The IMA Volumes in Mathematics and its Applications Volume 145
Series Editors Douglas N. Arnold Arnd Scheel
Institute for Mathematics and its Applications (IMA) The Institute for Mathematics and its Applications was established by a grant from the National Science Foundation to the University of Minnesota in 1982. The primary mission of the IMA is to foster research of a truly interdisciplinary nature, establishing links between mathematics of the highest caliber and important scientific and technological problems from other disciplines and industries. To this end, the IMA organizes a wide variety of programs, ranging from short intense workshops in areas of exceptional interest and opportunity to extensive thematic programs lasting a year. IMA Volumes are used to communicate results of these programs that we believe are of particular value to the broader scientific community. The full list of IMA books can be found at the Web site of the Institute for Mathematics and its Applications: http://www.ima.umn.edu/springer/volumes.html Presentation materials from the IMA talks are available at http://www.ima.umn.edu/talks/ Douglas N. Arnold, Director of the IMA * * * * * * * * * *
IMA ANNUAL PROGRAMS 1982–1983 1983–1984 1984–1985 1985–1986 1986–1987 1987–1988 1988–1989 1989–1990 1990–1991 1991–1992 1992–1993 1993–1994 1994–1995 1995–1996 1996–1997
Statistical and Continuum Approaches to Phase Transition Mathematical Models for the Economics of Decentralized Resource Allocation Continuum Physics and Partial Differential Equations Stochastic Differential Equations and Their Applications Scientific Computation Applied Combinatorics Nonlinear Waves Dynamical Systems and Their Applications Phase Transitions and Free Boundaries Applied Linear Algebra Control Theory and its Applications Emerging Applications of Probability Waves and Scattering Mathematical Methods in Material Science Mathematics of High Performance Computing
(Continued at the back)
Pao-Liu Chow Boris Mordukhovich George Yin Editors
Topics in Stochastic Analysis and Nonparametric Estimation
Pao-Liu Chow Department of Mathematics Wayne State University Detroit, MI 48202
Boris Mordukhovich Department of Mathematics Wayne State University Detroit, MI 48202 http://www.math.wayne.edu/~boris/
George Yin Department of Mathematics Wayne State University Detroit, MI 48202 http://www.math.wayne.edu/~gyin/
Series Editors Douglas N. Arnold Arnd Scheel Institute for Mathematics and its Applications University of Minnesota Minneapolis, MN 55455 USA
ISBN-13: 978-0-387-75110-8
e-ISBN-13: 978-0-387-75111-5
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FOREWORD
This IMA Volume in Mathematics and its Applications
TOPICS IN STOCHASTIC ANALYSIS AND NONPARAMETRIC ESTIMATION contains pap ers th at were pr esented at t he IMA P articipating Inst itut ion conference on "Asym pt ot ic An alysis in Stochas tic Processes, Nonpara metric Es timation , and Relat ed Problems" held on Septemb er 15-17, 2006 at Wayn e St ate University. T he confer ence, whi ch was one of a pproximate ly ten selected each year for partial support by th e IMA t hrough its affiliates pro gr am, was dedicated to Professor Rafail Z. Kh asminskii on th e occasion of his 75t h birthday, in recognition of his profound contributions t o the field of stochastic pro cesses and nonpar am etric est imation t heory. We ar e grateful t o t he participan ts and , esp ecially, to the conference organiz ers , for making th e event so successful. P ao-Liu Chow , Bori s Mordukhovich , and Geor ge Yin of the Dep artment of Mathem atics at Wayn e State University did a supe rb job organizing this first-ra t e event and in edit ing t hese pro ceedings. We t ake t his opportunity to t ha nk th e Nation al Science Foundation for its support of t he IMA.
