt + (1 − t)η)(ξ β1 ...βm − η β1 ...βm )dt,

and also the Cauchy-Schwartz inequality and the definition of the operators A, B. We then obtain 2

I(v1 , v2 ) ≡ 2hhv1 − v2 , A(v1 ) − A(v2 )ii + |B(v1 ) − B(v2 )|Q

= 2 Dα1 . . . Dαm (v1 − v2 ), Aα1 ...αm (Dβ1 . . . Dβm v1 ) − Aα1 ...αm (Dβ1 . . . Dβm v2 )



2 2 + B(Dβ1 . . . Dβm v1 ) − B(Dβ1 . . . Dβm v2 ) Q ≤ N kv1 − v2 kH 2  Z Z 1 X α
p−2 αm 1 −ε D ...D v2 + t(v1 − v2 ) dt |α1 |+...+|αm |=m

G




0

0

× |Dα1 . . . Dαm (v1 − v2 )|2 dx.

(4.10)

We have thus proved that the condition of monotonicity (A2 ) of Section 3.2 is satisfied. In fact, equality (4.10) enables us to verify the coercivity condition (A 3 ):

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From (4.10) and (4.7) we have, with v1 = v, v2 = 0, 2

2

2hhv, A(v)ii + |B(v)|Q ≤ I(v, 0) + 2hhv, A(0)ii − |B(0)|Q + 2 |B(v)|Q |B(0)|Q   2 1 p p q 2 + δ kf (t)k1 + N δ kvkV + 2δ kvkV + kA(0)kV ∗ + N kvkH ≤N δ δ X ε p kDα1 . . . Dαm vkp , − p−1 |α1 |+...+|αm |=m

where δ > 0 and N does not depend on δ. Using Theorems 4.6 and 4.7, noting that the growth condition (A4 ) has already been verified, and choosing δ sufficiently small, we conclude that the coercivity condition is satisfied. The proof of the theorem is complete.  Example 4.17. Let d = 1, E = R, and suppose that (4.1) has the form ! m p−2 m m ∂ ∂ ∂ u(t, x) dt du(t, x) = a(t, ω)(−1)m+1 m m u(t, x) ∂x ∂x ∂xm p/2 m ∂ + b(t, ω) m u(t, x) dw(t), ∂x

(4.11)

where a, b are appropriately measurable and also bounded processes. Here the algebraic condition of strong parabolicity becomes −2(p − 1)a +

p2 2 b ≤ −ε, 4

where ε > 0 is a constant. If this condition is satisfied, then by Theorem 4.15 we obtain assertions regarding the existence, uniqueness, stability with respect to the initial data, and the Markov property of solutions of (4.11) with the “boundary” ◦

condition u ∈ W m p (G). Equation (4.11) coincides with (3.2). Equations of the form (3.3) are studied in the next section. 4.3. Cauchy Problem for Linear Second-Order Equations In this section we continue the study of equation (4.1), assuming that m = 1, G = Rd , and A and B are linear functions of ξ which are generally not equal to zero for ξ = 0. Moreover, all assumptions of Section 4.2 are naturally assumed to be satisfied. Problem (4.1)–(4.3) becomes
du(t, x) = Dα aαβ (t, x)Dβ u(t, x) + fα (t, x) dt
(4.12) + bα (t, x)Dα u(t, x) + g(t, x) dw(t) + dz(t, x), u(t, ·) ∈ L2 (Rd ), u(0, x) = u0 (x), x ∈ Rd ,

(4.13)

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where aαβ , fα are real functions and bα and g are functions with values in E. Conditions (4.4) and (4.5) are equivalent to the boundedness of aαβ and |bα |E , together with the inequality Z T X Z T 2 2 E kfα k2 dt + E k |g|E k2 dt < ∞. 0

α

0

A solution of problem (4.12), (4.13) is understood in the sense of the integral identity (4.8); for a V -solution the identity is satisfied for almost all (t, ω), and for an H-solution, for each t with probability one.

Lemma 4.18. Suppose that, for all x, η ∈ Rd , t ∈ [0, T ], ω ∈ Ω, 2 d d X X 2 2 aij (t, x)η i η j − bi (t, x)η i ≥ ε |η| . i,j=1

i=1

(4.14)

Q

where ε > 0 is a constant, aij = aαβ , bi = bα if α is the i-th and β the j-th coordinate vectors. Then the algebraic condition of strong parabolicity is satisfied. The lemma is easily proved by means of inequalities of the type 2a0α η 0 η i ≤ |a0α ε |η i |2 + ε−1 |a0,α |η 0 |2 . The next result is a direct corollary of Lemma 4.18, Theorems 4.1 and 4.2, and also Corollary 3.7. Theorem 4.19. Suppose that condition (4.14) is satisfied. Then there exists a function u(t, ω) defined on [0, T ] × Ω with values in L2 (Rd ), strongly continuous in t in L2 (Rd ), Ft -consistent and such that (a) For almost all (t, ω), u ∈ W21 (Rd ); (b) 2 E sup ku(t)k2 t≤T

(c) for each η ∈

W21 (Rd ),

+E

Z

T 0

2

ku(t)k1,2 dt < ∞;

Z

t
(u(t), η)0 = (u0 , η)0 + (−1)α Dβ u(s), aαβ (s)Dα η 0 0 Z t + (−1)α (fα (s), Dα η)0 ds 0 Z t + (bα (s)Dα u(s) + g(s), η)0 dw(s) + (z(t), η)0 0

for all t ∈ [0, T ] (a.s.)

(4.15)

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Theorems 3.8 and 3.11 enable us to establish uniqueness, stability with respect to the initial data, and the Markov property of the function u considered in Theorem 4.19 [u is an H-solution of problem (4.12), (4.13)]. These properties of u are proved simply by referring to Theorems 3.8 and 3.11, and we are not going to discuss them. We now turn to a more important question, the question of raising the smoothness of a solution of problem (4.12), (4.13). One reason for addressing this question is that filtering of diffusion processes [86] leads to equations analogous to (4.15), but with the inner product in L2 (Rd ) replaced by the inner product in W2m (Rd ), i.e., the index 0 in the inner product in (4.15) is replaced by m so that (ϕ, ψ)m = (Dα1 . . . Dαm ϕ, Dα1 . . . Dαm ψ)0 . Thus, we denote the modified equation (4.15) by (4.15)m . For (4.15)m to be meaningful for sufficiently smooth u0 and coefficients aαβ , fα , bα , g, z, it suffices to restrict attention to functions u belonging to W2m+1 (Rd ) [for a.a. (t, ω)]. However, the filtering density is equal to the function (1 − ∆)m u multiplied by some function of time. Thus, assertions are needed regarding the membership of the solution of equation (4.15)m not in W2m+1 (Rd ) but in W22m (Rd ) [for a.a. (t, ω)]. However, merely an assertion on the existence of a solution of (4.15)m with values in W22m (Rd ) is of little use, since from the filtering theory it is known a priori only that the solutions belongs to W2m+1 (Rd ); if we wish to prove its smoothness, we must have not only a theorem on the existence of a solution of (4.15)m with values in W22m (Rd ) but also a theorem on the uniqueness of a solution with values in W2m+1 (Rd ). Thus, a theorem on raising the smoothness for equation (4.15)m is required. A proof of the corresponding result is given in Ref. [64]. Not to obscure the exposition with technical details, we here prove the theorem on raising smoothness only for equation (4.15). Throughout the remainder of the paper we fix an integer m > 0 and assume that z(t) is a square-integrable martingale with values in W2m (Rd ) which is continuous in t in W2m (Rd ) for all t, ω, the functions aαβ (resp. bα ) have m derivatives (resp. weak derivatives) in x, are continuous (weakly continuous) in x, and these derivatives of aαβ , bα are bounded (for bα in the norm of E) uniformly with respect to t, x, ω. Suppose that, for all (t, ω), the functions fα ∈ W2m (Rd ), u0 ∈ W2m (Rd ), and 2

E ku0 km,2 + E

Z

T 0

2

kfα (t)km,2 dt < ∞.

Let g ≡ 0. This condition involves no loss of generality, since the integral of g with respect to dw(t) can be included in z(t). Finally, we assume that condition (4.14) is satisfied. Theorem 4.20. There exists a set Ω0 ⊂ Ω such that P (Ω0 ) = 1 and for ω ∈ Ω0 the function u(t) of Theorem 4.19 belongs to W2m (Rd ) and is continuous in t in the

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norm of W2m (Rd ). Moreover, u ∈ W2m+1 (Rd ) a.e. in (t, ω), and Z T 2 2 Esup kukm,2 + E ku(s)km+1 ds < ∞. t≤T

0

Proof. We set H = W2m (Rd ) and identify H with its dual by means of (·, ·)m . It is then easily seen from the Parseval equality that W2m−1 is identified with V ∗ , the space dual to V = W2m+1 (Rd ). Thus, V ⊂ H ⊂ V ∗ , and it is obvious that each embedding is dense and continuous. As in the preceding section, it is easy to verify that the formulas
hhη, A(t)uii = aαβ (t)Dβ u + fα (t), (−1)α Dα η m , (B(t)u) e, η




m

= (bα (t), e)E Dα u, η ∗




m

define bounded linear operators A(t) : V → V , B(t)u : E → H. The reader can also verify without difficulty that the functions A(t)u, B(t)u satisfy conditions (A1 )–(A4 ) of Section 3.2 and also the measurability conditions of this section (see, e.g. Ref. [64]). Hence, by Corollary 3.7 in our case equation (4.1) has an H-solution u ˜(t). Application of Remark 4.12 shows that, for each η ∈ W2m (Rd ), Z t
(˜ u(t), η)m = (u0 , η)m + aαβ (s)Dβ u ˜(s) + f (s), (−1)|α| Dα η m 0 (4.16) Z t
α + bα (s)D u ˜(s), η m dw(s) + (z(t), η)m 0

for all t ∈ [0, T ] (a.s.) To avoid misunderstanding we note that (4.16) does not coincide with (4.15)m if m > 0 and aαβ , bα depend on x. If in (4.16) in place of η we substitute (1 − ∆)−m η and use the fact that by the Parseval equality (f, g)m = (f, (1 − ∆)m g)0 , where g ∈ W22m (Rd ), then we see that in (4.16) in place of m it is possible to write 0. After this, by Theorem 3.8 we obtain sup ku(t) − u ˜(t)k2 = 0 (a.s.), t≤T

and u(t) possesses the same properties as u ˜(t). The proof of the theorem is complete.



In conclusion, we discuss the significance of condition (4.14) for the validity of Theorem 4.19. For d = 1 and E = R we consider the equation du(t, x) =

1 ∂ 2 u(t, x) ∂u(t, x) dt + σ dw(t), 2 ∂x2 ∂x

where σ is a constant and the initial condition u0 (x) is nonrandom. If Theorem 4.19 is valid for this equation, then from (4.15) and the Parseval equality it follows

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that

Z

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 Z Z 1 t 2 u b(t, y)b η (y)dy = u b0 (y)b η (y)dy − y u b(s, y)b η (y)dy ds 2 0 R R R Z  Z t √ − −1 σ yb u(s, y)b η (y)dy dw(s), Z

0

R

where u b, ηb are the Fourier transforms of u, η. Here it is easy to interchange the integrals if ηb(y) is a compactly supported function, and we then find that for almost all (t, y, ω) Z t Z t √ 1 u b(s, y)ds − −1 σy u b(s, y)dw(s). (4.17) u b(t, y) = u0 (y) − y 2 2 0 0

We fix a y for which equation (4.17) holds for almost all (t, ω), and we denote the right side of (4.17) by ϕ(t, y). Then ϕ(t, y) satisfies (4.17) for all t (a.s.). The solution of the equation for ϕ is known: 1

ϕ(t, y) = e− 2 (1−σ

2

√ )y 2 t− −1 σyw(t)

u b0 (y).

Since u b(t, y) = ϕ(t, y) a.e. in (t, y, ω), it follows that Z T Z −(1−σ 2 )y 2 T −1 2 2 −2 e E ku(t)k1,2 dt = (1 + y ) u b0 (y) dy. 2 1 − σ 0 R

(4.18)

This implies that the left side of (4.18) is finite for all u0 ∈ L2 (R) if and only if |σ| < 1. The last condition in the present case is equivalent to (4.14). This example demonstrates the necessity of condition (4.14) for the validity of Theorem 4.19, and also the necessity of the coercivity condition (A3 ) of Section 3.2 for the validity of the results in that section. References [1] R. S. Liptser and A. N. Shiryaev, Statistics of Random Processes. (Nauka, Moscow, 1974). In Russian. [2] N. V. Krylov and B. L. Rozovskii, On conditional distribuitions of diffusion processes, Izv. Akad. Nauk SSSR, Ser. Mat. 42(2), 356–378, (1978). [3] N. T. J. Bailey, Stochastic birth, death, and migration processes for spatially distributed populations, Biometrika. 55(1), 189–198, (1968). [4] J. F. Crow and M. Kimura, An Introduction to Population Genetics Theory. (Harper and Row, New York, 1970). [5] G. Mal´ecot. Identical loci and relationship. In Proc. 5th Berkeley Symp. Math. Stat. Prob., IV, pp. 317–332. Calif. Univ. Press, (1967). [6] D. A. Dawson, Stochastic evolution equations, Math. Biosci. 15(3–4), 287–316, (1972). [7] W. H. Fleming, Distributed parameter stochastic systems in population biology, vol. 107, Lect. Notes Econ. Math. Syst., pp. 179–191. Springer, (1975). [8] W. Feller. Diffusion processes in genetics. In Proc. Second Berkeley Symp. 1. Math. Stat. Prob., pp. 227–246. Calif. Univ. Press, Berkeley.

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[9] I. M. Gel’fand and N. Y. Vilenkin, Generalized Functions: Applications of Harmonic Analysis. (Academic Press, 1964). [10] E. A. Novikov, Functionals and the method of random forces in the theory of turbulence, Zh. Eksp. Teor. Fiz. 47(5), 1919–1926, (1966). [11] A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics. (MIT Press, 1975). [12] V. I. Klyatskin, Stochastic Description of Dynamical Systems with Fluctuating Parameters. (Nauka, Moscow, 1975). In Russian. [13] B. Simon, The P (Φ)2 Model of Euclidean Qauntum Field Theory. (Mir, Moscow, 1976). Russian translation. [14] T. Hilda and L. Strett, On quantum theory in terms of white noise, Nagoya Math. J. 68(Dec.), 21–34, (1977). [15] D. A. Dawson, Stochastic evolution equations and related measure processes, J. Multivar. An. 5(1), 1–52, (1975). [16] S. Albeverio and R. Hoegh-Krohn, Dirichiet forms and diffusion processes on rigged Hilbert spaces, Z. Wahr. Verw. Geb. 40(1), 1–57, (1977). [17] Y. L. Daletskii, Infinite-dimensional elliptic operators and parabolic equations related to them, Usp. Mat. Nauk. 22(4), 3–54, (1967). [18] V. V. Baklan, The existence of solutions of stochastic equations in Hilbert space, Dopavidi Akad. Nauk Ukr. RSR. 10, 1299–1303, (1963). [19] V. V. Baklan, Equations in variational derivatives and Markov processes, Dokl. Akad. Nauk SSSR. 159(4), 707–710, (1964). [20] Y. L. Daletskii, Multiplicative operators of diffusion processes and differential equations in sections of vector bundies, Usp. Mat. Nauk. 30(2), 209–210, (1975). [21] Y. I. Belopol’skaya and Y. L. Daletskii, Diffusion processes in smooth Banach spaces and manifolds, Tr. Mosk. Mat. Obshch. 37, 78–79, (1978). [22] Y. I. Belopol’skaya and Z. I. Nagolkina, On multiplicative representations of solutions of stochastic equation, Dopovidi Akad, Nauk Ukr. RSR. 11, 977–969, (1977). [23] V. V. Baklan, The Cauchy problem for equations of parabolic type in infinitedimensional space, Mat. Fiz. Resp. Mezhved. Sb. 7, 18–25, (1970). [24] V. V. Baklan. On a class of stochastic partial differential equations. In The Behavior of Systems in Random Media, pp. 3–7. Kiev, (1976). In Russian. [25] A. V. Balakrishnan, Introduction to Optimization Theory in a Hilbert Space. (Springer, 1971). [26] A. V. Balakrishnan, Stochastic optimization theory in Hilbert spaces, I, Appl. Math. Opt. 1(2), 97–120, (1974). [27] A. V. Balakrishnan, Stochastic bilinear partial differential equations, vol. 111, Lect. Notes Econ. Math. Syst., pp. 1–43. Springer, (1975). [28] R. F. Curtain, Estimation theory for abstract evolution equations excited by general white noise processes, SIAM J. Cont. Optim. 14(6), 1124–1149, (1976). [29] R. F. Curtain, Stochastic evolution equations with general white noise disturbance, J. Math. Anal. Appl. 60(3), 570–595, (1977). [30] R. F. Curtain and P. L. Falb, Stochastic differential equations in Hilbert space, J. Diff. Eqs. 10(3), 412–430, (1971). [31] M. Metivier, Int´egrate stochastique par rapport a des processus a veleurs dans un espace de Banach reflexif, Teor. Veroyatn. Ee Primen. 19(4), 787–816, (1974). [32] M. Mativier and G. Pistone, Une formule d’isometrie pour l’int´egrale stochastique Hilbertienne et equations d’´evolution linˆeaires stochastiques, Z. Wahr. Verw. Geb. 33, 1–18, (1975). [33] A. Friedman, Partial Differential Equations. (Krieger, 1976). [34] K. Ito. On stochastic differential equations. In Matematika. Periodical Collection of

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equation, Dokl. Akad. Nauk SSSR. 225(1), 18–20, (1975). [59] M. I. Vishik and A. I. Komech, Infinite-dimensional parabolic equations connected with stochastic partial differential equations, Dokl. Akad. Nauk SSSR. 233(5), 769– 772, (1977). [60] M. I. Vishik and A. I. Komech. On the solvability of the Cauchy problem for the direct Kolmogrov equation corresponding to a stochastic equation of Navier-Stokes type. In Complex Analysis and Its Applications, pp. 126–136. Nauka, Moscow, (1978). In Russian. [61] M. Viot. Solutions faibles d’´equations aux derives partielles stochastiques non lin´eaires, (1976). Th´ese Doct. Sci. Univ. Pierre Marie Curie. [62] M. I. Vishik, Quasilinear strongly elliptic systems of differential equations having divergence form, Tr. Mosk.Mat. Obshch. 12, 125–184, (1963). [63] N. V. Krylov and B. L. Rozovskii. On Cauchy problem for superparabolic stochastic differential equation. In Proc. Third Soviet-Japanese Symposium on Probability Theory, Tashkent, pp. 77–79. (1975). [64] N. V. Krylov and B. L. Rozovskii, On the Cauchy problem for linear stochastic partial differential equations, Izv. Akad, Nauk SSSR, Ser. Mat. 41(6), 1329–1347, (1977). [65] I. I. Gihman and A. V. Skorokhod, Stochastic Differential Equations. (Springer, 1972). [66] H. Kunita, Stochastic integrals based on martingales taking values in Hilbert space, Nagoya Math. J. 38, 41–52, (1970). [67] P. A. Meyer. Notes sur les integrals stochastiques, I. Int´egrales Hilbertiennes. vol. 581, Lect. Notes Math., pp. 446–463. Springer, (1977). [68] L. Gross. Abstract wiener space. In Proc. 5th Berkeley Sympos. Math. Stat. Prob., 1965–1966, Vol. 2, Part 1, pp. 31–42. (1967). [69] L. Gross, Potential theory on Hilbert space, J. Funct. Anal. 1(2), 123–181, (1968). [70] H.-H. Kuo, Gaussian measures in Banach spaces. vol. 463, Lect. Notes Math., (Springer, 1975). [71] B. Gaveau, Int´egrate stochastique radonifiante, C. R. Acad. Sci. 276(8), A617–A620, (1973). [72] D. Lepingle and J. Y. Ouvrard, Martingales browniennes Hilbertinnes, C. R. Acad. Sci. 276(18), A1225–A1228, (1973). [73] M. Metivier. Integration with respect to process of linear functionals. Preprint. [74] M. Metivier, Reelle und Vektorwertige Quasimartingale und die Theorie der Stochastischen Integration. vol. 607, Lect. Notes Math., (Springer, 1977). [75] M. Metivier and G. Pistone, Su rune equation d’´evolution stochastique, Bull. Soc. Math. France. 104, 65–85, (1976). [76] P. A. Meyer. Un cours sur les integrals stochastiques. In Sem. Prob. X, vol. 511, Lect. Notes Math., pp. 249–400. Springer, (1976). [77] K. Kuratowski, Topology, Vol. 1. (Academic Press, 1966). [78] N. Kazamaki. Note on a stochastic integral equation. vol. 258, Lect. Notes Math., pp. 105–108. Springer, (1972). [79] C. Dol´eans-Dade, On the existence and unicity of solutions of stochastic integral equations, Z. Wahr. Verw. Geb. 36(2), 93–101, (1976). [80] P. E. Protter, On the existence, uniqueness, convergence and explosions of solutions of systems of stochastic integral equations, Ann. Probab. 5(2), 243–261, (1977). [81] L. I. Gal’chuk, On the existence and uniqueness of a solution for stochastic equations over a semimartingale, Teor. Veroyatn. Ee Primen. 23(4), 782–795, (1978). [82] E. Pardoux. Filtrage de diffusions avec conditiones frontiers: caracterisation de la densit´econditionelle. In Statistique Processus Stochastiques, Proceedings, Grenoble, vol. 636, Lect. Notes Math., pp. 163–188. Springer, (1977).

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[83] S. L. Sobolev, Applications of Functional Analysis in Mathematical Physics. (Amer. Math. Soc., 1969). [84] S. M. Nikol’skii, Approximation of Functions of Several Variables and Imbedding Theorems. (Nauka, Moscow, 1969). In Russian. [85] J.-L. Lions and E. Magenes, Probl`emes aux Limites non Homogen`es et Applications, Vol. 2. (Dunod, Paris, 1968). [86] S. G. Krein, Linear Differential Equations in Banach Space (Amer. Math. Soc., 1972).

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Chapter 2 Predictability of the Burgers Dynamics Under Model Uncertainty Dirk Bl¨ omker and Jinqiao Duan∗ Institut f¨ ur Mathematik RWTH Aachen, 52062 Aachen, Germany Complex systems may be subject to various uncertainties. A great effort has been concentrated on predicting the dynamics under uncertainty in initial conditions. In the present work, we consider the well-known Burgers equation with random boundary forcing or with random body forcing. Our goal is to attempt to understand the stochastic Burgers dynamics by predicting or estimating the solution processes in various diagnostic metrics, such as mean length scale, correlation function and mean energy. First, for the linearized model, we observe that the important statistical quantities like mean energy or correlation functions are the same for the two types of random forcing, even though the solutions behave very differently. Second, for the full nonlinear model, we estimate the mean energy for various types of random body forcing, highlighting the different impact on the overall dynamics of space-time white noises, trace class white-in-time and colored-in-space noises, point noises, additive noises or multiplicative noises.

Contents 1 2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Mean Energy . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Correlation Function . . . . . . . . . . . . . . . . . . . . . 3 Nonlinear Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Body forcing - Mean energy bounds . . . . . . . . . . . . . 3.2 Point forcing - Mean energy bounds . . . . . . . . . . . . . 3.3 Body forcing - Transient behavior . . . . . . . . . . . . . . 3.4 Trace class noise: Additive vs. multiplicative body noises References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction The Burgers equation has been used as a simplified prototype model for hydrodynamics and infinite dimensional systems. It is often regarded as a one-dimensional Navier-Stokes equation. Our motivation for considering this equation comes from the modeling of the hydrodynamics and thermodynamics of the coupled atmosphere∗ Department

of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA 71

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ocean system. At the air-sea interface, the atmosphere and ocean interact through heat flux and freshwater flux with a fair amount of uncertainty [1–3]. These translate into random Neumann boundary conditions for temperature or salinity. The Dirichlet boundary condition is also appropriate under other physical situations. The fluctuating wind stress forcing corresponds to a random body forcing for the fluid velocity field. The coupled atmosphere-ocean system is quite complicated and numerical simulation is the usual approach at this time. In this paper, we consider a simplified model for this system, i.e., we consider the Burgers equation with random Neumann boundary conditions and random body forcing. Although the stochastic Burgers equation is widely studied, most work we know are for Dirichlet boundary conditions or periodic boundary conditions [4–7]. The reference [8] studied the control of deterministic Burgers equation with Neumann boundary conditions. We consider the stochastic Burgers equation with boundary forcing on the interval [0, L] ∂t u + u · ∂x u = ν∂x2 u ∂x u(·, 0) = αη

∂x u(·, L) = 0.

(1.1) (1.2)

Here α > 0 denotes the noise strength and η is white noise, i.e., η is a generalized Gaussian process with Eη(t) = 0 and Eη(t)η(s) = δ(t − s). The restriction to noise on the left boundary is only for simplicity. Analogous results will be true, if forces act on both sides of the domain. We will see that boundary forcing coincides with point forcing at the boundary. Thus we also look at point forcing. As a simple example for point forcing, we consider ∂t v + v · ∂x v = ν∂x2 v + αδ0 η

(1.3)

u(·, −L) = u(·, L) = 0 ,

(1.4)

where δ0 is the Delta-distribution. We will compare solutions of (1.1) and (1.3) with solutions of the stochastic Burgers equation with body forcing. ∂t v + v · ∂x v = ν∂x2 v + σξ

(1.5)

either subject to Dirichlet or Neumann boundary conditions. Here the noise strength is denoted by σ > 0 and ξ is space-time white noise. I.e., ξ is a generalized Gaussian process with Eξ(t, x) = 0 and Eξ(t, x)ξ(s, y) = δ(t − s)δ(x − y). We will also consider trace class body noise, i.e., noise that is white in time but colored in space. For the linearized equations, we will compare statistical quantities of both solutions, which are frequently used. One of them is the mean energy Z 1 L E[u(t, x) − u(t)]2 dx, (1.6) L 0

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where u(t) =

1 L

Z

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L

u(t, x)dx. 0

Another important quantity, which gives information about the characteristic size of pattern, is the mean correlation function Z 1 L E[u(t, x) − u(t)] · [u(t, x + r) − u(t)]dx, C(t, r) := (1.7) L 0

which is usually averaged over all points r with a given distance from 0. We obtain the averaged mean correlation function b r) = 1 [C(t, r) + C(t, −r)] (1.8) C(t, 2 where we employ the canonical odd and 2L-periodic extension of u in order to define C(t, r) for any r ∈ R. For the linearized equation the main result states that mean energy and averaged mean correlation function are the same for solutions of (1.1) and (1.5). Nevertheless the solutions behave completely different. Furthermore, we give some qualitative properties like, for instance, the typical pattern size. This should carry over to a transient regime (i.e., small times) for the corresponding nonlinear equations. For the full nonlinear Burgers model, we estimate the mean energy for various types of random body forcing, highlighting the different impact on the overall dynamics of space-time white noises, trace class white-in-time and colored-in-space noises, point noises, additive noises or multiplicative noises. In the following, we discuss linear dynamics in §2 and nonlinear dynamics in §3. 2. Linear Theory Define A = ν∂x2 with D(A) = {w ∈ H 2 ([0, L]) : ∂x w(0) = 0, ∂x w(L) = 0} It is well-known (cf. e.g. [9]) that A has an orthonormal base of eigenfunctions {ek }k∈N0 in√L2 ([0, L]) withpcorresponding eigenvalues {λk }k∈N0 . In our situation e0 (x) = 1/ L, ek (x) = 2/L · cos(πkx/L) for k ∈ N, and λk = −ν(kπ/L)2 . Moreover A generates an analytic semigroup {etA }t≥0 . (cf. e.g. [10]). In fact, etA v0 is the solution of the following evolution problem ∂t v = Av,

∂x v(·, 0) = ∂x v(·, L) = 0,

v(0, x) = v0 (x).

(2.1)

t > 0, 0 < x < L,

(2.2)

The solution is etA v0 (x) := v(t, x) =

∞ X

k=0

< v 0 , e k > e λk t e k ,

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where < ·, · > is the usual scalar product in L2 (0, L). We now consider the following linearized problems. First ∂t u = Au,

∂x u(·, 0) = α∂t β,

∂x u(·, L) = 0.

(2.3)

Here the white noise η is given by the generalized derivative of a standard Brownian motion (cf. e.g. [11]), and α is the noise intensity. Secondly, ∂t v = Av + σ∂t W,

∂x v(·, 0) = 0,

∂x v(·, L) = 0,

(2.4)

where the space-time white noise is given by the generalized derivative of an IdP Wiener process. Namely, W (t) = k∈N0 βk (t)ek , where {βk }k∈N is a family of independent standard Brownian motions, and σ is the noise intensity. It is known (cf. e.g. [12]) that (2.4) has a unique weak solution given by the stochastic convolution (taking initial condition to be zero) Z t XZ t WA (t) = σ · e(t−τ )A dW (τ ) = σ · e(t−τ )λk dβk (τ )ek . (2.5) 0

0

k∈N0

We define the Neumann map D by (1 − A)Dγ = 0,

∂x Dγ(0) = γ, ∂x Dγ(L) = 0

for any γ ∈ R. It is known that D : R 7→ H 2 ([0, L]) is a continuous linear operator. In fact, we have explicit expression for this linear operator

D(γ) =

ex + e2L e−x γ. 1 − e2L

(2.6)

From Ref. [13] or Ref. [14] we immediately obtain, that (2.3) has a unique weak solution (taking initial condition to be zero) Z t Z(t) = (1 − A) e(t−τ )ADαdβ(τ ). (2.7) 0

In the next section we derive explicit formulas for Z in term of Fourier series. 2.1. Mean Energy To obtain the Fourier series expansion for Z, consider for e ∈ D(A) and γ ∈ R Z L < D(γ), (1 − A)e >L2 ([0,L]) = < D(γ), e > − D(γ) · exx dx = < D(γ), e > − = −γe(0),

Z

0

L

0

x=L

D(γ)xx · edx + D(γ)x · e|x=0

(2.8)

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by the definition of D. Hence, < Z(t), ek > = < Z

Z

t

75

e(t−τ )A Dαdβ(τ ), (1 − A)ek >

0

t

e(t−τ )λk < Dαdβ(τ ), (1 − A)ek > 0 Z t = αek (0) · e(t−τ )λk dβ(τ ).

=

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(2.9)

0

We now obtain Z(t) = α ·

X

ek (0)

k∈N0

Z

t

e(t−τ )λk dβ(τ )ek .

(2.10)

0

Finally, Z(t) = αe1 (0) ·

XZ

k∈N

t 0

e(t−τ )λk dβ(τ )ek + αe20 (0)β(t)

(2.11)

e(t−τ )λk dβk (τ )ek + σβ0 (t) .

(2.12)

and WA (t) = σ ·

XZ

k∈N

t 0

If we now choose σ = αe1 (0), we readily obtain that XZ t e2τ λk dτ = EkWA (t) − WA (t)k2 , EkZ(t) − Z(t)k2 = σ 2 · k∈N

0

where k · k is theRnorm in L2 ([0, T ]). Hence, the mean energy in both cases is given P t by σ 2 L−1 k∈N 0 e2τ λk dτ . For the mean energy we can prove the following theorem, which is similar to the results of [15] and [16]. Theorem 2.1. Fix σ 2 = α2 /L, then the mean energy CZ (t, 0) = CWA (t, 0) behaves p like C1 (α2 /L) t/ν for t  L2 /ν, and like C2 α2 /ν for t  L2 /ν.

The main difference to body forcing is the scaling in the length-scale L. The longtime scaling is independent of L, while the transient scaling is. 2.2. Correlation Function

To obtain results for the correlation function, we think of Z and WA to be periodic on [−L, L], and symmetric w.r.t. 0. I.e., we choose the standard 2L-periodic extension respecting the Neumann boundary conditions on [0, L]. To be more precise, we extend Z and WA in a Fourier series in the basis ek , which we then consider to be defined on the whole R.

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CZ (t, 0)

6

∼

∼

α2 L

q

t ν

∼ Fig. 2.1.

α2 ν

L2 ν

-t

The scaling of the mean energy for boundary forcing.

We consider firstly for k, l 6= 0  k+l  Z L  Lek (0)el (0) l((−1)2 2−1) sin(πlr/L) : k 6= l π(l −k ) . ek (x)el (x + r)dx =  0  Lek (0)2 cos(πkr/L) :k=l

Now relying on the independence of the Brownian motions, it is straightforward to verify 1 CWA (t, r) = E < WA (t, x) − WA (t), WA (t, x + r) − WA (t) > L Z α2 e21 (0) X t 2τ λk e dτ cos(πkr/L) , (2.13) = · L 0 k∈N

as

α2 e21 (0)

2

= σ . Furthermore, 1 (2.14) E < Z(t, x) − Z(t), Z(t, x + r) − Z(t) > L ∞ Z e2 (0) X t τ (λk +λl ) l((−1)k+l − 1) e dτ sin(πlr/L) . = CWA (t, r) + 1 π (l2 − k 2 ) 0

CZ (t, r) =

k,l=1 k6=l

Obviously, CZ and CWA do not coincide, but let us now look at the averaged correlation function b r) = 1 [C(t, r) + C(t, −r)] . C(t, 2 Then it is obvious that Now

bWA (t, r) = CWA (t, r) = C bZ (t, r) 6= CZ (t, r) . C

(2.15)

Theorem 2.2. For α2 e21 (0) = σ 2 the mean energy and the averaged mean correlabWA and C bZ for Z and WA coincide for any t ≥ 0. tion functions C

This is somewhat surprising, as realizations of Z and WA behave completely different, when the condition α2 e21 (0) = σ 2 is satisfied. See e.g. Figure 2.2 and Figure 2.3.

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u

5

0

−5 2 1

1.5 0.5

1 0 0.5

−0.5 0

t

−1

x

Fig. 2.2. Random boundary condition: One realization of the solution of the equation (2.3) for L = 1, ν = 1, α = 1 and initial condition u(x, 0) = 0.

It is even more surprising, as the scaling behavior of quantities like mean energy and mean correlation functions are an important tool in applied science, which for example is used to determine the size of characteristic length scales and the universality class the model belongs to. Here both linear models lie in the same class, although their behavior differs completely. The scaling behavior with respect to L and t of the mean energy can be described using the results of [15], where the mean surface width for very general models was discussed. Therefore we focus on the scaling properties of the mean correlation function. Here we also want to investigate the dependence on α and ν. bZ (t, r) or First we consider the scaling properties of the correlation function C CWA , as given in (2.14). We are especially interested in the smallest zero of the function, which gives information about characteristic length scales or pattern sizes. For this, we use the normalized correlation function. ρZ (t, r) =

bZ (t, r) C . b CZ (t, 0)

(2.16)

bZ (t, r). Note that CZ (t, 0) is the mean energy and the maximum of r 7→ C We begin with some technical results. For any continuously differentiable and integrable function f : R+ → R we obtain using the mean value theorem Z ∞ ∞ ∞ X X f (x)dx − f (k) ≤ sup |f 0 (η)|. (2.17) 0 η∈(k−1,k) k=1

k=1

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2 1.5 1

v

0.5 0 −0.5 −1 −1.5 −2 10 8

1 6

0.5 4

0 2

−0.5 0

t

−1

x

Fig. 2.3. Random body forcing: One realization of the solution of the equation (2.4) for L = 1, ν = 1, σ = 1 and initial condition v(x, 0) = 0.

For f (k) := e−2τ νk

2

π 2 /L2

cos(kπr/L) it is easy to verify that  √  4 τν 0 −τ νk2 π 2 /L2 π |f (k)| ≤ e · · r+ √ , L 2e

where we used that xs e−x ∞ X

sup

k=1 η∈(k−1,k)

and

Z tX ∞ 0 k=1

Moreover,

1 L

α

≤ (2αe)−1/2 for any x, α ≥ 0. Hence, ∞ X

π 4 √ τ ν] · [r + √ L 2e k=1 Z ∞ 2 2 2 π 4 √ √ ≤ [r + τ ν](1 + e−τ νk π /L dk) L 2e 0 Z ∞ 2 π 4 √ L = [r + √ τ ν](1 + √ · e−k dk) L π τν 0 2e

|f 0 (η)| ≤

e−τ ν(k−1)

0

∞

f (k)dkdτ =

0

= with G(x) :=

R∞ 0

2

1 sup |f (η)|dτ ≤ C L η∈(k−1,k)

Z tZ 0

2

1−e−2k 2k2

2

cos(kx)dk.

1 π 1 π

Z

r0

∞

π 2 /L2

r

·

√ √ t [r + tν][L + tν]. ν

(2.18)

(2.19)

2

1 − e−2tνk cos(kr)dk 2νk 2

t r · G( √ ), ν νt

(2.20)

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1.6 1.2 0.8 0.4 0

1

2

3

Fig. 2.4.

4

5

6

A sketch of G

Using (2.15) we immediately obtain r  2√  √ √ α2 1 r t α t √ CZ (t, r) = · G( √ ) + O [r + tν][L + tν] . L 2π ν L3 ν νt Note that the approximation with G is not L-periodic in r, while CZ (t, r) is. The solution is that the error term is O(1) for r near L. For the normalized correlation function we deduce √ √
G( √rνt ) + O L12 [r + tν][L + tν] bZ (t, r) C √ √
. ρZ (t, r) := = bZ (t, 0) G(0) + O L12 [ tν][L + tν] C From the properties of G we infer the following:

Theorem 2.3. Given δ ∈ (0, 1) and sufficiently small 2 > 0, there exists some 1 > 0 and three constants 0 < C1 < C2 < C3 depending only on δ and 2 such that for t < 1 L2 /ν the following holds: √ ρZ (t, r) ≥ δ for r ∈ [0, C1 tν] and |ρZ (t, r)| < 2

for

√ √ r ∈ [C2 tν, C3 tν].

Note that we did not show √ that the correlation function has a zero, but it is arbitrary small in a point r ≈ tν. Therefor the theorem says that the typical length-scale  √ is tν, at least for times t  L2 /ν. For t → ∞ we immediately obtain that ∞ 2 X α2 1 bZ (∞, r) = α C cos(kπr/L) =: F (r/L) 2 2 π ν k ν k=1

and

bZ (t, r) − C bZ (∞, r)| ≤ |C

∞

α2 −2tνπ2 /L2 X 1 e . π2 ν k2 k=1

We can look for the explicit representation of F , which is a 2-periodic function, and compute explicitly the zero, but all we need from F is, that for a given small

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0.15

0.1

0.05

–3

–2

0

–1

1

2

3

x

–0.05

Fig. 2.5.

A sketch of F

enough δ > 0 there is a xδ > 0 such that F > δ on [0, xδ ]. Moreover, there is some x0 such that F (x0 ) = 0. Consider the normalized correlation function 2 2 bZ (t, r) F (r/L) + O(e−2tνπ /L ) C = . ρZ (t, r) = bZ (t, 0) F (0) + O(e−2tνπ2 /L2 ) C Assume that tν  L2 (i.e., there is some small  > 0 such that tν > L2 ). Now, ρZ (t, x0 L) = O(e−2tνπ

2

/L2

)

and ρZ (t, xL) ≥

2 2 δ + O(e−2tνπ /L ) > 0 F (0)

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for any x < xδ . So for t  L2 /ν the first zero of ρZ should be of order L. A more precise formulation is: Theorem 2.4. Given δ ∈ (0, 0.8) and δ  2 > 0, there exists some 1 > 0, a constant C > 0, and a point xo > 0 depending only on δ and 2 such that for t > L2 /(ν1 ) we obtain the following: ρZ (t, r) ≥ δ

for

r ∈ [0, CL]

and |ρZ (t, x0 L)| < 2 . Thus the theorem tells us that for t  L2 /ν, the typical length scale is of order L, which is the size of system. This result is true for both boundary and body forcing. 3. Nonlinear Theory For the nonlinear results we leave the setting of boundary forcing. Mainly, due to the lack of a-priori estimates. Usually, for Neumann boundary conditions one relies on the maximum principle to bound solutions, but the solution for boundary forcing is quite rough. Therefore, we hardly get useful results. Only, transient bounds for small times are possible to establish. For the next sections, we focus first on body forcing and later on point forcing. We will see later that boundary forcing is actually just a point forcing in a point at the boundary. The main results of this sections are uniform bounds on the energy and thus on the correlation function C, as |C(t, r)| ≤ C(t, 0), and C(t, 0) is the energy. Furthermore, we show that for t → 0 the linear regime dominates. In Ref. [17] also H¨ older-continuity for the mean energy was shown for a quasigeostrophic model. We conclude this section by a qualitative discussion on upper bounds for the energy using additive and multiplicative trace-class noise. 3.1. Body forcing - Mean energy bounds Here we provide bounds on the mean energy for the body forcing case. We consider additive space-time white noise case first, and show that the mean energy and thus the correlation function is uniformly bounded in time. This result is known (cf. Ref. [18]) for Burgers equation using the celebrated Cole-Hopf transformation, but we provide here a simple proof for completeness. Furthermore, our proof is based on energy estimates and it is easily adapted to other types of equations and additional terms in the equation. In contrast to that Cole-Hopf transformation is strictly limited to the standard Burgers equation. For a long time for space-time white noise only uniform bounds for logarithmic moments were known. See Ref. [13, Lemma 14.4.1] or Ref. [19]. In Ref. [18] the

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transformation to a stochastic heat equation via the celebrated Cole-Hopf transformation was used to study finiteness of moments. Here we rely on a much simpler tool, which can also be applied to other equations. See for instance [17] for a quasigeostrophic model, where our analysis would apply, too. Consider ∂t u + u · ∂x u = ν∂x2 u + σ∂t W , u(·, −L) = u(·, L) = 0,

u(x, 0) = u0 (x).

(3.1) (3.2)

Here W is a Q-Wiener process with a continuous operator Q ∈ L(L2 ). Thus W might be cylindrical, and we include the case of space-time white noise. Using the semigroup et A the solution for this system is (see Ref. [13] or [4]): Z t 1 (3.3) u(t) = etA u0 − e(t−τ )A(λΦλ (τ ) + ∂x u(τ, x)2 )dτ + Φλ (t) , 2 0

where for some λ ≥ 0 fixed later

Φλ (t) = σα solves

Z

t

e(t−τ )(A−λ) dW (τ )

0

∂t Φ = ν∂x2 Φ − λΦ + σ∂t W subject to Φ(·, −L) = Φ(·, L) = 0,

Φ(x, 0) = 0 .

Our main result is now: Theorem 3.1. Consider initial conditions u0 with Eku0 k2 < ∞, which are independent of the Wiener process W (e.g. deterministic). Then the mean energy of the solution of (3.3) is uniformly bounded in time. I.e., sup Eku(t) − u(t)k2 < ∞ . t≥0

Remark 3.2. Actually, we prove that supt≥0 Eku(t)k2 < ∞. The main problem in the Rproof is that after applying Gronwall-type estimates we end up with terms t E exp{ 0 kΦλ (s)k2L∞ }. This might blow up in finite time, as second order exponential moments of the Gaussian may fail to exist, if t is too large. This is why we introduced artificially additional dissipation in the equation for Φλ , in order to get exponential moments small. For the proof of Theorem 3.1 define v(t) = u(t) − Φλ (t)

for t ≥ 0, λ ≥ 0 .

We see that v is a weak solution of 1 ∂t v + ∂x (v + Φλ )2 = ν∂x2 v + λΦλ 2

(3.4)

(3.5)

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v(·, −L) = v(·, L) = 0,

v(x, 0) = u0 (x).

(3.6)

The following calculation is now only formal, but it can easily be made rigorous using for instance spectral Galerkin approximations. Taking the scalar product in (3.5) yields Z L Z L 1 2 2 2 ∂t kvk = −kvx k + (v + Φλ ) vx dx + Φλ v dx 2 −L −L ≤ −kvx k2 + kΦλ kH −1 kvx k + |vx kkΦλ k2L4 + 2kvx kkΦλ kL∞ kvk 1 ≤ − c2p kvk2 + 4kvk2 kΦλ k2∞ + 2λ2 kΦλ k4L4 + 2kΦλ k2H −1 , 2

where we used Young inequality (ab ≤ 81 a2 + 2b2 ), and Poincare inequality kvk ≤ cp kvx k. Now, from Gronwall-type inequalities 2

Rt

2

kv(t)k2 ≤ e−cp t+8 0 kΦλ k∞ dτ ku(0)k2 Z t Rt 2 2 + e−cp (t−s)+8 s kΦλ k∞ dτ 4(λ2 kΦλ k4L4 + kΦλ k2H −1 )ds.

(3.7)

0

Now we use the following lemma, which is easily proved by Fernique’s theorem, if we consider Φλ as a Gaussian in L2 ([0, t0 ], L∞ ). Lemma 3.3. Fix K > 0 and t0 > 0, then there is a λ0 such that Z t sup E exp{16 kΦλ (s)k2L∞ ds} ≤ K 2 t∈[0,t0 ]

0

for all λ ≥ λ0 . Furthermore, we use that all moments of kΦλ kL∞ and kΦλ kH −1 are uniformly bounded in time. This is easily proven, using for instance the celebrated factorization method. 2 Now we first fix K > 0, and then t0 such that e−cp t K < 41 . This yields for t ∈ [0, t0 ] and λ sufficiently large Z t  1/2 2 2 −c2p t 2 Ekv(t)k ≤ e KEku(0)k + 4K e−cp (t−s) E(λ2 kΦλ k4L4 + kΦλ k2H −1 )2 ds , 0

using H¨ older, Lemma 3.3, and the independence of u(0) from Φλ . We now find a constant C depending on t0 and K such that sup Ekv(t)k2 ≤ KEku(0)k2 + C

and

t∈[0,t0 ]

Ekv(t0 )k2 ≤

1 Eku(0)k2 + C. 4

Using Eku(t)k2 ≤ 2Ekv(t)k2 + 2EkΦλ (t)k2 yields for a different constant C sup Eku(t)k2 ≤ KEku(0)k2 + C

t∈[0,t0 ]

and

Eku(t0 )k2 ≤

1 Eku(0)k2 + C. 2

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˜ λ (t), Now we repeat the argument for k ∈ N by defining v(t) = u(kt0 + t) − Φ ˜ where Φλ (t) has the same distribution than Φλ (t) due to a time shift of the Brownian motion. Now v solves again (3.5) with initial condition u(kt0 ). Note that by ˜ λ. construction u(kt0 ) is independent of Φ Repeating the arguments as before yields for k ∈ N0 sup Eku(t + kt0 )k2 ≤ KEku(kt0 )k2 + C

t∈[0,t0 ]

and 1 Eku(kt0 )k2 + C . 2 Now the following lemma, which is a trivial statement on discrete dynamical systems, finishes the proof. Eku((k + 1)t0 )k2 ≤

Lemma 3.4. Suppose for q < 1 and some C > 0 we have an+1 < qan + C, then an is bounded by an <

C + a0 . 1−q

3.2. Point forcing - Mean energy bounds Consider hyperviscous Burgers equation with point-forcing. We would like to proceed exactly the way, we did in the previous section, But we can not, as for point forcing, the solution of the linear equation might fail to be in L∞ . This is why we add additional damping. Hyperviscous Burgers equation has been studied in several occasions. See for example [20–22]. Consider for some  > 0 the operator A = −ν(−∆)1+ , where ∆ is the Laplacian subject to Dirichlet boundary conditions. Then the hyperviscous Burgers equation is given by ∂t u + u · ∂x u = A u + αδ0 β˙ u(·, −L) = u(·, L) = 0,

u(x, 0) = u0 (x).

(3.8) (3.9)

Here, β is a standard Brownian motion and δ0 the Delta-distribution. Using the semigroup etA the solution for this system is (see Ref. [13] or [4]): Z t 1 u(t) = etA u0 − e(t−τ )A (λΦλ (τ ) + ∂x u(τ, x)2 )dτ + Φλ (t) (3.10) 2 0

where for some λ ≥ 0 fixed later

Φλ (t) = α solves

Z

t

e(t−τ )(A −λ) δ0 (x)dβ(τ ) 0

∂t Φ = A Φ − λΦ + αδ0 β˙

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subject to Φ(·, −L) = Φ(·, L) = 0,

Φ(x, 0) = 0 .

Using the by p standard orthonormal basis {ek }k∈N of eigenfunctions of A given 2+2 ek (x) = 1/L · sin(−L + πkx ) with corresponding eigenvalues λ = −(πk/2L) , k 2L we see ∞ Z t X Φλ (t) = α e(t−τ )(λk −λ) dβ(τ )ek (0)ek . (3.11) k=1

0

Note that the Fourier-coefficients of that series are not at all independent. Thus we cannot rely on the better regularity results available for the stochastic convolution of the previous chapter. Especially, for  = 0 we cannot show that Φλ (t) ∈ L∞ ([−L, L]). Note that the series expansion of boundary and point forcing is very similar. Thus we can regard boundary forcing at a point forcing at the boundary, when the equation is subject to Neumann boundary conditions. Our main result is now: Theorem 3.5. For all  > 0 and all initial conditions u0 independent of β with Eku0 k2 < ∞ the solution of (3.10) satisfies that the mean energy is uniformly bounded in time. I.e., sup Eku(t) − u(t)k2 < ∞ . t≥0

We will proceed exactly as in the previous section. Now v = u − Φλ is a weak solution of 1 ∂t v + ∂x (v + Φλ )2 = ν∂x2 v + λΦλ , (3.12) 2 again subject to Dirichlet boundary conditions and initial condition v(0) = u0 . Now consider first the nonlinear term for some small δ > 0. Using H¨ older, 1 1 −p p 2 Sobolev embedding of H into L and the bound kukH 2γ ≤ CkAγ/(1+) uk  yields Z

L −L

vΦλ vx dx ≤ kvkL2+δ kΦλ kL(4+2δ)/δ kvx k 1

1

δ

1

1

≤ CkA4 1+ 2+δ vkkΦλ kL(4+2δ)/δ kA2 1+ vk , Now we can easily find an δ > 0 sufficiently small such that there is a p = p() ∈ (2, ∞) such that (using interpolation inequality) Z L 1 vΦλ vx dx ≤ CkvkL2 kΦλ kLp kA2 vk . −L

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Now we can use the same proof as in the section before. We only need that Φλ (t) ∈ L∞ (0, L). To be more precise, an easy calculation using the series expansion of (3.11) shows that for any  > 0 sup EkΦλ (t)k2L∞ ≤ C sup EkΦλ (t)k2 t≥0

H

t≥0

1+ 2

→ 0 as λ → ∞ .

It is now straightforward to prove an analog to Lemma 3.3. The remainder of the proof is analogous to the section before. Let us remark that we could even simplify that proof a little bit, by avoiding second order exponentials of Φλ . In that case we could work with λ = 0 3.3. Body forcing - Transient behavior Let us focus on Burgers equation with body forcing. The results for hyperviscous Burgers with point-forcing are completely analogous. We will prove: Theorem 3.6. Let u be a solution of (3.1) and consider for simplicity u(0) = 0. Denote by Eu (t) = Eku(t) − u(t)k2 the mean energy of u(t), then there is some δ0 such that 1

Eu (t) = EΦ0 (t) + O(t 2 +δ0 )

for t → 0 .

To be more precise, for some t0 > 0 sufficiently small there is a constant C > 0 1 such that |Eu (t) − EΦ0 (t)| ≤ Ct 2 +δ0 for all t ∈ [0, t0 ]. √ As we know from results like Theorem 2.1 that EΦ0 (t) behaves like t for small t, we can conclude that the linear regime dominates for small t. We could explicitly calculate δ0 , but omit this for simplicity of presentation. For the proof of Theorem 3.6 use |Eu (t) − EΦ0 (t)| ≤ CEkv(t)k2 , where we used Cauchy-Schwarz inequality and uniform bounds on Eku(t)k2 and EkΦ0 (t)k2 . Using (3.7) with λ = 0 and u(0) = 0 yields together with Lemma 3.3 Z t Ekv(t)k2 ≤ C (EkΦ0 (t)k4H −1 )1/2 dt . 0

It is now easy to show that EkΦ0 (t)k4H −1 behaves like t2δ0 for some δ0 > 0, which can be explicitly calculated using the methods of Theorem 2.1. Theorem 3.6 is now proved. A simple corollary using H¨ olders inequality is: Corollary 3.7. Under the assumptions of Theorem 3.6, we know for the mean correlation function 1

Cu (t, r) = CΦ0 (t, r) + O(t 2 +δ0 )

for t → 0 and all r .

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Notice that this result is only useful for small times and small r, as seen from the qualitative behavior of CΦ0 , which is similar to the results shown in section 2.2. 3.4. Trace class noise: Additive vs. multiplicative body noises Consider again a solution of the following Burgers equation: ˙ ∂t u + u · ∂x u = ν∂x2 u + σ W u(·, 0) = 0,

u(·, L) = 0,

u(x, 0) = u0 (x),

(3.13) (3.14)

where {W (t)}t≥0 is a Brownian motion, with covariance Q, taking values in the Hilbert space L2 (0, L) with the usual scalar product h·, ·i. We assume that the ˙ is noise colored in space but white in time. trace T r(Q) is finite. So W Applying the Itˆ o’s formula, we obtain 1 1 dkuk2 = hu, dW i + [hu, uxx − uux i + σ 2 T r(Q)]dt. 2 2

(3.15)

as before hu, uux i = 0. Thus d Ekuk2 = −2kuxk2 + σ 2 T r(Q). dt

(3.16)

By the Poincare inequality kuk2 ≤ ckux k2 for some positive constant depending only on the length L, we have d 2 Ekuk2 ≤ − kuk2 + σ 2 T r(Q). dt c Then using the Gronwall inequality, we finally get

(3.17)

1 2 2 (3.18) Ekuk2 ≤ Eku0 k2 e− c t + cσ 2 T r(Q)[1 − e− c t ]. 2 Note that the first term in this estimate involves with initial data, and the second term involves with the noise intensity σ as well as the trace of the noise covariance. We now consider multiplicative body noise forcing. ∂t u + u · ∂x u = ν∂x2 u + σuw, ˙

(3.19)

with the same boundary condition and initial condition as above, where wt is a scalar Brownian motion. So w˙ is noise homogeneous in space but white in time. By the Itˆ o’s formula, we obtain 1 1 dkuk2 = hu, σudwi + [hu, νuxx − uux i + σ 2 kuk2 ]dt. 2 2

(3.20)

Thus d Ekuk2 = −2νkuxk2 + σ 2 kuk2 dt 2ν ≤ (σ 2 − )kuk2 . c

(3.21)

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Therefore, Ekuk2 ≤ Eku0 k2 e(σ

2

− 2ν c )t

.

(3.22)

Note here that the multiplicative noise affects the mean energy growth or decay rate, while the additive noise affects the mean energy upper bound. Acknowledgments Part of this work was done at the Oberwolfach Mathematical Research Institute, Germany and the Institute of Applied Mathematics, the Chinese Academy of Sciences, Beijing, China. This work was partly supported by the NSF Grants DMS0209326 & DMS-0542450 and DFG Grant KON 613/2006. References [1] T. F. Stocker, D. G. Wright, and L. A. Mysak, A zonally averaged, coupled oceanatmosphere model for paleoclimate studies, J. Climate. 5, 773–797, (1992). [2] H. A. Dijkstra, Nonlinear Physical Oceanography. (Kluwer Academic Publishers, Boston, 2000). [3] J.-L. Lions, R. Temam, and S. H. Wang, On the equations of the large-scale ocean, Nonlinearity. 5(5), 1007–1053, (1992). ISSN 0951-7715. [4] G. Da Prato, A. Debussche, and R. Temam, Stochastic Burgers’ equation, NoDEA Nonlinear Differential Equations Appl. 1(4), 389–402, (1994). ISSN 1021-9722. [5] W. E and E. Vanden Eijnden, Statistical theory for the stochastic Burgers equation in the inviscid limit, Comm. Pure Appl. Math. 53(7), 852–901, (2000). ISSN 0010-3640. [6] I. D. Chueshov and P.-A. Vuillermot, Long-time behavior of solutions to a class of quasilinear parabolic equations with random coefficients, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire. 15(2), 191–232, (1998). ISSN 0294-1449. [7] I. D. Chueshov and P. A. Vuillermot, Long-time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise: Itˆ o’s case, Stochastic Anal. Appl. 18(4), 581–615, (2000). ISSN 0736-2994. [8] H. V. Ly, K. D. Mease, and E. S. Titi, Distributed and boundary control of the viscous Burgers’ equation, Numer. Funct. Anal. Optim. 18(1-2), 143–188, (1997). ISSN 0163-0563. [9] R. Courant and D. Hilbert, Methods of mathematical physics. Vol. II. Wiley Classics Library, (John Wiley & Sons Inc., New York, 1989). ISBN 0-471-50439-4. Partial differential equations, Reprint of the 1962 original, A Wiley-Interscience Publication. [10] A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems. Progress in Nonlinear Differential Equations and their Applications, 16, (Birkh¨ auser Verlag, Basel, 1995). ISBN 3-7643-5172-1. [11] L. Arnold, Stochastic differential equations: theory and applications. (WileyInterscience [John Wiley & Sons], New York, 1974). Translated from the German. [12] G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions. vol. 44, Encyclopedia of Mathematics and its Applications, (Cambridge University Press, Cambridge, 1992). ISBN 0-521-38529-6. [13] G. Da Prato and J. Zabczyk, Ergodicity for infinite-dimensional systems. vol. 229, London Mathematical Society Lecture Note Series, (Cambridge University Press, Cambridge, 1996). ISBN 0-521-57900-7.

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[14] G. Da Prato and J. Zabczyk, Evolution equations with white-noise boundary conditions, Stochastics Stochastics Rep. 42(3-4), 167–182, (1993). ISSN 1045-1129. [15] D. Bl¨ omker, S. Maier-Paape, and T. Wanner, Roughness in surface growth equations, Interfaces Free Bound. 3(4), 465–484, (2001). ISSN 1463-9963. [16] D. Bl¨ omker, S. Maier-Paape, and T. Wanner, Surface roughness in molecular beam epitaxy, Stoch. Dyn. 1(2), 239–260, (2001). ISSN 0219-4937. [17] D. Bl¨ omker, J. Duan, and T. Wanner, Enstrophy dynamics of stochastically forced large-scale geophysical flows, J. Math. Phys. 43(5), 2616–2626, (2002). ISSN 00222488. [18] J. A. Le´ on, D. Nualart, and R. Pettersson, The stochastic Burgers equation: finite moments and smoothness of the density, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3(3), 363–385, (2000). ISSN 0219-0257. [19] G. Da Prato and D. Gatarek, Stochastic Burgers equation with correlated noise, Stochastics Stochastics Rep. 52(1-2), 29–41, (1995). ISSN 1045-1129. [20] J. P. Boyd, Hyperviscous shock layers and diffusion zones: monotonicity, spectral viscosity, and pseudospectral methods for very high order differential equations, J. Sci. Comput. 9(1), 81–106, (1994). ISSN 0885-7474. [21] C. Gugg, H. Kielh¨ ofer, and M. Niggemann, On the approximation of the stochastic Burgers equation, Comm. Math. Phys. 230(1), 181–199, (2002). ISSN 0010-3616. [22] L. Machiels and M. O. Deville, Numerical simulation of randomly forced turbulent flows, J. Comput. Phys. 145(1), 246–279, (1998). ISSN 0021-9991.

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Chapter 3 Asymptotics for the Space-Time Wigner Transform with Applications to Imaging Liliana Borcea, George Papanicolaou∗ and Chrysoula Tsogka† Computational and Applied Mathematics, Rice University 6100 Main Street, Houston, TX 77005-1892 [email protected] We consider the space-time Wigner transform of the solution of the random Schr¨ odinger equation in the white noise limit and for high frequencies. We analyze in particular the strong lateral diversity limit in which the space-time Wigner transform becomes weakly deterministic. We also show how to use these asymptotic results in broadband array imaging in random media.

Contents 1 2 3 4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The parabolic approximation . . . . . . . . . . . . . . . . . . . . . . . . . . Scaling and the asymptotic regime . . . . . . . . . . . . . . . . . . . . . . . The Itˆ o-Liouville equation for the Wigner transform . . . . . . . . . . . . . 4.1 The white noise limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The high frequency limit and the space-time Wigner transform . . . 4.3 Statement of the strong lateral diversity limit . . . . . . . . . . . . . 4.4 The mean space-time Wigner transform . . . . . . . . . . . . . . . . 5 Self-averaging of the smoothed space-time Wigner transform, in the strong versity limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Application to imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Migration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Coherent interferometric imaging . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lateral . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . di. . . . . . . . . .

91 92 93 95 95 96 98 99 100 101 102 104 109

1. Introduction In this paper we analyze the self-averaging property of the space-time Wigner transform for solutions of the random Schr¨ odinger equation, in a particular asymptotic regime. We start with the wave equation in a random medium and then use the parabolic or paraxial approximation, which is valid when waves propagate primarily in a preferred direction and backscattering is negligible. This approximation is ∗ Department

of Mathematics, ([email protected]) † Department of Mathematics, ([email protected])

Stanford University

91

University, of

Chicago,

Stanford, Chicago,

CA

94305.

IL

60637.

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widely used in random wave propagation [1–5] and it is justified in some special cases in Ref. [6], in the regime that we consider here. The parabolic wave field satisfies a random Schr¨ odinger equation, which we consider in the white noise limit. White noise limits for random ordinary differential equations have been analyzed extensively [7–9]. For random partial differential equations, white noise limits are considered in [10] for diffusion equations and, more recently, in Refs. [11, 12], for the random Schr¨ odinger equation. The resulting Itˆ o-Schr¨ odinger equation for the limit wave field is a stochastic partial differential equation of independent interest that is analyzed in Ref. [5] and in a wider context in Ref. [13, 14]. We consider here the high frequency limit of this equation, using the space-time Wigner transform. This is a slight extension of the high frequency limits analyzed in Refs. [15, 16] and in Ref. [11], using the spatial Wigner transform. The limit process satisfies an Itˆ o-Liouville partial differential equation that arises from a stochastic flow [14, 17]. We analyze this Itˆ o-Liouville equation in the strong lateral diversity limit, where the propagating wave beam is wide with respect to the correlation length of the random inhomogeneities in the transverse direction, orthogonal to the axis of the beam. The importance of this limit in time reversal was pointed out in Ref. [18] and it was analyzed later in Refs. [15, 16, 19, 20], using the spatial Wigner transform. Applications to imaging are considered in Refs. [21–23], especially applications of the space-time Wigner transform, but the strong lateral diversity limit is not analyzed there. We dedicate this work to Boris Rozovskii on the occasion of his 60th birthday. 2. The parabolic approximation Let P (~x, t) be the solution of the acoustic wave equation 1 c2 (~x)

∂2P − ∆P = 0, ∂t2

t > 0, ~x ∈ R3 ,

(2.1)

with a given excitation source at time t = 0 and in a medium with sound speed c(~x) that is fluctuating about the mean value co , taken as constant for simplicity. We model the fluctuations of c(~x) as a random process   −1/2 ~x c(~x) = co 1 + σo µ (2.2) ` where µ is a normalized, bounded and statistically homogeneous random field, with mean zero and with smooth and rapidly decaying correlation function E{µ(~x + ~x0 )µ(~x0 )} = R(~x). Here the normalization means that R(~0) = 1,

Z

d~x R(~x) = 1,

(2.3)

(2.4)

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so that ` in (2.2) is the correlation length of the fluctuations. We consider a regime with weak fluctuations (σo  1) where backscattering of the waves by the medium can be neglected and where we can study P (~x, t) with the parabolic approximation [1]. For this, we take the z coordinate in the direction of propagation of the waves and we let ~x = (z, x), with x the two dimensional vector of coordinates transverse to the direction of propagation. In the parabolic approximation the wave field is given by Z 1 Pb (z, x, ω)e−iωt dω, Pb(z, x, ω) ≈ eikz ψ(z, x, k), (2.5) P (z, x, t) = 2π where k = ω/co is the wavenumber and ψ is a complex valued amplitude satisfying the Schr¨ odinger equation z x ∂ψ 2ik + ∆ x ψ + k 2 σo µ , ψ = 0, z > 0, (2.6) ∂z ` ` with ∆x denoting the Laplacian in x. This equation is obtained by substituting eikz ψ in the reduced wave equation for Pb ∆Pb + k 2 n2 (~x)Pb = 0,

with index of refraction n(~x) = co /c(~x) given by   ~x 2 n (~x) = 1 + σo µ , `

(2.7)

2

∂ ψ and by neglecting 2 the term ∂z2 under the hypothesis that ψ is slowly varying in z ∂ψ ∂ ψ (i.e., k ∂z  ∂z2 ). We now have an initial value problem for the wave amplitude ψ, governed by equation (2.6) with initial condition

ψ(0, x, k) = ψo (x, k).

(2.8)

We assume that ψo is a compactly supported function with frequency dependence in the positive interval   B B ω ∈ ωo − , ωo + , (2.9) 2 2 centered at ωo and with bandwidth B. The negative image of this interval is also included if the initial data is real. 3. Scaling and the asymptotic regime To carry out an asymptotic analysis of the wave field (2.5) we write the Schr¨ odinger equation (2.6) in dimensionless form   ∂ψ Lz L z z 0 L x x0 0 2 0 ∆ ψ + k L σ (k ) µ , ψ = 0, (3.1) 2ik 0 0 + x o z o ∂z ko L2x ` `

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with scaled variables x = L x x0 ,

z = Lz z 0 ,

ω = ωo ω 0 ,

k = ko k 0 ,

c = c o c0 .

(3.2)

Here ko = ωo /co is the central wavenumber, Lz quantifies the distance of propagation and Lx is a transversal length scale which we take to be the width of the propagating beam. Note that the scaled sound speed has constant mean < c0 >= 1. Therefore, since the scaled wavenumber k 0 is the same as the scaled frequency ω 0 , we shall replace ω 0 by k 0 from now on. To simplify notation we drop the primes on the scaled variables and we introduce three dimensionless parameters depending on the random medium ` ` , δ= , Lz Lx and the reciprocal of the Fresnel number =

θ=

1 Lz = 2 k o Lx 2π

3

σ = σo δ− 2 , 

λ0 L z Lx

Lx



.

(3.3)

(3.4)

Here λo is the central wavelength and the reciprocal of the Fresnel number is written as the ratio of the focusing spot size in time reversal imaging, λo Lz /Lx , and the transversal length scale Lx . The scaled form of equation (3.1) is ∂ψ 1  z x  σk 2 δ 2ik + θ∆x ψ + 1/2 µ , ψ = 0, z > 0 (3.5) ∂z  δ θ  and we study it in the asymptotic regime   δ  1,

θ  1,

σ = O(1).

(3.6)

Thus, we suppose that the waves travel many correlation lengths in the random medium (  1) and, to be consistent with the parabolic approximation, we take Lx  Lz (i.e.,   δ). We also take θ  1, which means that the time reversal imaging spot size is much smaller than Lx λo L z  Lx . Lx

(3.7)

Finally, we scale the strength of the fluctuations in (3.3) and (3.6) so that we can take the white noise limit  → 0 in (3.5). The asymptotic regime (3.6) can be realized with several scale orderings. In this paper we assume that   θ  δ  1,

(3.8)

which amounts to taking  → 0 as the first in a sequence of three limits. This leads to an Itˆ o-Schr¨ odinger equation for the limit ψ. The second limit θ/δ → 0 implies that we are in a high frequency regime  λo   1. ` δ

(3.9)

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In imaging, the spot size is small in this limit, when compared with the correlation length λo L z  `  Lx . Lx

(3.10)

This is a regime in which we can derive an Itˆ o-Liouville equation for the Wigner transform of ψ, under the additional assumption of isotropy of the fluctuations of the sound speed. Finally, we take the strong lateral diversity limit δ  1, which allows us to show that the appropriately smoothed Wigner transform is self-averaging. Other scale orderings consistent with (3.6) are θδ1

(3.11)

  δ ≤ θ  1.

(3.12)

and

The ordering (3.11) is considered in Ref. [15], in a study of statistical stability of time reversal in random media. It is a high frequency regime and it gives similar results to those obtained here. The scale ordering (3.12) is consistent with λo ∼ ` and it is used in numerical simulations in Refs. [21–24], in the context of array imaging of sources and reflectors. In the parabolic approximation this scaling is analyzed in Ref. [16]. The theory is not so well developed when the parabolic approximation does not apply. Nevertheless, it appears from the numerical simulations in Refs. [21–24] that the statistical stability that we have in regimes (3.8) or (3.11) is valid in the case (3.12). 4. The Itˆ o-Liouville equation for the Wigner transform In this section we give, without details, the Itˆ o-Liouville equation for the Wigner transform of ψ in the limits  → 0 and θ → 0. We then state the main result of this paper, which is that in the limit δ → 0 we have self-averaging for smooth linear functionals of the space-time Wigner transform. The proof is given in section 5. We consider an application of this self-averaging property in section 6, where we look at coherent interferometric imaging in random media, as introduced in Refs. [22, 23]. 4.1. The white noise limit Let us emphasize with the notation ψ  (z, x, k) the dependence on  of the wave amplitude satisfying (3.5). This amplitude depends on θ and δ as well, but since these are kept fixed in the first limit we suppress them from the notation. The initial wave field ψo is assumed independent of .

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It follows from [11, 12] that as  → 0, ψ  (z, x, k) converges weakly, in law, to the solution ψ(z, x, k) of Itˆ o-Schr¨ odinger equation    x k2 σ2 δ2 ikσδ iθ dψ = ∆x − R (0) ψdz + ψdB z, , z > 0, o 2 2k 8θ 2θ δ (4.1) ψ(0, x, k) = ψo (x, k), at z = 0. Here B(z, x) is a Brownian random field that is smooth in the transverse variable x. The mean of B is zero and its correlation is given by E{B(z1 , x1 )B(z2 , x2 )} = z1 ∧ z2 Ro (x1 − x2 ), where z1 ∧ z2 = min {z1 , z2 } and Ro (x) =

Z

∞

R(z, x)dz.

(4.2)

(4.3)

−∞

Because of our assumptions on R in section 2 we have that Ro is smooth and rapidly decaying. This is used in section 5 to deduce the statistical stability of the smoothed Wigner transform of ψ, in the limit θ/δ → 0 and δ → 0. 4.2. The high frequency limit and the space-time Wigner transform As in section 4.1, we now use the notation ψ θ (z, x, k) to emphasize the dependence of the solution of Itˆ o-Schr¨ odinger equation (4.1) on the parameter θ. We study the high frequency limit θ → 0 with the space-time Wigner transform W θ (z, x, k, q, r) ! ! Z Z e de x dk iq·ex−iekr θ θe x θe k θe x θe k θ = e ψ z, x + ,k + ψ z, x − ,k − , (2π)2 2π 2 2 2 2

(4.4)

where the bar on ψ θ denotes complex conjugate. The r variable in W θ is dual to e k and it represents the distance traveled by the waves in a medium with constant speed < c >= 1, during a travel time t = r/ < c >. The argument q in W θ is a e. two dimensional vector that is dual to x For an arbitrary but fixed z, the L2 norm of the Wigner transform W θ is determined by the space and frequency L2 norm of the initial wave function ψoθ Z  Z Z Z

θ

2 1/2

W (z, ·) 2 = dx dk dq dr W θ (z, x, k, q, r) L



! 2 ! 2 1/2 Z e e e de x dk θ θe x θk θ θe x θk  = dx dk ψ z, x + , k + x − , k − ψ z, (2π)2 2π 2 2 2 2

θ 2

θ

ψ (z, ·) 2 2

ψ 2 o L L = = , (2πθ)3/2 (2πθ)3/2 (4.5) Z

Z

Z

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because the Itˆ o-Schr¨ odinger equation (4.1) preserves the space and frequency L 2 norm of its solution [5]. This means that with a proper definition and scaling of the initial wave function ψoθ [25], we can bound the L2 norm of W θ (z, ·) uniformly with respect to θ. We formally obtain an Itˆ o-Liouville equation for the high frequency limit W as follows [12]. We use Itˆ o’s formula to get from (4.1) an equation for ψ1θ ψ2θ = ψ θ (z, x1 , k1 )ψ θ (z, x2 , k2 ), with θe x θe x , x2 = x − , (4.6) 2 2 θe k θe k k1 = k + , k2 = k − (4.7) 2 2 e and e and then we Fourier transform in x k and take the limit θ/δ → 0. The variables e are independent of the small parameters. We have e , k and k x, x   h

iθ 1 1 1 1 iθ 1 1 ∆ ∇ ∆ ∇ d ψ1θ ψ2θ = 2k+θ + · ∇ + − − · ∇ + 2 ∆x 2 ∆x x x x x e e e e x x e e 4 θ θ 4 θ θ k 2k−θ k x1 = x +

+

  2 2 2 k − θ4 e k σ2 δ2 4θ 2

θ θ + iσδ 2θ ψ1 ψ2

R0



θ|e x| δ



−

  2 2 2 k + θ4 e k σ2 δ2 4θ 2

h  k + θ2 e k dB z, xδ +

θe x 2δ




 R0 (0) ψ1θ ψ2θ dz

  k dB z, xδ − − k − θ2 e

θe x 2δ


i

(4.8) and, using the smoothness of B and Ro in the transverse variables, we have further   2    θ x θe x θe x x θe x dB z, + − dB z, − = · ∇x dB(z, x) + O (4.9) δ 2δ δ 2δ δ δ and Ro



θe x δ



 3 2 θ θ2 X 2 θ e · ∇Ro (0) + 2 = Ro (0) + x ∂ij Ro (0)˜ . xi x ˜j + O δ 2δ i,j=1 δ

(4.10)

e and e The equation for W follows by Fourier transforming (4.8) in x k, using the expansions (4.9) and (4.10) and letting θ/δ → 0. We simplify the result by assuming that the fluctuations are isotropic so that Ro (x) = Ro (|x|). This gives 00

2 ∇Ro (0) = 0 and ∂ij Ro (0) = R (0)δij

and we obtain for W the Itˆ o-Liouville equation h 2 2 ∂ dW = qk · ∇x − |q|2 ∂r + k 2Dκ ∆q + 2k

+ σk 2 ∇q W with initial condition

· ∇x dB z,

x δ




−

δ 2 Dr ∂ 2 2 ∂r 2

σδ ∂W 2 ∂r

i

dB z,

(4.11)

W dz x δ

W (z = 0, x, k, q, r) = Wo (x, k, q, r)




(4.12) ,

z>0

(4.13)

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and with the positive diffusion coefficients Dκ = −

σ 2 Ro00 (0) σ 2 Ro (0) and Dr = . 4 4

(4.14)

Equation (4.12) was also derived in Ref. [22] and we note that it is a stochastic flow equation [14, 17] that is the starting point of the analysis in this paper. We want to study the limit of the process W as δ → 0. 4.3. Statement of the strong lateral diversity limit Now that we have the Itˆ o-Liouville equation (4.12), we emphasize the dependence of the process on δ by writing W δ (z, x, k, q, r). We assume that the initial condition Wo is independent of δ. The mean W δ = E{W δ } is considered in section 4.4 and it follows, as is easily seen from (4.12), that as δ → 0, W δ converges weakly to the solution W of the phase space advection-diffusion equation ∂W = LW ∂z

(4.15)

W(0, x, k, q, r) = Wo (x, k, q, r),

(4.16)

with initial conditions

where 2

L=

q |q|2 ∂ k Dκ · ∇x − 2 + ∆q . ∂r 2 k 2k

(4.17)

This deterministic equation is solved explicitly in Ref. [22]. However, the point-wise variance of W δ is not zero for any δ and it does not vanish as δ → 0. This means that W δ is randomly fluctuating and it does not converge to a deterministic process as δ → 0 in the strong, point-wise sense. Nevertheless, we do have convergence in a weak sense as follows. Theorem 4.1. Suppose that Wo is in L2 and it does not depend on δ. Then, given any smooth and rapidly decaying test function ϕ(x, k, q, r), we have that Z Z Z Z δ < W , ϕ > (z) = dx dk dq dr ϕ(x, k, q, r)W δ (z, x, k, q, r) (4.18) converges in probability as δ → 0 to < W, ϕ > (z), for any z > 0. Theorem 4.1 is proved in section 5. It states that even though W δ does not have a deterministic point-wise limit, it is weakly self-averaging. That is, smooth linear functionals of W δ become deterministic in the limit δ → 0. This is the property that can be exploited in applications such as imaging in random media, as we explain in section 6.

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4.4. The mean space-time Wigner transform Taking expectations in (4.12) we get that W δ (z, x, q, r) = E{W δ (z, x, q, r)} satisfies the phase space advection-diffusion equation ∂W δ = Lδ W, ∂z

(4.19)

with initial condition Wo (x, k, q, r), where Lδ = L +

δ 2 Dr ∂ 2 . 2 ∂r2

(4.20)

Equivalently, W δ is given as an expectation 
W δ (z, x, κ, r) = E Wo Xδ (z), k, Qδ (z), Rδ (z) , (4.21)  δ where X (z), Qδ (z), Rδ (z) is the Itˆ o diffusion process with generator Lδ and with initial condition Xδ (0) = x,

Qδ (0) = q,

Rδ (0) = r.

(4.22)

For z > 0, the Itˆ o stochastic differential equations are 1 δ Q (z)dz, kp dQj δ (z) = k Dκ dBj (z), dXδ (z) =


j = 1, 2, Qδ = Qδ1 , Qδ2 , (4.23) δ 2 p |Q (z)| dz − δ Dr dB(z), dRδ (z) = − 2 2k where the driving is with three independent standard Brownian motions B(z) and  δ δ δ Bj (z), for j = 1, 2. The same process X (z), Q (z), R (z) also determines expectations of higher powers of W δ n n n o
n o E W δ (z, x, q, r) = E Wo Xδ (z), k, Qδ (z), Rδ (z) , n ≥ 1. (4.24)

As δ → 0, we see that W δ converges to the solution of (4.15), computed explicitly in Ref. [22]. Actually, all one point moments converge as δ → 0, n n n o
n o E W δ (z, x, q, r) → E Wo X(z), k, Q(z), R(z) , n ≥ 1, (4.25) where {X(z), Q(z), R(z)} is the δ independent Itˆ o diffusion 1 Q(z)dz, kp dQj (z) = k Dκ dBj (z), dX(z) =

dR(z) = −

|Q(z)| 2k

2

j = 1, 2,

Q = (Q1 , Q2 ) ,

(4.26)

2

dz,

z > 0,

with initial conditions X(0) = x,

Q(0) = q and R(0) = r.

(4.27)

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Clearly, W δ does not have a point-wise deterministic limit because the limit variance is not zero n
2 o 
2 E Wo X(z), k, Q(z), R(z) − E Wo X(z), k, Q(z), R(z) 6= 0. (4.28)

5. Self-averaging of the smoothed space-time Wigner transform, in the strong lateral diversity limit

In this section we prove Theorem 4.1. We begin by calculating the form of the infinitesimal generator Aδ of Itˆ o-Liouville process W δ (z, x, k, κ, r), considered as a process in the space of continuous functions in z with values in the space S 0 of Schwartz distributions w over R2 × R × R2 × R. Let F be a real valued test function on R and define for each test function ϕ in S over R2 × R × R2 × R the function f (w) by f (w) = F (< w, ϕ >).

(5.1)

We have that d E{F (< W δ (z), ϕ >)|W δ (0) = w}|z=0 dz 0 00 = Dδ (w)F (< w, ϕ >) + Mδ (w)F (< w, ϕ >)

Aδ f (w) =

(5.2)

where Dδ (w) =< w, L?δ ϕ >,

(5.3)

with L?δ the adjoint of Lδ in (4.20). The Mδ can be written as the sum of three terms Z Z Z Z Z Z Z Z σ2 δ2 0 0 dx dk dq dr dx0 dk dq0 dr0 w(x, k, q, r)w(x0 , k , q0 , r0 ) Mδ1 (w) = 8   0 x − x0 ∂ϕ(x, k, q, r) ∂ϕ(x0 , k , q0 , r0 ) , (5.4) ×Ro δ ∂r ∂r0 Mδ2 (w) Z Z Z Z Z Z Z Z σ2 δ 0 0 0 dx dk dq dr dx0 dk dq0 dr0 w(x, k, q, r)w(x 0 , k , q0 , r0 )k × =− 4   ∂ϕ(x, k, q, r) x − x0 0 · ∇q0 ϕ(x0 , k , q0 , r0 ) , (5.5) ∇x 0 R o δ ∂r Mδ3 (w) Z Z Z Z Z Z Z Z σ2 0 0 0 dx dk dq dr dx0 dk dq0 dr0 w(x, k, q, r)w(x 0 , k , q0 , r0 )kk × =− 8   0 2 X x − x0 ∂ϕ(x, k, q, r) ∂ϕ(x0 , k , q0 , r0 ) 2 . (5.6) ∂lj Ro δ ∂qj ∂ql0

j,l=1

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Now we get from (5.3) that as δ → 0, lim Dδ (w) = D(w) =< w, L? ϕ >,

(5.7)

δ→0

uniformly for w bounded in L2 . Here L? is the adjoint of L defined by (4.17). Furthermore, lim Mδj (w) = 0,

for j = 1, 2, 3,

δ→0

uniformly for w bounded in L2 , as we show next. From (5.4)-(5.6) we see that it is enough to show that Mδ3 (w) → 0. By the Cauchy-Schwartz inequality, we have |Mδ3 (w)| ≤ kwk2L2 J δ (ϕ),

(5.8)

where δ

2

[J (ϕ)] =

 

σ2 8

kk

0

2 Z 2 X

Z

dx dk 2 ∂lj Ro

j,l=1



Z

Z

dq dr

x − x0 δ



Z

dx

0

Z

dk

0

Z

dq

0

Z

dr0

2 0 ∂ϕ(x, k, q, r) ∂ϕ(x0 , k , q0 , r0 )  . ∂qj ∂ql0

Since Ro is rapidly decaying at infinity, we see that for any fixed test function ϕ, J δ (ϕ) tends to zero as δ → 0. We have shown therefore that for functions f (w) of the form (5.1), with ϕ in S fixed and uniformly for w bounded in L2 , 0

Aδ f (w) → Af (w) = D(w)F (< w, ϕ >),

(5.9)

D(w) =< w, L? ϕ > .

(5.10)

where

The operator A is the generator of the deterministic process W(z, x, k, q, r), that is the solution of (4.15). Since the limit process is deterministic, it follows that 0 convergence in law implies convergence in probability, weakly in S . It also follows that functions of the form (5.1) are sufficient again since the limit is deterministic 0 [14]. The moment condition needed for tightness for processes in C([0, Z], S ) or in 0 D([0, Z], S ) [26] is easily obtained as in Ref. [20], and we omit it here. 6. Application to imaging In this section we consider applications to imaging a source in a random medium from measurements of the wave field P at an array of transducers. The setup is shown in Figure 3.1, where we introduce a new coordinate system, with scaled range ζ = L − z measured from the array and with cross range (transverse) coordinates x defined with respect to the center ~y? = (L, 0) of the source, which can be small or

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~yS = (L + η S , ξS ) source

~xr = (0, xr )

~y? = (L, 0) ζ ζ=0

Fig. 3.1.

ζ=L

Setup for imaging a distributed source with a planar array of transducers

spatially distributed. The array is in the plane ζ = 0 and it consists of receivers at N discrete locations ~xr = (0, xr ), where we record the traces P (~xr , t) over a time window t ∈ [0, T ] that we suppose is long enough for P (~xr , t) ≈ 0

for t > T

to hold. This allows us to simplify the analysis by neglecting the effect of a finite measurement time window. The goal in imaging is to estimate the support of the source from the traces at the array and this is done very efficiently with Kirchhoff migration [27, 28], if there are no fluctuations of the sound speed. However, in random media Kirchhoff migration gives noisy and unpredictable results, in the sense that they lack statistical stability. As an alternative to Kirchhoff migration we introduced in Refs. [22, 23] a new, coherent interferometric (CINT) imaging functional, which is a statistically smoothed migration. The resolution analysis of CINT is given in Ref. [22]. Here we use the result of section 4.3 to show that it is statistically stable in the asymptotic regime (3.8). 6.1. Migration In this and the following sections all variables are scaled as in section 3. The wave field Pb at receiver location ~xr = (0, xr ) is 2 Pb(xr , ω) ≈ ei(δ/) k/θL ψ(L, xr , k),

(6.1)

where we used that ko Lz = δ 2 /(θ2 ) and we dropped the range coordinate in the argument of Pb , as it is always ζ = 0 at the array. We also kept the definition ψ = ψ(z, x, k), with z = L − ζ, which means that at the array the range coordinate in ψ is z = L. The parabolic amplitude ψ solves equation (3.5) with initial data ψo (x, k) and it depends on the three small parameters , θ and δ. In the previous

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sections we emphasized this dependence using superscripts, before taking limits. Here we don’t use the superscripts to simplify notation and we keep , θ and δ fixed until the very end where we apply the theoretical results of section 4.3. Classic Kirchhoff migration imaging [27, 28] estimates the support of the source by migrating (back propagating) the traces P (xr , t) to search points ~yS = (L + η S , ξ S ), in a fictitious, homogeneous medium, with scaled sound speed < c >= 1 and by summing over the receivers. The scaled distance from ~xr to ~yS is "

(L + η S )2 +



2 # 21 S 2 − x 2 ξ r  Lx S 2 S ≈L+η + 2 ξ − xr Lz 2δ (L + η S )

(6.2)

and it equals the scaled travel time τ (xr , ~yS ), since the scaled mean speed is < c >= 1. This gives the migration phase 2 S 2 ξ − xr k δ S S (ko Lz )ωτ (~xr , ~y ) ≈ 2 k(L + η ) +  θ 2θ (L + η S )

(6.3)

and the migrated wave field to ~yS

S Pb (xr , ω)e−i(ko Lz )ωτ (~xr ,~y )

2  S  k ξ − xr  δ2 k S . (6.4) ≈ ψ(L, xr , k) exp −i 2 η − i  2θ (L + η S )     θ   

The Kirchhoff migration image is given by I

KM

S

(~y ) =

N Z X r=1

S dω Pb (xr , ω)e−i(ko Lz )ωτ (~xr ,~y )

(6.5)

and, as shown in Refs. [21–23], it lacks statistical stability with respect to the realizations of the random medium and it gives noisy results that are difficult to interpret. We consider next coherent interferometric imaging, which is a statistically smoothed version of migration [22, 23]. Before describing this method, let us make the assumption that the array receivers are placed on a square mesh, in a square aperture of area a2 . The scaled mesh size is h and we suppose that it is small, so we can write N X r=1

≈

1 h2

Z

dx ∼

Z

dx,

(6.6)

with x varying continuously in the array aperture and with symbol ∼ denoting approximate, up to a multiplicative constant.

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6.2. Coherent interferometric imaging The coherent interferometric imaging technique was introduced in Refs. [22, 23] and it uses the coherence in the data traces P (x, t) to obtain reliable images in random media. There are two characteristic coherent parameters in the data: • The decoherence frequency Ωd , which is the difference in the frequencies ω1 and ω2 over which ψ(z, x, k1 ) and ψ(z, x, k2 ) become uncorrelated. • The decoherence length Xd , which is the distance |x1 −x2 | over which ψ(z, x1 , k) and ψ(z, x2 , k) become uncorrelated. These decoherence parameters depend on the statistics of the random medium and the range z and they are described in detail in the next section, for the asymptotic regime (3.8). Coherent interferometry (CINT) is a migration technique that works with cross correlations of the traces, instead of the traces themselves. These cross correlations are computed locally over space-time windows of size Xd × Ωd and they are called coherent interferograms. We give in section 6.2.2 the mathematical expression of the CINT imaging function and then we study its statistical stability. The CINT functional and its resolution properties are motivated and analyzed in Refs. [22, 23]. 6.2.1. The decoherence length and frequency e= The decoherence length and frequency can be determined from the decay over x (x1 − x2 )/θ and e k = (k1 − k2 )/θ of the expectation ! !+ * θe x θe k θe x θe k ,k + ,k − , ψ z, x − ψ z, x + 2 2 2 2

which we calculated explicitly in Ref. [22], by solving equation (4.19). The moment formula is given by ! !+ * θe x θe k θe x θe k ,k + ,k − ψ z, x − ψ z, x + 2 2 2 2 ( ) 2 2 e k 2 δ 2 Dr z k) k Dκ ϕ2 (z, e k)z|e x|2 −k ϕ1 (z, e exp − − ≈ 4π 2 z 2 2 6 ( Z Z ik ie k|x − ξ|2 2 ˜ + k 2 ϕ3 (z, e ˜2 + (x − ξ) · (e x − ξ) k)e x · ξ˜ − k ϕ4 (z, e k)|ξ| dξ dξ˜ exp 2z z ) 2 i k Dκ ϕ2 (z, e k)ϕ5 (z, e k)z h ˜ 2 e ˜ e · ξ + ϕ5 (z, k)|ξ| x − 6 ! ! θξ˜ θe k θξ˜ θe k . ψo ξ − , k − ψo ξ + , k + 2 2 2 2

(6.7)

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This result is obtained in the white noise limit and the approximation involves a simplification of the exact formula for small θ. The coefficients in this moment formula (6.7) are given by ϕ1 (z, e k) = ϕ2 (z, e k) = ϕ3 (z, e k) = ϕ4 (z, e k) =

ϕ5 (z, e k) =

q  q  z −ie kDκ 1/2 e  q  coth z −ikDκ , sinh1/2 z −ie kDκ q   −ie kDκ 3i  1 q − , e kzDκ tanh(z −ikD e κ) z q   kDκ i  3z −ie 1 q q − − 2 , 2e kz sinh(z −ie e kDκ ) cosh(z −ikDκ ) q q   e Dκ tanh(z −ikDκ ) kDκ ) tanh(z −ie 1 −  q q e e 2 −ikDκ z −ikDκ 1 q . cosh(z −ie kDκ )

(6.8)

(6.9)

(6.10)

(6.11) (6.12)

e → 0 and then study the decay over To determine the decoherence length we let k |e x| of the right hand side in (6.7). In the limit e k → 0, we get from (6.8)-(6.12) that ϕj (z, e k) =

and equation (6.7) simplifies to

(

1 + O(e k), j = 1, 2, 5 1/2 e O(k ), j = 3, 4

*  ( 2 )   + 2 θe x θe x −k k Dκ z|e x|2 ψ z, x + ≈ 2 2 exp − , k ψ z, x − ,k 2 2 4π z 6 ! ! Z Z ˜ ˜ ˜ o ξ + θ ξ , k ψo ξ − θ ξ , k dξ dξψ 2 2 ( ) 2 i ik k Dκ z h ˜ 2 ˜ ˜ e · ξ + |ξ| exp , (x − ξ) · (e x − ξ) − x z 6

(6.13)

(6.14)

with the explicit integration depending on the spatial support of the wave source function ψo . For example, in the case of a spatially distributed source, where ˜ k) ≈ ψo (ξ, k), ψo (ξ ± θξ/2,

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the integration over ξ˜ gives D
x ψ z, x + θe 2 , k ψ z, x −

θe x 2 ,k


E

≈

n 2 oZ 2 exp − k Dκ2z|ex| dξ |ψo (ξ, k)|2  2  ikz 2 Dk 3 e . exp − 2z3 Dκ x − ξ − 2 x

−3 2πz 3 Dκ

In the case of a small source with support of O(θ), where   ξ ψo θ−2 ψo ,k , θ

we can let ξ θξ in (6.14) and obtain ( 2 ) *    + 2 θe x θe x −k ik k Dκ z|e x|2 e , k ψ z, x − ,k ∼ exp − + x·x ψ z, x + 2 2 4π 2 z 2 8 z ! ! ( 2 ) 2 Z Z e k Dκ z|e x|2 ˜ x ik ˜ ξ˜ ξ˜ ˜ dξ dξ ψo ξ + , k ψo ξ − , k exp − ξ + 2 − z ξ · x . 2 2 6

e occurs as a Gaussian function, with standard deviation In either  case, the decay in x 1 of O k√zD . This means that the scaled decoherence length is κ

θ Xd (k) ∼ √ k zDκ

(6.15)

and it corresponds to the scaled expected time reversal spot size derived in Refs. [15, 22]. From the analysis in Refs. [15, 18, 22] we know that the effective aperture is given, in scaled variables, by p ae = Dκ z 3 (6.16) We can now write

Xd (k) ∼

θ kκd

(6.17)

where

p ae = Dκ z. (6.18) z The uncertainty in the direction of arrival of the waves in the random medium [23] is κd /θ. e → 0 in (6.7) Next, we estimate the decoherence frequency by setting x      oZ Z n 2 e e2 δ 2 Dr z −k ϕ1 (z,k) k θe k θe k ψ z, x, k + 2 ψ z, x, k − 2 = 4π2 z2 exp − 2 dξ dξ˜   2 ˜ 2 k Dκ ϕ2 (z,e k)ϕ25 (z,e k)z|ξ |2 ie k|x−ξ |2 ik 2 ˜ e ˜ exp − z (x − ξ) · ξ − k ϕ4 (z, k)|ξ| − 2z 6     ˜ ˜ e e θξ θξ ψo ξ + 2 , k + θ2k ψo ξ + 2 , k − θ2k κd =

(6.19)

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and by taking the large e k approximation in (6.19). We obtain from (6.8)-(6.12) that q q e κ − z2 kD e e 2 (1−i) , ϕ1 (z, k) ≈ z −2ikDκ e (6.20) ϕj (z, e k) = O(e k −1/2 ),

ϕ5 (z, e k) ≈ e−z

q

e κ kD 2

j = 2, 4

(1−i)

,

(6.21)

(6.22)

which means that as e k increases, the decay in (6.19) is determined by the factor s    z e e k 2 δ 2 Dr z  kDκ − exp −   2 2 2

and therefore, that the scaled decoherence frequency is   θ θ θ √ Ωd ∼ min , ≈ 2 as δ → 0. 2 z Dκ δ D r z z Dκ

(6.23)

In conclusion, both Xd and Ωd are small, of order θ in our scaling, which means that we can cover many decoherence lengths with an array aperture of O(1) and we can fit many frequency intervals of width Ωd in a broad bandwidth B/ωo = O(1). This is a key point for achieving the self-averaging property of the CINT imaging function discussed below. 6.2.2. The coherent interferometric imaging function as a smoothed spacetime Wigner transform Consider a smooth window χ(r; ρ) of length O(ρ), with Fourier transform Z e χ b(e k; ρ−1 ) = χ(r; ρ)eikr dr, supported in the wavenumber interval |e k| ≤ ρ−1 , where ρ∼

θ = O(1). Ωd

(6.24)

(6.25)

Let also Φ(κ; κd ) be a smooth function of two dimensional vectors κ, with support in a disk of radius O(κd ), with κd quantifying the uncertainty in the direction of arrival of the waves in the random medium, as explained in section 6.2.1. The Fourier transform of Φ is denoted by Z b x; κ−1 ) = Φ(κ; κd )e−ikκ·ex dκ Φ(ke (6.26) d and it is supported in the disk k|e x| ≤ κ−1 d .