Series Editors Dou glas N. Arnold, Dir ector of t he IMA Arnd Scheel, Depu ty Dir ect or of the IMA
v
DEDICATED TO PROFESSOR RAFAIL Z. KHASMINSKII ON THE OCCASION OF HIS SEVENTY-FIFTH BIRTHDAY
vii
PREFACE Research on stochastic analysis and nonparametric inference has witnessed tremendous progress in recent years resulting in significant impact on mathematics and its applications to diverse fields in science and technology. The progress in technology and the emerging applications have pr esented new challenges to the study of stochastic processes and the statistical estimation theory. To assess the past achievement and to provide a road map for future research, an IMA participating institution conference entitled "Conference on Asymptotic Analysis in Stochastic Processes, Nonparametric Estimation, and Related Problems" was held at Wayne State University, September 15-17, 2006. On the occasion of his seventy-fifth birthday, this conference was also held to honor Professor Rafail Z. Khasminskii for his fundamental contributions to many aspects of stochastic processes and the non parametric estimation theory. Presenting a concerted effort of researchers from multiple institutions, it assembled an impressive list of invited speakers, who are renowned leaders in the fields of probability theory, stochastic processes, stochastic differential equations, as well as in the non parametric estimation theory, and related fields. A number of invited speakers were early developers of the fields of probability and stochastic processes, establishing the foundation of the Modern probability theory. Their earlier works have stimulated much of subsequent progress in these fields. It is conceivable that the conference will have a lasting impact on th e further development of the conference subjects. After the conference, to commemorate this special event, we felt it was fitting to put together an IMA volume dedicated to Professor Rafail Z. Khasminskii for his outstanding scientific achievement. So we invited the distinguished speakers of the conference to contribute to this commemorative volume and most of them graciously agreed . It consists of nine papers on various topics in probability and statistics. They include authoritative expositions as well significant research papers of current interest. Roughly, they are grouped in three major areas: four papers in asymptotic analysis of stochastic differential equations; three papers on nonparametric estimation problems, and the remaining two concerning stochastic analysis of partial differential equations. The papers will be sequentially arranged accordingly. Without the help, assistance, support , and encouragement of many people, this volume could not have come into being. First of all, we would like to express our deep gratitude to the authors for their valuable contributions. Also we would like to take this opportunity to thank IMA , NSF, and Wayne State University for their financial support of the conference. We thank all of the conference participants, including the invited speakers, the poster presenters, and all attendees for making it a successful event. ix
x
PREFACE
Our thanks go to Douglas N. Arnold and Arnd Scheel for helping us with the preparations of the conference and this volume. We are grateful to the IMA staff, in particular to Patricia V. Brick and Dzung N. Nguyen for their help and assistance in putting the final product together as an IMA volume.
Pao-Liu Chow Boris Mordukhovich George Yin Department of Mathematics, Wayne State University
CONTENTS
Foreword
v
Dedi cated to Professor Rafail Z. Khasminskii on t he occasion of his seventy-fifth birthday . . . . . . . . . . . . . . . . . . . . . . . . . . vii Pr eface
IX
PART I : A SYMPTOTIC A NALYSIS I NVOLVING STOCHASTI C DIFFERENTIAL EQUATIO NS
Some recent result s on averagin g principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Mark Freidlin and Al exander Wentzell Cra mer's t heorem for nonn egative mult ivar iate point pro cesses with independent increments Fima Kleban er and Robert Lipts er
21
On bounded solutions of th e balan ced gene ra lized pa nto graph equ ation Leonid Boga chev, Gregory Derjel, Stanislav Molchanov, and John Ockendon
29
Nume rical methods for non-zero-sum stochastic differential games: Converge nce of t he Markov chain approxima tion method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Harold J. Kushner P ART II : NONPARAMET RIC E ST IMATION
On the est imation of a n ana lyt ic spectral density outs ide of t he observat ion band Ildar A . Ibm gim ov
85
On ora cle inequalities relat ed to high dim ensional linea r mod els . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Yuri Golubev Hypothesis testing under composite functions alt ernat ive. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Oleg V. Lepski and Christ ophe F. Pouet PART III : STOCHA STI C PARTIAL D IF FE RENTI AL EQUATIONS
On pa rab olic pd es and spdes in Sobo lev spaces with out and with weights Nicolai V. K rylov
W;
Stochastic par ab olic equations of full second ord er Sergey V. Lototsky and Boris L. Rozovskii xi
151 199
PART
I:
ASYMPTOTIC ANALYSIS INVOLVING
STOCHASTIC DIFFERENTIAL EQUATIONS
SOME RECENT RESULTS ON AVERAGING PRINCIPLE MARK FREIDLIN* AND ALEX AN DE R WENTZE LLt
1. Non-perturbed system. Aver aging principle is one of t he main method s in perturbation t heory. It came int o being more than two centuries ago in celest ial mechan ics, but even now there are man y ope n questions havin g to do wit h applications of this principle. And these questions are not just abo ut rigorous justificati on of procedures applied, bu t even it is unclear sometimes how to apply averaging procedures. On e ca n see some of t he difficulties in a relatively simple case of an oscillator with one degree of freedom . The non-perturbed syste m in this case is described by the equat ion
(1.1 ) If we int roduce the Hamiltonian H (q,p )
= ~ p2 + F (q), (q, p) = x
E lR. 2 ,
t he equa t ion (1.1) can be written as a system:
Xt = \1 H (X t ) ,
-\1 H (x ) = (OH H). op' - O oq
(1.2)
It is well known t hat t he energy H (X t ) is pr eserved: H (X t ) == H (X o), a nd t he flow X t preser ves the Leb esgue measure in lR.2 • Let us ass ume, at first, t hat t he Hamilt onian has j ust one well (t he p otenti al F (x ) has just one minimum , and lim lql--+oo F (q) = 00). The t rajectories X; are p er iod ic, and each one moves along t he corre spo nding level curve C (z) = { x E 1R2 : H (x ) = z} of t he Hamiltoni an (see Fi gure 1). The period of rot ation along t he curve C(z ) is equal to T (z ) =
i (z)1 \1~~X)I '
It is well known that if S(z) is t he area of the region
G(z ) C lR. 2 bounded by C(z), then T (z ) =
dS(z)
~. The flow X; ca n be con-
sidered on the curve C (z), z > min H (x ), and it has a unique normalized invariant measure J.L z on thi s curve with density m z(x ) = (T (z) !\1H(x) l)-l with resp ect to the curve length. If the Hamiltoni an H(x ) has more than one well, as in Fi gure 2, t hen t he level set C (z) = {x E lR. 2 : H (x) = z} can consist of several connect ed comp onents C(z , k ): C(z ) = u;~zi C (z , k ). In t his case, an invariant measure wit h density mz ,k (x ) = (T (z, k) I\1H( x )I)- l is con centrat ed on each
C (z , k ), where T (z , k) =
J
1\1 H (x )I- 1 df is t he period of rot ati on
J C (z , k )
' Department of Mathematics, Univ ersity of Maryland, Colleg e Par k, MD 20742 . t Depa rt ment of Mathematics, Tulane University, New Orleans, LA 70118. 1
2
MARK FREIDLIN AND ALEXANDER WENTZELL
H
---z ----
p
q
FIG .
1.
along O(z , k) . We'll see that non-uniqueness of the invariant measure on the level set O( z ) = Uk C( z , k) leads to essential changes in the averaging principle. If z is the value of the Hamiltonian H (x ) at a saddle point, the connected component of the level set O(z ) containing this saddle point consists of three traject ories (one is the equilibrium point). Let us denote with I' (F ig. 2) the graph that is homeomorphic to the set of connect ed components of all level sets (prov ided with the natural to po logy) . In the case of a one-well potential, I' consists of just one edge. Let us number the edges Jr, ..., In E I'. Each point of the graph T can be characterized by t he value z of t he Hamiltonian on the level set component correspo nd ing to this point and the number k of t he edge containing this point. The pair (z, k) forms a global coordinat e syst em in T. Let Y: ]R2 I---' I' be t he projection of ]R2: for x E ]R2 , Y (x ) is the point of I' wit h coordinat es (H(x) , k(x)) , where k (x ) is t he number of the edge in I' cont aining t he point corresponding to the level set component containing x. It is clear that not just H (x ) is pre served by the flow X t , but k(x ) as well. So k(x ) is a kind of addit ional first int egral for our syst em (a discret e one) . One can say that the whole Y (x ) , with its two coordinate s H (x ), k(x ) is a first inte gral. Let us denote with G(z , k) the region in ]R2 b ounded by the curve O(z, k ); let 5 (z , k ) be th e area of the region G(z, k) . An eight-shap ed curve (see Fi g. 2) is asso ciated to each saddle point of the Hamiltonian H (x ). 1 (Note that the Hamiltonian H (q, p) = "2 p2 + F (q) has no local maxima in our case; for general Hamiltoni an syste ms, H (x ) can have local maxima as well as minima .) Now let us consider a Hamiltoni an system wit h many degrees of freedom
x , = \l H (X t ) , \lH(x ) = \lH(q,p ) = (\lp H (q,p), - \lqH (q,p)) ,
q, p E R" .
SOME RECENT RESULTS ON AVERAGING PRINCIPLE
3
H
----1-----1i\-j ----------------- 0,
I,
0,
I, ---""">I'--:=::..---i---!!--__...--- ....--- . . - - - - --- - --
°1
, C1)GJ° 0,
q
0·
,
•
1
FIG. 2.
This system, in the case of n > 1, may have other smooth first integrals besides H(x): H 1(x) = H(x), ..., H1(x). If I = n , and the integrals are independent, the system is called completely integrable (some mild additional assumptions are usually made). Assume that all Hk(x) are smooth enough generic functions . Let us take C( z) = {x E JR2n : H 1(x) = Zl , ..., Hl(x) = Zl}, Z = (Zl, ... ,zz) E JR z, and assume that for every Z E JRz the set C(z) is compact . Some of the sets C(z) may consist of several connected components. After identifying all points of each connected component, we get a topological space which is called an open book. It consists of I-dimensional manifolds which can be glued together at some manifolds of smaller dimension. The points at which the matrix (\7H(x)) = (\7 HI(x), ..., \7 Hz(x)) has maximum rank I correspond to int erior points of the I-dimensional "pages" of this open book. The "binding" at which the pages are glued together correspond to the points where the rank is less than I. In particular, the graph I' considered in the case of one degree of freedom is an example of such an open book: its pages are edges, and the binding consists of the vertices. Another example: Consider n independent on e-degree-of-freedom oscillators
Xk(t) E JR 2,
k
= 1, ..., n .