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Using the windows (6.24), (6.26) and the migrated wave field (6.4), we define the coherent interferometric imaging function [23]

I

CINT

S

(~y ; ρ, κd ) ∼

Z

dk

Z

Z

2e S dx de kχ b(e k; ρ−1 ) e−i(δ/) kη

Z

b x; κ−1 ) de x Φ(ke d

! ! θe x θe k θe x θe k ψ L, x + ψ L, x − ,k + ,k − (6.27) 2 2 2 2  2 2  S  x x    (k + θek ) ξ S − x − θe θe k ξ − x + θe (k − 2 ) 2 2 2 + i exp −i  2θ (L + η S ) 2θ (L + η S )   

that becomes after simplifying the exponent,

I

CINT

Z −1 −i(δ/)2 e kη S e e b x; κ−1 ) b(k; ρ ) e de x Φ(ke (~y ; ρ, κd ) ∼ dk dx dk χ d ! ! θe x θe k θe x θe k ψ L, x + ,k + ψ L, x − ,k − (6.28) 2 2 2 2  2    S    x − ξS ξ − x  e . exp −ike x· − ik  (L + η S ) 2(L + η S )    Z

S

Z

Z

Now note that because the support of χ(r; ρ) is O(1), the imaging function is nonzero if the range offset satisfies
η S ≤ O 2 /δ 2  1.

We set then η S

2 S δ2 η

 and write approximately for ~yS = L +

I CINT (~yS ; ρ, κd ) ∼

Z

dk

Z

S 2 S δ2 η , ξ

Z Z b x; κ−1 ) dx de kχ b(e k; ρ−1 ) de x Φ(ke d



,

! ! θe x θe k θe x θe k ψ L, x + ,k + ψ L, x − ,k − (6.29) 2 2 2 2   2     S   ξS − x x −   S ξ e exp ike x· − ik η +  .   L 2L  

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Asymptotics for a Space-Time Wigner Transform

Next, we use the definition (4.4) of the Wigner transform in (6.29) and obtain Z Z Z Z I CINT (~yS ; ρ, κd ) ∼ dk dx dq dr W (L, x, k, q, r)    2 S Z ξ − x    de kχ b(e k; ρ−1 ) exp −ie k η S + − r (6.30) 2L Z

# " ξS − x −1 b e . de x Φ(ke − iq · x x; κd ) exp ike x· L

Finally, changing variables q = kκ, we get I CINT (~yS ; ρ, κd ) ∼

Z Φ

Z

Z



 dx dκ dr χ η S + ξS − x − κ; κd L

!Z

2 S ξ − x 2L



 − r; ρ

dk W (L, x, k, kκ, r).

(6.31)

In conclusion, the coherent imaging function is given by the Wigner transform, smoothed by convolution over directions κ and range r and by integration over the array locations x and wavenumbers k. The self-averaging of I CINT follows from Theorem 4.1. Acknowledgments The work of L. Borcea was partially supported by the Office of Naval Research, under grant N00014-02-1-0088 and by the National Science Foundation, grants DMS-0604008, DMS-0305056, DMS-0354658. It was also supported by INRIA in the group POEMS of P. Joly. The work of G. Papanicolaou was supported by grants ONR N00014-02-1-0088, 02-SC-ARO-1067-MOD 1 and NSF DMS-0354674001. The work of C. Tsogka was partially supported by the Office of Naval Research, under grant N00014-02-1-0088 and by 02-SC-ARO-1067-MOD 1. References [1] F. Tappert. The parabolic approximation method. In Wave Propagation and Underwater Acoustics, vol. 70, Lecture Notes in Physics, pp. 224–287. Springer-Verlag, Berlin, (1977). [2] V. I. Tatarskii, A. Ishimaru, and V. U. Zavorotny, Wave Propagation in Random Media (Scintillation). (SPIE and IOP, 1993). [3] B. J. Uscinski, Wave Propagation and Scattering. (Oxford Clarendon Press, 1986). [4] J. Fouque, G. Papanicolaou, and Y. Samuelides, Forward and markov approximation: The strong intensity fluctuations regime revisited, Waves in Random Media. 8, 303– 314, (1998).

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[5] D. A. Dawson and G. Papanicolaou, A random wave process, Appl. Math. Optim. 12, 97–114, (1984). [6] F. Bailly, J. Clouet, and J. Fouque, Parabolic and white noise approximation for waves in random media, SIAM Jour. Appl. Math. 56, 1445–1470, (1996). [7] R. Z. Khasminskii, A limit theorem for solutions of differential equations with random right hand side, Theory Probab. Appl. 11, 390–406, (1966). [8] G. Blankenship and G. C. Papanicolaou, Stability and control of stochastic systems with wide-band noise disturbances, SIAM J. Appl. Math. 34, 437–476, (1978). [9] H. Kushner, Approximation and Weak Convergence Methods for Random Processes, with Applications to Stochastic Systems Theory. (MIT Press, 1984). [10] R. Bouc and E. Pardoux, Asymptotic analysis of pdes with wide-band noise disturbances and expansion of the moments, Stochastic Analysis and Applications. 2, 369–422, (1984). [11] A. Fannjiang, White-noise and geometrical optics limits of wigner-moyal equation for wave beams in turbulent media, Comm. Math. Phys. 254, 289–322, (2005). [12] A. Fannjiang, White-noise and geometrical optics limits of wigner-moyal equation for beam waves in turbulent media. ii. two-frequency formulation, J. Stat. Phys. 120, 543–586, (2005). [13] D. A. Dawson. Measure-valued markov processes. In ed. P. L. Hennequin, Ecole d’ Ete de Probabilite de Saint-Flour XXI—1991. [Saint-Flour Summer School on Probability Theory XXI—1991], Lecture Notes in Mathematics, 1541. Springer-Verlag, (1993). [14] B. L. Rozovkii, Stochastic Evolution Systems: Linear Theory and Applications to Non-linear Filtering. (Kluwer Academic Press, 1990). [15] G. Papanicolaou, L. Ryzhik, and K. Solna, Statistical stability in time reversal, SIAM J. Applied Mathematics. 64(4), 1133–1155, (2004). [16] G. Papanicolaou, L. Ryzhik, and K. Solna, Self-averaging from lateral diversity in the Itˆ o-Schr¨ odinger equation, SIAM J. Multiscale Modeling and Simulation. (2007). To appear. [17] H. Kunita, Stochastic Flows and Stochastic Differential Equations. vol. 24, Cambridge Stud. Adv. Math., (Cambridge University Press, UK, 1997). [18] P. Blomgren, G. Papanicolaou, and H. Zhao, Super-resolution in time-reversal acoustics, Journal of the Acoustical Society of America. 111, 238–248, (2002). [19] G. Bal, On the self-averaging of wave energy in random media, Multiscale Model. Simul. 2, 398–420, (2004). [20] G.Bal, G. Papanicolaou, and L. Ryzhik, Self-averaging in time reversal for the parabolic wave equation, Stoch. Dyn. 2, 507–531, (2002). [21] L. Borcea, G. Papanicolaou, and C. Tsogka, Theory and applications of time reversal and interferometric imaging, Inverse Problems. 19, S134–164, (2003). [22] L. Borcea, G. Papanicolaou, and C. Tsogka, Interferometric array imaging in clutter, Inverse Problems. 21(4), 1419–1460, (2005). [23] L. Borcea, G. Papanicolaou, and C. Tsogka, Adaptive interferometric imaging in clutter and optimal illumination, Inverse Problems. 22(4), 1405–1436, (2006). [24] L. Borcea, G. C. Papanicolaou, C. Tsogka, and J. Berryman, Imaging and time reversal in random media, Inverse Problems. 18(5), 1247–1279, (2002). [25] G.Bal and L. Ryzhik, Time reversal and refocusing in random media, SIAM Jour. Appl. Math. 63, 1475–1498, (2003). [26] J. Fouque, La convergence en loi pour les processus a valeurs dans un espace nucleaire, Ann. Inst. H. Poincare Probab. Statist. 20, 225–245, (1984).

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[27] J. F. Claerbout and S. M. Doherty, Downward continuation of moveout-corrected seismograms, Geophysics. 37(5), 741–768, (1972). [28] G. Beylkin and R. Burridge, Linearized inverse scattering problems in acoustics and elasticity, Wave Motion. 12(1), 15–52, (1990).

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Chapter 4 The Korteweg-de Vries Equation with Multiplicative Homogeneous Noise Anne de Bouard and Arnaud Debussche∗ CNRS et Universit´e Paris-Sud, UMR 8628, Bˆ at. 425, 91405 ORSAY CEDEX, FRANCE [email protected]. We prove the global existence and uniqueness of solutions both in the energy space and in the space of square integrable functions for a Korteweg-de Vries equation with noise. The noise is multiplicative, white in time, and is the multiplication by the solution of a homogeneous noise in the space variable.

Contents 1 Introduction and statement of the results . . . . . . 2 Preliminaries and existence for a truncated equation 3 Global existence . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

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113 116 128 132

1. Introduction and statement of the results The aim of the paper is to prove the global existence and uniqueness of strong solutions for a Korteweg-de Vries equation with noise, which may be written in Itˆ o form as 1 du + (∂x3 u + ∂x (u2 ))dt = uϕdW 2

(1.1)

where u is a random process defined on (t, x) ∈ R+ × R, W is a cylindrical Wiener process on L2 (R) and ϕ is a convolution operator on L2 (R) defined by Z ϕf (x) = k(x − y)f (y)dy, for f ∈ L2 (R) R

where the convolution kernel k is an H 1 (R) ∩ L1 (R) function of x ∈ R. Here H 1 (R) is the usual Sobolev space of square integrable functions of the space variable x, having their first order derivative in L2 (R). Considering a complete orthonormal ∗ ENS

de Cachan, Antenne de Bretagne, Campus de Ker Lann, Av. R. Schuman, 35170 BRUZ, FRANCE, [email protected] 113

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system (ei )i∈N in L2 (R), we may alternatively write W as X W (t, x) = βi (t)ϕei (x),

(1.2)

i∈N

(βi )i∈N being an independent family of real valued Brownian motions. Hence, the correlation function of the process ϕW is E(ϕW (t, x)ϕW (s, y)) = c(x − y)(s ∧ t), where c(z) =

Z

x, y ∈ R,

s, t > 0,

k(z + u)k(u)du. R

The existence and uniqueness of solutions for stochastic KdV equations of the type (1.1) but with an additive noise have been studied in Refs. [1], [2], [3]. Here we extend those results to equation (1.1), that is the multiplicative case with homogeneous noise. Note that an equation of this form, but with an additional weak dissipation has been considered in Ref. [4]. Indeed, in this latter case where a dissipative term is added, such a noise may be viewed as a perturbation of the dissipation. Although our existence and uniqueness results would easily extend to the case where weak dissipation is added, the dissipative term is of no help in the existence proof, so we prefer stating the result for equation (1.1). Assuming k ∈ H 1 (R) ∩ L1 (R) will allow us to prove the global existence and uniqueness of solutions to equation (1.1) in the energy space H 1 (R), that is in the space where both quantities Z 1 m(u) = u2 (x)dx (1.3) 2 R

and

1 H(u) = 2

Z

1 (∂x u) dx − 6 R 2

Z

u3 dx

(1.4)

R

are well defined. Note that m and H are conserved quantities for the equation without noise, that is 1 ∂t u + ∂x3 u + ∂x (u2 ) = 0. (1.5) 2 It is important to solve equation (1.1) in the energy space, indeed most of the studies on the qualitative behavior of the solutions are done in this space. One of our aim in the future is to analyse the qualitative influence of a noise on a soliton solution of the deterministic equation, as we did in the additive case in Ref. [5], and this requires the use of the hamiltonian (1.4). However, our method of construction of solutions easily extends to treat the case of a kernel k ∈ L2 (R) ∩ L1 (R), obtaining global existence and uniqueness in L2 (R). It seems difficult to get a result with less regularity.

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It may be noted also that the use of a noise of the form given in (1.1) naturally brings some localization in the noisy part of the equation, at least in the limit where the amplitude of the noise goes to zero, and when the initial state is a solitary wave – or soliton – solution of the deterministic equation, that is a well localized solution which propagates with a constant shape and velocity. This localization in the noise was a missing ingredient in the study of the influence of an additive noise on the propagation of a soliton (see Ref. [5]). The precise existence result is the following, and the method we use to prove it closely follows the method in Ref. [2]. Theorem 1.1. Assume that the kernel k of the noise satisfies k ∈ H s (R) ∩ L1 (R), s = 0 or 1. Then for any u0 in H s (R), there is a unique adapted solution u with paths almost surely in C(R+ ; H s (R)) of equation (1.1). Moreover, u ∈ L2 (Ω; C(R+ ; L2 (R))). As in Refs. [2, 3], we use the functional framework introduced by Bourgain to study dispersive equations. Following [6], [7], [8], for s, b ∈ R, Xb,s denotes the space of tempered distributions f ∈ S 0 (R2 ) for which the norm Z Z 1/2 2s 3 2b b 2 kf kXb,s = (1 + |ξ|) (1 + |τ − ξ |) |f (τ, ξ)| dτ dξ R2

b ξ) stands for the space-time Fourier transform of f (t, x). In the is finite, where f(τ, same way, we set for b, s1 , s2 ∈ R, Z Z 1/2 2s2 2s1 3 2b b 2 kf kXeb,s ,s = |ξ| (1 + |ξ|) (1 + |τ − ξ |) |f (τ, ξ)| dτ dξ 1

2

R2

and

 eb,s1 ,s2 = f ∈ S 0 (R2 ), kf k e X Xb,s

1 ,s2

< +∞ .

eb,s1 ,s2 is necessary here. Indeed, since we work Note that the use of the space X with stochastic equations driven by white in time noises, we cannot require too much time regularity, and we have to choose 0 < b < 1/2. But then the bilinear estimate which allows to treat the KdV equation is not true in the space Xb,s as was already mentioned for the additive case in Ref. [2]. T eT For T ≥ 0, we also introduce the spaces Xb,s and X b,s1 ,s2 of restrictions to [0, T ] e of functions in Xb,s and Xb,s1 ,s2 . They are endowed with  e X , fe ∈ Xb,s and f |[0,T ] = f| e [0,T ] kf kXb,s T = inf kfk b,s kf kXe T

b,s1 ,s2

 e e = inf kfk Xb,s

1 ,s2

eb,s1 ,s2 and f |[0,T ] = fe|[0,T ] . , fe ∈ X

Because equation (1.1) is a multiplicative equation with a nonlinear deterministic part, we have to consider first a cut-off version of this equation (see Section 2). As we make use of the functional framework defined above, the cut-off will arise as a

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t . Moreover, this function function of the norm of the solution of the type k · kXb,s of the norm must be a regular function, in order to allow us to use a fixed point argument (i.e. in order that our mapping is a contraction mapping, see Section 2). The fact that the functional spaces we consider are nonlocal spaces in the time variable then brings a lot of technical difficulties, concerning points that would be obvious if we were dealing with more classical function spaces (see e.g. the proof of Lemma 2.1). The paper is organized as follows: Section 2 is devoted to the proof of several preliminary lemmas and propositions, which once brought together lead quite easily to the proof of global existence and uniqueness for the cut-off version of the equation – or to the local existence and uniqueness for equation (1.1). In Section 3 we prove that the solutions of equation (1.1) are global in time, by using estimates on the moments of the L2 -norm of the solution. Again, due to the spaces we consider for the local existence, the globalization argument is not obvious.

2. Preliminaries and existence for a truncated equation As is usual, we introduce the mild form of the stochastic Korteweg-de Vries equation 3 (1.1). We denote by U (t) = e−t∂x the unitary group on L2 (R) generated by the linear equation ∂u ∂ 3 u + 3 = 0. ∂t ∂x 3

Using Fourier transform, we have F(U (t)v)(ξ) = eitξ F(v)(ξ). We then rewrite (1.1) as follows u(t) = U (t)u0 −

1 2

Z

t

U (t−r)∂x (u2 (r))dr + 0

Z

t 0

U (t−r)(u(r)ϕdW (r)), t ≥ 0. (2.1)

eb,s1 ,s2 norms defined in the introduction have the nice property The Xb,s and X that they are increasing with T . However, it is more convenient to work with other norms, given by the multiplication by the function 1l[0,T ] . In the case we consider here, that is 0 ≤ b < 1/2, we can prove the following result, stating that the two norms are equivalent. Lemma 2.1. Let s ≥ 0 and 0 ≤ b < 1/2, then there exist two constants C1 , C2 depending on b but not on T such that for any f ∈ Xb,s T T . ≤ k1l[0,T ] (t)f kXb,s ≤ C2 kf kXb,s C1 kf kXb,s

Proof. The first inequality is clear and in fact we may choose C1 = 1. For the

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other inequality, let us set g(t) = 1l[0,T ] (t)U (−t)f (t) so that ZZ k1l[0,T ] (t)f k2Xb,s = (1 + |ξ|)2s (1 + |τ |)2b |b g(τ, ξ)|2 dτ dξ R2

=

ZZ

R2

(1 + |ξ|)2s k(Fx g)(·, ξ)k2H b dξ. t

The result follows from the following inequality k1l[0,T ] hkH b (R) ≤ CkhkH b (R) , h ∈ H b (R), which holds for a constant C ≥ 0 depending on 0 < b < 1/2. To prove this, we use the following equivalent norm on H b (R) (see for instance [9]): ZZ |h(t) − h(r)|2 2 dtdr + khk2L2 (R) . khkH b (R) = |t − r|1+2b R2 Clearly, k1l[0,T ] hk2L2 (R) ≤ khk2L2 (R) . Moreover ZZ

1l[0,T ] (t)h(t) − 1l[0,T ] (r)h(r) 2

R2

=2 =2 +2

|t − r|1+2b

1l[0,T ] (t)h(t) − 1l[0,T ] (r)h(r) 2

ZZ

Z

Z

r
0 0 ∞Z T

T

drdt

0

|t − r|1+2b

|h(t) − h(r)|2 drdt + 2 |t − r|1+2b 2

Z

0

T

Z

drdt 0

−∞

|h(t)|2 drdt |t − r|1+2b

|h(r)| drdt = I + II + III. |t − r|1+2b

The first term I is less than khkH b (R) . The second and third terms are equal to Z Z 1 T 1 T −2b |t| |h(t)|2 dt, III = |T − r|−2b |h(r)|2 dr. II = b 0 b 0 Both are bounded by CkhkH b (0,T ) . To see this, we note that it is obvious for b = 0 and results from Hardy inequality for b = 1 when H b (0, T ) is replaced by H01 (0, T ). The result follows by interpolation since, for 0 ≤ b < 1/2, H b (0, T ) = H0b (0, T ) .  T T eT We now define Yb,0 = Xb,0 ∩X b,0,−3/8 endowed with the norm kf kYb,0 T = max{kf kX T , kf k e T X b,0

b,0,−3/8

}.

T T eT We also use the space Yb,1 = Xb,1 ∩X b,1,−3/8 with a similar definition of its norm. From now on and thanks to Lemma 2.1, we will use the definition T kvkTYb,s = k1l[0,T ] vkYb,s each time we take a norm in Yb,s with 0 ≤ b < 1/2. For u0 ∈ L2 (Ω; H 1 (R)), we set

z(t) = U (t)u0 , and v(t) = u(t) − z(t).

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Then (2.1) is rewritten as v(t) = − 12 +

Z

Z

t 0

t 0

  U (t − r) ∂x (v 2 (r)) + ∂x (z 2 (r)) + 2∂x (z(r)v(r)) dr

(2.2)

U (t − r)((z(r) + v(r))ϕdW (r)), t ≥ 0.

Let θ be a cut-off function – θ(x) = 0 for x ≥ 2, θ(x) = 1 for 0 ≤ x ≤ 1, with θ ∈ C0∞ (R+ ) – and let θR = θ( R. ); we consider the cut-off version of (2.2) written for R > 0 as: Z  i h  1 t 2 2 r kvR kYb,0 ∂x (vR (r)) dr U (t − r) θR vR (t) = − 2 0 −

Z

0

Z

1 − 2 +

t

Z

h   i r U (t − r) θR kvR kYb,0 ∂x (z(r)vR (r)) dr t 0



U (t − r) ∂x (z 2 (r)) dr

t

0

(2.3)



U (t − r)((z(r) + vR (r))ϕdW (r)), t ≥ 0.

We find vR as a fixed point of the mapping TR , TR vR being defined by the right hand side above. Note that the cut-off is made in the L2 in space norm, even for the H 1 result. We will choose 0 < b < 1/2 and 1/2 < c. We use the following Lemma. T Lemma 2.2. For any 0 ≤ b < 1/2, R > 0, v ∈ Yb,1 , there exists C(R) such that





t v(t)

θR kvkYb,0

T Yb,0

≤ C(R)

and, for s = 0 or 1, there is a positive constant C, independent of R, such that





t v(t) T ≤ C kv(t)kY T .

θR kvkYb,0 b,s Yb,s

Proof. We use arguments similar to the proof of Lemma 2.1. Let w(t) = U (−t)v(t) then, using the same norm on H b (R) as in Lemma 2.1, Z 

2

2

  



t t (Fx w)(t, ξ) b dξ. v(t) T ≤ C

θR kvkYb,0

θR kvkYb,0 Xb,0

R

Ht ([0,T ])

The L2 part of the H b norm above is easily estimated, while the other part is

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bounded above by C

Z Z R

+C

0

Z Z R

Z

T

T 0

t 0

  |(F w)(t, ξ) − (F w)(r, ξ)|2 x x 2 t drdtdξ θR kvkYb,0 |t − r|1+2b

Z t 0

   2 |(F w)(r, ξ)|2 x r t drdtdξ θR kvkYb,0 − θR kvkYb,0 |t − r|1+2b

= I + II.   t t Next, we define τR = inf{t ≥ 0, kvkYb,0 ≥ 2R}; then θR kvkYb,0 = 0 for t ≥ τR and Z Z τR Z t 2 |(Fx w)(t, ξ) − (Fx w)(r, ξ)| drdtdξ I ≤C |t − r|1+2b R 0 0 ≤C

Z

R

k(Fx w)(·, ξ)k2H b (0,τR ) dξ ≤ CkvkX τR ≤ 2CR. b,0

In order to estimate II, we use the fact that for r < t,     2 C 2 r | r t t − kvkYb,0 θR kvkYb,0 − θR kvkYb,0 ≤ 2 |θ0 |2L∞ |kvkYb,0 R C C |k1l[0,t] vkYb,0 − k1l[0,r] vkYb,0 |2 ≤ 2 k1l[r,t] vk2Yb,0 R2 R Z C ≤ 2 (1 + |η|−3/4 )k1l[r,t] (Fx w)(., η)k2H b dη. R R ≤

We leave to the reader the estimate of the contribution to II of the L2 part of the H b norm above; indeed, it follows the same line as the estimate of the remaining contribution, which is bounded above by Z Z Z τR Z t Z t Z σ 2 |(Fx w)(σ2 , η) − (Fx w)(σ1 , η)|2 C −3/4 (1 + |η| ) dσ1 dσ2 R2 R R |σ2 − σ1 |1+2b 0 0 r r × ≤

C R2

Z Z R

R

(1 + |η|−3/4 )

Z

τR 0

Z

σ2 0

|(Fx w)(r, ξ)|2 drdtdηdξ |t − r|1+2b

Z

×

σ1 0

Z

τR σ2

dt |t − r|1+2b



|(Fx w)(r, ξ)|2 dr

|(Fx w)(σ2 , η) − (Fx w)(σ1 , η)|2 dσ1 dσ2 dηdξ |σ2 − σ1 |1+2b

where we have inverted the integrals in the time variables; this last term is in turn

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bounded above by  Z Z Z τR Z σ2 Z σ1 C −3/4 −2b 2 (1 + |η| ) |σ − r| |(F w)(r, ξ)| dr 1 x R2 R R 0 0 0 ×

|(Fx w)(σ2 , η) − (Fx w)(σ1 , η)|2 dσ1 dσ2 dηdξ , |σ1 − σ2 |1+2b

by the fact that |τR −r|−2b ≤ |σ2 −r|−2b ≤ |σ1 −r|−2b for 0 ≤ r ≤ σ1 ≤ σ2 ≤ t ≤ τR . Using then the same arguments as in the proof of Lemma 2.1, we finally get Z Z C 2 k(Fx w)(., ξ)kH b (0,τR ) dξ (1 + |η|−3/4 )k(Fx w)(., η)k2H b (0,τR ) dη II ≤ 2 R R R C kvk2Y τR kvk2X τR . b,0 b,0 R2 This, together with the estimate of I implies the first inequality of the Lemma for eb,0,−3/8 part, and the second inequality of the the Xb,0 part of the Yb,0 norm; the X Lemma are proved in the same way.  Next results state the estimates on the bilinear term appearing in (2.3). ≤

Proposition 2.3. Let a > 0, 0 < b < 1/2 < c < 1, with b + c > 1 and a, b, c T T sufficiently close to 1/2, then for any v ∈ Yb,s , z ∈ Xc,s , s = 0 or 1, we have k∂x (v 2 )kY−a,s T ≤ CkvkYb,0 T kvkY T , b,s

and

  k∂x (vz)kY−a,s T ≤ C kvkYb,0 T kzkX T + kvkY T kzkX T , c,s c,0 b,s k∂x (z 2 )kY−a,s T ≤ CkzkXc,s T kzkX T . c,0

Proof. These estimates are proved in Ref. [2], Proposition 2.2 and 2.3, for s = 0. These are easily extended to s = 1. It suffices to add a factor 1+|ξ| in the expression |hf, ∂x (gh)i| Z Z Z Z = ξ fb(τ, ξ) gb(τ − τ1 , ξ − ξ1 )b h(τ1 , ξ1 )dτ1 dξ1 dτ dξ , τ

ξ

τ1

ξ1

and to use the fact that 1 + |ξ| ≤ (1 + |ξ − ξ1 |) + (1 + |ξ1 |).



Remark 2.4. It does not seem possible to get rid of the homogeneous Sobolev eb,0,−3/8 , to get the result of Proposition 2.3, when b < 1/2, space, i.e. of X even in the case s = 1; indeed, a careful reading of the proof of Proposition 2.2 in Ref. [2] shows that the additional factor |ξ|−3/4 induced by the use eb,0,−3/8 is necessary in a region of the integral where |ξ1 |  |ξ|, so that of X |ξ − ξ1 | ∼ |ξ|, and with moreover |ξ| ≤ 1; hence the supplementary factor (1 + |ξ − ξ1 |)(1 + |ξ1 |)/(1 + |ξ|) is of no help there.

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It remains to derive the estimates on the stochastic integrals in (2.3). In order to be able to globalize the solutions in Section 3, we will need estimates on all the moments of the stochastic integrals. T Proposition 2.5. Let m ∈ N, s = 0 or 1, and v ∈ L2m (Ω, Xb,s ); then for any 0 ≤ b ≤ 1/2,

Z t

2m !  

2m 2m

E U (t − r) [v(r)ϕdW (r)] ≤ Ckkk E kvk T s X H (R)

T 0,s 0 Xb,s   2m ≤ CT bm kkk2m E kvk . T s X H (R) b,s

Moreover

Z t

2m

E U (t − r) [v(r)ϕdW (r)]

!

    2m 2m ≤ C kkk2m T H s (R) + kkkL1 (R) E kvkX0,s eT X b,s−3/8     2m 2m ≤ CT bm kkk2m + kkk E kvk . T s 1 X H (R) L (R)

0

b,s

Proof. We prove the Z t result for s = 1, the proof is exactly the same for s = 0. We U (t − r)[v(r)ϕdW (r)]. Let set w(t) = 1l[0,T ] (t) 0

g(t) = 1l[0,T ] (t)

Z

t

U (−r) [v(r)ϕdW (r)] , 0

then w(t) = U (t)g(t), t ≥ 0. We have

Z Z m    2m E kwkXb,1 = E (1 + |ξ|)2 (1 + |τ |)2b |b g (τ, ξ)|2 dτ dξ R2

Choosing Brownian motions (βk )k∈N , defined on R, we have (Fx g)(t, ξ) =

∞ X

1l[0,T ] (t)

∞ X

1l[0,T ] (t)

k=0

=

k=0

Z Z

t

3

eirξ Fx (v(r)ϕek ) (ξ)dβk (r)

0 t

−∞

3

1l[0,T ] (r)eirξ Fx (v(r)ϕek ) (ξ)dβk (r).

It follows gb(τ, ξ) =

=

∞ Z X k=0

∞ Z X k=0

1l[0,T ] (t) R

Z

t

3

−∞ 3

R

1l[0,T ] (r)eirξ Fx (v(r)ϕek ) (ξ)dβk (r)e−iτ t dt

1l[0,T ] (r)eirξ Fx (v(r)ϕek ) (ξ)

Z

∞ r

 1l[0,T ] (t)e−iτ t dt dβk (r).

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By Burkholder inequality, we deduce: Z Z

2

2b

m 

2

(1 + |ξ|) (1 + |τ |) |b g(τ, ξ)| dτ dξ R2  Z Z Z ∞ X 2 ≤ Cm E (1 + |ξ|)2 (1 + |τ |)2b 1l[0,T ] (r) |Fx (v(r)ϕek ) (ξ)| E

R3

k=0

Z ×

It is easy to see that Z

∞ r

∞

r

2 m  1l[0,T ] (t)e−iτ t dt drdξdτ .

2 1l[0,T ] (t)e−iτ t dt ≤ min{T 2 , 2τ −2 }.

Therefore, using Lemma 2.6 below,

E

Z Z

≤ CE

2

R2

2b

2

(1 + |ξ|) (1 + |τ |) |b g(τ, ξ)| dτ dξ

 Z Z Z Z

R4

m 

(1 + |ξ|)2 (1 + |τ |)2b min{T 2 , 2τ −2 }1l[0,T ] (r) |(Fx v(r))(ξ + η)| ×|b k(η)|2 dηdrdξdτ

2

m 

which in turn we bound from above, using the unitarity of U (t) in L2 and in H 1 , by CE

 Z Z Z

R3



 (1 + |ξ + η|)2 + (1 + |η|2 ) 1l[0,T ] (r) |(Fx v(r))(ξ + η)|2 ×|b k(η)|2 dηdrdξ

m 

 i   h 2m 2m 2m E k1 l vk + kkk E k1 l vk ≤ C kkk2m 2 1 [0,T ] [0,T ] X0,1 X0,0 H (R) L (R)

h    i 2m 2m 2m ≤ C kkk2m E kvk + kkk E kvk . 2 1 T T L (R) H (R) X X 0,1

0,0

For the second statement, we proceed similarly. However, the extra |ξ|−3/4 implies that a special treatment of the integral for |ξ| ≤ 1. On the region |ξ| ≥ 1, we simply

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use |ξ|−3/4 ≤ 1. The following estimate is thus sufficient to conclude.  Z m  ZZ 2 b 2 −3/4 2 E (1 + |ξ|) |ξ| 1l[0,T ] (r) |(Fx v(r))(ξ + η)| |k(η)| dηdrdξ R2

|ξ|≤1

≤ CE

 Z Z

R2

Z

|ξ|≤1

|ξ|

  ≤ Ckb kkL∞ (R) E kvk2m XT

−3/4

 m  2 2 b |k(η − ξ)| dξ 1l[0,T ] (r) |Fx (v(r))(η)| dηdr

0,0



≤ CkkkL1 (R) E kvk2m XT

0,0

 .



We now give the Lemma used in the above proof. Lemma 2.6. Let v ∈ X0,0 , then for any complete orthonormal system (ek )k∈N of L2 (R), we have Z ∞ X 2 2 |Fx (v(r)ϕek ) (ξ)| = |Fx (v(r))(ξ + η)| |b k(η)|2 dη. k=0

R

  Proof. We have Fx (v(r)ϕek ) (ξ) = Fx v(r, x)hk(x − y), ek (y)iL2y (ξ) = hFx (v(r, x)k(x − y)) (ξ), ek (y)iL2y . Therefore, by Parseval identity, ∞ X k=0

|Fx (v(r)ϕek ) (ξ)|2 = kFx (v(r, x)k(x − y)) (ξ)k2L2 , y

and by Plancherel theorem and an easy computation ∞ X k=0

¯ 2 2 kk2L2 , |Fx (v(r)ϕek ) (ξ)| = kFx,y (v(r, x)k(x − y)) (ξ, η)kL2 = kFx (v(r))(ξ + .)b η

which gives the conclusion.  The proof of the next proposition is left to the reader. It only makes use of classical arguments and ideas similar to those at the end of the proof of Proposition 2.5. Proposition 2.7. Let s = 0 or 1. For any T0 > 0, any R · stopping time τ and any predictable process v ∈ L2 (Ω; C([0, T0 ∧ τ ]; H s (R))), 0 U (· − r)[ϕv(r)dW (r)] has continuous paths with values in H s (R) ∩ H˙ −3/8 (R) and for any integer m, there is a constant Cm with !

Z t

2m  

2m

≤ C E sup kv(t)k E sup U (t − r)[ϕv(r)dW (r)] m Hs .

s ˙ −3/8 t≤T ∧τ t≤T ∧τ 0

0

H ∩H

0

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We are now able to prove the following existence theorem for the truncated equation. Theorem 2.8. Let s = 0 or 1 and assume that the convolution kernel of the operator ϕ satisfies k ∈ H s (R) ∩ L1 (R); then for any u0 in H s (R), equation (2.3) with T0 z(t) = U (t)u0 has a unique solution vR ∈ Yb,s , for any b with 0 < b < 1/2, and any 2 s T0 ≥ 0. Moreover vR ∈ L (Ω; C([0, T0 ]; H (R))). Proof. We use a fixed point argument on equation (2.3). The following lemma, whose first and third estimates were proved in Ref. [2], while the second one can be proved in the same way, is useful. Lemma 2.9. T • Let u0 ∈ H s (R), s = 0 or 1. For any T > 0 and c > 1/2, z = U (·)u0 ∈ Xc,s and

kzkXc,s T ≤ C(T )ku0 kH s (R) . T • For any u0 ∈ H s (R)∩ H˙ −3/8 (R), and any b with 0 ≤ b < 1/2, z = U (·)u0 ∈ Yb,s and T ≤ C(T )(ku0 kH s (R) + ku0 k ˙ −3/8 kzkYb,s H (R) ). R· T T • For any a, b ∈ (0, 1) with a + b ≤ 1, and any f ∈ Y−a,s , 0 U (· − r)f (r)dr ∈ Yb,s and

Z ·

U (· − r)f (r)dr ≤ CT 1−(a+b) kf kY−a,s T .

T Yb,s

0

We first assume that the hypothesis of Theorem 2.8 hold with s = 0. Let us fix a, b, c as in Proposition 2.3, with a + b < 1. Wefix T0 and take T ≤ T0 . Let T v1 , v2 ∈ Yb,0 , T being also fixed. We set v˜i (t) = θR |vi |Y t vi (t), i = 1, 2. Then, b,0

recalling that TR vR is defined by the right hand side of (2.3), we have   2 E kTR v1 − TR v2 kY T b,0

!

Z t h i

2 2 2 ≤ CE U (t − r)∂x (˜ v1 (r)) − (˜ v2 (r)) dr

T 0

Yb,0

Z t

2 !

+CE U (t − r)∂x [(˜ v1 (r) − v˜2 (r)) z(r)] dr

T 0

Yb,0

Z t

2 !

+CE U (t − r) [(v1 (r) − v2 (r)) ϕdW (r)]

T 0

Yb,0

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which, by Lemma 2.9 and Proposition 2.5, applied with m = 1, is bounded above by     2

2 2 CT 2(1−(a+b)) E ∂x (˜ v1 ) − (˜ v2 ) T Y−a,0     2 + CT 2(1−(a+b)) E k∂x ((˜ v1 − v˜2 ) z)kY T + CT b E kv1 − v2 k2X T . −a,0

b,0

By Proposition 2.3, it follows   2 E kTR v1 − TR v2 kY T b,0

   o n  2 2 2 2 ≤ CT 2(1−(a+b)) E k˜ v1 − v˜2 kY T k˜ v1 + v˜2 kY T + E k˜ v1 − v˜2 kY T kzkX T b,0

b,0

c,0

b,0

  2 +CT b E kv1 − v2 kX T . b,0

By Lemma 2.2,

2

k˜ v1 + v˜2 kY T ≤ C(R). b,0

Moreover, it is not difficult to use the arguments of the proof of Lemma 2.2 and prove 2

2

k˜ v1 − v˜2 kY T ≤ C(R) kv1 − v2 kY T . b,0

b,0

We deduce that for some α > 0,   2 2 E kTR v1 − TR v2 kY T ≤ C(R, T0 , ku0 kL2 (R) )T α E kv1 − v2 kY T . b,0

b,0

T Thus, TR has a unique fixed point vR ∈ L2 (Ω; Yb,0 ) for T ≤ T∗ where T∗ is chosen such that

C(R, T0 , ku0 kL2 (R) )T∗α ≤ 1/2. Moreover, using arguments similar to the proof of Proposition 2.5, it can be seen that Rt U (−r) [(z(r) + vR (r)) ϕdW (r)] is a square integrable martingale in L2 (R) ∩ 0 −3/8 ˙ H (R). Since (U (t))t∈R is strongly continuous on L2 (R) ∩ H˙ −3/8 (R), we deduce that Z t U (t − r) [(z(r) + vR (r)) ϕdW (r)] ∈ L2 (Ω; C([0, T∗ ]; L2 (R) ∩ H˙ −3/8 (R))). 0

Using then Lemma 2.9 with b > 1/2 and similar estimates as above, we deduce that vR is also in L2 (Ω; C([0, T∗ ]; L2 (R) ∩ H˙ −3/8 (R))).

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Then, we construct a solution in [T∗ , 2T∗ ]. First, we write equation (2.3) with t ≥ T∗ in the form

1 vR (T∗ + t) = U (t)vR (T∗ ) − 2 −

Z

−

1 2

+

Z

t

Z

t 0

h   i 2 2 U (t − r) θR kvR kY T∗ +r ∂x (vR (T∗ + r)) dr b,0

h   i U (t − r) θR kvR kY T∗ +r ∂x (z(T∗ + r)vR (T∗ + r)) dr b,0

0

Z t

0

t 0

  U (t − r) ∂x (z 2 (T∗ + r)) dr

U (t − r)((z(T∗ + r) + vR (T∗ + r))ϕdW (r)), t ≥ 0.

(2.4) T∗ Since vR (T∗ ) ∈ L2 (Ω; L2 (R) ∩ H˙ −3/8 (R)), the first term is in L2 (Ω; Yb,0 ). It is then T∗ easily seen that vR can be found on [T∗ , 2T∗ ] as a fixed point in L2 (Ω; Yb,0 ) in the same way as on the interval [0, T∗ ]. T0 Iterating this, we get a solution on [0, T0 ] which is in fact in L2 (Ω; Yb,0 ) and also 2 2 −3/8 ˙ in L (Ω; C([0, T0 ]; L (R) ∩ H (R))). This proves the result for s = 0. T Now suppose that the assumptions hold with s = 1. Let v ∈ L2 (Ω; Yb,1 ); we t )v(t), and using Lemma 2.9, have, setting v˜(t) = θR (kvkYb,0

h kTR vkY T ≤ CT 1−(a+b) k˜ v ∂x v˜kY T

−a,1

b,1

+ k∂x (˜ v z)kY T

Z t



+ U (t − r) [z(r)ϕdW (r)]

0

−a,1

T Yb,1


+ ∂ x z 2 Y T

−a,1

i

Z t



+ U (t − r) [v(r)ϕdW (r)]

0

, T Yb,1

so that by Proposition 2.3,

h kTR vkY T ≤ CT 1−(a+b) k˜ v kY T k˜ v kY T + k˜ v kY T kzkX T + k˜ v kY T kzkX T c,1 c,0 b,1 b,0 b,1 b,0 b,1

Z t i

+ kzkX T kzkX T + U (t − r) [z(r)ϕdW (r)]

T c,0 c,1 0 Yb,1

Z t

+ U (t − r) [v(r)ϕdW (r)] .

0

T Yb,1

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We then make use of Lemma 2.2 and Lemma 2.9 to get    kTR vkY T ≤ CT 1−(a+b) kvkY T + kzkX T0 C(R) + kzkX T0 b,1

b,1

c,1

Z t



+ U (t − r) [z(r)ϕdW (r)]

0

T Yb,1

c,0

Z t

+ U (t − r) [v(r)ϕdW (r)]

0

T Yb,1

  ≤ C(R, T0 , ku0 kL2 (R) )T 1−(a+b) kvkY T + ku0 kH 1 (R) b,1

Z t



+ U (t − r) [z(r)ϕdW (r)]

0

T Yb,1

Z t



+ U (t − r) [v(r)ϕdW (r)]

0

. T Yb,1

Thus, by Proposition 2.5 and Lemma 2.9,        E kTR vk2Y T ≤ C(R, T0 , ku0 kL2 (R) )T 2(1−(a+b)) + CT b E kvk2Y T + ku0 k2H 1 (R) . b,1

b,1

This shows that TR maps L

2

T (Ω; Yb,1 )

T into itself. Moreover, the ball in L2 (Ω; Yb,1 ) of 2(1−(a+b))

radius R0 is invariant by TR if T ≤ T∗∗ such that C(R, T0 , ku0 kL2 (R) )T∗∗ + b CT∗∗ ≤ 1/2 and R0 ≥ ku0 kH 1 (R) . Choosing T∗ ≤ T∗∗ in the construction of the solution vR of (2.3) in L2 , it follows T∗ that the solution vR is in L2 (Ω; Yb,1 ). We then use similar arguments as above to 2 prove that vR ∈ L (Ω; C([0, T∗ ]; H 1 (R)) and ! E

sup kvR k2H 1 (R)

[0,T∗ ]

2(1−(a+˜ b))

≤ CT∗

  2   2 2 E kvR kY T∗ + ku0 kH 1 (R) C(R) + ku0 kL2 (R) b,1

+E

Z t

2

sup U (t − r) [(z(r) + vR (r))ϕdW (r)]

[0,T∗ ]

2(1−(a+˜ b))

≤ CT∗

0

H 1 (R)

!

  2   2 2 E kvR kY T∗ + ku0 kH 1 (R) C(R) + ku0 kL2 (R) b,1

+CT∗ E

sup [0,T∗ ]

kvR k2H 1 (R)

!

2

+ CT∗ ku0 kH 1 (R) ,

with ˜b > 1/2 and a + ˜b < 1. Choosing a smaller T∗ if necessary, we deduce ! E

sup kvR k2H 1 (R)

[0,T∗ ]

≤ R02 ,

if R0 ≥ ku0 kH 1 (R) . On [T∗ , 2T∗ ], we use equation (2.4) and obtain by similar arguments     2 2 E kvR (T∗ + ·)kY T∗ ≤ C(T∗ )E kvR (T∗ )kH 1 (R)∩H˙ −3/8 (R) + ku0 k2H 1 (R) , b,1

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and E

sup [T∗ ,2T∗ ]

kvR k2H 1 (R)

!

  2 ≤ CE kvR (T∗ )kH 1 (R)∩H˙ −3/8 (R) + R02 .

We know from the L2 construction that vR ∈ L2 (Ω; C([0, T0 ]; H˙ −3/8 (R))), therefore ! !   2 2 2 2 E sup kvR kH 1 (R) ≤ CE kvR (T∗ )kH 1 (R) + R0 + E sup kvR kH˙ −3/8 (R) . [T∗ ,2T∗ ]

[0,T0 ]

It is now easy to iterate this argument and deduce that the solution vR is in T0 L2 (Ω; Yb,1 ) and also in L2 (Ω; C([0, T0 ]; H 1 (R))).  Theorem 2.8 gives the following local in time existence result for the non truncated equation. Corollary 2.10. Let s = 0 or 1 and assume that k ∈ H s (R) ∩ L1 (R); then for any u0 ∈ H s (R), there is a stopping time τ ∗ (u0 , ω) a.s. positive, such that (2.1) has an adapted solution u, defined a.s. on [0, τ ∗ (u0 )[, unique in some class, and with paths a.s. in C([0, τ ∗ (u0 )[; H s (R)). If s = 1, the uniqueness holds among solutions with paths in C([0, τ ∗ (u0 )[; H 1 (R)) a.s; moreover, the stopping time τ ∗ (u0 ) satisfies τ ∗ (u0 ) = +∞

or

t lim sup ku − U (·)u0 kYb,0 = +∞, a.s.

t%τ ∗ (u0 )

Remark 2.11. Let us explain what we mean by a solution on the random interval [0, τ ∗ (u0 )[. This means that u is defined on [0, τ ∗ (u0 )[ and is an adapted process such that for any stopping time τ < τ ∗ (u0 ) the following holds on [0, τ ]: Z Z t∧τ 1 t u(t) = U (t)u0 − U (t − r)∂x (u2 (r))dr + U (t − r)(u(r)ϕdW (r)). 2 0 0 T0 Proof. Let z(t) = U (t)u0 and let vR ∈ Yb,s for any b < 1/2 and any T0 > 0 be the t solution of (2.3) given by Theorem 2.8. We then set τR = inf{t ≥ 0, kvR kYb,0 ≥ R}; t ) = 1, hence vR is a solution of (2.2) on [0, τR ]. for t ∈ [0, τR ], we have θR (kvR kYb,0 It is not difficult to see that τR is non decreasing in R and that vR+1 = vR on [0, τR ]. Hence we may define u on [0, τ ∗ (u0 )[ with τ ∗ (u0 ) = limR→∞ τR by setting u(t) = vR (t) + z(t) for t ∈ [0, τR ] and u is then a solution of (2.1) on [0, τ ∗ (u0 )[. τR The uniqueness for u holds in the class z + Yb,0 for any R and it is not difficult to see that any solution u with paths in C([0, τ ∗ (u0 )[; H 1 (R)) is in this class. The last property of the lemma is an immediate consequence of the definition of τ ∗ (u0 ). 

3. Global existence As already seen, Theorem 2.8 gives a local in time existence result for the equation without cut-off. In the present section, we end the proof of Theorem 1.1 by showing that those solutions are globally defined in time. To that aim we need an estimate on kvkYb,0 T . We will use the following result.

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Proposition 3.1. Assume that k ∈ L2 (R). Let u ∈ C([0, τ ]); L2 (R)) be a solution of equation (2.1) with u0 ∈ L2 (R), where τ is a stopping time. Then, for any m ≥ 1, u ∈ L2m (Ω; C([0, τ ]; L2 (R))) and for any T > 0 ! sup

E

t∈[0,τ ∧T ]

|u(t)|2m L2 (R)

≤ C(T, ku0 kL2 (R) , m).

Proof. The result is a straightforward consequence of Itˆ o formula. We prove it for m = 2. For m ≤ 2 it then follows from H¨ older inequality. For m ≥ 2, the proof is similar. We apply Itˆ o formula to M (u) = kuk2L2(R) and obtain after a regularization argument and easy computations (see Ref. [1] for more details in the case of an additive noise or Ref. [10] for the case of the stochastic Schr¨ odinger equation): Z Z ∞ τ ∧r X M (u(τ ∧ r)) = M (u0 ) + 2 u2 (σ, x)ϕek (x)dxdβk (σ) + kkk2L2 (R)

Z

k=0 τ ∧r

0

R

M (u(σ))dσ.

0

We take the square of this identity and deduce: ! E

sup

r∈[0,τ ∧T ]

M 2 (u(r)) 

∞ Z 2  X r∧T Z ≤ 2M 2 (u0 ) + 4E  sup u2 (σ, x)ϕek (x)dxdβk (σ)  r∈[0,τ ∧T ] k=0 0 R  !2  Z τ ∧T +2kkk4 2 E  M (u(σ))dσ  L (R)

0

Z   ≤ 2M 2 (u0 ) + 4kkk2L2 (R) + 2T kkk4L2(R) E

τ ∧T

0

2

M (u(r))dr

!

,

thanks to Burkholder and H¨ older inequalities, and to Lemma 2.6. The result follows from Gronwall Lemma.  Let vR be the solution given by Theorem 2.8, let T0 > 0 be fixed, and let t τR = inf{t ∈ [0, T0 ], kvR kYb,0 ≥ R}; then on [0, τR ], vR + z is a solution to (2.1) 2 which is a.s. in C([0, τR ]; L (R)) and Proposition 3.1 applies: ! E

sup

t∈[0,τR ∧T0 ]

|vR (t) + z(t)|2m L2 (R)

≤ C(T0 , ku0 kL2 (R) , m).

(3.1)

T We now show that this implies an estimate on the Yb,0 norm of vR .

Lemma 3.2. Let vR be the solution of the truncated equation (2.3), then there exists a constant C(T0 , ku0 kL2 (R) ) independent of R such that   E kvR kY τR ≤ C(T0 , ku0 kL2 (R) ). b,0

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Proof. Step 1: Using similar arguments as in the beginning of the proof of Theorem 2.8, taking into account the fact that vR satisfies equation (2.3), we prove using Lemma 2.2 that for T0 ≥ T ≥ 0, i h kvR kY T ∧τR ≤ CT 1−(a+b) kvR k2Y T ∧τR + kzk2X T0 c,0

b,0

b,0

Z t



+ U (t − r) [u (r)ϕdW (r)] R

0

T

Yb,00

i h ≤ CT 1−(a+b) kvR k2Y T ∧τR + C(T0 ) ku0 k2L2 (R) b,0

Z t



+ U (t − r) [uR (r)ϕdW (r)]

0

with uR (t) = vR (t) + U (t)u0 . We set K1 = K1 (ω) = then

Z

1−(a+b) 2 CT0 C(T0 ) kuR kC([0,T0 ];L2 (R)) +

t 0

2

T

,

Yb,00

U (t − r) [uR ϕdW (r)]

T

,

Yb,00

CT 1−(a+b) kvR kY T ∧τR − kvR kY T ∧τR + K1 ≥ 0. b,0

b,0

Therefore, if we choose T = T (ω) such that T 1−(a+b) =

3 , we have 16CK1

kvR kY T ∧τR ≤ 2K1 . b,0

t Note indeed that vR (0) = 0 and that kvR k|Yb,0 is a continuous function of t. Similarly, for any k ≥ 0, we define k vR (t) = uR (t) − U (t − kT )uR (kT ),

t ∈ [kT, (k + 1)T ],

with T = T (ω) chosen above. Then the same argument shows that

k

vR [kT ∧τR ,(k+1)T ∧τR ] ≤ 2K1 , Yb,0

[T ,T ]

T where we use the space Yb,01 2 whose definition is exactly the same as Yb,0 but [0, T ] is replaced by [T1 , T2 ]. Step 2: Since uR is a solution of (2.1) on [0, τR ], we may write for any t ∈ [0, τR ], uR (t) as Z t Z t uR (t) = U (t)u0 − 21 U (t − r)∂x (u2R (r))dr + U (t − r) [uR (r)ϕdW (r)] 0 0 kt Z (k+1)T ∧t h
2 i 1X k = U (t)u0 − (r) + U (r − kT )uR (kT ) dr U (t − r)∂x vR 2 kT k=0 Z t + U (t − r) [uR (r)ϕdW (r)] , 0

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where kt is the integer part of t/T . Using this decomposition and the unitarity of U (σ) in H˙ −3/8 for any σ, we deduce that for any t ∈ [0, τR ], kuR (t) − U (t)u0 kH˙ −3/8 (R)

(k+1)T

Z ∧t kt

h i X
1 2

k U (t − r)∂x vR (r) + U (r − kT )uR (kT ) dr ≤

2 k=0 kT

˙ −3/8 H (R)

Z t

+ U (t − r) [uR (r)ϕdW (r)]

˙ −3/8 0 H (R)



(k+1)T ∧t Z k t

h i X
1 2

k U ((k + 1)T ∧ t − r)∂x vR (r) + U (r − kT )uR (kT ) dr ≤

2 k=0 kT

˙ −3/8 H (R)

Z t

+ U (t − r) [uR (r)ϕdW (r)] .

˙ −3/8 (R) H

0

Now, suppose that a is fixed with 0 < a < 1/2 in such a way that Proposition 2.3 [T ,T ] holds, and set ˜b = 1 − a, so that ˜b > 1/2. Then, using the fact that Y˜b,01 2 ⊂ C([[T1 , T2 ]; L2 (R) ∩ H˙ −3/8 (R)) for any positive T1 , T2 , we have for t ∈ [0, τR ] and k = 0, . . . , kt ,

Z

(k+1)T ∧t h
2 i

k dr U ((k + 1)T ∧ t − r)∂x vR (r) + U (r − kT )uR (kT )

kT

Z

≤

·

kT

Z

≤C

U (· − r)∂x ·

kT

h

U (· − r)∂x

k (r) vR

h

+ U (r − kT )uR (kT )


2 i

k vR (r) + U (r − kT )uR (kT )

dr


2 i

˙ −3/8 (R) H

˙ −3/8 (R)) C([kT ∧τR ,(k+1)T ∧τR ];H

dr

[kT ∧τR ,(k+1)T ∧τR ]

.

Y˜

b,0

By Lemma 2.9, the above term is majorized for each k ∈ {0, · · · , kt } by

h
2 i ˜

k CT 1−(a+b) ∂x vR + U (· − kT )uR (kT ) [kT ∧τR ,(k+1)T ∧τR ] Y−a,0 o n 2 k 2

≤ C vR Y [kT ∧τR ,(k+1)T ∧τR ] + kuR (kT )kL2 (R) , b,0

by Proposition 2.3 and Lemma 2.9 again, since a + ˜b = 1. By the result of step 1, we obtain kuR (t) − U (t)u0 kH˙ −3/8 (R) ≤ K2 ,

with

t ∈ [0, τR ],

h i 1 2 CT0 T −1 4K12 + kuR kC([0,T0 ];L2 (R))

2Z ·



+ U (· − r) [uR (r)ϕdW (r)]

˙ −3/8

K2 =

0

C([0,T0 ];H

. (R)

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Step 3: By Lemma 2.9, and the unitarity of U (kT ) on L2 (R) ∩ H˙ −3/8 (R) we have kU (t − kT )uR (kT ) − U (t)u0 kY [kT ∧τR ,(k+1)T ∧τR ] b,0

≤ C kU (−kT )uR (kT ) − u0 kL2 (R)∩H˙ −3/8 (R)

≤ C kuR (kT ) − U (kT )u0 kL2 (R)∩H˙ −3/8 (R) . Therefore, using step 2, kU (· − kT )uR (kT ) − U (·)u0 kY [kT ∧τR ,(k+1)T ∧τR ] ≤ K3 b,0   = C K2 + 2 kuR kC([0,T0 ];L2 (R)) .

Finally, for t ∈ [kT ∧ τR , (k + 1)T ∧ τR ], we have

k vR (t) = vR (t) + U (t − kT )uR (kT ) − U (t)u0 ,

and we may write, k0 being the integer part of T0 /T , kvR kY τR ≤ b,0

≤

k0 X

k0 X k=0

k

v R

k=0

≤

kvR kY [kT ∧τR ,(k+1)T ∧τR ] b,0

[kT ∧τR ,(k+1)T ∧τR ]

Yb,0



+ kU (· − kT )u(kT ) − U (·)u0 kY [kT ∧τR ,(k+1)T ∧τR ] b,0

T0 + 1 (2K1 + K3 ) . T 1/(1−(a+b))

Note that T −1 is proportional to K1 ; by Proposition 3.1 and Proposition 2.7, K1 and K3 have all moments finite, and it follows   E kvR kY τR ≤ c(T0 , ku0 kL2 (R) ) b,0

which concludes the proof of Lemma 3.2.  It is now straightforward to achieve the proof of Theorem 1.1. Indeed, due to Corollary 2.10, it suffices to see that lim supR→∞ τR ∧ T0 = T0 in probability as R goes to infinity. But this is an easy consequence of Markov inequality and Lemma 3.2, since   P (τR < T0 ) = P kvR kY T0 ∧τR ≥ R . b,0

References [1] A. de Bouard and A. Debussche, On the stochastic Korteweg-de Vries equation, J. Funct. Anal. 154(1), 215–251, (1998). ISSN 0022-1236. [2] A. de Bouard, A. Debussche, and Y. Tsutsumi, White noise driven Korteweg-de Vries equation, J. Funct. Anal. 169(2), 532–558, (1999). ISSN 0022-1236. [3] A. De Bouard, A. Debussche, and Y. Tsutsumi, Periodic solutions of the Korteweg-de Vries equation driven by white noise, SIAM J. Math. Anal. 36(3), 815–855 (electronic), (2004/05). ISSN 0036-1410.

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[4] R. L. Herman, The stochastic, damped KdV equation, J. Phys. A. 23(7), 1063–1084, (1990). ISSN 0305-4470. [5] A. de Bouard and A. Debussche, Random modulation of solitons for the stochastic Korteweg-de Vries equation, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire. 24, (2007). ISSN 0294-1449. To appear. [6] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal. 3(3), 209–262, (1993). ISSN 1016-443X. [7] C. E. Kenig, G. Ponce, and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J. 71(1), 1–21, (1993). ISSN 0012-7094. [8] C. E. Kenig, G. Ponce, and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc. 9(2), 573–603, (1996). ISSN 0894-0347. [9] R. A. Adams, Sobolev spaces. (Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975). Pure and Applied Mathematics, Vol. 65. [10] A. de Bouard and A. Debussche, The stochastic nonlinear Schr¨ odinger equation in H 1 , Stochastic Anal. Appl. 21(1), 97–126, (2003). ISSN 0736-2994.

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Chapter 5 On Stochastic Burgers Equation Driven by a Fractional Laplacian and Space-Time White Noise Zdzislaw Brze´zniak and Latifa Debbi∗ Department of Mathematics, University of York Heslington, York YO10 5DD, UK [email protected] We prove existence and uniqueness of a mild global solution to the Cauchy problem for the stochastic fractional Burgers equation on an interval with multiplicative space time white noise and periodic boundary conditions when the power of the Laplace operator is between 3/4 and 1.

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2 Existence of global solutions to approximating equations 3 Global solutions to Burgers equations . . . . . . . . . . 4 Proof of uniqueness . . . . . . . . . . . . . . . . . . . . A.1 Proof of Lemma 2.3 . . . . . . . . . . . . . . . . . . . . B.1 Gronwall Lemma . . . . . . . . . . . . . . . . . . . . . . C.1 Some estimates on stopped stochastic convolutions . . . D.1 Pointwise multiplication in Sobolev spaces . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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135 138 148 153 156 157 157 164 165

1. Introduction It is widely accepted that the classical Burgers equation can be used to model several physical phenomena and because of this it has been extensively studied. Recently its random generalizations have been introduced either by considering noisy force, see Refs. [1, 2] or by assuming randomness of the initial data, see a recent paper [3]. Moreover other types of generalizations have been studied. For example by considering an arbitrary algebraic nonlinearity instead of the quadratic, see Ref. [4] or by replacing the Laplace operator by a non local operator, see Ref. [5]). The first kind is called stochastic Burgers equation and the two latter ones are called generalized Burgers equation. Generalized Burgers equations with fractional differential operators are used to model some anomalous diffusions such as the far ∗ Institut

Elie Cartan, Nancy 1 B.P 239, 54506 Vandoeuvre-Les-Nancy cedex, France & Department of Mathematics, Faculty of Sciences, University Ferhat Abbas, El-Maabouda S´etif 19000, Algeria, [email protected] 135

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field (or long time) behavior of the acoustic waves propagating in a gas-filled tube and the wave propagation in viscoelastic medium (see Refs. [6–8] for the generalized Burgers equation and Refs. [9, 10] and the references therein for more details on anomalous diffusions and fractional differential equations). Generalized Burgers equations can also be found in works on continuum mechanics and hydrodynamics (shallow water, bubbly liquid) and in molecular biology. The equation dealt with in this paper is a generalization of Burgers equation which is characterized by the coexistence of the stochastic noise and the fractional power of the Laplacian. We are motivated by the tri-interaction between the wave steepening given by the nonlinearity and by the small dissipation given by the fractional power of the Laplacian and by the irregular random perturbation given by the cylindrical noise. In particular, the critical power for the Laplacian is the same as the one obtained in the deterministic fractional Burgers’ equation [5]. Some other stochastic fractional partial differential equations have been studied in several papers, see Refs. [11–15]. In this work, we are interested in the following Cauchy problem for a Burgers equation driven by a fractional Laplacian and space-time white noise: n du(t) = [−A u(t) + Bu2 (t)] dt + g(u(t)) dW (t), t > 0, α (1.1) u(0) = u0 , α

where 23 < α < 2, Aα = A 2 , A = −∆, D(A) = H 2,2 (0, 1) ∩ H01,2 (0, 1). Here by H k,p (0, 1), for k ∈ N, p ∈ [1, ∞) we denote the Banach space of all f ∈ Lp (0, 1) for which Dj f ∈ Lp (0, 1), j = 0, 1, . . . , k. The norm in H k,p (0, 1) is given by   p1 k X k f kH k,p (0,1) =  |Dj f |Lp (0,1)  . (1.2) j=0

We define the fractional order Sobolev space H β,p (0, 1), β ∈ R+ \ N by the complex interpolation method, i.e. H β,p (0, 1) = [H k,p (0, 1), H m,p (0, 1)]ϑ ,

(1.3)

where k, m ∈ N, ϑ ∈ (0, 1), k < m, are chosen to satisfy β = (1 − ϑ)k + ϑm.

(1.4)

One should bear in mind that the space on the LHS of formula (1.3) does not depend on k, m, ϑ provided they satisfy condition (1.4). In what follows by H0s,p (0, 1), s ≥ 0, p ∈ (1, ∞) we will denote the closure of C0∞ (0, 1) in the Banach space H s,p (0, 1). It is well known, see Theorem 11.1 in Ref. [16] and Theorem 1.4.3.2 on p.317 in Ref. [17] and Theorem 3.40 in Ref. [18] that H0s,p (0, 1) = H s,p (0, 1) iff s ≤ p1 . Since A is a self-adjoint operator in the Hilbert space H = L2 (0, 1), the fractional α power, see Ref. [19], Aα = A 2 is well defined and it follows from Theorem 1.15.3 on p. 103 in Ref. [17] that D(Aα ) = [H, D(A)] α2 , where [H, D(A)] α2 is the complex

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interpolation space of order α2 , see Refs. [16], [17] and Theorem 4.2 in Ref. [20]. Moreover, by Seeley [21], see also Theorem in section 4.4.3 in the monograph [17], we have that   H α,2 (0, 1) ∩ H01,2 (0, 1), if 1 < α ≤ 2,    H α,2 (0, 1), if 12 < α ≤ 1, 0 (1.5) D(Aα ) = ˜ α,2  if α = 12 ,  H (0, 1),   α,2 H (0, 1), if α < 21 ,

˜ α,2 (0, 1) is the space of all f ∈ H α,2 (R) such that supp(f ) ⊂ [0, 1]. where H Since the operator A is self-adjoint, A−1 exists and is compact, there exists an ONB {ej }j∈N and a sequence {λj }√ j∈N such that λj →∞ and Aej = λj ej , j ∈ N. In fact, in our special case, ek = 2 sin kπ· and λk = k 2 π 2 , k ∈ N. Then it is P∞ α 2 well known that D(Aα ) = D(Aα/2 ) = {v ∈ L2 (0, 1) : k=1 λk vk < ∞}, where √ R1 vk = hv, ek i = 2 0 v(x) sin kπx dx. Notice that B is the first order differential operator defined by ∂u ∂x and g : R→R, is a bounded Lipschitz continuous function on R. Let us note here that the Lipschitz condition is not restrictive. In fact, it will be easily seen from our proof that our method works when g is locally Lipschitz . {W (t), t ≥ 0} is a cylindrical Wiener process on the probability space (Ω, F, {F}t≥0 , P). The initial condition u0 is a L2 (0, 1)-valued F0 -measurable function. Let us denote by {Sα (t), t ≥ 0} the semigroup generated by −Aα . The following is now widely accepted definition, see Ref. [22]. Bu =

Definition 1.1. Suppose that 23 < α < 2. An Ft -adapted L2 (0, 1)-valued continuous process u = {u(t), t ≥ 0} is called a mild solution of equation (1.1) iff for some 2α p > α−1 E sup |u(t)|pL2 < ∞,

T >0

(1.6)

t∈[0,T ]

and for all t ≥ 0, a.s. the following identity holds Z t Z t 2 u(t) = Sα (t)u0 + Sα (t − s)Bu (s) ds + Sα (t − s)g(u(s)) dW (s). 0

(1.7)

0

In the representation of the stochastic integral in (1.7), with a slight abuse of notation, g is regarded as a nonlinear operator from H = L2 (0, 1) to L(H), the set of bounded linear operators on H, defined bya g(u)(h) = {(0, 1) 3 x 7→ g(u(x))h(x) ∈ R}. In other words, the nonlinear operator g is the Nemytski map associated with function g. For v ∈ H, g(v) is given as a multiplicative operator. From the hypothesis that g is bounded we have kg(v)k ≤ b0 , where b0 = supR |g(x)|. a Note

H.

that because g is bounded, the natural domain of this nonlinear operator is the whole space

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Remark 1.2. We will see later, in Lemmas 2.9 and 2.13 , that if an Ft -adapted L2 (0, 1)-valued continuous process u = {u(t), t ≥ 0} satisfies condition (1.6) then all terms on the RHS of (1.7) make sense. The main result of our paper is the following Theorem 1.3. Let u0 be a L2 (0, 1)-values F0 -measurable function such that E|u0 |pL2 < ∞

for some p > equation (1.1).

2α α−1

and let T > 0. Then there exists a unique mild solution of

The paper is organized in the following way. In section 2 we prove existence and uniqueness of a global solutions to the approximate versions of problem (1.1). In section 3 we construct a local maximal solution to problem (1.1) and then prove that it is in fact a global solution. 2. Existence of global solutions to approximating equations We begin with the following definition used throughout the whole paper. Definition 2.1. Let T > 0 and p ∈ [1, ∞] be fixed and X is a separable Banach space. By ZT,p (X) we denote the space of all X-valued continuous and Ft -adapted processes u = {u(t), t ∈ [0, T ]} such that kukpT,X,p := E sup |u(t)|pX < ∞.

(2.1)

t∈[0,T ]

When X = L2 (0, 1) we will usually denote the space ZT,p (L2 (0, 1)) by ZT,p and the norm k · kT,L2 (0,1),p by k · kT,p . We have the following simple but useful observation. Proposition 2.2. In the framework of Definition 2.1 the family spaces ZT,p (X) is decreasing with respect to parameter p. To be precise, if 1 ≤ p 1 ≤ p2 ≤ ∞, then ZT,p2 (X) ⊂ ZT,p1 (X) and kukT,X,p1 ≤ kukT,X,p2 , for all u ∈ ZT,X,p2 (X). Proof.

Follows immediately from Jensen’s Inequality.

We fix, for the time being, T > 0 and p > 2. Let πn be the projection from H onto the ball B(0, n) ⊂ L2 (0, 1) defined by ( v, if |v|L2 ≤ n, πn (v) = n if |v|L2 > n. |v| 2 v, L

The following result is well known fact, see Refs. [1, 23, 24].



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Lemma 2.3. The map πn : L2 (0, 1)→L2 (0, 1) is globally Lipschitz with Lipschitz constant 1. Moreover, |(πn u)2 − (πn v)2 |L1 ≤ 2n|u − v|L2 , for all u, v ∈ L2 , 2

|(πn u) |

(2.2)

2

≤ n|u| , for all u ∈ L ,

L1

L2

|(πn u)2 |L1 ≤ |u|2L2 , for all u ∈ L2 .

We fix n ∈ N and consider operators Hn , G : ZT,p → ZT,p defined by Z t Hn u(t) = Sα (t − s)B(πn u(s))2 ds

(2.3)

0

and

Gu(t) =

Z

t 0

Sα (t − s)g(u(s)) dW (s).

(2.4)

Let us recall that here g is regarded as a multiplicative operator. The above constitute a special part of the following result. Lemma 2.4. Suppose that q ∈ [2, ∞] and z ∈ Lq (0, 1). Let Z denotes the multiplication operator by z. Then, X α α 2 2 e−2π k t = 2 q |z|2Lq kSα (t)k2HS , t > 0, kSα (t)Zk2HS ≤ 2 q |z|2Lq k

where by k.kHS we denote the Hilbert-Schmidt norm. √ α/2 Proof. Let ek = 2 sin kπ· and µk = λk , where λk = k 2 π 2 , k ∈ N. Then, as explained earlier, {ek } is an ONB of H and Aα ek = µk ek , k ∈ N. Then, because both Sα (t) and Z are self-adjoint on H it follows from [25], section V.2.4, kSα (t)Zk2HS =

X k

|ZSα (t)ek |2L2 =

X k

|ze−µk t ek |2L2 =

X k

e−2µk t |zek |2L2 .

Let us observe that by the H¨ older inequality, |zek |L2 ≤ |z|Lq |ek |Lr , where r1 + q1 = 21 . Moreover, since |ek |L2 = 1 and |ek |L∞ = 21/2 it follows by applying the H¨ older 1/q inequality that |ek |Lr ≤ 2 . Therefore, X 2 e−2µk t , kSα (t)Zk2HS ≤ 2 q |z|2Lq k

what proves the result.



The above Lemma implies the following result. Corollary 2.5. Suppose that z ∈ Lq (0, 1) for q ∈ [2, ∞]. Let Z denotes the multiplication operator by z. Then, for β < 1 − α1 , one has Z t Z ∞ −β 2 s kSα (s)ZkHS ds ≤ s−β kSα (s)Zk2HS ds 0

0

2

≤ 2 q |z|2Lq (2π α )β−1 Γ(1 − β)

+∞ X

k=1

k −α(1−β) < ∞, t > 0. (2.5)

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Proof.

From the Fubini Theorem we have Z

∞ 0

2

s−β kSα (s)Zk2HS ds ≤ 2 q |z|2Lq =2 ≤2

β−1

2 q

2

2 q

Z

∞

s−β

0

|z|2Lq

+∞ X

α

e−2s(kπ) ds

k=1 +∞ X

(kπ)

αβ−α

k=1

|z|2Lq (2π α )β−1 Γ(1

− β)

Z

∞

τ −β e−τ dτ

0

+∞ X

k −α(1−β) .

k=1

Since by our assumptions α(1 − β) > 1 the series on the RHS above is convergent and the result follows.  Remark 2.6. us observe that the constant (2π α )β−1 Γ(1 − β) R ∞ Let −β is equal to 0 s kSα (s)k2HS ds.

P+∞

k=1

k −α(1−β)

Let us recall that for 0 < δ ≤ 1 the fractional power Aδα of Aα , see Ref. [19], is defined as the inverse of operator A−δ α defined by A−δ α =

1 Γ(δ)

Z

∞

tδ−1 Sα (t)dt.

0


δ −δ −1 Since, see Lemma 2.6.6 in Ref. [19], the operator A−δ is α is injective, Aα := Aα well defined. Furthermore, it is known that Aδα is a closed densely defined operator and that D(Aδα ) = R(A−δ α ), where R(Λ) denotes the range of an operator Λ. The following result is just Lemma 3.3 from [26], see also [22], [27] and [28]. We state and prove it here for the convenience of the reader. Lemma 2.7. Provided that γ > p−1 + δ the operator Rγ : Lp (0, T ; L2(0, 1)) → C([0, T ]; D(Aδα )) given by Rγ h(t) =

Z

t 0

(t − s)γ−1 Sα (t − s)h(s) ds, h ∈ Lp (0, T ; L2 (0, 1)),

is well defined, linear and bounded. Moreover, there exists a constant C p,γ−δ > 0 such that for all h ∈ Lp (0, T ; L2 (0, 1)) 1

|Rγ h|C([0,T ];D(Aδα )) ≤ Cp,γ−δ T γ−δ− p |h|Lp (0,T ;L2 (0,1)) . Proof.

Let us fix h ∈ Lp (0, T ; L2(0, 1)). Then for t ∈ (0, T ) we have

(2.6)

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|Aδα

|Rγ h(t)|D(Aδα ) =

Z

≤

Z

≤

t 0 t 0

Z

t 0

(t − s)γ−1 Sα (t − s)h(s) ds|L2

(t − s)γ−1 |Aδα Sα (t − s)h(s)|L2 ds (t − s)γ−1 kAδα Sα (t − s)k|h(s)|L2 ds.

Since Aα is self-adjoint it follows by Lemma 2.10 from [29] that kAδα Sα (s)k ≤ s−δ , s > 0. Hence by applying the H¨ older inequality we get Z t |Rγ h(t)|D(Aδα ) ≤ (t − s)γ−δ−1 |h(s)|L2 ds 0

≤ The integral

RT 0

Z

T

s

p (γ−δ−1) p−1

ds

0


p−1 p

Z

T

0

|h(s)|pL2 ds


p1

.

p

s(γ−δ−1) p−1 ds < ∞ when γ > p−1 + δ. Hence for all t ≥ 0

|Rγ h(t)|D(Aδα ) ≤ (

1 − 1/p 1− p1 γ−δ− p1 ) T |h|Lp (0,T ;L2 (0,1)) . γ − δ − 1/p



Remark 2.8. When δ = 0 then Rγ : Lp (0, T ; L2(0, 1))→C([0, T ]; L2 (0, 1)) is a bounded linear operator and because the semigroup {Sα (t)}t≥0 is contractive, Cp,1 = 1. Lemma 2.9. If g : R→R is linear growth, i.e. for some b0 , b1 ≥ 0 |g(x)| ≤ b0 + b1 |x|,

x ∈ R,

(2.7)

2α , then the nonlinear operator G : ZT,p → ZT,p is well defined α > 1 and p > α−1 and for u ∈ ZT,p one has

kGukT,p ≤ [Cp,γ b0 +

√

bp,γ T γ , 2Cp,γ b1 kukT,p ]C

(2.8)

where a positive number γ is such that 1p < γ < 21 (1 − α1 ) and Cp,γ is the constant
p −α(1−2γ) 2 b p := Cp (2π α )2γ−1 Γ(1 − 2γ) P . from inequality (2.6) and C p,γ k≥1 k Moreover, if g is a Lipschitz function, i.e. for some b2 ≥ 0 |g(x2 ) − g(x1 )| ≤ b2 |x2 − x1 |,

x1 , x2 ∈ R,

(2.9)

then G : ZT,p → ZT,p is Lipschitz continuous and for all u1 , u2 ∈ ZT,p the following inequality holds √ bp,γ b2 T γ ku2 − u1 kT,p . (2.10) kGu2 − Gu1 kT,p ≤ 2Cp,γ C

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Proof. We begin with an observation that a number γ such that p1 < γ < 21 (1− α1 ) 2α . We choose it and fix for the exists thanks to one of our assumptions that p > α−1 remaining parts of the proof. We first consider the case when g is bounded, i.e. g satisfies the condition (2.7) with b1 = 0. Let u ∈ ZT,p . By Lemma 2.4 we have Z

t 0

kSα (t − s)g(u(s))k2HS ds ≤

Z

t

kSα (t − s)k2HS kg(u(s))k2L∞ ds Z t ≤ b20 kSα (s)k2HS ds. 0

(2.11)

0

Since α > 1 and so 0 < 1 − α1 by Corollary 2.5 we infer that for all t > 0, the integral on the RHS of the inequality (2.11) is convergent. Hence the Itˆ o stochastic Rt integral 0 Sα (t − s)g(u(s)) dW (s) exists and thus the process G(u) is well defined. We shall prove that in fact Gu ∈ ZT,p . Employing the factorization method (see Refs. [1, 22, 27]), we can rewrite Gu(t) in the following form

Gu(t) = Rγ Y (t) = where sin πγ Y (s) = π

Z

s 0

Z

t

(t − s)γ−1 Sα (t − s)Y (s) ds, t ∈ (0, T ],

(2.12)

(s − r)−γ Sα (s − r)g(u(r)) dW (r), s ∈ [0, T ].

(2.13)

0

Arguing as above, we see that because 2γ < 1 − α1 , the RHS of (2.13) is a well defined L2 (0, 1)-valued stochastic Itˆ o integral. Furthermore, since γ > p−1 , by Lemma 2.7, a process Gu(t) = [Rγ Y ](t), t ∈ [0, T ] is continuous H-valued provided we can show that a.s. Y ∈ Lp (0, T ; H). In fact, we will show below that a stronger fact holds, namely that Y ∈ Lp (Ω; Lp (0, T ; H)). Indeed by the Burkholder’s inequality and Corollary 2.5 there exists a constant Cp > 0 such that

Z

T 0

p E Y (s) L2 ds ≤ Cp

Z

T

E 0

≤ Cp bp0

Z

s 0

Z T Z 0

(s − r)−2γ kSα (s − r)g(u(r))k2HS dr

s 0

r−2γ kSα (r)k2HS dr

 p2

 p2

ds

ds

+∞   p2 X b p bp T. =C ≤ Cp bp0 T (2π α )2γ−1 Γ(1 − 2γ) k −α(1−2γ) p,γ 0 k=1

(2.14)

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1 what implies that α(1 − 2γ) > 1 we infer that the last term Since 0 < γ < 12 − 2α is finite. Furthermore, by inequalities (2.6) and (2.14), we get


sup |Gu(t)|pL2 = E

E

t∈[0,T ]

≤



sup |Rγ Y (t)|pL2 = E |Rγ Y |pC(0,T ;L2 (0,1) ,

t∈[0,T ]

p Cp,γ T γp−1 E|Y |pLp (0,T ;L2 (0,1)

p bp ≤ Cp,γ Cp,γ bp0 T γp < ∞.

This concludes the proof of inequality (2.8) in the special case considered. Next, we should consider the case b0 = 0. However, the proof in this case is in fact a special case of the proof of the Lipschitz property of the map G given below and hence will be omitted. Moreover, the general case of inequality (2.8) is a consequence of the just proved special case and the proof of the inequality (2.10). Finally, we shall deal with the Lipschitz property of the map G. Take u2 , u1 ∈ ZT,p . From the 1st part of the proof we know that G(u2 ) and G(u1 ) both belong to ZT,p . However, in order to estimate the norm in of G(u2 ) − G(u1 ) we need to use Lemma 2.4 with q = 2 while in the 1st part of the proof we used it with q = ∞. 1 and introduce an auxiliary process h We choose γ such that p−1 < γ < 21 − 2α by Z sin πγ s (s − r)−γ Sα (s − r)(g(u2 (r)) − g(u1 (r)) dW (r), t ≥ 0. h(s) = π 0 Then we have Gu2 (t)−Gu1 (t) = Rγ h(t), where Rγ is the operator defined in Lemma 2.7. By the Burkholder inequality in view of Lemma 2.4 we have the following sequence of inequalities.

≤ π −p ≤ Cp

Z

Z

T

0 T

E 0

p

≤ Cp 2 2

p

Z

E

Z T

E 0

≤ Cp 2 2 bp2 p

Z

Z

T 0

Z

s

0 s

0

E

(2.15)

p (s − r)−γ Sα (s − r)(g(u2 (r)) − g(u1 (r)) dW (r) L2 ds

(s − r)

Z

T 0

E h(s)|pL2 ds −2γ

kSα (s − r)(g(u2 (r)) −

g(u1 (r))k2HS dr

s 0

 p2

ds

(s − r)−2γ kSα (s − r)k2HS |(g(u2 (r)) − g(u1 (r))|2L2 dr

Z

s 0

(s − r)−2γ kSα (s − r)k2HS |u2 (r) − u1 (r)|2L2 dr

≤ Cp 2 2 bp2 E[sup |u2 (r) − u1 (r)|pL2 ] [0,T ]

b p 2 p2 bp T ku − vkp < ∞. ≤C p,γ 2 T,p

Z T Z 0

s 0

r−2γ kSα (r)k2HS dr

 p2

 p2

ds

 p2

ds

ds

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Therefore, by Lemma 2.7 we have that p kGu − GvkpT,p = E|Rγ h|pC(0,T ;L2 ) ≤ Cp,γ T γp−1 E|h|pLp (0,T ;L2 ) p

p bp Cp,γ bp2 T γp ku − vkpT,p . ≤ 2 2 Cp,γ

(2.16) 

Remark 2.10. In the proof of the previous Lemma we used Lemma 2.7 with δ = 0. If instead we used it with δ ≥ 0 such that δ + p1 < 21 (1 − α1 ) then by choosing γ such that 1 1 1 (2.17) δ + < γ < (1 − ) p 2 α we would prove that G maps the space ZT,p (L2 ) into the space ZT,p (D(Aδα )). Moreover, under some appropriate assumptions the inequalities (2.8) and (2.10) take the following forms, √ γ−δ , kGukZT,p (D(Aδα )) ≤ [Cp,γ−δ b0 + 2Cp,γ−δ b1 kukT,p ]Cd p,γ T √ γ−δ bp,γ b2 T ku2 − u1 kT,p . kGu2 − Gu1 kZT,p (D(Aδα )) ≤ 2Cp,γ−δ C

Lemma 2.11. For each α > 32 there exists a constant Cα > 0 such that for all t > 0 and for any bounded and strongly-measurable function v : (0, t)→L 1 (0, 1) the following inequality holds Z t 3 ∂v (2.18) |Sα (t − s) (s)|L2 ds ≤ Cα t1− 2α sup |v(s)|L1 . ∂x s≤t 0

Proof.

We begin by showing that if v ∈ H 1,2 (0, 1) then  21 √ X α ∂v |L2 ≤ 2π 2πk 2 e−2s(kπ) |v|L1 , s > 0. ∂x +∞

|Sα (s)

(2.19)

k=1

By the Parseval Identity, the fact that | cos(kπ·)|L∞ ≤ 1 and by integration by parts formula we have |Sα (s)

+∞ X α ∂ ∂v 2 |L 2 = e−2s(kπ) h v, ek i2 ∂x ∂x k=1 Z 1 +∞ X
2 α √ ∂v (x) sin(kπx) dx 2π ≤ e−2s(kπ) 0 ∂x k=1 Z +∞ 1 X √
2 α v(x) cos kπx dx ≤ e−2s(kπ) k 2π 0

k=1

≤ 2π 2

+∞ X k=1

α

k 2 e−2s(kπ) |v|2L1 .

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This proves (2.19). Hence, if v ∈ L∞ (0, t; L1 (0, 1)), then Z t +∞  21 √ Z tX α ∂v k 2 e−2(t−s)(kπ) |v(s)|L1 ds |Sα (t − s) (s)|L2 ds ≤ 2π ∂x 0 0 k=1 Z tX +∞  21 √ α 1 k 2 e−2(t−s)(kπ) ds. ≤ 2π sup |v(s)|L s≤t

0

(2.20)

k=1

We need the following auxiliary result.

Lemma 2.12. Let us define a function h : (0, ∞)→R by formula h(s) =

∞ X

k=1

α

k 2 e−2s(kπ) , s ∈ (0, ∞).

Then there exists C > 0 such that

h(s) ≤ Cs−3/α ,

s > 0.

Proof. [Proof of Lemma 2.12] The function h is the Laplace transform of a purely P∞ P 2 1 atomic measure µ = k=1 k 2 δ2(kπ)α . Let U (σ) := µ((0, σ]) = 1 σ α k . Then, k≤ π ( 2 )

by Ref. [30], we have µ((0, σ]) =

X

k2 = 1

1 1 σ 1 1 σ 1 1 σ 1 [ ( ) α ]([ ( ) α ] + 1)(2[ ( ) α ] + 1) 6 π 2 π 2 π 2

1 σ α (2) ] k≤[ π 3

U (σ) ∼

3

2− α σ α 1 −1 σ 1 3 [π ( ) α ] = , 3 2 3π 3

as σ→∞,

where [·] denotes the integer part, we infer that for any y > 0, limσ→∞ Therefore the function L defined by L(x) =

U (x) 3

xα

U (σy) U (σ)

3

= yα.

, x > 0, varies slowly at infinity

and hence by the Abelian theorem, see Theorem XIII.5.2 in Ref. [31] we infer that h(s) ∼

Γ( α3

3 1 s− α L(s−1 ), as s → 0. + 1) −3

In particular, since limx→∞ L(x) = 23πα3 , there exists t0 > 0 and C0 > 3 3 0 such that h(s) ≤ C0 s− α , for s ∈ (0, t0 ]. Since the function h(s)s α , 3 3 s ∈ (0, ∞) is continuous and the lims→0 h(s)s α and lims→∞ h(s)s α = P∞ α 3 lims→∞ s α e−2s k=1 k 2 e−2s((kπ) −1) exist, then it is bounded on (0, ∞), hence it is on [0, T ].  Combining Lemma 2.12 with inequality (2.20) we infer that Lemma 2.11 is valid for v ∈ H 1,2 (0, 1). Hence the result follows by standard density argument.  Proof.

[An alternative proof of Lemma 2.12.] h(s) =

∞ X

k=1

α

k 2 e−2s(kπ) , s ∈ (0, ∞).

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Since for each N > 0 there exists CN > 0 such that e−x ≤ CN x−N , x > 0 we have, for some N > 0,

h(s) ≤

∞ X

k 2 (2s(kπ)α )−N =

k=1

∞ bN X C k 2−N α sN k=1

Taking N > α3 (so that the series exists C˜N > 0 such that

h(s) ≤

P∞

k=1

k 2−N α is convergent) we infer that there

C˜N , sN

s ∈ (0, ∞).

Inserting the last inequality into (2.20) yields Z

t 0

Z t ∂v 3 |Sα (t − s) (s)|L2 ds ≤ C sup |v(s)|L1 s− 2α ds. ∂x s≤t 0 3 1 1− 2α 1 ≤C . 3 sup |v(s)|L t 1 − 2α s≤t



Lemma 2.13. Assume that α > 32 and T > 0. Then the nonlinear operator Hn : ZT,p → ZT,p is well defined and globally Lipschitz. Moreover, for all n ∈ N and u, v ∈ ZT,p E sup t∈[0,T ]

E sup t∈[0,T ]

Z

t

0

Z

t 0

p 3 Sα (t − s)B(πn u(s))2 ds L2 ≤ np Cαp T p(1− 2α ) E sup |u(t)|pL2 ,

Sα (t − s)B(πn u(s))2 ds −

E sup t∈[0,T ]

≤ (2n) Z

t 0

p

Z

t∈[0,T ]

t

0

p Sα (t − s)B(πn v(s))2 ds L2

3 Cαp T p(1− 2α ) E

sup |u(t) − v(t)|pL2 ,

t∈[0,T ]

p/2 p 3 Sα (t − s)B(u(s))2 ds L2 ≤ Cαp/2 T 2 (1− 2α ) E sup |u(t)|pL2 , t∈[0,T ]

(2.21)

where Cα is the constant appearing in the inequality (2.18). Proof. Take u, v ∈ ZT,p . It is enough to assume that u, v are deterministic functions and u, v ∈ C([0, T ], L2 ). Then, by Lemma 2.11 and the second inequality

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in Lemma 2.3 we have Z Z t p t p Sα (t − s)B(πn u(s))2 2 ds sup Sα (t − s)B(πn u(s))2 ds L2 ≤ sup L t∈[0,T ]

t∈[0,T ]

0

≤

≤

3 Cαp T p(1− 2α )

0

sup |(πn u(s))2 |pL1 ,

s∈[0,T ]

3 Cαp T p(1− 2α ) np

sup |u(t)|pL2 ,

t∈[0,T ]

hence, we get the first inequality and that the operator Hn is well defined. To prove the second inequality, we use again Lemma 2.11 and the first inequality in Lemma 2.3 and arguing as above we get Z t
p sup Sα (t − s)B (πn u(s))2 − (πn v(s))2 ds L2 t∈[0,T ]

0

3

≤ Cαp T p(1− 2α ) sup |(πn u(s))2 − (πn v(s))2 |pL1 , s∈[0,T ]

≤

3 Cαp T p(1− 2α ) (2n)p

sup |u(t) − v(t)|pL2 .

t∈[0,T ]

Hence we conclude that Hn is globally Lipschitz. We finish with the proof of the third inequality. As before it is enough to assume that u ∈ C([0, T ], L2 ) is a deterministic function. Then, by Lemma 2.11 we have Z Z t p/2 t p/2 Sα (t − s)B(u(s))2 2 ds sup Sα (t − s)B(u(s))2 ds L2 ≤ sup L t∈[0,T ]

t∈[0,T ]

0

≤

Cαp/2 T

p 3 2 (1− 2α )

≤

Cαp/2 T

p 3 2 (1− 2α )

0

p/2

sup |(u(s))2 |L1 ,

s∈[0,T ]

sup |u(t)|pL2 .

t∈[0,T ]

 It follows from Lemmas 2.9 and 2.13 that a map Φn : ZT,p 3 u = Sα (·)u0 + [Hn u](·) + [Gu](·) ∈ ZT,p

(2.22)

is well defined. We are now ready to formulate the main result of this section. 2α and that the function g : R→R is Theorem 2.14. Assume that α > 23 , p > α−1 Lipschitz. Then the problem   du(t) = − Aα u(t) + B[(πn u(t))2 ] dt + g(u(t)) dW (t), t > 0, (2.23) u(0) = u0 ,

has a unique global mild solution un = {un (t), t ≥ 0} such that E sup |un (t)|pL2 < ∞, for each T > 0. t∈[0,T ]

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Proof. Let us fix n ∈ N. We will prove that the operator Φn is a strict contraction provided T > 0 is small enough. Then by the Banach Fixed Point Theorem there exists a unique un ∈ ZT,p such that Φn (un ) = un . It is standard to show that then un is a mild solution to problem 2.23 on the time interval [0, T ]. It is also standard, see Ref. [26] that this solution can be extended to the whole half-line [0, ∞) and that the extension, denoted also by un is a fixed point of Φn for each T > 0 and in consequence is a mild solution to problem 2.23 on the time interval [0, ∞). Let us take and fix u, v ∈ ZT,p . Then by Lemma 2.13 we have Z t p   kHn u − Hn vkp = E sup Sα (t − s)(B(πn u(s))2 − B(πn v(s))2 ) ds 2 T,p

t∈[0,T ]

L

0

3

≤ 2p Cαp np T p(1− 2α ) ku − vkpT,p .