(1.3)
4
MARK FREIDLIN AND ALEXANDER WENTZELL
FIG. 3.
Assume that all Hk(Xk) are smooth generic functions, lim\Xk!->OO Hk(Xk) = Then (1.3) considered as a system in jR2n is completely integrable: its first integrals are HI (Xl), ..., Hn(x n) . It is easy to see that the corresponding open book IT is the product of the graphs [k associated with Hk(Xk), k = 1, ... , n: IT = [1 X ... x [n. For instance, if n = 2, HI (Xl) has just one well as in Figure 1, and H 2(X2) has two as in Figure 2, then IT = [1 x [2 is shown in Figure 3: IT has three two-dimensional pages and the binding f l x {02}' Each page of the open book corresponding to system (1.3) is the product of edges of graphs fk associated with Hk(Xk), one edge from each graph. The edges of each f k are numbered, so a page of the open book IT can be identified by the set of integers iI, ..., in' where ik is the number of the edge taken from fk . Since in the example corresponding to Figure 3 one of the two graphs has just one edge, the pages are characterized just by the numbers 1, 2, 3. We can consider the mapping Y: jR2n I--> II such that y-l(y), y E IT, is the connected component of the level set C(z), z E ]Rl, correspond00.
SOME RECENT RESULTS ON AVERAGING PRINCIPLE
5
ing to y . In the case of system (1.3), the mapping Y is the product of mappings Yk : jR2 ......... fk associated to Hk(Xk): Y = Y 1 X ... X Yn . If y E II is an interior point of an n-dimensional page 7l'i l , ' '' ' in ' then y- 1(y) = C(Z1' i 1 ; ... ; Zn , in) is an n-dimensional manifold. The density const - (rr~=1 1\7H k(Xk)1) -1 on C(Z1' i 1; ...; Zn, in) defines an invariant measure for the system (1.3) on this n-dimensional surface. If the frequencies Wk(Zk, ik) , 1 ::; k ::; n, of rotations along the curves Ck(Zk, ik) (the ik-th component of the level set {x: Hk(x) = Zk}) are rationally independent, the normalized invariant measure on C(Z1' i 1 ; . .. ; Zn , in) is unique. If they are rationally dependent , an infinite set of invariant measures concentrated on tori of dimension smaller than n exists. In the generic case, the set of Z E jRn for which the frequencies are rationally dependent is dense in jRn. Existence of many normalized invariant measures leads to some difficulties in justification of the averaging principle. Most of our results can , actually, be generalized to dynamical systems with conservation laws, not necessarily Hamiltonian; but the Hamiltonian structure allows to simplify the results. Up to the present, we haven't spoken of any perturbations. Let us continue this for a little while . As the non-perturbed system we can consider a diffusion process. Let a process X; be defined by the stochastic differential equation in jR2n:
(1.4) Here W t is the Wiener process in jR2n, Wt the white noise ; the (2n x 2n)matrix o-(x) is assumed to have entries with bounded derivatives , and the stochastic term O'(Xt ) * Wt is understood in the sense that the process X t is governed by the generator L :
Lu(x),= (\7H(x), \7u(x))
+ ~ div(a(x)\7u(x)) ,
where a( x ) = 0'( x ) 0'* (x). It is easy to see that the Lebesgue me asure in jR2n is invariant for the process X t . Assume that a(x)\7H(x) == O. Then H(x) is a first integral for the process X t : H(X t ) == H(X o) (this follows from the Ito formula). Assume additionally that for every connected component C(z , k) of the level set C( z) = {x E jR2n : H(x) = z} such that C(z, k) does not contain critical points of the Hamiltonian there exists a positive constant ao such that e· a(x) e 2:: ao(e· e) for any vector e E jR2n such that e· \7H( x ) = 0, and x E C(z, k). Then on every connected component of this kind the process X, has a unique normalized invariant measure. This measure has a density mz,k (x) . It follows form the fact that the Lebesgue measure is invariant for
x, that
m z , k(X) =
(1\7 H(x)!J
!C(z,k)
1\7 H(x) I- 1 dS) -1 .
If C(z, k) does
contain a crit ical point of H(x) (we assume that it is at most one critical point), then the invariant measure is concentrated at this critical point.
6
MARK FREIDLIN AND ALEXANDER WE NTZELL
2. Perturbations. Perturbations of a dyn ami cal syst em ca n have vario us nature. They can be det erministic or random , and even t heir sm allness can be underst ood in different ways. Actually, the perturbed syst em is t he primary object. If the per turbed syst em depends on a small par am et er e > 0, t he non-perturbed system can be constru cte d as a simpler one that approximates the original system for small e. Sometimes we may suc ceed in findin g such an approxim ation, sometimes not. For a smooth vector field b(x , e), x E ]R2n , 0 ~ e « 1, we have: b(x, e) = b(x , 0) + e,6(X) + O(e). Therefore the classical per turbation theory, as a rul e, is dealing wit h addit ive small perturbations. Consider at first aut onomous deterministi c perturbati ons of the onedegree-of-freedom Hamiltoni an syste m (1.2):
Here ,6(x ) is a smooth vector field, 0 ~ e « 1. Under some mild additional assumptions , Xf is uniformly close to the non-perturbed traj ectory X; on any finit e time interval as e 1 o. However often we are interest ed in the behavior of Xf on long, of order c 1 , time intervals. To st udy Xf on long t ime intervals, it is convenient to rescale the time t aking Xf = X~/g ' then Xf satisfies t he equation
x~ = ~e \7H (Xn + ,6(Xn,
(2.1)
Displacements of XL for 0 < e « 1, have two comp onents: t he fast one , which is, actually, the moti on along the non-perturbed t rajectory at a spee d of ord er e- 1 , and th e slow compo nent that describes the displ acement in t he dir ect ion trans vers al to t he non-perturbed traject ories. In t he case of a one-well Hamilt oni an wit h one degree of freedom (see Fig. 1) t he slow mot ion can be cha ra cterized by the evolution of H (X f). Taking into account t he fact that \7H (x ) . \7H (x ) == 0, we can write :
H (Xn - H (X
o)=
it
\7H (X ; ) . ,6(X;) ds.
(2.2)
Before H (Xf) changes a little from its initial valu e z = H (X o), the traject ory X f makes many rotations along the loop C (z) = {x E ]R2 : H (x) = z} . This impli es that
r
e
E
Jo \7H (X s ) • ,6(X s ) ds =
[
t
1
1
T (z ) ! C (z )
\7H (x )· ,6(x )d£ ] 1\7 H (x )1 + Og(l) + o(t )
(we remember t hat (T(z) 1\7 H (x )1) -1 is t he invariant density on C(z )). We conclude from the divergence t heore m that
i
C( z)
\7H (x ). ,6(x ) d£ = \7H ( )I X 1
1 . G (z )
dlV,6(X) dx ,
SOME RECENT RESULTS ON AVERAGING PRINCIPLE
7
where G(z) is the region in]R2 bounded by the curve C(z). One can expect that in the one-well case lim IH(Xt) - Ztl = 0 dO
(2.3)
uniformly on any finite time interval, where Zt is the solution of the equation -(3(z) = T(1 )
z
1
G(z)
div(3(x) dx,
(2.4)
with the initial condition Zo = H(X o). The statement (2.3) is a classical manifestation of the averaging principle (see, e. g., [2], Section 52). Along with deterministic perturbations of the equation (1.2) of the form e(3(X), one can consider random perturbations leading, in the rescaled time, to the perturbed equation of the form
Xf = .!.e \lH(Xt) + (3(Xt) + iT(Xt) w,
(2.5)
Here Wt is the Gaussian white noise, and iT(x) a matrix with smooth entries. As before, the fast component of Xi is the motion along the nonperturbed trajectories, and the slow component is described by H(Xt). Applying Ito's formula, we obtain:
H(Xt) -
H(X~) =
it it
\lH(X:). (3(X:) ds
+
1
\lH(X:) . iT(X:) dWs
r .2:
+ "2 l«
(2.6)
(PH
2
aij(X:) 8x i8x j ds,
s, )=1
where a(x) = (aij(x)) = iT(x)iT*(x) . A one-dimensional Wiener process W(t) exists such that the stochastic term in (2.6) can be written as
W
(it
a(x:rv H(X:) . \lH(X:) dS).
Using this fact and the argu-
ments leading to (2.3) - (2.4) (see below), one can show that for Xi given by (2.5) the process H(Xt) converges weakly in the space of continuous functions e [O, T ] to the diffusion process defined by the equation
Zo = H(X o). Here (3(z) is the same as in (2.4), and
-
1
(3(z) = 2T(z) iT 2(z) = T(1 ) z
1
~H
!C(z)
~
2::., aij (x ) 8x i8xj I\lH(x)I'
1 .
)
dlv(a(x) \lH(x)· \lH(x)) dx.