Taking T ≤ (6Cα n)

2α 3−2α

(2.24)

, we get

1 ku − vkT,p . 3 Next, from inequality√ (2.10) in Lemma 2.9 we infer that G is Lipschitz continuous bp,γ b2 T γ , where b2 is the Lipschitz constant of the with Lipschitz constant 2Cp,γ C function g. √ bp,γ b2 )− γ1 , we infer that kGu − GvkT ≤ 1 ku − vkT . Taking T ≤ (3 2Cp,γ C 3 √ 2α bp,γ b2 )− γ1 , (6nCα ) 3−2α }, we get From (2.24) and (2.16) and for T ≤ min{(3 2Cp,γ C kΦn u − Φn vkT ≤ 23 ku − vkT . Thus Φn is a contraction on ZT,p and so there exists a unique fixed point un ∈ ZT,p of Φn .  kHn u − Hn vkT,p ≤

Concerning regularity of solutions to problem (2.23) we have the following result, a standard proof of which will be omitted. Proposition 2.15. The solution {un (t), t ≥ 0} of equation (2.23) belongs to the 3 < 1 and δ + p−1 < space C((0, T ]; D(Aδα )) ∩ C([0, T ]; L2(0, 1)) provided δ + 2α 1 1 2 (1 − α ). 3. Global solutions to Burgers equations In this section we assume that the function g is bounded and Lipschitz, i.e. condition (2.7) with b1 = 0 and condition (2.9) are satisfied. Let us define a stopping time τn by τn = inf {t > 0, |un (t)|L2 ≥ n}. Let n ≥ m, for t ≤ τm , P a.s. {um (s), s ≤ t} is solution of Equation (2.23). The same argument as in Ref. [26], compare with Lemma 4.11 and the proof of Theorem 4.10, based on the uniqueness part of Theorem 2.14, we getb P ⊗ ds a.s, for all 0 ≤ t ≤ τn , un (t) = um (t). Hence the sequence {τn }n>0 is increasing. Let τ∞ := lim τn . We n→∞

define the function u(t) for t ≤ τ∞ by u(t) = un (t), t ≤ τn . Then we also have that τn = inf {t > 0, |u(t)|L2 ≥ n}, n ∈ N.

b The

equality is in L2 (0, 1).

(3.1)

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Then, see Theorem 1.5 in Ref. [32], Theorem 4.10 in Ref. [26] , [33] or [34], u(t), t ≤ τ∞ is a unique maximal local (mild) solution to (1.1) with lifespan τ∞ , i.e. τ∞ is an accessible, strictly positive P-almost surely stopping time, (u(t), t < τ∞ ) is a progressively measurable process satisfying lim sup |u(t)|L2 = +∞ P-almost surely on {τ∞ < ∞}, t%τ∞

and u(t∧τn ) = Sα (t ∧ τn )u(0)+

Z

t∧τn

0

Sα (t ∧ τn − r)B(u(r)2 ) dr+Iτn (g(u))(t∧τn ), t ≥ 0 (3.2)

for all n ∈ N, where we define Iτn (g(u))(t) =

Z

t 0

1[0,τn ) (r)Sα (t − r)g(u(r ∧ τn )) dW (r), t ≥ 0.

For the choice of the process Iτn (g(u)) see Proposition C.1.1. Note that the process u has continuous paths, u ∈ C([0, τ∞ ) ; L2 ) P-almost surely. To prove that it is the global solution it is sufficient to prove that τ∞ = ∞. Let us consider, for n ∈ N, a process zn defined byc Z t zn (t) = 1[0,τn ) (s)Sα (t − s)g(un (s)) dW (s), t ≥ 0, (3.3) 0 Z t = 1[0,τn ) (s)Sα (t − s)g(u(s)) dW (s), t ≥ 0. 0

Let vn be a process defined pathwise as a solution to the following initial value problem (

dvn (t) dt

= −Aα vn (t) + B(vn (t) + zn (t))2 , t > 0, v(0) = u0 .

(3.4)

It follows from Corollary C.1.9 that the processes zn (t), t ≥ 0 defined by (3.3) satisfy the following. If q ∈ [2, ∞), p ∈ [1, ∞) and 1 1 1 < β < (1 − ) − δ, p 2 α

(3.5)

then there exists C > 0 such that for all T > 0 and all n ∈ N, E sup |zn (t)|pH αδ,q (0,1) ≤ CT pβ .

(3.6)

t∈[0,T ]

The following result follows directly from the definitions of the local solution and of the process zn . c The

second equality below follows as P-a.s. un (s) = u(s) for s ∈ [0, τn ).

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Lemma 3.1. If n ∈ N, then u(t ∧ τn ) = zn (t ∧ τn ) + vn (t ∧ τn ), t ≥ 0.

(3.7)

Let us also quote the following consequence of a much more general result from a monograph by Runst & Sickel [35], see Theorem 2 in section 4.4.4, Theorem 7.48 in Ref. [36] and section 2.3.5 and Remark 4.4.2/2 p. 324 in Ref. [37]. Proposition 3.2. Assume that 0 < s2 < 21 and s1 > max{s2 , q11 }, q1 ∈ [2, ∞). Then the pointwise multiplication between spaces H s1 ,q1 (0, 1) and H s2 ,2 (0, 1) is well defined and moreover, there exists C > 0 such that |uv|H s2 ,2 ≤ C|u|H s1 ,q1 |v|H s2 ,2 , u ∈ H s1 ,q1 (0, 1), v ∈ H s2 ,2 (0, 1).

(3.8)

We will also need the following results. Proposition 3.3. Assume that β ∈ ( 12 , 1). Then there exists a constant C > 0 such that for each u ∈ H0β,2 (0, 1) and each v ∈ H 1−β,2 (0, 1) the following inequality is satisfied Z 1 | u(x)Dv(x) dx| ≤ C|u|H β,2 |v|H 1−β,2 . (3.9) 0

0

Lemma 3.4. Suppose that T > 0, a function z : [0, T ]→H s,q (0, 1), with s and q α ,2 such that 1q < s < 21 , is continuous, and that a function v ∈ C(0, T ; H02 ) is a solution to ( dv(t) 2 dt = −Aα v(t) + B(v(t) + z(t)) , t > 0, (3.10) v(0) = u0 . Then, there exists a generic constant C > 0 such that α

α−1 ln+ |v(t)|L2 ≤ ln+ |v(0)|L2 + C(1 + ln+ t) + ln+ sup |z(τ )|H s,q

0≤τ ≤t

+ ln

+

α   α−1 sup |z(τ )|4H s,q + |z(τ )|4H 1−α/2,2 + sup |z(τ )|H s,q , (3.11)

0≤τ ≤t +

0≤τ ≤t

where ln x = max{0, ln x}.

Proof. [Proof of Proposition 3.3.] Let {en } be the ONB of L√2 (0, 1) as before and let {fn , n ≥ 0} be an ONB in L2 (0, 1), given by fn (x) = 2 cos nπx for n ≥ 1 and f0 (x) = 1. Assume that u ∈ H0β,2 (0, 1) and v ∈ C01 (0, 1), where C01 (0, 1) is the set of differentiable functions on (0, 1) with compact support. So that hen , Dvi = −nπhfn , vi. b the -Laplacian with the Neumann boundary conditions, i.e. Let us denote by A a linear self-adjoint operator defined by the following b = H 2,2 (0, 1) D(A) b A(u) = −∆u, u ∈ H 2,2 (0, 1).

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b k ) = (kπ)2 fk , k ∈ N and that D(A b 21 −β/2 ) = H 1−β,2 . Further, It is known that A(f β from formula (1.5) we have D(A 2 ) = H0β,2 (0, 1). Then, by the Plancherel Theorem we have the following sequence of inequalities |

Z

1 0

u(x)Dv(x) dx| = | =| ≤(

∞ X k=1 ∞ X k=1 ∞ X k=1

hu, ek ihek , Dvi| = |

∞ X k=1

−kπhu, ek ihv, fk i|

(kπ)β hu, ek i(kπ)1−β hv, fk i| 1

(kπ)2β hu, ek i2 ) 2 (

∞ X k=1

1

(kπ)2−2β hv, fk i2 ) 2

≤ C|u|H β,2 |v|H 1−β,2 (0, 1). 0

Using the density of

C01 (0, 1)

in

H01−β,2

= H 1−β,2 , we get the result.



Proof. [Proof of Lemma 3.4.] We begin with an observation that in view of α ,2 Proposition 3.3 (B(v 2 ), v)L2 = 0 for all v ∈ H02 . Indeed this is true for v ∈ α α ,2 ,2 C01 (0, 1), the space C01 (0, 1) is dense in H02 and, see Ref. [38], the space H02 is an algebra with pointwise multiplication. Next, applying Lemma III.1.2 from [39] the solution v of problem (3.10) we get 1 d |v(t)|2L2 = −hAα v(t), v(t)i + 2hB(z(t)v(t)), v(t)i + hBz 2 (t), v(t)i. 2 dt Since Aα = Aα/2 , where A is positive self-adjoint we infer that

(3.12)

hAα v, vi = hAα/2 v, vi = |Aα/4 v|2L2 , v ∈ D(Aα ).

Since α ∈ ( 32 , 2) so that α4 ∈ ( 83 , 21 ) in view of equality (1.5) we infer that there exists C > 0 such that |Aα/4 v|2L2 ≥ C|v|2 α2 ,2 , for v ∈ D(Aα/2 ). Therefored, H0

ν1 hAα v, vi ≥ |v|2 α2 ,2 , v ∈ D(Aα ). 2 H0

(3.13)

Applying Proposition 3.3 with β = α/2 we infer that |hB(zv), vi| ≤ C|v|H α/2,2 |zv|H 1−α/2,2 0

Let us choose positive numbers s and q such that q1 < s < 21 . Since α ∈ ( 23 , 2), we infer that 1 − α2 < 21 and therefore by Proposition 3.2, formula (3.8) we infer that that there exists a constant C > 0 such that 4|hB(zv), vi| ≤ C|v|H α/2,2 |z|H s,q |v|H 1−α/2,2 0

d One

α ,2

can choose an equivalent norm on H02 is equal to 1.

such that the constant ν1 > 0 in inequality (3.13)

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Moreover, since 0 < 1 − α2 < definition (1.3) we infer that

α 2

and H 0,2 (0, 1) = L2 (0, 1), by the interpolation and 2− 2

2

α

−1

,2

|v|H 1− α2 ,2 ≤ |v|L2 α |v| α α/2,2 , v ∈ H02 (0, 1). H0

Since α > 1 we infer by applying the classical Young inequality that for some generic constant C > 0 α ν1 α−1 2 (3.14) 4|hB(zv), vi| ≤ |v|2H α/2,2 + C|z|H s,q |v|L2 . 4 0 Applying again Proposition 3.3 with β = α/2 we infer that |hB(z 2 ), vi| ≤ C|v|H α/2,2 |z 2 |H 1−α/2,2 . 0

Arguing as before and with the same choice of constants we have 2|hB(z 2 ), vi| ≤ C|v|H α/2,2 |z|H s,q |z|H 1−α/2,2 0 ν1 2 (3.15) ≤ |v|H α/2,2 + C|z|4H s,q + C|z|4H 1−α/2,2 . 4 0 Combining inequalities (3.12) and (3.13) with inequalities (3.14) and (3.15) we infer that ν1 d |v(t)|2L2 ≤ − |v(t)|2H α/2,2 dt 2 α

α−1 2 4 4 +C|z(t)|H s,q |v(t)|L2 + C|z(t)|H s,q + C|z(t)|H 1−α/2,2 , t ≥ 0. (3.16)

Since there exists a constant ν2 > 0 such that |v|2

α/2,2

H0

α/2,2

H0

(0, 1), we infer that with ν :=

≥ ν2 |v|2L2 , u ∈

ν1 ν2 2 ,

α d α−1 2 |v(t)|2L2 ≤ (−ν + C|z(t)|H s,q )|v(t)|L2 dt + C|z(t)|4H s,q + C|z(t)|4H 1−α/2,2 , t ≥ 0.

(3.17)

Applying finally the Gronwall Lemma we obtain that for t ≥ 0, Rt

α α−1

|v(t)|2L2 ≤ |v(0)|2L2 e 0 (−ν+C|z(τ )|H s,q ) dτ Z t α   Rt α−1 +C |z(σ)|4H s,q + C|z(σ)|4H 1−α/2,2 e σ (−ν+C|z(τ )|H s,q ) dτ dσ, (3.18) 0

α α−1

≤ |v(0)|2L2 e−νt+Ct sup0≤τ ≤t |z(τ )|H s,q α   α−1 + Ct sup |z(τ )|4H s,q + |z(τ )|4H 1−α/2,2 eCt sup0≤τ ≤t |z(τ )|H s,q . 0≤τ ≤t

Applying the increasing function ln+ x = max{0, ln x} to both sides of the above inequality and then using the classical inequalities ln+ (a+b) ≤ ln+ (a)+ln+ (b)+ln 2 and ln+ (ab) ≤ ln+ (a) + ln+ (b), we get the result. 

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Proof. [Completion of the proof of Theorem 1.3.] To finish the proof we will employ the Hasminski’s method as in Ref. [34], see Ref. [40], Theorem III.4.1, for the finite-dimensional case. Let us recall that  τk = t ≥ 0; |u(t)|L2 ≥ k ,

k ∈ N.

In order to prove that τ∞ = +∞ P-almost surely it suffices to find a Lyapunov function V : L2 (0, 1) −→ R and a function C : [0, ∞)→(0, ∞) satisfying V ≥ 0 on L2 (0, 1),

qR :=

inf

|u|L2 ≥R

(3.19)

V (u)→∞ as R→∞,

EV (u(0)) < ∞

(3.20) (3.21)

and


EV (u(t ∧ τk )) ≤ C(t) 1 + EV (u(0)) for all t ≥ 0, k ∈ N.

(3.22)

Inequality (3.22) implies easily that

so that

 1 P τk < t ≤ E1{τk
k→∞

for each fixed t ≥ 0, and P{τ∞ < t} = 0 follows. Set V (u) = ln+ |u|L2 ,

u ∈ L2 (0, 1).

Obviously V is uniformly continuous on bounded sets and satisfies (3.19), (3.20). Moreover, (3.22) follows from (3.18) and (3.21) is obviously satisfied for initial condition u0 such that E ln+ |u0 |L2 < ∞.  4. Proof of uniqueness Suppose that two processes u1 (t), t ≥ 0 and u2 (t), t ≥ 0 are solutions to problem (1.1) with the same initial data u0 . We may suppose that both satisfy condition 2α . Let us define a stopping time τR = τR1 ∧ τR2 , (1.6) with the same constant p > α−1 where for i = 1, 2 and R > 0, τRi = inf {t > 0, |ui (t)|L2 ≥ R}, n ∈ N. Then for all R > 0 each ui satisfies the stopped version of equation (1.7), i.e.

(4.1)

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i

u (t ∧ τn ) = Sα (t ∧ τR )u0 +

Z

t∧τR 0

Sα (t ∧ τR − r)B(ui (r))2 dr

+ IτR (g(ui ))(t ∧ τR ), a.s., t ≥ 0 Z t i IτR (g(u ))(t) = 1[0,τR ) (r)Sα (t − r)g(ui (r ∧ τR )) dW (r), 0

(4.2) a.s., t ≥ 0.

Subtracting the equations (4.2) for i = 2, 1 and denoting by u the difference between u2 and u1 we infer that for t ≥ 0, a.s.

u(t ∧ τR ) =

Z

t∧τR 0

  Sα (t ∧ τR − r)B (u2 (r))2 − (u1 (r))2 dr

+ JτR (t ∧ τR ),

JτR (t) := IτR (g(u2 ) − g(u1 ))(t) Z t = 1[0,τR ) (r)Sα (t − r)g(u2 (r ∧ τR )) − g(u1 (r ∧ τR )) dW (r). 0

Because |u(t)|L2 ≤ R for t ∈ [0, τR ], the proof can be easily concluded by applying Gronwall Lemma in conjunction with Lemmata 2.11 and 2.13. Details of this idea are presented below. We shall prove that E|u(t ∧ τR )|pL2 = 0. i.e u(t) = 0 a.s on {t ≤ τR }. In fact, using Lemma A.1 from [34] and the semigroup property, we get, for some Cp > 0

  Rt E|u(t ∧ τR )|pL2 ≤ Cp E| 0 1(0,τR ) Sα (t − r)B (u2 (r ∧ τR ))2 − (u1 (r ∧ τR ))2 dr|pL2 Rt +Cp E| 0 1[0,τR ) (r)Sα (t − r)g(u2 (r ∧ τR )) − g(u1 (r ∧ τR )) dW (r)|pL2 . (4.3) Put the following locally integrable functions:

1

ϕ1 (s) := h 2 (s) =

X

k 2 e−2s(kπ)

k≥1

ϕ2 (r) :=

XZ

k≥1

r

α

 21

, s > 0, α

ξ −2γ e−2ξ(kπ) dξ, r > 0

0

ϕ := max{ϕ1 , ϕ2 }. Arguing as above, using Lemmata 2.6, 2.7 and 2.8 and thanks to the assumptions above, we get the following sequence of inequalities

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E|

Z

t

  1[0,τR ) Sα (t − r)B (u2 (r ∧ τR ))2 − (u1 (r ∧ τR ))2 dr|pL2 0 Z t p   ≤E |1[0,τR ) Sα (t − r)B (u2 (r ∧ τR ))2 − (u1 (r ∧ τR ))2 |L2 dr 0 Z tX  21  p α k 2 e−2(t−r)(kπ) 1[0,τR ) |u2 (r ∧ τR )) − u1 (r ∧ τR )|L2 ≤ 2 2 πp E 0

k≥1

 p |u2 (r ∧ τR )) + u1 (r ∧ τR )|L2 dr Z t p 3p ϕ1 (t − r)|u2 (r ∧ τR )) − u1 (r ∧ τR )|L2 dr ≤ 2 2 π p Rp E 0 Z t Z
p−1 t 3p ϕ1 (r)dr ϕ1 (t − r)E|u(r ∧ τR )|pL2 dr ≤ 2 2 π p Rp 0 0 Z t 3p 3 ϕ1 (t − r)E|u(r ∧ τR )|pL2 dr, ≤ C2 2 π p Rp t(1− 2α )(p−1)

(4.4)

0

and E|

Z

t 0

1[0,τR ) Sα (t − r)[g(u2 (r)) − g(u1 (r))] dW (r)|pL2 Z t
p ≤E (t − r)γ−1 |Y1 (r)|L2 dr , 0 p−1 Z t Z t p ) (γ−1)( p−1 E|Y1 (r)|pL2 dr, ≤ r 0

0

where Y1 (r) =

sin πγ π

Z

r 0


(r −s)−γ Sα (r − s)1[0,τR ) (s) g(u2 (s∧τR ))−g(u1 (s∧τR )) dW (s).

Furthermore, E|Y1 (r)|pL2 ≤

Cp bp2 E

Z

≤ Cp bp2 Z

r 0

(r − s)−2γ

XZ k≥1

∞

k≥1

α

e−2(r−s)(kπ) |u2 (s ∧ τR ) − u1 (s ∧ τR )|L2 α

s−2γ e−2s(kπ) ds

0

r 0

X

(r − s)−2γ

X

k≥1

p  2q

 α e−2(r−s)(kπ) E|u(s ∧ τR )|pL2 ds .

 p2

ds

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Hence

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Z t
E| 1[0,τR ) Sα (t − r) g(u2 (r)) − g(u1 (r)) dW (r)|pL2 0 Z tZ r X α ˜ ≤C (r − s)−2γ e−2(r−s)(kπ) E|u(s ∧ τR )|pL2 ds dr 0

≤ C˜ ≤ C˜

0

Z t Z 0

Z

t

0

k≥1

t−s

ξ −2γ

0

X

k≥1

 α e−2ξ(kπ) dξ E|u(s ∧ τR )|pL2 ds

(4.5)

ϕ2 (t − s)E|u(s ∧ τR )|pL2 ds.

Replacing (4.5) and (4.4) in (4.3) we get Z t E|u(t ∧ τR )|pL2 ≤ CR,p ϕ(t − r)E|u(r ∧ τR )|pL2 dr for all t ≥ 0. 0

Hence the result follows from the application of the Gronwall Lemma and Jensen inequality. This method can work for any exponent q ≥ p. A.1. Proof of Lemma 2.3 Lemma A.1.1. The map πn : L2 →L2 is globally Lipschitz with Lipschitz constant 1. Moreover, |(πn u)2 − (πn v)2 |L1 ≤ 2n|u − v|L2 , for all u, v ∈ L2 (0, 1), 2

(A.1)

2

|(πn u) |L1 ≤ n|u|L2 , for all u ∈ L (0, 1),

|(πn u)2 |L1 ≤ |u|2L2 , for all u ∈ L2 (0, 1).

Proof. Let us first consider the case when u, v ∈ L2 (0, 1) are such that |u|L2 > n, |v|L2 > n. Then we have |πn u − πn v|2L2 = |πn u|2L2 + |πn v|2L2 − 2hπn u, πn vi 2|u| |v| − 2hu, vi u v = n2 + n2 − 2n2 h , i = n2 |u| |v| |u| |v|   n2 |u|2L2 + |v|2L2 − 2hu, vi ≤ |u − v|2L2 . ≤ |u|L2 |v|L2 In the case when |u|L2 > n and |v|L2 ≤ n, we have  |v| 2 n  n|u|L2 + L |v|L2 |u|L2 − 2hu, vi |πn u − πn v|2L2 = |u|L2 n  n  2 2 |u|L2 + |v|L2 − 2hu, vi ≤ |u − v|2L2 . ≤ |u|L2

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The case |u|L2 < n, |v|L2 < n is trivial. The inequality (A.1) then easily follows. Indeed, by using the trivial identity a2 − b2 = (a + b)(a − b), the H¨ older inequality and the fact that the projection πn is both Lipschitz continuous and bounded we have, for u, v ∈ L2 (0, 1), the following inequalities, |(πn u)2 − (πn v)2 |L1 ≤ |(πn u) + (πn v)|L2 |(πn u) − (πn v)|L2 ≤ 2n|(πn u) − (πn v)|L2 .

In a similar way we can prove the second and the third inequalities.



B.1. Gronwall Lemma Lemma B.1.1. Assume that ϕ, ψ ∈ L1 (a, b) and y : [a, b]→R is an absolutely continuous function. Suppose that these functions satisfy dy(t) ≤ ϕ(t)y(t) + ψ(t) dt

for a.a. t ∈ (a, b).

(B.1)

Then we have y(t) ≤ y(a)e

Rt a

ϕ(τ ) dτ

+

Z

t

ψ(s)e a

Rt s

ϕ(τ ) dτ

ds,

for all t ∈ [a, b]. (B.2)

C.1. Some estimates on stopped stochastic convolutions The main technical tool is the following result stated in Ref. [41] and proven in Ref. [33], Theorem 4.2.7. We formulate the result in the Hilbert space framework as we do not require in this paper Banach space valued Itˆ o integrals. For a notion of γ-radonifying operator see Refs. [26] and [41] or Remark 6.1 in a recent paper [42]. The operator ideal of all gamma-radonifying operators from a separable Hilbert space H into a separable Banach space X will be denoted by R(H, X). Let us recall that with a naturally defined norm, R(H, X) is a separable Banach space, see Ref. [43]. Proposition C.1.1. Assume that H is a separable Hilbert space. Assume that X is an 2-smoothable Banach. Assume that a linear operator −A is an infinitesimal generator of an analytic semigroup (S(t))t≥0 on X. Assume finally that ξ is a progressively measurable operator-valued stochastic process such that S(t − r)ξ(r) ∈ R(H, X) for t > r ≥ 0, where R(H, X) is the Banach space of all γ-radonifying operators from H to X and there exists p > 2 and β0 ∈ ( p1 , 12 ) such that for some T ∈ (0, ∞), |||ξ|||pβ,p,T :=

Z

T 0

Z

s 0

(s − r)−2β kS(s − r)ξ(r)k2R(H,X) dr

p/2

ds < ∞, β < β0 .

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Assume that δ ∈ [0, β0 − 1p ). Then there exists an D(Aδ )-valued continuous stochastic process v(t), t ∈ [0, T ], such that Z t x(t) = S(t − s) ξ(s) dW (s), a.s. for each t ∈ [0, T ]. (C.1) 0

1 p

Moreover, if δ + < β < β0 then there exists a constant C > 0 independent of ξ and T such that for any accessible stopping time σ, E sup |x(t ∧ σ)|pD(Aδ ) ≤ C p T p(β−δ)−1 |||1[0,σ) ξ|||pβ,p,T

(C.2)

0≤t≤T

p

=C T

p(β−δ)−1

Z

T 0

Z

s∧σ 0

(s − r)

−2β

kS(s −

r)ξ(r)k2R(H,X)

dr

p/2

ds.

Corollary C.1.2. Assume that a progressively measurable L(H)-valued stochastic process ξ satisfies, for some M > 0 and T ∈ (0, ∞), |ξ(t)|L(H) ≤ M, t ∈ [0, T ], P − a.s., where L(H) is the space of all bounded linear operators in H. Assume that S(t) ∈ R(H, X) for t > 0 and there exists p > 2 and β0 ∈ ( p1 , 12 ) such that for T > 0 and β < β0 , p/2 Z T Z s r−2β kS(r)k2R(H,X) dr ds < ∞. 0

0

Then the assumptions of Proposition C.1.1 are satisfied and therefore the process x(t), t ≥ 0, satisfies the following inequality E sup |x(t ∧ σ)|pD(Aδ )

(C.3)

0≤t≤T

p

p

≤C M T

p(β−δ)−1

Z

T 0

Z

s∧σ 0

(s − r)

−2β

kS(s −

r)k2R(H,X)

dr

p/2

ds.

Proof. [Proof of Corollary C.1.2.] It is enough to use the fact, see Ref. [44], that R(H, X) an operator ideal and so for a constant C independent of ξ and {S(t)}t≥0 one has for all 0 ≤ r < s < ∞ kS(s − r)ξ(r)kR(H,X) ≤ CkS(s − r)kR(H,X) |ξ(r)|L(H) ≤ CM kS(s − r)kR(H,X) .



Proof. [Proof of Proposition C.1.1.] Let us choose β ∈ ( p1 +δ, β0 ). First we define a process y by Z t 1 y(t) = (t − s)−β S(t − s) ξ(s) dW (s), t ∈ [0, T ]. Γ(1 − β) 0 RT Then E 0 |y(t)|pX dt < ∞. Indeed, by the Fubini Theorem and the Burkholder and Young inequalities

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E

Z

T 0

y(s) p ds = X

Z

≤ Cp

T 0

Z

T

p E y(s) X ds E

0

Z

s

0

(s − r)−2β kS(s − r) ξ(r)k2R(H,X) dr

≤ Cp |||ξ|||pβ,p,T < ∞.

 p2

ds (C.4)

˜ such that y(·, ω) ∈ Lp (0, T ; X). For Hence, there exists a set of full measure Ω ˜ ω ∈ Ω we set Z t 1 (t − s)β−1 S(t − s) y(s, ω) ds, t ∈ [0, T ]. x(t, ω) = Γ(β) 0 ˜ be fixed. Let us recall without proof the following classical result. Let ω ∈ Ω

Lemma C.1.3. Assume that δ ≥ 0, p ∈ (1, ∞) and 1 > β > p1 + δ. Then the map Rβ : Lp (0, T ; X) → C([0, T ]; D(Aδ )) given by Z t Rβ h(t) = (t − s)β−1 S(t − s)h(s) ds, h ∈ Lp (0, T ; X), 0

is well defined, linear and bounded. Moreover, there exists a constant C p,β−δ > 0 1 such that the norm of kRβ k ≤ Cp,β−δ T β−δ− p . Since β > 1p + δ. By the above Lemma there exists a constant CT (independent of y (and hence ω)) such that

| x(t, ω)

|pD(Aδ ) ≤

In particular, | x(t, ω)

|pD(Aδ ) ≤

CT

Z

CT

Z

t 0

T ∧σ(ω) 0

| y(s, ω) |pX ds,

| y(s, ω) |pX ds,

t ∈ [0, T ].

t ∈ [0, T ∧ σ(ω)].

It follows that sup 0≤t≤T ∧σ(ω)

| x(t, ω)

|pD(Aδ ) ≤

CT

Z

T ∧σ(ω) 0

| y(s, ω) |pX ds.

Noting that sup | x(t ∧ σ) |pD(Aδ ) =

0≤t≤T

sup 0≤t≤T ∧σ

| x(t) |pX ,

we infer that sup | x(t ∧ σ) |pD(Aδ ) ≤ CT

0≤t≤T

Taking expectations gives

Z

σ∧T 0

| y(s) |pX ds a.s..

(C.5)

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E sup | x(t ∧ σ) |pD(Aδ ) ≤ CT E 0≤t≤T

Z

σ∧T 0

| y(s) |pX ds.

(C.6)

Note that

E

Z

σ∧T

|pX

| y(s)

0

ds = E =

Z

Z

T

0 T

0

| 1[0,σ) (s)y(s) |pX ds

E | 1[0,σ) (s)y(s) |pX ds.

(C.7)

Define next a process y˜(s) =

1 Γ(1 − β)

Z

s 0

(s − r)−β S(s − r)1[0,σ) (r)ξ(r) dW (r).

y˜ is well defined and for s ∈ [0, T ] ( y˜(s) 1[0,σ) (s)y(s) = 0

if 1[0,σ) (s) = 1, if 1[0,σ) (s) = 0.

Therefore, E | 1[0,σ) (s)y(s) |pX = ≤ + = +

Z

Z

Z

{1[0,σ) (s)=1} {1[0,σ) (s)=1}

{1[0,σ) (s)=0}

Z

Z

{1[0,σ) (s)=1}

{1[0,σ) (s)=0}

| 1[0,σ) (s, ω)y(s, ω) |pX dP(ω) | 1[0,σ) (s, ω)y(s, ω) |pX dP(ω) | y˜(s, ω) |pX dP(ω) | y˜(s, ω) |pX dP(ω) | y˜(s, ω) |pX dP(ω) = E | y˜(s) |pX .

Thus, integrating from 0 to T and then applying as earlier the Fubini Theorem and the Burkholder and Young inequalities we infer that

E ≤ Cp

Z

Z

T 0 T

| 1[0,σ) (s)y(s) E

0

Z

s 0

|pX

ds ≤ E

Z

T 0

| y˜(s) |pX ds

(s − r)−2β 1[0,σ) (r)kS(s − r) ξ(r)k2R(H,X) dr ≤ Cp |||1[0,σ) ξ|||pβ,p,T < ∞.

 p2

ds

(C.8) 

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Corollary C.1.4. Assume that the assumptions of Proposition C.1.1 are satisfied. Let 0 < T < ∞ and τ : Ω → [0, T ] be a stopping time. Furthermore assume that ξ(t), t < τ is an admissible R(H, X)-valued process with |||1[0,τ ) ξ|||β,p,T < ∞, if β < β0 . Set

z(t) =

Z

t 0

S(t − s) 1[0,τ ) (s)ξ(s) dW (s).

Then, there exists a modification z˜ of z such that z˜ ∈ Lp (Ω; C(0, T ; X)). Moreover, there exists C(T ) > 0, independent of ξ and τ , such that for each t ∈ [0, T ], E sup | z˜(s ∧ τ ) |pD(Aδ ) ≤ C(T )|||1[0,τ ) ξ|||pβ,p,t .

(C.9)

0≤s≤t

Proof.

Define the process η : [0, T ] × Ω → R(H, X) by η(s, ω) = 1[0,τ (ω))(s)ξ(s, ω).

As ξ is admissible and 1[0,τ ) is right continuous and adapted, then η right continuous and adapted. In particular, η has a progressively measurable modification. Moreover, as |||η|||β,p,T = |||ξ|||β,p,T < ∞, it follows, by Proposition C.1.1 that the process z is well defined and has a continuous modification z˜ with z˜ ∈ Lp (Ω; C(0, T ; X)). Furthermore, we can find C(T ) > 0, independent of ξ and τ , such that E sup | z˜(s ∧ τ ) |pD(Aδ ) ≤ C(T )|||1[0,τ ) (s)ξ|||pβ,p,t , 0≤s≤t

This completes the proof of Corollary C.1.4.

t ∈ [0, T ].

(C.10) 

Finally, we shall prove Corollary C.1.5. Assume that the assumptions of Proposition C.1.1 are satisfied. Let x(t), t ∈ [0, T ], be an D(Aδ )-valued continuous stochastic process such that Z t x(t) = S(t − s) ξ(s) dW (s), a.s. for each t ∈ [0, T ]. (C.11) 0

Let σ be an accessible [0, T ]-valued stopping time.

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Let y(t) = yσ (t), t ∈ [0, T ], be an X-valued continuous stochastic process such that Z t y(t) = S(t − s) 1[0,σ) (r)ξ(r ∧ σ) dW (s), a.s. for each t ∈ [0, T ] (C.12) 0

Then, for each t ∈ [0, T ],

x(t ∧ σ) = y(t ∧ σ), a.s.. Proof. If η(r) := 1[0,σ) (r)ξ(r ∧ σ), r ∈ [0, T ], then y depends on η in the same way as x depends on ξ and y − x on η − ξ. Therefore, by Proposition C.1.1 E sup |x(t ∧ σ) − y(t ∧ σ)|p ≤ C(T ) |||1[0,σ) (ξ − η)|||pβ,p,T 0≤t≤T

(C.13)

Hence, for each t ∈ [0, T ], E|x(t ∧ σ) − y(t ∧ σ)|p = 0. In particular, x(t ∧ σ) − y(t ∧ σ) = 0 a.s. as claimed.  We finish this section with describing applications of the results obtained here to the stochastic evolution equation driven by the fractional Laplace operator or, as called by some authors, stochastic fractional heat equation. The following result is stated Remark 6.1 in a recent paper [42]. It can be proved by applying Theorem 2.3 from Ref. [45]. Proposition C.1.6. Assume that K is a linear self-adjoint compact operator in ∞ e L2 (0, 1). Let {ej }∞ j=1 and {λj }j=1 be the corresponding sequence of eigenvectors 2 q and eigenvalues of K. Let q ∈ (1, ∞). Then K : L (0, 1)→L (0, 1) is γ-radonifying if and only if  q/2 Z 1 X ∞  λ2j |ej (x)|2  dx < ∞. 0

j=1

Moreover, the γ-radonifying norm of K, denoted by kKkR(L2,Lq ) , is equivalent to i1/q hR P 1 2 2 q/2 dx . 0 [ j λj |ej (x)| ] In particular, if supj |ej |L∞ < ∞, then there exists a constant Mq independent of K, such that kKkR(L2 ,Lq ) ≤ Mq kKkR(L2,L2 ) = Mq kKkHS(L2 ,L2 ) , where as before kKkHS(L2 ,L2 ) denotes the Hilbert-Schmidt norm of the operator K. Example C.1.7. Assume that q ∈ (1, ∞) and let Aq,α be the fractional power of order α2 of the operator Aq defined by  D(Aq ) = H 2,q (0, 1) ∩ H01,q (0, 1), (C.14) Au = −∆u, u ∈ D(Aq ). e Let

2 ∗ us here recall that {ej }∞ j=1 is an ONB of L (0, 1) and Kej = λj ej , j ∈ N .

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Briefly, Aq,α = Aq2 . Let us denote the semigroups generated by the operators Aq,α by one symbol {Sα (t)}t≥0 . Since |ej |L2 = 1 By Proposition C.1.6 we infer that there exists a constant C > 0 such that kSα (t)k2R(L2 ,Lq ) ≤ CkSα (t)k2HS(L2 ,L2 ) , t > 0.

(C.15) P

1 Assume that β < 12 − 2α =: β0 (α). Then, because kSα (t)k2HS(L2 ,L2 ) = k e−2π we infer that for s > 0 Z ∞ Z s r−2β kSα (r)k2HS(L2 ,L2 ) dr r−2β kSα (r)k2R(L2 ,Lq ) dr ≤ C 2

= C2

k t

0

0

Z

α α

∞

∞ X

r−2β

0

e−2π

α α

k r

dr = C 2

k=1

∞ Z X k=1

= C 2 Γ(1 − 2β)(2π α )2β−1

∞ X k=1

∞

r−2β e−2π

α α

k r

dr

0

2 k α(2β−1) =: Cα,β < ∞.

Therefore Z

T 0

Z

s

r 0

−2β

kSα (r)k2R(L2 ,Lq )

dr

p/2

p ds ≤ Cα,β T <∞

and so the semigroup {Sα (t)}t≥0 satisfies the assumptions of Corollary C.1.2 with 1 β0 = β0 (α) = 21 − 2α . Hence we proved the following result. Proposition C.1.8. Consider the framework of Example C.1.7 with α ∈ (1, 2] and q ∈ [2, ∞). Assume that a progressively measurable L(L2 )-valued stochastic process ξ satisfies, for some M > 0, |ξ(t)|L(L2 ) ≤ M, t ∈ [0, ∞), P − a.s..

(C.16)

Assume that p ∈ (2, ∞) and δ ≥ 0 satisfy the following inequality δ+

1 1 1 <β< − =: β0 (α). p 2 2α

(C.17)

Then there exists an D(Aδq,α )-valued continuous stochastic process x(t), t ∈ [0, T ] such that for t ≥ 0 Z t x(t) = S(t − s) ξ(s) dW (s), a.s. for each t ∈ [0, T ]. (C.18) 0

and there exists a constant C > 0 independent of ξ and T such that for any accessible stopping time σ, E sup |x(t ∧ σ)|pD(Aδ 0≤t≤T

q,α )

≤ C p M p T p(β−δ)

(C.19)

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Since D(Aδα,q ) = D(Aq

) ⊂ H αδ,q (0, 1) we infer the following corollaryllary.

Corollary C.1.9. Consider the framework of Example C.1.7 with α ∈ (1, 2] and q ∈ [2, ∞). Assume that p ∈ (2, ∞) and δ ≥ 0 satisfy condition (C.17). Then there exists a constant C > 0 such that for any T > 0, any accessible stopping time σ, any progressively measurable L(L2 )-valued stochastic process ξ satisfying (C.16), the process x(t), t ≥ 0 from Proposition C.1.8 satisfies the following E sup |x(t ∧ σ)|pH αδ,q ≤ C p M p T p(β−δ) .

(C.20)

0≤t≤T

Let us finish this section with the following two observations. The first is that given positive numbers α and δ satisfying inequality (C.17) we can always find q such that αδ > 1q . The second is that because of Jensen’s inequality, inequalities (C.19) and (C.20) is also valid for all p2 ≤ p. D.1. Pointwise multiplication in Sobolev spaces Since our proof relies on Theorems 1 and 2 from section 4.4.4 of Ref. [35] and the formulations of these theorems contain some errors we have decided to present them here for the convenience of the reader. Theorem 1. Assume that

n − s1 > p

0 < s 1 ≤ s2

(1)

1 1 1 ≤ + , p p1 p2

(2)

( ( pn1 − s1 )+ + ( pn2 − s2 )+ , maxi ( pni − si ) > 0, s1 + s 2 >

if maxi ( pni − si ) > 0,

otherwise

n n + − n, p1 p2

(3)

(4)

and either q ≥ q1 if s1 < s2

(5)

q ≥ max(q1 , q2 ) if s1 = s2 .

(6)

s1 Fps11,q1 · Fps22,q2 ,→ Fp,q

(7)

s1 Bps11 ,q1 · Bps22 ,q2 ,→ Bp,q

(8)

or

(i) Then

(ii) Also we have

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Theorem 2. Assume that the conditions (1), (2), (4) and (5) are satisfied. Assume also thatf ( ( pn1 − s1 )+ + ( pn2 − s2 )+ , if maxi ( pni − si ) > 0, n − s1 = (16) p maxi ( n − si ) > 0, otherwise pi

(i) If {i ∈ {1, 2} : si =

n and pi > 1} = ∅, pi

(17)

then (7) remains true. (ii) Suppose in addition thatg q ≥ q2 , if s1 −

n n = s2 − . p1 p2

(18)

If {i ∈ {1, 2} : si =

n and qi > 1} pi

∪{i ∈ {1, 2} : si <

n and qi > pi

n pi

n } = ∅, − si

(19)

then (8) remains true. References [1] G. Da Prato and D. Gatarek, Stochastic Burgers equation with correlated noise, Stochastic and Stochastic Reports. 52, 29–41, (1995). [2] W. E, K. Khanin, A. Mazel, and Y. Sinai, Invariant measures for Burgers equation with stochastic forcing, Ann. of Math. (2). 151(3), 877–960, (2000). ISSN 0003-486X. [3] J. Bertoin, C. Giraud, and Y. Isozaki, Statistics of a flux in Burgers turbulence with one-sided Brownian initial data, Comm. Math. Phys. 224(2), 551–564, (2001). ISSN 0010-3616. [4] J. D. Avrin, The generalized Burgers’ equation and the Navier-Stokes equation in Rn with singular initial data, Proceedings of the American Mathematical Society. 101(1), 29–40, (1987). [5] P. Biler, T. Funaki, and A. W. Wojbor, Fractal Burgers’ equations, J. Differential Equations. 148, 9–46, (1998). [6] N. Sugimoto, Generalized Burgers equations and fractional calculus, Nonlinear Wave Motion. (A. Jeffrey, Ed), 162–179, (1989). [7] N. Sugimoto, Burgers equations with a fractional derivative; hereditary effects on nonlinear acoustic waves, J. Fluid Mech. 225, 631–653, (1991). [8] N. Sugimoto and T. Kukatani, Generalized Burgers equations for nonlinear viscoelastic waves, Wave Motion. 7, 447–458, (1985). the book on the LHS of (16) one has n − s. But there is no s in the assertion and instead p of s we need to have s1 , i.e. n − s . Another error of this sort can be seen twice on the RHS of 1 p formula (16), where it is written: max i ( pn − s1 ). The correct form should be maxi ( pn − si ). i i g In the book instead of s − n below in (18) one has s − n . 1 1 p p f In

1

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[9] A. Le Mehaute, T. Machado, J. Trigeassou, and J. Sabatier, Fractional Differentiation and its Applications, FDA’04. vol. 2004-1, Procceedings of the first IFAC Workshop, (International Federation of Automatic Control, ENSEIRB, Bordeaux, France, July 19-21, 2004). [10] V. Uchaikin and V. M. Zolotarev, Chance and Stability, Stable Distributions and their Applications. Modern Probability and Statistics, (VSP., 1999). [11] J. Angulo, M. Ruiz-Medina, V. Anh, and W. Grecksch, Fractional diffusion and fractional heat equation, Adv.Appl.Prob. 32, 1077–1099, (2000). [12] R. Dalang and C. Mueller, Some non-linear s.p.d.e.’s that are second order in time, Electronic Journal of Probab. 8 no. 1, 1–21, (2003). [13] R. Dalang and M. Sanz-Sol´e, Regularity of the sample paths of a class of second-order spde’s., J. Funct. Anal. 227, no. 2, 304–337, (2005). [14] L. Debbi and M. Dozzi, On the solution of non linear stochastic fractional partial differential equations, Stochastic Processes and Their Applications. 115, 1764–1781, (2005). [15] C. Mueller, The heat equation with L´evy noise, Stoch. Proc. Appl. 74, 67–82, (1998). [16] J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I. (Springer-Verlag, New York, 1972). Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181. [17] H. Triebel, Interpolation theory, function spaces, differential operators. (Johann Ambrosius Barth, Heidelberg, 1995), second edition. ISBN 3-335-00420-5. [18] W. McLean, Strongly elliptic systems and boundary integral equations. (Cambridge University Press, Cambridge, 2000). ISBN 0-521-66332-6; 0-521-66375-X. [19] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. (Springer-Verlag, New York, 1983). [20] M. E. Taylor, Pseudodifferential Operators. Princeton University Press, (Princeton, New Jersey, 1981). ISBN 0-691-08282-0. [21] R. Seeley, Interpolation in Lp with boundary conditions, Studia Math. 44, 47–60, (1972). ISSN 0039-3223. Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, I. [22] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinte Dimensions. (Springer, Combridge university press, 1992). ISBN 0-521-38529-6. [23] G. Da Prato, A. Debussche, and R. Temam, Stochastic Burgers’ equation, NoDEA Nonlinear Differential Equations Appl. 1(4), 389–402, (1994). ISSN 1021-9722. [24] K. Twardowska and J. Zabczyk, A note on stochastic Burgers’ system of equations, Stochastic Analyis and Applications. 22 no. 6, 1641–1670, (2004). [25] T. Kato, Perturbation theory for linear operators. Die Grundlehren der mathematischen Wissenschaften, Band 132, (Springer-Verlag New York, Inc., New York, 1966). [26] Z. Brze´zniak, On stochastic convolution in Banach spaces and applications, Stochastics Stochastics Rep. 61(3-4), 245–295, (1997). ISSN 1045-1129. [27] G. Da Prato, S. Kwapien, and J. Zabczyk, Regularity of solutions of linear stochastic equations in Hilbert space, Stochastic. 23, 1–23, (1987). [28] D. Gatarek, A note on nonlinear stochastic equations in hilbert spaces, Statistics & Probability letters. 17, 387–394. [29] H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal. 16, 269–315, (1964). ISSN 0003-9527. [30] I. S. Gradshteyn, M. Ryzhik, A. Jeffrey, Yu. L. Geronimus, and M. Yu. Tseytlin, Table of integrals, series, and products. (Academic Press, Orlando, Florida/San Diego, California/New York/London, 1980). [31] W. Feller, An Introduction to Probability Theory and Its Applications : Vol. 2. (John

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Wiley and sons, New York, 1971). [32] J. Seidler, Da Prato-Zabczyk’s maximal inequality revisited. I, Math. Bohem. 118 (1), 67–106, (1993). ISSN 0862-7959. [33] A. Carroll, The stochastic nonlinear heat equation. (PhD thesis; The University of Hull, Hull, 1999). [34] Z. Brze´zniak, B. Maslowski, and J. Seidler, Stochastic nonlinear beam equations, Probab. Theory Related Fields. 132(1), 119–149, (2005). ISSN 0178-8051. [35] T. Runst and W. Sickel, Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations. vol. 3, de Gruyter Series in Nonlinear Analysis and Applications, (Walter de Gruyter & Co., Berlin, 1996). ISBN 3-11-015113-8. [36] R. A. Adams, Sobolev spaces. (Academic Press, New York/San Francisco/London, 1975). ISBN 0-12-044150-0. [37] H. Triebel, Theory of function spaces. vol. 78, Monographs in Mathematics, (Birkh¨ auser Verlag, Basel, 1983). ISBN 3-7643-1381-1. [38] H. Amann. Multiplication in Sobolev and Besov spaces. In Nonlinear analysis, Sc. Norm. Super. di Pisa Quaderni, pp. 27–50. Scuola Norm. Sup., Pisa, (1991). [39] T. Roger, Navier-Stokes equations. (AMS Chelsea Publishing, Providence, RI, 2001). ISBN 0-8218-2737-5. Theory and numerical analysis, Reprint of the 1984 edition. [40] R. Z. Has0 minski˘ı, Stochastic stability of differential equations. vol. 7, Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, (Sijthoff & Noordhoff, Alphen aan den Rijn, 1980). ISBN 90-286-0100-7. Translated from the Russian by D. Louvish. [41] Z. Brze´zniak and D. G¸atarek, Martingale solutions and invariant measures for stochastic evolution equations in Banach spaces, Stochastic Process. Appl. 84(2), 187–225, (1999). ISSN 0304-4149. [42] Z. Brze´zniak and Y. Li, Asymptotic compactness and absorbing sets for 2d stochastic navier-stokes equations on some unbounded domains, Trans. Amer. Math. Soc. to appear in. ISSN 0002-9947. [43] A. Neidhardt, Stochastic Integrals in 2-Uniformly Smooth Banach Spaces. (PhD thesis; University of Wisconsin, 1978). [44] P. Baxendale, Gaussian measures on function spaces, Amer. J. Math. 98(4), 891–952, (1976). ISSN 0002-9327. [45] Z. Brze´zniak and J. van Neerven, Space-time regularity for linear stochastic evolution equations driven by spatially homogeneous noise, J. Math. Kyoto Univ. 43(2), 261– 303, (2003). ISSN 0023-608X.

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Chapter 6 Stochastic Control Methods for the Problem of Optimal Compensation of Executives Abel Cadenillas, Jaksa Cvitani´c∗ and Fernando Zapatero† Department of Mathematical and Statistical Sciences, University of Alberta Edmonton, Alberta T6G 2G1, Canada. [email protected]. We consider the problem of an executive who receives call options as compensation. He can influence the mean return of the stock with his effort, and the level of volatility through his choice of projects. The executive wants to select the optimal effort and choice of projects to maximize the expected utility from the call option minus the disutility associated with the effort. We model this as a stochastic control problem, and present an analytical solution. In this framework, we also introduce the problem of the company that simultaneously wants to minimize overtime volatility and maximize final expected value of the price of the stock. We characterize (and compute numerically for the logarithmic case) the optimal strike price the company should choose. When the executive can affect the mean return of the stock, we find that options should be granted out-of-the-money. In a context of perfect information, the executive would not be able to affect the mean return of the stock independently of the volatility, and it would be optimal to grant options in-the-money. The mathematical techniques are those of stochastic control, convex analysis, and martingale theory.

Contents 1 2

3

4 5 6 7

Introduction . . . . . . . . . . . . . . . . . . The Executive . . . . . . . . . . . . . . . . . 2.1 Stock Dynamics . . . . . . . . . . . . . 2.2 The Problem of the Executive . . . . . 2.3 Optimal Effort and Choice of Projects The Company . . . . . . . . . . . . . . . . . 3.1 The Problem of the Company . . . . . 3.2 Optimal Strike Price . . . . . . . . . . Numerical Computations of the Strike Price . Price of the Options . . . . . . . . . . . . . . The Case of Additional Cash Compensation . Conclusions . . . . . . . . . . . . . . . . . . .

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170 171 171 173 173 176 176 176 179 182 186 194

and Social Sciences, Caltech, M/C 228-77, 1200 E. California Blvd. Pasadena, CA 91125, [email protected]. † FBE, Marshall School of Business, University of Southern California, Los Angeles, CA 900891427, [email protected]. 169

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

195

1. Introduction The optimal compensation of executives is currently one of the most active areas of research in mainstream finance, although it has not received much attention yet in the mathematical finance literature. The problem of the company consists in selecting the optimal compensation package to give the executive, and the problem of the executive consists in selecting the optimal strategy to take advantage of the compensation package. A call option gives its owner the right to buy one share of an underlying stock at a predetermined price. Options are similar to a leveraged portfolio invested in the underlying security. For that reason, options have become the main choice for companies to compensate executives in order to align their interests with those of the company stockholders. Hall and Leibman [1] and Murphy [2] report statistics about the number of companies that use stock options as the main component of the compensation of their executives. Carpenter [3] and Hall and Murphy [4] consider the problem of valuation of executive stock options, Detemple and Sundaresan [5] develop a pricing model for options that can be applied to executive options, and Hall and Murphy [6] study the problem of choosing the strike price of the options. Stoughton and Wong [7] consider the optimality of stock versus options in a context of industry competition. Aseff and Santos [8] address the broader problem of the optimality of stock options versus other contracts. Kadan and Swinkels [9, 10] compare options vs. stock when there is a possibility of bankruptcy. We develop a dynamic model in order to address some of the problems mentioned above. In particular, the existent literature considers the incentives of options in a static framework, which does not seem appropriate to address some of the questions considered. One of the potential problems of stock options compensation (see, for instance, Johnson and Tian [11, 12]) is the incentive to the executive to increase volatility. However, volatility is a dynamic concept and it is appropriate to consider problems like these in a dynamic setting. Another element that is missing in the current literature is the computation of the effort of the executive and the tradeoff between the amount of that effort and the disutility resulting from it. We assume that the executive receives call options as compensation and we study the optimal strike price of the options. The strike price of zero means that the compensation is in stock. In our setting, the effort of the executive affects the mean return of the stock. Besides, the executive can choose among a menu of projects with a tradeoff between expected return and volatility: projects with higher volatility offer a higher expected return (see Cadenillas, Cvitani´c and Zapatero [13] for a similar model in a different application). The executive will choose effort and volatility in order to maximize expected utility coming from compensation minus the effort disutility. The company cares both about the final value of the stock and the overtime volatility. The company

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chooses the strike price of the option so as to induce the executive to achieve an optimal mix of effort and volatility. Carpenter [14] studies, also in a dynamic setting, the optimal choice of volatility when the effort of the agent cannot affect the stock dynamics and considers several exogenously given strike prices and more general benchmark functions. Bolton and Harris [15] also study a model where the agent has control of the dynamics of the asset, for a different purpose. In Cadenillas Cvitani´c and Zapatero [16] we extend the model of this paper to the situation in which the type of the executive δ is unknown by the company, which has a prior distribution for it. For the executive that maximizes logarithmic utility from the options, minus the quadratic disutility from the effort, we compute the optimal level of effort and volatility in closed form. Based on that, we can compute the optimal strike price to be set by the company. We find that the strike price should be higher, the higher the type (quality) of the executive, and the more the company is interested in maximizing the expected value of the price of the stock (as opposed to minimizing overtime volatility). The relationship between the optimal strike price and the quality of the projects (defined as the additional expected return resulting from accepting an extra unit of risk) is not monotonic. We also compute the no-arbitrage (complete markets) price of the option. This would be the price of the option for the company and, therefore, would represent the cost the company incurs by granting options. The executive, on the other hand, faces incomplete markets since he cannot trade in the underlying. We compute the value he assigns to the option as the certainty equivalent associated with his utility. The price of the options is lower for the executive (facing incomplete markets) than for the company. We show that options seem to be more efficient when the executive is of a high type relative to the quality of the projects. Finally, we also compute the optimal effort of the executive when cash is a part of the compensation package, but we are not able to get explicit expressions for the optimal strike price in this case. The paper is structured as follows. In section 2 we present and solve explicitly the problem of the executive, and in section 3 we present and solve the problem of the company. In section 4 we perform some numerical computations and comparative statics. In Section 5 we compute the option prices from the perspectives of both the company and the executive. Section 6 considers the case of additional compensation in cash. We close the paper with some conclusions. 2. The Executive 2.1. Stock Dynamics Consider a probability space (Ω, F, P ) endowed with a filtration (Ft ), which is the P -augmentation of the filtration generated by a one-dimensional Brownian motion W . Our benchmark stock has a price that follows a geometric Brownian motion

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process, dSt = µSt dt + σSt dWt with starting value S0 . Here, µ ∈ [0, ∞) and σ ∈ (0, ∞) are exogenous constants. Definition 2.1. The level of effort is an (Ft )-adapted stochastic process u that satisfies "Z # T E |ut |2 dt < ∞. 0

The value ut is the amount of effort that the executive puts in the management of the company at time t. The choice of projects is an (Ft )-adapted stochastic process v that satisfies "Z # T 2 E |vt St | dt < ∞. 0

We assume that at each moment in time, the executive can choose different projects (or strategies) vt , that are characterized by a level of risk and expected return. We will denote by A the pairs (u, v) which satisfy the above technical conditions, and say that A is the class of admissible pairs of effort and choice of projects. When the company is managed by the executive, the dynamics of the stock price S is given by dSt = µSt dt + δut dt + αvt St dt + vt St dWt .

(2.1)

Here, δ ∈ [0, ∞) and α ∈ (0, ∞). The higher the value of u, the higher the expected value of the stock. On the other hand, the choice of projects v is equivalent to the choice of the volatility of the stock, although it also has an impact on the expected value. We assume that the executive can choose different projects or strategies that are characterized by a level of risk and expected return. Since α > 0, the higher the risk of a project, the higher its expected return. The parameter α is a characteristic of the type of the company. One possible interpretation of α (and, potentially, a way to estimate it empirically) would be the slope of the equivalent of the “Capital Market Line” resulting from all the projects available to the company (more about the distinction between firm-specific and market risk below). On the other hand, δ is a measure of the impact of the effort of the executive on the value of the company. It can be interpreted as an indicator of the quality of the executive. We emphasize that this is a partial equilibrium setting and we do not compare the dynamics of the stock of this company with the dynamics of other stocks. In the simplest setting, with complete markets, the coexistence of different stocks with different drifts (and maybe same standard deviations) would be consistent with incomplete information about the actions of the agent and its effects on the dynamics of the stock. In theory, in that setting, complete information would be

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consistent with δ = 0, which is a special case of the model that we consider in this paper. Without loss of generality, we will assume that µ = 0. We then rewrite equation (2.1) as dSt = δut dt + αvt St dt + vt St dWt .

(2.2)

2.2. The Problem of the Executive In our model, the executive is risk-averse and experiences disutility as a result of the effort. Problem 2.2. The objective of the executive is to select (b u, vb) ∈ A that maximizes the functional J defined by # " Z T  1 u2 dt . (2.3) J(u, v) := E log n(ST − K)+ − 2 0 t

Here, n is the number of call options the executive receives as a part of his compensation package. The second term of the objective function of the executive represents the disutility from effort. That disutility might be, for example, the result of spending more time working for the company. The other control, v, only involves the choice of projects the company will undertake and has no effect on the disutility of the executive, since it does not require any effort: the executive has a menu of projects and decides the level of risk to undertake. With this parameterization, the number of options n becomes irrelevant for the executive’s problem, because log {n(ST − K)+ } = log n + log(ST − K)+ . We will then assume that n = 1. It is obvious that a more general parameterization (such as power utility) would not have this property. Problem 2.1 of the executive of this paper is similar to the problem of the manager considered by Cadenillas, Cvitani´c and Zapatero [13]. In that paper, the manager controls the value of unlevered assets. 2.3. Optimal Effort and Choice of Projects First we introduce the auxiliary exponential martingale Z, defined by   1 2 Zt := exp − α t − αWt . 2

(2.4)

Also, consider the function of time T¯, defined by eα T¯t :=

2

(T −t)

−1 . (2.5) α2 Using the previous notation and given the following quadratic equation in z, δ 2 T¯0 z 2 + (S0 − K)z − 1 = 0,

(2.6)

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where δ is the parameter that measures the type of the executive and K is the strike price of the options, we denote by zˇ the positive solution of (2.6):   q 1 2 + 4δ 2 T ¯ (K − S ) + (K − S ) (2.7) zˇ = 0 0 0 2δ 2 T¯0

We now present the optimal controls of the executive:

Theorem 2.3. Consider the problem of the executive described in Section 2.2. Consider also the exponential martingale of (2.4), the positive number zˇ given by (2.7), and T¯, the time function of (2.5). The optimal effort u b of the executive is u bt = δˇ z Zt .

The optimal choice of projects vb is given by α + αˇ z δ 2 Zt T¯t . vbt St = zˇZt

(2.8)

(2.9)

The optimal effort and volatility determine that the price of the stock of the firm be given by St =

1 + K − zˇδ 2 Zt T¯t . zˇZt

(2.10)

If S0 > K the above formulas remain valid even for δ = 0, but with zˇ =

1 . S0 − K

If S0 ≤ K and δ = 0, then the utility of the executive is minus infinity, since it is impossible in this case to guarantee ST > K with probability one. Proof. This theorem can be proved applying the duality method to solve stochastic control problems. We omit the proof, because it is a special case of the proof of Theorem 6.1.  We observe that the optimal effort and volatility can also be written as functions of the price of the stock. Indeed, from equations (2.8)-(2.10), we obtain that when δ > 0,   q 1 2 + 4δ 2 T ¯ (K − S ) + (K − S ) (2.11) u bt = t t t 2δ T¯t

and

vbt St =

=

αδ + αδb ut T¯t u bt

2αδ 2 T¯t p (K − St ) + (K − St )2 + 4δ 2 T¯t   q α (K − St ) + (K − St )2 + 4δ 2 T¯t . + 2

(2.12)

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With respect to the optimal effort, as expected, u b is increasing in the strike price K: as K goes to infinity, the effort goes to infinity, as well. Besides, we note that zˇ is decreasing in T , the time to maturity of the option (T¯0 is increasing in T and zˇ is decreasing in T¯0 ). Therefore, the larger the maturity of the option, the lower the effort of the executive. The intuition is clear: a larger T has a similar effect on the executive as a reduction of the strike price. The effect of δ (the “type” of executive) depends on whether the option is in-, out-, or at-the-money. When the option is at-the-money, the optimal effort is independent of δ, as we can see by substituting (2.7) in (2.8). We can also check that when the option is in-the-money the effort is increasing in δ, and when the option is out-of-the-money the optimal effort decreases with δ. Since Z is a martingale, the initial expected value of the effort at any point in time is E[b ut ] = δˇ z. (2.13) With respect to the effect of α, we note that T¯0 is increasing in α and, therefore, zˇ is decreasing in α. Expected effort is, then, decreasing in α (everything else constant): the better the menu of projects the executive can choose, the lower the expected effort of the executive. The analysis of the volatility is more complicated. Since T¯· is a decreasing function of t and T¯T = 0, the second term of (2.9) decreases in expected value as we approach maturity, and will tend to be negligible relative to the first term. Therefore, for short maturities, optimal volatility will tend to decrease with higher strike price. For large maturities, the relation will tend to be the opposite. We also see that the volatility is increasing in the type δ of executive (ˇ z is decreasing in 2 δ, and zˇδ is increasing in δ). The economic intuition is straightforward: a hightype executive can afford more volatility because his effort will be more effective to counteract drops in the value of the stock. It is straightforward to see that the expected value of the volatility at a future date t is α 2 z δ 2 T¯t . (2.14) E[b vt St ] = eα t + αˇ zˇ Since T¯t is increasing in α, the expected volatility is increasing in α. In other words, the higher the expected return-risk tradeoff, the higher the risk the executive will be willing to undertake. Some of these comparative statics are illustrated in the results of Table 6.1 (that we analyze in detail in section 4). However, in that table the strike price is always the optimal one, and some of the previous conclusions hold for changes in a given parameter with constant strike price. It is also interesting to study the correlation between optimal effort and optimal choice of projects. By Itˆ o’s lemma, and equation (2.9), the dynamics of the optimal volatility are   1 − δ 2 zˇZt T¯t dWt . (2.15) d(b vt St ) = (. . .)dt + α2 zˇZt

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Equation (2.8) yields db ut = (. . .)dt − αb ut dWt .

(2.16)

It is clear that their correlation can be either positive or negative. Considering only the instantaneous correlation and ignoring the drift terms, we see that for a short maturity of the option they tend to be negatively correlated, and increases in optimal effort will be typically associated with decreases of the optimal level of volatility. 3. The Company 3.1. The Problem of the Company The company is interested in two things: the expected value and the volatility of the stock price. Specifically, we suppose that the objective of the company is to b that solves the following problem: select the strike price K ( ) Z T

max

K∈[0,∞)

λE[ST ] −

Var[St ]dt .

(3.1)

0

Here, T is the maturity of the European call options and λ is an exogenous constant that represents the tradeoff in the preferences of the company between the expected final value and the variance of the stock. The objective of the company has some similarity with quadratic preferences (in which the agent only cares about expected return and variance), but in our model the company cares about overtime variance. The objective function of the company is justified as a way to incorporate risk aversion in a tractable way. Another possible criterion is to select the number of options and the strike price to maximize the expected value of the price of the stock at maturity minus the price of the compensation package. In that case, the number of options and the strike price must be restricted so that the expected utility of the executive be greater or equal than a positive number. A similar problem has been studied by Cadenillas, Cvitani´c and Zapatero [13]. 3.2. Optimal Strike Price We assume that the company has full information about the parameters that characterize the dynamics of the stock, as well as the preferences of the executive. The objective of the company is given by (3.1). In order to characterize the optimal choice of the company we introduce the following auxiliary notation: 2

2

e3α T − 1 e2α T − 1 − , A := 2 3α 2α2    1 2α2 T 2 5 δ4 α2 T e + − 2T e − − T , B := 4 α 2α2 α2 2α2

(3.2) (3.3)

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and the function h defined by 2 1 1 h(z) := λ (eα T − 1) − 2 A − z 2 B + λzδ 2 T¯0 , z z

(3.4)

where λ is the parameter of (3.1). We also define √ ( −S0 + S02 +4δ T¯0 if δ > 0 2δ T¯0 η := 1 if δ = 0. S0 We will be interested in maximizing the function h in the interval [η, ∞). We now present the optimal strike price for the company. Theorem 3.1. Consider the company whose objective is given by (3.1). Let z¯ ∈ [η, ∞) be the quantity that maximizes (3.4), Then the optimal strike price is given by b = S0 − 1 + z¯δ 2 T¯0 . K z¯

(3.5)

Proof. Our first objective is to compute the variance and the expectation of St . Some preliminary computations give 2

E[Z 2 (t)] = eα t ,

2

E[Z(t)] = 1, E[Z −2 (t)] = e3α t ,

Thus, according to equation (2.10), E[St ] = and

1 α2 t e + K − zˇδ 2 T¯t zˇ



 1 2 ¯ V ar[St ] = V ar − zˇδ Zt Tt zˇZt " 2 #   2 1 1 2 =E − zˇδ Zt T¯t − E − zˇδ 2 Zt T¯t zˇZt zˇZt =

2

E[Z −1 (t)] = eα t .

2 2 2 1 3α2 t (e − e2α t ) + 2δ 2 T¯t (eα t − 1) + zˇ2 δ 4 T¯t2 (eα t − 1). zˇ2

Noting that T¯T = 0, K = S0 − 1/ˇ z + zˇδ 2 T¯0 from (2.6)-(2.7), we see that   1 α2 T +K λE[ST ] = λ e zˇ   1 α2 T 2¯ + S0 − 1/ˇ z + zˇδ T0 =λ e zˇ   1 α2 T 2¯ =λ e − 1/ˇ z + zˇδ T0 zˇ + an expression that involves neither zˇ nor K.

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Furthermore, Z T Z T Z T 2 1 3α2 t 2α2 t 2 (e −e )dt + 2δ T¯t (eα t − 1)dt Var[St ]dt = 2 z ˇ 0 0 (0 2 ) Z T 3α T 2α2 T 2 e − 1 e − 1 1 2 4 2 α t − + zˇ δ T¯t (e − 1)dt = 2 zˇ 3α2 2α2 0     1 2α2 T 2 5 δ4 α2 T e + − 2T e − − T +ˇ z2 4 α 2α2 α2 2α2 + an expression that involves neither zˇ nor K. Disregarding the terms that do not depend on zˇ or K, we see that (3.1) is equivalent to finding z¯ ∈ (0, ∞) that maximizes the function h defined in (3.4). Here, A and B are given in (3.2) and (3.3). The maximization has to be performed under the constraint that the corresponding K is non-negative. We can find K from (2.6) as K = S0 − 1/ˇ z + zˇδ 2 T¯0 , the equation given in (3.5). We observe that z¯ ∈ (0, ∞) satisfies the restriction S0 − 1/¯ z + z¯δ 2 T¯0 ≥ 0

if and only if δ T¯0 z¯2 + S0 z¯ − 1 ≥ 0. That is equivalent to z¯ ≥

(

−S0 +

√

1 S0

S02 +4δ T¯0 2δ T¯0

if δ > 0 = η. if δ = 0



We do not have a closed-form solution for the constant z¯. However, it can be found easily by applying the following necessary condition of optimality: 2

2B z¯4 − λδ 2 T¯0 z¯3 + λ(eα T − 1)¯ z − 2A = 0 if z¯ > η 4 2¯ 3 α2 T 2B z¯ − λδ T0 z¯ + λ(e − 1)¯ z − 2A ≤ 0 if z¯ = η,

(3.6) (3.7)

which is the result of differentiating (3.4) and the constraint z¯ ∈ [η, ∞) (which is b ∈ [0, ∞)). We will perform some numerical exercises in the next equivalent to K section. Next, we consider the case in which the company is only interested in variance minimization. Corollary 3.2. Consider a company that chooses the strike price that minimizes overtime variance, i.e., Z T min Var[St ]dt. (3.8) K≥0

0

Suppose also that δ = 0, i.e., that the executive cannot influence the drift independently of volatility. Then it is optimal to issue at the money options, i.e., b = S0 . K

(3.9)

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Proof. This corresponds to the case λ = 0. Since also δ = 0, then B = 0, and (3.4) implies we have to find z¯ that minimizes z12 A. Hence optimal z¯ = ∞, and (3.5) b = S0 . gives K 

If the company cares only about the variance and the executive can affect the drift of the stock only through the choice of the level of volatility, the corollary says that options should be optimally issued at-the-money (which is the customary practice). We will see later that in general we get a large range of possible optimal values for the strike price . Corollary 3.3. Consider a company that chooses the strike price that maximizes the expected value of the stock, i.e., max E[ST ].

(3.10)

K≥0

b = 0. Suppose also that δ = 0. Then, the optimal strike price is K

Proof. We note that in the case δ = 0, equation (2.10) gives 2

eα T + K. E[ST ] = zˇ Thus, according to (3.5), the company wants to find the value of K that maximizes E[ST ] = K(1 − eα

2

T

) + S 0 eα

b is zero. Since α > 0, the optimal K

2

T

. 

When the company only cares about the expected final price of the stock and the executive can influence the drift of the stock only through the choice of projects, this corollary says that it is optimal for the company to give all the compensation in stock. 4. Numerical Computations of the Strike Price In the previous section we derived the optimal exercise price for a company that cares about both the expected final value of the stock and the overtime volatility. It is expressed in equation (3.5). It depends on the solution to equations (3.6)-(3.7), which do not have an explicit solution. In this section we perform some numerical exercises in order to derive some properties of the optimal strike price. The results are included in Table 6.1. We study the strike price as a function of the parameters of the model. The effects of the time to maturity are obvious and we fix it at T = 5. The initial price of the stock is 100. Besides the optimal strike price, we also record the optimal initial effort level u b0 and the optimal initial choice of projects vb0 (expressed in %). The last two columns are helpful for intuitive purposes. The first general observation is that for strictly positive values of the parameters, the optimal strike price is out-of-the money. Next, we analyze the type of executive.

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Optimal strike price, effort, and volatility. S0 = 100; T = 5

δ

α

λ

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5

0.05 0.1 0.25 0.5 0.75 0.25 0.25 0.25 0.25 0.25 0.5 0.5 0.5 0.5 0.5 0.5 0.05 0.1 0.25 0.5 0.25 0.25 0.25 0.25 0.25 0.25 0.5 0.5 0.5 0.5 0.5 0.5

0.5 0.5 0.5 0.5 0.5 0.25 0.5 1 2 5 0.1 0.25 0.5 1 2 5 0.5 0.5 0.5 0.5 0.1 0.25 0.5 1 2 5 0.1 0.25 0.5 1 2 5

b K

140.01 111.77 103.78 108.70 143.14 103.40 103.78 104.70 107.88 120.16 108.60 108.64 108.70 108.85 109.16 110.35 146.35 116.24 116.02 142.98 115.52 115.70 116.02 116.67 118.12 123.88 142.87 142.91 142.98 143.11 143.39 144.24

u b0

9.56 2.38 0.85 0.98 1.57 0.79 0.85 0.99 1.46 3.28 0.97 0.97 0.98 0.99 1.02 1.13 1.94 0.85 0.76 0.95 0.75 0.76 0.76 0.78 0.82 0.98 0.96 0.96 0.96 0.97 0.97 0.99

vb0 (%) 2.41 1.26 1.54 5.38 33.31 1.48 1.54 1.70 2.31 5.18 5.33 5.35 5.35 5.43 5.56 6.06 2.57 2.79 7.26 26.66 7.19 7.22 7.26 7.35 7.56 8.50 26.62 26.64 26.66 26.72 26.83 27.17

The higher the type of the executive (higher δ), the higher the optimal strike price. A higher type can have more impact with lower effort and will also tend to choose more volatile projects, since she will be able to counteract negative shocks more easily. The quality of the projects α is very important and has an effect on the optimal strike price that is not monotonic: the optimal strike price is very high for low quality projects, then it is decreasing as α increases, up to a certain point. Beyond that point, the optimal strike price is increasing in α. For very low α, a high strike price is needed to force the executive to choose a high effort level. As the quality of the projects increases, less effort is required. However, when volatility becomes very effective (that is, large α), it is necessary again to set a higher strike price to induce higher effort in order to guarantee that the executive keeps an adequate mix

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Delta = 0.1

Delta = 1

Delta = 5

Delta = 10

160

600

150

500 140

400 130

300

K

120

K 110

200

100

100 90

0 0

0.1

0.2

0.3

0.4 Alpha

(a)

0.5

0.6

0.7

0.8

80 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Alpha

(b)

Fig. 6.1. The optimal strike price as a function of the quality of projects (a) Different values of δ (b) δ = 0.1

of effort u and choice of projects v, and does not rely on the choice of projects v exclusively. It is clear from our argument that the type of the executive plays a major role in this trade-off. This point is illustrated in Figures 6.1(a) and 6.1(b). In Figure 6.1(a) we observe that the pattern described above holds for any level of type of the executive, but the inflection point at which it is necessary to start raising the strike price again comes faster the higher the type of the executive. Figure 6.1(b) is the lowest of the four lines in Figure 6.1(a) measured in a different scale. The effect of λ in this range is clear: the less the company cares about overtime volatility, the higher the strike price it should set, regardless of the values of other parameters. In order to get a higher expected price of the stock, the executive can either exercise more effort or choose riskier projects. Since the company is concerned about volatility it will prefer more effort, but the only way to induce it is by setting a higher strike price. Of course, the impact of additional volatility on expected return (α) affects the effectiveness of the strike price to attain company goals. As we mentioned above, an important conclusion of our results is that for strictly positive values of the parameters, the optimal strike price is always out-of-themoney. Our results shed some light on the debate about whether options should be at-the-money or out-of-the-money (Hall and Murphy [6]). As we see in Table 6.2, options in-the-money would be optimal in the case of an executive whose effort does not have any effect on the stock return (δ = 0) and when the company is primarily concerned about the price of the stock (high λ). We also recall from our discussion about the dynamics of the stock of equation (2.1) that the case δ = 0 could correspond to the situation of perfect information about the effect of the actions of the executive on the dynamics of the stock. Options at-the-money or close to at-the-money are optimal (as we see asymptotically in Table 6.2) when the executive is of very low type (low δ) and the company is mainly interested in price

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volatility (low λ). We also see that for δ = 0 the optimal strike price gets lower as the company cares more about the expected return. Table 6.2.

Optimal strike price for “very small” δ. S0 = 100; T = 5

δ

α

λ

0 0 0 0 0 0.005 0.005 0.005

0.25 0.25 0.25 0.5 0.5 0.25 0.25 0.25

1 10 100 100 1000 100 10 0.1

b K

99.86 98.63 86.31 96.23 62.26 510.21 141.02 100.41

5. Price of the Options The executive of the problem considered in this paper faces incomplete markets as a result of the fact that he cannot sell the options and cannot trade in the underlying security. In practice, the executive will not be allowed to sell the option or take a short position in the underlying stock (as he would optimally do for diversification purposes). On the other hand, we can assume that the company faces complete markets and use, therefore, arbitrage arguments to price the option. There is no obvious way of computing the exact difference between the value of the option for the executive and for the company. The approach taken in the literature (that we follow here) is to compute the certainty equivalent of the executive: a constant amount of money instead of the option that would leave the executive at the same utility level as the option. We do not interpret the difference in the value of the noarbitrage price and the certainty equivalent as the actual dollar amount. Rather, we consider the ratio of those two numbers as meaningful: a higher ratio of the price for the executive to the price for the company indicates that the options are more appropriate as compensation. A low ratio will question the optimality of using options as incentive. Our setting allows us to compute the price of the option, both for the company and for the executive. We first introduce the price of the option for the company. Proposition 5.1. Consider the problem of the company described in section 3. Assume that the company faces complete markets, i.e., it can replicate the option with the underlying security and a risk-free asset that pays a constant interest rate r and whose price satisfies dBt = Bt rdt

and

B0 = 1.

(5.1)

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The price for the company of the call option granted to the executive as compensation is given by " (Z )# Z T 1 T 2 1 2 (α − θs )dWs + (α − 2r − θs )ds . (5.2) C0 = E exp z¯ 2 0 0 Here, θ is the market price of risk given by θt := (¯ µt − r)/¯ νt ,

(5.3)

where r ∈ [0, ∞) is the interest rate of (5.1), and µ ¯ and ν¯ are defined by ¯ t α2 + z¯δ 2 Zt eα2 (T −t) H ¯ t + K − z¯δ 2 Zt T¯t H ¯ t + α¯ αH z δ 2 Zt T¯t ν¯t := ¯ . Ht + K − z¯δ 2 Zt T¯t

µ ¯t :=

(5.4) (5.5)

In addition, Z and T¯ are defined in (2.4) and (2.5), z¯ is defined in Theorem 3.1, ¯ is defined by and H ¯ t := H

1 . z¯Zt

(5.6)

Proof. The company faces complete markets and prices the option by arbitrage. We introduce the notation ¯t = H

1 z¯Zt

where Z is given by equation (2.4) and z¯ is defined in Theorem 3.1. Applying Itˆ o’s formula, we see that Z t 2 Z t 1 α α = 1+ ds + dWs , Zt Z Z s s 0 0 or equivalently ¯t = 1 + H z¯

Z

t

¯ s ds + α2 H

0

Z

t

¯ s dWs . αH

0

Furthermore, eα Zt T¯t =

2

T

−1

α2

−

Z

t

Zs e α

2

(T −s)

0

ds −

Z

t

αT¯s Zs dWs .

0

Thus, according to (2.10), ¯ t + K − z¯δ 2 Zt T¯t ) = (H ¯ t α2 + z¯δ 2 Zt eα dSt = d(H

2

(T −t)

¯ t + αδ 2 z¯Zt T¯t )dWt , )dt + (αH

where T¯ is given by equation (2.5). According to equations (5.4)-(5.5), we have dSt /St = µ ¯t dt + ν¯t dWt .

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The market price of risk is then given by (5.3), where µ ¯ and ν¯ are defined in (5.4)(5.5). The risk neutral density is ) ( Z Z T 1 T 2 ∗ θ ds . ZT = exp − θs dWs − 2 0 s 0 The no-arbitrage price is then ¯T ] C0 = E[e−rT ZT∗ (ST − K)+ ] = E[e−rT ZT∗ H h i R R T T 2 2 1 1 = E e 0 (α−θs )dWs + 2 0 (α −2r−θs )ds . z¯ The executive, however, cannot replicate the option given his trading constraints. In order to compute the value of the option for him, we use the certainty equivalent suggested in Hall and Murphy [6]. This is the fixed amount of money, paid instead of the option, that would provide the executive the same level of utility as the option. Proposition 5.2. Consider the problem of the executive described in section 2. The fixed amount of money CE that would provide the executive the same utility level as the option is  h i 1 2 1 2 −2 α2 T 2 α T − δ α (e − 1)¯ z . (5.7) CE = exp z¯ 2 Proof. We compute the fixed amount x that provides the executive with the same utility level as the option. The certainty equivalent CE = x is computed from # " Z 1 T 2 + u b dt . log x = E log(ST − K) − 2 0 t

¯ T + K, u Substituting ST = H bt = δ¯ z Zt , and computing expectations, we obtain 2 1 z 2 ]. log x = − log z¯ + [α2 T − δ 2 α−2 (eα T − 1)¯ 2



We now perform some comparative statics. The numerical results are presented in Table 6.3. It is similar to Table 6.1, but it also includes the prices of the option (we do not report the optimal effort level in Table 6.3). We assume that the company b C0 is the price of the chooses the optimal strike price, recorded in the column K. option for the company, that we compute using equation (5.2). The expression of equation (5.2) does not have a closed form solution, but is easy to compute numerically using Monte Carlo simulation (see Boyle, Broadie and Glasserman [17] for a review of the method). Column CE is the price of the option for the executive, the certainty equivalent, that we compute using equation (5.7). We compare the values in the columns C0 and CE. The options are worth less for the executive than for the company. However, the ratio will vary depending on the values of the parameters. When the ratio of the price of the option for the executive and the price

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of the option for the company is too low, options might not be the optimal form of compensation. Finally, in the column denoted BS we record the Black and Scholes price (Black and Scholes [18]). For volatility, we use the optimal initial volatility, that is, vb0 as given by equation (2.9) and reported in the column “Vol.”The only extra parameter required for Table 6.3 is the interest rate of equation (5.1). For concreteness, we assume that the interest rate is zero.

Table 6.3.

Option price.

S0 = 100; T = 5 δ

α

λ

0.1 0.1 0.1 1 1 1 1 1 1 1 1 5 5 5 5 5

0.25 0.5 0.75 0.25 0.5 0.75 0.25 0.25 0.25 0.5 0.5 0.25 0.5 0.25 0.25 0.25

0.5 0.5 0.5 0.5 0.5 0.5 0.25 1 2 0.5 2 0.5 0.5 0.1 0.25 1

b K

102.02 101.35 104.40 103.78 108.70 143.14 103.40 104.70 107.88 108.70 109.16 116.02 142.98 115.52 115.70 116.67

Vol

C0

CE

BS

0.52 0.61 3.39 1.54 5.38 33.31 1.48 1.70 2.31 5.38 5.56 7.26 26.66 7.19 7.22 7.35

0 0.061 0.173 0.589 0.713 1.720 0.323 0.120 0.216 0.713 0.685 1.800 3.655 1.854 1.762 1.607

0 0 0 0.165 0.016 0 0.237 0.068 0.002 0.016 0.010 1.372 0.110 1.453 1.412 1.221

0.020 0.118 1.375 0.25 1.807 17.52 0.275 0.206 0.170 1.807 1.839 1.704 11.745 1.740 1.730 1.658

Table 6.3 complements nicely our conclusions from Table 6.1. It shows that when the quality of the projects is very high (large α) and the executive is of a low quality (small δ), the optimal strike price will be high in order to force the executive to exercise more effort. As a result, the value of the option for the executive will be low relative to the complete market price of the option. In that case it might not be optimal to grant options as an incentive mechanism. In summary, it seems that options are more appropriate the higher the quality of the executive and the lower the quality of the projects. We also observe that in the cases in which the ratio of the certainty equivalent with respect to the price of the option is high, the Black and Scholes price is a relatively good approximation to the true price of the option, with our choice of volatility. However, in general, the quality of the approximation using Black and Scholes is questionable.

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6. The Case of Additional Cash Compensation In this section we consider the case in which the executive receives, in addition to the stock options, some cash compensation. The problem of the executive is to select (b u, vb) ∈ A that maximize the criterion J˜ defined by # " Z T  1 ˜ v) := E log w + n(ST − K)+ − u2 dt . (6.1) J(u, 2 0 t

Here w > 0 represents some cash compensation independent of the price of the stock of the company. The rest of the parameters are as in section 2. Alternatively, this case can be interpreted as the situation in which the executive has some wealth whose value is independent of the price of the stock. The main difference between this case and the case considered in section 2 is that now the number of options received in the compensation package becomes relevant. The problem becomes considerably more difficult, but we can still compute numerically the optimal effort and volatility. We include in this section the main result and some numerical examples. We start by considering the auxiliary function g defined by  w N (d2 (t, y)) + yN (d1 (t, y)), (6.2) g(t, y) := K − n where yn } + 21 α2 (T − t) log{ cw √ , (6.3) d1 (t, y) := α T −t yn } − 21 α2 (T − t) log{ cw √ , (6.4) d2 (t, y) := α T −t and  2 Z x z 1 √ exp − dz. (6.5) N (x) := 2 2π −∞ We observe that    1 w 1 1 1 ∂ 2 √ √ exp − (d2 (t, y)) g(t, y) = K − ∂y n 2 y α T −t 2π   1 1 1 √ + N (d1 (t, y)). + √ exp − (d1 (t, y))2 2 α T −t 2π

Let us consider the function f : [1, ∞) 7→ R defined by   1 nK −1 . f (x) := log{x} − 1 − w x

We observe that f (1) = − nK w < 0, f (∞) = ∞, and f is strictly increasing (indeed, 1 2 for every y > 1: f 0 (y) = y1 + ( nK w − 1) y 2 > 1/y − 1/y > 0). Thus, there exists a constant c = c(n, K, w) ∈ (1, ∞) which satisfies the following two properties: f (c) = 0

and

y > c ⇐⇒ f (y) > 0.

(6.6)

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Let z˜ be the positive solution of the nonlinear equation   1 S0 = g 0, − zδ 2 T¯0 . z

(6.7)

Theorem 6.1. Consider an executive who wants to solve (6.1). Consider the stochastic process Z and the function T¯ defined in section 2. Then the optimal effort is given by u bt = δ˜ z Zt

(6.8)

and the optimal choice of projects is given by

˜ is defined by Here, H

˜t vbt St = αH

∂ ˜ t ) + α˜ g(t, H z δ 2 Zt T¯t . ∂y

˜t = H

(6.9)

1 . z˜Zt

(6.10)

Proof. We want to find u b and vb that solve the problem " # Z T max E F (ST ) − G(us )ds , (u,v)∈A

0

where F (s) = log{w + n(s − K)+ }

and

G(u) =

u2 . 2

We observe that the stochastic process Z of (2.4) satisfies dZt = −αZt dWt

and

Z(0) = 1.

Applying the formula of integration by parts, we see that Z t Z t Z t St = S 0 + Zs δus ds + Ss Zs (vs − α)dWs . 0

0

Let us consider the stochastic process M defined by Z t Z t Mt := Zt St − δ Zs us ds = S0 + Ss Zs (vs − α)dWs . 0

0

Obviously, M is a local martingale, but we would like to prove that M is also a martingale. For that purpose, it is sufficient to verify the condition   E sup |Mt | < ∞. (6.11) 0≤t≤T

According to the Burkholder-Davis-Gundy inequality (see, e.g., Theorem 3.3.28 of Karatzas and Shreve [19]), it is sufficient to verify  !1/2  Z T  < ∞. (6.12) E (vt − α)2 St2 Zt2 dt 0

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We observe that, according toR Theorem 1.6.16 of Yong and Zhou [20],  T E sup0≤t≤T Zt2 < ∞. Since E[ 0 |vt St |2 dt] < ∞ by Definition 2.1, that the orem applied to equation (2.1) or (2.2), gives E sup0≤t≤T St2 < ∞. Applying H¨ older’s inequality (see, e.g., Theorem 4.2 of Chow and Teicher [21]) and again the RT condition E[ 0 |vt St |2 dt] < ∞, we note that   !1/2  !1/2  Z T Z T 2 2  ≤ E  sup Zt2  E (vt St Zt ) dt (vt St ) dt 0≤t≤T

0



=E



0

sup Zt2

0≤t≤T

1/2

Z

  1/2 ≤ E sup Zt2 0≤t≤T

T

2

(vt St ) dt 0

E

"Z

T



E

Z

T 0



2

(vt St ) dt 0

< ∞. This implies that

!1/2 

#!1/2 (6.13)

!1/2   < ∞, ((vt − α)St Zt ) dt 2

(6.14)

and therefore M is a martingale. Thus,

E[MT ] = E[M0 ] = S0 .

(6.15)

Next, consider the dual function F : (0, ∞) 7→ R defined by F˜ (z) = max[F (s) − sz]. s≥0

The maximum is attained at the points of the form sb = sb(z, a)    1 w − ,0 I{log{w+max( nz −w,0)}−zK−max(1− wz = K + max n ,0)>log{w}} z n +aI{log{w+max( nz −w,0)}−zK−max(1− wz n ,0)=log{w}}    1 w − ,0 I{log{w+n max( z1 − wn ,0)}−zK−z max( z1 − wn ,0)>log{w}} = K + max z n +aI{log{w+n max( z1 − wn ,0)}−zK−z max( z1 − wn ,0)=log{w}}
where a is either 0 or K + max z1 − w n , 0 . Consider also the dual function G : (0, ∞) 7→ R defined by ˜ G(z) = max[−G(u) + δuz], u

where the maximum is attained at u b=u b(z) = δz.

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By definitions, we get " # " Z T Z ˜ E F (ST ) − G(us )ds ≤ E F (zZT ) + 0

T

#

˜ G(zZ s )ds + zE[MT ].

0

Since M is a martingale, E[M (T )] = S0 . Therefore the above inequality gives an upper bound for our maximization problem. The upper bound will be attained if the maximums are attained, and if E[MT ] = S(0). In other words, the optimal solution is given by SbT = sb(˜ z ZT , A)

and

u bt = δ˜ z Zt ,

where A and z˜ are chosen so that A is any FT measurable random variable taking
, 0 , and so that E[MT ] = S(0). only two possible values, 0 and K + max z1 − w n The optimal vb is obtained from the martingale representation of " # Z T Z t b Mt = E[MT |Ft ] = E ZT ST − δ Zs u bs ds|Ft = S0 + (b vs − α)Ss Zs dWs . 0

0

For the case α > 0 that we are considering, we will set A ≡ 0. Introduce the ˜ defined by stochastic process W ˜ t = Wt + αt. W

˜ is a Brownian motion under the measure P˜ defined by dP˜ /dP = We observe that W ZT . Introduce also the notation Z t 1 2 ˜ ˜ H(t) := and Mt := St − z˜δ Zs ds. z˜Zt 0 We observe that 1 ˜ H(t) = exp z˜



α2 t + αW (t) 2



=

  1 α2 ˜ (t) . exp − t + αW z˜ 2

Thus, ˜t = H ˜ t αdW ˜. dH Hence, for every 0 ≤ t ≤ s:

  1 2 ˜ ˜ ˜ ˜ Hs = Ht exp − α (s − t) + α(Ws − Wt ) . 2

We also note that Z t Z t Z t Z t Z t ˜ s −˜ ˜ s. St −˜ zδ2 Zs ds = S0 + δus ds+ v s Ss d W zδ2 Zs ds = S0 + v s Ss d W 0

0

0

0

˜ is a P˜ −martingale. Hence, Thus, the stochastic process M ˜ t = E[ ˜M ˜ T | Ft ], M

0

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or equivalently "

˜ sb(˜ St = E z ZT , 0) − z˜δ

2

Z

#

T

Zs ds | Ft .

t

We note that for every 0 ≤ t ≤ s:   2   α α2 ˜ s − αW (s) , Zs = exp − s − αW (s) = exp 2 2 so for every 0 ≤ t ≤ s:

Denoting

    2 ˜ (s) − W ˜ (t)) | Ft ˜ s | Ft ] = E ˜ Zt exp α (s − t) − α(W E[Z 2  2 = Zt exp α (s − t) . eα T˜t =

2

(T −t)

α2

−1

,

we obtain ˜ s(˜ St = E[b z ZT , 0)| Ft ] − z˜δ 2 Zt T˜t . Here, sb(˜ z ZT , 0) =



K + max



w 1 − ,0 z˜ZT n



× I{log{w+n max( z˜Z1 − wn ,0)}−˜zZT K−˜zZT max( z˜Z1 − wn ,0)>log{w}} T T    w ˜ )− ,0 = K + max H(T n × I{log{w+n max(H(T ˜ )− w ,0)>log{w}} . ˜ )− w ,0)}− 1 K− 1 max(H(T n

˜ H(T )

˜ H(T )

n

Thus, ˜ s(˜ E[b z ZT , 0)| Ft ]  ˜ ˜ ) − w , 0)} − 1 K = K P log{w + n max(H(T ˜ ) n H(T  w 1 ˜ max(H(T ) − , 0) > log{w}| Ft − ˜ ) n H(T  h  w ˜ max H(T ˜ ) − ,0 +E n ×I{log{w+n max(H(T ˜ )− w ,0)}− n

1 1 K− H(T ˜ ˜ H(T ) )

˜ )− w ,0)>log{w}} |Ft max(H(T n

i

.