G (z )
8
MARK FREIDLIN AND ALEXANDER WENTZELL
R.Z.Khasminskii was the first who rigorously considered aver aging for stochastic differential equations [16], [17]. On e can consider perturbed systems of the form
x~ = ~c V'H (X i) + ,B1(X~ , Y/) , yt
(2.7)
= ,B2 (X~, Y/) ,
with the component Xi two-dimensional, and ~£ one-dimensional. In this case, the fast component is the same as before , but the phase space of the slow component is an open book being the dir ect product of the slowcomponent phase space of the system (2.1) and the y-axis , since now the coordinate y is a first integral of the corresponding non-perturbed system. Another type of perturbations is systems with a small delay. Let us consider a perturbation of (1.1) of the following form:
t ? h.
(2.8)
This equation should be solved with the initial condition qf given for t between 0 and h. Suppose f ( , ) is a smooth function ; let us introduce the function f (q) of one variable that is equal to the valu e of the function f ( , ) on t he "diagonal" : f (q) = f (q, q). Then for small positive h we can write ijf = f (qf ,qf-h ) = f (qf ) - h· h (qf , qf ) qf + o(h ). So, to some extent, we ca n t reat small delay as a small additive perturbation. Let us give a sket ch of the pr oof of (2.3). Consider an auxiliary function E lR 2 , that satisfies the equ ation
u(x ), x
Lu(x ) := V'H (x ) • V'u(x)
=
V'H (x ) . (3( x ) - ;B(z )
(2.9)
on each curve C (z ) = {x: H(x ) = z }, z > min x H (x ). We assume t ha t H (x ) has jus t one well as in Figure 1, so that C (z ) has one connected component, and the non-perturbed system has just one normalized invari ant measure concentrated on C (z) . For each z , the equation (2.9) has a solution, since the right-hand side of (2.9) is orthogonal to this measure. This solution is unique up to an addit ive constant a = a(z) ; and the constants a(z) can be chosen so that u(x) has continuous derivatives in lR 2 . If Xi is the solution of the equation (2.1), then
From t his equalit y and (2.9) we conclude that for any T > 0 there exists a const an t C 1 such that (2.10)
SOME RECENT RESULTS ON AVERAGING PRINCIPLE
9
It follows from (2.2), (2.4) and (2.10) that IH(Xt) - Ztl :::; K
it
IH(X:) - Zsl ds + C 1c,
0:::; t :::;
T,
where K is the Lipschitz constant for (3(z). This inequality implies (2.3). At the first glance, using such an auxiliary function u(x) is just a trick, but actually, similar proofs of the averaging principle work in many situations. In the example mentioned above, deterministic motion along the loops
C(z) played the part of the fast motion. Consider now perturbations of the stochastic equation (1.4) in ]R2n. After an appropriate time change, the perturbed equation takes the form
x; = ~c V H(Xt) + yc~O"(Xt) * Wt + f3(Xf)
(2.11)
(for brevity's sake, we consider only deterministic perturbations). Let the assumptions made at the end of Section 1 be satisfied, and the Hamiltonian H(x), x E ]R2n, has just one well. On each (2n -I)-dimensional manifold C(z) = {x E ]R2n: H(x) = z} the process defined by (1.4) has a unique invariant density with respect to the (2n -I)-dimensional surface area d.S
T(z)
=
1 IV H(y) I- 1 as. JC(z)
Let G(z) be the region in (3(z) by
]R2n
bounded by C(z), and define the function
-f3(z) = T(1 ) z
1
divf3(x) dx.
G(z)
For each z , let us consider the problem (similar to (2.9)):
Lu(x) = VH(x). f3(x) -(3(z), Here Lu(x)
x
E
C(z).
(2.12)
= ~div(a(x)Vu(x)) + VH(x). Vu(x) is the generator of
the non-perturbed diffusion process on C(z). Since the right-hand side of (2.12) is orthogonal to mz(x) (which is the solution of the adjoint equation L*mz(x) = 0, x E C(z), unique up to a constant factor), the problem (2.12) is solvable (by the Fredholm alternative). The solution of (2.12) is unique up to an additive constant a = a(z) . Using the uniqueness of the normalized invariant density, one can prove that the constants a = a(z) can be chosen so that the function u(x) has continuous second derivatives in ]R2n. Then, applying Ito's formula to u(Xt) one can obtain bounds allowing to prove convergence of H(Xt) in probability to the solution Zt
10
MARK FREIDLIN AND ALEXANDER WENTZELL
Y2 , -,
,,
-c
,
,, \
\
\
\
\
\
\ \ \
\ \ I
, ,
\
\ I I
I I
I I I I I I I
I I I I I
I
I
I
I
I
X
I
I
,,
I
I
I
I
FIG. 4.