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To continue with the computations, we are going to use c = c(n, K, w) ∈ (1, ∞) defined in (6.6). We note that  n o  ˜ ) − w,0 − 1 K log w + n max H(T ˜ ) n H(T    1 ˜ ) − w , 0 > log{w} max H(T − ˜ ) n H(T  n o  ˜ ) − w,0 − 1 K = log w + n max H(T ˜ ) n H(T    1 ˜ ) − w , 0 > log{w} max H(T − ˜ n H(T ) o n w ˜ ∩ H(T ) − > 0 n  o n  [ ˜ ) − w,0 − 1 K log w + n max H(T ˜ ) n H(T    1 ˜ ) − w , 0 > log{w} max H(T − ˜ ) n H(T o n ˜ )− w ≤0 ∩ H(T n   n o  1 1 ˜ w ˜ )− w − > log{w} K− H(T ) − = log w + n H(T ˜ ) ˜ ) n n H(T H(T  o n ˜ )− w >0 ∩ H(T n  [  1 1 K− (0) > log{w} log{w + n(0)} − ˜ ) ˜ ) H(T H(T  o n ˜ )− w ≤0 ∩ H(T n  n  o w w 1 ˜ ˜ > log{w} ∩ H(T ) − > 0 K−1+ = log{nH(T )} − ˜ ) ˜ ) n nH(T H(T  n o [  w 1 ˜ )− ≤0 K > log{w} ∩ H(T log{w} − ˜ ) n H(T    ( )  o ˜ ) w  n˜ w nK nH(T ∩ H(T ) − > 0 >  ˜  +1− = log ) ˜ )   w n nH(T w nH(T w ! ) ) (( n o ˜ ) w nH(T ˜ )− >0 > 0 ∩ H(T = f w n ) ) (( n o ˜ ) w nH(T ˜ > c ∩ H(T ) − > 0 = w n o n w ˜ )>c . = H(T n

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Thus,

o h  i n ˜ H ˜T − w I ˜ ˜ s(˜ ˜ T > c w | Ft + E | Ft E[b z ZT , 0)| Ft ] = K P˜ H {HT >c w } n n n   Z ∞ 1 1 x2 p exp − dx =K cw 1 1 1 2 (T − t) 2π(T − t) ˜ }+ 2 α(T −t) α log{ n H t     Z ∞ 1 2 w ˜ Ht exp − α (T − t) + αx − + 1 cw 1 1 2 n ˜ }+ 2 α(T −t) α log{ n H t   2 1 x 1 p exp − dx. 2 (T − t) 2π(T − t)

To continue with the computations, we are going to return to the original probability measure P . We can write   ˜ dP ˜ T − 1 α2 T = H(T ) = exp αW ˜ 2 dP˜ H(0)

and

˜ t − αt. Wt = W We note that according to Bayes’ formula,   i h 1 2 ˜ ˜ ˜ E HT I{H˜ T >c w } | Ft = exp αWt − α t n 2   1 ˜T o I{H˜ T >c w } | Ft  n × E H n ˜ exp αWT − 21 α2 T i h ˜tE I ˜ | Ft =H } {HT >c w n o n w ˜ ˜ = Ht P HT > c | F t    n ˜tP H ˜ t exp α(WT − Wt ) + 1 α2 (T − t) > c w | Ft =H 2 n     ˜ t P α(WT − Wt ) + 1 α2 (T − t) > log 1 cw | Ft . =H ˜t n 2 H We know that under P , the random variable WT − Wt has, given Ft , a normal distribution with mean zero and variance T − t. Hence, o h i  w w ˜ n ˜ ˜ H ˜T I ˜ ˜ s(˜ P HT > c | F t + E | Ft E[b z ZT , 0)| Ft ] = K − {HT >c w } n n n  w ˜ ˜ ˜ N (d2 (t, Ht )) + Ht N (d1 (t, Ht )), = K− n where ˜

Ht n 1 2 ˜ t ) := log{ cw }√+ 2 α (T − t) d1 (t, H α T −t ˜

˜ t ) := d2 (t, H

tn } − 21 α2 (T − t) log{ Hcw √ , α T −t

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and N (x) := This means that we can write

Z

x −∞

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 2 z 1 √ exp − dz. 2 2π

˜ s(˜ ˜ t ), E[b z ZT , 0)| Ft ] = g(t, H where the function g is defined by Z ∞ g(t, y) := K

Thus,

  1 1 x2 p exp − dx cw 1 1 1 2 (T − t) 2π(T − t) α log{ n y }+ 2 α(T −t)   Z ∞ 1 2 w y exp{− α (T − t) + αx} − + cw 1 1 1 2 n α log{ n y }+ 2 α(T −t)   1 x2 1 p exp − dx 2 (T − t) 2π(T − t)  w N (t, d2 (y)) + yN (t, d1 (y)). = K− n ˜ t ) − z˜δ 2 Zt T¯t . St = g(t, H

(6.16)

If we take t = 0 in the above equation, we get S0

  1 2 ˜ ¯ − z˜δ 2 T¯0 . = g(0, H0 ) − z˜δ Z0 T0 = g 0, z˜

(6.17)

Although this does not give an explicit solution for z˜, it is possible to obtain it numerically. Applying Itˆ o’s formula in (6.16), and comparing equations (2.2) and (6.16), we obtain

Here,

˜t vbt St = αH

∂ ˜ t ) + α˜ g(t, H z δ 2 Zt T¯t . ∂y

(6.18)

   1 w 1 1 1 ∂ 2 √ √ exp − (d2 (t, y)) g(t, y) = K − ∂y n 2 yα T −t 2π   1 1 1 √ + N (d1 (t, y)). + √ exp − (d1 (t, y))2 2 α T −t 2π observe that the u b and above are adapted stochastic processes with R T vb defined RWe T E[ 0 |b ut |2 dt] < ∞ and E[ 0 |b vt St |2 dt] < ∞. Therefore, if w > 0 and α > 0, the optimal effort u b and optimal choice of projects vb are given by equations (6.8)-(6.9).  From the discussion above, it is clear that both the optimal drift u b and the optimal choice of projects vb are completely determined by z˜. This has to be found numerically from equation (6.7). We present several examples in Table 6.4. We see that the optimal effort increases with the number of options granted; it does not change much when the strike price changes; and it may be decreasing or increasing with respect to the time to maturity, depending on the number of options granted.

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Optimal effort.

S0 = 100, w = 50, δ = 0.5, α = 0.5 n 1 5 1 5 1 5 1 5 1 5 1 5

K

T

z˜

90 90 90 90 100 100 100 100 110 110 110 110

1 1 5 5 1 1 5 5 1 1 5 5

0.0062743 0.0114899 0.0071296 0.0106822 0.0061745 0.0113005 0.0071088 0.0106929 0.0060784 0.0111076 0.0070867 0.0106937

7. Conclusions We present a model where an executive is granted stock options as compensation. His decisions can affect the dynamics of the stock of the company in two ways: first, he can increase the expected return of the stock through his effort; second, he can choose the level of risk of the stock price by selecting different projects. The effort produces disutility, and higher level of risk will result in higher expected return. The executive is risk averse. We obtain closed form solutions for both the optimal effort and choice of projects. The company chooses the strike price of the call options. The company cares both about the mean and the volatility of the stock price. We find that there is a large range of optimal strike prices, depending on the values of the parameters of the model, but for most interesting cases it will be optimal to issue options out-of-the-money. We find that the optimal strike price is increasing with the type of the executive and the emphasis on the mean stock price, rather than volatility. The relationship between the optimal strike price and the quality of the projects is not monotonic. In our setting, we can price the options for the company (assuming complete markets) and for the executive (through the certainty equivalent). Although options are always worth less for the executive than for the company, we show that the ratio is more favorable when the projects are of low quality and the executive is of high quality. In order to derive closed form solutions for the optimal policies of the executive we have to assume that the utility is logarithmic and the only source of wealth of the executive is the package of options granted as compensation. As a result, the number of options is irrelevant. It is possible to extend our method to more general utility functions, as done in the proof of Theorem 6.1. In that case, the optimal solution would typically depend on the number of options. That would allow us to

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address some important issues such as policies for resetting the option contracts. However, it seems that in the general case we cannot get results as explicit as for the logarithmic utility. We can also extend some of the results to the case in which the executive receives cash, besides options. In this case, we can verify that the number of options becomes relevant. However, a procedure to derive the optimal strike price does not seem to be explicit. Another possible extension of the present work is to include tax effects in the model. Acknowledgements The research of A. Cadenillas was supported by the Natural Sciences and Engineering Research Council of Canada grant 194137. The research of J. Cvitani´c was supported in part by the National Science Foundation, under NSF grant DMS 04-03575. References [1] B. Hall and J. Leibman, Are CEOs really paid like bureaucrats?, Quarterly Journal of Economics. 113(3), 653–691, (1998). [2] K. J. Murphy. Executive compensation. In Handbook of Labor Economics, Vol III, pp. 2485–2563. North Holland, (1999). [3] J. Carpenter, The exercise and valuation of executive stock options, Journal of Financial Economics. 48(2), 127–158, (1998). [4] B. Hall and K. J. Murphy, Stock options for undiversified executives, Journal of Accounting and Economics. 33, 3–42, (2002). [5] J. Detemple and S. Sundaresan, Nontraded asset valuation with portfolio constraints: a binomial approach, Review of Financial Studies. 12(4), 835–872, (1999). [6] B. Hall and K. J. Murphy, Optimal exercise prices for executive stock options, American Economic Review. 2, 209–214, (2000). [7] N. Stoughton and K. Wong. Option compensation, accounting choice and industrial competition, (2003). Working Paper, University of California. [8] J. G. Aseff and M. S. Santos, Stock options and managerial optimal contracts, Econom. Theory. 26(4), 813–837, (2005). ISSN 0938-2259. [9] O. Kadan and J. Swinkels. Moral hazard with bounded payments, (2005). Working paper. [10] O. Kadan and J. Swinkels. Stocks or options? moral hazard, firm viability and the design of compensation contracts, (2005). Working paper. [11] S. Johnson and Y. Tian, The value and incentive effects of nontraditional executive stock option plans, Journal of Financial Economics. 57, 3–34, (2000). [12] S. Johnson and Y. Tian, Indexed executive stock options, Journal of Financial Economics. pp. 35–64, (2000). [13] A. Cadenillas, J. Cvitani´c, and F. Zapatero, Leverage decision and manager compensation with choice of effort and volatility, Journal of Financial Economics. 73(1), 71–92, (2004). [14] J. Carpenter, Does option compensation increase managerial risk appetite?, Journal of Finance. 55, 2311–2331, (2000).

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[15] P. Bolton and C. Harris. The continuous-time principal-agent problem: Frequentmonitoring contracts, (2001). Working Paper, Princeton University, Princeton. [16] A. Cadenillas, J. Cvitani´c, and F. Zapatero. Executive stock options as a screening mechanism. Working paper. [17] P. Boyle, M. Broadie, and P. Glasserman, Monte Carlo methods for security pricing, Journal of Economic Dynamics and Control. (1997). [18] F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy. 3, 637–654, (1973). [19] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus. (Springer, New York, 1991). [20] J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations. (Springer, New York, 1999). [21] Y. S. Chow and H. Teicher, Probability Theory: Independence, Interchangeability, Martingales. Second Edition. (Springer, New York, 1988).

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Chapter 7 The Freidlin-Wentzell LDP with Rapidly Growing Coefficients

Pavel Chigansky and Robert Liptser∗ Department of Mathematics, The Weizmann Institute of Science Rehovot 76100, Israel [email protected] The Large Deviations Principle (LDP) is verified for a homogeneous diffusion process whose coefficients are locally Lipschitz functions with super linear growth. It is assumed that the drift is directed towards the origin and the growth rates of the drift and diffusion terms are properly balanced. Nonsingularity of the diffusion matrix is not required.

Contents 1 2 3 4

Introduction . . . . . . . . . . . . . . . . . . . Notations and the main result . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . The proof of C-exponential tightness . . . . . . 4.1 Auxiliary lemma . . . . . . . . . . . . . 4.2 The proof of (3.1) . . . . . . . . . . . . . 4.3 The proof of (3.2) . . . . . . . . . . . . . 5 Local LDP upper bound . . . . . . . . . . . . . 6 Local LDP lower bound . . . . . . . . . . . . . 6.1 Nonsingular a(x) . . . . . . . . . . . . . 6.2 General a(x) . . . . . . . . . . . . . . . A.1 Exponential estimates for martingales . . . . . A.2 Pseudoinverse of nonnegative definite matrices A.3 Exponential negligibility of Xtε,β − Xtε . . . . References . . . . . . . . . . . . . . . . . . . . . . .

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197 198 200 201 201 202 203 204 208 208 211 213 214 214 217

1. Introduction In this paper we extend the set of conditions, under which Freidlin-Wentzell’s Large Deviation Principle (LDP) for a homogeneous diffusion process remains valid. We consider a family {(Xtε )t≥0 }ε→0 of diffusions, where Xtε ∈ Rd , d ≥ 1 is defined by ∗ Department

of Electrical Engineering Systems, Tel Aviv University, 69978 Tel Aviv, Israel, [email protected] 197

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the Itˆ o equation Xtε = x0 +

Z

t 0

b(Xsε )ds + ε

Z

0

t

σ(Xsε )dBs ,

(1.1)

with respect to a standard Brownian motion Bt , where b(x) and σ(x) are vector and matrix valued continuous functions of dimensions d and d × d respectively, guaranteeing existence of the unique weak solution. The classical Freidlin-Wentzell setting [1] (see also Dembo and Zeitouni [2]) is applicable to the model (1.1) with bounded b(x) and σ(x) and uniformly positive definite diffusion matrix a(x) = σσ ∗ (x). Various LDP versions can be found in Dupuis and Ellis [3], Feng [4], Feng and Kurtz [5], Friedman [6], Liptser and Pukhalskii [7], Mikami [8], Narita [9], Stroock [10], Ren and Zhang [11]. In the recent paper [12], Puhalskii extends LDP to (1.1) with continuous and unbounded coefficients and singular a(x), assuming b(x) and a(x) are Lipschitz continuous functions (concerning singular σ(x) see also Liptser et al, [13]). Being Lipschitz continuous, the entries of b, σ grow not faster than linearly and, thereby, automatically guarantee one of the necessary conditions for LDP (k · k denotes the Euclidean norm in Rd )   (1.2) lim lim ε2 log P sup kXtε k > C = −∞, ∀ T > 0. C→∞ ε→0

t≤T

Relinquishing the linear growth condition for b, σ would require additional assumptions providing (1.2). This paper is inspired by Puhalskii’s remark in Ref. [12]: If the drift is directed towards the origin, then no restrictions are needed on the growth rate of the drift coefficient.

In particular, in this case the LDP holds, regardless of the growth rate of b(x), for a constant diffusion matrix (not necessarily nonsingular). In this paper, we show that in fact LDP remains valid for (1.1) with non-constant diffusion term, if its growth rate is properly balanced relatively to the drift (see (2.1) of Theorem 2.1 below). Our result is formulated in terms of KhasminskiiVeretennikov’s condition (2.1) (see Refs. [14–16]) The rest of the paper is organized as follows. In Sections 2 and 3, the main result, notations and preliminary facts on the LDP are given. Sections 4 - 6 contain the proof of the main result. Auxiliary technical details are gathered in Appendices A.1 - A.3. 2. Notations and the main result The following notations and conventions are used throughout the paper. - ∗ denotes the transposition symbol - all vectors are columns (unless explicitly stated otherwise) - |x| and kxk denote the `1 and `2 (Euclidean) norms of x ∈ Rd

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-

-

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(x, y) denotes the scalar product of x, y ∈ Rd kxk2Γ = (x, Γx) with an nonnegative definite matrix Γ a(x) = σ(x)σ ∗ (x) a⊕ (x) denotes the Moore-Penrose pseudoinverse matrix of a(x) (see Ref. [17]) ∇V (x) is the gradient (row) vector of V (x):  ∂V (x) ∂V (x)  ,..., ∇V (x) := ∂x1 ∂xd hM, N it is the joint quadratic variation process of continuous martingales Mt and Nt ; for brevity hM, M it = hM it a.s. abbreviates “almost surely”; when the corresponding measure is not specified the Lebesgue measure on R+ is understood % is the locally uniform metric on C[0,∞) (Rd ) I denotes d × d identity matrix the convention 0/0 = 0 is kept throughout X ε = (Xtε )t≥0 inf{∅} = ∞.

We study the LDP for the family {X ε }ε→0 (C[0,∞) (Rd ), %) with
in the metric space P∞ −k d %(x, y) = k=1 2 1 ∨ supt≤k kxt − yt k , x, y ∈ C[0,∞) (R ). Recall that {X ε }ε→0 satisfies the LDP with the good rate function J(u) : C[0,∞) (Rd ) 7→ [0, ∞] and the rate ε2 , if the level sets of J(u) are compact and for any closed set F and open set G in C[0,∞) (Rd ),
lim ε2 log P X ε ∈ F ≤ − inf J(u), u∈F
lim ε2 log P X ε ∈ G ≥ − inf J(u).

ε→0

ε→0

Our main result is

u∈G

Theorem 2.1. Assume: (H-1) the entries of b(x) and σ(x) are locally Lipschitz continuous functions, (x, b(x)) = −∞, (H-2) limkxk→∞ kxk (x, a(x)x) ≤ K, ∀ kxk > L. (H-3) for some positive constants K and L, kxk |(x, b(x))|

Then {Xtε }ε→0 obeys the LDP in the metric space (C[0,∞) (Rd ), %) with the rate ε2 and the rate function ( R∞ 1 ku˙ t − b(ut )k2a⊕ (ut ) dt, u ∈ Γ J(u) = 2 0 ∞, u 6∈ Γ, where

n

Γ = u ∈ C[0,∞)

R∞ u0 = x0 , dut  dt, 0 ku˙ t k2 dt < ∞ o : . a(ut )a⊕ (ut )[u˙ t − b(ut )] = [u˙ t − b(ut )] a.s.

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Remark 2.2. In the scalar case (recall 0/0=0)  R 2  1 ∞ (u˙ t − b(ut )) dt, du = u˙ dt, u = x , R ∞ u˙ 2 dt < ∞ t t 0 0 t 0 0 2 σ (ut ) J(u) = 2  ∞, otherwise.

Example 2.3. A typical example within the scope of Theorem 2.1 is Z t Z t ε ε 3 Xt = x 0 − (Xs ) ds + ε |Xsε |3/2 dBs . 0

0

3. Preliminaries We follow the framework, set up by A. Puhalskii (see Refs. [18, 19]):  Exponential tightness ⇐⇒ LDP Local LDP The exponential tightness in the metric space (C[0,∞) , %) is convenient to verify in terms of, so called, C-exponential tightness conditions introduced by A. Puhalskii (see e.g. Ref. [7]), based on the stopping times technique introduced by D. Aldous in Refs. [20, 21]). To this end, let us assume that the diffusion processes are defined on a stochastic basis (Ω, F, Fε = (Ftε )t≥0 , P), satisfying the usual conditions, where the filtration Fε may depend on ε. Recall (see Ref. [7]) that the family of diffusion processes is C-exponentially tight if for any T > 0, η > 0 and any Fε -stopping time θ,   (3.1) lim lim ε2 log P sup kXtε k > C = −∞, C→∞ ε→0

t≤T

  ε − Xθε k > η = −∞. lim lim ε2 log sup P sup kXθ+t

4→0 ε→0

θ≤T

(3.2)

t≤4

The family of diffusion processes obeys the local LDP in (C[0,∞) (Rd ), %) if for any T > 0 there exists a local rate function JT (u) such that   lim lim ε2 log P sup kXtε − ut k ≤ δ ≤ −JT (u) (3.3) δ→0 ε→0



t≤T

 lim lim ε log P sup kXtε − ut k ≤ δ ≥ −JT (u). 2

δ→0 ε→0

(3.4)

t≤T

Under the conditions (3.1)-(3.4), the family of diffusion processes obeys the LDP with the rate ε2 and the good rate function J(u) = sup JT (u), u ∈ C[0,∞) (Rd ), T

where JT (u) =

( RT 1 2

∞,

0

ku˙ t − b(ut )k2a⊕ (ut ) dt,

u ∈ ΓT

u 6∈ ΓT ,

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with n

ΓT = u ∈ C[0,T ]

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RT o u0 = x0 , dut  dt, 0 ku˙ t k2 dt < ∞ : . ⊕ a(ut )a (ut )[u˙ t − b(ut )] = [u˙ t − b(ut )] a.s.

Thus the proof of Theorem 2.1 reduces to establishing (3.1)–(3.4). 4. The proof of C-exponential tightness 4.1. Auxiliary lemma

Let D be a nonlinear operator acting on continuously differentiable functions V (x) : Rd →R as follows: 1 DV (x) = (∇V (x), b(x)) + (∇V (x), a(x)∇V (x)). 2 Lemma 4.1. Assume there exists twice continuously differentiable nonnegative function V (x) such that (a-1) limC→∞ inf kxk≥C V (x) = ∞ (a-2) for some L > 0, DV (x) ≤ 0, ∀ kxk > L. Then (3.1) holds. Proof.

Notice that (3.1) is equivalent to
lim lim ε2 log P ΘC ≤ T = −∞,

C→∞ ε→0

where

ΘC = inf{t : kXtε k ≥ C},

C>0

(4.1)

(4.2)

are stopping times relative to Fε . We use (A.1.1) of Proposition A.1.1 to estimate log P(ΘC ≤ T ). An appropriate martingale Mtε is constructed with the help of function V (x). Let Ψ(x) be the 2 (x) . By the Itˆ o formula Hessian of V , namely a matrix with the entries Vij (x) = ∂∂xVi ∂x j ε

−2

V

ε (XΘ ) C ∧t

+

Z

=ε

ΘC ∧t 0

−2

V (x0 ) +

Z

ΘC ∧t 0

ε−2 (∇V (Xsε ), b(Xsε ))ds

ε−1 (∇V (Xsε ), σ(Xsε )dBs ) + Rt

Z

ΘC ∧t 0

  1 trace Ψ(Xsε )a(Xsε ) ds. 2

We choose Mtε = 0 ε−1 (∇V (Xsε ), σ(Xsε )dBs ), which has the variation process Rt hM ε it = 0 ε−2 (∇V (Xsε ), a(Xsε )∇V (Xsε ))ds. Clearly ε MΘε β ∧t = ε−2 V (XΘ ) − ε−2 V (x0 ) C ∧t C Z ΘC ∧t Z − ε−2 (∇V (Xsε ), b(Xsε ))ds − 0

ΘC ∧t 0

  1 trace Ψ(Xsε )a(Xsε ) ds. 2

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Hence, by the definition of D, one gets 1 ε MΘε C ∧T − hM ε iΘC ∧T = ε−2 V (XΘ ) − ε−2 V (x0 ) C ∧T 2 Z ΘC ∧T Z ΘC ∧T   1 ε ε − trace Ψ(Xs )a(Xs ) ds − ε−2 DV (Xsε )ds. (4.3) 2 0 0

On the set {ΘC ≤ T }, we have

ε ε−2 V (XΘ ) − ε−2 V (x0 ) ≥ ε−2 inf V (x) − ε−2 V (x0 ), C ∧T kxk≥C

and

Z

ΘC ∧T 0


T
1 trace Ψ(Xsε )a(Xsε ) ds ≤ sup trace Ψ(x)a(x) , 2 2 kxk≤C

and, by (a-2), Z Θ∧T − ε−2 DV (Xsε )ds 0

Z ≥ −

ΘC ∧T

0

ε−2 I{kXsε k≤L} DV (Xsε )ds ≥ −ε2 T sup |DV (x)|. kxk≤L

These inequalities and (4.3) imply

1 MΘε C − hM ε iΘC ≥ ε−2 inf V (x) − ε−2 V (x0 ) 2 kxk≥C
T − sup trace Ψ(x)a(x) − ε−2 T sup DV (x) 2 kxk≤C kxk≤L

on the set {ΘC ≤ T }. Hence, due to (A.1.1) of Proposition A.1.1
ε2 log P ΘC ≤ T ≤
T ε2 − inf V (x) + V (x0 ) + sup trace Ψ(x)a(x) + T sup DV (x) kxk≥C 2 kxk≤C kxk≤L −−−→ − inf V (x) + V (x0 ) + T sup DV (x) ε→0

kxk≥C

kxk≤L

and it remains to recall that by (a-1) limC→∞ inf kxk≥C V (x) = ∞.



4.2. The proof of (3.1) We apply Lemma 4.1 to V (x) =

ckxk2 , 1 + kxk

1 with a positive parameter c ≤ K for K from (2.1) of Theorem 2.1. The function V (x) is twice continuously differentiable and satisfies (a-1). It is left to show that V (x) satisfies (a-2) as well.

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x Direct computations give ∇V (x) = c (2+kxk)kxk (1+kxk)2 kxk . Denote

r(x) :=

(2 + kxk)kxk (1 + kxk)2

and notice that r(x) ≤ 1. By assumption (2.1) of Theorem 2.1, one can choose L > 0 sufficiently large so that (x, b(x)) < 0 for any kxk ≥ L. On the other hand, 1 by assumption (2.1) of Theorem 2.1, −1 + 2c kxk(x,a(x)x) |(x,b(x))| ≤ − 2 for kxk ≥ L and ! c2 r2 (x) (x, a(x)x) r(x) (x, b(x)) + DV (x) = c kxk 2 kxk2 ! c2 r2 (x) (x, a(x)x) r(x) (x, b(x)) + = −c kxk 2 kxk2 ! c (x, a(x)x) |(x, b(x))| − 1 + r(x) = cr(x) kxk 2 kxk |(x, b(x))| ! c (x, a(x)x) |(x, b(x))| −1+ ≤ cr(x) kxk 2 kxk |(x, b(x))| |(x, b(x))| 1 ≤ − cr(x) 2 kxk

and (a-2) follows.



4.3. The proof of (3.2) The obvious inclusion n o n o[n o ε ε sup kXθ+t − Xθε k > η ⊆ sup kXθ+t − Xθε k > η, ΘC = ∞ ΘC ≤ T t≤4

t≤4

reduces the proof to verifying

  ε lim lim ε2 log sup P sup kXθ+t − Xθε k > η, ΘC = ∞ = −∞

4→0 ε→0

θ≤T

(4.4)

t≤4

for any fixed C. Indeed if (4.4) holds, then   ε − Xθε k > η lim lim ε2 log sup P sup kXθ+t 4→0 ε→0

θ≤T

t≤4

  ε − Xθε k > η, ΘC = ∞ ≤ lim lim ε log sup P sup kXθ+t 2

4→0 ε→0

θ≤T

t≤4

_

lim lim ε2 log P(ΘC ≤ T

C→∞ ε→0




and, thus, (3.2) is implied by (4.4) and (4.1). So, it is left to check (4.4) for any entry xεt of Xtε :   lim lim ε2 log sup P sup |xεθ+t − xεθ | > η, ΘC = ∞ = −∞. 4→0 ε→0

θ≤T

t≤4

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An entry of Xtε satisfies xεt

=

xε0

+

Z

t 0

γsε ds + εmεt ,

ε where γtε is Fε -adapted R t ε continuous random process and mt is F -continuous martinε gale with hm it = 0 µs ds. Since b and σ are locally Lipschitz continuous functions, ε there is a constant lC , such that |γΘ | ≤ lC and µεΘC ∧t ≤ lC . Taking into account C ∧t that Z θ+t n o  sup γsε ds ≥ η, ΘC = ∞ ⊆ lC 4 ≥ η = ∅, for 4 < η/lC , t≤4

θ

it is left to verify

  lim lim ε2 log sup P sup |εmεθ+t − εmεθ | > η, ΘC = ∞ = −∞.

4→0 ε→0

θ≤T

t≤4

Due to the obvious inclusion n o sup |εmεθ+t − εmεθ | > η, ΘC = ∞ = t≤4

n

sup |εmεΘC ∧(θ+t) − εmεΘC ∧θ | > η, ΘC = ∞

t≤4

⊆

n

o

o sup |εmεΘC ∧(θ+t) − εmεΘC ∧θ | > η ,

t≤4

we shall verify   lim lim ε2 log sup P sup |εmεΘC ∧(θ+t) − εmεΘC ∧θ | > η = −∞.

4→0 ε→0

θ≤T

t≤4

Notice that nεt := εmεΘC ∧(θ+t) − εmεΘC ∧θ is a continuous martingale relative to R Θ ∧(θ+t) ε (FΘε C ∧θ+t )t≥0 (see e.g. Ch. 4, §7 in Ref. [22]) with hnε it = ε2 ΘCC∧θ µs ds ≤
ε2 lC t. By the statement (A.1.1) of Proposition A.1.1, P supt≤4 |nεt | ≥ η ≤
2 2 2 2e−η /(2lC ε 4) , so that limε→0 ε2 log P supt≤4 |nεt | ≥ η ≤ − 2lηC 4 −−−→ −∞.  4→0 5. Local LDP upper bound We start with the observation that (3.3) holds if for any T > 0   lim lim ε2 log P sup kXtε − ut k ≤ δ, ΘC = ∞ ≤ −JT (u), δ→0 ε→0

t≤T

since by the inclusion n o n o[n o sup kXtε − ut k ≤ δ ⊆ sup kXtε − ut k ≤ δ, ΘC = ∞ ΘC ≤ T t≤T

t≤T

(5.1)

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we have   lim lim ε2 log P sup kXtε − ut k ≤ δ

δ→0 ε→0

t≤T

 _
lim ε2 log P(ΘC ≤ T , ≤ lim lim ε2 log P sup kXtε − ut k ≤ δ, ΘC = ∞ δ→0 ε→0

ε→0

t≤T

and, by (4.1), the last term goes to −∞ as C→∞. The proof for u0 6= x0 or dut 6 dt is standard (see e.g. Ref. [2]) and is omitted. The rest of the proof is split into two R T steps. Step 1: u0 = x0 , dut  dt, 0 ku˙ s k2 ds < ∞. Define the set n o A = sup kXtε − ut k ≤ δ, ΘC = ∞ . t≤T

With a continuously differentiable vector-valued Rfunction λ(s) of dimension d, let t us introduce a continuous local martingale Ut = 0 (λ(s), εσ(Xsε )dBs ) and its martingale exponential zt = eUt −0.5hU it , where Z t hU it = ε2 (λ(s), a(Xsε )λ(s))ds. 0

It is well known that zt is a continuous positive local martingale, as well as a supermartingale. Consequently, EzT ≤ 1 and, therefore, 1 ≥ EI{A} zT .

(5.2)

The required R tupper bound for P(A) is obtained by estimating zT from below on A. Since Ut = 0 (λ(s), dXsε − b(Xsε )ds), UT − 0.5hU iT = Z Th i ε2 (λ(s), dXsε − b(Xsε )ds) − (λ(s), a(Xsε )λ(s))ds = 2 0 Z Th i 2 ε (λ(s), u˙ s − b(us )) − (λ(s), a(us )λ(s)) ds 2 0 Z T Z T + (λ(s), dXsε − u˙ s ds) + (λ(s), b(us ) − b(Xsε )ds 0

+

Z

0

T

0

(5.3)

ε2 (λ(s), [a(us ) − a(Xsε )]λ(s))ds. 2

We derive lower bounds on the set A for each term in the right hand side of (5.3). Applying the Itˆ o formula to (λ(t), Xtε − ut ), and taking into account that X0ε = u0 , we find that Z T Z T ˙ (λ(T ), XTε − uT ) = (λ(s), dXsε − u˙ s ds) + (λ(s), Xsε − us )ds. 0

0

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Therefore, Z T (λ(s), dXsε − u˙ s ds) 0

Z ≥ − (λ(T ), XTε − uT ) −

T 0

˙ (λ(s), Xsε − us )ds ≥ −r1 δ,

with r1 := r1 (λ, T, C) ≥ 0, independent of ε. Further, with ri := ri (λ, T, C) ≥ 0, i = 2, 3, due to the local Lipschitz continuity of σ and a, we find that Z T (λ(s), b(us ) − b(Xsε ))ds ≥ −r2 (λ, C, T )δ 0

Z

T

0

ε2 (λ(s), [a(us ) − a(Xsε )]λ(s))ds ≥ −ε2 r3 (λ, C, T )δ. 2

Hence with r := r1 + r2 + ε2 r3 , Z Th i ε2 (λ(s), u˙ s − b(us )) − (λ(s), a(us )λ(s)) ds − r(λ, T, C)δ. log zT ≥ 2 0

Set ν(s) = ε2 λ(s) and rewrite the above inequality as: Z i ν  1 Th 1 log zT ≥ 2 (ν(s), u˙ s − b(us )) − (ν(s), a(us )ν(s)) ds − r 2 , T, C δ. ε 0 2 ε

This lower bound, along with (5.2), provides the following upper bound Z Th i
1 2 ε log P A ≤ − (ν(s), u˙ s − b(us )) − (ν(s), a(us )ν(s)) ds 2 0 ν  + ε2 r 2 , T, C δ. ε
ν 2 Clearly limε→0 ε r ε2 , T, C < ∞ and, hence, Z Th i
1 lim lim ε2 log P A ≤ − (ν(s), u˙ s − b(us )) − (ν(s), a(us )ν(s)) ds. (5.4) δ→0 ε→0 2 0 Since the left hand side of (5.4) is independent of ν(s), (5.1) is derived by minimizing the right hand side of (5.4) with respect to ν(s). Two difficulties arise on the way to direct minimization: - the matrix a(us ) may be singular - the entries of ν(s) should be continuously differentiable functions. Assume first a(us ) is a positive definite matrix, uniformly in s, and write 1 1 (ν(s), u˙ s − b(us )) − (ν(s), a(us )ν(s)) = ku˙ s − b(us )k2a−1 (us ) 2 2
1

1/2

2 − a (us ) ν(s) − a−1 (us )[u˙ s − b(us )] . 2

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If the entries of a−1 (us )[u˙ s − b(us )] are continuously differentiable functions, then, by taking ν(s) ≡ −a−1 (us )[u˙ s − b(us )] we find that Z
1 T ku˙ s − b(us )k2a−1 (us ) ds. (5.5) lim lim ε2 log P A ≤ − δ→0 ε→0 2 0 RT In the general case, due to 0 ku˙ s k2 ds < ∞, the entries of a−1 (us )[u˙ s − b(us )] are square integrable with respect to the Lebesgue measure on [0, T ]. Choose a maximizing sequence νn (s), n ≥ 1, of continuously differentiable functions such

2 RT that limn→∞ 0 νn (s) − a−1 (us )[u˙ s − b(us )] ds = 0. Since all the entries of a(us ) are uniformly bounded on [0, T ] Z T

1/2


a (us ) νn (s) − a−1 (us )[u˙ s − b(us )] 2 ds = 0 lim n→∞

0

and (5.5) holds too. Now we drop the uniform nonsingularity assumption of a(us ). The upper bound in (5.5) remains valid with a(us ) replaced by aβ (us ) ≡ a(us ) + βI, where β is a positive number and I is (d × d)-unit matrix: Z
1 T ku˙ s − b(us )k2[a(us )+βI]−1 ds. lim lim ε2 log P A ≤ − δ→0 ε→0 2 0 For any fixed s, the function ku˙ s − b(us )k2[a(us )+βI]−1 increases with β ↓ 0 and by Lemma A.2.1 possesses the limit   a(us )a⊕ (us )[u˙ s − b(us )]  ku˙ − b(u )k2 , ⊕ s s a (us ) lim ku˙ s − b(us )k2[a(us )+βI]−1 = = [u˙ s − b(us )]  β→0  ∞, otherwise.

Thus the required upper bound    RT
− 0 21 ku˙ s − b(us )k2a⊕ (us ) ds, 2 lim lim ε log P A ≤ δ→0 ε→0   ∞,

a(us )a⊕ (us )[u˙ s − b(us )]

= [u˙ s − b(us )], a.s.

otherwise

follows by the monotone convergence theorem. RT Step 2. u0 = x0 , dut  dt, 0 ku˙ s k2 ds = ∞. RT We emphasize that dut  dt on [0, T ] implies 0 ku˙ s kds < ∞ and return to the upper bound from (5.4). Since b and σ are locally Lipschitz, one can choose a constant L (depending on u(s)), so that, |(ν(s), b(us ))| ≤ kb(us )kkν(s)k ≤ Lkν(s)k and (ν(s), a(us )ν(s)) ≤ Lkν(s)k2 . Then, (5.4) implies Z Th i
L lim lim ε2 log P A ≤ − (ν(s), u˙ s ) − Lkν(s)k − kν(s)k2 ds. δ→0 ε→0 2 0

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Let νn (s) be a sequence of continuously differentiable functions, approximating the bounded (for each fixed p > 0) function L−1 u˙ s I{ku˙ s k≤p} in the sense that RT limn→∞ 0 k L1 u˙ s I{ku˙ s k≤p} − νn (s)k2 ds = 0. Thus,
1 lim lim ε2 log P A ≤ − δ→0 ε→0 2L

Z

|0

T

Z T + ds s k≤p} {z } |0

2

u˙ s I{ku˙

↑∞ as p ↑ ∞



u˙ s ds −−−→ −∞ {z

<∞

}

p→∞



6. Local LDP lower bound
If limδ→0 limε→0 ε2 log P supt≤ΘC ∧T |Xtε − ut | ≤ δ ≤ −JT (u) = −∞, then the corresponding local LDP lower bound is −∞ as well and hence only the case JT (u) < ∞ is to be considered, i.e. we may restrict ourselves to analyzing test functions with the properties: (i) u0 = x0 (ii) dut  dt

(iii) a(ut )a⊕ (ut )[u˙ t − b(ut )] = [u˙ t − b(ut )] a.s. Z T (iv) ku˙ t − b(ut )k2a⊕ (ut ) dt < ∞, ∀ T > 0 (v)

Z

0 T

0

ku˙ t k2 dt < ∞.

Another helpful observation is that (3.4) holds if for any C > 0   sup kXtε − ut k ≤ δ ≥ −JT (u) lim lim ε2 log P δ→0 ε→0

(6.1)

(6.2)

t≤ΘC ∧T

due to n and (4.1).

sup t≤ΘC ∧T

o n o[n o kXtε − ut k ≤ δ ⊆ sup kXtε − ut k ≤ δ ΘC ≤ T t≤T

6.1. Nonsingular a(x) In this section, the matrix a(x) is assumed to be uniformly nonsingularin x ∈ R, in the sense that a(x) ≥ βI for a positive number β. Let λ(s) := σ −1 (Xsε ) u˙ s − b(Xsε ) R Θ ∧t and introduce a martingale Ut = 0 C 1ε (λ(s), dBs ) and its martingale exponential R Θ ∧t zt = eUt −0.5hU it , t ≤ T , where hU it = 0 C ε12 kλ(s)k2 ds. By (iv) and (v) of (6.1), hU iT ≤ const. and hence EzT = 1. We use this fact in order to define a new probability measure Qε by dQε = zT dP. Since zT is positive ε P-a.s., P  Qε as well and dP = z−1 T dQ .

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We proceed with the proof of (6.2) by applying Z ε e = P(A) z−1 T dQ

(6.3)

e A

 ε e = sup to the set A t≤ΘC ∧T kXt − ut k ≤ δ , and estimating from below the right hand side in (6.3). In order to realize this program, it is convenient to have a ε semimartingale description of the process XΘ under Qε . Recall that the random C ∧t process BΘC ∧t is a martingale under P with the variation process hBiΘC ∧t ≡ (ΘC ∧ t)I. It is well known (see e.g. Theorem 2, Ch. 4, §5 in Ref. [22]) that BΘC ∧t is et + AB , a continuous semimartingale under Qε with the decomposition BΘC ∧t = B t ε e e where Bt is a martingale (under Q ) with hBit ≡ hBiΘC ∧t and, by the Girsanov theorem, Z ΘC ∧t 1 −1 ε σ (Xs )[u˙ s − b(Xsε )]ds. AB = t ε 0 In particular, ε XΘ C ∧t

= uΘC ∧t + ε

Z

0

ΘC ∧t

es , t ≤ T, σ(Xsε )dB

Qε -a.s.

As the next preparatory step we derive the semimartingale decomposition of U t under Qε . As before, the continuous martingale Ut under P is transformed to a semimartingale under Qε : et + AU Ut = U t

et , having the variation process hU e it ≡ hU it , P- and with continuous Qε -martingale U ε U Q -a.s., and a continuous drift At ≡ hU it . et + hU it , t ≤ T, Qε -a.s. and, thereby, z−1 = e−UeT − 21 hU iT . ConseThus, Ut = U T quently, (6.3) is transformed to Z   e = eT − 1 hU iT dQε P(A) exp − U 2 e A Z ΘC ∧T Z   ε 2 eT − 1 k u ˙ − b(X )k ds dQε . = exp − U −1 ε s s a (Xs ) 2ε2 0 e A We are now in the position to derive a lower for the right hand side. Replacing  bound e with a smaller set A e ∩ B, where B = ε2 U eT ≤ η , write A e ≥ P(A)

Z

Z ΘC ∧T   1 η ku˙ s − b(Xsε )k2a−1 (Xsε ) ds dQε . exp − 2 − 2 ε 2ε 0 e A∩B

By the local Lipschitz continuity of b, σ and the uniform nonsingularity of a(x), ku˙ s − b(Xsε )k2a−1 (Xsε ) − ku˙ s − b(us )k2a−1 (us ) ≤ lC (ku˙ s k + 1)2 δ, δ ≤ 1,

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e ∩ B for any s ≤ ΘC ∧ T . Then, on the set A Z Z  η δlC T e P(A) ≥ exp − 2 − 2 (ku˙ s k + 1)2 ds ε ε e A∩B 0 Z ΘC ∧T  1 − 2 ku˙ s − b(us )k2a−1 (us ) ds dQε 2ε 0 Z Z T  η δlC ≥ exp − 2 − 2 (ku˙ s k + 1)2 ds ε ε e A∩B 0 Z T  1 − 2 ku˙ s − b(us )k2a−1 (us ) ds dQε . 2ε 0

Consequently,

e ≥ −η − δlC lim ε log P(A) 2

ε→0

Z

T 0


e ∩B . (ku˙ s k + 1)2 ds − JT (u) + lim ε2 log Qε A ε→0




e ∩ B = 0 by showing We prove now that limε→0 ε2 log Qε A

e = 0 and lim Qε Ω \ B = 0. lim Qε Ω \ A ε→0

ε→0

To this end, recall that

Z n e = ε sup Ω\A

t≤T

n Z

Ω \ B = ε

ΘC ∧t

0 ΘC ∧T

0

σ

o es σ(Xsε )dB

>δ

−1

(Xsε )[u˙ s

−

es b(Xsε )]dB

o

(6.4)

>η .

R Θ ∧t es or We verify (6.4) componentwise. Let Lεt denote any entry of 0 C σ(Xsε )dB R ΘC ∧t −1 ε es . We show that σ (Xs )[u˙ s − b(Xsε )]dB 0

lim Qε ε sup Lεt > δ = 0 and lim Qε ε LεT > δ = 0. (6.5) ε→0

ε→0

t≤T

Rt In both cases, Lεt is a continuous Qε -martingale with hLε it = 0 g(s)ds R RT g(s)dsdQε < ∞. Then (6.5) holds by Doob’s inequality: Ω 0 Z Z T
4ε2 lim Qε ε sup Lεt > δ ≤ 2 g(s)dsdQε −−−→ 0. ε→0 ε→0 δ t≤T Ω 0

and

Now, for any fixed δ and η,

e ≥ −η − δlC lim ε log P(A) 2

ε→0

Z

T 0

(ku˙ s k + 1)2 ds − JT (u).

The required lower bound

e ≥ −JT (u) lim lim ε2 log P(A)

δ→0 ε→0

follows by taking limη→0 limδ→0 .



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6.2. General a(x) This part of the proof requires perturbation arguments. The idea is to use the already obtained local LDP lower bound for the uniformly nonsingular a(x). Let Wt be a standard d dimensional Brownian motion, independent of Bt , defined on the same stochastic basis. Since b and σ are assumed to be locally Lipschitz continuous, one can introduce the perturbed diffusion process controlled by a free parameter β ∈ (0, 1]: Z t Z t p ε,β ε,β (6.6) Xt = x 0 + b(Xs )ds + ε [σ(Xsε,β )dBs + βdWs ]. 0

0

Rt The process Xtε,β , defined in (6.6), solves the Itˆ o equation Xtε,β = x0 + 0 b(Xsε,β )ds+ Rt ε 0 [a(Xsε,β ) + βI]1/2 dBsβ with respect to a standard Brownian motion Btβ = Rt √ ε,β −1/2 [σ(Xsε,β )dBs + βdWs ]. Then the family {(Xtε,β )t≤T }ε→0 satis0 [a(Xs )+βI]

fies the local LDP lower bound. Indeed, the matrix aβ (x) is uniformly nonsingular, its entries are locally bounded and satisfy the assumption (2.1) of Theorem 2.1 since (x, a(x)x) kxk (x, aβ (x)x) = +β kxk |(x, b(x))| kxk |(x, b(x))| |(x, b(x))| kxk |(x,b(x))| converges to zero as kxk→∞ by (2.1). In particular, RT kXtε,β k ≥ C} and u0 = x0 , dut  dt, 0 ku˙ t k2 dt < ∞, we have

and

lim lim ε2 log P

δ→0 ε→0



sup

β t≤ΘC ∧T

β with ΘC = inf{t :

 kXtε,β − ut k ≤ δ ≥ −

1 2

Z

T

0

ku˙ s − b(us )k2(a(us )+βI)−1 ds. (6.7)

Further, we will use (6.7) to establish Z   1 T ku˙ s − b(us )k2a⊕ (us ) ds. lim lim ε2 log P sup kXtε − ut k ≤ δ ≥ − 2 0 t≤T δ→0 ε→0

(6.8)

To this end, we introduce the filtration Gε = (Gtε )t≥0 , with the general condiβ are tions, generated by (Xtε , Xtε,β )t≥0 and notice that both ΘC (see (4.2)) and ΘC ε stopping times relative to G . Hence, β τCβ = ΘC ∧ ΘC

(6.9)

is a stopping time as well relative to Gε . Obviously,
lim lim ε2 log P τCβ ≤ T = −∞. C→∞ ε→0

However, the proof of (6.8) requires a stronger property:   lim lim ε2 log sup P τCβ ≤ T = −∞. C→∞ ε→0

β∈(0,1]

(6.10)

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β It is clear, that (6.10) is valid if it is valid with τCβ replaced by ΘC . The latter is verified along the lines of Lemma 4.1 proof:


β ε2 log sup P ΘC ≤ T ≤ − inf V (x) + V (x0 ) kxk≥C

β∈(0,1]

+


T ε2 sup sup trace Ψ(x)[a(x) + βI] + T sup sup Dβ V (x) 2 β∈(0,1] kxk≤C β∈(0,1] kxk≤L −−−→ − inf V (x) + V (x0 ) + T sup sup Dβ V (x) −−−−→ −∞, ε→0

kxk≥C

C→∞

β∈(0,1] kxk≤L

where Dβ V (x) = (∇V (x), b(x)) + 12 (∇V (x), aβ (x)∇V (x)). We are now in the position to prove (6.8). With δ ≤ β 1/4 , write n

sup kXtε,β − ut k ≤ δ

β t≤τC ∧T

=

n

sup kXtε,β − ut k ≤ δ

β t≤τC ∧T

[n ⊆

n

⊆

n

n

o\n

sup kXtε,β − ut k ≤ δ

β t≤τC ∧T

β t≤τC ∧T

sup kXtε − ut k ≤ 2β 1/4

β t≤τC ∧T

sup kXtε − ut k ≤ 2β 1/4 t≤T

β t≤τC ∧T

o

o[n

o[n

o

sup kXtε − Xtε,β k > β 1/4

β t≤τC ∧T

o\n

sup kXtε − Xtε,β k > β 1/4

β t≤τC ∧T

sup kXtε − Xtε,β k ≤ β 1/4

o\n

sup kXtε,β − ut k ≤ β 1/4

[n ⊆

o

o

sup kXtε − Xtε,β k ≤ β 1/4

β t≤τC ∧T

sup kXtε − Xtε,β k > β 1/4

β t≤τC ∧T

sup kXtε − Xtε,β k > β 1/4

β t≤τC ∧T

[n

o

o

o

τCβ ≤ T

o

Hence,

P



sup β t≤τC ∧T

kXtε,β

(

  − ut k ≤ δ ≤ 3 P sup kXtε − ut k ≤ 2β 1/4 

t≤T

) _ _ 
β ε,β 1/4 ε P sup kXt − Xt k > β P τC ≤ T . β t≤τC ∧T

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β Clearly, ΘC can be replaced by τCβ , and so

−

1 2

Z

T 0

  ku˙ s − b(us )k2(a(us )+βI)−1 ds ≤ lim ε2 log P sup kXtε − ut k ≤ 2β 1/4 _

2

lim ε log P

ε→0



ε→0

sup β t≤τC ∧T

t≤T

kXtε _

−

Xtε,β k

> β 1/4



  lim ε2 log sup P τCβ ≤ T . (6.11)

ε→0

β∈(0,1]

Next we use the following facts: (1) by Lemma A.2.1 and (6.1), lim

β→0

Z

T 0

ku˙ s −

b(us )k2(a(us )+βI)−1 ds

=

Z

T 0

ku˙ s − b(us )k2a⊕ (us ) ds;

(2) by Lemma A.3.2, lim lim ε2 log P

β→0 ε→0



 sup kXtε − Xtε,β k > β 1/4 = −∞;

β t≤τC ∧T

  (3) by (6.10), limC→∞ limε→0 ε2 log supβ∈(0,1] P τCβ ≤ T = −∞. Hence, passing to the limit β→0 and then C→∞ in (6.11) and taking into account (1)-(3), one gets the required lower bound 2

lim lim ε log P β→0 ε→0



sup kXtε t≤T

− ut k ≤ 2β

1/4



1 ≥− 2

Z

T 0

ku˙ s − b(us )k2a⊕ (us ) ds. 

A.1. Exponential estimates for martingales

Proposition A.1.1. (Lemma A.1 in Ref. [23]) Let M = (Mt )t≥0 , Mt ∈ R, be a continuous local martingale with M0 = 0 and the predictable variation process hM it defined on some stochastic basis with general conditions. Let τ be a stopping time, α and B positive constants and A some measurable set. (PA-1) if Mτ − 12 hM iτ ≥ α on A, then P(A) ≤ e−α ;

α2

(PA-2) if Mτ ≥ α and hM iτ ≤ B on A, then P(A) ≤ e− 2B ; α2

(PA-3) P(supt≤T |Mt | ≥ α, hM iT ≤ B) ≤ 2e− 2B ; α2 W (PA-4) P(supt≤T |Mt | ≥ α) ≤ 2e− 2B P(hM iT > B).

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A.2. Pseudoinverse of nonnegative definite matrices Let A⊕ be the Moore-Penrose pseudoinverse matrix of A (see Ref. [17]). Lemma A.2.1. For d × d nonnegative definite matrix A and x ∈ Rd , ( kxk2A⊕ , AA⊕ x = x −1 lim (x, (A + βI) x) = β→0 ∞, otherwise. Proof. Let S be an orthogonal matrix, S ∗ S = I, such that D := S ∗ AS is a diagonal matrix. Then, due to S ∗ (A + βI)S = D + βI, we have S ∗ (A + βI)−1 S = (D + βI)−1 and S(D + βI)−1 S ∗ = (A + βI)−1 . Write (y := S ∗ x) (x, (A + βI)−1 x) = (x, S(D + βI)−1 S ∗ x) = (S ∗ x, (D + βI)−1 S ∗ x) = (y, (D + βI)−1 y) = (y, (D + βI)−1 DD⊕ y) + (y, (D + βI)−1 (I − DD⊕ )y). Since limβ→0 (D + βI)−1 DD⊕ = D⊕ , one gets lim (y, (D + βI)−1 DD⊕ y) = kyk2D⊕ = kxk2A⊕

β→0

while limβ→0 (y, (D + βI)−1 (I − DD⊕ )y) 6= ∞ only if (I − DD ⊕ )y = 0. Since the latter condition is nothing but (I − AA⊕ )x = 0, the desired statement holds.  A.3. Exponential negligibility of Xtε,β − Xtε We start with an auxiliary result. Proposition A.3.1. Let Yt be a nonnegative continuous semimartingale defined on a stochastic basis (with general conditions): Yt =

Z

t

h1 (s)Ys ds + ε 0

Z

t 0

h2 (s)Ys dMs0 Z t p p Z t 00 2 h3 (s) Ys dMs + ε β h4 (s)ds, +ε β 0

(A.1)

0

where hi (s), i = 1, . . . , 4, are bounded predictable processes and Mt0 , Mt00 are continuous martingales, dhM 0 it = m0 (t)dt, dhM 00 it = m00 (t)dt, hM 0 , M 00 it ≡ 0 with bounded m0 (t) and m00 (t). Assume that for any T > 0 and β > 0,   p (A.2) lim lim ε2 log P sup Yt > L = −∞. L→∞ ε→0

t≤T

Then, for any T > 0,

  lim lim ε2 log P sup Yt > β 1/4 = −∞.

β→0 ε→0

t≤T

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Proof.

Obviously Yt solves the integral equation Z t i h p p Yt = E t Es−1 ε βh3 (s) Ys dMs00 + ε2 βh4 (s)ds , 0


Rt where Et = exp 0 [h1 (s)−ε2 0.5h22 (s)]ds+ 0 εh2 (s)dMs0 . Let for definiteness |hi | ≤ r, where r is a constant. Then, with ε ≤ 1, Z t 2 h2 (s)dMs0 . sup | log Et | ≤ T (r + 0.5r ) + sup ε Rt

t≤T

t≤T

0

Hence the random variable supt≤T | log Et | is bounded on the set Z t  sup ε h2 (s)dMs0 ≤ C . t≤T

0

Moreover, it is exponentially tight in the sense that   lim lim ε2 log P sup | log Et | > C = −∞. C→∞ ε→0

(A.3)

t≤T

The latter is implied by

Z t   h2 (s)dMs0 > C = −∞ lim lim ε2 log P sup ε

C→∞ ε→0

t≤T

(A.4)

0

Rt since the martingale Nt = ε 0 h2 (s)dMs0 has the quadratic variation process hN it = Rt ε2 0 h2 (s)m0 (s)ds and, with some positive number r1 ,
we have ε2 h2 (s)m0 (s) ≤ ε2 r1 . Then, by taking into account that P hN iT > ε2 r1 T = 0 and applying the state
2 2 ment (PA-4) of Proposition A.1.1, we obtain P supt≤T |Nt | > C ≤ 2e−C /(2ε r1 T ) providing (A.4).  Now we estimate supt≤T |Yt | on the set supt≤T log Et ≤ C . Write Z t p p Es−1 ε βh3 (s) Ys dMs00 . sup |Yt | ≤ eC T rε2 β + eC sup t≤T

t≤T

0

This upper bound and (A.2), (A.3) reduce the proof of Proposition A.3.1 to: Z t  p p 2 Es−1 ε βh3 (s) Ys dMs00 > β 1/4 , lim lim ε log P sup β→0 ε→0

t≤T

0

sup

t≤T

 p Yt ≤ L, sup | log Et | ≤ C = ∞ t≤T

for any C > 0 and L > 0. Introduce the martingale Z t Z t p p Es−2 ε2 βh23 (s)Ys m00 (s)ds Nt00 = Es−1 ε βh3 (s) Ys dMs00 with hN 00 it = 0

and denote C = we find that



supt≤T

√

0

Yt ≤ L, supt≤T | log Et | ≤ C . With r2 ≥ h23 (s)Lm00 (s), hN 00 iT ≤ e2C r2 T ε2 β.

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Hence,     P sup |Nt00 | > β 1/4 , C = P sup |Nt00 | > β 1/4 , hN 00 iT ≤ e2C r2 T ε2 β, C t≤T

t≤T

  ≤ P sup |Nt00 | > β 1/4 , hN 00 iT ≤ e2C r2 T ε2 β . t≤T

By (PA-3) from Proposition A.1.1 the latter term is upper bounded by 2 exp



 β 1/2 . 2e2C r2 T ε2 β

Then we obtain   lim ε2 log P sup |Nt00 | > β 1/4 , C ≤ −

ε→∞

t≤T

1 −−−→ −∞. 2e2C r2 T β 1/2 β→0 

We apply Proposition A.3.1 in order to prove Lemma A.3.2. For any T > 0 and C > 0, lim lim ε2 log P

β→0 ε→0





sup Xtε,β − Xtε > β 1/4 = −∞.

β t≤τC ∧T

Proof. Recall that Xtε and Xtε,β solve (1.1) and (6.6) respectively and τCβ is given in (6.9). Set 4ε,β = Xτε,β − Xτεβ ∧t . By (1.1) and (6.6), β t ∧t C

C

4ε,β t

=

Z

β τC ∧t


b(Xτε,β ) − b(Xτεβ ∧s ) ds+ β ∧s C

C

0

+ε

Z

β τC ∧t

0

p
ε ) − σ(X ) dB + ε σ(Xτε,β βWτ β ∧t . β s β τ ∧s ∧s C

C

C

Due to the local Lipschitz continuity of b and σ and with 0/0 = 0, the vector-valued and matrix-valued functions:



− σ Xτεβ ∧s b X ε,β − b Xτεβ ∧t σ X ε,β β β τC ∧s τC ∧s C C f (s) = and g(s) = k4ε,β k4ε,β s k s k are well defined and their entries are bounded by a constant depending on C. Hence 4ε,β = t

Z

β ∧t τC

0

k4ε,β s kf (s)ds + ε

Z

β τC ∧t

0

k4ε,β s kg(s)dBs + ε

p βWτ β ∧t . C

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ε,β ε,β 2 Since k4ε,β o formula, we find that t k = (4t , 4t ), by the Itˆ Z t 2 ε,β k4ε,β 2k4ε,β t k = s k(4s , f (s))ds 0

+ε

Z

β τC ∧t

0

p Z +ε β + ε2

Z

ε,β 2k4ε,β s k(4s , g(s)dBs )

β τC ∧t

0

β τC ∧t

0

(A.5)

2(4ε,β s , dWs )

  2 ∗ k4ε,β s k trace g(s)g (s) ds

+ ε2 β(τCβ ∧ t)d. Now, by letting ϕ(s) = 2 k4ε,β t k =

Z

β τC ∧t

0

+ε

Z

2(4ε,β s ,f (s)) k4ε,β s k

bs = and dB

2(4ε,β s ,g(s)dBs ) , k4ε,β s k

we rewrite (A.5) as:

  2 2 ∗ k4ε,β s k ϕ(s) + ε trace[g(s)g (s)] ds

β τC ∧t

0

2 b k4ε,β s k dBs + ε

p Z β

β τC ∧t

0

k4ε,β s k

2(4ε,β s , dWs ) k4ε,β s k

+ ε2 β(τCβ ∧ t)d.

(A.6)

With the notations -

2 Yt = k4ε,β t k  h1 (s) = I{τ β ≤s} ϕ(s) + ε2 trace[g(s)g ∗ (s)] C h2 (s) ≡ 1 h4 (s) = I{τ β ≤s} d

-

Mt0

C

bt , m0 (s) = =B

- Mt00 =

∗ ε,β 4(4ε,β s , g(s)g (s)4s ) 2 k4ε,β s k

R τCβ ∧t (4ε,β , dWs ) , m00 (s) ≡ 4, 2 s ε,β 0 k4s k

the equation (A.6) is in the form of (A.1). Since hi (s), i = 1, . . . , 4 are bounded √ − Xτεβ ∧t k ≤ kX ε,β k + kXτεβ ∧t k ≤ 2C, i.e., (A.2) holds too, the and Y t ≡ kX ε,β β β τC ∧t

C

τC ∧t

C

statement of the lemma follows from Proposition A.3.1.



Acknowledgement Research of P. Chigansky is supported by a grant from the Israel Science Foundation. References [1] M. I. Freidlin and A. D. Wentzell, Random perturbations of dynamical systems. vol. 260, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of

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[2]

[3]

[4] [5]

[6]

[7] [8] [9] [10] [11] [12] [13]

[14]

[15] [16] [17] [18] [19]

[20] [21]

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Mathematical Sciences], (Springer-Verlag, New York, 1984). ISBN 0-387-90858-7. Translated from the Russian by Joseph Sz¨ ucs. A. Dembo and O. Zeitouni, Large deviations techniques and applications. vol. 38, Applications of Mathematics (New York), (Springer-Verlag, New York, 1998), second edition. ISBN 0-387-98406-2. P. Dupuis and R. S. Ellis, A weak convergence approach to the theory of large deviations. Wiley Series in Probability and Statistics: Probability and Statistics, (John Wiley & Sons Inc., New York, 1997). ISBN 0-471-07672-4. , A Wiley-Interscience Publication. J. Feng, Martingale problems for large deviations of Markov processes, Stochastic Process. Appl. 81(2), 165–216, (1999). ISSN 0304-4149. J. Feng and T. G. Kurtz, Large deviations for stochastic processes. vol. 131, Mathematical Surveys and Monographs, (American Mathematical Society, Providence, RI, 2006). ISBN 978-0-8218-4145-7; 0-8218-4145-9. A. Friedman, Stochastic differential equations and applications. Vol. 2. (Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1976). Probability and Mathematical Statistics, Vol. 28. R. S. Liptser and A. A. Pukhalskii, Limit theorems on large deviations for semimartingales, Stochastics Stochastics Rep. 38(4), 201–249, (1992). ISSN 1045-1129. T. Mikami, Some generalizations of Wentzell’s lower estimates on large deviations, Stochastics. 24(4), 269–284, (1988). ISSN 0090-9491. K. Narita, Large deviation principle for diffusion processes, Tsukuba J. Math. 12(1), 211–229, (1988). ISSN 0387-4982. D. W. Stroock, An introduction to the theory of large deviations. Universitext, (Springer-Verlag, New York, 1984). ISBN 0-387-96021-X. J. Ren and X. Zhang, Freidlin-Wentzell’s large deviations for homeomorphism flows of non-Lipschitz SDEs, Bull. Sci. Math. 129(8), 643–655, (2005). ISSN 0007-4497. A. A. Puhalskii, On some degenerate large deviation problems, Electron. J. Probab. 9, no. 28, 862–886 (electronic), (2004). ISSN 1083-6489. R. Liptser, V. Spokoiny, and A. Y. Veretennikov, Freidlin-Wentzell type large deviations for smooth processes, Markov Process. Related Fields. 8(4), 611–636, (2002). ISSN 1024-2953. R. Z. Has0 minski˘ı, Stochastic stability of differential equations. vol. 7, Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, (Sijthoff & Noordhoff, Alphen aan den Rijn, 1980). ISBN 90-286-0100-7. Translated from the Russian by D. Louvish. E. Pardoux and A. Y. Veretennikov, On the Poisson equation and diffusion approximation. I, Ann. Probab. 29(3), 1061–1085, (2001). ISSN 0091-1798. ` Pardoux and A. Y. Veretennikov, On Poisson equation and diffusion approximaE. tion. II, Ann. Probab. 31(3), 1166–1192, (2003). ISSN 0091-1798. A. Albert, Regression and the Moore-Penrose pseudoinverse. (Academic Press, New York, 1972). Mathematics in Science and Engineering, Vol 94. A. Puhalskii. On functional principle of large deviations. In New trends in probability and statistics, Vol. 1 (Bakuriani, 1990), pp. 198–218. VSP, Utrecht, (1991). A. Puhalskii, Large deviations and idempotent probability. vol. 119, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, (Chapman & Hall/CRC, Boca Raton, FL, 2001). ISBN 1-58488-198-4. D. Aldous, Stopping times and tightness, Ann. Probability. 6(2), 335–340, (1978). D. Aldous, Stopping times and tightness. II, Ann. Probab. 17(2), 586–595, (1989). ISSN 0091-1798.

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[22] R. S. Liptser and A. N. Shiryayev, Theory of martingales. vol. 49, Mathematics and its Applications (Soviet Series), (Kluwer Academic Publishers Group, Dordrecht, 1989). ISBN 0-7923-0395-4. Translated from the Russian by K. Dzjaparidze [Kacha Dzhaparidze]. [23] A. Guillin and R. Liptser, MDP for integral functionals of fast and slow processes with averaging, Stochastic Process. Appl. 115(7), 1187–1207, (2005). ISSN 0304-4149.

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Chapter 8 On the Convergence Rates of a General Class of Weak Approximations of SDEs Dan Crisan and Saadia Ghazali∗ Department of Mathematics, Imperial College London 180 Queen’s Gate, London SW7 2BZ [email protected] In this paper, the convergence analysis of a class of weak approximations of solutions of stochastic differential equations is presented. This class includes recent approximations such as Kusuoka’s moment similar families method and the Lyons-Victoir cubature of Wiener Space approach. We show that the rate of convergence depends intrinsically on the smoothness of the chosen test function. For smooth functions (the required degree of smoothness depends on the order of the approximation), an equidistant partition of the time interval on which the approximation is sought is optimal. For functions that are less smooth (for example Lipschitz functions), the rate of convergence decays and the optimal partition is no longer equidistant. Our analysis rests upon Kusuoka-Stroock’s results on the smoothness of the distribution of the solution of a stochastic differential equation. Finally the results are applied to the numerical solution of the filtering problem.

Contents 1 Introduction . . . . . . . . 2 Preliminaries . . . . . . . . 3 The Main Theorem . . . . . 4 An Application to Filtering 5 Some Auxiliary Results . . References . . . . . . . . . . . .

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221 224 227 239 243 246

1. Introduction Stochastic differential equations (SDEs) constitute an ideal mathematical model for a multitude of phenomena and processes encountered in areas such as filtering, optimal stopping, stochastic control, signal processes and mathematical finance, most notably in option pricing (see for example Oksendal [1] and Kloeden & Platen [2]). Unlike their deterministic counterparts, SDEs do not have explicit solutions, apart from in a few exceptional cases, hence the necessity for a sound theory of their numerical approximation. ∗ [email protected]

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In this paper we will be concerned with SDEs written in Stratonovich form, in other words, we will look at equations of the form, Xt = X 0 +

Z

t

V0 (Xs )ds + 0

k Z X j=1

t 0

Vj (Xs ) ◦ dWsj ,

(1.1)

where the last term is a stochastic integral of Stratonovich type. There are two classes of numerical methods for approximating SDEs. The objective of the first is to produce a pathwise approximation of the solution (strong approximation). The second method involves approximating the distribution of the solution at a particular instance in time (weak approximation). For example when one is only interested in the expectation E[ϕ(Xt )] for some function ϕ, it is sufficient to have a good approximation of the distribution of the random variable Xt rather than of its sample paths. This observation was first made by Milstein [3] who showed that pathwise schemes and L2 estimates of the corresponding errors are irrelevant in this context since the objective is to approximate the law of Xt . This paper contains approximations that belong to this second class of algorithms. Classical results in this area concentrate on solving numerically SDEs for which the so-called ‘ellipticity condition’, or more generally the ‘Uniform H¨ ormander condition’ (UH), is satisfied. For a survey of such schemes see, for example, Kloeden & Platen [2] or Burrage, Burrage & Tian [4]. Under this condition, for any bounded measurable function ϕ, the semigroup of operators {Pt }t∈[0,∞) defined, (Pt ϕ)(x) = E[ϕ(Xt (x))],

(1.2)

where X (x) = {Xt (x)}t∈[0,∞) solves (1.1) with initial condition X0 = x, is smooth for any t > 0. It is this property upon which the majority of these schemes rely. For example, the classical Euler-Maruyama scheme requires Pt ϕ to be four times differentiable in order to obtain the optimal rate of convergence. Talay( [5], [6]) and, independently, Milstein [7] introduced the appropriate methodology to analyse this scheme. They express the error as a difference including a sum of terms involving the solution of a parabolic PDE. Their analysis also shows the relationship between the smoothness of ϕ and the corresponding error. Talay & Tubaro [8] prove an even more precise result showing that, under the same conditions, the errors corresponding to the Euler-Maruyama and many other schemes can be expanded in terms of powers of the discretisation step. Furthermore, Bally & Talay [9] show the existence of such an expansion under a much weaker hypothesis on ϕ: that ϕ need only be measurable and bounded (even the boundedness condition can be relaxed). Higher order schemes require additional smoothness properties of Pt ϕ (see for example, Platen & Wagner [10]). In the eighties, Kusuoka & Stroock( [11], [12], [13]) studied the properties of Pt ϕ under a weaker condition, the so-called UFG condition (see (2.3) in Section 2). Essentially, this condition states that the Lie algebra generated by the vector fields {Vi }ki=1 ∈ Cb∞ (Rd ; Rd ) is finite dimensional as a Cb∞ (Rd )-module. Kusuoka &

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Stroock conclude Malliavin’s undertaking to recover H¨ ormander’s hypo-ellipticity theory for degenerate second order elliptic operators. They show that under the UFG condition, Pt ϕ retains certain regularity properties, in particular, that Vi1 Vi2 ...Vin Pt ϕ is well defined for any vector fields Vir , where ir ∈ {1, ..., k}. They also give upper bounds for the supremum norm of Vi1 Vi2 ...Vin Pt ϕ. Besides the UFG condition, our analysis rests upon a second assumption which we call the V0 condition. It states that V0 can be expressed in terms of {V1 , ...Vk } ∪ {[Vi , Vj ] , 1 ≤ i < j ≤ k}. This premise is weaker than the ellipticity assumption and has been used, for example, by Jerison and S´ anchez-Calle ( [14], [15]) to obtain estimates for the heat kernel. This second condition enables one to control the supremum norm of Vi1 Vi2 ...Vin Pt ϕ for ir ∈ {0, 1, ..., k} (see Corollary 5.4 in Section 5). A number of schemes have recently been developed to work under these weaker conditions rather than the ellipticity condition, their convergence depending intrinsically on the above estimates of Vi1 Vi2 ...Vin Pt ϕ. A further advantage of this new generation of schemes is a consequence of the the sup ϕclassical resultd stating that ϕ port of X (x) is the closure of the set S = x : [0, ∞) → R where x solves the ODE, xϕ t =x+

Z

t 0

V0 (xϕ s )ds +

k Z X j=1

t 0

Vj (xϕ s )ϕ (s) ◦ ds,

and ϕ : [0, ∞) → Rd is an arbitrary smooth function (see Stroock & Varadhan [16], [17], [18], Millet & Sanz-Sole [19]). These schemes attempt to keep the support of the approximating process on the set S. In this way, stability problems that are known to affect classical schemes can be avoided. For example, Ninomiya & Victoir [20] give an explicit example where the Euler-Maruyama approximation fails whilst their algorithm succeeds (see Example 3.4 below for the algorithm). Their example involves an SDE related to the Heston stochastic volatility model in finance. In this paper we give a general criterion for the convergence of a class of weak approximations incorporating this new category of schemes. This criterion is based upon the stochastic Stratonovich-Taylor expansion of Pt ϕ and demonstrates how the rate of convergence depends on the smoothness of the test function selected. Our plan is as follows. In section 2 we set out some essential terminology, adopted partly from Kusuoka [21], which is required to index the stochastic Stratonovich-Taylor expansions that follow and to state the UFG and V0 conditions imposed on the given vector fields (see (2.3) & (2.4) below). Section 3 contains our main results on the convergence analysis of a class of weak approximations of solutions of SDEs, characterised by introducing the concept of an m-perfect family (Definition 3.1). The Lyons-Victoir and Ninomiya-Victoir approximations are both members of this class. Although the Kusuoka approximation is not within this family, it can effectively be categorised in the same way (see Example 3.6 and the

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subsequent comment). In our main Theorem 3.7, we show that the rate of convergence depends intrinsically on the smoothness of the chosen test function; the higher the order of the approximation, the smoother the test function required. For a smooth function, an equidistant partition of the time interval on which the approximation is sought is optimal. For less smooth functions, this is no longer true. We emphasise that the UFG+V0 conditions are not required for a smooth test function. Furthermore, the Kusuoka approximation does not require the V0 condition. Finally, in Section 4 we present an application of this theory to the numerical solution of the filtering problem. 2. Preliminaries Let (Ω, F, P) be a probability space satisfying the usual conditions and W = {Wt }t∈[0,∞) be a k-dimensional Brownian motion defined on it. We also set Wt0 = t for t ∈ [0, ∞) . Let Cb∞ (Rd , Rd ) denote the space of smooth functions ϕ : Rd → Rd with bounded derivatives, that is, bounded partial derivatives (of all orders) of the d component functions {ϕi : Rd → R}di=1 exist. We have a dual interpretation of this space, in that we also regard its elements as vector fields. In other words, a smooth function V : Cb∞ (Rd ) → Cb∞ (Rd ) is equivalent to an operator on Cb∞ (Rd ) ≡ Cb∞ (Rd , R) defined by, Vϕ=

d X i=1

i

Cb∞ (Rd )

Vi

∂ϕ ∂xi

where V ∈ is the i-th component of L for i = 1, . . . , d. We next consider the Stratonovich SDE with drift vector {V0i (x)}di=1 and disd ∞ d d persion matrix {Vji (x)}d,k i,j=1 for x ∈ R , for some V0 , . . . , Vk ∈ Cb (R , R ). This is written componentwise as, Z t k Z t X i i i Xt = X 0 + V0 (Xs )ds + Vji (Xs ) ◦ dWsj (2.1) 0

j=1

0

for i = 1, . . . , d. The following essential terminology has been adopted from Kusuoka [21]. Let A be the set of multi-indices, A = {∅} ∪

∞

∪ {0, 1, . . . , k}m

m=1

where k is the dimension of the Brownian Motion introduced above. This set will be used to index the Stratonovich-Taylor expansions that follow. Let |·| and k·k be the following two norms defined on A by r

|∅| = 0, |α| = r if α = (i1 , . . . , ir ) ∈ {0, 1, . . . , k} for r ∈ N and kαk = |α| + card {1 ≤ j ≤ |α| : ij = 0} . Furthermore, let A0 = A\ {∅}, A1 = A\ {∅, (0)} and correspondingly, A(m) = {α ∈ A : kαk ≤ m}, A0 (m) = {α ∈ A0 :

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kαk ≤ m} and A1 (m) = {α ∈ A1 : kαk ≤ m} where the integer m ∈ N above corresponds to the level of the truncation in the Stratonovich-Taylor expansions that follow (see (3.4)). We also require the following concatenation on A, (i1 , . . . , ir ) ∗ (j1 , . . . , js ) = (i1 , . . . , ir , j1 , . . . , js ) and need to define a further operation on the vector fields {Vj ∈ Cb∞ (Rd , Rd )}kj=0 introduced above. Definition 2.1. For α ∈ A , the vector field V[α] is defined inductively by, V[ϕ] = 0; V[(j)] = Vj ; V[α∗(j)] = [Vα , Vj ] for j = 0, 1, . . . , k where [Vi , Vj ] := Vi Vj − Vj Vi for Vi , Vj ∈ Cb∞ (Rd , Rd ). In the following we will make use of the semi-norm, kϕkV,i =

i X

X

u=1 α1 ,...,αu ∈A0 kα1 ∗...∗αu k=i



V[α ] · · · V[α ] ϕ . 1 u ∞

for i ∈ N. Furthermore, we introduce the semi-norm, kϕkp = for p ∈ N, ϕ ∈ Cbp (Rd ) where,

i

∇ ϕ

∞

=

p X

i

∇ ϕ i=1

(2.2)

∞



∂iϕ

j1 ,...,ji ∈{1,...,d} ∂xj1 . . . ∂xji ∞ max

|α|

and note that it can easily be deduced from the chain rule, for α ∈ A0, ϕ ∈ Cb (Rd ), that kVα ϕk∞ ≤ C kϕk|α| and hence kϕkV,i ≤ C kϕki for i ∈ N. Also let k·kp,∞ be the norm kϕkp,∞ := kϕk∞ + kϕkp for ϕ ∈ Cbp (Rd ). We define the space, n o CbV,i (Rd ) = ϕ : kϕkV,i < ∞

which appears in the definition of an m-perfect family below (see Definition 3.1). We now introduce the two conditions on the vector fields {Vj ∈ Cb∞ (Rd , Rd )}kj=0 required to treat the case when the test function ϕ is not smooth. We emphasise that the smooth case requires neither the UFG nor the V0 condition. Furthermore, the Kusuoka approximation (Example 3.6) does not require the V0 condition as the corresponding error is controlled in terms of {V[β] : β ∈ A1 } alone.

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The UFG condition [Kusuoka & Stroock [13, 21]]. There exists some l ∈ N such that for any α ∈ A1 we have, X V[α] = ϕα,β V[β] . (2.3) β∈A1 (l)

where ϕα,β ∈

Cb∞ (Rd )

for all β ∈ A1 (l).

The V0 condition. There exist ϕβ ∈ Cb∞ (Rd ), β ∈ A1 (2) such that, X V0 = ϕβ V[β] .

(2.4)

β∈A1 (2)

As mentioned in the introduction, the UFG condition states that the Lie algebra generated by the vector fields {Vi }ki=1 ∈ Cb∞ (Rd ; Rd ) is finite dimensional as a Cb∞ (Rd )-module. It is implied by the Uniform H¨ ormander Condition which states that, X

2 ∃l ∈ N and c > 0 s.t. V[α] (x), ξ ≥ c |ξ|2 (2.5) α∈A1 (l)

Pd

for all x, ξ ∈ Rd , where hV, ξi = i=1 V i ξ i for V ∈ Cb∞ (Rd , Rd ). Under conditions (2.3)+(2.4), one can show that for any r ∈ Z+ , {αi ∈ A0 }ri=1 and p = 1, ..., kα1 ∗ . . . ∗ αr k, there exists a constant CpT > 0 such that,

V[α ] . . . V[α ] Pt ϕ ≤ CpT t(p−kα1 ∗...∗αr k)/2 kϕk , t ∈ [0, T ] . (2.6) 1 r p ∞

where {Pt }t∈[0,∞) is the semigroup associated with X defined in (1.2). We remark that condition (2.3) will only give us (2.6) for αi ∈ A1 , i = 1, ...r. Under both (2.3)+(2.4) we have, kV0 Pt ϕk∞ ≤ Ct−1 kϕk∞ . However, under condition (2.3) alone, kV0 Pt ϕk∞ may be of higher order. Kusuoka has given an explicit example in which, l

l

ct− 2 kϕk∞ ≤ kV0 Pt ϕk∞ ≤ Ct− 2 kϕk∞ for some constants c, C > 0 and where l is the constant appearing in (2.3) (see Proposition 14 and Proposition 16 in Ref. [22]). This coarser bound on kV0 Pt ϕk∞ results in lower rates of convergence. The authors believe that (2.4) is the most general condition required to preserve the same rates of convergence as those obtained when {Vi }ki=1 satisfy the ellipticity condition (for non-smooth test functions ϕ). Result (2.6) is a corollary of a certain representation theorem proved in KusuokaStroock [13]. For completeness, we state the representation theorem (Theorem 5.2) in Section 5 where we also sketch a proof of inequality (2.6) in Corollary 5.4. We note that the case p = 1 has been proved in Ref. [13] for {αi ∈ A1 }ri=1 . Inequality (2.6) proves to be crucial in obtaining upper bounds on the error of of the class algorithms that we study below (see Theorem 3.7).

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3. The Main Theorem In this section we introduce the concept of an m-perfect family. Such families correspond to various weak approximations of SDEs, including the Lyons-Victoir and Ninomiya-Victoir schemes. The main result appears in Theorem 3.7 and Corollary 3.8. For α = (i1 , . . . , ir ) ∈ A0 and ϕ ∈ Cbr (Rd ) let fα,ϕ be defined as f(i1 ,...,ir ),ϕ := Vi1 ...Vir ϕ. We also need to define the iterated Stratonovich integral Z sr−2  Z t Z s0 1 Ifα,ϕ (t) := ··· fα,ϕ (Xsr−1 ) ◦ dWsir−1 ◦ · · · ◦ dWsi1r−1 ◦ dWsi0r , 0

0

0

for t ≥ 0. If i1 = 0 then Ifα,ϕ (t) is well defined for ϕ ∈ Cbr (Rd ). However, if i1 6= 0 then Ifα,ϕ (t) is well defined provided ϕ ∈ Cbr+2 (Rd ), since the semimartingale Rproperty of fα,ϕ (X) is required in the definition of the first Stratonovich intes 1 . Note that the Stratonovich integrals are evaluated gral 0 r−2 fα,ϕ (Xsr−1 ) ◦ dWsir−1 innermost first. Finally let  Z sr−2 Z t Z s0 i1 1 ◦ dWsr−1 ◦ · · · ◦ dWsi1r−1 ◦ dWsi0r . ··· Iα (t) := 0

0

0

Let α = (i1 , . . . , ir ) ∈ A0 be an arbitrary multi-index such that kαk = m ∈ N (and |α| = r ∈ N). If m is odd, then E[Iα (t)] = 0 and if m is even then ( m t2 if α ∈ Am,r r− m m 0 2 2 ( 2 )! , (3.1) E[Iα (t)] = 0 otherwise where Am,r is the set of multi-indices α = α1 ∗ · · · ∗ α m2 ∈ A0 (m) such that each 0 αi = (0) or (j, j) for some j ∈ {1, . . . , k}. Note that r − m 2 is equal to the number of pairs of indices (j, j) occurring in α. A proof of this result can be found in Ref. [23]. The set of iterated Stratonovich integrals plays a central role in the theory of approximation of solutions of SDEs and there are numerous papers that study its structure. Here, we adopt the hierarchical set approach introduced by Kloeden & Platen [24]. An alternative method can be found in Gaines [25] where it is shown how Lyndon words provide a basis for iterated Stratonovich integrals and also how shuffle products may be used to obtain moments of stochastic integrals. Pettersson [26] gives a notationally and computationally convenient StratonovichTaylor expansion. Furthermore, Burrage & Burrage [27] use rooted-tree theory to describe the aforementioned set and Burrage & Burrage [28] presents an approach based on B-series. We state three further results in (3.2), (3.3) and (3.5). The proofs are all elementary and can be found in Ref. [23]. The first two give an upper bound on the L2 norm of Ifα,ϕ (t) for smooth ϕ. The third provides an explicit form for the remainder of ϕ(Xt ) when expanded in terms of iterated integrals.

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For ϕ ∈ Cb we have

(Rd ) and any multi-index α = (i1 , . . . , ir ) ∈ A0 such that i1 6= 0,

k X

kαk kαk+1

Ifα,ϕ (t) ≤ c1 kfα,ϕ k t 2 + c2 kVi fα,ϕ k∞ t 2 ∞ 2

(3.2)

i=1

||α||

for some constants c1 ≡ c1 (α), c2 ≡ c2 (α) ≥ 0. For ϕ ∈ Cb index α = (i1 , . . . , ir ) ∈ A0 such that i1 = 0 we have

kαk

Ifα,ϕ (t) ≤ c1 kfα,ϕ k t 2 . ∞ 2

(Rd ) and any multi-

For m ∈ N, ϕ ∈ Cbm+3 (Rd ) and x ∈ Rd , we define the truncation, X ϕm fα,ϕ (x)Iα (t). t (x) := ϕ(x) +

(3.3)

(3.4)

α∈A0 (m)

Then for t ≥ 0 the remainder is Rm,t,ϕ (x) := ϕ(Xt ) − ϕm t (x) = (

X

kαk=m+1

X

+

)Ifα,ϕ (t). (3.5)

kαk=m+2,α=0∗β,kβk=m

In the following, we define a class of approximations of X expressed in terms of ¯ (x) = {X ¯ t (x)}t∈[0,∞) for x ∈ Rd , which certain families of stochastic processes, X are explicitly solvable. In particular, we can explicitly compute the operator, ¯ t (x))]. (Qt ϕ)(x) = E[ϕ(X

(3.6)

m m The semigroup PT will then be approximated by Qm hn Qhn−1 . . . Qh1 where {hj := tj − tj−1 }nj=1 and πn = {tj := ( nj )γ T }nj=0 for n ∈ N, is a sufficiently fine partition of the interval [0, T ]. In particular hj ∈ [0, 1) for j = 1, ..., n. The underlying idea is that Qt ϕ will have the same truncation as Pt ϕ. ¯ (x) = {X ¯ t (x)}t∈[0,∞) , where x ∈ Rd , be a family of progressively meaSo let X ¯ t (y) = X ¯ t (x0 ) P−almost surely, surable stochastic processes such that, limy→x0 X d for any t ≥ 0 and x0 ∈ R . As a result, the operator Qt defined in (3.6) has the property that Qt ϕ ∈ Cb (Rd ) for any ϕ ∈ Cb (Rd ). In particular, Qt : Cb (Rd ) → Cb (Rd ) is a Markov operator.

¯ (x) = {X ¯ t (x)}t∈[0,∞) where x ∈ Rd , is Definition 3.1. For m ∈ N, the family X said to be m-perfect for the process X if there exist constants C > 0 and M ≥ m+1 such that for ϕ ∈ CbV,M (Rd ), sup |Qt ϕ(x) − E[ϕm t (x)]| ≤ C

x∈Rd

M X

i=m+1

ti/2 kϕkV,i .

(3.7)

As we can see from (3.7), the quantity E[ϕm t (x)] plays the same role as the classical truncation in the standard Taylor expansion of a function. Using (3.1) we

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deduce that, E[ϕ0t (x)] = ϕ(x) E[ϕ2t (x)]

= ϕ(x) + V0 ϕ(x)t +

k X

Vi2 ϕ(x)

i=1

E[ϕ4t (x)] = E[ϕ2t (x)] +

k X

V02 ϕ(x)

i=1

+

k X

Vi2 V0 ϕ(x)

i=1

t = ϕ(x) + Lϕ(x)t 2

k

t2 t2 X + V0 Vi2 ϕ(x) 2 4 i=1

k X t2 t2 + Vj2 Vi2 ϕ(x) 4 i,j=1 8

= ϕ(x) + Lϕ(x)t + L2 ϕ(x)

t2 , 2

Pk where L = V0 + 12 i=1 Vi2 . Furthermore, since E[Iα (t)] = 0 for odd kαk, it follows that E[ϕ1t (x)] = E[ϕ0t (x)], E[ϕ3t (x)] = E[ϕ2t (x)] and E[ϕ5t (x)] = E[ϕ4t (x)]. There now follow some examples of m−perfect families corresponding to {Pt }t∈[0,∞) as described in (1.2), the Lyons-Victoir method and the NinomiyaVictoir algorithm. Example 3.2. The family of stochastic processes {Xt (x)}t∈[0,∞) , where x ∈ Rd , is m−perfect. More precisely there exists a constant c3 > 0 such that for ϕ ∈ CbV,m+2 (Rd ), sup |Pt ϕ(x) − E[ϕm t (x)]| ≤ c3 x

Proof.

m+2 X

i=m+1

ti/2 kϕkV,i ,

(3.8)

For ϕ ∈ CbV,m+3 (Rd ),

X X m + )Ifα,ϕ (t)] |Pt ϕ(x)−E[ϕt (x)]|=|E[Rm,t,ϕ (x)]|= E[( kαk=m+1 kαk=m+2,α=0∗β,kβk=m

Applying inequality (3.2) to the first sum, X

kαk=m+1



Ifα,ϕ (t) ≤ 2

X

kαk=m+1

≤ c4

m+2 X

i=m+1

{c1 (α) kfα,ϕ k∞ t

m+1 2

+ c2 (α)

k X i=1

kVi fα,ϕ k∞ t

m+2 2

ti/2 kϕkV,i

(3.9)

for some constant c4 > 0. Applying result (3.3) to the second sum, X X

Ifα,ϕ (t) ≤ c1 (α) kfα,ψ k∞ t 2 kαk=m+2,α=0∗β,kβk=m

}

kαk=m+2,α=0∗β,kβk=m

≤ c5 kϕkV,m+2 t

m+2 2

(3.10)

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for some c5 > 0. The result for ϕ ∈ CbV,m+3 (Rd ) follows from combining (3.9) and (3.10). Since none of the terms in (3.8) depend on partial derivatives of order m+3, the inequality is also valid for any ϕ ∈ CbV,m+2 (Rd ) (a standard approximation method can be used).  ¯ (x) = {X ¯ t (x)}t∈[0,1] , where In the following example, the family of processes X d x ∈ R , corresponds to the Lyons-Victoir approximation (see Ref. [29]). The example involves a set of l finite variation paths, ω1 , . . . , ωl ∈ C00 ([0, 1], Rk ), for some l P l ∈ N, together with some weights λ1 , . . . , λl ∈ R+ such that λj = 1. These paths j=1

are said to define a cubature formula on Wiener Space of degree m if, for any α ∈ A0 (m), l X E[Iα (1)]= λj Iαωj (1) j=1

where, ω I(ij1 ,...,ir ) (1)

:=

Z

1 0

Z

s0 0

···(

Z

sr−2 0

i

dωji1 (sr−1 )) · · · dωjr−1 (s1 )dωjir (s0 ).

From the scaling properties of the Brownian motion we can deduce, for t ≥ 0, E[Iα (t)]=

l X

λj Iαωt,j (t)

j=1


√ where ωt,1 , . . . , ωt,l ∈ C00 ([0, t], Rk ) is defined by ωt,j (s) = tωj st , s ∈ [0, t]. In other words, the expectation of the iterated Stratonovich integrals Iα (t) is the same under the Wiener measure as it is under the measure, Qt :=

l X

λj δωt,j .

j=1

¯ to satisfy the evolution equation (2.1) but with the Example 3.3. If we choose X driving Brownian motion replaced by the paths ωt,1 , . . . , ωt,l defined above then the family of processes, {X t (x)}t∈[0,1] , with corresponding operator (Qt ϕ)(x) := EQt [ϕ(X t (x))], is m-perfect. More precisely, there exists a constant c6 > 0 such that for ϕ ∈ CbV,m+2 (Rd ), m+2 X ti/2 kϕkV,i sup Qt ϕ(x) − E[ϕm t (x)] ≤ c6 x

i=m+1

For example, if (λj , ωt,j ) are chosen such that for l = 2k the paths are ωt,j : t 7→ t(1, zj1 , .., zjk ) for j = 1, . . . , 2k with points zj ∈ {−1, 1}k and weights λj = 2−k , we obtain a cubature formula of degree 3 and a corresponding 3-perfect family.

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|α|

ω

Proof. Let us first observe that Iα t,j (t) = t 2 Iα j (1) Hence, for ϕ ∈ CbV,m+2 (Rd ), Qt ϕ(x) − E[ϕm t (x)] = |EQt [Rm,t,ϕ (x)]| X X ≤( + ) kfα,ϕ k∞ kEQt [Iα (t)]k2 kαk=m+1

≤( ≤( where kα =

l P

j=1

X

kαk=m+2,α=0∗β,kβk=m

kαk=m+1

X

kαk=m+1

X

+

kαk=m+2,α=0∗β,kβk=m

X

+

kαk=m+2,α=0∗β,kβk=m

) kfα,ϕ k∞

l X j=1

)kα kfα,ϕ k∞ t

ω

λj Iα j (1) 2 .

λj kIαωt ,j (t)]k2 |α| 2

.



Remarks (i) There has been no change to the underlying measure in the example above. Merely a representation in terms of the measure Qt has been introduced  to ease the computation of Qt . More precisely, the family of processes X t (x) t∈[0,1] where x ∈ Rd is constructed as follows. We take,

X 0 (x) = x and then randomly choose a path ωt,r from the set {ωt,1 , . . . , ωt,l } with corresponding probabilities (λ1 , . . . , λl ). Each process then follows a deterministic trajectory driven by the solution of the ordinary differential equation, dX t = V0 (X t )dt +

k X

j Vj (X t )dωt,k

j=1

for some V0 , . . . , Vk ∈ Cb∞ (Rd , Rd ) as in (2.1). We can therefore compute the expected values of a functional of Xt (x) as integrals on the path space with respect to the Radon measure Qt . Hence the identities,     Qt ϕ(x) = E ϕ(X t (x)) = EQt ϕ(X t (x))

(ii) The approach adopted by Lyons and Victoir to construct the above approximation resembles the ideas developed by Clark and Newton in a series of papers ( [30], [31], [32], [33]). Heuristically, Clark and Newton constructed strong approximations of SDEs using flows driven by vector fields which were measurable with respect to the filtration generated by the driving Wiener process. In a similar vein, Castell & Gaines [34] provide a method of strongly approximating the solution of an SDE by means of exponential Lie series. For the following example, we will denote by exp(V t)f the value at time t of the solution of the ODE y 0 = V (y) , y (0) = f where V ∈ Cb∞ (Rd , Rd ). In particular,

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exp(V t) (x) is exp(V t)f for f being the identity function. The family of processes Y (x) = {Yt (x)}t∈[0,1] below corresponds to the Ninomiya-Victoir approximation (see Ref. [20]). Example 3.4. Let Λ and Z be two independent random variables such that Λ is Bernoulli distributed P(Λ = 1) = P(Λ = −1) = 21 and Z = (Z i )ki=1 is a standard normal k− dimensional random variable. Consider the family of processes Y (x) = {Yt (x)}t∈[0,1] defined by

Yt (x) =

 k Q    exp( V20 t) exp(Z i Vi t1/2 ) exp( V20 t)(x)    exp( V20 t)

i=1 k Q i=1

if Λ = 1

exp(Z k+1−i Vk+1−i t1/2 ) exp( V20 t)(x) if Λ = −1

with the corresponding operator (Qt ϕ)(x) := E[ϕ(Yt (x))]. Then there exists a constant c7 > 0 such that for ϕ ∈ CbV,8 (Rd ) sup Qt ϕ(x) − E[ϕ5t (x)] ≤ c7 t3 kϕkV,6 x

Hence {Yt (x)}t∈[0,1] is 5-perfect. Proof.

We first consider the case Λ = 1. Let {Y i }k+1 i=0 be defined, dϕ 0 (Y ) ds s dϕ 1 (Y ) ds s dϕ i (Y ) ds s dϕ k+1 (Y ) ds s

t = V0 ϕ(Ys0 ) for s ∈ [0, ], Y00 = x 2 √ = Z 1 V1 ϕ(Ys1 ) for s ∈ [0, t], Y01 = Y t0 2

√ = Z i Vi ϕ(Ysi ) for s ∈ [0, t], Y0i = Y√i−1 , i = 2, . . . , k t t = V0 ϕ(Ysk+1 ) for s ∈ [0, ], Y0k+1 = Y√kt 2

It follows from the definition of the algorithm and by Itˆ o’s Formula that, t t2 ϕ(Y tk+1 ) = ϕ(Y√kt )+V0 ϕ(Y√kt ) +V02 ϕ(Y√kt ) + 2 2 8

Z

t/2 0

Z

s1 0

Z

s2 0

V03 ϕ(Ysk+1 )ds3 ds2 ds1 . 3

(3.11) We need to expand the right hand side of (3.11) and divide the resulting expansion into two parts: the required truncation and a remainder whose expected value should be bounded by Ct3 kϕkV,6 . The final term in (3.11) belongs to the remainder and indeed, # " Z t/2 Z s1 Z s2

(t/2)3 t3 3 k+1 ≤ kϕkV,6 . (3.12) E V0 ϕ(Ys3 )ds3 ds2 ds1 ≤ V03 ϕ ∞ 0 6 48 0 0

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Convergence Rate of Weak Approximations

Expanding the third term in (3.11), Z √t k−1 2 2 k Z k Vk V02 ϕ(Ysk )ds V0 ϕ(Y√t ) = V0 ϕ(Y√t ) + 0

=

) V02 ϕ(Y√k−1 t

+Z

k

Vk V02 ϕ(Y√k−1 ) t

√

t+

Z

√ t

0

= ...

Z

s1 0

Zk


2

Vk2 V02 ϕ(Ysk2 )ds2 ds1

k X √ √ ) Z i Vi V02 ϕ(Y√i−1 t = V02 ϕ(Y t0 ) + Z 1 V1 V02 ϕ(Y t0 ) t + t 2

+

k Z X i=1

So

2

√ t

0

V02 ϕ(Y√kt ) = V02 ϕ(x) + +

k Z X i=1

Z

√ 0

Z

t

s1

Zi

0

t 2

0

Z

Vi2 V02 ϕ(Ysi2 )ds2 ds1

k X √ √ Z i Vi V02 ϕ(Y√i t ) t V03 ϕ(Ys0 )ds + Z 1 V1 V02 ϕ(Y t0 ) t + 2

s1 0


2

i=2

Zi


2

i=2

Vi2 V02 ϕ(Ysi2 )ds2 ds1 .

h √i = Now the last four terms are all O(t) since E Z 1 V1 V02 ϕ(Y t0 ) t i 2 h R t h

t √i i−1 3 0 3 i 2 E Z Vi V0 ϕ(Y√t ) t = 0 because Z is normal, E 02 V0 ϕ(Ys )ds ≤ V0 ϕ ∞ 2 i h R √ R


2 t s and finally E 0 0 1 Z i Vi2 V02 ϕ(Ysi2 )ds2 ds1 ≤ Vi2 V02 ϕ ∞ 2t . So for the third term in (3.11) we have established, !   k 2 3 2 X

3

2 2 t t t 2 2 k

V 0 ϕ +

V i V 0 ϕ E V0 ϕ(Y√ ) − V0 ϕ(x) ≤ t 8 ∞ ∞ 8 16 i=1

t3 kϕkV,6 . (3.13) 16 Similarly, for the second term on the RHS of (3.11),  !  k t X 2 t t t3 2 k t √ V V0 ϕ(x) ≤ kϕkV,6 E V0 ϕ(Y t ) − V0 ϕ(x) + V0 ϕ(x) + 2 2 i=1 i 2 2 8 (3.14) Finally, for the first term on the RHS of (3.11), ! Pk 2 2 h i ϕ(x) + V0 ϕ(x) 2t + V02 ϕ(x)t8 + i=1 Vi4 ϕ(x) t4! k  P P Pk 2 2 E ϕ(Y√t ) − k i−1 + i=1 Vi2 ϕ(x) 2t + V0 Vi2 ϕ(x) t4 + i=2 j=1 Vj2 Vi2 ϕ(x) t4 ≤

t3 kϕkV,6 (3.15) 16 Substituting (3.12), (3.13), (3.14) and (3.15) in (3.11) gives the bound for the case Λ = 1. An analogous bound is then established for the case Λ = −1. The final result is obtained by taking the average of the two cases. See [23] for details.  ≤

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The following Lemma is required to prove the main theorem below. Lemma 3.5. For 0 < s ≤ t ≤ 1 and any m−perfect family {X t (x)}t∈(0,1] with corresponding operator Q = {Qt }t∈(0,1] we have, kPt (Ps ϕ) − Qt (Ps ϕ)k∞ ≤ c8 kϕkp

M X tj/2

j=m+1 s

j−p 2

,

(3.16)

where ϕ ∈ Cbp (Rd ) for 0 ≤ p < ∞ and some constant c8 > 0. In particular, for ϕ ∈ CbM (Rd ), kPt (Ps ϕ) − Qt (Ps ϕ)k∞ ≤ c8 kϕkp t

m+1 2

.

(3.17)

Proof. Since Cb∞ (Rd ) is dense in Cbp (Rd ) in the topology generated by the norm ||·||p,∞ it suffices to prove (3.16) and (3.17) only for a function ϕ ∈ Cb∞ (Rd ). By Corollary 5.4 in Section 5, kPt ϕkV,j =

j X

X

≤

j X

X

i=1 α1 ,...,αi ∈A0 kα1 ∗...∗αi k=j

i=1 α1 ,...,αi ∈A0 kα1 ∗...∗αi k=j



V[α ] · · · V[α ] Pt ϕ 1 i ∞ CpT kϕkp

t(kα1 ∗...∗αi k−p)/2

≤

c9 kϕkp t

j−p 2

for some c9 ≡ c9 (j, p) ≥ 0. Then (3.16) and (3.17) follow from the definition of an m−perfect family.  ¯ (x) = {X ¯ t (x)}t∈[0,∞) below corresponds to the The family of processes X Kusuoka approximation. We recall that Kusuoka’s result requires only the UFG condition. Example 3.6. A family of random variables {Zα : α ∈ A0 } is said to be mr moment similar if E[ |Zα | ] < ∞ for any r ∈ N, α ∈ A0 and Z(0) = 1 with, E[Zα1 . . . Zαj ] = E[Iα1 . . . Iαj ] for any j = 1, . . . , m and α1 , . . . , αj ∈ A0 such that kα1 k + · · · + kαj k ≤ m where Iα is defined as above. Let {Zα : α ∈ A0 } be a family of m−moment similar random variables and let ¯ (x) = {X ¯ t (x)}t∈[0,∞) be the family of processes, X ¯ t (x) = X

m X 1 j! j=0

X

α1 ,...,αj ∈A0 , kα1 k+···+kαj k≤m

t

kα1 k+···+kαj k 2

(Pα01 . . . Pα0j )(V[α1 ] . . . V[αj ] H)(x) (3.18)

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where H : Rd → Rd is defined H(x) = x and Pα0 := |α|−1

|α| X (−1)j+1 j=0

j

X

Zβ1 . . . Zβj

β1 ∗...∗βj =α

with the corresponding operator Q = {Qt }t∈(0,1] in Cb (Rd ) defined by, ¯t (x))] Qt ϕ(x) = E[ϕ(X

for ϕ ∈ Cb∞ (Rd ) Then, kPt+s ϕ − Qt Ps ϕk∞ ≤ c10 k∇ϕk∞ for some constant c10 > 0. Proof.

m+1 mX

tj/2

j=m+1 s

j−1 2

(3.19)

See Definition 1, Theorem 3 and Lemma 18 in Kusuoka [35] for (3.19). 

¯ (x) , x ∈ R as defined in (3.18) is not m-perfect. However, The family X inequality (3.19) is a particular case of (3.16) where p = 1 and M = mm+1 . Since (3.16) is the only result required to obtain (3.20), we deduce from the proof of Theorem 3.7 that (3.20), with p = 1, holds for Kusuoka’s method as well. Similarly part (ii) of Corollary 3.8 holds for Kusuoka’s method. The set of vector fields appearing in (3.18) belong to the Lie algebra generated by the original vector fields {V0 , V1 , ..., Vk } . Ben Arous [36] and Burrage & Burrage [27] employ the same set of vector fields to produce strong approximations of solutions of SDEs. Notably, the same ideas appear much earlier in Magnus [37], in the context of approximations of the solution of linear (deterministic) differential equations. Castell [38] also gives an explicit formula for the solution of an SDE in terms of Lie brackets and iterated Stratonovich integrals. We now prove our main result on m-perfect families, the gist of which can be conveyed by the concept of local and global order of an approximation. Local order measures how close an approximation is to the exact solution on a sub-interval of the integration, given an exact initial condition at the start of that subinterval. The global order of an approximation looks at the build up of errors over the entire integration range. The theorem below states that, in the best possible case, the global order of an approximation obtained using an m-perfect family is one less than the local order. More precisely, for a suitable partition, the global error is of order m−1 whilst the local error is of order m+1 2 2 . Let us define the function,  − 1 min(γp,(m−1)) if γp 6= m − 1 n 2 Υp (n) = n−(m−1)/2 ln n for γp = m − 1 d

In the following,



m m E γ,n (ϕ) := PT ϕ − Qm hn Qhn−1 . . . Qh1 ϕ

∞

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for γ ∈ R, n ∈ N. Theorem 3.7. Let T, γ > 0 and πn = {tj = ( nj )γ T }nj=0 be a partition of the interval [0, T ] where n ∈ N is such that {hj = tj − tj−1 }nj=1 ⊆ (0, 1]. Then for any m-perfect family {X t (x)}t∈[0,T ] with corresponding operator Q = {Qt }t∈(0,1] we have, for ϕ ∈ Cbp (Rd ) where p = 1, ..., m,

E γ,n (ϕ) ≤ c11 Υp (n) kϕk + Ph1 ϕ − Qm (3.20) h ϕ p

1

∞

for some constant c11 ≡ c11 (γ, M, T ) > 0 where M ≥ m + 1, as in Definition 3.1. then, In particular, if γ ≥ m−1 p E γ,n (ϕ) ≤

Proof.

c11 n

m−1 2

We have,



kϕkp + Ph1 ϕ − Qm h1 ϕ ∞

E γ,n (ϕ) = Phn (PT −hn ϕ) − Qm hn (PT −hn ϕ) +

n−1 X

m m Qm hn . . . Qhj+1 (PT −hj+1 −···−hn ϕ − Qhj PT −hj −···−hn ϕ)

n−1 X



m m Qm hn . . . Qhj+1 (Phj Ptj−1 ϕ − Qhj Ptj−1 ϕ ).

j=1

= Phn (Ptn−1 ϕ) − Qm hn (Ptn−1 ϕ) +

j=1

By Lemma 3.5, there exists a constant c8 > 0 such that,



Phn (Ptn−1 ϕ) − Qm h (Ptn−1 ϕ) n

∞

≤ c8 kϕkp

M l/2 X hn l−p 2

.

l=m+1 tn−1

Since P is a semigroup and Qm hj is a Markov operator for j = 2, . . . , n − 1,







m

m m (P P ϕ − Q P ϕ ) ≤ P ϕ − Q P ϕ

Qhn . . . Qm

P

h t t h t t j j−1 j−1 j j−1 j−1 hj+1 hj hj ∞

≤ c12 kϕkp

l/2 M X hj

∞

l−p 2

l=m+1 tj−1

for some c12 > 0. Finally, since Qm hj is a Markov operator, it follows from (3.24) that,

m

m m

Qh . . . Qm

h2 (Ph1 ϕ − Qh1 ϕ) ∞ ≤ Ph1 ϕ − Qh1 ϕ ∞ . n Combining these last four results gives,

E

γ,n

l/2 n M X X



hj m m m

(ϕ) = PT ϕ − Qhn . . . Qh1 ϕ ∞ ≤ Ph1 ϕ − Qh1 ϕ ∞ +c12 ||ϕ||p l−p . 2 j=2 l=m+1 tj−1

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It follows, almost immediately from the definition of hj that,  γ−1 Z u γT (j − 1)γ−1 j du hj = nγ j −1 j−1 j u γ−1 ) ≤ max[( j−1 )γ−1 , 1] ≤ max[2γ−1 , 1]. Hence for but for j ∈ {2, . . . , n}, ( j−1 l = m + 1, . . . , M , l/2

hj

(l−p)/2

tj−1

( γT (j−1) nγ ≤ 

≤ c13 (

γ−1

max[2γ−1 , 1])l/2
(l−p)/2 j−1 γ T n

(γ−1)l γ(l−p) γp−l T p T l − (l−p) 2 )2 (j − 1) 2 − 2 = c13 ( γ ) 2 (j − 1) 2 γ n n

where c13 ≡ c13 (γ, M ) = max[1, (γ max[2γ−1 , 1])M/2 ]. It follows that, l/2

M X

hj

M X

hj

(l−p)/2 l=m+1 tj−1

M γp−l 1 γp X ≤ c14 ( ) 2 (j − 1) 2 n l=m+1

PM γp−(m+1) γp−l 2 where c14 ≡ c14 (γ, M, T ) = T p/2 c13 and since l=m+1 (j − 1) 2 = (j − 1) PM −(m+1) γp−(m+1) l 2 M we have, × l=0 (j − 1)− 2 ≤ (j − 1) l/2

(l−p)/2 l=m+1 tj−1

γp−(m+1) 1 γp 2 ≤ c14 M ( ) 2 (j − 1) . n

(3.21)

We now consider (3.21) for three different ranges of γ.  P  P∞ γp−(m+1) γp−(m+1) n m−1 2 2 ≤ j=2 (j − 1) and since the For γ ∈ 0, p , j=2 (j − 1) series on the RHS is convergent, we have, n−

γp 2

n X j=2

(j − 1)

γp−(m+1) 2

≤ c15 n−

γp 2

for some constant c15 ≡ c15 (γ, M ) > 0. Pn −1 ≤ c16 ln n for some constant c16 ≡ c16 (γ, M ) > 0 For γ = m−1 j=2 (j − 1) p , so we have, n−

γp 2

n X j=2

(j − 1)

γp−(m+1) 2

≤ c16 n−

(m−1) 2

ln n

R 1 γp−(m+1) R1 Pn j−1 γp−(m+1) 1 γp−(m−1) 2 2 2 dx = c17 0 x−1+ dx For γ > m−1 j=2 ( n ) p , n ≤ c17 0 x ≤ c18 (like a Riemann integral) for some constants c17 ≡ c17 (γ, M ), c18 ≡ c18 (γ, M ) so, n

− γp 2

n X j=2

(j − 1)

γp−(m+1) 2

=n

− m−1 2

 n  X j −1 j=2

n

γp−(m+1) 2

m−1 1 ≤ c18 n− 2 n



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We observe that the rate of convergence is the controlled by the maximum

between Υ (n) and the rate at which Ph1 ϕ − Qm h1 ϕ ∞ converges to 0. We define γk2 ¯ k1 ,k2 (n) = Υk1 (n) + n− 2 . Hence we have the following corollary: Υ Corollary 3.8. (i) For any ϕ ∈ CbM (Rd ), ¯ m+1,m+1 (n) kϕk . E γ,n (ϕ) ≤ c19 Υ M for some constant c19 > 0. In particular, if γ ≥ 1, then E γ,n (ϕ) ≤ (ii) For any ϕ ∈ Cb1 (Rd ),

c19 n

m−1 2

kϕkM .

¯ 1,1 (n) kϕk E γ,n (ϕ) ≤ c21 Υ 1

for some constant c21 > 0, if there exists a constant c20 > 0 independent of t such that, √ ¯ t (x) − x ≤ c20 t. sup X (3.22) x∈Rd

In particular, if γ ≥ m − 1, then E γ,n (ϕ) ≤ (iii) For any ϕ ∈

Cbl (Rd )

c21

n

m−1 2

where 1 < l < M ,

kϕk1 .

¯ l,c23 (n) kϕk E γ,n (ϕ) ≤ c24 Υ l for some constant c24 > 0, if there exist two constants c22 , c23 > 0 independent of t such that, kPt ϕ − Qm t ϕk∞ ≤ c22 t In particular, if γ ≥ m − 1, then E γ,n (ϕ) ≤

c24 n

c23 2

m−1 2

kϕkl .

(3.23)

kϕkl .

Proof. (i) The result follows from the theorem and the definition of an m−perfect family. (ii) If ϕ ∈ Cb (Rd ) is Lipschitz then, √ |Qt ϕ(x) − ϕ (x)| ≤ c20 ||∇ϕ||∞ t (3.24) hence,

P h1 ϕ − Q m

h ϕ 1

∞

≤ c20 ||ϕ||1

√

t.

(iii) The result follows from the theorem and (3.23). Finally we define µt to be the law of Xt : µt (ϕ) = E [ϕ (Xt )] for ϕ ∈ Cb (Rd ). We also define µN t to be the probability measure defined by, h i Z m m m m m µN (ϕ) = E Q Q . . . Q ϕ (X ) = Qm 0 t hn hn−1 h1 hn Qhn−1 . . . Qh1 ϕ (x) µ0 (dx) Rd



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for ϕ ∈ Cb (Rd ) and need to introduce the family of norms on the set of signed measures: n o |µ|l = sup |µ (ϕ)| , ϕ ∈ Cbl (Rd ), kϕkl,∞ ≤ 1 , l ≥ 1.

Obviously, |µ|l ≤ |µ|l0 if l ≤ l0 . In other words, the higher the value of l, the coarser the norm. We have the following. Corollary 3.9. N ¯ m+1,m+1 (n). In particular, if γ ≥ 1, then (i) For Nl ≥ M , cwe have µt − µt l ≤ c19 Υ 19 µt − µt ≤ m−1 . l n 2 ¯ 1,1 (n). In particular, if γ ≥ m − 1, (ii) If (3.22) is satisfied then µt − µN t l ≤ c21 Υ c 21 then µt − µN t l ≤ m−1 . n 2 ¯ l,c23 (n). In particular, if γ ≥ m−1, (iii) If (3.23) is satisfied then µt − µN t l ≤ c24 Υ c24 N then µt − µt l ≤ m−1 . n

2

Throughout, the constants c19 , c21 , c23 , c24 > 0 correspond to those found in Corollary 3.8.

Remark We deduce that there is a payoff between the rate of convergence and the coarseness of the norm employed: the finer the norm the slower the rate of convergence. Hence intermediate results such as part (iii) of Corollaries 3.8 and 3.9 may prove useful in subsequent applications. The additional constraint (3.23) holds, for example, for the Lyons-Victoir method, as a cubature formula of degree m is also a cubature formula of degree m0 for m0 ≤ m. Similarly, it holds for Kusuoka’s approximation since an m−similar family is also m0 −similar for any m0 ≤ m. 4. An Application to Filtering We begin with a short description of the filtering problem. Let (X, Y ) be a system of partially observed random processes. The process X satisfies the stochastic differential equation (1.1) and is the unobserved component. The process Y is the observable component and satisfies the evolution equation, Z t Yt = h(Xs )ds + Bt , 0

where {Bt }t∈[0,∞) is an l-dimensional Brownian motion independent of X and
l
h = hi i=1 ∈ Cb∞ Rd , Rl . Let (Yt )t≥0 be the filtration generated by Y , Yt = σ (Ys , 0 ≤ s ≤ t). The problem of stochastic filtering for the partially observed system (X, Y ) involves the construction of πt (ϕ), where π = {πt , t ≥ 0} is the conditional distribution of Xt given Yt and ϕ belongs to a suitably large class of functions. If ϕ is square integrable with respect to the law of Xt then, πt (ϕ) = E [ϕ (Xt ) |Yt ] , P − almost surely.

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˜ absolutely Using Girsanov’s theorem, one can find a new probability measure P continuous with respect to P (and vice versa), so that Y is a Brownian motion ˜ independent of X and, almost surely, under P, πt (ϕ) =

ρt (ϕ) , ρt (1)

(4.1)

where, "

˜ ϕ(Xt ) exp ρt (ϕ) = E

l Z X i=1

t 0

l

1X hi (Xs )dYsi − 2 i=1

Z

t 0

! # hi (Xs )2 ds Yt

(4.2)

˜ is the expectation with respect to P. ˜ The measure ρt is called the unand E normalised conditional distribution of the signal. The identity (4.1) is called the Kallianpur-Striebel Formula. In the following, we will denote by ||·||p , the Lp 1 ˜ |ξ| = E ˜ [|ξ|p ] p , for any rannorm with respect to the probability measure P, p

dom variable ξ. The laws of the families X (x) = {Xt (x)}t∈[0,∞) , x ∈ Rd and ¯ (x) = {X ¯ t (x)}t∈[0,∞) , x ∈ Rd are not affected by the change of measure, hence, X ˜ we can write, to avoid working with both P and P ˜ ˜ ¯ (Pt ϕ)(x) = E[ϕ(X t (x))], (Qt ϕ)(x) = E[ϕ(Xt (x))].

In the following, we will only consider equidistant partitions and smooth functions. The method of approximation and the results closely follow the application of the classical Euler method as described in Picard [39] and Talay [5]. Let yr , r = 1, ..., n be the observation process increments yr = Y (r+1)t − Y rt and n n
∞ d hr ∈ Cb R , r = 0, ..., n − 1, be the (observation dependent) functions defined
2

Pl ¯ r : C ∞ Rd → C ∞ Rd , r = 0, 1, ..., n by hr = (hi y i − t hi ). Let Rr , R i=1

r

s

2n

s

b

b

be the following operators,


¯ n ϕ (x) = Qs ϕ (x) for ϕ ∈ C ∞ Rd , x ∈ Rd Rsn ϕ (x) = Ps ϕ (x) , R s b
and, for r = 0, 1, ..., n − 1, and for ϕ ∈ Cb∞ Rd , x ∈ Rd , ˜ [ ϕ (Xs (x)) exp (hr (Xs (x)))| Ys ] = Ps ϕr (x) Rsr ϕ (x) = E


  ¯ s (x) Ys = Qs ϕr (x) , ¯ r ϕ (x) = E ¯ s (x) exp hr X ˜ ϕ X R s

where ϕr = ϕ exp (hr ) and s ∈ [0, 1]. Firstly, one approximates ρ by replacing the (continuous) with  it observation path a discrete version. We choose the equidistant partition n , i = 0, 1, ...n of the interval [0, t] and consider only the observation data {yr , r = 0, 1, ..., n}. We define the measure, " ! # n−1  X  n ˜ ρt (ϕ) = E ϕ(Xt ) exp hi X it (4.3) Yt n i=0 i h
˜ R0t R1t ...Rnt ϕ (X0 ) Yt for ϕ ∈ Cb Rd . =E n

n

n

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Following Theorem 1 from Picard [39], for any ϕ ∈ Cb∞ (Rd ) there is a constant c ≡ c (t, ϕ) such that, ||ρt (ϕ) − ρnt (ϕ)||2 ≤

c n Qn

Qn ¯ it . For this we need to The next step is to approximate i=0 Rit with i=0 R n n adapt the definition of an m−perfect family so that we may use functions which are parametrized by the observation path Y . Let CbY,∞ (Rd ) be the set of measurable
functions, f : Rd × C [0, T ], Rl → R with the following properties:
i. for any y ∈ C [0, T ], Rl the function x → f (x, y) belongs to Cb∞ (Rd ). ii. for any multi-index α ∈ A, any x ∈ Rd and p ≥ 1, |Dα f (x, Y )|p < ∞. iii. for any multi-index α ∈ A and p ≥ 1, |||Dα f |||p,∞ = supx∈Rd |Dα f (x, Y )|p < ∞. P For f ∈ CbY,∞ (Rd ) we define the norm |||f |||m |||Dα f |||p,∞ . We note that p = α∈A(m)
if f : Rd × C [0, T ], Rl → R is constant then ||Dα f (x, Y )||p =

1 in the y variable,

∇ f + ... k∇m f k . |Dα f (x, Y )| and |||f |||m = ||f || + p ∞ ∞ ∞ ¯ (x) that satisfies the equivalent of (3.7) We now consider an m-perfect family X extended to functions in CbY,∞ (Rd ); more precisely we will assume that for any f ∈ CbY,∞ (Rd ), ˜ m |Y ] Qt f − E[f t

p,∞

≤C

M X

i

i=m+1

ti/2 |||f |||p ,

(4.4)

for some constants C > 0 and M ≥ m + 1, where ftm is the truncation defined in (3.4). Note that the original definition (3.7) implies (4.4) due to the inequality kϕkV,i ≤ C kϕki for i ∈ N first established in Section 2. Indeed, if f ∈ Cb∞ (Rd ), in other words it is constant in the y variable, then (3.7) and (4.4) actually coincide. The original Markov family X (x), the family generated by the Lyons-Victoir method and the one generated by the Ninomiya-Victoir algorithm satisfy (4.4). The following two lemmas are required to prove the main theorem below. ¯ (x) be an m-perfect family X ¯ (x) that satisfies (4.4). Then there Lemma 4.1. Let X is a constant c25 = c25 (t, m, p) > 0 such that for any ϕ ∈ Cb∞ (Rd ) and r = 1, ..., n, we have, ¯ r−1 r r+1 r+1 r n R R ...R ϕ ≤ c25 n−(m+1)/2 ||ϕ||M . R t R t R t ...Rnt ϕ − Rr−1 t t t t n

n

n

n

n

n

n

n

p,∞

Proof. Again, using the variational argument in Friedman [40] p.122 (5.17) one can check that for any ϕ ∈ Cb∞ (Rd ) and r = 1, ..., n the functions, exp (hr−1 ) Rrt Rr+1 ...Rnt ϕ ∈ CbY,∞ (Rd ). t n

n

n

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Moreover there is a constant c26 ≡ c26 (M, p) such that for any ϕ ∈ Cb∞ (Rd ) and r = 1, ..., n, M ...Rnt ϕ ≤ c26 kϕkM (4.5) exp (hr−1 ) Rrt Rr+1 t n

p

n

n

¯ (x) be an m-perfect family that satisfies (4.4). Then since X (x) also satisfies Let X (4.4), it follows by the triangle inequality that for s ∈ [0, 1] , |||Qs f − Ps f |||p,∞ ≤ Cs(m+1)/2 |||f |||M p

Hence,

...Rnt ϕ − P nt exp (hr−1 ) Rrt Rr+1 ...Rnt ϕ Q nt exp (hr−1 ) Rrt Rr+1 t t n

n

n

n

n

n

 (m+1)/2 M t ...Rnt ϕ ≤C exp (hr−1 ) Rrt Rr+1 t n n n n p

p,∞

(4.6)

¯ r−1 Rrt Rr+1 ...Rnt ϕ = The result now follows from (4.5) and (4.6) and the fact that R t t n

n

n

n

...Rnt ϕ and Rr−1 Rrt Rr+1 ...Rnt ϕ = P nt exp (hr−1 ) Rrt Rr+1 Q nt exp (hr−1 ) Rrt Rr+1 t t t t n n n n n n n n n n  ...R t ϕ. n

¯ (x) be an m-perfect family that satisfies (4.4). Then there is a Lemma 4.2. Let X constant c27 ≡ c27 (t, m, p) > 0 such that for any ϕ ∈ Cb∞ (Rd ) we have, 0 1 ¯ 0t R ¯ 1t ...R ¯ nt ϕ ≤ c27 n−(m−1)/2 ||ϕ||M . R t R t ...Rnt ϕ − R n

Proof.

n

n

n

n

p,∞

n

Let us observe that,

¯ 0t R ¯ 1t ...R ¯ nt ϕ = R0t R1t ...Rnt ϕ − R ¯ 0t R1t ...Rnt ϕ R t R t ...Rnt ϕ − R 0

1

n

n

n

n

n

n

n

+

n−1 X j=1

n

n

n



n

n

¯ jt Rj+1 ¯ j−1 ¯ 0t ...R R ...Rnt ϕ − Rjt Rj+1 ...Rnt ϕ R t t t n

n

n



n

n

¯ nt ϕ − Rnt ϕ ¯ n−1 ¯ 0t ...R R +R t n

n

n

n

n

Also note that for p ≥ 1 and r = 1, ..., n,  p  ¯ 0 ¯ j−1 ¯ j j+1 R t R t ...Rnt ϕ − Rjt Rj+1 ...Rnt ϕ R t ...R t t n n n n n n n n p,∞  p r+1 r−1 r r+1 r n n ˜ R ¯ r−1 R ...R R ...R R ϕ − R R ϕ ≤ c28 E t t t t t t t t n

n

≤ c28 n

−(m+1)/2

n

n

n

n

n

n

n



n

||ϕ||M .

p,∞





where c28 ≡ c28 (t, m, p) > 0 is a constant independent of ϕ and r. The claim follows.  Finally let us define now the measures, i h
¯ 1t ...R ¯ nt ϕ (X0 ) Yt for ϕ ∈ Cb Rd ˜ R ¯ 0t R ρ¯nt (ϕ) = E n

and let

π ¯tn

=

ρ¯nt /¯ ρnt (1)

n

n

be its normalized version.

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¯ (x) be an m-perfect family that satisfies (4.4). Then there is Theorem 4.3. Let X a constant c29 ≡ c29 (t, m, p) > 0 such that for any ϕ ∈ Cb∞ (Rd ) we have, ||ρnt (ϕ) − ρ¯nt (ϕ)||p ≤ c29 n−(m−1)/2 ||ϕ||M . Proof.

We have for p ≥ 1,

ip i p1  h h ¯ 0t R ¯ 1t ...R ¯ nt ϕ (X0 ) Yt ˜ E ˜ R0t R1t ...Rnt ϕ − R ||ρnt (ϕ) − ρ¯nt (ϕ)||p ≤ E n n n n n n 0 1 n 0 1 n ¯tR ¯ t ...R ¯ t ϕ , ≤ R t R t ...R t ϕ − R n

n

n

n

n

n

hence the result.

p,∞



¯ (x) be an m-perfect family that satisfies (4.4) with m ≥ 3 Corollary 4.4. Let X and assume that X0 has all moments finite. Then for any ϕ ∈ Cb∞ (Rd ) there is a constant c30 ≡ c30 (t, m, p, ϕ) > 0 such that, ¯tn (ϕ)||p ≤ ||πtn (ϕ) − π Proof.

c30 (ϕ) . n

Since,

||ϕ|| 1 |ρt (ϕ) − ρ¯nt (ϕ)| + |ρt (1) − ρ¯nt (1)| ρt (1) ρt (1) the Corollary follows by applying Theorem 4.3 and the fact that ρt (1)−1 p is finite for any p ≥ 0.  |πt (ϕ) − π ¯tn (ϕ)| ≤

5. Some Auxiliary Results The main aim of this section is to deduce inequality (5.3). In the following we will adopt the framework of Kusuoka-Stroock [13]. Let (Θ, B, W) be the standard Wiener space with continuous paths θ : [0, ∞) → Rd satisfying θ(0) = 0. Then Θ with the topology of uniform convergence on compact intervals is a Polish space. Also let H ⊂ ΘR be the Hilbert space of absolutely continuous functions h ∈ Θ such ∞ 2 that khkH = ( 0 |h0 (t)| dt)1/2 < ∞. Let W 1 (R) denote the space of measurable Φ : Θ → R with the following two properties: (i) For all h ∈ H, there exists a measurable function Φh : Θ → R such that Φh = Φ, W − a.s. and t ∈ R → Φh (θ + th) ∈ R is strictly absolutely continuous for all θ. (ii) There exist a measurable map, DΦ : Θ → L(H; R) such that, for all h ∈ H and ε > 0,   Φ(θ + th) − Φ(t) − DΦ(θ)(h) ≥ ε = 0. lim W θ : t |t|↓0

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On W 1 (R) the norm k·kq;R is defined as follows,

(n) (n) kΦkq;R := kΦkR

Lq (W)

,

n P (n) where kΦkR = kDm ΦkHm (R) for q ∈ [2, ∞) . Note that D 0 Φ = Φ with m=0

0 (n)

D Φ 0 = |Φ| so E[|Φ|] ≤ kΦkq;R . Finally let G(L) be the set of all Φ ∈ H (R) W 1 (R) to which D and the Ornstein-Uhlenbeck operator L as defined in Ref. [11] can be applied infinitely often.

Definition 5.1. We say that f ∈ ηr (Rd ; R) for r ∈ Z if f is a measurable map from (0, ∞) × Rd × Ω into R such that,

(i) f (t, ·, ω) : Rd → R is smooth for each t ∈ (0, ∞) and W − a.e ω ∈ Ω. (ii) f (·, x, ·) : (0, ∞) × Ω → R is progressively measurable for each x ∈ Rd . ∂α d (iii) ∂x α f (t, x, ·) ∈ G(L) and is continuous in t ∈ (0, ∞) for any α ∈ A 1 , x ∈ R

α (k) 1 ∂

(iv) sup sup r/2 < ∞ for any α ∈ A1 , k ∈ N, T > 0 and α f (t, x, ω) 0
∂x

t

q;R

2 ≤ q < ∞.

For a proof of the following Theorem see Kusuoka-Stroock [13], Theorem 2.15 (p. 405). Theorem 5.2. (Kusuoka-Stroock) Under the UFG condition, for any Φ ∈ ¯ α ∈ ηr−kαk (Rd ; R) such that, ηr (Rd ; R), r ∈ Z and α ∈ A1 there exists Φ ¯ α (t, x)f (Xt (x))] V[α] E[Φ(t, x)f (Xt (x))] = E[Φ

(5.1)

for any f ∈ Cb∞ (Rd ), t > 0, x ∈ Rd . The following immediate extension of Theorem 5.2 holds under the additional condition V0. Lemma 5.3. Under the UFG+V0 condition, for any Φ ∈ ηr (Rd ; R) there exists ¯ α ∈ ηr−2 (Rd ; R) such that, Φ ¯ 0 (t, x)f (Xt (x))] V0 E[Φ(t, x)f (Xt (x))] = E[Φ

for any f ∈

Cb∞ (Rd ),

(5.2)

d

t > 0, x ∈ R .

Proof. From Theorem 5.2 we get that for any Φ ∈ ηr (Rd ; R), r ∈ Z and β ∈ A1 (2) ¯ β ∈ ηr−kβk (Rd ; R) such that, there exists Φ ¯ β (t, x)f (Xt (x))] V[β] E[Φ(t, x)f (Xt (x))] = E[Φ

for any f ∈ Cb∞ (Rd ), t > 0, x ∈ Rd . Hence, X V0 E[Φ(t, x)f (Xt (x))] = ϕβ V[β] E[Φ(t, x)f (Xt (x))] β∈A1 (2)

¯ 0 (t, x)f (Xt (x))] = E[Φ

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The following Corollary is a generalization of Corollary 2.19 (p.407) in KusuokaStroock [13]. Corollary 5.4. For any r ∈ Z+ , {αi ∈ A0 }ri=1 and kα1 ∗ . . . ∗ αr k, there exists a constant CpT > 0 such that,

V[α ] . . . V[α ] Pt ϕ ≤ CpT t(p−kα1 ∗...∗αr k)/2 kϕk , t ∈ [0, T ] . (5.3) 1 r p ∞

for ϕ ∈ Cb∞ (Rd ) where k·kp for p ∈ N is defined in (2.2).

Proof. For the case p = 1, let Y (x) be the matrix valued process Yti,j (x) := ∂ i ∂xj (X (x)) where i, j = 1, ..., d. (see Ikeda & Watanabe [41], Chapter V, for details). Then for any i, j = 1, ..., d we have Y i,j (t, x) ∈ η0 (Rd ; R), in particular, " d # X sup sup E Y i,j (x) ≤ c(T ). (5.4) t∈[0,T ]x∈Rd

i=1

Differentiating under the integral sign (see Friedman [40], p.122 (5.17)) gives,   d X ∂ ∂ϕ i,j E[ϕ(Xt (x))] = E (Xt (x))Yt (x) . (5.5) ∂xj ∂xi i=1 Hence,

V[αr ] Pt ϕ (x) =

  d X ∂ϕ (Xt (x))Φi (t, x) E ∂xi i=1

P j (x) Yti,j (x) ∈ η0 (Rd ; R) for i = 1, ..., d. Then by Theorem with Φi (t, x) = dj=1 V[α r] ¯ iα ∈ η−(kα ∗···∗α k) (Rd ; R), i = 1, ..., d such that, 5.2 & Lemma 5.3 there exists Φ 1 r−1 for any ϕ ∈ Cb1 (Rd ),   d X ∂ϕ i V[α1 ] . . . V[αr−1 ] V[αr ] Pt ϕ (x) = V[α1 ] . . . V[αr−1 ] E (Xt (x))Φ (t, x) ∂xi i=1   d X ∂ϕ i ¯ = E Φα (t, x) (Xt (x)) ∂xi i=1

for any ϕ ∈ Cb∞ (Rd ). Hence,

d X V[α ] . . . V[α ] Pt ϕ (x) ≤ ||ϕ|| ¯ iα (t, x)|] E[|Φ 1 r 1 i=1

¯ iα (t, x), i = and (5.3) follows from the uniform bound (in (t, x)) on the L1 -norm of Φ 1, ..., d.

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For p > 1 one can obtain an analogue of (5.5) with higher derivatives of Pt ϕ in terms of derivatives of ϕ and a set of processes analogous to Y . The proof follows in a similar manner.  Acknowledgments The authors wish to thank the anonymous referee for his comments. We would also like to thank Alan Bain for carefully reading the paper and his many pertinent remarks. References [1] B. Øksendal, Stochastic differential equations. Universitext, (Springer-Verlag, Berlin, 1998), fifth edition. ISBN 3-540-63720-6. An introduction with applications. [2] P. E. Kloeden and E. Platen, Numerical solution of stochastic differential equations. vol. 23, Applications of Mathematics (New York), (Springer-Verlag, Berlin, 1992). ISBN 3-540-54062-8. [3] G. N. Milstein, A method with second order accuracy for the integration of stochastic differential equations, Theory Probab. Appl. 23(2), 396–401, (1978). ISSN 0040-361x. [4] K. Burrage, P. M. Burrage, and T. Tian, Numerical methods for strong solutions of stochastic differential equations: an overview, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460(2041), 373–402, (2004). ISSN 1364-5021. Stochastic analysis with applications to mathematical finance. [5] D. Talay. Efficient numerical schemes for the approximation of expectations of functionals of the solution of a SDE and applications. In Filtering and control of random processes (Paris, 1983), vol. 61, Lecture Notes in Control and Inform. Sci., pp. 294– 313. Springer, Berlin, (1984). [6] D. Talay, Discr´etisation d’une ´equation diff´erentielle stochastique et calcul approch´e d’esp´erances de fonctionnelles de la solution, RAIRO Mod´el. Math. Anal. Num´er. 20 (1), 141–179, (1986). ISSN 0764-583X. [7] G. N. Milstein, Weak approximation of solutions of systems of stochastic differential equations, Theory Probab. Appl. 30(4), 750–766, (1985). ISSN 0040-361X. [8] D. Talay and L. Tubaro, Expansion of the global error for numerical schemes solving stochastic differential equations, Stochastic Anal. Appl. 8(4), 483–509 (1991), (1990). ISSN 0736-2994. [9] V. Bally and D. Talay, The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function, Probab. Theory Related Fields. 104(1), 43–60, (1996). ISSN 0178-8051. [10] E. Platen and W. Wagner, On a Taylor formula for a class of Itˆ o processes, Probab. Math. Statist. 3(1), 37–51 (1983), (1982). ISSN 0208-4147. [11] S. Kusuoka and D. Stroock. Applications of the Malliavin calculus. I. In Stochastic analysis (Katata/Kyoto, 1982), vol. 32, North-Holland Math. Library, pp. 271–306. North-Holland, Amsterdam, (1984). [12] S. Kusuoka and D. Stroock, Applications of the Malliavin calculus. II, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32(1), 1–76, (1985). ISSN 0040-8980. [13] S. Kusuoka and D. Stroock, Applications of the Malliavin calculus. III, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34(2), 391–442, (1987). ISSN 0040-8980.

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[14] D. S. Jerison and A. S´ anchez-Calle, Estimates for the heat kernel for a sum of squares of vector fields, Indiana Univ. Math. J. 35(4), 835–854, (1986). ISSN 0022-2518. [15] A. S´ anchez-Calle, Fundamental solutions and geometry of the sum of squares of vector fields, Invent. Math. 78(1), 143–160, (1984). ISSN 0020-9910. [16] D. W. Stroock and S. R. S. Varadhan, Diffusion processes with continuous coefficients. I, Comm. Pure Appl. Math. 22, 345–400, (1969). ISSN 0010-3640. [17] D. W. Stroock and S. R. S. Varadhan. On the support of diffusion processes with applications to the strong maximum principle. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III: Probability theory, pp. 333–359, Berkeley, Calif., (1972). Univ. California Press. [18] D. Stroock and S. R. S. Varadhan, On degenerate elliptic-parabolic operators of second order and their associated diffusions, Comm. Pure Appl. Math. 25, 651–713, (1972). ISSN 0010-3640. [19] A. Millet and M. Sanz-Sol´e. A simple proof of the support theorem for diffusion processes. In S´eminaire de Probabilit´es, XXVIII, vol. 1583, Lecture Notes in Math., pp. 36–48. Springer, Berlin, (1994). [20] S. Ninomiya and N. Victoir. Weak approximations of stochastic differential equations and application to derivative pricing. www.crest.fr/pageperso/elie/bachelier\ _fichiers/Programme\_Fichiers/oxf%ord2004.pdf. [21] S. Kusuoka. Approximation of expectation of diffusion process and mathematical finance. In Taniguchi Conference on Mathematics Nara ’98, vol. 31, Adv. Stud. Pure Math., pp. 147–165. Math. Soc. Japan, Tokyo, (2001). [22] S. Kusuoka, Malliavin calculus revisited, J. Math. Sci. Univ. Tokyo. 10(2), 261–277, (2003). ISSN 1340-5705. [23] S. Ghazali. The Global Error in Weak Approximations of Stochastic Differential Equations. PhD thesis, Imperial College London, (2006). [24] P. E. Kloeden and E. Platen, Stratonovich and Itˆ o stochastic Taylor expansions, Math. Nachr. 151, 33–50, (1991). ISSN 0025-584X. [25] J. G. Gaines, The algebra of iterated stochastic integrals, Stochastics Stochastics Rep. 49(3-4), 169–179, (1994). ISSN 1045-1129. [26] R. Pettersson, Stratonovich-Taylor expansion and numerical methods, Stochastic Anal. Appl. 10(5), 603–612, (1992). ISSN 0736-2994. [27] K. Burrage and P. M. Burrage, High strong order methods for non-commutative stochastic ordinary differential equation systems and the Magnus formula, Phys. D. 133(1-4), 34–48, (1999). ISSN 0167-2789. Predictability: quantifying uncertainty in models of complex phenomena (Los Alamos, NM, 1998). [28] K. Burrage and P. M. Burrage, Order conditions of stochastic Runge-Kutta methods by B-series, SIAM J. Numer. Anal. 38(5), 1626–1646 (electronic), (2000). ISSN 00361429. [29] T. Lyons and N. Victoir, Cubature on Wiener space, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460(2041), 169–198, (2004). ISSN 1364-5021. Stochastic analysis with applications to mathematical finance. [30] J. M. C. Clark. An efficient approximation scheme for a class of stochastic differential equations. In Advances in filtering and optimal stochastic control (Cocoyoc, 1982), vol. 42, Lecture Notes in Control and Inform. Sci., pp. 69–78. Springer, Berlin, (1982). [31] J. M. C. Clark. Asymptotically optimal quadrature formulae for stochastic integrals. In Proceedings of the 23rd Conference on Decision and Control, Las Vegas, Nevada, pp. 712–715. IEEE, (1984). [32] N. J. Newton, An asymptotically efficient difference formula for solving stochastic

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Chapter 9 Flow Properties of Differential Equations Driven by Fractional Brownian Motion Laurent Decreusefond and David Nualart∗ GET, Ecole Nationale Sup´erieure des T´el´ecommunications LTCI-UMR 5141, CNRS, 46, rue Barrault, 75634 Paris, France [email protected] We prove that solutions of stochastic differential equations driven by fractional Brownian motion for H > 1/2 define flows of homeomorphisms on Rd .

Contents 1 Introduction . . . . . . . . . . 2 Preliminaries . . . . . . . . . . 3 Stochastic differential equations 4 Flow of homeomorphisms . . . References . . . . . . . . . . . . . .

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249 251 253 258 261

1. Introduction Suppose that B H = {BtH , t ≥ 0} is an m-dimensional fractional Brownian motion with Hurst parameter H ∈ (0, 1), defined in a complete probability space (Ω, F, P ). That is, the components B H,j , j = 1, . . . , m, are independent zero mean Gaussian processes with the covariance function  1  2H t + s2H − |t − s|2H . (1.1) RH (t, s) = 2 For H = 12 , the process B H is an m-dimensional ordinary Brownian motion. On the other hand, from (1.1) it follows that 2 E( BtH,j − BsH,j ) = |t − s|2H .

As a consequence, the processes B H,j have stationary increments, and for any α < H we can select versions with H¨ older continuous trajectories of order α on a compact interval [0, T ]. ∗ Department

of Mathematics, University of Kansas, 1460 Jayhawk Blvd, Lawrence, Kansas 660457523, [email protected] 249

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This process was first studied by Kolmogorov [1] and later by Mandelbrot and Van Ness [2], where a stochastic integral representation in terms of an ordinary Brownian motion was established. The fractional Brownian motion  has Hthe following self-similar property: For any constant a > 0, the processes a−H Bat , t ≥ 0 and BtH , t ≥ 0 have the same distribution. For H = 21 the process B H has independent increments, but for H 6= 21 , this property is no longer true. In particular, if H > 12 , the fractional Brownian motion has the long range dependence property, which means that for each j = 1, . . . , m ∞ X

n=1

H,j Corr(Bn+1 − BnH,j , B1H,j ) = ∞.

The self-similar and long range dependence properties make the persistent fractional Brownian motion a convenient model for some input noises in a variety of topics from finance to telecommunication networks, where the Markov property is not required. This fact has motivated the recent development of the stochastic calculus with respect to the fractional Brownian motion. See Refs. [3–5], and the references therein. In this paper we are interested in stochastic differential equations on Rd driven by a multi-dimensional fractional Brownian motion with Hurst parameter H > 21 , that is, equations of the form Z t m Z t X Xti = X0i + σ i,j (s, Xs )dBsH,j + bi (s, Xs )ds, (1.2) j=1

0

0

i = 1, . . . , d. The stochastic integral appearing in (1.2) is a path-wise RiemannStieltjes integral. In fact, under suitable conditions on σ, the processes σ(s, Xs ) and BsH have trajectories which are H¨ older continuous of order strictly larger than 1 and we can use the approach introduced by Young [6]. A first result on the 2 existence and uniqueness of a solution for this kind of equations was obtained by Lyons [7], using the notion of p-variation. On the other hand, the theory of rough path analysis introduced by Lyons [7] (see also the monograph by Lyons and Qian [8]), has allowed Coutin and Qian [9] to establish the existence of strong solutions and a Wong-Zakai type approximation limit for the stochastic differential equations of the form (1.2) driven by a fractional Brownian motion with parameter H > 14 . In Ref. [9] sufficient conditions on the vector fields b and σ are given to ensure existence and uniqueness of the solution of (1.2) even when vector fields do not commute. Z¨ ahle [10] has introduced a generalized Stieltjes integral using the techniques of fractional calculus. This integral is expressed in terms of fractional derivative Rb operators and it coincides with the Riemann-Stieltjes integral a f dg, when the functions f and g are H¨ older continuous of orders λ and µ, respectively and λ+µ > 1. Using this formula for the Riemann-Stieltjes integral, Nualart and R˘ a¸scanu [11] have

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obtained the existence of a unique solution for the stochastic differential equations (1.2) under general conditions on the coefficients. Later on, Nualart and Saussereau [12] have studied the regularity in the sense of Malliavin Calculus of the solution of Equation (1.2), and they have established the absolute continuity of the law of the random variable Xt under some non-degeneracy conditions on the coefficient σ. The main result of this paper is the flow and homeomorphic properties of X as a function of the initial condition x. Since the solution of (1.2) is defined path-wise, ordinary (i.e., deterministic) methods are in use here. Namely, we use the estimates found in Ref. [11] and approximate fractional Brownian motion by a sequence of regular processes to prove that the solution {Xrt (x), 0 ≤ r ≤ t ≤ T, x ∈ Rd } of Z t Z t Xrt (x) = x + σ (s, Xrs (x)) dB H (s) + b(s, Xrs (x))ds, r

r

d

defines a flow of R -homeomorphisms. The paper is organized as follows. Section 2 contains some preliminaries on fractional calculus. In Section 3 we review some results on the properties of the solution of Equation (1.2) and we establish some continuity estimates as a function of the initial condition and the driven input, which are needed later. Finally in Section 4 we show that Equation (1.2) defines a flow of homeomorphisms. 2. Preliminaries Let a, b ∈ R, a < b. Let f ∈ L1 (a, b) and α > 0. The left-sided and right-sided fractional Riemann-Liouville integrals of f of order α are defined for almost all x ∈ (a, b) by Z x 1 α−1 α (x − y) f (y) dy Ia+ f (x) = Γ (α) a and

α Ib− f

−α

(−1) (x) = Γ (α)

Z

b x

(y − x)

α−1

f (y) dy,

R∞ −α respectively, where (−1) = e−iπα and Γ (α) = 0 rα−1 e−r dr is the Euler funcα α tion. Let Ia+ (Lp ) (resp. Ib− (Lp )) be the image of Lp (a, b) under the action of α α α α operator Ia+ (resp. Ib− ). If f ∈ Ia+ (Lp ) (resp. f ∈ Ib− (Lp )) and 0 < α < 1 then the Weyl derivative ! Z x 1 f (x) f (x) − f (y) α Da+ f (x) = +α dy 1(a,b) (x) (2.1) Γ (1 − α) (x − a)α (x − y)α+1 a resp.

α Db− f

(−1)α (x) = Γ (1 − α)

f (x) +α (b − x)α

Z

b x

f (x) − f (y) (y − x)

α+1

dy

!

1(a,b) (x)

!

(2.2)

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is defined for almost all x ∈ (a, b) . For any 0 < λ ≤ 1, denote by C λ (0, T ) the space of λ-H¨ older continuous functions f : [0, T ] → R, equipped with the norm kf k∞ + kf kλ , where kf kλ =

|f (t) − f (s)|

sup

(t − s)λ

0≤s
.

Recall from Ref. [13] that we have: p 1 α α and q = then Ia+ (Lp ) = Ib− (Lp ) ⊂ Lq (a, b) . p 1 − αp 1 1 α α • If α > then Ia+ (Lp ) ∪ Ib− (Lp ) ⊂ C α− p (a, b) . p

• If α <

α The linear spaces Ia+ (Lp ) are Banach spaces with respect to the norms

α

α kf kI α (Lp ) = kf kLp + Da+ f Lp ∼ Da+ f L p , a+

α and the same is true for Ib− (Lp ). Suppose that f ∈ C λ (a, b) and g ∈ C µ (a, b) with λ + µ R> 1. Then, from b the classical paper by Young [6], the Riemann-Stieltjes integral a f dg exists. The following proposition can be regarded as a fractional Rintegration by parts formula, b and provides an explicit expression for the integral a f dg in terms of fractional derivatives (see Ref. [10]).

Proposition 2.1. Suppose that f ∈ C λ (a, b) and g ∈ C µ (a,R b) with λ + µ > 1. Let b λ > α and µ > 1 − α. Then the Riemann-Stieltjes integral a f dg exists and it can be expressed as Z b Z b 1−α α α f dg = (−1) Da+ f (t) Db− gb− (t) dt, (2.3) a

a

where gb− (t) = g (t) − g (b). Z¨ ahle [10] introduced a generalized Stieltjes integral of f with respect to g defined by the right-hand side of (2.3), assuming that f and g are functions such that g(b−) 1−α α exists, f ∈ Ia+ (Lp ) and gb− ∈ Ib− (Lq ) for some p, q ≥ 1, 1/p+1/q ≤ 1, 0 < α < 1. Let α < 21 and d ∈ N∗ . Denote by W0α,∞ (0, T ; Rd ) the space of measurable functions f : [0, T ] → Rd such that kf kα,∞ := sup

t∈[0,T ]

|f (t)| +

Z

t 0

|f (t) − f (s)| (t − s)

α+1

ds

!

< ∞.

We have, for all 0 < ε < α C α+ε (0, T ; Rd) ⊂ W0α,∞ (0, T ; Rd ) ⊂ C α−ε (0, T ; Rd ).

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Denote by WT1−α,∞ (0, T ; Rm ) the space of measurable functions g : [0, T ] → Rm such that   Z t |g(y) − g(s)| |g(t) − g(s)| + dy < ∞. kgk1−α,∞,T := sup (t − s)1−α (y − s)2−α 0<s 0 we have C 1−α+ε (0, T ; Rm) ⊂ WT1−α,∞ (0, T ; Rm ) ⊂ C 1−α (0, T ; Rm) .

(2.4)

Moreover, if g belongs to WT1−α,∞ (0, T ; Rm ), its restriction to (0, t) belongs to 1−α It− (L∞ (0, t; Rm )) for all t and 1−α
1 sup Dt− gt− (s) Γ(1 − α) 0<s
Λα (g) :=

(2.5)

Rt The integral 0 f dg can be defined for all t ∈ [0, T ] if g belongs to WT1−α,∞ (0, T ) and f satisfies kf kα,1 :=

Z

T 0

|f (s)| ds + sα

Z

T 0

Z

s 0

|f (s) − f (y)| dy ds < ∞. (s − y)α+1

Furthermore the following estimate holds Z T f dg ≤ Λα (g) kf kα,1 . 0

3. Stochastic differential equations driven by an fBm

We are going to consider first the case of a deterministic equation. Let 0 < α < 21 be fixed. Let g ∈ WT1−α,∞ (0, T ; Rm ). Consider the deterministic differential equation on Rd Z t m Z t X ξti = xi0 + bi (s, ξs )ds + σ i,j (s, ξs ) dgsj , t ∈ [0, T ] , (3.1) 0

j=1

0

i,j i d i = 1, ..., d, where x0 ∈ Rd , and
the coefficients
σ , b : [0, T ] × R → R are measur
i,j i able functions. Set σ = σ d×m , b = b d×1 and for a matrix A = ai,j d×m
P 2 P 2 and a vector y = y i d×1 denote |A|2 = i,j ai,j and |y|2 = i y i . Let us consider the following assumptions on the coefficients.

(H1) σ(t, x) is differentiable in x, and there exist some constants

0 < β, δ ≤ 1,

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M1 , M2 , M3 > 0 such that the following properties hold:  i) Lipschitz continuity     |σ(t, x) − σ(t, y)| ≤ M1 |x − y|, ∀ x ∈ Rd , ∀ t ∈ [0, T ]      older continuity   ii) H¨   |∂ σ(t, x) − ∂ σ(t, y)| ≤ M |x − y|δ , xi xi 2  ∀ |x| , |y| ∈ Rd , ∀t ∈ [0, T ] , i = 1, . . . , d,      iii) H¨ older continuity in time      |σ(t, x) − σ(s, x)| + |∂xi σ(t, x) − ∂xi σ(s, x)| ≤ M3 |t − s|β   ∀ x ∈ Rd , ∀ t, s ∈ [0, T ] .

(H2) There exist constants L1 , L2 > 0 such that the following properties hold:  i) Local Lipschitz continuity     |b(t, x) − b(t, y)| ≤ L1 |x − y|, ∀ |x| , |y| ∈ Rd , ∀t ∈ [0, T ] , Set

 ii) Linear growth    |b(t, x)| ≤ L2 (1 + |x|), ∀x ∈ Rd , ∀t ∈ [0, T ] . α0 = min



δ 1 , β, 2 1+δ



.

The following existence and uniqueness result has been proved in Ref. [11]. Theorem 3.1. Suppose that the coefficients σ(t, x) and b(t, x) satisfy assumptions (H1) and (H2). Then, if
α < α0 there exists a unique solution of Equation (3.1) in the space C 1−α 0, T ; Rd .

Actually, these conditions can be slightly relaxed. For instance, the H¨ older continuity of the partial derivatives of σ and the H¨ older continuity of the coefficient b may hold only locally (see Ref. [11] for the details). We now state two theorems which are consequences of the estimates found in Ref. [11].
For any λ ≥ 0 we introduce the equivalent norm in the space W0α,∞ 0, T ; Rd defined by   Z t |f (t) − f (s)| kf kα,λ = sup e−λt |f (t)| + ds . (t − s)α+1 t∈[0,T ] 0

Theorem 3.2. Let ξt (x0 ) denote the solution of (3.1) at time t with initial condition x0 . Fix R > 1. Then there exists a constant C such that for any x0 and x1 in the ball 1 h  i 1−2α 1 B(0, R) = {x ∈ Rd , |x| ≤ R} and for any λ > R exp C (1 + Λα (g)) 1−2α we have    −1 1 |x0 − x1 |. kξ(x0 ) − ξ(x1 )kα, λ ≤ 1 − R exp C (1 + Λα (g)) 1−2α λ2α−1

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Proof. It is proved in Ref. [11] that there exists a constant C1 such that if λ0 = 1 C1 (1 + Λα (g)) 1−2α then for any initial condition x0 in the ball of radius R we have kξ(x0 )kα,λ0 ≤ 2 (1 + |x0 |) ≤ 4R.

(3.2)

Given a function f ∈ W0α,∞ (0, T ; Rd) we define as in Ref. [11] Z t (b) Ft (f ) = b(s, f (s))ds, 0 Z t (σ) Gt (g, f ) = σ(s, f (s))dg(s), 0 Z r |f (r) − f (s)|δ ds. ∆(f ) = sup (r − s)α+1 r∈[0,T ] 0 If f, h ∈ W0α,∞ (0, T ; Rd ) (see Ref. [11]) there exist constants C2 and C3 such that

(b)

≤ C2 λα−1 kf − hkα,λ , (3.3)

F (f ) − F (b) (h) α,λ

(σ)

≤ C3 Λα (g)λ2α−1

G (g, f ) − G(σ) (g, h) α,λ



(σ)

G (g, f )

α,λ

× (1 + ∆ (f ) + ∆ (h)) kf − hkα,λ ,   ≤ C4 Λα (g)λ2α−1 1 + kf kα,λ

Also, if f ∈ W0α,∞ (0, T ; Rd) and h is a bounded measurable function, then

(b) ≤ C5 (1 + khk∞ ),

F (h)

1−α  

(σ)

≤ C6 Λα (g) 1 + kf kα,∞ .

G (g, f ) 1−α

(3.4) (3.5)

(3.6) (3.7)

We have the following estimate

∆(ξ(x0 )) = sup r∈[0,T ]

Z

r 0

|ξr (x0 ) − ξs (x0 )|δ ds (r − s)α+1

δ−α(1+δ)

≤

T kξ(x0 )k1−α , δ − α (1 + δ)

and using (3.6), (3.7), and (3.2) we obtain



kξ(x0 )k1−α ≤ |x0 | + F (b) (ξ(x0 ))



+ G(σ) (ξ(x0 )) 1−α 1−α   ≤ |x0 | + C5 (1 + kξ(x0 )k∞ ) + Λα (g)C6 1 + kξ(x0 )kα,∞ ≤ C7 eλ0 T (1 + |x0 |)(1 + Λα (g))

≤ 2C7 eλ0 T R(1 + Λα (g))   1 = 2C7 exp T C1 (1 + Λα (g)) 1−2α R(1 + Λα (g)).

(3.8)

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Hence, from (3.8) we get   1 ∆(ξ(x0 )) ≤ C8 exp T C1 (1 + Λα (g)) 1−2α R(1 + Λα (g)).

(3.9)

Using (3.3), (3.4), and (3.9) we obtain for x0 and x1 in the ball of radius R kξ(x0 ) − ξ(x1 )kα, λ ≤ |x0 − x1 | + kF (b) (ξ(x0 )) − F (b) (ξ(x1 ))kα, λ



+ G(σ) (g, ξ(x0 )) − G(σ) (g, ξ(x1 )) α,λ

≤ |x0 − x1 | + C1 λ

+ C2 Λα (g)λ

As a consequence,

2α−1

α−1

kξ(x0 ) − ξ(x1 )kα,λ

(1 + ∆ (ξ(x0 )) + ∆ (ξ(x1 ))) kξ(x0 ) − ξ(x1 )kα,λ

≤ |x0 − x1 | + C1 λα−1 kξ(x0 ) − ξ(x1 )kα,λ    1 + C2 Λα (g)λ2α−1 1 + 2C8 exp T C1 (1 + Λα (g)) 1−2α  × R(1 + Λα (g)) kξ(x0 ) − ξ(x1 )kα,λ .

kξ(x0 ) − ξ(x1 )kα, λ

    1 ≤ |x0 − x1 | + λ2α−1 exp C (1 + Λα (g)) 1−2α R kξ(x0 ) − ξ(x1 )kα,λ

for some constant C, which implies the result.



Theorem 3.3. The map ξ : WT1−α,∞ (0, T ; Rm ) −→ W0α,∞ (0, T ; Rd) g 7−→ ξ,

where ξ is the solution of (3.1) with x0 ∈ B(0, R), is continuous. Namely, for λ > 0 large enough we have kξ(g) − ξ(h)kα, λ ≤ Proof.

C1 λ2α−1 kξ(g)kα, λ Λα (g − h). 1 − C2 λ2α−1 (1 + Λα (h))

With the previous notations, we have ξ(g) = x0 + F (b) (ξ(g)) + G(σ) (g, ξ(g))

and ξ(h) = x0 + F (b) (ξ(h)) + G(σ) (h, ξ(h)). Thus, ξ(g) − ξ(h) = F (b) (ξ(g)) − F (b) (ξ(h))   + G(σ) (g − h, ξ(g)) + G(σ) (h, ξ(g)) − G(σ) (h, ξ(h)) .

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According to (3.5), for λ sufficiently large, kξ(g) − ξ(h)kα, λ ≤ C2 λα−1 kξ(g) − ξ(h)kα,λ

+ C4 λ2α−1 (1 + Λα (g − h))kξ(g)kα, λ + C3 λ2α−1 Λα (h)kξ(g) − ξ(h)kα, λ .

Hence the result.



Consider Equation (1.2) on Rd , for t ∈ [0, T ], where X0 is a d-dimensional random variable, and the coefficients σ i,j , bi : Ω × [0, T ] × Rd →R are measurable functions. Theorem 3.4. Suppose that X0 is an Rd -valued random variable, the coefficients σ(t, x) and b(t, x) satisfy assumptions (H1) and (H2), where the constants might depend on ω, with β > 1−H, δ > 1/H −1. Then if α ∈ (1 − H, α0 ), then there exists a unique stochastic process, whose trajectories belong to the space W 0α,∞ (0, T ; Rd), solution of the stochastic equation (1.2) and, moreover, for P-almost all ω ∈ Ω
X (ω, ·) ∈ C 1−α 0, T ; Rd .

Consider the particular case where b = 0 and σ is time independent, that is, Z t Xt = X 0 + σ(Xs )dBsH . (3.10) 0

By the above theorem this equation has a unique solution provided σ is continuously differentiable, and σ 0 is bounded and H¨ older continuous of order δ > H1 − 1. Hu
and Nualart [14] have established the following estimates. Choose θ ∈ 1 2 , H . Then, the solution to Equation (3.10) satisfies sup |Xt | ≤ 21+kT (kσ

0

k∞ ∨|σ(0)|)

0≤t≤T

1/θ

1/θ

kB H kθ

(|X0 | + 1) ,

(3.11)

where k is a constant depending only on θ. Moreover, if σ is bounded and kσ 0 k∞ 6= 0 this estimate can be improved in the following way   1−θ 1 sup |Xt | ≤ |X0 | + kkσk∞ T θ kB H kθ ∨ T kσ 0 k∞θ kB H kθθ , (3.12) 0≤t≤T

where again k is a constant depending only on θ. These estimates improve those obtained by Nualart and R˘ a¸scanu [11] based on a suitable version of Gronwall’s lemma. The estimates (3.11) and (3.12) lead to the following integrability properties for the solution of Equation (3.10).

Theorem 3.5. Consider the stochastic differential equation (3.10), and assume that E(|X0 |p ) < ∞ for all p ≥ 2. If σ 0 is bounded and H¨ older continuous of order 1 δ > H − 1, then   p E sup |Xt | < ∞ (3.13) 0≤t≤T

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for all p ≥ 2. If furthermore σ is bounded and E (exp(λ|X0 |γ )) < ∞ for any λ > 0 and γ < 2H, then    γ <∞ (3.14) E exp λ sup |Xt | 0≤t≤T

for any λ > 0 and γ < 2H. Nualart and Saussereau [12] have proved that the random variable Xt belongs locally to the space D∞ if the function σ is infinitely differentiable and bounded together with all its partial derivatives. As a consequence, they have derived the absolute continuity of the law of Xt for any t > 0 assuming that the initial condition is constant and the vector space spanned by {(σ i,j (x0 ))1≤i≤d , 1 ≤ j ≤ m} is Rd . Applying Theorem 3.5 Hu and Nualart have proved [14] that if the function σ is infinitely differentiable and bounded together with all its partial derivatives, then for any t ∈ [0, T ] the random variable Xt belongs to the space D∞ . As a consequence, if the matrix a(x) = σσ T (x) is uniformly elliptic, then, for any t > 0 the probability law of Xt has a C ∞ density. In a recent paper, Baudoin and Coutin [15] have extended this result and derived the regularity of the density under H¨ ormander hypoellipticity conditions. 4. Flow of homeomorphisms Let π = {0 = t0 < t1 < · · · < tn = T } be the uniform partition of the interval [0, T ]. n,H That is tk = kT the polygonal approximation of n , k = 0, . . . , n. We denote by B the fractional Brownian motion defined by Btn,H =

n−1 X

BtHk +

k=0

  n (t − tk ) BtHk+1 − BtHk 1(tk ,tk+1 ] (t). T

In order to get a precise rate for these approximations we will make use of the following exact modulus of continuity of the fractional Brownian motion (see Ref. [16]). There exists a random variable G such that almost surely for any s, t ∈ [0, T ] we have p H B − B H ≤ G|t − s|H log (|t − s|−1 ). (4.1) t s Fix θ < H. We have the following result, which provides the rate of convergence of these approximations in H¨ older norm. Lemma 4.1. There exist a random variable CT,β such that p kB H − B n,H kC θ (0,T ;Rm ) ≤ CT,β nθ−H log n.

(4.2)

Proof. To simplify the notation we will assume that m = 1. Fix 0 < s < t < T and assume that s ∈ [tl , tl+1 ] and t ∈ [tk , tk+1 ]. Let us first estimate h1 (s, t) =

1 |B n,H − BtH − (Bsn,H − BsH )| . (t − s)θ t

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T n,

then using (4.1) we obtain h   n |h1 (s, t)| ≤ T −β nβ BtHk − BtH + (t − tk ) BtHk+1 − BtHk T   i n + BtHl − BsH + (s − tl ) BtHl+1 − BtHl T p ≤ 4GT −θ+H n−H+θ log (n/T ).

If t − s < Tn , then there are two cases. Suppose first that s, t ∈ [tk , tk+1 ]. In this case, if n is large enough we obtain using (4.1) H H |BtH − BsH | n |Btk+1 − Btk | + (t − s) (t − s)θ T (t − s)θ p p ≤ G|t − s|H−θ log |t − s|−1 + GT −1+H log(n/T ) n1−H (t − s)1−θ p ≤ 2GT −θ+H n−H+θ log (n/T ).

|h1 (s, t)| ≤

On the other hand, if s ∈ [tk−1 , tk ] and t ∈ [tk , tk+1 ] we have, again if n is large enough   1 n H H H H |h1 (s, t)| ≤ − B + (t − t ) B − B B k t tk+1 tk (t − s)θ tk T  n o n − BtHk − BsH − (tk − s) BtHk − BtHk−1 T h  i 1 n H H H H H H ≤ |B (t − s) |B − B | + |B − B | − B | + t tk tk−1 tk+1 tk s (t − s)θ T     p G n Hp H −1 + 2(t − s) ≤ |t − s| log |t − s| log (n/T ) (t − s)θ T p −θ+H −H+θ ≤ 3GT n log (n/T ). This proves (4.2).



Corollary 4.2. For any α ∈ (1 − H, 1/2), we have: sup Λα (B n,H ) < +∞ and n

lim Λα (B n,H − B H ) = 0.

n→+∞

Proof. Choose η > 0 in such a way that 1 − α + η < H. According to (2.4) and (2.5), we have Λα (B n,H ) ≤ cη kB n,H kC 1−α+η (0,T ;Rm ) and Λα (B n,H − B H ) ≤ cη kB n,H − B H kC 1−α+η (0,T ;Rm ) . Then, Lemma 4.1 implies that the sequence B n,H converges to B H in the norm of C 1−α+η (0, T ; Rm) which yields the results. 

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Consider for any 0 ≤ r ≤ t ≤ T and any natural number n ≥ 1 the following equations Z t Z t n n n,H n Xrt (x) = x + σ (s, Xrs (x)) dB (s) + b(s, Xrs (x))ds, (4.3) r

r

and Yrtn (x)

=x+

Z

t r

σ (s, Ystn (x)) dB n,H (s)

+

Z

t r

b(s, Ystn (x))ds.

(4.4)

We know from standard results on ordinary differential equations that for any n ≥ 1, (1) (2) (3) (4)

Equations (4.3) and (4.4) have a unique solution. n n For any x ∈ Rd , for any 0 ≤ r ≤ τ ≤ t ≤ T , Xτnt (Xrτ (x)) = Xrt (x). d n n n For any x ∈ R , for any 0 ≤ r ≤ τ ≤ t ≤ T , Yrτ (Yτ t (x)) = Yrt (x). n The maps (x 7→ Xrt (x)) and (x 7→ Yrtn (x)) are Rd -homeomorphisms inverse of each other: n n Xrt (Yrtn (x)) = x and Yrtn (Xrt (x)) = x.

We are then in position to prove our main theorem: Theorem 4.3. Assume that Hypotheses (H1) and (H2) hold. Then, claims 1, 2, 3 and 4 also hold for the equations Z t Z t H Xrt (x) = x + σ (s, Xrs (x)) dB (s) + b(s, Xrs (x))ds, (4.5) r

r

and Yrt (x) = x + Proof.

Z

t H

σ (s, Yst (x)) dB (s) + r

Z

t

b(s, Yst (x))ds.

(4.6)

r

Point 1 is proved in Ref. [11]. As to the second claim, proceed as follows: n n Xτnt (Xrτ (x)) − Xτ t (Xrτ (x)) = Xτnt (Xrτ (x)) − Xτnt (Xrτ (x))

+ (Xτnt − Xτ t )(Xrτ (x)).

Fix ε > 0 and α such that 1−H < α < 21 . Fix a trajectory ω ∈ Ω. Choose n0 so that Λα (B n,H −B H ) ≤ ε for all n ≥ n0 and choose λ such that λ2α−1 C2 supn Λα (B n,H ) ≤ 1 2 . Then, according to Theorem 3.3, for any n ≥ n0 , n kXr· − Xr· kα,λ ≤

C1 λ2α−1 kXr· kα, λ Λα (B n,H − B H ) 1 − C2 λ2α−1 (1 + Λα (B n,H ))

≤ 2C1 λ2α−1 kXr· kα, λ ε. Hence, for n ≥ n0 ,

|(Xτnt − Xτ t )(Xrτ (x))| ≤ cε.

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n The convergence of Xr· implies that there exists R such that for any τ ∈ [r, t] and n for any n ≥ n0 , Xrτ (x) ∈ B(0, R). Then, Theorem 3.2 implies that for λ large enough n |Xτnt (Xrτ (x)) − Xτnt (Xrτ (x))| ! !−1 1   1−2α λ2α−1 ≤ 1 − R exp C 1 + sup Λα (B n,H ) n

n × |Xrτ (x) − Xrτ (x)| n ≤ c|Xrτ (x) − Xrτ (x)|.

We have thus proved that n 0 = lim Xτnt (Xrτ (x)) − Xτ t (Xrτ (x)) n→+∞

n (x) − Xτ t (Xrτ (x)) = lim Xrt n→+∞

= Xrt (x) − Xτ t (Xrτ (x)). Other points are handled similarly.



Acknowledgement This work was carried out during a stay of Laurent Decreusefond at Kansas University, Lawrence KS. He would like to thank KU for warm hospitality and generous support. References [1] A. N. Kolmogorov, Wienershe Spiralen und einige andere interessante Kurven im Hilbertschen Raum, C. R. (Doklady) Acd. Sci. URSS (N. S.). 26, 115–118, (1940). [2] B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev. 10, 422–437, (1968). ISSN 1095-7200. ¨ unel, Stochastic analysis of the fractional Brownian [3] L. Decreusefond and A. S. Ust¨ motion, Potential Anal. 10(2), 177–214, (1999). ISSN 0926-2601. [4] D. Feyel and A. de La Pradelle, On fractional Brownian processes, Potential Anal. 10(3), 273–288, (1999). ISSN 0926-2601. [5] D. Nualart. Stochastic integration with respect to fractional Brownian motion and applications. In Stochastic models (Mexico City, 2002), vol. 336, Contemp. Math., pp. 3–39. Amer. Math. Soc., Providence, RI, (2003). [6] L. C. Young, An inequality of the H¨ older type connected with Stieltjes integration, Acta Math. 67, 251–282, (1936). [7] T. Lyons, Differential equations driven by rough signals. I. An extension of an inequality of L. C. Young, Math. Res. Lett. 1(4), 451–464, (1994). ISSN 1073-2780. [8] T. Lyons and Z. Qian, System control and rough paths. Oxford Mathematical Monographs, (Oxford University Press, Oxford, 2002). ISBN 0-19-850648-1. Oxford Science Publications.

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[9] L. Coutin and Z. Qian, Stochastic analysis, rough path analysis and fractional Brownian motions, Probab. Theory Related Fields. 122(1), 108–140, (2002). ISSN 0178-8051. [10] M. Z¨ ahle, Integration with respect to fractal functions and stochastic calculus. I, Probab. Theory Related Fields. 111(3), 333–374, (1998). ISSN 0178-8051. [11] D. Nualart and A. R˘ a¸scanu, Differential equations driven by fractional Brownian motion, Collect. Math. 53(1), 55–81, (2002). ISSN 0010-0757. [12] D. Nualart and B. Saussereau. Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion, (2005). Preprint. [13] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional integrals and derivatives. (Gordon and Breach Science Publishers, Yverdon, 1993). ISBN 2-88124-864-0. Theory and applications, Edited and with a foreword by S. M. Nikol0 ski˘ı, Translated from the 1987 Russian original, Revised by the authors. [14] Y. Hu and D. Nualart. Differential equations driven by H¨ older continuous functions of order greater than 1/2, (2006). Preprint. [15] F. Baudoin and M. Hairer. A version of h¨ ormander’s theorem for the fractional brownian motion, (2006). Preprint. [16] W. Wang, On a functional limit result for increments of a fractional Brownian motion, Acta Math. Hungar. 93(1-2), 153–170, (2001). ISSN 0236-5294.

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Chapter 10 Regularity of Transition Semigroups Associated to a 3D Stochastic Navier-Stokes Equation Franco Flandoli and Marco Romito∗ Dipartimento di Matematica Applicata, Universit di Pisa via Bonanno Pisano, 25/b, 56126 Pisa, Italy [email protected] A 3D stochastic Navier-Stokes equation with a suitable non degenerate additive noise is considered. The regularity in the initial conditions of every Markov transition kernel associated to the equation is studied by a simple direct approach. A by-product of the technique is the equivalence of all transition probabilities associated to every Markov transition kernel.

Contents 1 2

Introduction . . . . . . . . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . . . 2.1 Notations . . . . . . . . . . . . . . . . . . . . 2.2 Definitions, assumptions and known results . 3 The Log-Lipschitz estimate . . . . . . . . . . . . . . 3.1 Probability of blow-up . . . . . . . . . . . . . 3.2 Derivative of the regularised problem . . . . . 4 Equivalence of all transition probabilities . . . . . . 5 Conclusion and remarks . . . . . . . . . . . . . . . . A.1 An exponential tail estimate for the Stokes problem A.2 The deterministic equation . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction An old dream in stochastic fluid dynamics is to prove the well posedness of a stochastic version of the 3D Navier-Stokes equations, taking advantage of the noise, as one can do for finite dimensional stochastic equations with non regular drift (see for instance Stroock & Varadhan [1]). The problem is still open, although some intriguing results have been recently proved, see for instance Da Prato & Debussche [2], Mikulevicius & Rozovski [3], Flandoli & Romito [4, 5]. We recall here the framework constructed in Ref. [4] and prove some additional results. ∗ Dipartimento

di Matematica, Universit di Firenze, viale Morgagni 67/a, 50134 Firenze, Italy, [email protected] 263

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We consider a viscous, incompressible, homogeneous, Newtonian fluid described 3 by the stochastic Navier-Stokes equations on the torus T = [0, L] , L > 0, ∞ X · ∂u + (u · ∇) u + ∇p = ν4u + σi hi (x) β i (t) ∂t i=1

(1.1)

with div u = 0 and periodic boundary conditions, with suitable fields hi (x) and independent Brownian motions βi (t). The 3D random vector field u = u (t, x) is the velocity of the fluid and the random scalar field p = p (t, x) is the pressure. To simplify the exposition, we avoid generality and focus on one of the simplest set of assumptions: σi2 = λ−3 i where λi are the eigenvalues of the Stokes operator (see the next section). This assumption also allows us to compare more closely the results in Da Prato & Debussche [2] and Flandoli [6]. However, following Flandoli & Romito [4], we could treat any power law for σi . Under this assumption, one can associate a transition probability kernel P (t, x, ·) to equation (2.2), which is the abstract version of (1.1), in D(A) (see the definitions in Section 2.1 below), satisfying the ChapmanKolmogorov equation. In other words, there exists a Markov selection in D(A) for equation (2.2). To avoid misunderstandings, this does not mean that equation (2.2) has been solved in D(A) with continuous trajectories: this would imply well posedness. What has been proved is that the law of weak martingale solutions is supported on D(A) for all times, with a number of related additional properties, but a priori the typical trajectory may sometimes blow-up in the topology of D(A). The transition probabilities P (t, x, ·) are irreducible and strong Feller, hence equivalent, in D(A). These results and the existence of P (t, x, ·) have been proved first in Da Prato & Debussche [2] and Debussche & Odasso [7] by a careful selection from the Galerkin scheme. Then another proof by an abstract selection principle and the local-in-time regularity of equation (2.2) has been given in Flandoli & Romito [4]. More precisely, first one proves the existence of a Markov kernel P (t, x, ·) by means of a general and abstract method, then one proves that any such kernel is irreducible and strong Feller, hence equivalent, in D(A). We complement here the approach of Ref. [4] with two results. First, the simple idea used in Ref. [4] to prove the strong Feller property is here developed further, to show a weak form of Lipschitz continuity of P (t, x, ·) in x ∈ D(A). More precisely, we prove the estimate CT (1 + |Ax0 |6 )|Ah| log(|Ah|−1 ) (1.2) t∧1 for t ∈ (0, T ], x0 , h ∈ D(A), with |Ah| ≤ 1. This result has been proved in a stronger version in Da Prato & Debussche [2] for the transition kernel constructed from the Galerkin scheme, and also in Flandoli [6] for any Markov kernel associated to equation (2.2). In both cases the proof is based on the very powerful approach |P (t, x0 + h, Γ) − P (t, x0 , Γ)| ≤

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introduced in Ref. [2] which however requires a considerable amount of technical work. Here we give a rather elementary proof along the lines of Flandoli & Romito [4], based on the following simple idea: given x0 , h ∈ D(A), for a short random time the solution is regular, unique and differentiable in the initial conditions; then the propagation of regularity in x from small time to arbitrary time is due to the Markov property. Unfortunately we cannot prove in this way the stronger estimate obtained in Ref. [2] (where the right-hand-side of (1.2) has the form t−1+ε (1 + |Ax0 |2 )|Ah|), so our first result here has mostly a pedagogical character, since the proof is conceptually very easy. The second result, which follows from the same main estimates used to prove (1.2), is the equivalence P (1) (t, x, ·) ∼ P (2) (t0 , x0 , ·) for any t, t0 > 0 and x, x0 ∈ D(A), when P (i) (t, x, ·), i = 1, 2, are any two Markov transition kernels associated to equation (2.2) in D(A). We have not proved yet the existence of invariant measures associated to such kernelsa , but if we assume to have such invariant measures, it also follows that they are equivalent. This result and the gradient estimates discussed above could be steps to understand better the open question of well posedness for equation (2.2). In particular, it seems to be not so easy to produce examples of stochastic differential equations without uniqueness but where all Markov solutions are equivalent. Among the open problems related to this research we mention the relation between the regularity results for P (t, x, ·) in the initial condition discussed above and the properties of Malliavin derivatives, investigated for stochastic 3D Navier-Stokes equations by Mikulevicius and Rozovsky [3, 8]. 2. Preliminaries 2.1. Notations Denote by T = [0, 1]3 the three-dimensional torus, and let L2 (T ) be the space of vector fields u : T → R3 with L2 (T )-components. For every α > 0, let Hα (T ) be the space of fields u ∈ L2 (T ) with components in the Sobolev space H α (T ) = W α,2 (T ). Let D∞ be the space of infinitely differentiable divergence free periodic fields u on T , with zero mean. Let H be the closure of D ∞ in the topology of L2 (T ): it is the space of all zero mean fields u ∈ L2 (T ) such that div u = 0 and u · n on the boundary is periodic. We denote by h., .iH and |.|H (or simply by h., .i and |.|) the usual L2 -inner product and norm in H. Let V (resp. D(A)) be the closure of D ∞ in the topology of H1 (T ) (in the topology of H2 (T ), respectively): it is the space of divergence free, zero mean, periodic elements of H1 (T ) (respectively of H2 (T )). a This

is apparently due to technical reasons and it is the subject of a work in progress.

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The spaces V and D(A) are dense and compactly embedded in H. From Poincar´e R 2 2 inequality we may endow V with the norm kukV := T |Du (x)| dx. Let A : D(A) ⊂ H → H be the operator Au = −4u (component wise). Since A is a selfadjoint positive operator in H, there is a complete orthonormal system (hi )i∈N ⊂ H of eigenfunctions of A, with eigenvalues 0 < λ1 ≤ λ2 ≤ . . . (that is, Ahi = λi hi ). The fields hi in equation (2.2) will be these eigenfunctions. We have hAu, uiH = kuk2V for every u ∈ D(A). Let V 0 be the dual of V ; with proper identifications we have V ⊂ H ⊂ V 0 with continuous injections, and the scalar product h·, ·iH extends to the dual pairing h·, ·iV,V 0 between V and V 0 . We may enlarge this scheme to D(A) ⊂ V ⊂ H ⊂ V 0 ⊂ D(A)0 . Let B (·, ·) : V × V → V 0 be the bi linear operator defined as hw, B (u, v)iV,V 0 =

3 Z X

i,j=1

ui T

∂vj wj dx ∂xi

for every u, v, w ∈ V . We shall repeatedly use the following inequality: 1/2 A B (u, v) ≤ C0 |Au| |Av| H

(2.1)

for u, v ∈ D(A). The proof is elementary (see Flandoli [9]). 2.2. Definitions, assumptions and known results

We (formally) rewrite equations (1.1) as an abstract stochastic evolution equation in H, du(t) + [νAu(t) + B (u(t), u(t))] dt =

∞ X

σi hi dβi (t) .

(2.2)

i=1

Let us set Ω = C ([0, ∞); D(A)0 ) and denote by (ξt )t≥0 the canonical process on Ω, defined as ξt (ω) = ω (t), by F the Borel σ-algebra in Ω and by Ft the σ-algebra generated by the events {ξs ∈ Γ} with s ∈ [0, t] and Γ a Borel set of D(A)0 . Finally, denote by B(D(A)) the Borel σ-algebra of D(A) and by Bb (D(A)) the set of all real valued bounded measurable functions on D(A)). Definition 2.1. Given a probability measure µ0 on H, we say that a probability measure P on (Ω, F) is a solution to the martingale problem associated to equation (2.2) with initial law µ0 if 2 (MP1) P [ξ ∈ L∞ loc ([0, ∞); H) ∩ Lloc ([0, ∞); V )] = 1,

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(MP2) for each ϕ ∈ D ∞ the process (Mtϕ , Ft , P )t≥0 , defined P -a. s. on (Ω, F) as Z t Z t Mtϕ := hξt − ξ0 , ϕiH + νhξs , AϕiH ds − hB(ξs , ϕ), ξs iH ds 0

0

is a continuous square integrable martingale with quadratic variation X [M ϕ ]t = t σi2 |hϕ, hi i|2 , i∈N

(MP3) the marginal of P at time 0 is µ0 .

Remark 2.2. Among all test functions in property [M P 2], we can choose ϕ = hi . Set for all i, βi (t) = σ1i Mthi (and 0 if σi = 0). The (βi )i∈N are a sequence of P independent standard Brownian motions. Under the assumption i σi2 < ∞, the P∞ series i=1 σi hi βi (t) defines an H-valued Brownian motion on (Ω, F, Ft , P ), that we shall denote by W (t). The canonical process (ξt ) is a weak martingale solution of (2.2), in the sense that it satisfies (2.2) in the following weak form: there exists a Borel set Ω0 ⊂ Ω with P (Ω0 ) = 1 such that on Ω0 for every ϕ ∈ D ∞ and t ≥ 0 we have Z t Z t hB (ξs , ϕ) , ξs iH ds = hW (t) , ϕiH . (2.3) ν hξs , AϕiH ds − hξt − ξ0 , ϕiH + 0

0

The following theorem is well known, see for instance the survey paper of Flandoli [9] and the reference therein. P 2 Theorem 2.3. Assume i σi < ∞. Let µ be a probability measure on H such R 2 that H |x|H µ (dx) < ∞. Then there exists at least one solution to the martingale problem with initial condition µ. Definition 2.4. We say that P (·, ·, ·) : [0, ∞) × D(A) × B (D(A)) → [0, 1] is a Markov kernel in D(A) of transition probabilities associated to equation (1.1) if P (·, ·, Γ) is Borel measurable for every Γ ∈ B (D(A)), P (t, x, ·) is a probability measure on B (D(A)) for every (t, x) ∈ [0, ∞) × D(A), the Chapman-Kolmogorov equation Z P (t + s, x, Γ) = P (t, x, dy) P (s, y, Γ) D(A)

holds for every t, s ≥ 0, x ∈ D(A), Γ ∈ B (D(A)), and for every x ∈ D(A) there is a solution Px on (Ω, F ) of the martingale problem associated to equation (2.2) with initial condition x such that P (t, x, Γ) = Px [ξt ∈ Γ] for all t ≥ 0. We recall the following result from Da Prato & Debussche [2], Debussche & Odasso [7] or Flandoli & Romito [4]: Theorem 2.5. There exists at least one Markov kernel P (t, x, Γ) in D(A) of transition probabilities associated to equation (1.1).

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We recall that a P (t, x, Γ) is called irreducible in D(A) if for every t > 0, x0 , x1 ∈ D(A), ε > 0, we have P (t, x0 , BA (x1 , ε)) > 0, where BA (x1 , ε) is the ball in D(A) of centre x1 and radius ε. We say that P (t, x, Γ) is strong Feller in D(A) if Z x 7→ ϕ (y) P (t, x, dy) D(A)

is continuous on D(A) for every bounded measurable function ϕ : D(A) → R and for every t > 0. It is well known (see for example Da Prato & Zabczyk [10, Proposition 4.1.1]) that irreducibility and strong Feller in D(A) imply that the laws P (t, x, ·) are all mutually equivalent, as (t, x) varies in (0, ∞)×D(A). Because of this equivalence property, we say that P (t, x, Γ) is regular. We recall also that P (t, x, Γ) is called stochastically continuous in D(A) if limt→0 P (t, x, BA (x, ε)) = 1 for every x ∈ D(A) and ε > 0. In Da Prato & Debussche [2], the transition probability kernel constructed by Galerkin approximations is proved to be stochastically continuous, irreducible and strong Feller in D(A), hence regular. More generally (see Flandoli & Romito [4]): Theorem 2.6. Every Markov kernel P (t, x, Γ) in D(A) of transition probabilities associated to equation (1.1) is stochastically continuous, irreducible and strong Feller in D(A), hence regular. 3. The Log-Lipschitz estimate Theorem 3.1. Let P (t, x, Γ) be a Markov kernel in D(A) of transition probabilities associated to equation (1.1). Then, given T > 0, there is a constant C T such that the inequality CT (1 + |Ax0 |6 )|Ah| log(|Ah|−1 ) t∧1 holds for every t ∈ (0, T ], x0 , h ∈ D(A), with |Ah| ≤ 1, and Γ ∈ B (D(A)). |P (t, x0 + h, Γ) − P (t, x0 , Γ)| ≤

We explain here only the logical skeleton of the proof, which is very simple. The two main technical ingredients will be treated in the next two separate subsections. The first idea is to decompose: P (t, x0 + h, Γ) − P (t, x0 , Γ) Z = [P (ε, x0 + h, dy) − P (ε, x0 , dy)] P (t − ε, y, Γ) . D(A)

To shorten some notation, let us write Z (Pt ϕ) (x) =

D(A)

ϕ (y) P (t, x, dy)

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so, with the function ϕ (x) = 1{x∈Γ} the previous identity reads (Pt ϕ) (x0 + h) − (Pt ϕ) (x0 ) = (Pε (Pt−ε ϕ)) (x0 + h) − (Pε (Pt−ε ϕ)) (x0 ) .

(3.1)

It is now sufficient to estimate (Pε ψ) (x0 + h) − (Pε ψ) (x0 ) uniformly in ψ ∈ Bb (D(A)). The value of ε has to be chosen depending on the size of x0 and h, as we shall see. The second idea is to use an initial coupling: we introduce the equation with 2 cut-off χR (|Au| ), where χR (r) : [0, ∞) → [0, 1] is a non-increasing smooth function equal to 1 over [0, R], to 0 over [R + 2, ∞), and with derivative bounded by 1. The equation is h  i P∞ 2 du + Au + B(u, u)χR |Au| dt = i=1 σi hi dβi (t) , (3.2) u (0) = x. The definition of martingale problem for this equation is the same (with obvious adaptations) as the definition given above for equation (1.1). Let τR : Ω → [0, ∞] be defined as τR (ω) = inf {t ≥ 0 : |Aω (t)| ≥ R} . We recall the following result from Flandoli & Romito [4, Lemma 5.11]: (R)

Lemma 3.2. For every x ∈ D(A) there is a unique solution Px of the martingale problem associated to equation (3.2), with the additional property Px(R) [ξ ∈ C ([0, ∞) ; D(A))] = 1. Let Px be any solution on (Ω, F ) of the martingale problem associated to equation (2.2) with initial condition x. Then    (R)  EPx ϕ (ξt ) 1{τR ≥t} = EPx ϕ (ξt ) 1{τR ≥t} for every t ≥ 0 and ϕ ∈ Bb (D(A)). Introduce the notation (R)

(Pt

(R)

ϕ) (x) = EPx

[ϕ (ξt )] .