of the equation (2.4) with the initial condition Zo = H(Xeo), uniformly in every finite time interval (we assume that the initial condition for the process X is taken the same for all e » 0) . Let us mention one more averaging problem (see [4]). Let D be a set in jR3 shown in Figure 4: D contains the x-axis; for each x E jRl, the crosssection D x = {y = (Yl' Y2) E jR2: (x, us, Y2) E D is a bounded connected region; D has a smooth boundary aD, and the normal n(x, y) to is not parallel to the x-axis for any (x, y) E aD. Let us make the region D narrower in the y-directions by the factor s: D" = {(x ,y) : x E jRl, Y E jR2, (x, e- 1y) ED} . Consider the Wiener process Zf = (X"(t) ,Y{(t), Y2"(t)) in D" with normal reflection at the boundary aD", Let If(x,y), (x,y) ED", be the projection of the unit inward normal vector to aD" on the x-axis, and l2(x, y) its projection onto the (Yl' Y2)-plane. Then the process Zf can be described by the stochastic equations
o
' t1 + 11"(X"t , v'")l''' X, te = W L t t, {T" L t
= W't2 + 12"(X"t , v'")l''' L t t,
(2.13)
where Wl, Wp are, respectively, a one-dimensional and a two -dimensional independent Wiener processes, and If is the local time of the process Z[ on BD": Here the fast component of Z[ is a two -dimensional Wiener process in the cross-section with reflection at its boundary, while Xi is the slow component. One can expect that the slow component converges
SOME RECENT RES ULTS ON AVERAGI NG PRINCIPLE
11
as € 1 0 to a one-dimensional diffusion pr ocess X t ; the local- time term in the first of the equations (2.13) leads to a drift t erm in the process X t . To calculat e the drift t erm we can consider the following pr oblem (t hat corresponds, in this cas e, to the problem (2.12)) :
6.u(y) = A ,
y E Dx ,
fJu l fJn y E D",
,f(x, y) I, i(x, y)1 '
(2.14)
where x E ]Rl is a par ameter, and n = n(y) is the unit inward normal vect or to fJDx . The const ant A can be chosen in a unique way t o make t he problem (2.14) solvable:
A = __ 1_ 1 I~(X , y) ae, S(x) JaD", 1J2(x, y)1 where S(x) is the area of the cross-section D x . Using the Ito formula, one can obt ain that the component X; converges in distribution as e 1 0 to the diffusion process X; in ]R1 governed by the operator
1 d2v S'(x ) dv Lv(x) ='2 dx2 + 2S(x) da: ' 3. Multiwell Hamiltonians. In the pre vious section we mentioned severa l examples in which using the averaging principle one can calculat e the characterist ics of t he limiting slow motion. The fast motion in t hose examples had exactly one normalized invariant measure on each level set C(z ) = {x E ]R2n: H (x) = z }: we assumed that H (x ) had just one well (if t he fast motion was deterministic, we assumed that n = 1, and for n > 1 we considered the case of the fast motion being a non-degener at e diffusion pr ocess on each manifold C(z)). Now we are going to consider t he case of non-unique invari ant measure. Let , at first , the system have one degree of freedom, and suppose that the Hamiltoni an has more than one well like in Fig. 2. Consider a perturbed syst em describ ed by equation (2.1). In this case, the fast motion is again the motion along the non-perturbed trajectories . But now the fast motion, at least for some z , has several normalized invariant measures. For example, in the case shown in Fig . 2, at least two such measures exist for H(03) ::; z < H (02 ): one concentrated on C(z, 1), another on C (z , 2). The slow component Y/ is the projection of X; onto the graph r associat ed with H (x ): ~e = Y (Xt) = (H (Xt), k(Xt) ). One can show that lim elo ~e , in gener al, may not exist (see, for instance, [3]). So in the multi well case, in general , the averaging principle does not hold at least in th e classical sense . To establish a weaker version of t he averaging principle in t he mul ti well case , one can cons ider a stochastic regularization of the pr oblem . Let us denote by X:' e the solu tion of (2.1) wit h init ial conditio n X;' e = X E ]R2 . Suppose the initi al point is stochastically perturbed: we consider t he
12
MARK FREIDLIN A ND ALEXANDER WENTZELL