The previous lemma implies that for every ψ ∈ Bb (D(A)) we have |(Pε ψ)(x) − (Pε(R) ψ)(x)| ≤ 2Px [τR < ε] kψk∞ .

Summarising: Corollary 3.3. For every x0 , h ∈ D(A) and ψ ∈ Bb (D(A)) we have |(Pε ψ)(x0 + h) − (Pε ψ)(x0 )| ≤ 2 (Px0 +h [τR < ε] + Px0 [τR < ε]) kψk∞ + (Pε(R) ψ)(x0 + h) − (Pε(R) ψ)(x0 ) .

(3.3)

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Let us give now the proof of Theorem 3.1. Assume t ∈ (0, T ], x0 , h ∈ D(A) be given, with |Ah| ≤ 1. Let K > 0 be such that |Ax0 | + 1 ≤ K. We have |A(x0 + h)| ≤ K, so we may apply Proposition 3.5 below to both x0 and x0 + h. We thus get, for ε ∈ (0, 5C ∗1K 2 ), where C ∗ > 0 is the constant defined by (A.3) in the Appendix, we have Px0 +h [τ2K < ε] + Px0 [τ2K < ε] ≤ 2C# e−η#

K2 4ε

.

1 5C ∗ K 2 )

Given h, K and t as above, let us look for a value ε ∈ (0, such that ε ≤ t and the latter exponential quantity is smaller than |Ah|. We impose η#

K2 ≥ log(|Ah|−1 ) 4ε

hence it is sufficient to take ε≤

η# K 2 t 1 ∧ ∧ . 4 log(|Ah|−1 ) 2 5C ∗ K 2

(3.4)

We have proved so far the first claim of the following lemma. The second claim is a simple consequence of (3.1) and the previous corollary. Lemma 3.4. Given t > 0, x0 , h ∈ D(A), with |Ah| ≤ 1, and Γ ∈ B (D(A)), if ε is chosen as in (3.4), then Px0 +h [τ2K < ε] + Px0 [τ2K < ε] ≤ 2C# |Ah| and for ϕ(x) = 1{x∈Γ} and ψ = Pt−ε ϕ,

|Pt ϕ(x0 + h) − Pt ϕ(x0 )| ≤ 4C# |Ah| kϕk∞ + |Pε(2K) ψ(x0 + h) − Pε(2K) ψ(x0 )|.

Finally, from Proposition 3.6 below, renaming the constant C, with ϕ (x) = 1{x∈Γ} and ψ = Pt−ε ϕ, C 6 (2K) ψ) (x0 + h) − (Pε(2K) ψ) (x0 ) ≤ |Ah| eCK ε . (Pε ε Thus, for ε as in (3.4), we get |Pt ϕ (x0 + h) − Pt ϕ (x0 )| ≤ 4C# |Ah| +

6 C |Ah|eCK ε . ε

Let us further restrict ourselves to η# K 2 t 1 1 ε≤ ∧ ∧ ∧ 6, −1 ∗ 2 4 log(|Ah| ) 2 5C K K so that we have |Pt ϕ(x0 + h) − Pt ϕ(x0 )| ≤ 4C# |Ah| +

C |Ah|. ε

The choice ε=C

t∧1 K 6 log(|Ah|−1 )

is admissible for a suitable constant C > 0, and we finally get (1.2). The proof of Theorem 3.1 is complete.

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3.1. Probability of blow-up Proposition 3.5. Let K ≥ 1 and assume that x0 ∈ D(A) and ε > 0 are given such that |Ax0 | ≤ K and ε ≤ 5C ∗1K 2 , where C ∗ is the constant defined in (A.3). Then Px0 [τ2K < ε] ≤ C# e−η#

K2 4ε

,

for suitable universal constants η# > 0 and C# > 0. Proof. one has

From Corollary A.2.2 we know that if ε ≤ θε2 ≤

1 2 K 4

⇒

1 5C ∗ K 2

|Au(s)| < 2K for s ∈ [0, ε]

⇒

and |Ax0 | ≤ K, then τ2K ≥ ε,

where θε is defined in (A.2) in the Appendix. Therefore, with the constraints |Ax0 | ≤ K and ε ≤ 5C ∗1K 2 , by Proposition A.1.1 one gets Px0 [τ2K < ε] ≤ Px0



Θ2ε

 1 2 K2 > K ≤ C# e−η# 4ε . 4



3.2. Derivative of the regularised problem Here we show the regularity of the transition semigroup associated to the regularised problem (3.2). Proposition 3.6. For every R ≥ 1 and x0 , h ∈ D(A),  C kψk    6 (R) ∞ |Ah| eCR ε , Pε ψ (x0 + h) − Pε(R) ψ (x0 ) ≤ ε

where C is a universal constant.

Proof. We write the following computations for the limit problem but the understanding is that we do it on the Galerkin approximations. For every ψ ∈ Bb (H), ε > 0, from the Bismut-Elworthy-Li formula (see Da Prato & Zabczyk [10]),     (R) Pε ψ (x0 + h) − Pε(R) ψ (x0 )

C kψk∞ ≤ sup E ε η∈[0,1]

"Z

ε 0

# 3 2  12 2 (R) , A Dh ux0 +ηh (s) ds

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where, for each R ≥ 1 and x ∈ D(A), ux is the solution, starting at x, of problem (3.2). From the regularised equation we have 2 3 2 1 d 2 (R) ADh u(R) x (t) + A Dh ux (t) 2 dt D E 2 (R) (R) (R) (R) (R) ≤ χR (|Au(R) (t)| ) AD u , AB(D u , u ) + AB(u , D u ) h h h x x x x x x 2 (R) (R) (R) (R) (R) + 2χ0R (|Au(R) x (t)| )hAux , ADh ux i|hADh ux , AB(ux , ux )i|  3  (R) (R) 2 2 ≤ CχR |Au(R) A Dh u(R) x (t) ADh ux (t) Aux (t) x (t)|  2  3 (R) 3 2 (R) (R) D u (t) + Cχ0R Au(R) (t) (t) u (t) Au AD A h h x x x x   2 2 1 3 (R) 2 (R) 2 2 (R) ≤ A 2 Dh u(R) (t) + Cχ (t) u (t) (t) Au AD Au h x R x x x 2 2  2 (R) 6 2 + Cχ0R |Au(R) ADh u(R) x (t)| x (t) Aux (t) 2 2 1 3 6 (R) ≤ A 2 Dh u(R) x (t) + CR ADh ux (t) . 2

Thus

2 1 3 2 2 1 d 2 6 (R) ADh u(R) A Dh u(R) x (t) + x (t) ≤ CR ADh ux (t) . 2 dt 2 This implies 2 6 2 (t) ADh u(R) ≤ eCR t |Ah| x

and

Z

Thus

ε

0

 Z 2 3 2 (R) 2 A Dh ux0 +ηh (s) ds ≤ |Ah| 1 +

ε

6

CR6 eCR s ds 0



2

6

= |Ah| eCR ε .

 C kψk    6 (R) ∞ |Ah| eCR ε . Pε ψ (x0 + h) − Pε(R) ψ (x0 ) ≤ ε The proposition is proved.



4. Equivalence of all transition probabilities To make the following statement independent of previous results, we shall assume stochastic continuity, irreducibility and the strong Feller property in the theorem below, but we recall that these properties have been proved for every Markov kernel in D(A) associated to equation (1.1), under the assumptions of the introduction. Theorem 4.1. Let P (i) (t, x, Γ) be two Markov kernels in D(A) of transition probabilities associated to equation (1.1). Assume they are stochastically continuous, irreducible and strong Feller in D(A). Then the probability measures P (1) (t, x, ·) and P (2) (t0 , x0 , ·) are equivalent, for any t, t0 > 0 and x, x0 ∈ D(A).

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Proof. Step 1. Let us recall the following fact (see Ref. [10]): if a Markov kernel of transition probabilities P (t, x, Γ) in D(A) (or any Polish space) is stochastically continuous, irreducible and strong Feller, then the laws P (t1 , x1 , .) and P (t2 , x2 , .) are equivalent, for every t1 , t2 > 0, x1 , x2 ∈ D(A). Therefore, under our assumptions, it is sufficient to prove that, given t0 > 0, x0 ∈ D(A), the laws P (1) (t0 , x0 , .) and P (2) (t0 , x0 , .) are equivalent. Since the argument is symmetric between the two kernels, it is sufficient to prove the following statement: given t0 > 0, x0 ∈ D(A), if Γ is a Borel set in D(A) such that P (2) (t0 , x0 , Γ) = 0 then P (1) (t0 , x0 , Γ) = 0. In the following steps we prove such claim. Let us remark that, again by the equivalence property for P (2) , the assumption that P (2) (t0 , x0 , Γ) = 0 implies that P (2) (t, x, Γ) = 0 for every t > 0, x ∈ D(A). Therefore what we are going to prove is the following claim: given t0 > 0, x0 ∈ D(A), let Γ be a Borel set in D(A) such that P (2) (t, x, Γ) = 0 for every t > 0, x ∈ D(A); then P (1) (t0 , x0 , Γ) = 0. Step 2. Since both P (1) (·, ·, ·) and P (2) (·, ·, ·) satisfy (3.3), P (1) (t, x, Γ) = |P (1) (t, x, Γ) − P (2) (t, x, Γ)| ≤ 2(Px(1) [τR < t] + Px(2) [τR < t]). Now, for every pair (ε, x), with ε > 0 and x ∈ D(A), such that 5C ∗ (1 + |Ax|)2 ε ≤ 1 (the constant C ∗ is defined in (A.3), in the appendix), Proposition 3.5 implies that P (1) (ε, x, Γ) ≤ 2C# e−η#

(1+|Ax|)2 4ε

1

≤ 2C# e− 4ε η# .

Step 3. For every ε < 5C1 ∗ , set Aε = {x ∈ D(A) : 5C ∗ (1 + |Ax|)2 ε ≤ 1}, then by the Markov property and the previous step, Z P (1) (ε, x, Γ) P (1) (t0 , x0 , dx) P (1) (t0 + ε, x0 , Γ) = Acε

+

Z

P (1) (ε, x, Γ) P (1) (t0 , x0 , dx) Aε 1

≤ 2C# e− 4ε η# + P (1) (t0 , x0 , Acε ) Since P (1) (s, x0 , D(A)) = 1, we have P (1) (t0 , x0 , Acε ) −→ 0, as ε → 0, and thus lim P (1) (t0 + ε, x0 , Γ) = 0.

ε→0

Step 4. Again by the Markov property, for every neighborhood G of x0 in D(A), Z P (1) (t0 + ε, x0 , Γ) = P (1) (t0 , y, Γ) P (1) (ε, x0 , dy) ≥ P (1) (ε, x0 , G) inf P (1) (t0 , y, Γ). y∈G

Since the kernel P (1) is stochastically continuous, it follows that P (1) (ε, x0 , G) → 1, as ε → 0, and so, by the previous step, inf y∈G P (1) (t0 , y, Γ) −→ 0 as ε → 0. By the strong Feller property, the map y 7→ P (1) (t0 , y, Γ) is continuous, hence in conclusion P (1) (t0 , x0 , Γ) = 0. The proof is complete. 

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5. Conclusion and remarks We have proved that the transition probabilities associated to any Markov selection are all equivalent to each other. However, the problem of uniqueness of Markov selections remains open. We stress that it would imply uniqueness of solutions to the martingale problem, by the argument of Stroock & Varadhan [1, Theorem 12.2.4]. The estimates proved in this work allows us at least to state a sufficient condition for uniqueness of Markov selections. The proof is inspired by a well known proof in semigroup theory as well as by the proof of uniqueness given by Bressan and co-authors (see for instance Ref. [11]). Proposition 5.1. Assume that a Markov selection (Px )x∈D(A) has the following property: for every t > 0 and x ∈ D(A),  r c ! n X n k =0 lim P t − t, x, BA 0, n→∞ n t k=1

where BA (0, n) is the ball in D(A) of radius n. Then (Px )x∈D(A) coincides with any other Markov selection. Proof. Let (Qx )x∈D(A) be another Markov selection. Let us rewrite, for ϕ ∈ Cb (D(A)): Pt ϕ − Qt ϕ = Pt− nt P nt ϕ − Pt− nt Q nt ϕ +Pt− nt Q nt ϕ − Qt− nt Q nt ϕ and so on iteratively until we have Pt ϕ − Q t ϕ =

n X k=1

  Pt− kt P nt ψ (k−1)t − Q nt ψ (k−1)t n

n

n

where ψs = Qs ϕ. We have, by using (3.3) and Proposition 3.5,   P nt ψ (k−1)t − Q nt ψ (k−1)t (x) Pt− kt n n n i  h ξt− kt = EPx P nt ψ (k−1)t − Q nt ψ (k−1)t n n n h    i ≤ EPx P nt ψ (k−1)t − Q nt ψ (k−1)t ξt− kt 1{ξt− k t ∈A t } n n n n n      +EPx P nt ψ (k−1)t − Q nt ψ (k−1)t ξt− kt 1{ξt− k t ∈Act } n

n

n

n

n

n

+ 2Px [ξt− k t ∈ Act ] n n  r c ! k n t − t, x, BA 0, , n t

≤ 4C# e

≤ 4C# e− t η# + 2P

−n t η#

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where we have set At = {5C ∗ t(1 + |Ax|)2 ≤ 1} and, roughly, A nt ≈ BA 0, Hence  r c ! n X k n η −n P t − t, x, BA 0, |Pt ϕ(x) − Qt ϕ(x)| ≤ 4nC# e t # + 2 n t

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pn
t

.

k=1

which completes the proof of the proposition.



The criterion of this proposition is apparently not really useful at the present stage of our understanding. Indeed, if we apply Chebichev inequality we get the sufficient condition  n  1+ε 2(1+ε)  X t Px E =0 lim Aξt− kt n n→∞ n k=1

with is implied by the condition Z t  2(1+ε) EPx |Aξs | ds < ∞ 0

which however would easily imply the well posedness of the 3D Navier-Stokes equation by direct estimates of the difference of two solutions. A.1. An exponential tail estimate for the Stokes problem

Consider the following Stokes problem 3

dZ + AZ dt = A− 2 dW, Z(0) = 0, and set Θt = sups∈[0,t] |AZ(s)|. The next result is well known, but we give a proof to keep track of the dependence on the constants of interest in this paper. Proposition A.1.1. There exist η# > 0 and C# > 0 such that for every K ≥ and ε > 0,  K2 P Θε ≥ K] ≤ C# e−η# ε .

1 2

1

Proof. Step 1. Set y(t) = ε− 2 Z(εt), then it is easy to see that y solves the equa1 tion dy + εAy dt = Q 2 dW . Next, fix a value α ∈ ( 61 , 41 ), then by the factorisation method (see Da Prato & Zabczyk [12, Chapter 5]), Z t Z t −ε(t−s)A y(t) = e dWs = Cα e−ε(t−s)A (t − s)α−1 Y (s) ds, Rs

0

0

−ε(s−r)A

−α

where Y (s) = 0 e (s−r) dWr and Cα denotes a generic constant depending only on α (it will keep changing value along the proof). For every t ∈ (0, 1], older’s inequality that since α > 16 , it follows from H¨ Z 1  61 Z t α−1 6 . |Ay(t)|H ≤ Cα (t − s) |AY (s)|H ds ≤ Cα |AY (s)|H ds 0

0

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In conclusion, since ε−1 Θ2ε = supt∈[0,1] |Ay(t)|2H , it follows by the above inequality and standard arguments that " Z  2 K −a ˜ P Θε ≥ K] ≤ e ε E exp a

1 0

6

|AY (s)| ds

 31 !#

,

(A.1)

with a constant a that will be specified later (and a ˜ = a Cα ). Step 2. In order to estimate the expectation in (A.1), notice that exp a ˜

Z

≤a ˜

1

|AY (s)| ds

0

Z

6

1 0

|AY (s)|

10

 31 !

ds

 15

Z 1  n3 ∞ X a ˜n 6 |AY (s)|H ds = (A.2) n! 0 n=0 Z 1  n3 Z 1 X ∞ 2 X a ˜n a ˜n 6 |AY (s)|H |AY (s)|2n + ≤ H n! n! 0 0 n=3 n=0 a ˜2 + 2

Z

1 0

8

|AY (s)| ds

 21

+

Z

1

2

ea˜|AY (s)|H ds

0

Step 3. Now, AY (s) is a centered Gaussian process with covariance (cfr. proof of Theorem 5.9 in Da Prato & Zabczyk [12]) Z s ˜ Qs = (s − r)−2α A−1 e−2ε(s−r)A dr, 0

so that, by Proposition 2.16 in Ref. [12], 2

˜

1

E[ea˜|AY (s)|H ] = e− 2 Tr[log(1−2˜aQs )] , 1 ˜ s ) is the spectrum of Q ˜ s . Similarly, , where σ(Q provided that a ˜ ≤ inf λ∈σ(Q˜ s ) 2λ ˜ s ))p , for all integers p. E|AY (s)|2p = Cp (Tr(Q ˜ s ), then there is a eigenvalue In order to choose a suitable value of a, let µ ∈ σ(Q λ of A such that µ = µ(λ) is given by

µ = λ−1

Z

s

r−2α e−2rλε dr = λ−2+2α (2ε)−(1−2α)

0

Z

2λεs 0

r−2α e−r dr ≤ Cα λ−1 0 ,

where λ0 is the smallest eigenvalue of A. Hence a can be chosen as Cα λ0 , for a suitable Cα . Step 4. We conclude the proof: since a is small enough, we have that −Tr[log(1 − ˜ s )] ≤ Cα Tr[Q ˜ s ] and, as in step 3, 2˜ aQ T r[Qs ] =

X

λ∈σ(A)

λ

−2+2α

(2ε)

−(1−2α)

Z

2λεs 0

r−2α e−r dr ≤ Cα ε−(1−2α) ,

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2

where the sum in λ converges since α < 41 and λn ≈ n 3 . Hence, by (A.1) and (A.2), " Z # Z 1  21 Z 1  51 1 2  2 a ˜2 8 10 a ˜ |AY (s)| − aK H |AY (s)| ˜ |AY (s)| + + e P Θε ≥ K] ≤ e ε E a 2 0 0 0   −(1−2α) aK 2 + ε−(1−2α) + ε−2(1−2α) ≤ C α e− ε eCα ε ≤ C# e−η#

K2 ε

,

where η# and C# can be easily found, since K ≥ 21 .



A.2. The deterministic equation The basic ingredient of our approach is the bunch of regular paths that every weak solution has for a positive local (random) time, when the initial condition is regular. It was called regular jet in Flandoli [6]. It is based on the solutions of the following deterministic equation Z t u(t) + (Au (s) + B (u, u)) ds = x + w (t) . (A.3) 0

We say that u ∈ C ([0, ∞; Hσ ) ∩ L2loc ([0, ∞); V ) is a weak solution of (A.3) if Z t hu(t), ϕi + (hu (s) , Aϕi − hB (u (s) , ϕ) , u (s)i) ds = hx, ϕi + hw(t), ϕi 0

∞

for every ϕ ∈ D . Notice that all terms in the above definition are meaningful, included the quadratic one in u due to the estimate |hB (u, v) , zi| ≤ C |Dv|L∞ |u|L2 |z|L2 . We take w ∈ Ω∗ where Ω∗ =

\

β∈(0, 21 )

C β ([0, ∞); D(Aα )) .

α∈(0, 43 )

Consider also the auxiliary Stokes equations Z t z(t) + Az (s) ds = w (t) 0

having the unique mild solution z(t) = e−tA w (t) −

Z

t 0

Ae−(t−s)A (w (s) − w (t)) ds.

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From elementary arguments based on the analytic estimates Aα e−tA ≤ t ∈ (0, T ), we have (see for instance Flandoli [13] for details)

Cα,T tα

for

z ∈ C ([0, ∞); D(A)) .

Let us set θT = sup |Az (t)| .

(A.4)

t∈[0,T ]

Let C0 > 0 be the constant of inequality (2.1) and let C ∗ := 4C02 .

(A.5)

Lemma A.2.1. Given x ∈ D(A) and w ∈ Ω∗ , let K ≥ |Ax| and ε > 0 be such that  
1 ∗ + C ε <1 K 2 + θε2 2K 2 Then there exists a solution u ∈ C ([0, ε] ; D(A)), which is unique in the class of weak solutions, and |Au (s)| < 2K for s ∈ [0, ε].

Proof. We show only the quantitative estimate, the other statements being standard in the theory of Navier-Stokes equations. For simplicity, all computations will be made on the limit problem, although they should be made on its Galerkin approximations. The uniqueness of local solution ensures that the procedure is nevertheless correct. Set v = u − z, then dv + Av + B (u, u) = 0 dt

and, by using (2.1), d 2 2 |Av| + 2 kAvkV ≤ 2 |hAv, AB (u, u)i| ≤ 2 kAvkV A1/2 B (u, u) dt ≤ 2C0 kAvkV |Au|2 ≤ kAvk2V + C02 |Au|4 2

≤ kAvkV + C ∗ (|Av|2 + |Az|2 )2 .

Hence on [0, ε] we have that d 2 |Av| ≤ C ∗ (|Av|2 + θε2 )2 , dt 2

and so, if we set y(t) = |Av (t)| + θε2 , it follows that dy ≤ C ∗ y2 , dt

on [0, ε] .

Consequently, since y > 0 (except for the irrelevant case w ≡ 0), we have y(s) ≤

y(0) , 1 − C ∗ sy(0)

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namely, |A (u (s) − z (s))|2 + θε2 ≤ for s ∈ [0, ε]. Therefore

2

|Ax| + θε2   2 1 − C ∗ s |Ax| + θε2

2

2(|Ax| + θε2 ) 2(K 2 + θε2 )  ≤ . 2 1 − C ∗ s (K 2 + θε2 ) 1 − C ∗ s |Ax| + θε2
This result is true until 1 − C ∗ s K 2 + θε2 > 0, namely for s ∈ [0, C ∗ (K12 +θ2 ) ). The ε assumption of the lemma ensures that [0, ε] is included in this interval. Thus the last inequality is true at least on [0, ε]. Moreover, again by the assumption of the lemma, 2

|Au (s)| ≤

that implies


2(K 2 + θε2 ) < 1 − C ∗ s K 2 + θε2 2 4K 2(K 2 + θε2 ) < 4K 2 , 1 − C ∗ s (K 2 + θε2 )

and thus |Au(s)|2 < 4K 2 , for s ∈ [0, ε].



Corollary A.2.2. Assume there are K > 0 and ε > 0 such that ε≤

1 5C ∗ K 2

and

θε2 ≤

1 2 K , 4

then, for every x ∈ D(A) such that |Ax| ≤ K, we have |Au (s)| < 2K for s ∈ [0, ε]. References [1] D. W. Stroock and S. R. S. Varadhan, Multidimensional diffusion processes. vol. 233, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (Springer-Verlag, Berlin, 1979). ISBN 3-540-90353-4. [2] G. Da Prato and A. Debussche, Ergodicity for the 3D stochastic Navier-Stokes equations, J. Math. Pures Appl. (9). 82(8), 877–947, (2003). ISSN 0021-7824. [3] R. Mikulevicius and B. L. Rozovskii, Global L2 -solutions of stochastic Navier-Stokes equations, Ann. Probab. 33(1), 137–176, (2005). ISSN 0091-1798. [4] F. Flandoli and M. Romito. Markov selections for the 3D stochastic Navier-Stokes equations. http://www.arxiv.org/abs/math.PR/0602612. [5] F. Flandoli and M. Romito, Markov selections and their regularity for the threedimensional stochastic Navier-Stokes equations, C. R. Math. Acad. Sci. Paris. To appear. [6] F. Flandoli, On the method of Da Prato and Debussche for the 3D stochastic Navier Stokes equations, J. Evol. Eq. To appear. [7] A. Debussche and C. Odasso. Markov solutions for the 3d stochastic NavierStokes equations with state dependent noise. http://www.arxiv.org/abs/math.AP/ 0512361.

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[8] R. Mikulevicius and B. L. Rozovskii, Stochastic Navier-Stokes equations for turbulent flows, SIAM J. Math. Anal. 35(5), 1250–1310 (electronic), (2004). ISSN 0036-1410. [9] F. Flandoli. An introduction to 3D stochastic fluid dynamics. In SPDE in hydrodynamics: recent progress and prospects (CIME course), Lecture Notes in Mathematics. Springer. http://www.cime.unifi.it. [10] G. Da Prato and J. Zabczyk, Ergodicity for infinite-dimensional systems. vol. 229, London Mathematical Society Lecture Note Series, (Cambridge University Press, Cambridge, 1996). ISBN 0-521-57900-7. [11] A. Bressan, Hyperbolic systems of conservation laws. vol. 20, Oxford Lecture Series in Mathematics and its Applications, (Oxford University Press, Oxford, 2000). ISBN 0-19-850700-3. The one-dimensional Cauchy problem. [12] G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions. vol. 44, Encyclopedia of Mathematics and its Applications, (Cambridge University Press, Cambridge, 1992). ISBN 0-521-38529-6. [13] F. Flandoli, Stochastic differential equations in fluid dynamics, Rend. Sem. Mat. Fis. Milano. 66, 121–148, (1996). ISSN 0370-7377.

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Chapter 11 Rate of Convergence of Implicit Approximations for Stochastic Evolution Equations Istv´ an Gy¨ ongy and Annie Millet∗ School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh King’s Buildings, Edinburgh, EH9 3JZ, United Kingdom [email protected] Stochastic evolution equations in Banach spaces with unbounded nonlinear drift and diffusion operators are considered. Under some regularity condition assumed for the solution, the rate of convergence of implicit Euler approximations is estimated under strong monotonicity and Lipschitz conditions. The results are applied to a class of quasilinear stochastic PDEs of parabolic type.

Contents 1 2 3 4

Introduction . . . . . . . . . . . . . Preliminaries and the approximation Convergence results . . . . . . . . . Examples . . . . . . . . . . . . . . . 4.1 Quasilinear stochastic PDEs . 4.2 Linear stochastic PDEs . . . . References . . . . . . . . . . . . . . . . .

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1. Introduction Let V ,→ H ,→ V ∗ be a normal triple of spaces with dense and continuous embeddings, where V is a separable and reflexive Banach space, H is a Hilbert space, identified with its dual by means of the inner product in H, and V ∗ is the dual of V . Thus hv, hi = (v, h) for all v ∈ V and h ∈ H ∗ = H, where hv, v ∗ i = hv ∗ , vi denotes the duality product of v ∈ V , v ∗ ∈ V ∗ , and (h1 , h2 ) denotes the inner product of h1 , h2 ∈ H. Let W = {W (t) : t ≥ 0} be a d1 -dimensional Brownian motion carried ∗ Laboratoire

de Probabilit´es et Mod`eles Al´eatoires (CNRS UMR 7599), Universit´es Paris 6Paris 7, Boˆıte Courrier 188, 4 place Jussieu, 75252 Paris Cedex 05, [email protected], and Centre ´ d’Economie de la Sorbonne (CNRS UMR 8174), Equipe SAMOS-MATISSE, Universit´e Paris 1 Panth´eon Sorbonne, 90 Rue de Tolbiac, 75634 Paris Cedex 13, [email protected] 281

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by a stochastic basis (Ω, F, (Ft )t≥0 , P ). Consider the stochastic evolution equation Z t d1 Z t X u(t) = u0 + A(s, u(s)) ds + Bk (s, u(s)) dW k (s) , (1.1) 0

k=1

0

where u0 is a V -valued F0 -measurable random variable, A and B are (non-linear) adapted operators defined on [0, ∞[×V ×Ω with values in V ∗ and H d1 := H ×...×H, respectively. It is well-known, see Refs. [1–3], that this equation admits a unique solution if the following conditions are met: There exist constants λ > 0, K ≥ 0 and an (Ft )-adapted non-negative locally integrable stochastic process f = {ft : t ≥ 0} such that (i) (Monotonicity) There exists a constant K such that 2hu − v, A(t, u) − A(t, v)i + (ii) (Coercivity) 2hv, A(t, v)i +

d1 X

k=1

(iii) (Linear growth)

d1 X k=1

|Bk (t, u) − Bk (t, v)|2H ≤ K|u − v|2H ,

(1.2)

|Bk (t, v)|2H ≤ −λ|v|2V + K|v|2H + f (t),

|A(t, v)|2V ∗ ≤ K|v|2V ∗ + f (t), (iv) (Hemicontinuity) lim hw, A(t, v + λu)i = hw, A(t, v)i

λ→0

hold for all for u, v, w ∈ V , t ∈ [0, T ] and ω ∈ Ω. Under these conditions equation (1.1) has a unique solution u on [0, T ]. (See Definition 2.3 below for the definition of the solution.) Moreover, if E|u0 |2H < ∞ RT and E 0 f (t) dt < ∞, then Z T E sup |u(t)|2H + E |u(t)|2V dt < ∞. t≤T

0

In Ref. [4] it is shown that under these conditions, approximations defined by various implicit and explicit schemes converge to u. Our aim is to prove rate of convergence estimates for these approximations. To achieve this aim we require stronger assumptions: a strong monotonicity condition on A, B and a Lipschitz condition on B in v ∈ V . In the present paper we consider implicit time discretizations. Note that without space discretizations, in general, explicit time discretizations do not converge. Consider, for example, the heat equation du(t) = ∆u(t), with initial condition u(0) = u0 ∈ L2 (Rd ). Then the explicit time discretization on the grid {k/n}nk=0 gives the approximai d tion un (k/n) := (I + ∆/n)k u0 at time t = k/n. Hence clearly, if u0 ∈ / ∩∞ i=1 W2 (R ),

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then u(k/n) does not belong to the Sobolev space W2l (Rd ), with any fixed negative index l, when k is sufficiently large. The study of various space-time discretization schemes will be done in the continuation of the present paper. We require also the following time regularity from the solution u (see condition (T2)): E|u0 |2V < ∞, almost surely ut ∈ V for every t ∈ T , and there exist some constants C and ν > 0 such that E|u(t) − u(s)|2V ≤ C |t − s|2ν , for all s, t ∈ [0, T ]. Note that unlike the solutions to stochastic differential equations, the solutions to stochastic PDEs can satisfy this condition with a variety of exponents ν, different from 1/2, due to the interplay between space and time regularities of the solutions. (See Ref. [5] for space and time regularity of the solutions to stochastic parabolic PDEs of second order.) Note also that our general setting allows us to cover a large class of stochastic parabolic PDEs of order 2m for any m ≥ 1 (see Ref. [1] for the class of stochastic parabolic SPDEs of order 2m and Ref. [6] for the stochastic Cahn-Hilliard equation). In the case of time independent operators A and B we obtain the rate of convergence for the implicit approximation uτ corresponding to the mesh size τ = T /m of the partition of [0, T ] X E max |u(iτ ) − uτ (iτ )|2H + E |u(iτ ) − uτ (iτ )|2V τ ≤ Cτ ν , i≤m

i≤m

where C is a constant independent of τ . If in addition to the above assumptions A is also Lipschitz continuous in v ∈ V then the order of convergence is doubled, X E max |u(iτ ) − uτ (iτ )|2H + E |u(iτ ) − uτ (iτ )|2V τ ≤ Cτ 2ν . i≤m

i≤m

In the case of time dependent A and B it is natural to assume that they are H¨ older continuous in t in order to control the error due to their discretization in time. However, it is possible to control this discretization error when the operator A is not even continuous in t, if we discretize it by taking the average of A(s) over the intervals [ti , ti+1 ]. This explains the discretization of A(t) and condition (T1) below. If both operators A and B are H¨ older continuous in time then we use also τ τ the obvious discretization: Ati = A(ti+1 , .) and Bk,t = B(ti , .). i As examples we present a class of quasi-linear stochastic partial differential equations (SPDEs) of parabolic type, and show that it satisfies our assumptions. Thus we obtain rate of convergence results also for implicit approximations of linear parabolic SPDEs, in particular, for the Zakai equation of nonlinear filtering. We refer to Refs. [3, 7–9] for basic results for the stochastic PDEs of nonlinear filtering. We will extend these results to degenerate parabolic SPDEs, and to space-time explicit and implicit schemes for stochastic evolution equations in the continuation of this paper.

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In Section 2 we give a precise description of the schemes and state the assumptions on the coefficients which ensure the convergence of these schemes to the solution u of (1.1). In Section 3 estimates for the speed of convergence of time implicit schemes are stated and proved. Finally, in the last section, we give a class of examples of quasi-linear stochastic PDEs for which all the assumptions of the main theorem, Theorem 3.4, are fulfilled. As usual, we denote by C a constant which can change from line to line. 2. Preliminaries and the approximation scheme Let (Ω, F, (Ft )t≥0 , P ) be a stochastic basis, satisfying the usual conditions, i.e., (Ft )t≥0 is an increasing right-continuous family of sub-σ-algebras of F such that F0 contains every P -null set. Let W = {W (t) : t ≥ 0} be a d1 -dimensional Wiener martingale with respect to (Ft )t≥0 , i.e., W is an (Ft )-adapted Wiener process with values in Rd1 such that W (t) − W (s) is independent of Fs for all 0 ≤ s ≤ t. Let T be a given positive number. Consider the stochastic evolution equation (1.1) for t ∈ [0, T ] in a triplet of spaces V ,→ H ≡ H ∗ ,→ V ∗ ,

satisfying the following conditions: V is a separable and reflexive Banach space over the real numbers, embedded continuously and densely into a Hilbert space H, which is identified with its dual H ∗ by means of the inner product (·, ·) in H, such that (v, h) = hv, hi for all v ∈ V and h ∈ H, where h·, ·i denotes the duality product between V and V ∗ , the dual of V . Such triplet of spaces is called a normal triplet. Let us state now our assumptions on the initial value u0 and the operators A, B in the equation. Let A : [0, T ] × V × Ω → V ∗ ,

B : [0, T ] × V × Ω → H d1

be such that for every v, w ∈ V and 1 ≤ k ≤ d1 , hA(s, v), wi and (Bk (s, v), w) are adapted processes and the following conditions hold: (C1) The pair (A, B) satisfies the strong monotonicity condition, i.e., there exist constants λ > 0 and L > 0 such almost surely 2 hu − v, A(t, u) − A(t, v)i +

d1 X k=1

|Bk (t, u) − Bk (t, v)|2H

+λ |u − v|2V ≤ L |u − v|2H

(2.1)

for all t ∈]0, T ], u and v in V . (C2) (Lipschitz condition on B) There exists a constant L1 such that almost surely d1 X k=1

|Bk (t, u) − Bk (t, v)|2H ≤ L1 |u − v|2V

(2.2)

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for all t ∈ [0, T ], u and v in V . (C3) (Lipschitz condition on A) There exists a constant L2 such that almost surely |A(t, u) − A(t, v)|2V ∗ ≤ L2 |u − v|2V

(2.3)

for all t ∈ [0, T ], u and v in V . (C4) u0 : Ω → V is F0 -measurable and E|u0 |2V < ∞. There exist non-negative random variables K1 and K2 such that EKi < ∞, and d1 X

k=1

|Bk (t, 0)|2H ≤ K1 ,

(2.4)

|A(t, 0)|2V ∗ ≤ K2

(2.5)

for all t ∈ [0, T ] and ω ∈ Ω. Remark 2.1. If λ = 0 in (2.1) then one says that (A, B) satisfies the monotonicity condition. Notice that this condition together with the Lipschitz condition (2.3) on A implies the Lipschitz condition (2.2) on B. Remark 2.2. (1) Clearly, (2.3)–(2.5) and (2.2)–(2.4) imply that A and B satisfy the growth condition d1 X j=1

|Bk (t, v)|2H ≤ 2L1 |v|2V + 2K1 ,

(2.6)

and |A(t, v)|2V ∗ ≤ 2L2 |v|2V + 2K2

(2.7)

respectively, for all t ∈ [0, T ], ω ∈ Ω and v ∈ V . (2) Condition (2.3) obviously implies that the operator A is hemicontinuous: lim hA(t, u + εv), wi = hA(t, u), wi

ε→0

(2.8)

for all t ∈ [0, T ] and u, v, w ∈ V . (3) The strong monotonicity condition (C1), (C2), (2.4) and (2.5) yield that the pair (A, B) satisfies the following coercivity condition: there exists a non-negative random variable K3 such EK3 < ∞ and almost surely 2 hv, A(t, v)i +

d1 X k=1

|Bk (t, v)|2H +

λ 2

|v|2V ≤ L|v|2H + K3

(2.9)

for all t ∈]0, T ], ω ∈ Ω and v ∈ V . Proof.

We show only (3). By the strong monotonicity condition (C1)

2 hv, A(t, v)i +

d1 X k=1

|Bk (t, v)|2H + λ |v|2V ≤ L|v|2H + R1 (t) + R2 (t)

(2.10)

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with R1 (t) = 2 hv, A(t, 0)i, R2 (t) =

d1 X k=1

|Bk (t, 0)|2H + 2

Using (C2) and (2.5), we have

d1  X k=1

 Bk (t, v) − Bk (t, 0) , Bk (t, 0) .

2 |R1 (t)| ≤ λ4 |v|2V + 4K λ ,   21 d1 X |R2 (t)| ≤ 2  |Bk (t, v) − Bk (t, 0)|2H 

d1 X

j=1

k=1

≤ λ4 |v|2V + CK1 .

|Bk (t, 0)|2H

! 21

+ K1

Thus, (2.10) concludes the proof of (2.9).



Definition 2.3. An H-valued adapted continuous process u = {u(t) : t ∈ [0, T ]} is a solution to equation (1.1) on [0, T ] if almost surely : u(t) ∈ V for almost every t ∈ [0, T ], Z T |u(t)|2V dt < ∞ , (2.11) 0

and

(u(t), v) = (u0 , v) +

Z

t 0

hA(s, u(s)), vi ds +

d1 Z X

k=1

t

(Bk (s, u(s)), v) dW k (s)

(2.12)

0

holds for all t ∈ [0, T ] and v ∈ V . We say that the solution to (1.1) on [0, T ] is unique if for any solutions u and v to (1.1) on [0, T ] we have   P sup |u(t) − v(t)|H > 0 = 0. t∈[0,T ]

The following theorem is well-known (see Refs. [1–3]). Theorem 2.4. Let A and B satisfy the monotonicity, coercivity, linear growth and hemicontinuity conditions (i)-(iv) formulated in the Introduction. Then for every H-valued F0 -measurable random variable u0 ,Requation (1.1) has a unique solution T u on [0, T ]. Moreover, if E|u0 |2H < ∞ and E 0 f (t) dt < ∞, then Z T
2 |u(t)|2V dt < ∞ (2.13) E sup |u(t)|H + E t∈[0,T ]

0

holds.

Hence by the previous remarks we have the following corollary. Corollary 2.5. Assume that conditions (C1), (C2) hold. Then for every H-valued random variable u0 equation (1.1) has a unique solution u, and if E|u0 |2H < ∞, then (2.13) holds.

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Approximation scheme. For a fixed integer m ≥ 1 and τ := T /m we define the approximation uτ for the solution u by an implicit time discretization of equation (1.1) as follows: uτ (t0 ) = u0 , uτ (ti+1 ) = uτ (ti ) + τ Aτti uτ (ti+1 ) +

d1 X

τ Bk,t uτ (ti ) i

k=1







W k (ti+1 ) − W k (ti )

1 τ

Z




for 0 ≤ i < m, (2.14)

where ti := iτ and Aτti (v) =

τ Bk,0 (v) = 0,

ti+1

A(s, v) ds ,

(2.15)

ti

τ Bk,t (v) = i+1

1 τ

Z

ti+1

Bk (s, v) ds

(2.16)

ti

for i = 0, 1, 2, ..., m. A random vector uτ := {uτ (ti ) : i = 0, 1, 2, ..., m} is called a solution to scheme (2.14) if uτ (ti ) is a V -valued Fti -measurable random variable such that E|uτ (ti )|2V < ∞ and (2.14) holds for every i = 0, · · · , m − 1. We use the notation κ1 (t) := iτ for t ∈ [iτ, (i + 1)τ [, and κ2 (t) := (i + 1)τ for t ∈]iτ, (i + 1)τ ] (2.17) for integers i ≥ 0, and set At (v) = Ati (v),

Bk,t (v) = Bti (v)

for t ∈ [ti , ti+1 [, i = 0, 1, 2, ...m − 1 and v ∈ V . Another possible choice is Aτti (u) = A(ti+1 , u)

τ and Bk,t (u) = Bk (ti , u) i

for i = 0, 1, · · · , m − 1.

(2.18)

The following theorem establishes the existence and uniqueness of uτ for large enough m, and provides estimates in V and in H. We remark that in practice (2.14) should also be solved numerically. This is possible for example by Galerkin’s approximations and by finite elements methods. In the continuation of this paper we consider explicit and implicit time discretization schemes together with simultaneous ‘space discretizations’, and we estimate the error of the corresponding approximations for (1.1). Theorem 2.6. Assume that A and B satisfy the monotonicity, coercivity, linear growth and hemicontinuity conditions (i)–(iv). Assume also that (C4) holds. Let Aτ and B τ be defined either by (2.15) and (2.16), or by (2.18). Then there exist an

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integer m0 and a constant C, such that for m ≥ m0 equation (2.14) has a unique solution {uτ (ti ) : i = 0, 1, ..., m}, and m X 2 τ u (iτ ) 2 τ ≤ C . E max uτ (iτ ) H + E V 0≤i≤m

(2.19)

i=1

Proof. For the sake of simplicity, we only give the proof in the case Aτ and B τ are defined by (2.15) and (2.16). This theorem with estimate m X 2 τ u (iτ ) 2 τ ≤ C max E uτ (iτ ) H + E V

0≤i≤m

(2.20)

i=1

in place of (2.19) is proved in Ref. [4] for a slightly different implicit scheme. For the above implicit scheme the same proof can be repeated without essential changes. For the convenience of the reader we recall from Ref. [4] that the existence and uniqueness of the solution {uτ (ti ) : i = 0, 1, 2, ..., m} to (2.14)–(2.16) is based on the following proposition (Proposition 3.4 from Ref. [4]): Let D : V → V ∗ be a mapping such that (a) D is monotone, i.e., for every x, y ∈ V , hD(x) − D(y), x − yi ≥ 0; (b) D is hemicontinuous, i.e., lim hD(x + εy), zi = hD(x), zi for every x, y, z ∈ V ; ε→0

(c) there exist positive constants K, C1 and C2 , such that |D(x)|V ∗ ≤ K (1 + |x|V ),

hD(x), xi ≥ C1 |x|2V − C2 ,

∀x ∈ V.

Then for every y ∈ V ∗ , there exists x ∈ V such that D(x) = y and |x|2V ≤

C1 + 2 C 2 1 + 2 |y|2V ∗ . C1 C1

If there exists a positive constant C3 such that hD(x1 ) − D(x2 ), x1 − x2 i ≥ C3 |x1 − x2 |2V ∗ ,

∀x1 , x2 ∈ V ,

(2.21)

then for any y ∈ V ∗ , the equation D(x) = y has a unique solution x ∈ V . Note that for each i = 1, 2, ...m−1 equation (2.14) for x := uτ (ti+1 ) can be rewritten as Dx = y with D := I −

τ Aτti ,

τ

y := u (ti ) +

d1 X

τ Bk,t uτ (ti ) i

k=1




W k (ti+1 ) − W k (ti )




where I denotes the identity on V . It is easy to verify that due to conditions (i)–(iv) and (C4) the operator D satisfies the conditions (a), (b) and (c) for sufficiently large m. Thus a solution {uτ (ti ) : i = 0, 1, ..., m} can be obtained by recursion on i for all m greater than some m0 . To show the uniqueness we need only verify (2.21). By (2.15) and by the monotonicity condition (i) we have Z ti+1 hD(x1 ) − D(x2 ), x1 − x2 i = |x1 − x2 |2H − hA(s, x1 ) − A(s, x2 ), x1 − x2 i ds ti

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≥ |x1 − x2 |2H − Kτ |x1 − x2 |2H = (1 − Kτ )|x1 − x2 |2H , where the constant K is from (1.2). Hence it is clear that (2.21) holds if m is sufficiently large. Now we show (2.20). From the definition of uτ (ti+1 ) we have

|uτ (tj )|2H = |u0 |2H + I(tj ) + J (tj ) + K(tj ) −

j X i=1

|Aτti (uτ (iτ ))|2H τ

(2.22)

for tj = jτ , j = 0, 1, 2, ...m, where Z tj I(tj ) := 2 huτ (κ2 (s)), A(s, uτ (κ2 (s)))i ds, 0 X X τ J (tj ) := | Bk,t (uτ (iτ ))(W k (ti+1 ) − W k (ti ))|2H , i 1≤i<j

K(tj ) := 2

XZ k

k

tj

0


τ uτ (κ1 (s)), Bk,s (uτ (κ1 (s))) dW k (s),

and κ1 , κ2 are piece-wise constant functions defined by (2.17). By Itˆ o’s formula for every k, l = 1, 2, ..., d1 , (W k (ti+1 ) − W k (ti ))(W l (ti+1 ) − W l (ti )) = δkl (ti+1 − ti ) + M kl (ti+1 ) − M kl (ti ), where δkl = 1 for k = l and 0 otherwise, and kl

M (t) :=

Z

t k

0

k




l

W (s) − W (κ1 (s) ) dW (s) +

Z

t 0


W l (s) − W l (κ1 (s)) dW k (s).

Thus we get J (tj ) = J1 (tj ) + J2 (tj ), with J1 (tj ) := J2 (tj ) :=

X X

1≤i<j

Z

tj

0

k

X k,l

τ |Bk,t (uτ (ti ))|2H τ , i

τ τ (Bk,s (uτ (κ1 (s))), Bl,s (uτ (κ1 (s)))) dM kl (s).

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By the Davis inequality we have

E max |J2 (tj )| ≤ 3 j≤m

X

(Z

E

k,l

X

≤ C1

k,l

X

≤ C1

k,l

× ≤ d 1 C1

E

h

τ τ |Bk,s (uτ (κ1 (s)))|2H |Bl,s (uτ (κ1 (s)))|2H

0

(Z

T 0

τ |Bk,s (uτ (κ1 (s)))|4H |W l (s)

τ √ E max Bk,t (uτ (tj )) H τ j

τ X k

+ C1 τ −1

kl

dhM i(s)

2 − W (κ1 (s)) ds l

)1/2

)1/2

j≤m

n1 Z



T

T

0

2 o1/2 i τ |Bk,s (uτ (κ1 (s)))|2H W l (s) − W l (κ1 (s)) ds

τ 2 τ E max Bk,t (uτ (tj )) H j j≤m

X

E

k,l

≤ C2 1 + E

X

j≤m

Z

T

0

2 τ |Bk,s (uτ (κ1 (s)))|2H W l (s) − W l (κ1 (s)) ds

 |uτ (jτ )|2V τ ,

where C1 and C2 are constants, independent of τ . Here we use that by Jensen’s inequality for every k X

1≤i<j

τ |Bk,t (uτ (iτ ))|2H τ ≤ i

Z

tj 0

|Bk (s, uτ (κ2 (s))|2H ds,

and that the coercivity condition (ii) and the growth condition on (iii) imply the growth condition (2.6) on B with some constant L1 and random variable K1 satisfying EK1 < ∞. Hence by taking into account the coercivity condition we obtain   E max I(tj ) + J (tj ) j≤m Z tj h i

 X ≤ E max 2 uτ (κ2 (s)) , A(s, uτ (κ2 (s))) + |Bk (s, uτ (κ2 (s))|2H ds j≤m

0

k

+ E max |J2 (tj )| j≤m



≤ C 1 + max E|uτ (jτ )|2H + E j≤m

m X j=1

|uτ (jτ )|2V τ



(2.23)

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with a constant C independent of τ . By using the Davis inequality again we obtain (Z )1/2 T X
2 τ τ τ E max K(tj ) ≤ 6 E u (κ1 (s)), Bk,s u (κ1 (s)) ds j≤m



0

k

≤ 6 E max uτ (jτ ) ≤

≤

1 2

1 2

j≤m

H

(Z

T

0

k

E max |uτ (jτ )|2H + 18 E j≤m

X

Z



T

0

)1/2  τ  |Bk,s (uτ (κ1 (s)))|2H ds

X 2 τ Bk,s (uτ (κ1 (s))) H ds k

E max |uτ (jτ )|2H + C 1 + E j≤m

X

j≤m



|uτ (jτ )|2V τ 

(2.24)

with a constant C independent of τ . From (2.20)–(2.24) we get
E max |uτ (jτ )|2H ≤ E|u0 |2H + E max I(tj ) + J (tj ) + E max |K(tj )| j≤m j≤m j≤m X τ 2 τ 2 1 |uτ (jτ )|2V τ ) ≤ 2 E max |u (jτ )|H + C (1 + max E|u (jτ )|H + E j≤m

≤

1 2

E max |u j≤m

j≤m

τ

(jτ )|2H

j≤m

+ C (1 + L) < ∞

by virtue of (2.20), which proves the estimate (2.19).



3. Convergence results In order to obtain a speed of convergence, we require further properties from B(t, v) and from the solution u of (1.1). We assume that there exists a constant ν ∈]0, 1/2] such that: (T1) The coefficient B satisfies the following time-regularity: There exists a constant C and a random variable η ≥ 0 with finite first moment, such that almost surely d1 X

k=1

|Bk (t, v) − Bk (s, v)|2H ≤ |t − s|2ν (η + C|v|2V )

(3.1)

for all s ∈ [0, T ] and v ∈ V .

(T2) The solution u to equation (1.1) satisfies the following regularity property: almost surely u(t) ∈ V for all t ∈ [0, T ], and there exists a constant C > 0 such that E|u(t) − u(s)|2V ≤ C |t − s|2ν

(3.2)

for all s, t ∈ [0, T ]. Remark 3.1. Clearly, (3.2) implies sup E|u(t)|2V < ∞.

t∈[0,T ]

(3.3)

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Finally, in order to prove a convergence result in the H norm uniformly in time, we also have to require the following uniform estimate on the V -norm of u: (T3) There exists a random variable ξ such that Eξ 2 < ∞ and sup |u(t)|V ≤ ξ

(a.s.).

t≤T

To establish the rate of convergence of the approximations we first suppose that the coefficients A and B satisfy the Lipschitz property. Theorem 3.2. Suppose that the conditions (C1)-(C4), (T1) and (T2) hold. Let Aτ and B τ be defined by (2.15) and (2.16). Then there exist a constant C and an integer m0 ≥ 1 such that sup E|u(lτ ) − uτ (lτ )|2H + E

0≤l≤m

m X j=0

|u(jτ ) − uτ (jτ )|2V τ ≤ C τ 2ν

(3.4)

for all integers m ≥ m0 . The following proposition plays a key role in the proof. Proposition 3.3. Assume assumptions (i) through (iv) from the Introduction and let Aτ and B τ be defined by (2.15) and (2.16). Suppose, moreover condition (C4). Then

|u(tl ) − uτ (tl )|2H = 2 +



ti

k=1

0

tl 0

d1 l−1 Z ti+1 X X  i=0

+2

−

Z

d 1 Z tl X

l−1 Z ti+1 X i=0



ti

k=1



 u(κ2 (s)) − uτ (κ2 (s)), A(s, u(s)) − A(s, uτ (κ2 (s))) ds

2 τ Bk (s, u(s)) − Bk,s (uτ (ti )) dW k (s) 

H


τ Bk (s, u(s)) − Bk,s (uτ (ti )) , u(κ1 (s)) − uτ (κ1 (s)) dW k (s) 

 2 A(s, u(s)) − A(s, uτ (ti+1 )) ds

holds for every l = 1, 2, ..., m.

H

(3.5)

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Using (2.14) we have for any i = 0, · · · , m − 1

Proof.

|u(ti+1 ) − uτ (ti+1 )|2H − |u(ti ) − uτ (ti )|2H = Z ti+1

 2 u(ti+1 ) − uτ (ti+1 ), A(s, u(s)) − A(s, uτ (ti+1 )) ds ti

d1 Z X

+2

k=1 ti+1

Z −

=2

Z

ti

+



ti+1

ti

ti



d1 Z X

ti+1

ti



2  τ Bk (s, u(s)) − Bk,s (uτ (ti )) dW k (s)

H

 u(ti+1 ) − uτ (ti+1 ), A(s, u(s)) − A(s, uτ (ti+1 )) ds

d Z 2 1 X ti+1   τ τ k + Bk (s, u(s)) − Bk,s (u (ti )) dW (s) ti k=1

d1 Z ti+1 X

+2

ti

k=1

Z −



 A(s, u(s)) − A(s, uτ (ti+1 )) ds

k=1 ti+1

  τ Bk (s, u(s)) − Bk,s (uτ (ti )) dW k (s) , u(ti+1 ) − uτ (ti+1 )

ti+1

ti



H

  τ Bk (s, u(s)) − Bk,s (uτ (ti )) dW k (s) , u(ti ) − uτ (ti )

2 A(s, u(s)) − A(s, u (ti+1 )) ds τ





H

Summing up for i = 1, · · · , l − 1, we obtain (3.5).



Proof of Theorem 3.2. Taking expectations in both sided of (3.5) and using the strong monotonicity condition (C1), we deduce that for l = 1, · · · , m, E|u(tl ) − uτ (tl )|2H

≤E

Z

tl 0

+

 2 u(κ2 (s)) − uτ (κ2 (s)), A(s, u(κ2 (s))) − A(s, uτ (κ2 (s))) ds

d1 X

E

k=1

≤ −λ E

Z

+ Lτ

0

tl

Z

tl−1

0

|Bk (s, u(κ2 (s))) − Bk (s, uτ (κ2 (s)))|2H ds +

3 X

Rk

k=1

|u(κ2 (s)) − uτ (κ2 (s))|2V ds + Lτ E|u(tl ) − uτ (tl )|2H

l−1 X i=1

E|u(ti ) − uτ (ti )|2H ds +

3 X k=1

Rk ,

(3.6)

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where

Z

R1 =E R2 =

tl 0

d1 X

E

k=1

R3 =

 2 u(κ2 (s)) − uτ (κ2 (s)), A(s, u(s)) − A(s, u(κ2 (s))) ds ,

Z

d1 X l−1 X

τ

|Bk (s, u(s))|2H ds ,

0

hZ

E

k=1 i=1

ti

Z

−

Z 2 1 ti Bk (t, uτ (ti )) dt ds Bk (s, u(s)) − τ ti−1 H i |Bk (t, u(ti )) − Bk (t, uτ (ti ))|2H dt .

ti+1

ti ti−1

The Lipschitz property of A imposed in (2.3), (3.2) and Schwarz’s inequality imply Z tl |u(κ2 (s)) − uτ (κ2 (s))|V |u(s) − u(κ2 (s))|V ds , |R1 | ≤ L2 E 0

 21  Z tl  21  Z tl |u(s) − u(κ2 (s))|2V ds |u(κ2 (s)) − uτ (κ2 (s))|2V ds E ≤ L2 E

λ ≤ E 3

Z

0

0

tl

0

|u(κ2 (s)) − uτ (κ2 (s))|2V ds + Cτ 2ν .

(3.7)

A similar computation based on (2.2) yields |R3 | ≤

d1 X l−1 X

E

k=1 i=1

λ ≤ E 3

Z

Z

ti

dt ti−1

1 τ

Z

ti+1 ti

 ds |Bk (s, u(s)) − Bk (t, uτ (ti ))|2H

− |Bk (t, u(ti ))) − Bk (t, uτ (ti )))|2H

tl−1



|u(κ2 (t)) − uτ (κ2 (t))|2V dt + C R30

0

where R30 =

d1 X

E

k=1

1 τ

Z

tl

ds t1

Z

κ1 (s) κ1 (s)−τ

dt |Bk (s, u(s)) − Bk (t, u(κ2 (t)))|2H .

Hence, using (2.2), (3.1) and (3.2) we have R30

≤

d1 X k=1

1 E τ

Z

tl

ds t1

Z

κ1 (s) κ1 (s)−τ

h dt |Bk (s, u(s)) − Bk (t, u(s))|2H

+ |Bk (t, u(s)) − Bk (t, u(t))|2H + |Bk (t, u(t)) − Bk (t, u(κ2 (t)))|2H Z tl Z Z κ2 (t)+τ h 1 tl−1 ≤E τ 2ν |u(s)|2V ds + CE dt ds |u(s) − u(t)|2V τ t1 0 κ2 (t) i 2ν 2 + |u(t) − u(κ2 (t)|V ≤ C τ .

i

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Hence |R3 | ≤ C τ 2ν +

λ E 3

Z

tl 0

|u(κ2 (s)) − uτ (κ2 (s))|2V ds .

(3.8)

Furthermore (2.6) and (3.3) imply |R2 | ≤ Cτ

(3.9)

with a constant C independent of τ . By inequalities (3.6)–(3.9), for sufficiently large m, Z tl λ |u(κ2 (s)) − uτ (κ2 (s))|2V ds E|u(tl ) − uτ (tl )|2H + E 3 0 ≤

l−1 X i=1

L τ E|u(ti ) − uτ (ti )|2H + Cτ 2ν .

(3.10)

Pm Since supm i=1 L τ < +∞, a discrete version of Gronwall’s lemma yields that there exists C > 0 such that for m large enough sup E|u(tl ) − uτ (tl )|2H ≤ Cτ 2ν .

0≤l≤m

This in turn with (3.2) implies E

Z

T 0

|u(s) − uτ (κ2 (s))|2V ds ≤ Cτ 2ν ,

which completes the proof of the theorem.  Assume now that the solution u of equation (1.1) satisfies also the condition (T3) Then we can improve the estimate (3.4) in the previous theorem. Theorem 3.4. Let (C1)-(C4) and (T1)–(T3) hold, and let Aτ and B τ be defined by (2.15) and (2.16). Then for all sufficiently large m E max |u(jτ ) − u 0≤j≤m

τ

(jτ )|2H

+E

m X j=0

|u(jτ ) − uτ (jτ )|2V τ ≤ C τ 2ν

holds, where C is a constant independent of τ . Proof.

For k = 1, · · · , d1 , set τ Fk (t) = Bk (t, u(t)) − Bk,t (uτ (κ1 (t)))

m(t) =

d1 Z X

k=1

t 0

Fk (s) dW k (s)

and

G(s) = m(s) − m(κ1 (s)).

(3.11)

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Then by Itˆ o’s formula |m(ti+1 ) − m(ti )|2H = 2

Z

+

ti+1 ti

X

d1 Z X k=1

(G(s) , Fk (s)) dW k (s)

k

ti+1

ti

|Fk (s)|2H ds

for i = 0, ..., m − 1. Hence by using (3.5) we deduce that for l = 1, · · · , m |u(tl ) − uτ (tl )|2H ≤ I1 (tl ) + I2 (tl ) + 2M1 (tl ) + 2M2 (tl )

(3.12)

with I1 (t) := 2

Z

0

I2 (t) :=

t

hu(κ2 (s)) − uτ (κ2 (s)) , A(s, u(s)) − A(s, uτ (κ2 (s)))i ds,

d1 Z X k=1

M1 (t) :=

d1 Z X k=1

M2 (t) :=

d1 Z X k=1

t τ |Bk (s, u(s)) − Bk,s (uτ (κ1 (s)))|2H ds,

0 t 0 t 0


G(s) , Fk (s) dW k (s),


Fk (s) , u(κ1 (s)) − uτ (κ1 (s)) dW k (s).

By (C3) sup |I1 (tl )| ≤

0≤l≤m

Z

T 0

|u(κ2 (s)) − uτ (κ2 (s))|2V ds + L2

≤ (1 + 2L2 )

m X i=1

Z

T 0

|u(ti ) − uτ (ti )|2V τ + 2L2

Z

|u(s) − uτ (κ2 (s))|2V ds T

0

|u(s) − u(κ2 (s))|2V ds.

Hence by Theorem 3.2 and by condition (T2) E sup |I1 (tl )| ≤ Cτ 2ν ,

(3.13)

0≤l≤m

where C is a constant independent of τ . Using Jensen’s inequality, (2.6) and condition (T1) we have for s ≤ τ X k

|Fk (s)|2H =

X k

|Bk (s, u(s))|2H ≤ 2 L1 |u(s)|2H + 2 K1 ,

(3.14)

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while for s ∈ [ti , ti+1 ], 1 ≤ i ≤ m, one has for some constant C independent of τ Z X 1 X ti |Fk (s)|2H ≤ |Bk (s, u(s)) − Bk (r, uτ (ti ))|2H dr τ t i−1 k k Z 1 X ti h ≤3 |Bk (s, u(s)) − Bk (r, u(s))|2H τ t i−1 k i + |Bk (r, u(s)) − Bk (r, u(ti ))|2H + |Bk (r, u(ti )) − Bk (uτ (ti ))|2H dr i h
≤ C τ 2ν η + |u(s)|2V + |u(s) − u(ti )|2V + |u(ti ) − uτ (ti )|2V . (3.15) Thus, (3.14) and (3.15) yield sup |I2 (tl )| ≤ C

0≤l≤m

Z

τ 0

+C

|u(s)|2V Z

ds + Cτ + C τ

T 0

|u(s) −

2ν

u(κ2 (s))|2V

Z

T


η + C |u(s)|2V ds

0

ds + C

m X i=1

|u(ti ) − uτ (ti )|2V τ.

Hence by Theorem 3.2 and by condition (T2) E sup |I2 (tl )| ≤ Cτ 2ν ,

(3.16)

0≤l≤m

P where C is a constant independent of τ . Since sup0≤s≤T k |Fk (s)|2H need not be measurable, we denote by Γ the set of random variables ζ satisfying X sup |Fk (s)|2H ≤ ζ (a.s.). 0≤s≤T

k

For ζ ∈ Γ, the Davis inequality, and the simple inequality ab ≤ τ2 a2 + E sup |M1 (tl )| ≤ 3 E 1≤l≤m

Z



0

d1 T X

≤ 3 E ζ 1/2

k=1


| Fk (s) , G(s) |2 ds

"Z

T 0

|G(s)|2H ds

3 3 ≤ τ inf Eζ + E ζ∈Γ 2 2τ

Z

T 0

# 12 

1 2 2τ b

yield

! 12



|G(s)|2H ds.

(3.17)

By (2.6) and (3.15) we deduce X
sup |Fk (s)|2H ≤ C τ 2ν sup |u(s)|2V + η + 1 + C max |u(ti ) − uτ (ti )|2V 0≤s≤T

k

0≤s≤T

1≤i≤m

  ≤ C 1 + ξ + max |u(ti ) − uτ (ti )|2V , 1≤i≤m

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where ξ is the random variable from condition (T3) and C is a constant, independent of τ . Hence Theorem 3.2 yield τ inf Eζ ≤ τ C (Eη + Eξ) + C τ ζ∈Γ

m X i=1

E|u(ti ) − uτ (ti )|2V ≤ C1 τ 2ν ,

(3.18)

where C1 is a constant, independent of τ . Similarly, due to conditions (T1)-(T2) and Theorem 3.2 Z T   XZ T |u(s)|2V ds E |Fk (s)|2H ds ≤ C τ 2ν 1 + E k

0

0

+ C τ 2ν + C τ E

m X i=1

|u(ti ) − uτ (ti )|2V ≤ C τ 2ν

(3.19)

with a constant C, independent of τ . Furthermore, the isometry of stochastic integrals and (3.22) yield 2 Z T Z T Z t X 1 1 2 k E |G(t)|H dt ≤ E Fk (s) dW (s) dt τ τ 0 0 κ1 (t) k H Z T Z t X 1 2 ≤ E dt |Fk (s)|H ds ≤ C τ 2ν . (3.20) τ 0 κ1 (t) k

Thus from (3.17) by (3.18) and (3.20) we have E sup |M1 (tl )| ≤ Cτ 2ν

(3.21)

1≤l≤m

Finally, the Davis inequality implies E sup |M2 (tl )|H ≤ 3 E 1≤l≤m

Z

T 0

X k


| Fk (s) , u(κ1 (s)) − u (κ1 (s)) |2 ds


1 ≤ E sup |u(κ1 (s)) − uτ (κ1 (s)) |2H + 18 E 4 1≤l≤m

τ

Z

tj

0

! 21

|Fk (s)|2H ds.

(3.22)

Thus, from (3.12) by inequalities (3.13), (3.16), (3.21) and (3.22) we obtain 1 E sup |u(tl ) − uτ (tl )|2H ≤ C τ 2ν , 2 1≤l≤m with a constant C, independent of τ , which with (3.4) completes the proof of the theorem.  We now prove that if the coefficient A does not satisfy the Lipschitz property (C3) but only the coercivity and growth conditions (2.7)-(2.9), then the order of convergence is divided by two. Theorem 3.5. Let A and B satisfy the conditions (C1), (C2) and (C4). Suppose that conditions (T1) and (T2) hold, and let Aτ and B τ be defined by (2.15) and

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(2.16). Then there exists a constant C, independent of τ , such that for all sufficiently large m m X sup E|u(jτ ) − uτ (jτ )|2H + E |u(jτ ) − uτ (jτ )|2V τ ≤ C τ ν . (3.23) 0≤j≤m

j=1

Proof. Using (3.5), taking expectations and using (C1) with u(s) and uτ (κ2 (s)), we obtain for every l = 1 · · · , m Z tl E|u(tl ) − uτ (tl )|2H ≤ −λE |u(s) − uτ (κ2 (s)|2V ds +E where ¯1 = R

r X

Z

2E

j=1

¯2 = R

d1 X

E

k=1

¯3 = R

Z

tl 0

τ 0

d1 X l−1 X

E

k=1 i=1



Z

0

tl

0

K1 |u(s) − uτ (κ2 (s)|2H ds +

3 X

¯i , R

(3.24)

k=1

 u(κ2 (s)) − u(s) , A(s, u(s)) − A(s, uτ (κ2 (s))) ds ,

|Bk (s, u(s))|2H ds , 1 τ

Z

ti+1

ds ti

Z

ti ti−1

h dt |Bk (s, u(s)) − Bk (t, uτ (ti )))|2H

i − |Bk (t, u(t)) − Bk (t, uτ (ti ))|2H .

Using (2.7), (3.2), (3.3) and Schwarz’s inequality, we deduce Z tl   ¯ |u(κ2 (s)) − u(s)|V |u(s)|V + |uτ (κ2 (s))|V + K2 ds |R 1 | ≤ C E 0

 Z tl  21  Z tl   12 2 2 2 |u(s) − u(κ2 (s))|V ds ≤C E |u(s)|V + |u(κ2 (s))|V ds E 0

 Z +C E

tl

0

|u(s) − u(κ2 (s))|2V ds

≤ Cτ ν .

 21

0

(3.25)

Furthermore, Schwarz’s inequality, (C2) and computations similar to that proving (3.8) yield for any δ > 0 small enough Z tl−1 d1 X ¯3 | ≤ δ |R E |Bk (t, u(t)) − Bk (t, uτ (κ2 (t)))|2H dt k=1

+C ≤

λ E 2

0

d1 X l−1 X

k=1 i=1 tl−1

Z

0

1 E τ

Z

ti+1

ds ti

Z

ti

ti−1

dt |Bk (s, u(s))) − Bk (t, u(t))|2H

|u(s) − uτ (κ2 (s))|2V ds + Cτ 2ν .

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This inequality and (3.25) imply that Z tl λ τ 2 E|u(tl ) − u (tl )|H + E |u(s) − uτ (κ2 (s))|2V ds 2 0 Z tl ≤ K1 E|u(s) − uτ (κ2 (s))|2H ds + C τ ν . 0

Hence for any t ∈ [0, T ], E|u(t) − uτ (κ2 (t)|2H ≤ 2 E|u(κ2 (t)) − uτ (κ2 (t))|2H + 2 E|u(t) − u(κ2 (t))|2H Z κ2 (t) ≤ 2 K1 E|u(s) − uτ (κ2 (s))|2H ds + C τ ν + 2 E|u(t) − u(κ2 (t))|2H 0 Z t ≤ 2 K1 E|u(s) − uτ (κ2 (s))|2H ds + C τ ν + 2 E|u(t) − u(κ2 (t))|2H 0 h
i + C τ sup E |u(s)|2H + |uτ (κ2 (s))|2H . s

Itˆ o’s formula and (2.9) imply that for any t ∈ [0, T ], E|u(t) −

u(κ2 (t))|2H

≤ K1 E

Z

=E

κ2 (t) t

Z

κ2 (t) t

h

2hA(s, u(s)) , u(s)i +

d1 X

k=1

i |Bk (s, u(s))|2H ds

|u(s)|2H ds ≤ K1 τ sup E|u(s)|2H . 0≤s≤T

Hence (2.13) and (2.19) imply that E|u(t) − u

τ

(κ2 (t))|2H

≤ 2 K1

Z

t 0

E|u(s) − uτ (κ2 (s))|2H ds + C τ ν

and Gronwall’s lemma yields sup E|u(t) − uτ (κ2 (t))|2H ≤ Cτ ν .

(3.26)

0≤t≤T

Therefore, E

Z

T 0

|u(t) − uτ (κ2 (t)|2V dt < Cτ ν

(3.27)

follows by (3.24). Finally taking into account that by (T2) there exists a constant C such that E|u(t) − u(κ2 (t)|2V ≤ Cτ 2ν from (3.26) and (3.27) we obtain (3.23).

for all t ∈ [0, T ], 

Using the above result one can easily obtain the following theorem in the same way as Theorem 3.2 is obtained from Theorem 3.4.

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Theorem 3.6. Let A and B satisfy the conditions (C1), (C2) and (C4). Suppose that conditions (T1)–(T3) hold and let Aτ and B τ be defined by (2.15) and (2.16). Then there exists a constant C such that for m large enough, E max |u(jτ ) − uτ (jτ )|2H + E 0≤j≤m

m X j=0

|u(jτ ) − uτ (jτ )|2V τ ≤ C τ ν .

(3.28)

Remark 3.7. By analyzing their proof, it is not difficult to see that Theorems 3.2, 3.4, 3.5 and 3.6 remain true, if instead of (2.15) and (2.16), one uses (2.18) in the definition of the implicit scheme, and requires furthermore that A satisfies the following time-regularity similar to (T1): there exist a constant C ≥ 0 and a random variable η ≥ 0 with finite expectation, such that almost surely
|A(t, u) − A(s, u)|2V ∗ ≤ |t − s|2ν η + Ckuk2V for 0 ≤ s ≤ t ≤ T and u ∈ V .

4. Examples 4.1. Quasilinear stochastic PDEs Let us consider the stochastic partial differential equation   du(t, x) = Lu(t, x) + F (t, x, ∇u(t, x), u(t, x) dt

d1 X   + Mk u(t, x) + Gk (t, x, u(t, x)) dW k (t),

(4.1)

k=1

for t ∈ (0, T ], x ∈ Rd with initial condition u(0, x) = u0 (x),

x ∈ Rd ,

(4.2)

where W is a d1 -dimensional Wiener martingale with respect to the filtration (Ft )t≥0 , F and Gk are Borel functions of (ω, t, x, p, r) ∈ Ω × [0, ∞) × Rd × Rd × R and of (ω, t, x, r) ∈ Ω × [0, ∞) × Rd × R, respectively, and L, Mk are differential operators of the form X X α L(t)v(x) = Dα (aαβ (t, x)Dβ v(x)), Mk (t)v(x) = bα k (t, x)D v(x), |α|≤1,|β|≤1

|α|≤1

(4.3) d with functions aαβ and bα of (ω, t, x) ∈ Ω × [0, ∞) × R , for all multi-indices α= k P (α1 , ..., αd ), β = (β1 , ..., βd ) of length |α| = i αi ≤ 1, |β| ≤ 1. Here, and later on D α denotes D1α1 ...Ddαd for any multi-indices α = (α1 , ..., αd ) ∈ ∂ {0, 1, 2, ...}d, where Di = ∂x and Di0 is the identity operator. i We use the notation ∇p := (∂/∂p1 , ..., ∂/∂pd). For r ≥ 0 let W2r (Rd ) denote the space of Borel functions ϕ : Rd →R whose derivatives up to order r are square

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integrable functions. The norm |ϕ|r of ϕ in W2r is defined by X Z |ϕ|2r = |Dγ ϕ(x)|2 dx. Rd

|γ|≤r

In particular, W02 (Rd ) = L2 (Rd ) and |ϕ|0 := |ϕ|L2 (Rd ) . Let us use the notation P for the σ-algebra of predictable subsets of Ω × [0, ∞), and B(Rd ) for the Borel σ-algebra on Rd . We fix an integer l ≥ 0 and assume that the following conditions hold. Assumption (A1) (Stochastic parabolicity). There exists a constant λ > 0 such that ! d1 X X X αβ α β 1 a (t, x) − 2 bk bk (t, x) z α z β ≥ λ |z α |2 (4.4) k=1

|α|=1,|β|=1

|α|=1

for all ω ∈ Ω, t ∈ [0, T ], x ∈ Rd and z = (z 1 , ..., z d ) ∈ Rd , where z α := z1α1 z2α2 ...zdαd for z ∈ Rd and multi-indices α = (α1 , α2 , ..., αd ). Assumption (A2) (Smoothness of the linear term). The derivatives of aαβ and d bα k up to order l are P ⊗ B(R ) -measurable real functions such that for a constant K |Dγ aαβ (t, x)| ≤ K,

|Dγ bα k (t, x)| ≤ K,

for all |α| ≤ 1, |β| ≤ 1, k = 1, · · · , d1 , (4.5) for all ω ∈ Ω, t ∈ [0, T ], x ∈ Rd and multi-indices γ with |γ| ≤ l. Assumption (A3) (Smoothness of the initial condition). Let u0 be a W2l -valued F0 -measurable random variable such that E|u0 |2l < ∞.

(4.6)

Assumption (A4) (Smoothness of the nonlinear term). The function F and their first order partial derivatives in p and r are P ⊗ B(Rd ) ⊗ B(Rd ) ⊗ B(R)measurable functions, and gk and its first order derivatives in r are P ⊗B(Rd)⊗B(R) -measurable functions for every k = 1, .., d1 . There exists a constant K such that |∇p F (t, x, p, r)| +

∂ | ∂r F (t, x, p, r)|

+

d1 X k=1

∂ | ∂r Gk (t, x, r)| ≤ K

(4.7)

for all ω ∈ Ω, t ∈ [0, T ], x ∈ Rd , p ∈ Rd and r ∈ R. There exists a random variable ξ with finite first moment, such that |F (t, ·, 0, 0)|20 + for all ω ∈ Ω and t ∈ [0, T ].

d1 X k=1

|Gk (t, ·, 0)|20 ≤ ξ

(4.8)

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Definition 4.1. An L2 (Rd )-valued continuous (Ft )-adapted process u = {u(t) : t ∈ [0, T ]} is called a generalized solution to the Cauchy problem (4.1)-(4.2) on [0, T ] if almost surely u(t) ∈ W21 (Rd ) for almost every t, Z T |u(t)|21 dt < ∞, 0

and

n d(u(t), ϕ) =

X



o (−1)|α| aαβ (t) Dβ u(t) , Dα ϕ + F (t, ∇u(t), u(t)) , ϕ dt

|α|≤1,|β|≤1

+

d1 n X X k=1

|α|≤1



o α bα (t)D u(t) , ϕ + G (t, u(t)) , ϕ dW k (t) k k

holds on [0, T ] for every ϕ ∈ C0∞ (Rd ), where (v, ϕ) denotes the inner product of v and ϕ in L2 (Rd ). Set H = L2 (Rd ), V = W21 (Rd ) and consider the normal triplet V ,→ H ,→ V ∗ based on the inner product in L2 (Rd ), which determines the duality h , i between V and V ∗ = W2−1 (Rd ). By (4.5), (4.7) and (4.8) there exist a constant C and a random variable ξ with finite first moment, such that X
(−1)|α| aαβ (t) Dβ v , Dα ϕ ≤ C|v|1 |ϕ|1 , |α|≤1,|β|≤1

d1 X

k=1


2 α 2 2 |(bα k (t)D v , ϕ | ≤ C|v|1 |ϕ|0 ,


| F (t, ∇v, v) , ϕ |2 ≤ C|v|21 |ϕ|21 + ξ|ϕ|20 ,

d1 X

k=1


|(Gk (t, v(t)) , ϕ |2 ≤ C|v|21 |ϕ|20 + ξ|ϕ|20

for all ω, t ∈ [0, T ] and v, ϕ ∈ V . Therefore the operators A(t), Bk (t) defined by X

(−1)|α| aαβ (t) Dβ v , Dα ϕ + F (t, ∇v, v) , ϕ , hA(t, v), ϕi = |α|≤1,|β|≤1



α (Bk (t, v) , ϕ) = bα k (t)D v , ϕ + Gk (t, v) , ϕ ,

v, ϕ ∈ V

(4.9)

are mappings from V into V ∗ and H, respectively, for each k and ω, t, such that the growth conditions (2.6) and (2.7) hold. Thus we can cast the Cauchy problem (4.1)– (4.2) into the evolution equation (1.1), and it is an easy exercise to show that Assumptions (A1), (A2) with l = 0 and Assumption (A4) ensure that conditions (C1) and (C2) hold. Hence Corollary 2.5 gives the following result. Theorem 4.2. Let Assumptions (A1)-(A4) hold with l = 0. Then problem (4.1)(4.2) admits a unique generalized solution u on [0, T ]. Moreover, Z T   2 E sup |u(t)|0 + E |u(t)|21 dt < ∞. (4.10) t∈[0,T ]

0

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Next we formulate a result on the regularity of the generalized solution. We need the following assumptions. Assumption (A5) The first order derivatives of Gk in x are P ⊗ B(Rd ) ⊗ B(R)measurable functions, and there exist a constant L, a P ⊗ B(R)-measurable function K of (ω, t, x) and a random variable ξ with finite first moment, such that d1 X

k=1

|Dα Gk (t, x, r)| ≤ L|r| + K(t, x),

|K(t)|20 ≤ ξ

for all multi-indices α with |α| = 1, for all ω ∈ Ω, t ∈ [0, T ], x ∈ Rd and r ∈ R. Assumption (A6) The first order derivatives of F in x are P ⊗ B(Rd ) ⊗ B(Rd ) ⊗ B(R)-measurable functions, and there exist a constant L, a P ⊗ B(R)-measurable function K of (ω, t, x) and a random variable ξ with finite first moment, such that |∇x F (t, x, p, r)| ≤ L(|p| + |r|) + K(t, x),

|K(t)|20 ≤ ξ

for all ω, t, x, p, r. Assumption (A7) There exist P ⊗ B(R)-measurable functions gk such that Gk (t, x, r) = gk (t, x)

for all k = 1, 2, ..., d1 , t, x, r,

and the derivatives in x of gk up to order l are P ⊗ B(R)-measurable functions such that d1 X k=1

|gk (t)|2l ≤ ξ,

for all (ω, t), where ξ is a random variable with finite first moment. Theorem 4.3. Let Assume (A1)-(A4) with l = 1. Then for the generalized solution u of (4.1)-(4.2) the following statements hold: (i) Suppose (A5). Then u is a W21 (Rd )-valued continuous process and Z T   2 E sup |u(t)|1 + E |u(t)|22 dt < ∞ ; (4.11) t≤T

0

(ii) Suppose (A6) and (A7) with l = 2. Then u is a W22 (Rd )-valued continuous process and Z T   E sup |u(t)|22 + E |u(t)|23 dt < ∞ . (4.12) t≤T

Proof.

0

Define ψ(t, x) = F (t, x, ∇u(t, x), u(t, x)),

ϕk (t, x) = Gk (t, x, u(t, x))

for t ∈ [0, T ], ω ∈ Ω and x ∈ Rd , where u is the generalized solution of (4.1)-(4.2). Then due to (4.10) Z T XZ T E |ψ(t)|20 dt < ∞, E |ϕk (t)|21 dt < ∞. 0

k

0

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Therefore, the Cauchy problem   dv(t, x) = Lv(t, x) + ψ(t, x) dt +

d1 X  k=1

v(0, x) = u0 (x),

 Mk v(t, x) + ϕk (t, x) dW k (t),

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t ∈ (0, T ] , x ∈ Rd ,

x ∈ Rd

(4.13) (4.14)

has a unique generalized solution v on [0, T ]. Moreover, by Theorem 1.1 in Ref. [10], v is a W21 -valued continuous Ft -adapted process and Z T   |v(t)|22 dt < ∞. E sup |v(t)|21 + E t≤T

0

Since u is a generalized solution to (4.13)–(4.14), by virtue of the uniqueness of the generalized solution we have u = v, which proves (i). Assume now (A6) and (A7) with l = 2. Then obviously (A5) holds, and therefore due to (4.11) Z T XZ T E |ψ(t)|21 dt < ∞, E |ϕk (t)|22 dt < ∞. 0

k

0

Thus by Theorem 1.1 in Ref. [10] the generalized solution v = u of (4.13)–(4.14) is a W22 (Rd )-valued continuous process such that (4.12) holds. The proof of the theorem is complete.  Corollary 4.4. Let (A1)-(A4) hold with l = 2. Assume also (A6) and (A7). Then there exists a constant C such that for the generalized solution u of (4.1)–(4.2) we have E|u(t) − u(s)|21 ≤ C|t − s| Proof.

for all s, t ∈ [0, T ].

By the theorem on Itˆ o’s formula (see Refs. [1] or [11]) from almost surely Z t d1 Z t X     u(t) = u0 + Lu(s) + ψ(s) ds + Mk u(s) + gk (s) dW k (s) 0

k=1

0

holds, as an equality in L2 (Rd ), for all t ∈ [0, T ], where

ψ(s, ·) := F (s, ·, ∇u(s, ·), u(s, ·)). Due to (ii) from Theorem 4.3 Z t Z t 2  2 E L(r)u(r) + ψ(r) dr ≤ E |Lu(r) + ψ(r)|1 dr 1 s s Z t ≤ |t − s| E |Lu(r) + ψ(r)|21 dr s ! Z Z T

≤ C |t − s|

E

0

|u(t)|23 dt + E

T

0

|ψ(t)|21 dt

≤ C|t − s|

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for all s, t ∈ [0, T ], where C is a constant. Furthermore, by Doob’s inequality Z t 2 Z t   E|Mk u(r) + gk (r)|21 dr E Mk u(r) + gk (r) dW k (r) ≤ 4 s

s

1

h

≤ C1 |t − s| 1 + E

Z

T

0

i |u(t)|22 dt ≤ C2 |t − s|

for all s, t ∈ [0, T ], where C1 and C2 are constants. Hence Z t
2 E|u(t) − u(s)|21 ≤2E Lu(r) + ψ(r) dr 1

s

d1 Z t X 2   + 2E Mk u(r) + gk (r) dW k (r) ≤ C|t − s|, k=1

1

s

and the proof of the corollary is complete.



The implicit scheme (2.14) applied to problem (4.1)-(4.2) reads as follows. uτ (t0 ) = u0 ,


uτ (ti+1 ) = uτ (ti ) + Lτti uτ (ti+1 ) + Ftτi (uτ (ti+1 ) τ +

d1 X k=1

for 0 ≤ i < m , where


τ Mk,t uτ (ti ) + Gτk,ti (uτ (ti )) (W k (ti+1 ) − W k (ti )) , (4.15) i X

Lτti v : =

β Dα (aαβ ti (x)D v),

τ Mk,t := i

aαβ ti (x) : =

1 τ

ti+1

Ftτi (x, p, r) : = Gτk,0 (x, r)

1 τ

: = 0,

aαβ (s, x) ds,

ti

bα k,0 (x) = 0,

bα k,ti+1 (x) = Z

α bα k,ti D v,

|α|≤1

|α|≤1,|β|≤1

Z

X

ti+1

1 τ

Z

(4.16) ti+1

bk (s, x) ds,

(4.17)

ti

F (s, x, p, r) ds, ti

Gτk,ti+1 (x, r)

:=

Z

ti+1

Gk (s, x, r) ds. ti

Definition 4.5. A random vector {uτ (ti ) : i = 0, 1, 2, ..., m} is a called a generalized solution of the scheme (4.15) if uτ (t0 ) = u0 , uτ (ti ) is a W21 (Rd )-valued Fti -measurable random variable such that E|uτ (ti )|21 < ∞

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and almost surely X (uτ (ti ), ϕ) =

β τ α τ τ τ (−1)|α| (aαβ ti D u (ti ), D ϕ)τ + (Fti−1 (∇uti−1 , uti−1 ), ϕ)τ

|α|≤1,|β|≤1

+

Xh X k

|α|≤1

i
k τ k α τ bα ti−1 D uti−1 + Gk,ti−1 (uti−1 ), ϕ (W (ti ) − W (ti−1 )

for i = 1, 2, ..., m and all ϕ ∈ C0∞ (Rd ), where (·, ·) is the inner product in L2 (Rd ). From this definition it is clear that, using the operators A, Bk defined by (4.9), we can cast the scheme (4.15) into the abstract scheme (2.14). Thus by applying Theorem 2.6 we get the following theorem. Theorem 4.6. Let (A1)-(A4) hold with l = 0. Then there exists an integer m 0 such that (4.15) has a unique generalized solution {uτ (ti ) : i = 0, 1, ..., m} for every m ≥ m0 . Moreover, there exists a constant C such that E max |uτ (ti )|20 + E 0≤i≤m

for all integers m ≥ m0 .

m X i=1

|uτ (ti )|21 ≤ C

To ensure condition (T1) to hold we impose the following assumption. Assumption (H) There exists a constant C and a random variable ξ with finite first moment such that for k = 1, 2, ..., d1 α 1/2 |Dγ (bα k (t, x) − bk (s, x))| ≤ C|t − s|

|gk (s) − gk (s)|2l ≤ ξ|t − s|

for all ω ∈ Ω, x ∈ Rd and |γ| ≤ l,

for all s, t ∈ [0, T ].

Now applying Theorem 3.4 we obtain the following result.

Theorem 4.7. Let (A1)-(A4) and (A6)-(A7) hold with l = 2. Assume (H) with l = 0. Then (4.1)–(4.2) and (4.15) have a unique generalized solution u and u τ = {uτ (ti ) : i = 0, 1, 2, ..., m}, respectively, for all integers m larger than some integer m0 . Moreover, for all integers m > m0 E max |u(iτ ) − uτ (iτ )|20 + E 0≤i≤m

where C is a constant, independent of τ .

m X i=1

|u(iτ ) − uτ (iτ )|21 τ ≤ Cτ,

(4.18)

Proof. By Theorems 4.2 and 4.6 the problems (4.1)–(4.2) and (4.15) have a unique solution u and uτ , respectively. It is an easy exercise to verify that Assumption (H) ensures that condition (T1) holds. By virtue of Corollary 4.4 condition (T2) is valid with ν = 1/2. Condition (T3) clearly holds by statement (i) of Theorem 4.3. Now we can apply Theorem 3.4, which gives (4.18). 

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4.2. Linear stochastic PDEs Let Assumptions (A1)–(A3) and (A7) hold and impose also the following condition on F . Assumption (A8) There exist a P ⊗ B(R)-measurable function f such that F (t, x, p, r) = f (t, x),

for all t, x, p, r,

and the derivatives in x of f up to order l are P ⊗ B(R)-measurable functions such that |f (t)|2l ≤ ξ, for all (ω, t), where ξ is a random variable with finite first moment. Now equation (4.13) has become the linear stochastic PDE d1 X     du(t, x) = Lu(t, x) + f (t, x) dt + Mk u(t, x) + gk (t, x) dW k (t),

(4.19)

k=1

and by Theorem 3.4 we have the following result.

Theorem 4.8. Let r ≥ 0 be an integer. Let Assumptions (A1)–(A3) and (A7)– (A8) hold with l := r + 2, and let Assumption (H) hold with l = r. Then there is an integer m0 such that (4.19)–(4.2) and (4.15) have a unique generalized solution u and uτ = {uτ (ti ) : i = 0, 1, 2, ..., m}, respectively, for all integers m > m0 . Moreover, E max |u(iτ ) − uτ (iτ )|2r + E 0≤i≤m

m X i=1

|u(iτ ) − uτ (iτ )|2r+1 τ ≤ Cτ

holds for all m > m0 , where C is a constant independent of τ . Proof. For r = 0 the statement of this theorem follows immediately from Theorem 4.7. For r > 0 set H = W2r (Rd ) and V = W2r+1 (Rd ) and consider the normal triplet V ,→ H ≡ H ∗ ,→ V ∗ based on the inner product (· , ·) := (· , ·)r in W2r (Rd ), which determines the duality h· , ·i between V and V ∗ . Using Assumptions (A3), (A7) and (A8) with l = r, one can easily show that there exist a constant C and a random variable ξ such that Eξ 2 < ∞ and X
(−1)|α| aαβ (t) Dβ v , Dα ϕ r ≤ C|v|r+1 |ϕ|r+1 , |α|≤1,|β|≤1

d1 X

k=1


2 2 2 α |(bα k (t)D v , ϕ r | ≤ C|v|r+1 |ϕ|r ,


| f (t) , ϕ r |2 ≤ ξ|ϕ|2r ,

d1 X

k=1


|(gk (t) , ϕ r |2 ≤ ξ|ϕ|2r

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for all ω, t ∈ [0, T ] and v, ϕ ∈ W2r+1 (Rd ). Therefore the operators A(t, ·), Bk (t, ·) defined for v, ϕ ∈ V by X

hA(t, v), ϕi = (−1)|α| aαβ (t) Dβ v , Dα ϕ r + f (t) , ϕ r , |α|≤1,|β|≤1

α (Bk (t, v) , ϕ) = bα k (t)D v , ϕ




r


+ gk (t) , ϕ r ,

(4.20)

are mappings from V into V ∗ and H, respectively, for each k and ω, t, such that the growth conditions (2.6) and (2.7) hold. Thus we can cast the Cauchy problem (4.19)– (4.2) into the evolution equation (1.1), and it is an easy to verify that conditions (C1)–(C4) hold. Thus this evolution equation admits a unique solution u, which clearly a generalized solution to (4.19)– (4.2). Due to assumptions (A1)– (A3) and (A7)–(A8) by Theorem 1.1 in Ref. [10] u is a W r+2 (Rd )-valued stochastic process such that E

sup |u(t)|2r+2 t≤T

+E

Z

T 0

|u(t)|2r+3 dt < ∞.

Hence it is obvious that (T3) holds, and it is easy to verify (T2) with ν = 21 like it is done in the proof of Corollary 4.4. Finally, it is an easy exercise to show that (T1) holds. Now we can finish the proof of the theorem by applying Theorem 3.4.  From the previous theorem we obtain the following corollary by Sobolev’s embedding from W2r to C q . Corollary 4.9. Let q be any non-negative number and assume that the assumptions ¯ and u ¯τ of of Theorem 4.8 hold with r > q + d2 . Then there exist modifications u τ γ γ τ u and u , respectively, such that the derivatives D u ¯ and D u ¯ in x up to order q are functions continuous in x. Moreover, there exists a constant C independent of τ such that X
E max sup |Dγ u ¯(iτ, x) − u ¯τ (iτ, x) |2 0≤i≤m x∈Rd

|γ|≤q

+E

m X

sup

X

d i=1 x∈R |γ|≤q+1

2 |Dγ u ¯(iτ, x) − u ¯τ (iτ, x) τ ≤ Cτ.

(4.21)

Acknowledgments This paper was written while I. Gy¨ ongy was visiting the University of Paris 1. His research is partially supported by EU Network HARP. The research of A. Millet is partially supported by the research project BMF2003-01345 The authors wish to thank the referee for helpful comments.

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References [1] N. Krylov and B. Rosovskii, Stochastic evolution equations, J.Soviet Mathematics. 16, 1233–1277, (1981). ´ [2] E. Pardoux. Equations aux d´eriv´ees partielles stochastiques nonlin´eares monotones. ´ Etude de solutions fortes de type Itˆ o, (1975). Th´ese Doct. Sci. Math. Univ. Paris Sud. [3] B. L. Rozovski˘ı, Stochastic evolution systems. vol. 35, Mathematics and its Applications (Soviet Series), (Kluwer Academic Publishers Group, Dordrecht, 1990). ISBN 0-7923-0037-8. Linear theory and applications to nonlinear filtering, Translated from the Russian by A. Yarkho. [4] I. Gy¨ ongy and A. Millet, On discretization schemes for stochastic evolution equations, Potential Anal. 23(2), 99–134, (2005). ISSN 0926-2601. [5] N. V. Krylov, On Lp -theory of stochastic partial differential equations in the whole space, SIAM J. Math. Anal. 27(2), 313–340, (1996). ISSN 0036-1410. [6] C. Cardon-Weber, Cahn-Hilliard stochastic equation: existence of the solution and of its density, Bernoulli. 7(5), 777–816, (2001). ISSN 1350-7265. [7] N. V. Krylov and B. L. Rozovski˘ı, Conditional distributions of diffusion processes, Izv. Akad. Nauk SSSR Ser. Mat. 42(2), 356–378, 470, (1978). ISSN 0373-2436. ´ Pardoux. Filtrage non lin´eaire et ´equations aux d´eriv´ees partielles stochastiques [8] E. ´ ´ e de Probabilit´es de Saint-Flour XIX—1989, vol. 1464, Lecture associ´ees. In Ecole d’Et´ Notes in Math., pp. 67–163. Springer, Berlin, (1991). [9] E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes, Stochastics. 3(2), 127–167, (1979). ISSN 0090-9491. [10] N. V. Krylov and B. L. Rozovski˘ı, The Cauchy problem for linear stochastic partial differential equations, Izv. Akad. Nauk SSSR Ser. Mat. 41(6), 1329–1347, 1448, (1977). ISSN 0373-2436. [11] I. Gy¨ ongy and N. V. Krylov, On stochastics equations with respect to semimartingales. II. Itˆ o formula in Banach spaces, Stochastics. 6(3-4), 153–173, (1981/82). ISSN 0090-9491.

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Chapter 12 Maximum Principle for SPDEs and Its Applications

Nicolai V. Krylov School of Mathematics, University of Minnesota, 127 Vincent Hall, Minneapolis, MN, 55455, [email protected]. The maximum principle for SPDEs is established in multidimensional C 1 domains. An application is given to proving the H¨ older continuity up to the boundary of solutions of one-dimensional SPDEs.

Contents 1 Introduction . . . . . . . . . . . 2 The maximum principle . . . . . 3 Auxiliary results . . . . . . . . . 4 Proof of Theorems 2.5 and 2.6 . 5 Auxiliary functions . . . . . . . . 6 Continuity of solutions of SPDEs References . . . . . . . . . . . . . . .

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311 312 315 321 325 329 337

1. Introduction The maximum principle is one of the most powerful tools in the theory of secondorder elliptic and parabolic partial differential equations. However, until now it did not play any significant role in the theory of SPDEs. In this paper we show how to apply it to one-dimensional SPDEs on the half line R+ = (0, ∞) and prove the H¨ older continuity of solutions on [0, ∞). This result was previously known when the coefficients of the first order derivatives of solution appearing in the stochastic term in the equation obeys a quite unpleasant condition. On the other hand, if they just vanish, then the H¨ older continuity was well known before (see, for instance, Ref. [1] and the references therein). To the best of our knowledge the maximum principle was first proved in Ref. [2] (see also Ref. [3] for the case of random coefficients) for SPDEs in the whole space by the method of random characteristics introduced there and also in Ref. [4]. Later the method of random characteristics was used in many papers for various purposes, for instance, to prove smoothness of solutions (see, for instance, Refs. [5–8] and the references therein). It was very tempting to try to use this method for proving the maximum principle for SPDEs in domains. However, the implementation of 311

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the method turns out to become extremely cumbersome and inconvenient if the coefficients of the equation are random processes. Also, it requires more regularity of solutions than actually needed. Here in Section 2 we state the maximum principle in domains under minimal assumptions. We prove it in Section 4 by using methods taken from PDEs after we prepare some auxiliary results in Section 3. Section 6 contains an application of the maximum principle to investigating the H¨ older continuity up to the boundary of solutions of one-dimensional SPDEs. Note that, for instance, in Refs. [5, 6] and in many other papers that can be found from our list of references the regularity properties are proved only inside domains. Quite sharp regularity for solutions of SPDEs in multidimensional domains is established in Ref. [9], it is stated in terms of appropriate weighted Sobolev spaces and, unfortunately, do not imply even the pointwise continuity up to the boundary. It is worth saying that we only deal with one-dimensional case and coefficients independent of the space variable. In a subsequent paper we intend to treat the general case. In Section 5 we introduce some auxiliary functions used in Section 6. We denote by Rd the Euclidean space of points x = (x1 , ..., xd ), Di =

∂ . ∂xi

For a domain D ⊂ Rd and we set W21 (D) to be the closure of the set of infinitely differentiable functions ϕ having finite norm kϕk2W 1 (D) = kϕk2L2 (D) + kϕx k2L2 (D) 2

0

with respect to this norm. Here ϕx is the gradient of ϕ. By W 12 (D) we denote the closure of C0∞ (D) with respect to the norm k · kW21 (D) . Our way to say that u ≤ v 0

on ∂D is that (u − v)+ ∈ W 12 (D). As usual, the summation convention is enforced and writing N (....) is to say that the constant N depends and depends only on the contents of the parentheses. Such constants may change from line to line. 2. The maximum principle 1 Let D be a domain in Rd of class Cloc and let (Ω, F, P ) be a complete probability space with a given filtration (Ft , t ≥ 0) of σ-fields Ft ⊂ F complete with respect to F, P . We are investigating some properties of a function ut (x) = ut (ω, x) satisfying Z t (ϕ, ut ) = (ϕ, u0 ) + (ϕ, σsik Di us + νsk us + gsk ) dmks 0

+

Z

0

t i i i (ϕ, Di (aij s Dj us ) + bs Di us + Di (as us ) − cs us + fs + Di fs ) dVs .

(2.1)

for all t ∈ [0, ∞) and any ϕ ∈ C0∞ (D). Here mkt , k = 1, 2, ..., are one-dimensional continuous local Ft -martingales, starting at zero, Vt is a nondecreasing continuous

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Ft -adapted process starting at zero, (ϕ, ·) is the pairing between a generalized function on D and a test function ϕ, the summation convention over repeated indices is enforced, and the meaning of the remaining objects and further assumptions are described below. We need some real-valued functions ξti (x), K1 (t) > 0, and K2 (t) ≥ 0 defined for i = 1, ..., d, t ∈ [0, ∞), x ∈ Rd and also depending on ω. i i ik k k We assume that aij t (x), bt (x), at (x), ct (x), σt (x), νt (x), and gs are real-valued d functions defined for i, j = 1, ..., d, k = 1, 2, ..., t ∈ [0, ∞), x ∈ R and also depending on ω ∈ Ω. Assumption 2.1. We suppose that, for any ω, hmi , mj it = 0 if i 6= j, and for any k we have dhmk it ≤ dVt . Assumption 2.2. For all values of the arguments (i) σ i := (σ i1 , σ i2 , ...), ν := (ν 1 , ν 2 , ...), g := (g 1 , g 2 , ...) ∈ `2 ; (ii) for all λ ∈ Rd X | λi ξ i |2 ≤ K1 (2aij − αij )λi λj , i

where αij = (σ i , σ j )`2 . The case ξ ≡ 0 is not excluded and in this case Assumption 2.2 (ii) is just the usual parabolicity assumption. Assumption 2.3. i i ik k i (i) The functions aij t (x), bt (x), at (x), ct (x), σt (x), νt (x), ξt (x), K1 (t), and K2 (t) are measurable with respect to (ω, t, x) and Ft -adapted for each x; i i ik k i (ii) the functions aij t (x), bt (x), at (x), ct (x), σt (x), νt (x), and ξt (x) are bounded; (iii) for each ω, t the functions ηti := ait − bit − (σti , νt )`2 − ξti are once continuously differentiable on D, have bounded derivatives, and satisfy Di η i − 2c + |ν|2`2 ≤ K2

(2.2)

for all values of arguments; (iv) for each ϕ ∈ C0∞ (D) the processes ϕft , ϕft1 ,..., ϕftd are L2 (D)-valued and ϕgt is an L2 (D, `2 )-valued Ft -adapted and jointly measurable; for all t ∈ [0, ∞) and ω∈Ω Z t X
kϕfs k2L2 (D) + kϕfsi k2L2 (D) + kϕgs k2L2 (D,`2 ) + K1 (s) + K2 (s) dVs < ∞. 0

i

Assumption 2.4. For each ϕ ∈ C0∞ (D) (i) the process ϕut = ϕut (ω) is L2 (D)-valued, Ft -adapted, and jointly measurable; (ii) for any ω ϕut ∈ W21 (D)

(dVt -a.e.);

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(iii) for each t ∈ [0, ∞) and ω Z t kϕus k2W 1 (D) dVs < ∞. 2

0

The above assumptions are supposed to hold throughout this section. Here is the maximum principle saying, in particular, that if g k = f i = 0, f ≤ 0 and u ≤ 0 on the parabolic boundary of [0, T ] × D, then u ≤ 0 in [0, T ]. By the way, our solutions are L2,loc (D)-valued functions of ω and t, so that for each ω and t an equivalence class is specified. Naturally, if we write ut (ω) ≤ 0, or ut ≤ 0 we mean that in the corresponding class there is a nonpositive function. Theorem 2.5. Let τ2 ≥ τ1 be stopping times, τ1 < ∞ for any ω. Suppose that, for any ω, i = 1, ..., d, k = 1, 2, ..., Iut >0 gtk = Iut >0 fti = 0,

0

1 u+ t ∈ W 2 (D),

Iut >0 ft ≤ 0

dVt -almost everywhere on (τ1 , τ2 ) and suppose that uτ1 ≤ 0 for any ω. Then almost surely ut ≤ 0 for all t ∈ [τ1 , τ2 ] ∩ [τ1 , ∞). The following comparison principle is a generalization of Theorem 2.5. Theorem 2.6. Let τ2 ≥ τ1 be stopping times, τ1 < ∞ for any ω. Let ρt ≥ 0, t ∈ [0, ∞), be a nondecreasing continuous Ft -adapted process and let f¯t , f¯t1 ,..., f¯td , and g¯t satisfy Assumption 2.3 (iv). Let u ¯t be a process satisfying Assumption 2.4 and such that equation (2.1) holds for all t ∈ [0, ∞) and any ϕ ∈ C0∞ (D) with f¯t , f¯t1 ,..., f¯td , and g¯t in place of ft , ft1 ,..., ftd , and gt , respectively. Assume that, for any ω, (dVt -a.e.) on [τ1 , τ2 ] we have Iut >ρ¯ut (gt − ρt g¯t ) = Iut >ρt u¯t (fti − ρt f¯ti ) = 0, Iut >ρt u¯t (ft − ρt f¯t ) ≤ 0,

Iut >ρt u¯t u ¯t ≥ 0,

i = 1, ..., d, 0

(ut − ρt u ¯t )+ ∈ W 12 (D).

Finally, assume that uτ1 ≤ ρτ1 u ¯τ1 for any ω. Then almost surely ut ≤ ρt u ¯t for all t ∈ [τ1 , τ2 ] ∩ [τ1 , ∞). Corollary 2.7. Assume that, for any ω, (dVt -a.e.) on (τ1 , τ2 ) × D we have Iut >1 (νtk + gtk ) = Iut >1 (fti + ait ) = 0, Iut >1 f ≤ Iut >1 c,

i = 1, ..., d, k = 1, 2, ... 0

(ut − 1)+ ∈ W 12 (D).

Also assume that uτ1 ≤ 1 for any ω. Then almost surely ut ≤ 1 for all t ∈ [τ1 , τ2 ] ∩ [τ1 , ∞).

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Indeed, it suffices to take u ¯t ≡ 1, ρt ≡ 1 and observe that u ¯t satisfies (2.1) with f¯ti = −ait , f¯t = ct , and g¯t = −νt in place of fti , ft , and gt , respectively. This corollary generalizes the corresponding results of Refs. [2] and [3], where k ν = g k = f i = ai = 0. Remark 2.8. Our equation has a special structure, which may look quite restrictive. In particular, we assume that the martingales mkt are mutually orthogonal. The general case, actually, reduces to this particular one after using the fact that one can always orthogonalize the martingales by using, for instance, the GrammSchmidt procedure. This, of course, would change σ, ν, and g, and writing the corresponding general conditions would only obscure the matter. Then passing from mkt to (no summation in k) Z t
1/2 dt ρks dmks , ρks = k dt + dhm it 0

allows one to have dhmk it ≤ dt and adding after that t to Vt allows one to have dhmk it ≤ dVt . Again we should modify our coefficients but we will see in the proof of Theorem 2.6 that this modification does not affect Assumption 2.2, which is an assumption about parabolicity of our equation and not strict nondegeneracy. 3. Auxiliary results In this section the notation ut is sometimes used for different objects than in Section 2. Denote by R the set of real-valued functions convex r(x) on R such that (i) r is continuously differentiable, r(0) = r 0 (0) = 0, (ii) r0 is absolutely continuous, its derivative r 00 is bounded and left continuous, that is usual r00 which exists almost everywhere is bounded and there is a left-continuous function with which r00 coincides almost everywhere. For r ∈ R by r00 we will always mean the left-continuous modification of the usual second-order derivative of r. Remark 3.1. For each r ∈ R there exists a sequence rn ∈ R of infinitely differentiable functions such that |rn (x)| ≤ N |x|2 , |rn0 (x)| ≤ N |x|, and |rn00 | ≤ N with N < ∞ independent of x and n, rn , rn0 , rn00 →r, r0 r00 on R. Indeed, let ζ ∈ C0∞ (R) be a nonnegative function with support in (0, 1) and unit integral. For ε > 0 define ζε (x) = ε−1 ζ(x/ε) and rε (x) = r ∗ ζε (x) − r ∗ ζε (0) − xr0 ∗ ζε (0). Then rε is infinitely differentiable, rε (0) = rε0 (0) = 0, |rε00 | = |r00 ∗ ζε | ≤ sup |r00 | < ∞. In particular, |rε0 (x)| = |

Z

x 0

rε00 (y) dy| ≤ N |x|,

|rε (x)| = |

Z

x 0

rε0 (y) dy| ≤ N |x|2 .

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Finally, the convergences rε →r and rε0 →r0 follow by the continuity of r and r 0 and the convergence rε00 →r00 follows from the dominated convergence theorem, the left continuity of r00 and the formula Z 1 00 rε (x) = r00 (x − εy)ζ(y) dy. 0

In the following lemma the assumption that D is a locally smooth domain is not used. Lemma 3.2. Let ut = ut (ω) be an L2 (D)-valued process such that u0 is F0 -measurable. Let ft and gt = (gt1 , gt2 , ...) be Ft -adapted and jointly measurable processes with values L2 (D) and L2 (D, `2 ), respectively. Assume that for each t ∈ [0, ∞) we have Z t (kfs k2L2 (D) + kgs k2L2 (D,`2 ) ) dVs < ∞ (3.1) 0

and for any ϕ ∈

C0∞ (D)

(ϕ, ut )L2 (D) = (ϕ, u0 )L2 (D) +

Z

t 0

(ϕ, fs )L2 (D) dVs +

Z

t 0

(ϕ, gsk )L2 (D) dmks .

(3.2)

Then (i) ut is a continuous L2 (D)-valued function (a.s.); (ii) for any r ∈ R (a.s.) for all t ∈ [0, ∞) Z t kr1/2 (ut )k2L2 (D) = kr1/2 (u0 )k2L2 (D) + hs dVs + mt , (3.3) 0

where hs := (r0 (us ), fs )L2 (D) + (1/2)k(r 00 )1/2 (us )ˇ gs k2L2 (D,`2 ) , gˇsk

:=



dhmk is dVs

1/2

gsk ,

mt :=

Z

and mt is a local martingale; (iii) (a.s.) for t ∈ [0, ∞) + 2 2 ku+ t kL2 (D) = ku0 kL2 (D) +

t 0

Z

(r0 (us ), gsk )L2 (D) dmks

t

hs dVs + mt , 0

where gs Ius >0 k2L2 (D,`2 ) , hs := 2(u+ s , fs )L2 (D) + kˇ mt := 2

Z

0

and mt is a local martingale.

(3.4)

t k k (u+ s , gs )L2 (D) dms

(3.5)

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Proof. (i) Recall that the operation of stochastic integration of Hilbert space valued processes is well defined. Therefore, the process Z t Z t u bt = u0 + fs dVs + gsk dmks 0

0

is well defined as a continuous L2 (D)-valued process. We also recall how the scalar product interacts with integrals. Then it is seen that for any t and ϕ ∈ C0∞ (D) we have (ϕ, ut ) = (ϕ, u bt ) (a.s.). Since both parts are continuous in t, the equality holds for all t at once (a.s.), and since C0∞ (D) is dense in L2 (D), we have that ut = u bt for all t (a.s.). This proves (i). As a corollary we obtain that sup kut kL2 (D) < ∞, t≤T

∀T < ∞ (a.s.).

(3.6)

(ii) It suffices to prove (3.3) for infinitely differentiable r ∈ R. Indeed, for rn from Remark 3.1, passing to the limit in all term in (3.3) apart from mt presents no problem at all in light of (3.6) and the dominated convergence theorem. Also Z t mt (n) := (rn0 (us ), gsk )L2 (D) dmks →mt 0

uniformly in t on finite intervals in probability because Z tX hm(n) − mit = (rn0 − r0 )(us ), gsk )2L2 (D) dhmk is 0

≤

≤

Z

t 0

Z tX 0

k

k

(rn0 − r0 )(us ), gsk )2L2 (D) dVs

k(rn0 − r0 )(us )k2L2 (D) kgs k2L2 (D,`2 ) dVs →0

again owing to (3.6) and the dominated convergence theorem. Thus, we may concentrate on the case that r is infinitely differentiable. Take a symmetric ζ ∈ C0∞ (Rd ) with support in the unit ball centered at the origin and unit integral. For ε > 0 set ζε (x) = ε−d ζ(x/ε) and for functions v = v(x) define v (ε) = v ∗ ζε . Also set Dε = {x ∈ D : dist (x, ∂D) < ε}. According to (3.2) for any x ∈ Dε and t ≥ 0 we have Z t Z t (ε) (ε) ut (x) = u0 (x) + fs(ε) (x) dVs + gsk(ε) (x) dmks . 0

0

By Itˆ o’s formula we have that on Dε Z t (ε) (ε) (ε) 00 (ε) r(ut ) = r(u0 ) + [r0 (u(ε) gs(ε) |2`2 ] dVs s )fs + (1/2)r (us )|ˇ 0

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+

Z

t 0

k(ε) r0 (u(ε) dmks . s )gs

(3.7)

Here, for each ε > 0, the integrands are smooth functions of x and their magnitudes along with the magnitudes of each of their derivatives in Dε are majorized by a constant (possibly depending on ε) times or kgs k2L2 (D,`2 )

kfs kL2 (D) sup kus kL2 (D) s≤t

or kgsk kL2 (D) sup kus kL2 (D) . s≤t

This and (3.1) and (3.6) allow us to use Fubini’s theorem while integrating through (3.7) and conclude Z t 1/2 (ε) 2 1/2 (ε) 2 (ε) kr (ut )kL2 (Dε ) = kr (u0 )kL2 (Dε ) + [(r0 (u(ε) s ), fs )L2 (Dε ) 0

+(1/2)k(r 00 )1/2 (u(ε) gs(ε) k2L2 (Dε ,`2 ) ] dVs s )ˇ +

Z

t 0

k(ε) )L2 (Dε ) dmks . (r0 (u(ε) s ), gs

(3.8)

Now we let ε ↓ 0. We use that for any function v ∈ L2 (D) kv (ε) kL2 (Dε ) ≤ k(vID )(ε) kL2 (Rd ) ≤ kvID kL2 (Rd ) = kvkL2 (D) k(ε)

and v (ε) IDε →v in L2 (D). In particular, gs (ε) that r0 (us )IDε →r0 (us ) in L2 (D) and

(ε)

IDε →gsk and us IDε →us implying

k(ε) (r0 (u(ε) )L2 (Dε ) →(r0 (us ), gsk )L2 (D) s ), gs

for each k and dP × dVs -almost all (ω, s). We also use (3.1) and (3.6) to assert that ∞ Z t X k(ε) sup |(r0 (u(ε) )L2 (Dε ) |2 dVs s ), gs k=1

0 ε∈(0,1)

≤ N sup kus k2L2 (D) s≤t

Z

t 0

kgs k2L2 (D,`2 ) dVs < ∞.

As is easy to see this implies that the local martingale part in (3.8) converges to m t as ε ↓ 0 in probability locally uniformly with respect to t. Similar manipulations with other terms in (3.8) allow us to get (3.3). Since (3.5) is just a particular case of (3.3), the lemma is proved. Remark 3.3. Lemma 3.2 remains true if in the definition of R instead of requiring r00 to have a left-continuous modification we required it to have a right-continuous one, and of course, in (3.4) used this right-continuous modification. This is seen after replacing u with −u.

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In case r(x) = (x+ )2 the function r00 has both right- and left-continuous modifications, so that in the definition of mt one can use 2Ius >0 or 2Ius ≥0 . It follows that (a.s.) for any t Z t kˇ gs Ius =0 k2L2 (D,`2 ) dVs = 0. 0

Furthermore, since, for any v ∈ L2 (D), ut + v has the same form as ut , Z t kˇ gs Ius =v k2L2 (D,`2 ) dVs = 0. 0

Let ut be an L2 (D)-valued F0 -

Lemma 3.4. Let D be an arbitrary domain. measurable process such that for any ω ut ∈ W21 (D)

(dVt -a.e.) and for each T ∈ [0, ∞) and ω Z T kut k2W 1 (D) dVt < ∞.

(3.9)

2

0

Let ft , ft1 , ..., ftd , and gt = (gt1 , gt2 , ...) be Ft -adapted and jointly measurable processes with values in L2 (D) and L2 (D, `2 ), respectively. Assume that for each t ∈ [0, ∞) we have Z t X (kfs k2L2 (D) + kfsi k2L2 (D) + kgs k2L2 (D,`2 ) ) dVs < ∞, (3.10) 0

i

and for each t ∈ [0, ∞), ϕ ∈

C0∞ (D),

(ϕ, ut ) = (ϕ, u0 ) +

Z

and ω

t

(ϕ, fs + 0

Di fsi ) dVs

+

Z

t 0

(ϕ, gsk ) dmks .

(3.11)

Finally, assume that there is a compact set G ⊂ D such that ut (x) = ft (x) = fti (x) = gtk (x) = 0 outside G. Then (a) ut is a continuous L2 (D)-valued function (a.s.); (b) (a.s.) for all t ∈ [0, ∞) Z t + 2 2 ku+ k = ku k + hs dVs + mt , (3.12) t L2 (D) 0 L2 (D) 0

where i gs Iϕus >0 k2L2 (D,`2 ) , hs := 2(u+ s , fs )L2 (D) − 2(Ius >0 Di us , fs )L2 (D) + kˇ

mt := 2

Z

t 0

k k (u+ s , gs )L2 (D) dms .

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Proof. Observe that (3.11) holds for all infinitely differentiable functions ϕ. Fur0

thermore, since ut ∈ W 12 (D) (dVt -a.e.) assertion (a) is well known (see, for instance, Ref. [10], the references therein, and Remark 2.8). To prove (b), take ε smaller than the distance between G and ∂D. Notice that, owing to the symmetry of ζ, for ϕ ∈ C0∞ (D) (ε)

(ε)

(ϕ(ε) , ut ) = (ϕ(ε) , ut )L2 (Rd ) = (ϕ, ut )L2 (Rd ) = (ϕ, ut )L2 (D) .

Therefore, it follows from (3.11) that for any ϕ ∈ C0∞ (D) Z t Z t (ε) (ε) (ϕ, ut ) = (ϕ, u0 ) + (ϕ, f˜sε ) ds + (ϕ, gsk(ε) ) dmks , 0

0

where

f˜sε := fs(ε) + Di fsi(ε) is an L2 (D)-valued function with norm that is locally square integrable against dVs . By Lemma 3.2 for any r ∈ R Z t (ε) (ε) kr1/2 (ut )k2L2 (D) = kr1/2 (u0 )k2L2 (D) + hεs dVs + mεt , (3.13) 0

where

mεt

:=

Z

t 0

(r0 (u(ε) ˇsk(ε) )L2 (D) dmks , s ), g

00 1/2 (ε) (ε) 2 ˜ε hεs := (r0 (u(ε) (us )ˇ gs kL2 (D,`2 ) s ), fs )L2 (D) + (1/2)k(r ) (ε) 00 (ε) (ε) i(ε) = (r0 (u(ε) )L2 (D) +(1/2)k(r 00 )1/2 (u(ε) gs(ε) k2L2 (D,`2 ) . s ), fs )L2 (D) −(r (us )Di us , fs s )ˇ

If r is infinitely differentiable, then by using (3.9) and (3.10) one easily passes to the limit in (3.13) as ε→0. The argument is quite similar to the corresponding argument in the proof of Lemma 3.2 and, for smooth r ∈ R, yields Z t kr1/2 (ut )k2L2 (D) = kr1/2 (u0 )k2L2 (D) + hs dVs + mt , 0

where

mt =

Z

t 0

(r0 (us ), gsk )L2 (D) dmks ,

hs = (r0 (us ), fs )L2 (D) − (r00 (us )Di us , fsi )L2 (D) + (1/2)k(r 00 )1/2 (us )ˇ gs k2L2 (D,`2 ) . Finally, as in the proof of Lemma 3.2 one easily passes from smooth r ∈ R to arbitrary ones and gets (3.12) by taking r(x) = (x+ )2 . The lemma is proved. Lemma 3.4 serves as an auxiliary tool to prove a deeper result. Lemma 3.5. Let D be an arbitrary domain. Assume that for each ϕ ∈ C0∞ (D) (i) ϕut is an L2 (D)-valued process such that ϕu0 is F0 -measurable;

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(ii) for any ω ϕut ∈ W21 (D) (dVt -a.e.) and for each T ∈ [0, ∞) and ω Z T kϕut k2W 1 (D) dVt < ∞. 2

0

(iii) Let ft , ft1 , ..., ftd , and gt = (gt1 , gt2 , ...) be Ft -adapted and jointly measurable processes with values in L2 (D) and L2 (D, `2 ), respectively. Assume that for each t ∈ [0, ∞) and ϕ ∈ C0∞ (D) we have Z t X (kϕfs k2L2 (D) + kϕfsi k2L2 (D) + kϕgs k2L2 (D,`2 ) ) dVs < ∞, 0

i

(ϕ, ut ) = (ϕ, u0 ) +

Z

t

(ϕ, fs + 0

Di fsi ) dVs

+

Z

t 0

(ϕ, gsk ) dmks .

Then, for any ϕ ∈ C0∞ (D), (a) ϕut is a continuous L2 (D)-valued function (a.s.); (b) (a.s.) for all t ∈ [0, ∞) Z t hs dVs + mt , k(ϕut )+ k2L2 (D) = k(ϕu0 )+ k2L2 (D) + 0

where

hs := 2((ϕus )+ , ϕfs − fsi Di ϕ)L2 (D) − 2(Iϕus >0 Di (ϕus ), ϕfsi )L2 (D) +kϕˇ gs Iϕus >0 k2L2 (D,`2 ) ,

mt := 2

Z

0

t k k (ϕu+ s , ϕgs )L2 (D) dms .

C0∞ (D)

Proof. Clearly, for any ϕ, η ∈ we have Z t Z t i i (ϕ, ηut ) = (ϕ, ηu0 ) + (ϕ, ηfs − fs Di η + Di (ηfs )) dVs + (ϕ, ηgsk ) dmks . (3.14) 0

0

Therefore, ηut satisfies the assumptions of Lemma 3.4 with ηfs − fsi Di η, ηfsi , and ηgsk in place of fs , Di fsi , and gsk , respectively. By applying Lemma 3.4 to ηut in place of ut we get the result with η in place of ϕ. This certainly proves the lemma. 4. Proof of Theorems 2.5 and 2.6

In this section the assumptions stated in Section 2 are supposed to be satisfied. We use the fact that due to our hypothesis that D ∈ C 1 , there exist sequences ζn and ζ¯n of nonnegative C0∞ (D)-functions such that 0 ≤ ζn , ζ¯n ≤ 1, ζn , ζ¯n →1 in D as 0

n→∞ and for any v ∈ W 12 (D), i = 1, ..., d,

kvDi ζn kL2 (D) ≤ N (k(1 − ζ¯n )vkL2 (D) + k(1 − ζ¯n )DvkL2 (D) ),

(4.1)

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where N is independent of n and v (see, for instance, the proof of Theorem 5.5.2 in Ref. [11]). We also know (see, for instance, the proof of Lemma 2.3.2 in Ref. [12] or Problem 17, Chapter 5 in Ref. [11]) that if v ∈ W21 (D), then v + ∈ W21 (D) and Di v + = Iv>0 Di v. Proof of Theorem 2.5. Set K = K1 + K2 ,

ϕt =

Z

t

K(s) ds. 0

Take the sequences of nonnegative ζn , ζ¯n ∈ C0∞ (D) from above. By Itˆ o’s formula and Lemma 3.5 Z t + 2 2 −ϕt kζn u+ k e = kζ u k + hns dVs + mt (n), n 0 L2 (D) t L2 (D) 0

where 2 eϕs hns = I1s + I2s + I3s − K(s)kζn u+ s kL2 (D) , i i i ij I1s = 2 ζn u+ s , ζn [fs + bs Di us − cs us ] − [us as + as Dj us + fs ]Di ζn i I2s = −2 Iζn us >0 Di (ζn us ), ζn [us ais + aij s Dj u s + f s ]




L2 (D)

,




L2 (D)

,

I3s = kζn Iζn us >0 [ˇ σsi Di us + νˇs us + gˇs ]k2L2 (D,`2 ) , mt (n) = 2

Z

t 0

ik k k e−ϕs ζn u+ s , ζn [σs Di us + νs us + gs ]

Since u+ τ1 = 0 we have

2 e−ϕτ2 ∧t∨τ1 kζn u+ τ2 ∧t∨τ1 kL2 (D) =

Z

t 0




L2 (D)

dmks .

Iτ2 >s>τ1 hns dVs + m ¯ t (n),

(4.2)

where m ¯ t (n) := mτ2 ∧t∨τ1 (n) − mτ1 (n) is a local martingale. Next we use the assumptions of the theorem and see that for dVs -almost all s ∈ (τ1 , τ2 ) we have i i ij I1s ≤ 2(ζn u+ s , ζn [bs Di us − cs us ] − [us as + as Dj us ]Di ζn )L2 (D) i + + = 2(ζn2 u+ s , bs Di us − cs us )L2 (D) + I4s

with + i ij + I4s = −2(ζn u+ s Di ζn , us as + as Dj us )L2 (D) .

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At this moment we recall (4.1) and observe that 2 + + (ζn |Di ζn |, (u+ s ) )L2 (D) ≤ N kζn us kL2 (D) kus Di ζn kL2 (D) .

Then we see that
+ + ¯ I4s ≤ N k(1 − ζ¯n )u+ s kL2 (D) + k(1 − ζn )Dus kL2 (D) kus kW21 (D) ,

where and below by N we denote various finite constants. In I2s

+ + Iζn us >0 Di (ζn us ) = Di (ζn u+ s ) = u s Di ζ n + ζ n Di u s ,

so that + i ij + I2s = −2(ζn2 Di u+ s , us as + as Dj us )L2 (D) + I4s .

Next, ζn Iζn us >0 = ζn Ius >0 , ij + I3s ≤ kζn Ius >0 [σsi Di us + νs us ]k2L2 (D,`2 ) = (ζn2 Di u+ s , αs Dj us )L2 (D) + i + 2 +2(ζn2 Di u+ s , us (σs , νs )`2 )L2 (D) + kζn |νs |`2 us kL2 (D) .

Also observe that certain parts of I2s and I3s can be combined if we use that ij + 2 + ij + −2(ζn2 Di u+ s , as Dj us )L2 (D) + (ζn Di us , αs Dj us )L2 (D) 2 ≤ −K1−1 (s)kζn ξsi Di u+ s kL2 (D) .

It follows that for dVs -almost all s ∈ (τ1 , τ2 ) Z + 2 + 2 2 eϕs hns ≤ [ζn2 (bis − ais + (σsi , νs )`2 )2u+ s Di us + ζn (us ) (|νs |`2 − 2cs )] dx D


+ + ¯ +N k(1 − ζ¯n )u+ s kL2 (D) + k(1 − ζn )Dus kL2 (D) kus kW21 (D) 2 + 2 −K1−1 (s)kζn ξsi Di u+ s kL2 (D) − K(s)kζn us kL2 (D) .

Here bis − ais + (σsi , νs )`2 = −ξsi − ηsi and we transform the integral of + i 2 + 2 ζn2 (−ηsi )2u+ s Di us = −ηs ζn Di (us )

by integrating by parts. Then we get that for dVs -almost all s ∈ (τ1 , τ2 ) Z + 2 + 2 2 i eϕs hns ≤ [−ζn2 ξsi 2u+ s Di us + ζn (us ) (|νs |`2 − 2cs + Di ηs )] dx D

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Z

D


+ 2 i + ¯ + ¯ (u+ s ) ζn ηs Di ζn dx + N k(1 − ζn )us kL2 (D) + k(1 − ζn )Dus kL2 (D) kus kW21 (D) 2 + 2 −K1−1 (s)kζn ξsi Di u+ s kL2 (D) − K(s)kζn us kL2 (D) .

We also use the fact that −1 + i + 2 2 + 2 | − ζn2 ξsi 2u+ s Di us | ≤ K1 (s)|ζn ξs Di us | + K1 (s)ζn (us ) ,

|νs |2`2 − 2cs + Di ηsi ≤ K2 (s). Then we easily see that for dVs -almost all s ∈ (τ1 , τ2 )

+2

Z


+ + ¯ eϕs hns ≤ N k(1 − ζ¯n )u+ s kL2 (D) + k(1 − ζn )Dus kL2 (D) kus kW21 (D)

D


+ 2 i + ¯ + ¯ (u+ s ) ζn ηs Di ζn dx ≤ N k(1− ζn )us kL2 (D) +k(1− ζn )Dus kL2 (D) kus kW21 (D) .

Now (4.2) yields

2 e−ϕτ2 ∧t∨τ1 ku+ ¯ t (n) τ2 ∧t∨τ1 kL2 (D) ≤ m

+N

Z

t 0

+ + ¯ (k(1 − ζ¯n )u+ s kL2 (D) + k(1 − ζn )Dus kL2 (D) )kus kW21 (D) dVs .

(4.3)

The integrals against dVs in (4.3) tend to zero as n→∞ by the dominated convergence theorem. Since the sum of them with continuous local martingales is nonnegative, the local martingales and the right-hand side of (4.3) tend to zero uniformly on finite time intervals in probability (see, for instance, Ref. [13]). So does the left-hand side and the theorem is proved. Proof of Theorem 2.6. Obviously, u bt = ρ t u ¯t satisfies Z t Z t (ϕ, u bt ) = (ϕ, u b0 ) + (ϕ, σsik Di u bs + νsk u bs + ρs g¯sk ) dmks + (ϕ, u ¯s ) dρs 0

+

Z

t

0

0

(ϕ, Di (aij bs ) + bis Di u bs + Di (ais u bs ) − cs u bs + ρs f¯s + ρs Di f¯si ) dVs . s Dj u

We rewrite this equation introducing Vbt = Vt + ρt ,

pt =

dρt , dVbt

i i (b aij ait , bbit , b ct ) = qt (aij t ,b t , at , bt , ct ),

We also set

fbt = qt ρt f¯t + pt u ¯t , m b kt =

Z

t

0

dVt , dVbt

1/2

(b σtik , νbtk ) = qt (σtik , νtk ),

fbti = qt ρt f¯ti ,

qs−1/2 dmks

qt =

1/2

gbtk = qt ρt g¯tk .

(0−1/2 := 0).

(4.4)

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Notice that since dhmk it ≤ dVt = qt dVbt the last integral makes sense. In this notation (2.1) and (4.4) are rewritten as Z t (ϕ, ut ) = (ϕ, u0 ) + (ϕ, σ bsik Di us + νbsk us + qs1/2 gsk ) dm b ks 0

+

Z

t

0

+

bi (ϕ, Di (b aij ais us ) − b cs us + qs fs + qs Di fsi ) dVbs , s Dj us ) + bs Di us + Di (b

Z

t 0

(ϕ, u bt ) = (ϕ, u b0 ) +

Z

t

0

(ϕ, σ bsik Di u bs + νbsk u bs + gbsk ) dm b ks

(ϕ, Di (b aij bs ) + bbis Di u bs + Di (b ais u bs ) − b cs u bs + fbs + Di fbsi ) dVbs , s Dj u

respectively. We subtract these equations, denote vt = ut − u bt , and observe that b for any ω we have dVs -almost everywhere on (τ1 , τ2 ) that Ivs >0 (qs1/2 gsk − gbsk ) = Ivs >0 (qs fsi − fbsi ) = 0,

Ivs >0 (qs fs − fbs ) = qs Ivs >0 (fs − ρs f¯s ) − ps u ¯s Ivs >0 ≤ 0.

We also use the fact that the above versions of equations (2.1) and (4.4) satisfy the same Assumptions 2.1, 2.2, 2.3, and 2.4 with qs ξsi and qs Ki (s) in place of ξsi and Ki (s), respectively. Then we the desired result directly from Theorem 2.5. The theorem is proved. 5. Auxiliary functions Let C[0, ∞) be the set of real-valued continuous functions on [0, ∞). For x· ∈ C[0, ∞) set xs = x0 for s ≤ 0 and for n = 0, 1, 2, ... and t ≥ 0 introduce n/2 ∆− n (x· , t) = 2

osc

[t−2−n ,t]

x· .

If c ∈ (0, ∞), then define Mn− (x· , c, t) = #{k = 0, ..., n : ∆− k (x· , t) ≤ c}. For n negative we set Mn− (x· , c, t) := 0. For c ≥ 0, d > 0, δ > 0 introduce √ √ γ(c, d, δ) = 1 − P ( min wt ≤ −c − d/ 2, max wt ≤ d − d/ 2). t≤δ/2

t≤δ/2

As is easy to see γ(c, d, δ) ≥ P (wt

√ reaches d − d/ 2 before reaching √ √ c + d/ 2 = > 1/ 2, c+d

√ − c − d/ 2)

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so that 2 log2 γ(c, d, δ) > −1. Set, for m = 0, 1, 2, ..., Qm := Qm (x· ) := {(s, y) : s ≥ 0, xs < y < xs + 2−m/2 }. Lemma 5.1. For m = 0, 1, 2, ..., t ≥ 0 and x ∈ (0, 2−m/2 ) introduce √ rm (t, x) = rm (x· , t, x) = P (xt + x + wτ δ = xt−τ + 2−m/2 ), √ where τ = inf{s > 0 : (t − s, xt + x + ws δ) 6∈ Qm }. Then −

−

rm (t, x) ≤ [γ(c, d, δ)]Mm+n (x· ,c,t)−Mm−1 (x· ,c,t)−k ,

where n = n(2

m/2

(5.1)

x/d), k = k(c + d), and

n(y) = [(−2 log2 y)+ ],

k(d) = 2 + [(2 log2 d)+ ].

Proof. Define t¯ = 2m t,

x ¯ = 2m/2 x,

w ¯s = 2m/2 ws2−m ,

x ¯s = 2m/2 xs2−m .

Then as is easy to see rm (t, x) is rewritten as √ P (¯ xt¯ + x ¯+w ¯τ¯ δ = x¯t¯−¯τ + 1), where

(5.2)

√ x· )} = 2m τ. τ¯ = inf{s > 0 : (t¯ − s, x ¯t¯ + x ¯+w ¯s δ) 6∈ Q0 (¯

Since w ¯· is a Wiener process, by Corollary 3.4 in Ref. [14] expression (5.2) is less than −

¯

[γ(c, d, δ)]Mn¯ (¯x· ,c,t)−k , where n ¯ = n(¯ x/d). Here Mn¯− (¯ x· , c, t¯) = #{j = 0, ..., n ¯ : 2j/2 = #{j = 0, ..., n ¯ : 2(j+m)/2

osc

[ t¯−2−j ,t¯ ]

osc

[t−2−j−m ,t ]

x¯· ≤ c}

x· ≤ c}

− − = #{j = m, ..., m + n ¯ : ∆− n (x· , c, t) − Mm−1 (x· , c, t) j (x· ) ≤ c} = Mm+¯

and the result follows. The lemma is proved. Lemma 5.2. Let T ∈ (0, ∞). Assume that 1 − inf Mm (x· , c, t) > α > 0. lim m→∞ m + 1 t∈[0,T ] Take constants p > 0 and ν so that 1 < νp < pχ + 1 < 0, where χ = −2α log2 γ(c, d, δ). Then, for rm from Lemma 5.1 it holds that Z 2−m/2 1 p −m(νp−1)/(2α) r (t, x) dx < ∞. sup sup 2 νp m x m≥0 t∈[0,T ] 0

(5.3)

(5.4)

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Proof. By Lemma 5.1 for a constant N and γ = γ(c, d, δ) −

rm (t, x) ≤ N γ Mm+n −m ,

− − where x ≤ 2−m/2 , n = n(2m/2 x/d), Mm+n = Mm+n (x· , c, t). Furthermore,

n(2m/2 x/d) ≥ (−2 log2 (2m/2 x) + 2 log2 d)+ − 1

≤ (−2 log2 (2m/2 x))+ − N = −m − 2 log2 x − N, where N is a constant. Hence, m + n ≥ −2 log2 x − N . Since obviously rm ≤ 1 we have that rm (t, x) ≤ 1 ∧ (N γ

− −m+M−2 log

2 x−N

).

By the assumption if x is small enough − M−2 log2 x−N > α(−2 log2 x).

Therefore, for x ∈ (0, 2−m/2 ]

rm (t, x) ≤ 1 ∧ (N γ −m−2α log2 x ) = 1 ∧ (N γ −m xχ ).

(5.5)

Next, Z

2−m/2 0

Z ∞ 1 p 1 r (t, x) dx ≤ (1 ∧ (N γ −m xχ ))p dx νp xνp m x 0 Z ∞ 1 (1 ∧ (N xχ ))p dx, = γ m(1−νp)/χ νp x 0

where the last integral is finite owing to (5.3). This proves the lemma. Let wt be a Wiener process with respect to a filtration {Ft , t ≥ 0} of complete σfields and let at and σt be bounded real-valued processes predictable with respect to {Ft , t ≥ 0} and such that at − σt2 ≥ δσt2 , where δ ∈ (0, ∞) is a constant, at − σt2 > 0 for all (ω, t) and for all ω Z ∞ [at − σt2 ] dt = ∞. 0

Set Dx = ∂/∂x. For m = 0, 1, 2, ... we will be dealing with the SPDE dv(t, x) = (1/2)at Dx2 v(t, x) dt + σt Dx v(t, x) dwt

in Bm = (0, ∞) × (0, 2−m/2 ) with boundary conditions v(t, 0) = 0,

v(t, 2−m/2 ) = 1,

v(0, x) = 0,

t > 0,

0 < x < 1.

(5.6) (5.7)

Recall that by Theorem 2.1 in Ref. [14] there is a deterministic function α0 (c), c > 0, such that α0 (c)→1 as c→∞ and with probability one for any T ∈ (0, ∞) lim

inf

n→∞ t∈[0,T ]

1 M − (w· , c, t) = α0 (c). n+1 n

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Theorem 5.3. For each m = 0, 1, 2, ... there is a function vm (t, x) = vm (ω, t, x) ¯m such that defined on Ω × B ¯m , (i) vm (t, x) is Ft -measurable for each (t, x) ∈ B ¯ (ii) vm (t, x) is bounded and continuous in Bm \ {(0, 2−m/2)} for each ω, (iii) derivatives of vm (t, x) of any order with respect to x are continuous in Bm ∪ ({0} × (0, 2−m/2 )) for each ω, (iv) equations (5.6) and (5.7) hold for each ω, (v) almost surely, for any (t, x) ∈ Bm Z t Z t vm (t, x) = (1/2)as Dx2 vm (s, x) ds + σs Dx vm (s, x) dws , 0

0

√ (vi) for any T ∈ (0, ∞), c, d > 0, p > 0, α > 0 such that α0 (c δ) > α, and ν satisfying 1 < νp < χp + 1, where χ = −2α log2 γ(c, d, 1), we have that with probability one Z 2−m/2 1 p −m(νp−1)/(2α) v (t, x) dx < ∞. πT := sup sup 2 νp m x m≥0 t∈[0,T ] 0

(5.8)

(5.9)

Proof. In Lemma 5.1 take δ = 1 and set v˜m (x· , xt + x, t) = rm (x· , t, x), where rm is introduced in that lemma. Set Z t Z ϕt ψt = (as − σs2 ) ds, ξt = σs dws , F˜t = Fϕt , 0

0

v¯m (t, x) = v¯m (ω, t, x) = v˜m (ξ· , t, x),

vm (t, x) = vm (ω, t, x) = v¯m (ψt , x + ξψt ),

where ϕt = inf{s ≥ 0 : ψs ≥ t} is the inverse function to ψt . It is proved in Theorem 4.1 of Ref. [14] that v0 possesses properties (i)-(v). The proof that this is also true for any m is no different. Furthermore, it is well known that √ Z t δ σs dws = w ˜ψ(t) ˜ , 0

where w ˜t is a Wiener process and ψ˜t = δ

Z

t 0

σs2 ds.

Hence ξt = δ −1/2 w ˜ψ(ϕ ˜ t ) with ˜ t ))0 = δσ 2 /(as − σ 2 )|s=ϕ ≤ 1. (ψ(ϕ t s s It follows that for n = 0, 1, 2, ... we have √ ˜ t )), Mn− (ξ· , c, t) ≥ Mn− (w ˜· , c δ, ψ(ϕ

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√ inf Mn− (ξ· , c, t) ≥ inf Mn− (w˜· , c δ, t),

t≤T

t≤T

and with probability one lim inf

n→∞ t≤T

√ 1 Mn− (ξ· , c, t) ≥ α0 (c δ) > α. n+1

Finally, for M = supω,t (at − σt2 ) we have Z 2−m/2 Z 2−m/2 1 p 1 p v (t, x) dx ≤ sup v¯ (t, x + ξt ) dx sup xνp m xνp m t≤M T 0 t≤T 0 = sup t≤M T

Z

2−m/2 0

1 p v˜ (ξ· , t, x + ξt ) dx = sup xνp m t≤M T

Z

2−m/2 0

1 p r (ξ· , t, x) dx. xνp m

After this it only remains to use Lemma 5.2. The theorem is proved. Remark 5.4. Obviously, for any ε ∈ (0, 2−(m+2)/2 ) we have vm (t, ·) ∈ W21 (ε, 2−m/2 − ε)

for any t ∈ [0, ∞) and for any T ∈ (0, ∞) we have Z T kvm (t, ·)kW21 (ε,2−m/2 −ε) dt < ∞. 0

Furthermore, by using the deterministic and stochastic versions of Fubini’s theorem one easily proves that for any ϕ ∈ C0∞ (0, 2−m/2 ) with probability one for all t ∈ [0, ∞) Z t Z t (ϕ, as Dx2 vm (s, ·)) ds + (ϕ, σs Dx vm (s, ·)) dws . (ϕ, vm (t, ·) = (1/2) 0

0

6. Continuity of solutions of SPDEs We take the processes at , σt as before Theorem 5.3 but impose stronger assumptions on them. Assume that there exist constants δ0 , δ1 ∈ (0, 1] such that, for every (ω, t) δ0 ≤ δ1 at ≤ at − σt2 ≤ δ0−1 ,

We will be dealing with solutions ut (x) of dut = ((1/2)at Dx2 ut + ft ) dt + (σt Dx ut + gt ) dwt

(6.1)

on R+ with zero initial condition. To specify the assumptions on f, g and the class of solutions we borrow the Banach spaces Hγp,θ (τ ) and Lp,θ (τ ) from [15]. We also denote by M the operator of multiplying by x. Recall that, for p ≥ 2, 0 < θ < p, the norms in Hγp,θ (τ ), γ = 1, 2, and Lp,θ (τ ) are given by Z τZ ∞ kvkpLp,θ (τ ) = E xθ−1 |v(t, x)|p dxdt, 0

0

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kvkH1p,θ (τ ) = kvkLp,θ (τ ) + kM DxvkLp,θ (τ ) , kvkH2p,θ (τ ) = kvkH1p,θ (τ ) + kM 2 Dx2 vkLp,θ (τ ) . Given p ≥ 2, θ ∈ [p−1, p), any stopping time τ , f ∈ M −1 Lp,θ (τ ), and g ∈ H1p,θ (τ ) by Theorem 3.2 in Ref. [15] equation (6.1) with zero initial condition has a unique solution u ∈ M H2p,θ (τ ) and kM −1 ukH2p,θ (τ ) ≤ N (kM f kLp,θ (τ ) + kgkH1p,θ (τ ) ),

where N = N (p, θ, δ0 , δ1 ). We will also use Theorem 4.7 in Ref. [16], which implies that if u is a solution of (6.1) of class M H2p,θ (τ ) with zero initial condition and f ∈ M −1 Lp,θ (τ ), and g ∈ H1p,θ (τ ) and if there are numbers T ∈ (0, ∞) and β such that 2/p < β ≤ 1,

τ ≤ T,

then for almost any ω the function ut (x) is continuous in (t, x) (that is, has a continuous modification) and E sup sup |xβ−1+θ/p ut (x)|p ≤ N T βp/2 (kM −1 ukH2p,θ (τ ) + kM f kLp,θ (τ ) + kgkH1p,θ (τ ) ), t≤τ x>0

where N = N (p, θ, β, δ0 ). Everywhere below we take p > 2. Theorem 6.1. Let T ∈ (0, ∞), c > 0, α ∈ (0, 1), θ > 0, µ be some constants such √ that α0 (c δ1 ) > α, θ0 < θ < p,

µ < p(1 + 2 log2 γ(c)) − 2 = θ0 − 2 + 2p(1 − α) log2 γ(c),

where γ(c) = γ(c, 1, 1),

θ0 = p(1 + 2α log2 γ(c))

(> 0).

Let f ∈ M −1 Lp,p−1 (T ), g ∈ H1p,p−1 (T ), and let u ∈ M H2p,p−1 (T ) be a solution of (6.1) with zero initial condition. Finally, assume that ft (x) = gt (x) = 0 for x ≥ 1 and f ∈ M −1 Lp,µ (T ), g ∈ H1p,µ (T ). Then there exist stopping times τn ↑ T , defined independently of f and g such that, for each n, u ∈ M H2p,θ (τn ) and kM −1 ukpH2

p,θ (τn )

≤ n(kM f kpLp,µ(τn ) + kgkpH1

p,µ (τn )

)

(6.2)

Here is the result about the continuity of ut (x) we were talking about in the introduction. Remark 6.2. By Theorem 4.7 in Ref. [16] and Theorem 6.1 if we have a number β ∈ (2/p, 1], then there exists a sequence of stopping times τn ↑ T such that E sup sup |x−ε ut (x)|p < ∞, t≤τn x>0

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where ε = 1 − β − θ/p. Due to the freedom of choosing α, β, and θ, the number ε can be made as close from the right as we wish to p 1− lim√ (2 + θ0 )/p = −2/p − 2α0 (c δ1 ) log2 γ(c). α→α0 (c δ1 )

If we allow arbitrary p, then the rate of convergence of ut (x) to zero as x ↓ 0 is almost p xε0 , ε0 = −2α0 (c δ1 ) log2 γ(c) > 0,

which is the same as we obtained for vm (t, x) (see (5.5)). Hence, the presence of f and g does not spoil the situation too much. It is also worth noting that f and g still may blow up near zero even if p is large. When p is large we can take (µ − 1)/p as close to 1 + 2 log2 γ(c) as we wish and then the integral Z 1 xµ−1 |xft (x)|p dx 0

converges if |ft (x)| blows up near x = 0 slightly slower than x−2(1+log2 γ(c)) . Here log2 γ(c)→0 as c→∞ and one can allow |ft (x)| to blow up almost as x−2 . However, when f and g become more irregular near 0, the rate with which the solution goes to zero at 0 deteriorates. In connection with this it is interesting to investigate what happens with ε0 as δ1 ↓ 0. Take an m so large that α0 (m) > 1/2 √ −1/2 and set c = mδ1 − 1/2. Then for δ1 small we have α0 (c δ1 ) > 1/2 and √ √ ε0 ≥ − log2 [1 − P ( min ws ≤ −c − 1/ 2, max ws ≤ 1 − 1/ 2)), s≤1/2

s≤1/2

√ √ ε0 ln 2 ≥ − ln[1 − P ( min ws ≤ −c − 1/ 2, max ws ≤ 1 − 1/ 2)) s≤1/2

s≤1/2

√ √ ∼ P ( min wt ≤ −c − 1/ 2, max ws ≤ 1 − 1/ 2) s≤1/2

s≤1/2

√ √ √ = P ( min wt ≤ −c − 1/ 2) − P ( min wt ≤ −c − 1/ 2, max ws ≥ 1 − 1/ 2) s≤1/2

s≤1/2

s≤1/2

and √ √ P ( min wt ≤ −c − 1/ 2, max ws ≥ 1 − 1/ 2) ≤ 2P ( min wt ≤ −c − 1), s≤1/2

s≤1/2

s≤1/2

so that √ √ P ( min wt ≤ −c − 1/ 2, max ws ≤ 1 − 1/ 2) s≤1/2

s≤1/2

√ ≥ P ( min wt ≤ −c − 1/ 2) − 2P ( min wt ≤ −c − 1). s≤1/2

s≤1/2

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Next, as a→∞ 2 P ( min ws ≤ −a) = P (|w1/2 | ≥ a) = √ π s≤1/2

Z

∞ a

2 2 1 e−x dx ∼ √ a−1 e−a π

and √ lim [P ( min wt ≤ −c − 1/ 2)

δ1 ↓0

s≤1/2

√ 2 √ 1 −2P ( min wt ≤ −c − 1)](c + 1/ 2)e(c+1/ 2) = √ . π s≤1/2

Hence −1/2 m2 /δ1

lim [mδ1

e

δ1 ↓0

1 ε0 ] ≥ √ . π ln 2

This result may seem unsatisfactory since the guaranteed value of ε0 is extremely small when δ1 is small. However, recall that by Remark 4.2 in Ref. [14] the best possible rate with which the solutions go to zero for small δ1 is less than (1 + κ)(2πδ1 )−1/2 e−1/(2δ1 ) , where κ > 0 is any number. To prove Theorem 6.1, first we prove the following. Lemma 6.3. Assume that, for an m = 0, 1, 2, ... we have ft (x) = gt (x) = 0 if x ≤ 2−m/2 . Then almost surely for all t ≤ T and x ∈ (0, 2−m/2 ) |ut (x)| ≤ vm (t, x) sup |us (2−m/2 )|.

(6.3)

s≤t

Proof. By Theorem 4.7 in Ref. [16] the function ut (x) is continuous in [0, T ] × (0, 2−m/2 ) (a.s.) and therefore to prove (6.3) it suffices to prove that for each ε ∈ (0, 2−(m+2)/2 ) almost surely for all t ≤ T and x ∈ D := (ε, 2−m/2 − ε) |uεt (x)| ≤ vm (t, x) sup |uεs (2−m/2 − ε)| =: vm (t, x)ρεt ,

(6.4)

s≤t

where uεt (x) = ut (x − ε). The function uεt satisfies (6.1) with f = g = 0 in (0, T ) × (ε, 2−m/2 + ε) and in (0, T ) × D. Furthermore, (a.s.) for almost any t ∈ (0, T ) we have Dx ut ∈ Lp (D) implying that the limit of uεt (x) as x ↓ ε exists. Since (a.s.) for almost all t ∈ (0, T ) also (x − ε)−1 uεt ∈ Lp (D), the limit is zero. As x ↑ 2−m/2 − ε the situation is simpler and we see that (a.s.) for almost all t ∈ (0, T ) we have lim

D3x→∂D

(uεt (x) − vm (t, x)ρεt )+ = 0.

Furthermore, (a.s.) for almost all t ∈ (0, T ) it holds that uεt ∈ W21 (D) and Z T kuεt k2W 1 (D) dt < ∞. 0

2

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Combining this with Remark 5.4 we see that (a.s.) for almost all t ∈ (0, T ) we have 0

(uεt − vm (t, ·)ρεt )+ ∈ W21 (D), (uεt − vm (t, ·)ρεt )+ ∈ W 12 (D) and Z T k(uεt − vm (t, ·)ρεt )+ k2W 1 (D) dt < ∞. 2

0

By Theorem 2.6 we conclude that almost surely for all t ≤ T and x ∈ D uεt (x) ≤ vm (t, x)ρεt .

By combining this with similar inequality for −uεt we obtain (6.4). The lemma is proved. Proof of Theorem 6.1. Clearly, we only need prove Theorem 6.1 for f and g such that ft (x) = gt (x) = 0 for all ω, t if x is small. Then f ∈ M −1 Lp,ϑ (T ),

g ∈ H1p,ϑ (T )

(6.5)

for any ϑ. According to Lemma 3.6 in Ref. [15], for each stopping time τn ≤ T , we have u ∈ M H2p,θ (τn ) if u ∈ M Lp,θ (τn ) and under this condition the left-hand side of (6.2) is dominated by a constant N = N (θ, p, δ0 , δ1 ) times Z τn Z ∞ p −1 xθ−1 |Ft (x)|p dxdt, (6.6) kM ukLp,θ (τn ) + E 0

0

where

Ft (x) := |xft (x)| + |gt (x)| + |xDx gt (x)|. Observe that obviously (α, γ(c) ≤ 1) θ > θ0 > µ

(6.7)

and since ft (x) = gt (x) = 0 for x ≥ 1, the integral involving Ft will increase if we replace θ with µ. It follows that to prove the theorem, it suffices to estimate only the lowest norm of u, that is to prove the existence of τn ↑ T such that kM −1 ukpLp,θ (τn ) ≤ n[kM f kpLp,µ(τn ) + kgkpH1

p,µ (τn )

].

(6.8)

Next, take a ϑ ∈ [p − 1, p) such that ϑ > θ. For any stopping time τ ≤ T , by Lemma 4.3 in Ref. [15] we have u ∈ M H2p,ϑ (τ ) and by Theorem 3.2 of [15] Z τZ ∞ (6.9) E xϑ−1 |ut (x)/x|p dxdt ≤ N [kM f kpLp,ϑ(τ ) + kgkpH1 (τ ) ], 0

p,ϑ

0

where N = N (p, ϑ, δ0 , δ1 ). As before on the right we can replace ϑ with µ. On the left one can replace ϑ with θ if one restricts the domain of integration with respect to x to x ≥ 1. Therefore (6.8) will be proved if we prove the existence of appropriate stopping times τn such that Z τn Z 1 E xθ−1 |ut (x)/x|p dxdt ≤ n[kM f kpLp,µ(τn ) + kgkpH1 (τn ) ]. (6.10) 0

0

p,µ

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Take a nonnegative η ∈ C0∞ (R+ ) with support in (1, 4) such that the (1/2)periodic function on R ∞ X

η(2x+k/2 )

k=−∞

is identically equal to one. Introduce,

ηm (x) = η(2m/2 x),

(fmt , gmt ) = (ft , gt )ηm .

Also introduce umt as solutions of class M H2p,p−1 (T ) of (6.1) with zero initial condition and fmt and gmt in place of ft and gt , respectively. Since only finitely many fmt and gmt are not zero, we have ∞ X

ut (x) =

umt (x) = I1 (t, x) + I2 (t, x),

m=1

where I1 (t, x) :=

∞ X

umt (x)Ix≤2−m/2 ,

I2 (t, x) :=

m=1

∞ X

umt (x)Ix>2−m/2 .

m=1

Estimating I2 . Take a ϑ as above, set ε = (ϑ−θ)/(2p) and use H¨ older’s inequality to obtain ∞ X |I2 (t, x)|p ≤ 2εpm upmt (x)J p/q (x), m=1

where

J(x) :=

∞ X

m=1

2−εqm Ix>2−m/2 ≤ N x2εq ,

J p/q (x) ≤ N xϑ−θ .

Then use (6.9) again to get E

≤N

E

m=1

≤N where

∞ X

∞ X

m=1

Z

Z

E

τ 0

0

Z

Z

τ Z τ

0

1 0 ∞

0

Z

xθ−1 |I2 (t, x)/x|p dxdt 2m(ϑ−θ)/2 xϑ−1 |umt (x)/x|p dxdt

∞ 0

2m(ϑ−θ)/2xϑ−1 |Fmt (x)|p dxdt,

Fmt (x) = |xfmt (x)| + |gmt (x)| + |xDx gmt (x)|. Here we notice few facts, which will be also used in the future, that on the supports of fmt (x) and gmt (x) we have x ∼ 2−m/2 , 2m(ϑ−θ)/2 Fmt (x) ∼ xθ−ϑ Fmt (x) and Fmt (x) ≤ Ft (x)¯ ηm (x)

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where η¯m (x) = ηm (x) + x2m/2 |η 0 (2m/2 x)|. Notice that the (1/2)-periodic function ∞ X

p η¯m (2y )

m=−∞

is bounded on R. Then we see that Z τZ 1 E xθ−1 |I2 (t, x)/x|p dxdt 0

Z

≤ NE

τ 0

0

Z

∞ 0

xθ−1 |Ft (x)|p

∞ X

p η¯m (x) dxdt

m=1

≤ N [kM f kpLp,θ (τ ) + kgkpH1

p,θ (τ )

]

for any τ ≤ T with a constant N under control. As above we can reduce θ in the last expression to µ. Estimating I1 . Here we will see how τn appear and how we get a substantial drop from θ to µ. We have seen above that the smaller µ is the weaker the statement of the theorem becomes. Therefore, we may concentrate on µ so close to p(1 + 2 log2 γ(c)) − 2 from below that 2 < βp := p(1 + 2 log2 γ(c)) − µ ≤ p. Then 2/p < β ≤ 1. Observe that |I1 (t, x)|p ≤

X ∞

m−q

m=1

m=1

≤ N | log2 x|p

≤N where

∞ X

m=1

E

Z

τ 0

J(τ ) :=

Z

τ 0

Z

2−m/2 0

∞ X

m=1

E

Z

∞ X

m=1

0

It follows that for any θ < θ Z E

p/q X ∞

mp |umt (x)|p Ix≤2−m/2

|umt (x)|p Ix≤2−m/2 .

1 0

xθ−1 |I1 (t, x)/x|p dxdt

| log2 x|p xθ−1 |umt (x)/x|p dxdt ≤ N J(τ ),

τ 0

Z

2−m/2 0

0

xθ −1 |umt (x)/x|p dxdt.

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By Theorem 4.7 in Ref. [16] and Theorem 3.3 in Ref. [17], for any τ ≤ T Z τZ ∞ ε p βp/2 E sup sup |x umt (x)| ≤ N T E xθ1 −1 |Fmt (x)|p dxdt, (6.11) t≤τ x>0

0

0

where N = N (p, δ0 , δ1 , β) and θ1 := p − 1,

ε := β − 1 + θ1 /p = β − 1/p > 0.

Therefore, E sup |umt (2−m/2 )|p ≤ 2m(βp−p+θ1 )/2 E sup sup |xε umt (x)|p t≤τ x>0

t≤τ

Z

≤ NE

τ 0

Z

∞ 0

2m(βp−p+θ1 )/2 xθ1 −1 |Fmt (x)|p dxdt

≤ N 2m(βp−p+1)/2 E

Z

τ 0

Z

∞ 0

|Fmt (x)|p dxdt.

Next, observe that, by Lemma 6.3 for x ∈ [0, 2−m/2 ] and t ≤ T , |umt (x)| ≤ vm (t, x) sup |ums (2−m/2 )|. s≤t

Hence J(τ ) ≤

∞ X

m=1

≤

E sup |umt (2−m/2 )| t≤τ

∞ X

2

m(νp−1)/(2α)

Z

τ 0

Z

2−m/2 0

E sup |umt (2

0

xθ −1 |vm (t, x)/x|p dxdt −m/2

)|

t≤τ

m=1

Z

τ

πt dt, 0

where ν is defined according to νp = p − θ0 + 1 and πt is introduced in Theorem 5.3. So far θ 0 was only restricted to θ 0 < θ, so that νp > 1. Due to the assumption that θ > θ0 one can satisfy θ0 < θ0 < θ in which case (5.8) holds. Then in light of Theorem 5.3 one can find stopping times τn ↑ T such that Z τn πt dt ≤ n. 0

Then

J(τn ) ≤ nN

∞ X

2m(νp−1)/(2α) 2m(βp−p+1)/2 E

m=1

Z

τn 0

As is easy to see the inequalities θ 0 > θ0 and βp − p + (νp − 1)/α < −µ

Z

∞ 0

|Fmt (x)|p dxdt.

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are equivalent. Hence, E

Z

τn 0

≤ nN

Z

∞ X

1

xθ−1 |I1 (t, x)/x|p dxdt ≤ N J(τn )

0

2

m(1−µ)/2

E

m=1

≤ nN

∞ X

E

m=1

≤ nN E

Z

Z

τn 0

τn 0

Z

Z

τn 0

∞ 0

∞ 0

Z

Z

∞ 0

|Fmt (x)|p dxdt

xµ−1 |Fmt (x)|p dxdt

xµ−1 |Ft (x)|p dxdt.

By combining this estimate with the estimate of I2 , noticing that the above constants N are independent of f and g and, if necessary, renumbering the sequence τn we come to (6.10). This proves the theorem. Acknowledgements Several typos in the original version of the article were kindly pointed out by Kyeong-Hun Kim and Hongjie Dong. The author is sincerely grateful for that. Author’s work was partially supported by NSF Grant DMS-0140405. References [1] K.-H. Kim, Lq (Lp ) theory and H¨ older estimates for parabolic SPDEs, Stochastic Process. Appl. 114(2), 313–330, (2004). ISSN 0304-4149. [2] N. V. Krylov and B. L. Rozovski˘ı. On the first integrals and Liouville equations for diffusion processes. In Stochastic differential systems (Visegr´ ad, 1980), vol. 36, Lecture Notes in Control and Information Sci., pp. 117–125. Springer, Berlin, (1981). [3] N. V. Krylov and B. L. Rozovsky, On the characteristics of degenerate second order parabolic itˆ o equations, J. Soviet Math. 32(4), 336–348, (1986). [4] H. Kunita, On backward stochastic differential equations, Stochastics. 6(3-4), 293– 313, (1981/82). ISSN 0090-9491. [5] S. Bonaccorsi and G. Guatteri, Stochastic partial differential equations in bounded domains with Dirichlet boundary conditions, Stoch. Stoch. Rep. 74(1-2), 349–370, (2002). ISSN 1045-1129. [6] S. Bonaccorsi and G. Guatteri. Classical solutions for SPDEs with Dirichlet boundary conditions. In Seminar on Stochastic Analysis, Random Fields and Applications, III (Ascona, 1999), vol. 52, Progr. Probab., pp. 33–44. Birkh¨ auser, Basel, (2002). [7] G. Da Prato and L. Tubaro, Fully nonlinear stochastic partial differential equations, SIAM J. Math. Anal. 27(1), 40–55, (1996). ISSN 0036-1410. [8] L. Tubaro, Some results on stochastic partial differential equations by the stochastic characteristics method, Stochastic Anal. Appl. 6(2), 217–230, (1988). ISSN 0736-2994. [9] K.-H. Kim, On stochastic partial differential equations with variable coefficients in C 1 domains, Stochastic Process. Appl. 112(2), 261–283, (2004). ISSN 0304-4149.

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[10] N. Krylov and B. Rosovskii, Stochastic evolution equations, J.Soviet Mathematics. 16, 1233–1277, (1981). [11] L. C. Evans, Partial differential equations. vol. 19, Graduate Studies in Mathematics, (American Mathematical Society, Providence, RI, 1998). ISBN 0-8218-0772-2. [12] O. A. Ladyzhenskaya and N. N. Ural0 tseva, Linear and quasilinear elliptic equations. Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, (Academic Press, New York, 1968). [13] N. V. Krylov, Introduction to the theory of diffusion processes. vol. 142, Translations of Mathematical Monographs, (American Mathematical Society, Providence, RI, 1995). ISBN 0-8218-4600-0. Translated from the Russian manuscript by Valim Khidekel and Gennady Pasechnik. [14] N. V. Krylov, One more square root law for Brownian motion and its application to SPDEs, Probab. Theory Related Fields. 127(4), 496–512, (2003). ISSN 0178-8051. [15] N. V. Krylov and S. V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients on a half line, SIAM J. Math. Anal. 30(2), 298–325 (electronic), (1999). ISSN 0036-1410. [16] N. V. Krylov, Some properties of traces for stochastic and deterministic parabolic weighted Sobolev spaces, J. Funct. Anal. 183(1), 1–41, (2001). ISSN 0022-1236. [17] N. V. Krylov and S. V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients in a half space, SIAM J. Math. Anal. 31(1), 19–33 (electronic), (1999). ISSN 0036-1410.

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Chapter 13 On Delay Estimation and Testing for Diffusion Type Processes

Yury A. Kutoyants Laboratoire de Statistique et Processus, Facult des Sciences, Universit du Maine Avenue Olivier Messiaen 72085 Le Mans CEDEX 9 [email protected] We present a review of results concerning one example of parameter estimation problem, which is the same time regular and singular depending on the chosen asymptotics. The observed process satisfies the linear stochastic differential equation with the delayed trend coefficient and we are interested in the estimation of this delay. The trend coefficient depends on the unknown parameter (delay) as smooth as the Wiener process depends on time, i.e., we have a continuous but not differentiable w.r.t. parameter statistical model. We study the asymptotic behavior of the estimators and tests in two asymptotics: the first one corresponds to the small noise perturbation, when the diffusion coefficient tends to zero and the second is large samples limit, when the time of observation tends to infinity. We show that the first asymptotics corresponds well to regular statistical experiments situation and the second one is typical for non regular (discontinuous type) statistical models.

Contents 1 2

Introduction . . . . . . . . . . . . . . . . . . . . . Estimation (small noise asymptotics) . . . . . . 2.1 Asymptotic expansion . . . . . . . . . . . . 2.2 Generalizations . . . . . . . . . . . . . . . . 3 Hypotheses Testing (small noise asymptotics) . . 4 Estimation (large samples asymptotics) . . . . . 5 Hypotheses Testing (large samples asymptotics) 6 Discussion . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

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339 343 345 346 348 349 352 353 354

1. Introduction We consider several statistical problems concerning a one dimensional parameter ϑ (called delay parameter) of the stochastic differential equation by the continuous time observations X = {Xt , 0 ≤ t ≤ T } of the solution of stochastic differential equation dXt = S(Xt−ϑ ) dt + σ(Xt ) dWt , 339

Xs , s ≤ 0,

0 ≤ t ≤ T.

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Here S (x) and σ (x) are trend and diffusion coefficients, Wt , 0 ≤ t ≤ T is standard Wiener process and Xs are independent of the Wiener process initial values. We are interested in asymptotic estimation and testing problems in two different asymptotics. The first one corresponds to the dynamical system with small noise (σ → 0), also called perturbed dynamical system and the second asymptotics is typical for classical statistics large samples asymptotics (T → ∞). The first type of asymptotics we realize on the linear model dXt = −γ Xt−ϑ dt + σ dWt ,

Xs = xs , −β ≤ s ≤ 0,

0 ≤ t ≤ T,

(1.1)

and the second asymptotics is studied for the linear ergodic model (T → ∞) dXt = −γ Xt−ϑ dt + σ dWt ,

Xs = xs , −β ≤ s ≤ 0,

0 ≤ t ≤ T.

Therefore we have exactly the same statistical model in both cases. The unknown parameter (delay) ϑ ∈ (α, β) = Θ, where 0 < α < β, γ > 0, and xs , −β ≤ s ≤ 0 is a deterministic function (initial values). Note that this equation has a unique strong solution for all γ, ϑ and σ ( [1], Theorem 4.6). We study the properties of the maximum likelihood ϑbσ,T and Bayesian ϑ˜σ,T

(for quadratic loss function and positive continuous prior density p (·)) estimators defined by the usual equations  
sup L ϑ, X T = L ϑbσ,T , X T ϑ∈Θ

and

ϑ˜σ,T =

Z


R ϑp (ϑ) L ϑ, X T dϑ ϑp (ϑ|X) dϑ = RΘ p (ϑ) L (ϑ, X T ) dϑ Θ Θ

where the log-likelihood ratio is ( [1], Theorem 7.7) Z T Z T
γ γ2 T 2 ln L ϑ, X = − 2 Xt−ϑ dXt − 2 Xt−ϑ dt. σ 0 2σ 0

The properties of the estimators are studied as σ → 0 and T → ∞ separately. We present here a review of some results concerning delay parameter estimation obtained in the works [2, 3] and [4–9]. The further results on parameter estimation for linear stochastic differential equations with delay (in large samples asymptotics) can be found in the works by Gushchin and K¨ uchler [10, 11]. The nonparametric estimation problems in the case of integral-type drift are treated by Reiss [12, 13]. What indeed is surprising with these problems is the possibility to have regular and singular estimation and testing problems for the same model. The first limit (σ → 0) provides us statistically regular estimation and testing problems and the large samples asymptotics (T → ∞) will be similar to the problems of change-point type. Let us remind the difference between regular and singular statistical problems. We recall the well-known facts from classical estimation theory concerning regularity conditions and properties of estimators [6]. Let X n = {X1 , . . . , Xn } be i.i.d. r.v.’s

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and Xj has density function f (ϑ, x), where ϑ ∈ Θ = (α, β). We denote by Pϑ the distribution of X n . If we have to estimate ϑ ∈ Θ by observations X n and describe the properties of estimators as n → ∞, then there are at least two situations: regular and non regular. p In regular situation the function f (ϑ, x) is supposed to be differentiable w.r.t. ϑ in L2 -sense and the Fisher information 2 Z ∞  ∂f (ϑ, x) −1 f (ϑ, x) dx I (ϑ) = ∂ϑ −∞

is function of ϑ. Then the corresponding family of measures n positive continuous o (n) Pϑ , ϑ ∈ Θ is locally asymptotically normal (LAN), i.e., the normalized likelihood ratio admits the representation (n)

Zn (u) =

dPϑ+ √u

n

(n)

dPϑ

  u2 (X n ) = exp u∆n (ϑ) − I (ϑ) + rn , 2

where ∆n (ϑ) ⇒ ∆ (ϑ) ∼ N (0, I (ϑ)) and rn → 0 (see Ref. [14], [15]). Hence   u2 I (ϑ) . Zn (u) =⇒ Z (u) = exp u∆ (ϑ) − 2

The notion of LAN is equivalent to the notion of regular statistical experiment. If the family of measures of any statistical model is LAN, then we have immediately the minimax lower bound on the risk of all estimators (Hajek-Le Cam bound). For the quadratic loss function it is an asymptotically correct version of Rao-Cram´er bound: for all estimators ϑ¯n and ϑ0 ∈ Θ
2 −1 sup n Eϑ ϑ¯n − ϑ ≥ I (ϑ0 ) . (1.2) lim lim δ→0 n→∞ |ϑ−ϑ0 |<δ

In this regular case (under some additional conditions) the maximum likelihood estimator (MLE) ϑbn and the wide class of Bayesian estimators (BE) ϑ˜n are consistent, asymptotically normal   √  √  n ϑbn − ϑ =⇒ u b, n ϑ˜n − ϑ =⇒ u ˜ (1.3)

where

sup Z (u) = Z (b u) , u

and

u b=

  ∆ (ϑ) −1 ∼ N 0, I (ϑ) , I (ϑ)

R   uZ (u) du ∆ (ϑ) −1 ∼ N 0, I (ϑ) . = u ˜= R I (ϑ) Z (u) du

Moreover both are asymptotically efficient in the sense of the bound (1.2) (see Ref. [15]), i.e., for the MLE we have  2 −1 lim lim sup n Eϑ ϑbn − ϑ = I (ϑ0 ) δ→0 n→∞ |ϑ−ϑ0 |<δ

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for all ϑ ∈ Θ. In non regular cases the situations are different. Note that there are many types of non regularity (see, for example, Ref. [15], Chapters 5 and 6). Here we remind one case only. Suppose that we have the shift parameter f (ϑ, x) = f (x − ϑ), where the function f (·) has a jump at point x∗ , i.e., f+ = f (x∗ +), f− = f (x∗ −) and f+ − f− 6= 0. Then the normalized likelihood ratio (for u ≥ 0) (n)

Zn (u) =

dPϑ+ u

n

(n)

dPϑ

  f− (X n ) =⇒ Z (u) = exp ln π+ (u) − u (f− − f+ ) f+

where π+ (u) , u ≥ 0 is Poisson process of intensity f+ . For u ≤ 0 we have ln Z (u) = + π− (−u) − u (f− − f+ ), where π− (u) , u ≥ 0 is Poisson process of intensity ln ff− f− . The MLE and a wide class of Bayesian estimators are consistent, their limit distributions are different     ˜ (1.4) b, n ϑ˜n − ϑ =⇒ u n ϑbn − ϑ =⇒ u where the random variables u b and u ˜ are defined by the relations R uZ (u) du Z (b u) = sup Z (u) , u ˜= R Z (u) du u

and we have the convergence of their moments. The minimax lower bound for all estimators ϑ¯n and ϑ0 ∈ Θ is lim lim

sup

δ→0 n→∞ |ϑ−ϑ0 |<δ

n2 Eϑ ϑ¯n − ϑ


2

≥ E ϑ0 u ˜2 .

(1.5)

Moreover, asymptotically efficient are Bayesian estimators only because we have the strict inequality Eb u2 > E˜ u2 (see Ref. [15], Chapter 5 for details). Having the problem of delay parameter estimation by the observations (1.1) we can ask the first question: to what statistical situation corresponds it, regular or some non regular? Surprising, the answer depends on the type of the limit we have and is the following: if we study the properties of estimators as σ → 0, then the problem of estimation is regular (like (1.2), (1.3)) and if we are interested by large samples properties (T → ∞) then the problem of parameter estimation is non regular in statistical sense (like (1.4),(1.5)). The particularity of this problem can be seen from the integral representation of the trend coefficient Z t−ϑ Xt−ϑ = X0 − γ Xs−ϑ ds + σ Wt−ϑ . 0

Hence Xt−ϑ it is as smooth w.r.t. ϑ as Wiener process w.r.t. time, i.e., we have trend coefficient continuous but not differentiable w.r.t. ϑ for all t ∈ (0, T ].

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2. Estimation (small noise asymptotics) Let us fix the time of observation T and consider the problem of delay estimation by observations dXt = −γ Xt−ϑ dt + σ dWt ,

Xs = xs , −β ≤ s ≤ 0,

0≤t≤T

(2.1)

where the diffusion coefficient σ 2 is supposed to be small, i.e. we study the properties of estimators ϑbσ,T = ϑbσ and ϑ˜σ,T = ϑ˜σ as T is fixed and σ → 0. The model (2.1) can be considered as a small perturbation of the deterministic linear system dxt = −γ xt−ϑ , dt

xs , −β < s ≤ 0,

0 ≤ t ≤ T.

(2.2)

by white Gaussian noise of intensity σ 2 . The behavior of such stochastic systems (without delays) statistical problems for them are well studied (see, e.g., [16], [5]). Remind that if the observed process is dXt = S (ϑ, Xt ) dt + σ dWt ,

X0 = x 0 ,

0≤t≤T

(2.3)

where S (ϑ, x) is a smooth function, then the corresponding family of measures is LAN. The lower (Hajek-Le Cam’s) bound is ¯ 2 Z T ϑσ − ϑ −1 2 lim lim sup Eϑ ≥ I (ϑ0 ) , I (ϑ) = S˙ ϑ (ϑ, xt ) dt. (2.4) σ δ→0 σ→0 |ϑ−ϑ0 |<δ 0 The behavior of estimators is entirely similar to that of the regular case (1.2),(1.3), i.e., the MLE ϑbσ and Bayesian estimators ϑ˜σ are consistent, asymptotically normal   ϑbσ − ϑ −1 =⇒ N 0, I (ϑ) , σ

  ϑ˜σ − ϑ −1 =⇒ N 0, I (ϑ) σ

(2.5)

and asymptotically efficient in the sense of the bound (2.4).

If the trend coefficient is a discontinuous function of ϑ, say, h i dXt = S1 (Xt ) 1{t<ϑ} + S2 (Xt ) 1{t≥ϑ} dt + σdWt , X0 = x0 , 0 ≤ t ≤ T

where S2 (xt ) − S1 (xt ) = Vt 6= 0 and S1 (x) , S2 (x) are smooth functions of x, then the situation changes. The normalized likelihood ratio   2   σ , X L ϑ + uσ 2 |u| Vϑ Zσ (u) = =⇒ Z (u) = exp W (u) − , (2.6) L (ϑ, X σ ) 2 where W (·) is two-sided Wiener process. Let us introduce two random variables u b and u ˜ by the equations R u Z(u) du Z (b u) = sup Z(u), u ˜ = RR . (2.7) u∈R R Z(u) du

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The lower bound on the risk of all estimators is lim lim

sup

δ→0 σ→0 |ϑ−ϑ0 |<δ

σ −4 Vϑ40 E ϑ¯σ − ϑ


2

≥ E˜ u2 .

(2.8)

The MLE ϑbσ and BE ϑ˜σ are consistent, have the following limit distributions   Vϑ2  ˜ Vϑ2  b ϑ − ϑ =⇒ u b , ϑ − ϑ =⇒ u ˜, (2.9) σ σ σ2 σ2 and BE are asymptotically efficient in the sense of this bound (see Ref. [5], Section 5.1). Note, that the random process Z (u) appeared as well in the change point problem for the model of signal in white Gaussian noise dXt = S (t − ϑ) dt + σ dWt ,

X0 = 0,

0≤t≤T

where S (t) has a jump S (t∗ +) − S (t∗ −) 6= 0. In the asymptoics of small noise (σ → 0) it was shown that the bound like (2.8) is valid and the estimators have behavior similar (2.9) (see Ref. [15], Section 7.2). The values Eb u2 and E˜ u2 were 2 calculated by several authors. Terent’ev [17]showed that Eb u = 26 (see also Ref. [15], Section 7.3). Ibragimov and Khasminski [15] obtained the first approximation of the value E˜ u2 by Monte-Carlo simulation. Then Golubev [18] found an integral representation of E˜ u2 , which provided by numerical calculation the value E˜ u2 with higher precision. Finally, Rubin and Song [19] showed that E˜ u2 = 16 ζ (3) , P∞ where ζ (s) = n=1 n−s is Riemann’s zeta function. As 16 ζ (3) = 19, 276 ± 0, 006 we see that Eb u2 > E˜ u2 and the MLE is not asymptotically efficient. In our case of observations (2.1) the derivative w.r.t. ϑ of the trend coefficient −γXt−ϑ does not exist. Nevertheless, the family of measures is LAN, i.e.,   L (ϑ + uσ, X σ ) u2 Zσ (u) = = exp u ∆σ (ϑ) − I (ϑ) + rσ , L (ϑ, X σ ) 2 where ∆σ (ϑ) ∼ N (0, I (ϑ)) and rσ → 0 and the following quantity plays the role of Fisher information: Z T 2 I (ϑ) = γ x2t−2ϑ dt 0

where we suppose for simplicity that the initial values xs , s ≤ 0 are given for s ∈ [−2β, 0]. As the family is LAN we have the following minimax lower bound on the mean square error of all estimators ϑ¯σ (like (1.2),(2.4)). ¯ 2 ϑσ − ϑ −1 lim lim sup E ≥ I (ϑ0 ) . (2.10) σ δ→0 σ→0 |ϑ−ϑ0 |<δ

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Comparison with (1.2), (2.6) shows that I (ϑ) here indeed plays the role of Fisher information. Then we define the asymptotically efficient estimators as estimators for which we have equality in this inequality for all ϑ0 ∈ Θ. Theorem 2.1. The MLE ϑbσ and BE ϑ˜σ are consistent, asymptotically normal   ϑbσ − ϑ −1 =⇒ N 0, I (ϑ) , σ

  ϑ˜σ − ϑ −1 =⇒ N 0, I (ϑ) , σ

(2.11)

we have the convergence of all moments of these estimators and both estimators are asymptotically efficient. Proof can be found in Ref. [4], (see also Ref. [5], Section 2.4). The proofs of this theorem and the Theorems 4 and 5 below consist in the verification of the conditions of two general theorems by Ibragimov and Khasminski [15]. These conditions are given in terms of the normalized likelihood ratios Zσ (·) and ZT (·) respectively. Comparison of (2.6),(2.7) with (2.4),(2.5) shows that we have the regular problem of parameter estimation. 2.1. Asymptotic expansion The asymptotic normality (2.6) can be proved in probability too, i.e., the estimators can be written as ξ1 ϑbσ = ϑ + p σ + o (σ) , I (ϑ)

ξ1 ϑ˜σ = ϑ + p σ + o (σ) , I (ϑ)

where σ −1 o (σ) → 0 in probability and the r.v. Z T γ ξ1 = − p xt−2ϑ dWt ∼ N (0, 1) I (ϑ) 0

is defined on the same probability space. The proof of these representations can be obtained using the method developed in Ref. [5], Section 3.1. It is interesting to see the next after Gaussian terms of these expansions by the powers of σ. In regular problems of parameter estimation by observations of the process (2.3) with sufficiently smooth w.r.t. ϑ and x trend coefficient S (ϑ, x) such expansion for the MLE has the form

ξ1 ϑbσ = ϑ + p σ + a1 ξ12 + a2 ξ1 ξ2 + a3 ξ1 ξ3 + a4 ξ4 σ 2 + o σ 2 , I (ϑ)

where ξ1 , ξ2 , ξ3 are Gaussian r.v.’s and ξ4 is an Itˆ o integral of a Gaussian process. We see that the next after Gaussian term have to be of order σ 2 (see Ref. [5], Section 3.1). The following theorem shows that this term in the case of the model (2.3) is of the order σ 3/2 only.

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Theorem 2.2. The MLE ϑbσ admits the representation √   γ T ζσ sgn (ξ1 ) 3/2 ξ1 b p σ− σ + o σ 3/2 , ϑσ = ϑ + p 3 I (ϑ) |ξ1 | 2 I (ϑ) 4

(2.12)

where ζσ ∼ N (0, 1) and is asymptotically independent of ξ1 ∼ N (0, 1). Proof. The proof can be found in Ref. [6]. It is based on the expansion of the likelihood ratio by the powers of σ
L ϑ + uσ, X T ln Zσ (u) = ln L (ϑ, X T ) Z T Z T γ2 γ 2 [Xt−ϑ−uσ − Xt−ϑ ] dWt − 2 [Xt−ϑ−uσ − Xt−ϑ ] dt =− σ 0 2σ 0 p p u2 I (ϑ) − γ |u| T ζσ σ 1/2 + O (σ) , = u I (ϑ) ξ1 − 2 where Z τ Wt−ϑ−uσ − Wt−ϑ p ζσ = η (t, u, σ) dWt , η (t, u, σ) = T |u| σ 0

and τ is a stopping time

τ = inf



t:

Z

t 0

2



η (s, u, σ) ds ≥ 1 .

For construction of the r.v. ζσ we define η (s, u, σ) = 1 for s ∈ [T, T + 1] and we introduce an independent Wiener process Ws for s ∈ [T, T + 1] (see Ref. [6], proof of the theorem 1.19). Note that in Ref. [7] we used in the expansion the random variable Z T ζ= η (t, u, σ) dWt , 0

which is asymptotically normal, but using the convergence Z T 2 η (t, u, σ) dt −→ 1 0

proved there it is easy to see that ζ − ζσ → 0 in probability. The similar expansion can be found for Bayesian estimator as well, but the final expression is more cumbersome. 2.2. Generalizations

There are several direct generalizations of Theorem 2.1. We can call the model (2.3) as shift-delay and consider as well the problem of scale-delay estimation for the model of observations dXt = −γ Xϑt dt + σ dWt ,

X0 = x 0 ,

0 ≤ t ≤ T,

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where ϑ ∈ (0, 1). The study is quite close to (2.3) and yields the same properties of MLE and BE with the Fisher information Z T t2 x2ϑ2 t dt, I (ϑ) = γ 2 0

i.e., these estimators are consistent, asymptotically normal (2.7) and asymptotically efficient. The proof can be found in Ref. [4], [5], Section 2.4. The asymptotic expansion like (2.8) can be obtained as well. Another generalization can be done for nonlinear system dXt = S (Xt−ϑ , t) dt + σ b (Xt , t) dWt ,

Xs = xs , −β < s ≤ 0,

0≤t≤T

where S (·) is a smooth function, then we have for the same estimators the asymptotic normality (2.7) with the Fisher information Z T 2 0 2 S (xt−2ϑ , t − ϑ) S (xt−ϑ , t) I (ϑ) = dt. 2 b (xt , t) 0 Here xt is solution of the equation dxt = S (xt−ϑ ) , dt

xs , −2β < s ≤ 0,

0≤t≤T

0

∂ and S (x, t) = ∂x S (x, t). The proof can be found in Ref. [2] (see also Ref. [5]). It is interesting to see what happens if we have multiple delays in the linear system

dXt =

k X

λj Xt−τj dt + σ dWt ,

j=1

Xs = xs , −β < s ≤ 0,

0 ≤ t ≤ T,

where ϑ = (λ1 , . . . , λk , τ1 , . . . , τk ) is unknown parameter. bT is consistent, asymptotically normal and asympIt is shown that the MLE ϑ totically efficient [20]. Let us consider the linear stochastic differential equation with integral-type delay ! Z δ dXt = Xt−s µ (ds) dt + σ dWt , 0 ≤ t ≤ T. (2.13) 0

We suppose that the deterministic initial values Xs = xs , −β < s ≤ 0 are given. Here the measure µ (·) is unknown. Then we can study the nonparametric estimation of the function Z δ f (t) = xt−s µ (ds) , 0 ≤ t ≤ T 0

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where xt = xt (µ) is solution of the equation (2.13) with σ = 0. We show that the kernel-type estimator  Z T  1 τ −t b fσ (t) = K dXt ψσ 0 ψσ

with ψσ = σ 2/3 is uniformly consistent lim

sup

 2 sup Eµ fbσ (t) − f (t) = 0

σ→0 µ∈Θ (L) a≤t≤b δ

and asymptotically normal (see Ref. [9] for details). Here 0 < a < b < T and Θδ (L) = {µ : supp (µ) ⊂ [0, δ]

kµk ≤ L} .

Using this nonparametric estimator we can estimate the measure µ (·) in the case Pk (2.4), i.e., when µ = i=1 λi δτ −i . Note that the minimum distance estimator of the parameter ϑ = (λ1 , . . . , λk , τ1 , . . . , τk ) is consistent and asymptotically normal (see Refs. [8, 9]). 3. Hypotheses Testing (small noise asymptotics) The observed process is always dXt = −γ Xt−ϑ dt + σ dWt ,

Xs = x s , s ≤ 0

0≤t≤T

and we have to test the following two hypotheses H0 :

H1 :

ϑ = 0,

(no delay)

ϑ > 0.

Therefor the observed process under hypothesis H0 is Ornstein-Uhlenbeck. The contiguous alternatives correspond to ϑ = uσ and we can rewrite the hypotheses testing problem as : H0 :

H1 :

u = 0,

(no delay)

u > 0,

Let us fix some ε ∈ (0, 1) and denote by Kε the class of tests of asymptotic level 1 − ε. The direct calculation shows that the limit (σ → 0) of the power function of Neyman-Pearson test ϕσ ∈ Kε is n o 1/2 βN P (u) = P ζ > zε − u I0 ,

where ζ ∼ N (0, 1), zε is 1−ε quantil of N (0, 1) and the Fisher information I0 = I (0) is I (0) =


x20 γ 3 1 − e−2γT . 2

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We call a test ϕ◦σ ∈ Kε locally asymptotically uniformly most powerful if its power function βσ◦ (u) = Eu ϕ◦σ (X σ ) satisfies the relation: for any test ϕ¯σ ∈ Kε and any K > 0   lim inf βσ◦ (u) − β¯σ (u) ≥ 0. σ→0 0≤u≤K

Here β¯σ (u) = Eu ϕ¯σ (X σ ). Let us introduce three tests. The first one is score-function test ϕ∗σ (X σ ) = χ{∆

σ >zε }

,

where γ ∆σ = − √ σ I0

Z

T

xt [dXt + γXt dt] . 0

Note that under H0 the statistic ∆σ ∼ N (0, 1). The second test is the likelihood ratio test ϕ˜σ (X σ ) = χ{δ





σ >bε }

,

δσ (X σ ) = sup L (ϑ, X σ ) , ϑ>0

where bε = exp zε2 /2I0 . The third is Wald’s test based on the MLE ϑbσ : ϕ bσ (X σ ) = χnδb

−1/2 σ >zε I0

o,

ϑbσ δbσ (X σ ) = . σ

Theorem 3.1. The tests ϕ∗σ , ϕ˜σ , ϕ bσ belong to Kε and are locally asymptotically uniformly most powerful.

Proof. The proof follows from the local asymptotical normality of the family of measures at the point ϑ = 0 and the standard arguments applied to regular statistical experiments (see, e.g. [21]). It consists in the verification that the power functions of these tests converge uniformly on compacts [0, K] to the limit power function βN P (u) of the Neyman-Pearson test. 4. Estimation (large samples asymptotics) Let us consider the same linear stochastic differential equation dXt = −γ Xt−ϑ dt + σ dWt ,

Xs = xs , −β < s ≤ 0,

0≤t≤T

but now we study the properties of MLE ϑbσ,T = ϑbT and BE ϑ˜σ,T = ϑ˜T of the delay ϑ inthe asymptotics of large samples, i.e., as T → ∞. Suppose that γ > 0, π ϑ ∈ 0, 2γ = Θ, (β = π/2γ), then the process Xt has ergodic properties [22]. Remind the properties of parameter estimators for ergodic diffusion processes dXt = S (ϑ, Xt ) dt + dWt ,

X0 ,

0 ≤ t ≤ T.

(4.1)

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If the function S (ϑ, x) is smooth w.r.t. ϑ, then the family of measures is LAN and we have the lower bound on the risks of all estimators : Z
2 −1 2 sup T Eϑ ϑ¯T − ϑ ≥ I (ϑ0 ) , I (ϑ) = S˙ (ϑ, x) f (ϑ, x) dx, lim lim δ→0 T →∞ |ϑ−ϑ0 |<δ

where f (ϑ, x) is the density of invariant law. The properties of estimators are typical for regular experiments, i.e., the MLE ϑbT and BE ϑ˜T are consistent, asymptotically normal       √  √  −1 −1 T ϑbT − ϑ =⇒ N 0, I (ϑ) , T ϑ˜T − ϑ =⇒ N 0, I (ϑ) ,

and asymptotically efficient (see Ref. [6]). The properties of estimators change if the trend coefficient is not a smooth function. For example, if S (ϑ, x) = S(x − ϑ), where S (x) has a finite jump at some point x∗ , i.e.; S (x√ ∗ +) − S (x∗ − +) 6= 0, then the rate of convergence of estimators is T and not T . To illustrate this let us consider a simple switching ergodic diffusion process dXt = −sgn (Xt − ϑ) dt + dWt ,

X0 ,

0 ≤ t ≤ T.

(4.2)

To describe the properties of estimators we reintroduce the random process (2.6)   1 u ∈ R, Z(u) = exp W (u) − |u| , 2 where W (·) is two-sided Wiener process and corresponding two random variables u b and u ˜ defined by the equations (2.7). Note that the normalized likelihood ratio
u , XT L ϑ + 4T =⇒ Z (u) (4.3) ZT (u) = L (ϑ, X T ) and this (together with some other estimates on the likelihood ratio) provides us the lower bound like (2.8)
2 sup 16 T 2 Eϑ ϑ¯T − ϑ ≥ E˜ u2 . (4.4) lim lim δ→0 T →∞ |ϑ−ϑ0 |<δ

Moreover, the MLE and BE are consistent, have the following limit distributions     4 T ϑbT − ϑ =⇒ u b, 4 T ϑ˜T − ϑ =⇒ u ˜, (4.5) and the BE are asymptotically efficient in the sense of this bound (see [6], Section 3.4).

Let us return to delay estimation problem. First we verify that the normalized likelihood ratio     L ϑ + γ 2uT , X T |u| =⇒ Z (u) = exp W (u) − . (4.6) ZT (u) = L (ϑ, X T ) 2

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Then we obtain the lower bound: for all estimators ϑ¯T we have
2 lim lim sup γ 4 T 2 Eϑ ϑ¯T − ϑ ≥ E˜ u2 .

(4.7)

δ→0 T →∞ |ϑ−ϑ0 |<δ

Therefore we can define the asymptotically efficient estimators as estimators for which we have equality in this inequality for all ϑ0 ∈ Θ.

Theorem 4.1. The MLE ϑbT and BE ϑ˜T are consistent, have the following limits in distribution     γ 2 T ϑbT − ϑ =⇒ u b, γ 2 T ϑ˜T − ϑ =⇒ u ˜ (4.8)

and BE are asymptotically efficient.

For the proof see K¨ uchler and Kutoyants [3] and Kutoyants [6], Section 3.3. Remind how the weak convergence of the random function ZT (·) to Z (·) yields the weak convergence (4.8) of the MLE [15]. Let us denote ϑ0 the true value of the parameter. Then we can write    o n  z (T ) (T ) 2 b b P ϑ0 γ T ϑT − ϑ 0 < z = P ϑ0 ϑT < ϑ 0 + 2 γ T ( )

(T ) T T = P ϑ0 sup L ϑ, X > sup L ϑ, X ϑ<ϑ0 + γ 2zT

ϑ≥ϑ0 + γ 2zT


) L ϑ, X L ϑ, X T = sup > sup T T ϑ<ϑ0 + γ 2zT L (ϑ0 , X ) ϑ≥ϑ0 + γ 2zT L (ϑ0 , X )     (T ) (T ) = Pϑ0 sup ZT (u) > sup ZT (u) = Pϑ0 sup ZT (u) − sup ZT (u) > 0 (T ) P ϑ0

(



u

T

u≥z

−→ P sup Z (u) − sup Z (u) > 0 u
u≥z



u
u≥z

= P {b u < z} ,

where we changed the variable ϑ = ϑ0 + γ 2uT and used the convergence (4.6). Comparison of (4.7), (4.8) with (4.4), (4.5) shows that the delay estimation problem is singular and similar to space change point estimation problem. Let us suppose that the observed process 2π < s ≤ 0, 0 ≤ t ≤ T β   has two unknown parameters γ ∈ (α, β) , α > 0 and τ ∈ 0, 2π and we have to estiβ mate the both parameters simultaneously, i.e. we have two-dimensional parameter ϑ = (γ, τ ). We need the further notation. dXt = −γ Xt−τ dt + σ dWt ,

• normalizing matrix ϕT =

Xs = x s , −



T −1/2 , 0 0 , T −1



.

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• Fisher information of the component γ Z ϑ) = I (ϑ

∞ 0

x2t dt

where xt is fundamental solution of the deterministic equation 2π dxt = −γ xt−τ , xs = 0, − < s < 0, x0 = 1, dt β     b = vb, γub2 and w ˜ = vb, γu˜2 where the random variables • the random vectors w u b and ˜ are the b and u ˜ Gaussian r.v.  u  same as before and vb is independent of u −1 ϑ) N 0, I (ϑ

 The  lower bound is given by the following inequality: for any ϑ 0 ∈ Θ = (α, β) × 2π ¯ 0, β and any estimator ϑ T lim lim

sup

δ→0 T →∞ |ϑ−ϑ0 |<δ

 2  E˜ u2 −1 ¯ − ϑ ≥ I (ϑ ϑ ) + ϑ Eϑ ϕ−1 . T T γ4

We call an estimator ϑ∗ asymptotically efficient if for all ϑ0 ∈ Θ T

  2 E˜ u2 ϑ)−1 + 4 . Eϑ ϕ−1 ϑ∗ − ϑ = I (ϑ T T δ→0 T →∞ |ϑ−ϑ0 |<δ γ   b = γ ˜ = (˜ The asymptotic properties of the MLE ϑ bT , τbT and BE ϑ γT , τ˜T ) T T are described in the following theorems. lim lim

sup

b and BE ϑ ˜ of the parameter ϑ are consistent, have Theorem 4.2. The MLE ϑ T T different limit distributions     −1 ˜ b − ϑ =⇒ w, b ˜ ϕ−1 ϑ ϕ ϑ − ϑ =⇒ w, T T T T     e − ϑ p converge. b − ϑ p and Eϑ ϕ−1 ϑ for any p > 0 the moments Eϑ ϕ−1 ϑ T T T T Moreover, the BE are asymptotically efficient. For the proof see Ref. [6], Section 3.3. From this theorem it follows that the parameters γ and τ are estimated with the different rates, say, √ u b T (b γT − γ) =⇒ vb, T (b τT − τ ) =⇒ 2 . γ

5. Hypotheses Testing (large samples asymptotics) The observed process is dXt = −γ Xt−ϑ dt + σ dWt ,

Xs = x s , s ≤ 0

0≤t≤T

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and we have to test the following two hypotheses H0 : H1 :

ϑ = 0,

(no delay)

ϑ > 0,

The contiguous alternatives correspond to ϑ = u/γT : H0 : H1 :

u = 0,

(no delay)

u > 0,

We have no asymptotically uniformly most powerful test and describe the likelihood ratio test only. This test is defined as follows


L ϑ, X T T T δT X = sup . ϕ˜T X = χ δ > 1 , T { T ε} ϑ>0 L (0, X )

Its properties are given in the following theorem.

Theorem 5.1. The likelihood ratio test ϕ˜T belongs to Kε and its power function   1 u + o (1) , (5.1) βT (u, ϕ bT ) = P ζ + max [η, ξ] > ln − ε 2 where the random variables ζ, η and ξ are independent and

ζ ∼ N (0, u) , Fξ (x) = 1 − e−x , x ≥ 0    √  √  x u u x + e−x 1 − Φ √ − , Fη (x) = Φ √ + 2 2 u u

(5.2) (5.3)

where Φ (x) is Gaussian N (0, 1) distribution function. Proof. Remind that the normalized likelihood ratio ZT (u) converges to the process Z (u) (see (4.6)), hence    
1 1 T = P0 sup ZT (u) > P 0 δT X > ε ε u≥0      h ui 1 1 = P sup W (u) − > ln =ε −→ P sup Z (u) > ε 2 ε u≥0 u≥0   because the random variable ξ = supu≥0 W (u) − u2 has exponential distribution. Therefore ϕ˜T ∈ Kε . The proof of (5.1)-(5.3) can be found in Ref. [6], Section 5.2. 6. Discussion The difference of the properties of estimators and tests in these two different limits can be explain as follows. The properties of estimators depend strongly of the local

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structure of the likelihood ratio. Let us write it as (  Z T  Xt−ϑ−ϕσ,T u − Xt−ϑ L (ϑ + ϕσ,T u, X) = exp −γ dWt Zσ,T (u) = L (ϑ, X) σ 0 2 ) Z  γ 2 T Xt−ϑ−ϕσ,T u − Xt−ϑ dt , − 2 0 σ where ϕσ,T → 0. Using equation (1.1) we obtain the representation Z t−ϑ−ϕσ,T u Xt−ϑ−ϕσ,T u − Xt−ϑ γ =− Xs−ϑ ds + Wt−ϑ−ϕσ,T u − Wt−ϑ . σ σ t−ϑ If we put ϕσ,T = σ → 0, then

Xt−ϑ−ϕσ,T u − Xt−ϑ −→ γ u xt−2ϑ σ because the stochastic process Xt converges uniformly in t ∈ [0, T ] to the deterministic function xt = xt (ϑ), which is solution of the differential equation dxt = −γ xt−ϑ , xs , s ≤ 0. dt Therefore the normalized likelihood ratio converges ) ( Z Z T u2 γ 2 T 2 xt−2ϑ dt Zσ (u) =⇒ Z (u) = exp uγ xt−2ϑ dWt − 2 0 0 and we have regular estimation and testing problems. The contribution of the term Wt−ϑ−ϕσ,T u − Wt−ϑ in this case is asymptotically negligible. If we fix σ and consider T → ∞ with ϕσ,T = T −1 , then Z t−ϑ− Tu Xt−ϑ− Tu − Xt−ϑ γ =− Xs−ϑ ds + Wt−ϑ− Tu − Wt−ϑ σ σ t−ϑ r u γu Xt−2ϑ + η (t, ϑ, u) , ∼− T T 2

where Eϑ η (t, ϑ, u) = 1. We see that the main part of the contribution is due to the Wiener process and as the Wiener process is not differentiable w.r.t. time, the statistical problems became non regular. References [1] R. S. Liptser and A. N. Shiryaev, Statistics of random processes. I. vol. 5, Applications of Mathematics (New York), (Springer-Verlag, Berlin, 2001), expanded edition. ISBN 3-540-63929-2. General theory, Translated from the 1974 Russian original by A. B. Aries, Stochastic Modelling and Applied Probability. [2] G. T. Apoyan, An estimate for the parameter of a nondifferentiable drift coefficient, Erevan. Gos. Univ. Uchen. Zap. Estestv. Nauki. 161(1), 33–42, (1986). ISSN 01320173.

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[3] U. K¨ uchler and Y. A. Kutoyants, Delay estimation for some stationary diffusion-type processes, Scand. J. Statist. 27(3), 405–414, (2000). ISSN 0303-6898. [4] Y. A. Kutoyants, An example of an estimate for the parameter of a nondifferentiable drift coefficient, Teor. Veroyatnost. i Primenen. 33(1), 188–192, (1988). ISSN 0040361X. [5] Y. Kutoyants, Identification of dynamical systems with small noise. vol. 300, Mathematics and its Applications, (Kluwer Academic Publishers Group, Dordrecht, 1994). ISBN 0-7923-3053-6. [6] Y. A. Kutoyants, Statistical inference for ergodic diffusion processes. Springer Series in Statistics, (Springer-Verlag London Ltd., London, 2004). ISBN 1-85233-759-1. [7] Y. A. Kutoyants, On delay estimation for stochastic differential equations, Stoch. Dyn. 5(2), 333–342, (2005). ISSN 0219-4937. [8] Y. Kutoyants and T. Mourid, Estimation dans un mod`ele autor´egressif avec retards, C. R. Acad. Sci. Paris S´er. I Math. 315(4), 455–458, (1992). ISSN 0764-4442. [9] Y. Kutoyants and T. Mourid, Estimation par la distance minimale pour un processus de type diffusion avec retards, Publ. Inst. Statist. Univ. Paris. 38(2), 3–18, (1994). [10] A. A. Gushchin and U. K¨ uchler, Asymptotic inference for a linear stochastic differential equation with time delay, Bernoulli. 5(6), 1059–1098, (1999). ISSN 1350-7265. [11] A. A. Gushchin and U. K¨ uchler, On parametric statistical models for stationary solutions of affine stochastic delay differential equations, Math. Methods Statist. 12 (1), 31–61, (2003). ISSN 1066-5307. [12] M. Reiß, Minimax rates for nonparametric drift estimation in affine stochastic delay differential equations, Stat. Inference Stoch. Process. 5(2), 131–152, (2002). ISSN 1387-0874. [13] M. Reiss, Adaptive estimation for affine stochastic delay differential equations, Bernoulli. 11(1), 67–102, (2005). ISSN 1350-7265. [14] J. H´ ajek. Local asymptotic minimax and admissibility in estimation. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. I: Theory of statistics, pp. 175–194, Berkeley, Calif., (1972). Univ. California Press. [15] I. A. Ibragimov and R. Z. Has0 minski˘ı, Statistical estimation. vol. 16, Applications of Mathematics, (Springer-Verlag, New York, 1981). ISBN 0-387-90523-5. Asymptotic theory, Translated from the Russian by Samuel Kotz. [16] M. I. Freidlin and A. D. Wentzell, Random perturbations of dynamical systems. vol. 260, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (Springer-Verlag, New York, 1984). ISBN 0-387-90858-7. Translated from the Russian by Joseph Sz¨ ucs. [17] A. S. Terent’yev, Probability distribution of a time location of an absolute maximum at the output of a sinchronized filter, Radioengineering and Electronics. 13(4), 652– 657, (1968). [18] G. K. Golubev, Computation of efficiency of maximum-likelihood estimate when observing a discontinuous signal in white noise, Problems Inform. Transmission. 15(3), 61–69, (1979). [19] H. Rubin and K. S. Song, Exact computation of the asymptotic efficiency of maximum likelihood estimators of a discontinuous signal in a Gaussian white noise, Ann. Statist. 23(3), 732–739, (1995). ISSN 0090-5364. [20] Y. A. Kutoyants, T. Mourid, and D. Bosq, Estimation param´etrique d’un processus de diffusion avec retards, Ann. Inst. H. Poincar´e Probab. Statist. 28(1), 95–106, (1992). ISSN 0246-0203. [21] G. G. Roussas, Contiguity of probability measures: some applications in statistics.

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(Cambridge University Press, London, 1972). Cambridge Tracts in Mathematics and Mathematical Physics, No. 63. [22] U. K¨ uchler and B. Mensch, Langevin’s stochastic differential equation extended by a time-delayed term, Stochastics Stochastics Rep. 40(1-2), 23–42, (1992). ISSN 10451129.

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Chapter 14 On Cauchy-Dirichlet Problem for Linear Integro-Differential Equation in Weighted Sobolev Spaces Remigijus Mikulevicius and Henrikas Pragarauskas∗ Department of Mathematics, University of Southern California Los Angeles, CA 90089-2532 [email protected] We study the Cauchy-Dirichlet problem in a smooth bounded domain for linear parabolic integro-differential equations. Sufficient conditions are derived under which the problem has a unique solution in weighted Sobolev classes. The result can be used in the regularity analysis of certain functionals arising in the theory of Markov processes.

Contents 1 2 3

Introduction . . . . . . . . Notation and main result . Proof of the main results . 3.1 Proof of Theorem 2.1 3.2 Proof of Theorem 2.2 References . . . . . . . . . . . .

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357 359 361 365 368 374

1. Introduction The paper is devoted to the Cauchy-Dirichlet problem for integro-differential equations associated with d-dimensional Markov process Xts,x , t ≥ s, defined by Ito stochastic differential equation dXt = σ(t, Xt ) dWt + b(t, Xt ) dt +

Z

y q(dt, dy) + |y|≤1

Z

y p(dt, dy), (1.1) |y|>1

Xs = x, where Wt is a standard d-dimensional Wiener process, p(dt, dy) is the jump measure of Xt with the compensator π(t, Xt , dy)dt, and q(dt, dy) = p(dt, dy) − π(t, Xt , dy)dt is the corresponding martingale measure. A very simple example is the process Xt ∗ Institute

of Mathematics and Informatics, Akademijos 4, Vilnius, Lithuania, [email protected] 357

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satisfying the equation dXt = σ(t, Xt ) dWt + b(t, Xt ) dt + dZt ,

(1.2)

Xs = x, where Zt is a d-dimensional α-stable process, α ∈ (0, 2). In this case π(t, x, dy) = dy/|y|d+α . In many problems arising in the theory of Markov processes it is important to study the smoothness properties of the functionals Z τ s,x ∧T v(s, x) = E f (t, Xts,x ) dt, s

where τ s,x is the first exit time of the process Xts,x from a domain G ⊆ Rd . If v is a sufficiently smooth function, it is a solution to Cauchy-Dirichlet problem ∂t u + Au + f = 0, in (0, T ) × G, u(T, x) = 0, x ∈ Rd ,

(1.3)

u(t, x) = 0, t ∈ [0, T ], x ∈ / G, where 2 Au(t, x) = aij (t, x)∂ij u(t, x) + bi (t, x)∂i u(t, x)

+

Z

[u(t, x + y) − u(t, x) − ∂i u(t, x)yi 1{|y|≤1} ] π(t, x, dy),

aij (t, x) = (1/2)σ(t, x)σ ∗ (t, x), 1{|y|≤1} is the indicator function of {y ∈ Rd : |y| ≤ 1}, and the implicit summation convention over repeated indices is assumed. The problem of such type (including the case of nonlinear equations) was considered by a number of authors (see e.g. Refs. [1–4], and references therein) in Sobolev and H¨ older spaces under certain restrictive assumptions on π(t, x, dy). In [3], only finite number of jumps can occur outside of D in finite time, and in Ref. [2] very restrictive assumptions on the order of jumps are imposed. In Ref. [5], general results were obtained in weighted Sobolev classes for a differential operator L. We extend some results in Ref. [5] for the integro-differential operator Lu and show for a bounded domain G ⊆ Rd that the assumptions on π(t, x, dy) can be considerably relaxed. The results in Refs. [3] and [2] do not apply for the equation (1.2) which is obviously covered by our Theorem 2.1. The results obtained can be also used in the analysis of non-linear equations arising in the optimal control theory of Markov processes. The main result of the paper is presented in Section 2 and proved in Section 3.

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2. Notation and main result We will consider a bounded domain G in Rd of class Cu1 . In other words (see Ref. [5]), there are two numbers r0 , K0 ∈ (0, ∞) and an increasing function ω0 (ε) on [0, ∞) such that ω0 (ε) ↓ 0 as ε ↓ 0 so that for each x0 ∈ ∂G, there exists a oneto-one continuously differentiable mapping Ψ of Br0 (x0 ) = {y ∈ Rd : |y − x0 | < r0 } onto a domain D ⊆ Rd having the following properties: (i) D+ = Ψ (Br0 (x0 ) ∩ G) ⊆ Rd+ = {x ∈ Rd : x1 > 0} and Ψ(x0 ) = 0; (ii) Ψ (Br0 (x0 ) ∩ ∂G) = D ∩ {x ∈ Rd : x1 = 0}; (iii) |Ψ(x)| + |∂Ψ(x)| ≤ K0 on Br0 (x0 ) and |Ψ−1 (y1 ) − Ψ−1 (y2 )| ≤ K0 |y1 − y2 | for all y1 , y2 ∈ D; (iv) for x1 , x2 ∈ Br0 (x0 ), we have |∂Ψ(x1 ) − ∂Ψ(x2 )| ≤ ω0 (|x1 − x2 |). Denote ρ(x) =dist(x, ∂G), x ∈ G; ρ(x) = 0, if x ∈ / G. For p > 1, θ ∈ R, define Lp,θ (G) = Lp (G; ρθ−d (x) dx) the space of all measurable functions u(x) on G such that Z |u(x)|p ρθ−d (x) dx < ∞. G

γ Hp,θ (G) be γ γ

Let the space of all measurable function u(x) on G such that u, ρ∂u, . . . ρ ∂ u ∈ Lp,θ (G), where γ = 0, 1, 2. R p p |u(x)| dx. We mostly are interested in the spaces , |u| = Denote δ = θ−d p,G p G 0 1 2 Hp,θ+p (G), Hp,θ (G), Hp,θ−p (G) with the norms 1+δ 0 |u|Hp,θ+p |p,G , (G) = |uρ δ 1+δ 1 (G) = |uρ |p,G + |∂uρ |u|Hp,θ |p,G , −1+δ 2 |u|Hp,θ−p |p,G + |∂uρδ |p,G + |∂ 2 uρ1+δ |p,G . (G) = |uρ γ Let T > 0. We introduce the following spaces Hp,θ (T, G), γ = 0, 1, 2, of functions u = u(t, x) with the norm (Z )1/p T

γ |u|Hp,θ (T,G) =

0

|u(t, ·)|pH γ

p,θ (G)

dt

.

1 2 Let Hp,θ (T, G) be the space of all functions u ∈ Hp,θ−p (T, G) such that

u(t, x) =

Z

t 0

f (s, x) ds, 0 ≤ t ≤ T,

0 in a generalized sense, where f ∈ Hp,θ+p (T, G), which means that for all ϕ ∈ C0∞ (G),

Z

u(t, x)ϕ(x) dx =

Z tZ 0

f (s, x)ϕ(x) dxds, t ≤ T.

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1 For u ∈ Hp,θ (T, G), we define 2 0 |u|H1p,θ (T,G) = |u|Hp,θ−p (T,G) + |f |Hp,θ+p (T,G) .

Let us introduce the operators 2 Lu(t, x) = aij (t, x)∂ij u(t, x) + bi (t, x)∂i u(t, x) − r(t, x)u(t, x), (t, x) ∈ [0, T ] × G,

where a = (aij )1≤i,j≤d is symmetric and nonnegative definite, and Z ˜(t, x)π(t, x, dy), (t, x) ∈ [0, T ] × G, Iu(t, x) = ∇2y u where u ˜(t, x) =



u(t, x), (t, x) ∈ [0, T ] × G, 0, 0 ≤ t ≤ T, x ∈ / G,

and ∇2y u ˜(t, x) = u ˜(t, x + y) − u ˜(t, x) − ∂i u ˜(t, x)yi 1{|y|≤1} . The summation convention that repeated indices indicate summation from 1 to d is followed here as it will throughout. We will need the following assumptions. A1. The domain G is bounded of class Cu1 (see Ref. [5]) A2. There exist two positive numbers 0 < c < C such that c|ξ|2 ≤ aij ξi ξj ≤ C|ξ|2 for all ξ ∈ Rd , and lim

sup |aij (t, x) − aij (t, y)| = 0.

|x−y|→0 0≤t≤T

A3. The functions bi (t, x), r(t, x) are measurable and bounded; A4. Let p > 1, d − 1 < θ < d − 1 + p. The following two types of assumptions will be used for the measurable family of non-negative measures π(t, x, dy) on Rd \{0}. B. (The case of a dominating measure). There are measures π ¯ (t, dy), t ∈ [0, T ] such that π(t, x, dy) ≤ π ¯ (t, dy), t ∈ [0, T ], x ∈ G, and Z sup |y|2 ∧ 1¯ π (t, dy) < ∞, 0≤t≤T

lim sup

ε→0 0≤t≤T

Z

|y|≤ε

|y|2 π ¯ (t, dy) = 0.

C. (“general case”) There is a constant N such that Z |y|2 ∧ 1 π(t, x, dy) ≤ N

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and lim

sup

ε→0 0≤t≤T,x∈G

Z

|y|≤ε

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|y|2 π(t, x, dy) = 0.

We consider a linear equation ∂t u(t, x) = Lu(t, x) + Iu(t, x) + h(t, x), (t, x) ∈ [0, T ] × G,

(2.1)

u(0, x) = 0.

1 A function u ∈ Hp,θ (T, G) is called a solution of (2.1), if for every ϕ ∈ C0∞ (G) and all t ∈ [0, T ], Z Z tZ u(t, x)ϕ(x) dx = [Lu(s, x) + Iu(s, x) + h(s, x)]ϕ(x) dxds. 0

Our main results are the following two statements. The first one holds for the case of a dominating measure. Theorem 2.1. Assume the assumptions A1-A4 and B are satisfied. Then for each 0 1 h ∈ Hp,θ+p (T, G) there is a unique solution u ∈ Hp,θ (T, G) of (2.1). Moreover, there is a constant C independent of h such that 2 0 |u|Hp,θ−p (T,G) ≤ C|h|Hp,θ+p (T,G) .

The next statement holds under the assumption C for π(t, x, dy). Theorem 2.2. Assume p > d, θ ≤ p and the assumptions A1-A4 and C hold. 0 1 Then for each h ∈ Hp,θ+p (T, G) there is a unique solution u ∈ Hp,θ (T, G) of (2.1). Moreover, there is a constant C independent of h such that 2 0 |u|Hp,θ−p (T,G) ≤ C|h|Hp,θ+p (T,G) .

3. Proof of the main results Consider a partial differential equation ∂t u(t, x) = Lu(t, x) + h(t, x), (t, x) ∈ [0, T ] × G,

(3.1)

u(0, x) = 0, x ∈ G.

According to Theorem 2.10 in Ref. [5], the following statement holds. 0 Theorem 3.1. Assume A1- A4 are satisfied. Then for each h ∈ Hp,p+θ (T, G) there 1 is a unique solution u ∈ Hp,θ (T, G) of (3.1). Moreover, NT 2 0 |u|Hp,θ−p |h|Hp,θ+p (T,G) ≤ N e (T,G) ,

where N is independent of T and h.

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2 0 Assume V t : Hp,θ−p (G) → Hp,p+θ (G), t ≥ 0, is a family of linear operators such that the following assumption holds. 2 L. (i) For any u ∈ Hp,θ−p (T, G), the function V u(t, x) = V t u(t, x) is a measurable function on GT = [0, T ] × G (ii) For each ε > 0 there is a constant µε such that 0 2 1 (G) , |V t u|Hp,θ+p (T,G) ≤ ε|u|Hp,θ−p (G) + µε |u|Hp,θ

2 for all u ∈ Hp,θ−p (G), t ≥ 0. We will show that under the assumptions A1-A4 and L the equation

∂t u(t, x) = Lu(t, x) + V t u(t, x) + h(t, x), (t, x) ∈ [0, T ] × G,

(3.2)

u(0, x) = 0, x ∈ G.

1 has a unique solution u ∈ u ∈ Hp,θ (T, G). According to Lemma 2.6 in Ref. [5], there is a function ψ ∈ C ∞ (G) such that ¯ |∂ψ(x)| ≥ 1 on ∂G; (i) ψ and ∂ψ are continuous on G, (ii) ψ(x) > 0 for any x ∈ G, and ψ = 0 on ∂G; for each ε > 0 the function ψ is bounded away from zero on the set {x ∈ G : ρ(x) ≥ ε}; (iii) for any integer k ≥ 0 we have

sup ρk (x)|∂ k+1 ψ(x)| < ∞.

(3.3)

G

(iv) there are two constants 0 < N1 < N2 such that N1 ρ(x) ≤ ψ(x) ≤ N2 ρ(x), x ∈ G.

(3.4)

Remark 3.2. a) Since ρ and ψ are comparable ((3.4) holds), we obtain equivalent norms of the spaces if the distance ρ is replaced by its regularized version ψ. b) The inequality (3.4) implies that for each ε > 0 there is a constant Cε such that ε 1 ≤ + Cε ψ. (3.5) ψ 1 We will need the following estimate of Hp,θ (G) norm. 2 Lemma 3.3. For each ε > 0 there is a constant Cε such that for all v ∈ Hp,θ−p (G) 1 (G) ≤ ε|v|H 2 |v|Hp,θ + Cε |vρ1+δ |p,G . p,θ−p (G)

Proof.

2 Let v ∈ Hp,θ−p (G) and u = vψ 1+δ . We have

∂i u = ∂i vψ 1+δ + (1 + δ)vψ δ ∂i ψ, 2 2 ∂ij u = ∂ij vψ 1+δ + (1 + δ)∂i vψ δ ∂j ψ + (1 + δ)∂j vψ δ ∂i ψ

2 +(1 + δ)δvψ δ−1 ∂j ψ∂i ψ + (1 + δ)vψ δ ∂ij ψ.

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For each ε > 0 there is a constant Cε such that |∂u|p,G ≤ ε|∂ 2 u|p,G + Cε |u|p,G . Therefore, using the properties of ψ, for each ε we can find a constant Cε > 0 such that |∂i u|p,G ≤ ε |∂ 2 vψ 1+δ |p,G + |∂i vψ δ |p,G + |vψ δ−1 |p,G




+Cε |vψ 1+δ |p,G . By (3.5), for each ε > 0 there is a constant Cε such that δ 1+δ 1 (G) ≤ C(|vψ |p,G + |∂vψ |v|Hp,θ |p,G ) ≤ C(|vψ δ |p,G + |∂u|p,G ) 1+δ 2 ≤ ε|u|Hp,θ−p |p,G , (G) + Cε |uψ

and the statement follows.



Now we prove the existence and uniqueness of the solution to (3.2). Proposition 3.4. Assume A1- A4 and L are satisfied. Then for each h ∈ 0 1 Hp,p+θ (T, G) there is a unique solution u ∈ Hp,θ (T, G) of (3.2). Moreover, 2 0 |u|Hp,θ−p (T,G) ≤ C|h|Hp,θ+p (T,G) ,

where C is independent of h. Proof. For τ ∈ [0.1] define the operators Aτ u = Lu + τ V u. 1 Hp,θ (T, G) solves the equation

Assume u ∈

∂t u(t, x) = Aτ u(t, x) + h(t, x), (t, x) ∈ [0, T ] × G,

(3.6)

u(0, x) = 0, x ∈ G,

0 where h ∈ Hp,p+θ (T, G). We will prove that there is a constant independent of h, u and τ such that 2 0 |u|Hp,θ−p (T,G) ≤ C|h|Hp,θ+p (T,G) ,

(3.7)

Then by Theorem 3.1 , there is a constant C independent of τ, h and u such that for all t ≤ T |u|pH 2

p,θ−p (t,G)

≤ C(|h|pH 0

p,θ+p (t,G)

+ |V u|pH 0

p,θ+p (t,G)

).

Therefore, by Lemma 3.3, there is a constant C such that for all t ≤ T, |u|pH 2

p,θ−p (t,G)

≤ C(|h|pH 0

p,θ+p (t,G)

+ |ρ1+δ u|pp,Gt ),

(3.8)

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RtR where |f |pp,Gt = 0 G |f (s, x)|p dxds. On the other hand, multiplying both sides of the equation (3.6) by ρ1+δ we have, by Lemma 3.3, (3.8) and assumption L, |ρ1+δ u(t, ·)|pp,G ≤C

Z

t

[

2 X

0 k=0

≤C ≤C

Z

Z

|ρ1+δ ∂ k u(s, ·)|pp,G + |ρ1+δ V s u(s, ·)|pp,G + |ρ1+δ h(s, ·)|pp,G ] ds t

p,θ−p (G)

0 t 0

[u(s, ·)|pH 2

+ |ρ1+δ u(s, ·)|pp,G + |ρ1+δ h(s, ·)|pp,G ] ds

[|ρ1+δ u(s, ·)|pp,G + |ρ1+δ h(s, ·)|pp,G ] ds,

t ≤ T. Therefore (3.7) follows by Gronwall’s lemma and (3.8). 1 0 Consider the mappings Tτ : Hp,θ (T, G) → Hp,θ+p (T, G) defined by Z t u(t, x) = f (s, x) ds 7→ f − Aτ u, 0

where f ∈

0 Hp,θ+p (T, G).

Obviously, there is a constant independent of τ such that 0 |Tτ u|Hp,θ+p (T,G) ≤ C|u|H1p,θ (T,G) .

On the other hand, there is a constant C independent of τ such that for all u ∈ 1 Hp,θ (T, G), 0 |u|H1p,θ (T,G) ≤ C|Tτ u|Hp,θ+p (T,G) .

Rt 0 1 (T, G), u ∈ Hp,θ (T, G)) we have Indeed, for u(t, x) = 0 f (s, x) ds (f ∈ Hp,θ+p ∂t u = Aτ u + (f − Aτ u) and by (3.7), 0 0 |u|H1p,θ (T,G) ≤ C(|f − Aτ u|Hp,θ+p (T,G) + |Aτ u|Hp,θ+p (T,G) )

0 2 ≤ C(|f − Aτ u|Hp,θ+p (T,G) + |u|Hp,θ−p (T,G) )

0 0 ≤ C|f − Aτ u|Hp,θ+p (T,G) = C|Tτ u|Hp,θ+p (T,G) .

Since by Theorem 3.1, T0 is an onto map, all Tτ , τ ∈ [0, 1] are onto maps as well by Theorem 5.2 in Ref. [6].  We will show that under the assumption B or C, Z 0 ˜(x)π(t, x, dy) ∈ Hp,θ+p (G), I t u(x) = Iu(t, x) = ∇2y u 2 if u ∈ Hp,θ−p (G), and I t u, t ≥ 0, satisfies the assumption L.

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3.1. Proof of Theorem 2.1 The following lemma shows that under B, the assumptions L for holds for Iu. Therefore Theorem 2.1 follows immediately by Proposition 3.4. 2 Lemma 3.5. Assume B holds and u ∈ Hp,θ−p (G). Then for each ε > 0 there is Cε such that for all t ≥ 0 2 1 (G) . |ρ1+δ I t u|p,G ≤ ε|u|Hp,θ−p (G) + Cε |u|Hp,θ

Proof.

2 Assume u ∈ Hp,θ−p (G). Then

Iu(x) =

Z

... + Γ1ε

Z

...+ Γ2ε

Z

. . . = (I1 + I2 + I3 )u(x), |y|>ε

where Γ1ε = {y ∈ Rd : x + y ∈ G, |y| ≤ ε}, Γ2ε = {y ∈ Rd : x + y ∈ / G, |y| ≤ ε}. 10 . We start with I1 u and split it into two integrals I1 u(x) =

Z

... = Γ1ε

Z

... + Γ11 ε

Z

... = (I11 + I12 )u(x), Γ12 ε

1 12 1 where Γ11 ε = Γε ∩ {y : |y| ≤ ρ(x)}, Γε = Γε ∩ {y : |y| > ρ(x)}. For any τ ∈ (0, 1), y ∈ Γ11 ε we have ρ(x + τ y) ≥ ρ(x) − τ |y| ≥ (1 − τ )ρ(x). Therefore

ρ(x) ≤ (1 − τ )−1 . ρ(x + τ y) If ε is sufficiently small, ρ1+δ (x)|∇2y u(t, x)| = ρ1+δ (x)| ≤ N |y|2

Z

Z 1

0

1 0

2 (1 − τ )∂ij u(x + τ y)yi yj dτ |

(1 − τ )|(ρ1+δ ∂ 2 u)(x + τ y)|



ρ(x) ρ(x + τ y)

1+δ

dτ.

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By H¨ older inequality, |ρ1+δ I11 u(x)|pp,G ≤N ≤N ×

Z (Z G

Z Z G

(Z

Γ11 ε

≤ N |ρ where δq =

Γ11 ε

Γ11 ε

Z

1 0

1 0

(1 − τ )

(1 − τ )

0

∂

u|pp,G

and

−δ

|(ρ

1+δ 2

2

∂ u)(x + τ y)| dτ |y| π ¯ (t, dy)

)p

dx

|(ρ1+δ ∂ 2 u)(x + τ y)|p dτ |y|2 π ¯ (t, dy)

1

1+δ 2

θ−d p

Z

Z

p p−1

−δq

2

dτ |y| π ¯ (t, dy)

(Z

|y|≤ε

=

θ−d p−1

2

) pq

|y| π ¯ (t, dy)

dx

)1+ pq Z

1 0

(1 − τ )

−δq

dτ

 pq

,

< 1, because θ < d − 1 + p. Thus

2 |ρ1+δ I11 u|p,G ≤ N γ(ε)|ρ1+δ ∂ 2 u|p,G ≤ N γ(ε)|u|Hp,θ−p (G) ,

where γ(ε) = sup t>0

Z

2

|y|≤ε

|y| π ¯ (t, dy)

Z

1 0

(1 − τ )

−δq

dτ

 pq

→ 0,

as ε → 0, by assumption B. If y ∈ Γ12 ε , we have ρ(x + y) ≤ ρ(x) + |y| ≤ 2|y|, and the estimate is straightforward: ρ1+δ (x)|u(x + y)| ≤ ρ1+δ (x)ρ1−δ (x + y)|(ρδ−1 u)(x + y)| ≤ 2|y|2 |(ρδ−1 u)(x + y)|, ρ1+δ (x)|u(x)| ≤ ρ2 (x)|(ρδ−1 u)(x)| ≤ |y|2 |(ρδ−1 u)(x)|, ρ1+δ (x)|∂u(x)||y| ≤ |y|2 |(ρδ ∂u)(x)|, and |ρ

1+δ

I12 u(x)|pp,G

≤

C|u|pH 2 p,θ−p (G)

Z

2

|y|≤ε

20 . Consider now I2 u. Obviously, |I2 u(x)| ≤

|u(x)|¯ π (t, Γ2ε )

+ |∂u(x)|

Z

Γ2ε

|y| π ¯ (t, dy)

!p

|y| π ¯ (t, dy).

.

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If y ∈ Γ2ε , then ε ≥ |y| > ρ(x) and Z

1+δ −1+δ δ | ρ I2 u (x)| ≤ C |(ρ u)(x)| + |(ρ ∂u)(x)| sup t

|y|≤ε

|y|2 π ¯ (t, dy).

Thus

2 |ρ1+δ I2 u|p,G ≤ Cκ(ε)|u|Hp,θ−p (G) ,

R where κ(ε) = supt |y|≤ε |y|2 π ¯ (t, dy). 0 3 . Finally, we handle I3 u. It follows that Z |I3 u(x)| ≤ |u(x + y)|¯ π (t, dy) + |u(x)|¯ π (t, {|y| > ε}) |y|>ε

+|∂u(x)|

Z

First we estimate Ju(x) =

1≥|y|>ε

Z

|y|>ε

|y|¯ π (t, dy).

|u(x + y)|¯ π (t, dy).

For ε0 > 0 and z ∈ G we have

|u(z)| = |u(z)ρ−1+δ (z)|ρ1−δ (z)1{ρ(z)≤ε0 } + |u(z)ρδ (z)|ρ−δ (z)1{ρ(z)>ε0 } δ ≤ ε1−δ |u(z)ρ−1+δ (z)| + ε−δ 0 0 |u(z)ρ (z)|.

Therefore, Ju(x) ≤

Z

|y|>ε


δ |u(x + y)| ε1−δ ρ−1+δ (x + y) + ε−δ ¯ (t, dy). 0 0 ρ (x + y) π

By Minkowski’s inequality, Z |Ju|p,G ≤ sup t>0

|y|>ε

δ π ¯ (t, dy) ε1−δ |uρ−1+δ |p,G + ε−δ 0 0 |uρ |p;G


δ ≤ N ε−2 ε1−δ |uρ−1+δ |p,G + ε−δ 0 0 |uρ |p;G .




Since G is bounded,

|ρ1+δ Ju|p,G ≤ N |Ju|p,G . The remaining terms J2 u(x) = |u(x)|¯ π (t, {|y| > ε} , J3 u(x) = |∂u(x)|

Z

1≥|y|>ε

|y|¯ π (t, dy)

are estimated in the following way: ρ1+δ (x)J2 u(x) ≤ N ρδ (x)|u(x)|¯ π (t, {|y| > ε}) ≤ N ε−2 |(ρδ u)(x)|, ρ1+δ (x)J3 u(x) ≤ N ε−1 |(ρ1+δ ∂u)(x)|.

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Therefore, 1 (G) , |ρ1+δ J2 u|p,G ≤ N ε−2 |u|Hp,θ

1 (G) . |ρ1+δ J3 u|p;G ≤ N ε−1 |u|Hp,θ

By choosing in these estimates a small ε > 0 and then a small ε0 we obtain the statement  So, the statement of Theorem 2.1 follows by Proposition 3.4. 3.2. Proof of Theorem 2.2 We will use the maximal functions. So, the function u needs to be bounded. The following embedding theorem holds in weighted Sobolev spaces. Proposition 3.6. (see Proposition 2.2 in Ref. [7]) If γp > d, then θ

γ |ρ p u|∞,G ≤ N |u|Hp,θ (G) .

In particular, θ

2 |ρ p −1 u|∞,G ≤ N |u|Hp,θ−p (G) ,

¯ if p ≥ θ. and u is bounded on G, Denote Ty u(x) =

|∇2y u(x)| , x ∈ G, y ∈ Rd , |y|2

and define the maximal function Mf (x) = sup R−d R>0

Lemma 3.7. Let u ∈ C0∞ (G), p > all x ∈ G sup |y|≤ 14 ρ(x)

Proof.

d 2

Z

f (x + z) dz. |z|≤R

∨ 1. Then there is a constant N such that for



|Ty u(x)|p ≤ N ρ−(1+δ)p (x)[M |∂ 2 uρ1+δ |p (x) + M |∂uρδ |p (x)
+M |uρ−1+δ |p (x)].

According to [4] and [1], for all x, y ∈ Rd ,
p (Ty u(x)) ≤ N M |∂ 2 u|p (x).

Fix x ∈ G. Let ϕ ∈ C0∞ (Rd ), ϕ(z) = 1, if |z| ≤ 12 , ϕ(z) = 0, if |z| > 1 and ϕr (z) = ϕ(

z−x ). r

(3.9)

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Obviously, Ty u(x) = Ty (uϕr ) (x), if |y| ≤ 1 4 ρ(x)

1 2 r.

By (3.9), we have for |y| ≤

1 2r

≤


p p (Ty u(x)) = (Ty (uϕr )(x)) ≤ N M |∂ 2 (uϕr |p )(x)


≤ N M |∂ 2 uϕr |p + |∂u · ∂ϕr |p + |u∂ 2 ϕr |p (x).

For r = 12 ρG (x), 2

M |∂ uϕr |

p




(x) = sup R

−d

R>0

≤ sup R−d R>0

Z

Z

|z|≤R

|∂ 2 u(x + z)|p ϕpr (x + z) dz

|z|≤R∧r

|(∂ 2 uρ1+δ )(x + z)|p ρ−(1+δ)p (x + z) dz


≤ N ρ−(1+δ)p (x)M |∂ 2 uρ1+δ |p (x),

because ρ(x + z) ≥ 21 ρ(x), |z| ≤ r = 12 ρ(x). Similarly we get Z M (|∂u · ∂ϕr |p ) (x) ≤ N r−p sup R>0

|(∂uρδ )(x + z)|p ρ−δp (x + z) dz

|z|≤R∧r


≤ N ρ−(1+δ)p (x)M |∂uρδ |p (x),

and


M |u∂ 2 ϕr |p (x) ≤ N r−2p sup R−d R>0

Z

|z|≤R∧r

|(uρ−1+δ )(x + z)|p ρ(1−δ)p (x + z) dz


≤ N ρ−(1+δ)p (x)M |uρ−1+δ |p (x).



Using the properties of maximal functions (see Ref. [8]) we obtain the following obvious statement. 2 Corollary 3.8. If u ∈ Hp,θ−p (G), p >

d 2

∨ 1, then

2 |ρ1+δ T u|p,G ≤ N |u|Hp,θ−p (G) ,

where T u(x) =

sup |y|≤ 14 ρ(x)

Proof. For each p > By Lemma 3.7, ρ(1+δ)p¯(x)

d 2

sup |y|≤ 14 ρ(x)

∨ 1, there are p¯ >

d 2

Ty u(x).

∨ 1 and ε > 0 such that p = p¯(1 + ε).



|Ty u(x)|p¯ ≤ N [M |∂ 2 uρ1+δ |p¯ (x) + M |∂uρδ |p¯ (x)
+M |uρ−1+δ |p¯ (x)].

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Therefore Z

(ρ(1+δ)p¯(x)

|y|≤ 41 ρ(x)

G

≤ C(

Z

sup

2

G

|∂ uρ

1+δ p(1+ε) ¯

|

|Ty u(x)|p¯)1+ε dx

dx +

Z

δ p(1+e) ¯

G

|∂uρ |

dx +

Z

G

¯ |uρ−1+δ |p(1+ε) dx).

 In order to prove that the assumption L holds for I t u, the following estimate is needed as well. Lemma 3.9. Assume θ ≤ p, r > 0, u ∈ C0∞ (G). Then there is a constant N such that for all x ∈ G sup |y|>r, ρ(x+y)>0


|u(x + y)|p ≤ N r−(1+δ)p [M |∂ 2 uρ1+δ |p (x) 2p |y|

+M |∂uρδ |p (x) + M |uρ−1+δ |p (x)].

Proof.

Fix x ∈ G and denote U (y) = u(x + y). Then sup |y|>r, ρ(x+y)>0

1 |u(x + y)| = sup 2 sup |U (ty)|, |y|2 t>r t y∈Γt |y|=1

where Γt = {y : ρ(x + ty) > 0}. For y ∈ Γt , let ρt (y) = dist(y, ∂Γt ). Notice that dist(y, ∂Γt ) =

inf

z,ρ(x+tz)=0

|y − z| =

1 inf |ty − tz| t z,ρ(x+tz)=0

=

1 inf |(x + ty) − (x + tz)| t z,ρ(x+tz)=0

=

1 ρ(x + ty). t

(3.10)

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Let ϕ ∈ C0∞ (Rd ), ϕ ≥ 0, ϕ(y) = 1, if |y| ≤ 1, and ϕ(y) = 0, if |y| > 2. By Proposition 3.6 for U (ty)ϕ(y), y ∈ Γt , we have 2 |U (t·)|∞;B1 ∩Γt ≤ |U (t·)ϕ|∞,Γt ≤ N |U (t·)ϕ|Hp,θ−p (Γt )

≤ N [|∂ 2 (U (t·)ϕ)ρ1+δ |p,Γt + |∂(U (t·)ϕ)ρδt |p,Γt t +|U (t·)ϕρ−1+δ |p,Γt ] t ≤ N [|∂ 2 U (t·)ρ1+δ |p,B2 ∩Γt + |∂U (t·)ρδt |p,B2 ∩Γt t +|U (t·)ρ−1+δ |p,B2 ∩Γt ], t where BR = {y : |y| < R} . Using the change of variable ty = z, we obtain |∂

2

U (t·)ρ1+δ |pp,B2 ∩Γt t

=

Z

B2 ∩Γt

t2p |∂ 2 u(x + ty)|p t−(1+δ)p ρ(1+δ)p (x + ty) dy

= t(1−δ)p t−d

and

B2t ∩G

|∂ 2 u(x + z)|p ρ(1+δ)p (x + z) dz


≤ N t(1−δ)p M |∂ 2 uρ1+δ |p (x),

|∂U (t·)ρδt |pp,B2 ∩Γt =

Z

=t

Also,

Z

tp |∂u(x + ty)|p t−δp ρδp (x + ty) dy

B2 ∩Γt

(1−δ)p −d

t

Z

B2t ∩G

|∂u(x + z)|p ρδp (x + z) dz


≤ N t(1−δ)p M |∂uρδ |p (x).

|U (t·)ρ−1+δ |pp,B2 ∩Γt = t

Z

=t

B2 ∩Γt

|u(x + ty)|p t(1−δ)p ρ(−1+δ)p (x + ty) dy

(1−δ)p −d

t

Z

B2t ∩G

|u(x + z)|p ρ(−1+δ)p (x + z) dz


≤ N t(1−δ)p M |uρ−1+δ |p (x).

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These estimates and (3.10) imply that sup |y|>r, ρ(x+y)>0

|u(x + y)|p 1 = sup 2p |U (t·)|p∞;S1 ∩Γt |y|2p t>r t ≤ sup t>r

1 N t(1−δ)p (M... + M... + M...) t2p

≤ N r−(1+δ)p (M... + M... + M...) and the statement follows.



The Lemma below completes the proof of Theorem 2.2. It shows that I t u satisfies the assumption L. 2 Lemma 3.10. Assume u ∈ Hp,θ−p (G), p > d, θ ≤ p and the assumption C holds. Then for each ε > 0 there is a constant Cε > 0 such that 2 1 (G) . |ρ1+δ Iu|p,G ≤ ε|u|Hp,θ−p (G) + Cε |u|Hp,θ

Proof.

Fix ε > 0 and split the integral Z Z ∇2y u(x)π(t, x, dy) + Iu(x) = |y|≤ε∧ 41 ρ(x)

... + ε≥|y|>ε∧ 14 ρ(x)

Z

... |y|>ε

= I1 u(x) + I2 u(x) + I3 u(x). 10 . Estimate of I1 u. Obviously, |I1 u(x)| ≤ γ(ε) R

|∇2y u(x)| , |y|2 |y|≤ε∧ 1 ρ(x) sup 4

|y|2 π(t, x, dy). By Lemma 3.7, |y|≤ε

p |ρ1+δ (x)I1 u(x)|p ≤ (N γ(ε)) [M |∂ 2 uρ1+δ |p (x) + M |∂uρδ |p (x)

where γ(ε) = supt>0,x∈G


+M |uρ−1+δ |p (x)]

for p > d2 ∨ 1. Therefore using properties of the maximal functions (see Ref. [8] and the proof of Corollary 3.8 above) 2 |ρ1+δ I1 u|p;G ≤ N γ(ε)|u|Hp,θ−p (G) .

20 . Estimate of I2 u. We have Z |I2 u(x)| ≤ |˜ u(x + y)|π(t, x, dy)| + |u(x)|π(t, x, Γε (x)) Γε (x)

+|∂u(x)|

Z

Γε (x)

|y|π(t, x, dy)

= I21 u(x) + I22 u(x) + I23 u(x),

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where Γε (x) = {y : ε ≥ |y| > 41 ρ(x)}. We have Z |u(x + y)| |y|2 π(t, x, dy) I21 u(x) ≤ sup |y|2 y∈Γε (x), |y|≤ε ρ(x+y)>0

≤ N γ(ε)

sup |y|> 41 ρ(x), ρ(x+y)>0

|u(x + y)| . |y|2

Therefore by Lemma 3.9,



|ρ1+δ (x)I21 u(x)|p ≤ (N γ(ε))p [M |∂ 2 uρ1+δ |p (x) + M |∂uρδ |p (x)
+M |uρ−1+δ |p (x)],

and by properties of maximal functions (see Ref. [8] and the proof of Corollary 3.8 above)) 2 |ρ1+δ I21 u|p;G ≤ N γ(ε)|u|Hp,θ−p (G) .

Obviously,

|I22 u(x)| + |I23 u(x)| ≤ N γ(ε) ρ−2 (x)|u(x)| + ρ−1 (x)|∂u(x)| and therefore




|ρ1+δ I22 u|p,G + |ρ1+δ I23 u|p,G ≤ N γ(ε) |ρ−1+δ u|p,G + |ρδ ∂u|p,G




2 ≤ N γ(ε)|u|Hp,θ−p (G) .

Thus

2 |ρ1+δ I2 u|p;G ≤ N γ(ε)|u|Ho,θ−p (G) .

30 . Estimate of I3 u. We have Z |I3 u(x)| ≤ |u(x + y)|π(t, x, dy) + |u(x)|π(t, x, {|y| > ε}) |y|>ε

+|∂u(x)|

Z

1≥|y|>ε

|y|π(t, x, dy)

(3.11)

≤ N (ε−2 |u|∞;G + ε−1 |∂u(x)|).

For ε0 > 0 and z ∈ G,

θ

θ

|u(z)| = |u(z)ρ p −1 (z)|ρ1− p (z)1{ρ(z)≤ε0 } θ

θ

+|u(z)ρ p (z)|ρ− p (z)1{ρ(z)>ε0 } θ 1− p

≤ ε0

θ

−θ

θ

|uρ p −1 |∞;G + ε0 p |uρ p |∞;G .

(3.12)

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It follows from (3.11) and (3.12) that 1− θp

|ρ1+δ (x)I3 u(x)| ≤ N ε−2 ε0

θ

|uρ p −1 |∞;G ρ1+δ (x)

−θ

θ

+N ε−2 ε0 p |uρ p |∞;G ρ1+δ (x) +N ε−1 |∂u(x)|ρ1+δ (x). By embedding theorem (we use p > d when γ = 1), θ 1− p

|ρ1+δ I3 u|p;G ≤ N ε−2 ε0

−θ

−2 2 1 (G) |u|Hp,θ−p ε0 p |u|Hp,θ (G) + N ε

1 (G) . +N ε−1 |u|Hp,θ

Choosing a small ε > 0 and then a small ε0 we obtain our statement.



Acknowledgement We are very grateful to our referee for valuable comments. References [1] J.-M. Bony, Probl`eme de Dirichlet et semi-groupe fortement fell´erien associ´es a ` un op´erateur int´egro-differentiel, C. R. Acad. Sci. Paris S´er. A-B. 265, A361–A364, (1967). [2] F. Gimbert and P.-L. Lions, Existence and regularity results for solutions of secondorder, elliptic integro-differential operators, Ricerche Mat. 33(2), 315–358, (1984). [3] M. G. Garroni and J.-L. Menaldi, Green functions for second order parabolic integrodifferential problems. vol. 275, Pitman Research Notes in Mathematics Series, (Longman Scientific & Technical, Harlow, 1992). ISBN 0-582-02156-1. [4] R. Mikuleviˇcius and H. Pragarauskas, On the Cauchy problem for certain integrodifferential operators in Sobolev and H¨ older spaces, Liet. Mat. Rink. 32(2), 299–331, (1992). ISSN 0132-2818. [5] K.-H. Kim and N. V. Krylov, On the Sobolev space theory of parabolic and elliptic equations in C 1 domains, SIAM J. Math. Anal. 36(2), 618–642 (electronic), (2004). ISSN 0036-1410. [6] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order. vol. 224, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (Springer-Verlag, Berlin, 1983), second edition. ISBN 3-54013025-X. [7] S. V. Lototsky, Dirichlet problem for stochastic parabolic equations in smooth domains, Stochastics Stochastics Rep. 68(1-2), 145–175, (1999). ISSN 1045-1129. [8] E. M. Stein, Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30, (Princeton University Press, Princeton, N.J., 1970).

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Chapter 15 Strict Solutions of Kolmogorov Equations in Hilbert Spaces and Applications Giuseppe Da Prato Scuola Normale Superiore 56126, Pisa, Italy [email protected] Consider a stochastic differential equation in a Hilbert space H, with the associated differential operator K0 determined by the coefficients in the equation, and the corresponding Markov semigroup Pt . The paper investigates the problem of constructing a core for the (weak) generator K of Pt in the space of real, uniformly continuous, and bounded functions on H, and studies the relation between K and K0 . The paper proposes an approach based on the concept of a strict solution of the corresponding Kolmogorov equation: a suitable assumption of existence of strict solutions imply good results on the core, and the assumption is verified in an SPDE example.

Contents 1 2 3 4

Introduction . . . . . . . . . . . . Existence of cores . . . . . . . . . Invariant measures . . . . . . . . . Application . . . . . . . . . . . . . 4.1 Estimates for Xx (t, x) . . . 4.2 Estimates for Xxx (t, x) . . . 4.3 Estimates of T R[Xx,x (t, x)] 4.4 Estimates of Pt ϕ . . . . . . References . . . . . . . . . . . . . . . .

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375 379 381 383 384 386 387 388 390

1. Introduction We are concerned with a stochastic differential equation in a separable Hilbert space H (norm | · |, inner product h·, ·i) of the following form   dX(t) = (AX(t) + b(X(t))dt + σ(X(t))dW (t), t ≥ 0, (1.1)  X(0) = x ∈ H,

where A : D(A) ⊂ H→H is the infinitesimal generator of a strongly continuous semigroup etA in H and the mappings b : D(b) ⊂ H→H and σ : H→L(H) are (generally) nonlinear. Moreover, W (t) is a cylindrical Wiener process in H. 375

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We shall assume that problem (1.1) has a unique mild solution X(t, x) (a concept that has to be made precise in any specific situation) which is adapted and continuous in mean square. We shall denote by Pt the transition semigroup Pt ϕ(x) = E[ϕ(X(t, x))],

ϕ ∈ Cb (H), x ∈ H,

(1.2)

where Cb (H) denotes the Banach space of all real uniformly continuous and bounded functions in H endowed with the norm kϕk0 = sup |ϕ(x)|, x∈H

ϕ ∈ Cb (H).

The Kolmogorov equation corresponding to (1.1) looks like  1   ut (t, x) = Tr [(σσ ∗ )(x)uxx (t, x)] + hAx + b(x), ux (t, x)i,   2          

for x ∈ D(A) ∩ D(b),

u(0, x) = ϕ(x),

(1.3)

x ∈ H.

As well known, a candidate for the solution of (1.3) is given by u(t, x) = E[ϕ(X(t, x))],

t ≥ 0, x ∈ H.

Let us give some notations. We shall denote by Cb1 (H) the space of all real functions in H which are continuous together with their Fr´echet derivatives of the first order. It is a Banach space with the norm kϕk1 = kϕk0 + kϕx k0 . Spaces Cbk (H), k > 1, are defined similarly. Moreover we denote by {ek } a complete orthonormal system in H. Definition 1.1. A strict solution of (1.3) is a function u : [0, ∞) × H→R such that (i) u is continuous in [0, ∞) × H and u(0, x) = ϕ(x), x ∈ H. (ii) For all t ≥ 0, u(t, ·) ∈ Cb1 (H) and there exist the second derivatives of u, ux,x(t, ·)(h, k) in all directions h, k ∈ H. (iii) For all t > 0 the function ∞ X H→R, x 7→ Tr [(σσ ∗ )(x)uxx (t, x)] = uxx (t, x)(σ(x)ek , σ(x)ek ) k=1

belongs to Cb (H). (iv) For all x ∈ H, u(·, x) ∈ C 1 ((0, +∞)). (v) For all (t, x) ∈ (0, +∞) × D(A) ∩ D(b), equation (1.3) is fulfilled.

We are also interested in the elliptic equation 1 λϕ − Tr [(σσ ∗ )(x)ϕxx (t, x)] − hAx + b(x), ϕx (t, x)i = f (x) (1.4) 2 where λ > 0 and f ∈ Cb (H) are given. The following definition is the elliptic counterpart of Definition 1.1. Definition 1.2. A strict solution of (1.4) is a function ϕ : H→R such that

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(i) ϕ ∈ Cb1 (H) and there exist the second derivatives of ϕ, ϕx,x (x)(h, k) in all directions h, k ∈ H. (ii) For all x ∈ D(A) ∩ D(b), equation (1.4) is fulfilled. For this equation a candidate solution is given, as well known, by Z +∞ ϕ(x) = e−λt Pt f (x)dt, x ∈ H.

(1.5)

0

Finally, we define the Kolmogorov operator K0 setting  1   K0 ϕ(x) = Tr [(σσ ∗ )(x)ϕxx (x)] + hAx + b(x), ϕx (x)i, ϕ ∈ D(K0 ), 2    D(K0 ) = ϕ ∈ Cb1 (H) : conditions (i) of Definition 1.2 holds .

(1.6)

It is well known that the semigroup Pt is not strongly continuous in Cb (H) in general (in fact in all interesting situations). However, we can define its infinitesimal generator by proceeding as in Ref. [1]. For any λ > 0 and any f ∈ Cb (H) we set Z ∞ Fλ (f )(x) = e−λt Pt f (x)dt, x ∈ H. (1.7) 0

Then the following result holds.

Proposition 1.3. For any f ∈ Cb (H) and any λ > 0 we have Fλ (f ) ∈ Cb (H) and

1 kf k0 . (1.8) λ Moreover there exists a unique closed operator K : D(K) ⊂ Cb (H)→Cb (H) such that for any λ > 0 and any f ∈ Cb (H) we have Fλ (f ) = (λ − K)−1 f. kFλ (f )k0 ≤

Proof. Let first take f in Cb1 (H). Then for all x, y ∈ H we have Z ∞ e−λt E(|f (X(t, x)) − f (X(t, y))|)dt |Fλ (f )(x) − Fλ (f )(y)| ≤ 0

≤ kf k1

Z

∞ 0

e−λt E|X(t, x) − X(t, y)|dt.

Since X(t, x) is mean square continuous, we see that Fλ (f ) ∈ Cb (H). Moreover, it is obvious that (1.8) holds. Since Cb1 (H) is dense in Cb (H) we can conclude, by a straightforward argument, that Fλ (f ) ∈ Cb (H) for all f ∈ Cb (H) and that (1.8) holds. Now, by a direct computation, we see that Fλ fulfills the resolvent identity Fλ − Fµ = (µ − λ)Fλ Fµ , Since for every f ∈ Cb (H) lim λFλ (f )(x) = lim

λ→∞

λ→∞

Z

+∞ 0

λ, µ > 0.

e−τ P λτ f (x)dτ = f (x),

x ∈ H,

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F is one-to-one. So, by a classical result, see e.g. Ref. [2], there exists a unique closed operator K : D(K) ⊂ Cb (H)→Cb (H) such that for any λ > 0 and any f ∈ Cb (H) we have Fλ (f ) = (λ − K)−1 f.  The goal of this paper is to study strict solutions both of equations (1.3) and (1.4) and to investigate the relationship between the “abstract” generator K and the “concrete” differential operator K0 defined by (1.6). We shall consider situations where K is an extension of K0 (see Remark 2.4 below) and K0 determines K, more precisely that D(K0 ) is a core for K (in a sense to be made precise). This fact is important in order to prove several properties of the operator K (starting from the analogous ones for the more accessible operator K0 ), as we shall explain in section 3 below. A natural idea to find strict solutions of equation (1.4) when f ∈ Cb3 (H) is to check, by successive differentiations, that the candidate function ϕ given by (1.5) fulfills (1.4). This method works easily provided b and σ are of class C 3 , since in this case X(t, ·) is of class C 2 for any t ≥ 0, see Ref. [3]. But in all interesting applications b and σ are not even of class C 1 so that, proving that u is a strict solution, can be much more involved. In section 4 we shall present an application to a very simple but non trivial stochastic partial differential equation. We plan to take in consideration more interesting stochastic PDEs in successive papers. We quote in this direction the paper [4] where this method was used for the 3D-stochastic Navier-Stokes. However, in that case a much weaker notion (than Definition 1.1) of strict solution was used. We end this section by giving a useful characterization of D(K) similar to that proved in Ref. [5]. Proposition 1.4. Let ϕ ∈ D(K). Then we have lim+

h→0

1 (Ph ϕ(x) − ϕ(x)) = Kϕ(x), h

for all x ∈ H

(1.9)

and



1

< +∞. (P ϕ − ϕ) sup

h h

h∈(0,1]

(1.10)

0

Conversely, if there exists ϕ, g ∈ Cb (H) such that lim

h→0+

1 (Ph ϕ(x) − ϕ(x)) = g(x), h

and (1.10) holds we have ϕ ∈ D(K) and Kϕ = g.

for all x ∈ H

(1.11)

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Proof. Assume that ϕ ∈ D(K). Fix λ > 0 and set f = λϕ − Kϕ. Then Fλ (f ) = ϕ and for any h > 0 and any x ∈ H we have Z +∞ λh e−λs Ps f (x)ds. Ph ϕ(x) = Ph Fλ (f )(x) = e h

It follows that Dh Ph ϕ(x)|h=0 = λ

Z

+∞ 0

e−λs Ps f (x)ds − f (x)

= λFλ (f )(x) − f (x) = KFλ (f )(x) = Kϕ(x), and so (1.9) follows. Moreover, since Z |Ph ϕ(x) − ϕ(x)| ≤ (eλh − 1)

+∞ h

e−λs Ps f (x)ds

Z  λh  h e − 1 −λh 1 − e−λh −λs + e Ps f (x)ds ≤ kf k0 e + ≤ ch, 0 λ λ

where c is a suitable positive constant, we see that (1.10) follows as well. Assume now that there exists ϕ, g ∈ Cb (H) such that (1.11) is fulfilled. Since clearly for any x ∈ H d 1 Pt ϕ(x) = lim (Pt+h ϕ(x) − Pt ϕ(x)) = Pt g(x), h→0 h dt we have Fλ (ϕ)(x) = − =

=

1 1 ϕ(x) + λ λ

1 λ Z

Z

t

Pt ϕ(x)de−λt

0

t

e−λt Pt g(x)dt

0

1 1 ϕ(x) + Fλ (g)(x). λ λ

Therefore Fλ (ϕ)(x) =

1 1 ϕ(x) + Fλ (g)(x), λ λ

which implies ϕ ∈ D(K) and Kϕ = g.



2. Existence of cores When the dimension of H is infinite it is well known that Cb2 (H) is not dense in Cb (H), see Ref. [6]. For this reason we introduce a notion of convergence weaker

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than that of Cb (H), see Ref. [5]. We say that a sequence {ϕn } ⊂ Cb (H) is πconvergent to ϕ ∈ Cb (H) if lim ϕn (x) = ϕ(x) for all x ∈ H and there exists c > 0 n→∞

such that |ϕn (x)| ≤ c for all n ∈ N and x ∈ H.

Proposition 2.1. Let {fn } ⊂ Cb (H) be π-convergent to a function f in Cb (H). Then {Fλ (fn )} is π-convergent to f . Proof. The conclusion follows from the mean square continuity of X(t, x) and the dominated convergence theorem.  In the next definition we follow Ref. [5]. Definition 2.2. (i) A subspace Z of Cb (H) is said to be π-dense in Cb (H) if for any ϕ ∈ Cb (H) there exists a sequence {ϕn } ⊂ Z which is π–convergent to ϕ. (ii) A subspace Y of D(K) is a π-core for K if for any ϕ ∈ D(K) there exists a sequence {ϕn } ⊂ Y which is π–convergent to ϕ and such that the sequence {Kϕn } is π–convergent to Kϕ. It is easy to see that for any k ∈ N, Cbk (H) is π-dense in Cb (H), see e.g. Refs. [5] and [3]. We notice that there are many π-cores for K. It is enough to choose a π-dense linear subspace R(H) in Cb (H) and set [ Γ := (λ − K)−1 R(H). (2.1) λ>0

In fact Γ is π-dense in Cb (H) and consequently in D(K), endowed with the graph norm, by Proposition 2.1. Concerning existence of π-cores, we shall make the following basic assumption. Hypothesis 2.3. (i) K is an extension of K0 . (ii) There exists a linear subspace R(H) which is π-dense in Cb (H) and such that for any f ∈ R(H), ϕ = Fλ (f ) is a strict solution of (1.4). Then we define Γ by (2.1). Remark 2.4. Assumption 2.3-(i) holds provided the following Itˆ o formula is fulfilled for any ϕ ∈ Cb2 (H), Z t E[ϕ(X(t, x))] = ϕ(x) + E K0 ϕ(X(s, x))ds, x ∈ H, t ≥ 0. 0

In fact, from Proposition 1.4 it follows that D(K0 ) ⊂ D(K) and K0 ϕ = Kϕ.

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Let ϕ ∈ Γ; we do not know if ϕ2 ∈ Γ; however we can show that ϕ2 ∈ D(K0 ). We have in fact Lemma 2.5. Assume that Hypothesis 2.3 is fulfilled. Then for all ϕ ∈ Γ we have ϕ2 ∈ D(K0 ) and K0 (ϕ2 )(x) = 2ϕ(x) K0 ϕ(x) + |σ ∗ (x)Dϕ(x)|2 ,

x ∈ D(A) ∩ D(b).

(2.2)

Proof. Let ϕ ∈ Γ, λ > 0. Then f = λϕ − Kϕ belongs to R(H) by assumption. We know by Hypothesis 2.3 that ϕ ∈ D(K0 ). Let now x ∈ D(A) ∩ D(b). Then we have hAx, D(ϕ2 (x))i = 2ϕ(x)hAx, Dϕ(x)i, hb(x), D(ϕ2 (x))i = 2ϕ(x)hb(x), Dϕ(x)i, and Tr [(σσ ∗ )(x)D2 (ϕ2 (x))] = 2ϕ Tr [(σσ ∗ )(x)D2 (ϕ(x))] + 2|σ ∗ (x)Dϕ(x)|2 . Consequently ϕ2 ∈ D(K0 ) and (2.2) follows from a straightforward computation.  3. Invariant measures In this section we assume, besides Hypothesis 2.3 that there exists an invariant measure ν for Pt , that is a probability measure in (H, B(H)) a such that Z Z Pt f dν = f dν, f ∈ Cb (H). (3.1) H

H

We shall prove the basic integration by parts formula (3.4) below. We recall that this formula is the main tool in order to define the Sobolev space W 1,2 (H, ν), for studying regularity properties of the domain D(K) of K and also to prove the Poincar`e and log-Sobolev inequalities. We will not deal with these problems in the present paper; several results in this direction can be found in Refs. [7], [8] and references therein. Let λ > 0. Multiplying both sides of (3.1) by e−λt and integrating with respect to t over (0, +∞), yields Z Z 1 f dν. (3.2) R(λ, K)f dν = λ H H Let now ϕ ∈ D(K) and λ > 0. Set f = λϕ − Kϕ so that ϕ = R(λ, K)f. Then, taking into account (3.2), it follows that Z Z Z Z 1 1 f dν = (λϕ − Kϕ)dν, ϕdν = R(λ, K)f dν = λ H λ H H H a B(H)

is the σ-algebra of all Borel subsets of H.

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which implies Z

for all ϕ ∈ D(K).

Kϕdν = 0, H

(3.3)

It is well known that Pt can be uniquely extendible to a strongly continuous semigroup of contractions in L2 (H, ν) still denoted Pt ; let K2 be its infinitesimal generator. Remark 3.1. Assume that Hypothesis 2.3 is fulfilled. Then, by the dominated convergence theorem and a standard monotone classes argument, it follows that R(H) is dense in L2 (H, ν). Consequently, Γ is a core for K2 . We now consider the mapping γ : Γ ⊂ D(K2 )→L2 (H, ν), ϕ 7→ |σ ∗ Dϕ|2 , where D(K2 ) is endowed with the graph norm. Proposition 3.2. The mapping γ is continuous and uniquely extendible to D(K2 ). Moreover the following identity holds Z

H

ϕ Kϕdν = −

1 2

Z

H

|σ ∗ Dϕ|2 dν,

ϕ ∈ D(K2 ).

(3.4)

Proof. Let first ϕ ∈ Γ. Since the measure ν is invariant for Pt we have by Lemmas 2.5 and identity (3.3) Z

K0 (ϕ2 )dν = 0. H

Then, integrating both sides of identity (2.2) yields Z

H

ϕ K0 ϕdν = −

1 2

Z

H

|σ ∗ Dϕ|2 dν,

ϕ ∈ Γ.

(3.5)

Now let ϕ ∈ D(K2 ). Since Γ is a core for K2 , there exists a sequence {ϕn } ⊂ Γ convergent to ϕ in L2 (H, ν) and such that {Kϕn } is convergent to Kϕ in L2 (H, ν). Then by (3.5) we have Z

∗

H

2

|σ (x)D(ϕn − ϕm )| dν = −2

Z

H

(ϕn − ϕm ) K0 (ϕn − ϕm )dν,

so the sequence {σ ∗ Dϕn } is Cauchy in L2 (H, ν) and the conclusion follows.



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4. Application We consider here the following reaction-diffusion equation in the Hilbert space H = L2 (0, 2π).  dX(t, ξ) = [(Dξ2 X(t, ξ) − X(t, ξ)) + g(X(t, ξ))]dt + dW (t, ξ),          t ≥ 0, ξ ∈ [0, 2π],  (4.1)    X(t, 0) = X(t, 2π), D X(t, 0) = D X(t, 2π), t ≥ 0, ξ ξ        X(0, ξ) = x(ξ), ξ ∈ [0, 2π], where g ∈ Cb3 (R) and W is a cylindrical Wiener process in H which will be specified below. Let us write problem (4.1) in the abstract form (1.1). Define  2  Ax(ξ) = Dξ x(ξ) − x(ξ), ξ ∈ [0, 2π], x ∈ D(A)

and



D(A) = {x ∈ H 2 (0, 2π) : x(0) = x(2π), Dξ x(0) = Dξ x(2π)} b(x)(ξ) = g(x(ξ)),

x ∈ H, ξ ∈ [0, 2π].

Denote by (ek )k∈Z the complete orthonormal system of L2 (0, 2π) b , 1 ek (ξ) = √ eikξ , ξ ∈ [0, 2π], k ∈ Z 2π and define W (t) =

X

βk (t)ek ,

k∈Z

where {βk (t)}k∈Z is a family of standard Brownian motions mutually independent in a filtered probability space (Ω, F, (Ft )t≥0 , P). We have Aek = −(1 + k 2 )ek ,

k∈Z

and ketA k ≤ e−t ,

t ≥ 0.

It is well known that problem (4.1) has a unique mild solution X(t, x), see e.g. Ref. [9]. We recall that a mild solution of equation (4.1) is a stochastic process b Here

we are dealing with the usual complexification of L2 (0, 2π).

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X ∈ CW ([0, T ], H) such that Z t Z t X(t, x) = etA x + e(t−s)A b(X(s, x))dW (s) + e(t−s)A dW (s), 0

0

t ≥ 0, x ∈ H.

(4.2) Here CW ([0, T ], H) is the space of all mean square continuous adapted stochastic process X(·) defined in [0, T ] and taking values in H, endowed with the norm !1/2 kXkCW ([0,T ],H) =

sup E(|X(t)|2 )

.

t∈[0,T ]

Notice that, in spite of the fact that the function g is of class C 3 , the Nemitskii operator b is not of class C 1 (except when g is constant). However, b is differentiable in all directions of L2 (0, 2π) and it is twice differentiable in all directions of L4 (0, 2π). So, the Kolmogorov equation cannot be solved by the standard method from Chapter 7 in Ref. [3]. We notice that the existence of a strict solutions of the Kolmogorov equation corresponding to (4.1) have been studied (in a more general setting) in Ref. [10] using the Bismut-Elworthy formula. 4.1. Estimates for Xx (t, x) For any h ∈ H, t ≥ 0, x ∈ H, we denote by Xx (t, x)h : = η h (t, x) the directional derivative of Xx (t, x) in the direction of h. It is well known that η h (t, x) does exist, (see Ref. [11]) and that it is the mild solution of the problem d h η (t, x) = Aη h (t, x) + f 0 (X(t, x))η h (t, x), dt that is η h (t, x) = etA h +

Z

η h (0, x) = h,

(4.3)

t

e(t−s)A f 0 (X(s, x))η h (s, x)ds.

(4.4)

0

Proposition 4.1. We have |η h (t, x)|2L2 (0,2π) ≤ e−2(1−kgk1 )t |h|2L2 (0,2π) ,

t ≥ 0, h ∈ H.

(4.5)

Proof. In fact, multiplying scalarly both sides of (4.3) by η h (t, x) and integrating over [0, 2π] yields, 1 d h |η (t, x)|2L2 (0,2π) = hAη h (t, x), η h (t, x)iL2 (0,2π) 2 dt +hg 0 (X(t, x))η h (t, x), η h (t, x)iL2 (0,2π) ≤ (−1 + kgk1 )|η h (t, x)|2L2 (0,2π) and so, the conclusion follows from a classical comparison result.



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Assume that one has to estimate the series ∞ X |η h (t, x)|2L2 (0,2π) . k=1

To this purpose the estimate (4.5) will not be enough. For this reason we prove another estimate. Proposition 4.2. We have |η

h

(t, x)|2L2 (0,2π)

tA

≤ |e h|

L2 (0,2π)

+ kgk1

Z

t

e(kgk1 −1)(t−s) |esA h|L2 (0,2π) ds.

0

(4.6)

Proof. By (4.4) we have |η h (t, x)|L2 (0,2π) ≤ |etA h|L2 (0,2π) +

Z

t 0

|e(t−s)A g 0 (X(s, x))η h (s, x)|L2 (0,2π) ds

tA

≤ |e h|L2 (0,2π) + kgk1 Now, setting

Z

t 0

e−(t−s) |η h (s, x)|L2 (0,2π) ds.

ϕ(t) = et |η h (t, x)|L2 (0,2π) , we have ϕ(t) ≤ et |etA h|L2 (0,2π) + kgk1

Z

t

ϕ(s)ds. 0

The conclusion follows from the Gronwall lemma.



We are going now to estimate, for further use, the L2 norm of [η h (t, x)]2 . Write 1 d h [η (t, x)]2 = Aη h (t, x) η h (t, x) + g 0 (X(s, x))[η h (s, x)]2 . 2 dt Since A([η h (t, x)]2 ) = 2Aη h (t, x) η h (t, x) + |Dη h (t, x)|2 , we deduce that d h [η (t, x)]2 ≤ A([η h (t, x)]2 ) − |Dη h (t, x)|2 + 2kgk1[η h (t, x)]2 dt ≤ A([η h (t, x)]2 ) + 2kgk1[η h (t, x)]2 . Therefore h

2

tA

2

[η (t, x)] ≤ e (h ) + 2kgk1

Z

t

e(t−s)A [η h (s, x)]2 ds. 0

Arguing as in the proof of Proposition 4.2, we find the following result.

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Proposition 4.3. We have h

2

tA

2

|[η (t, x)] |L2 (0,2π) ≤ |e (h )|L2 (0,2π) +

Z

t 0

e(2kgk1 −1)(t−s) |esA (h2 )|L2 (0,2π) ds. (4.7)

Using the well known ultracontractivity type estimate |etA f |L2 (0,2π) ≤ ct−1/4 e−t |f |L1 (0,2π) ,

(4.8)

we find the result Corollary 4.4. There exist c1 > 0, κ1 > 0 such that |[η h (t, x)]2 |L2 (0,2π) ≤ c1 (1 + t−1/4 )eκ1 t |h|2L2 (0,2π) .

(4.9)

Proof. We have in fact by (4.8) |etA (h2 )|L2 (0,2π) ≤ ct−1/4 e−t |h2 |L1 (0,2π) = ct−1/4 e−t |h|2L2 (0,2π) , and the conclusion follows from (4.7).



4.2. Estimates for Xxx (t, x) We set now ζ h (t, x) = hXxx (t, x)h, hi,

t ≥ 0, x, h ∈ H.

ζ h does exist and fulfills the equation (see e.g. Ref. [11])  d  ζ h (t, x) = Aζ h (t, x) + g 0 (X(t, x))ζ h (t, x) + g 00 (X(t, x))[η h (t, x)]2 ,  dt   h ζ (0, x) = 0.

(4.10)

Proposition 4.5. There exist c2 > 0 and κ2 > 0 such that |ζ h (t, x)|2L2 (0,2π) ≤ c2 eκ2 t |h|4L2 (0,2π) .

(4.11)

Proof. Multiplying both sides of (4.10) by ζ h (t, x) and integrating with respect to ξ over [0, 2π] yields 1 d h |ζ (t, x)|2L2 (0,2π) ≤ kgk1 |ζ h (t, x)|2L2 (0,2π) 2 dt Z 1 +kgk2 [η h (t, x)]2 |ζ h (t, x)|dξ 0

≤ (kgk1 | + +

1 kgk22 )|ζ h (t, x)|2L2 (0,2π) 2

1 h |[η (t, x)]2 |L2 (0,2π) . 2

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It follows that |ζ

h

≤

Z

Z

e(2kgk1 |+kgk2 )(t−s) (1 + s−1/4 )2 e2κ1 t |h|4L2 (0,2π) ds

(t, x)|2L2 (0,2π)

t 0

2

e(2kgk1 |+kgk2 )(t−s) |[η h (s, x)]2 |2L2 (0,2π) ds.

Now, using (4.9), we find |ζ h (t, x)|2L2 (0,2π) ≤ c21

t 0

2

and the conclusion follows. 4.3. Estimates of T R[Xx,x (t, x)] We want here to estimate the norm of the vector trace of Xx,x (t, x), namely TR[Xx,x (t, x)] : =

∞ X

k=1

Let us set

hXx,x (t, x)ek , ek i =

∞ X

ζ ek (t, x).

k=1

t ≥ 0, x ∈ H.

T (t, x) = TR[Xx,x (t, x)],

Setting in (4.10) h = ek and summimg up on k we see that T (t, x) fulfills the equation  d  T (t, x) = AT (t, x) + g 0 (X(t, x))T (t, x) + g 00 (X(t, x))Z(t, x),  dt (4.12)   T (0, x) = 0, where

Z(t, x) =

∞ X

[η ek (t, x)]2 .

k=1

Proposition 4.6. There exist c3 > 0 and κ3 > 0 such that |TR (Xx,x (t, x))|2L2 (0,2π) ≤ c3 eκ3 t . (4.13) √ Proof. By (4.7) we have, taking into account that e2k = 2π e2k , Z t ek 2 tA 2 |[η (t, x)] |L2 (0,2π) ≤ |e (ek )|L2 (0,2π) + e(2kgk1 −1)(t−s) |esA (e2k )|L2 (0,2π) ds 0

2

≤ (2π)2 e−4k t + (2π)2

Z

t

2

e(2kgk1 −1)(t−s) e−4k s ds.

0

Summing up on k yields |Z(t, x)|2 ≤ (2π)2

∞ X k=1

2

e−4k t + (2π)2

Z

t 0

e(2kgk1 −1)(t−s)

∞ X

k=1

2

e−4k s ds.

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Since, for a suitable constant c > 0, ∞ X

e−4k

k=1

the conclusion follows.

2

t

≤ ct−1/2 , 

4.4. Estimates of Pt ϕ In this subsection we show that Hypothesis 2.3 is fulfilled by choosing R(H) = EA (H), where EA (H) is the linear span of all real and imaginary parts of the exponential functions ϕh , ϕh (x) := eihh,xi ,

x ∈ H,

h ∈ D(A).

It is easy to see that the space EA (H) is dense in L2 (H, ν), see e.g. Ref. [8]. Let ϕ ∈ EA (H) and set u(t, x) = Pt ϕ(x). Then for any h ∈ H we have   hux (t, x), hi = E hDϕ(X(t, x)), η h (t, x)i , (4.14)   huxx (t, x)h, hi = E hD2 ϕ(X(t, x))η h (t, x), η h (t, x)i

By (4.5) it follows that

  +E hDϕ(X(t, x)), ζ h (t, x)i .

|hux (t, x), hi| ≤ kϕk1 e−(1−kgk1 )t |h|, for all h ∈ H. Moreover by (4.11) we deduce that 1/2

1

|huxx (t, x)h, hi| ≤ c2 e 2

κ2 t

|h|2 ,

(4.15)

(4.16)

(4.17)

for all h ∈ H. Now by (4.17) it follows easily that ϕ ∈ Cb1 (H) and |ux (t, x)| ≤ kϕk1 e−(1−kgk1 )t .

(4.18)

It remains to estimate TR[uxx (t, x)]. We have TR [uxx (t, x)] = Tr [E(Xx∗ (t, x)D2 ϕ(X(t, x))Xx∗ (t, x))] (4.19) +E [hDϕ(X(t, x)), TR[Xxx (t, x)i] . It follows, by (4.5) and (4.13) that TR [uxx (t, x)] = kTr D2 ϕk0 + kϕk1 c3 eκ3 t .

(4.20)

Moreover, recalling that for all ϕ ∈ E(H) we have that Dϕ(x) ∈ D(A) for any x ∈ H and that ADϕ ∈ Cb (H), we have   TA (u(t, x)) = hAx, ux (t, xi = E hADϕ(X(t, x)), η h (t, x)i , x ∈ D(A).

So u(t, x) fulfills conditions (i)–(iii) of Definition 1.1. We can now prove Proposition 4.7. u(t, x) is a strict solution of (1.3).

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Proof. Let us introduce Galerkin approximations. For any n ∈ N we denote by Pn the projector Pn x =

n X k=1

hx, ek iek ,

x∈H

and set An = APn . Then we consider the approximating equation   dX n (t, x) = (Pn AX n (t, x) + Pn g(Pn X n (t, x)))dt + Pn dW (t), 

(4.21)

X n (0, x) = Pn x,

which has a unique solution X n (t, x). We define also ηnh (t, x) = Xxn (t, x)h,

n ζnh (t, x) = Xxx (t, x)(h, h).

The following results are standard. lim X n (·, x) = X(·, x),

in CW ([0, T ], H),

n→∞

lim η h (t, x) n→∞ n

= η h (t, x),

in CW ([0, T ], H),

lim ζ h (t, x) n→∞ n

= ζn (t, x),

in CW ([0, T ], H),

and

for any h ∈ H. Moreover un (t, x) = E[ϕ(X n (t, x))] is a strict solution of the approximating Kolmogorov equation  1 N   unt (t, x) = Tr [unxx (t, x)] + hAPn x, uN x (t, x)i + hPn g(Pn x), ux (t, x)i, 2 (4.22)   n u (0, x) = ϕ(Pn x), x ∈ H.

Letting n→∞ we see that

lim un (t, x) = u(t, x),

n→∞

uniformly in t, x ∈ H,

lim hPn Ax, un (t, x)i = hx, Au(t, x)i,

n→∞

uniformly in t, x ∈ H,

and lim Tr [unxx (t, x)] = Tr uxx (t, x),

n→∞

uniformly in t, x ∈ H.

Thus lim

n→∞

and (1.3) is fulfilled.

d d n u (t, x) = u(t, x) dt dt

uniformly in t, x ∈ H 

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In an analogous way, using inequalities (4.16), (4.17), (4.20), we can prove the result. Proposition 4.8. Set ϕ(x) =

Z

∞

e−λt u(t, x)dt.

0

Then ϕ is a strict solution of (1.4). References [1] S. Cerrai, A Hille-Yosida theorem for weakly continuous semigroups, Semigroup Forum. 49(3), 349–367, (1994). ISSN 0037-1912. [2] K. Yosida, Functional analysis. Die Grundlehren der Mathematischen Wissenschaften, Band 123, (Academic Press Inc., New York, 1965). [3] G. Da Prato and J. Zabczyk, Second order partial differential equations in Hilbert spaces. vol. 293, London Mathematical Society Lecture Note Series, (Cambridge University Press, Cambridge, 2002). ISBN 0-521-77729-1. [4] G. Da Prato and A. Debussche, Ergodicity for the 3D stochastic Navier-Stokes equations, J. Math. Pures Appl. (9). 82(8), 877–947, (2003). ISSN 0021-7824. [5] E. Priola, On a class of Markov type semigroups in spaces of uniformly continuous and bounded functions, Studia Math. 136(3), 271–295, (1999). ISSN 0039-3223. [6] A. S. Nemirovski and S. M. Semenov, The polynomial approximation of functions in hilbert spaces, Mat. Sb. (N.S.). 92, (1973). [7] G. Da Prato, A. Debussche, and B. Goldys, Some properties of invariant measures of non symmetric dissipative stochastic systems, Probab. Theory Related Fields. 123 (3), 355–380, (2002). ISSN 0178-8051. [8] G. Da Prato, Kolmogorov equations for stochastic PDEs. Advanced Courses in Mathematics. CRM Barcelona, (Birkh¨ auser Verlag, Basel, 2004). ISBN 3-7643-7216-8. [9] G. Da Prato and J. Zabczyk, Ergodicity for infinite-dimensional systems. vol. 229, London Mathematical Society Lecture Note Series, (Cambridge University Press, Cambridge, 1996). ISBN 0-521-57900-7. [10] S. Cerrai. Classical solutions for Kolmogorov equations in Hilbert spaces. In Seminar on Stochastic Analysis, Random Fields and Applications, III (Ascona, 1999), vol. 52, Progr. Probab., pp. 55–71. Birkh¨ auser, Basel, (2002). [11] S. Cerrai, Second order PDE’s in finite and infinite dimension. vol. 1762, Lecture Notes in Mathematics, (Springer-Verlag, Berlin, 2001). ISBN 3-540-42136-X. A probabilistic approach.

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Author Index

Bl¨ omker, D., 71 Borcea, L., 91 de Bouard, A., 113 Brze´zniak, Z., 135

Liptser, R., 197

Cadenillas, A., 169 Chigansky, P., 197 Crisan, D., 221 Cvitani´c, J., 169

Nualart, D., 249

Mikulevicius, R., 357 Millet, A., 281

Papanicolaou, G., 91 Pragarauskas, H., 357 Da Prato, G., 375

Debbi, L., 135 Debussche, A., 113 Decreusefond, L., 249 Duan, J., 71

Romito, M., 263 Rozovskii, B. L., 1 Tsogka, C., 91

Flandoli, F., 263 Zapatero, F., 169 Ghazali, S., 221 Gy¨ ongy, I., 281 Krylov, N. V., 1, 311 Kutoyants, Yu. A., 339

391

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Subject Index

Bismut-Elworthy-Li formula, 271

LDP, 199 locally asymptotical normality, 341

Chapman-Kolmogorov equation, 267 coercivity, 282 coherent interferometry, 104 comparison principle for SPDEs, 314 cubature formula, 230

Markov kernel, 267 irreducible, 268 stochastically continuous, 268 strong Feller, 268 martingale problem, 266 maximum principle for SPDEs, 314 mean energy, 72 measurability, 10 m-perfect family, 228 mild solution, 137 monotonicity, 30, 35, 282 Moore-Penrose pseudoinverse, 214

decoherence parameters, 104 effort, 172 exponential tightness, 200 filtering problem, 2, 239 fractional Brownian motion, 249 fractional Laplacian, 140

normal triple, 281

γ-radonifying operator, 157 generalized Stieltjes integral, 252 Gronwall’s lemma, 157

pointwise multiplier, 164 project, 172 Rao-Cram´er bound, 341 rate function, 199

Hadamard’s formula, 35, 60 Hajek-Le Cam bound, 341 hemicontinuity, 282 homogeneous noise, 113 hyperviscous Burgers equation, 84

soliton, 114 stochastic parabolicity, 302 Stokes problem, 275 strict solution, 376 strong parabolicity, 59

incomplete market, 182 Itˆ o-Liouville equation, 97

Wald’s test, 349 Weyl derivative, 251 Wigner transform, 96

Kirchhoff migration, 103 Kolmogorov equation, 5 393

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