solution X~+8< ,€ of (2.1) with the initial condition X~+8C€ = x+o(, where ( is uniformly distributed in a unit circle centered at the origin, and 0 is a positive parameter (that we'll take eventually to 0). Then the solution is a stochastic process. We can consider, first , the limiting behavior of the slow component Y(X~+8<,€) of X~+8( ,€ as e 1 0, and then the limit as o 1 o. If such a double limit exists in some sense, it can be considered as the limiting slow component in our original problem. This approach is natural since the initial point is usually known with some error. This way of stochastic regularization of a deterministic problem was, actually, used by D.Anosov [1] for justification of the averaging principle for a class of multidimensional problems, in particular, for perturbations of completely integrable systems in a region in which the angle-action coordinates can be introduced. The strongest results for perturbations of systems in the angle-action coordinates were obtained by A.Neistadt [19]. It was shown in [3] that this approach can be used for justification of the averaging principle for a class of one-degree of freedom multiwell systems. In particular , it works if the Hamiltonian has just one saddle point. But in the case of more than one saddle point, the double limit does not exist for a broad class of perturbations. Let us explain what the double limit passage yields in the case of a Hamiltonian with just one saddle point. Suppose H(x) has one saddle point, as shown in Fig. 2; r is the graph corresponding to H. To be specific , assume that div ,6(x) < 0 for x E ]R2. Then for every small positive 0 and every x E ]R2 with H (x) > H (O 2 ) there exists a to = to(x) such that y:~8 = Y(X:a+8(,€) belongs alternately to the edges h or h as E --+ 0. Consider a stochastic process Y/, t ~ 0, on T that , while insid e the edges hiE {I , 2, 3}, moves in accordance with the equation (2.4) with T( z) and 7J(z) replac ed with Ti(z) , 7Ji (Z) calculated by integrating, respectively, over C(z, i) and G(z, i). When the trajectory Y/ comes to the vertex O 2 (from the edge It) , it goes without any delay to 12 or
h with probabilities Pi =
r
i:
div,6(x) dX/
r
div ,6(x) dx , i
i..: are shown in Fig. 2).
= 2, 3,
respectively (the regions G 2 and G 3 8 converges weakly as e 1 0 One can prove [3] that the slow motion in the space
Y:'
SOME RECENT RESULTS ON AVERAGING PRI NCIPLE
13
As was mentioned above , if t he Hamiltonian has more than one saddle po int , such a regularization works only for a special class of pertur bations (see [3]). So to ju st ify t he averag ing principle in t he multi well case we should consider some stochastic perturbat ions of a different kind. It turns out t hat for t his end it 's good to consider whit e-noise type stochastic perturbations (of course, such perturbations are of interest by t hemselve s and not as a device for regularizati on of deterministic perturbations) . Let X ~ ' sc be defined by t he equ ation
so th at the generator y:, sc of
L e , Xu(x ) a(x )
=
X ~' sc
is given by
1\lH - (x ) . \lu(x ) + f3(x ) . \lu(x ) + "2 x div (a(x) \lu(x)) ,
~
= o-(x )o-* (x ).
Let us consider t he case of f3( x) == O. It follows from [12]' Ch. 8 that in t his case t he pr ocesses Y: ' x = Y (X ~ ' X) on r converge weakl y in e [O, T ] as IE ! 0 to a diffusion process on the graph r . For each sc > 0, t he process yt is determined by a set of operators Li defined by x d dVi(Z) Li vi(z ) = - r ( ) -d (ai(z ) - d- ) ' one operat or on each edge of r , and 2 i Z Z Z by t he gluing conditions at the vertices; Ti(z ) are t he sa me as befor e, and
Yt
ai(z ) =
r
1 ( ) i Z
r
div (a(x )\lH(x )) dx. The operators Li determine t he
} C (z ,i )
behavior of ~x inside t he edges , while t he gluin g cond itions det ermine it at t he vert ices. The vert ices of the graph r are classified into interior and exterio r ones. For example, for t he Hamilt oni an shown in Fi gur e 2, ju st one inte rior vertex exist s, nam ely, O 2 . Exterior verti ces are inaccessible, and no gluing condit ions are given at them ; at every int erior vertex a gluing condition is pres crib ed , in addit ion to the requirement of continuity of Vi(Z) and Livi(z) as fun ctions of y = (z , i) . For the example of F ig. 2, the gluing condit ion has t he form O:i
=
r div (a (x )\l H (x )) dx, i = 2,3 , and
i:
2:i:1 O:i ~~i (0 2 ) = 0: 1 = - 0: 2 - 0:3.
0, where
If f3( x) =1= 0 in (3 .1) , the gluing conditions for yt remain t he sam e (assuming th at det a(x ) =f. 0) ; t his is pr oved using the Girsan ov formula. d The operators Li in this case have t he form Li = Li + !3 i (z ) dz
Now we can look for t he limit of ~x , 0 ::; t ::; T , as x ! O. It t urns out t hat the weak limit of ~x exists, and it is the same process yt as was
14
MARK FREIDLIN AND ALEXANDER WENTZELL
considered above . So if the regularization by random perturbation of the initial point is possible, its result coincides with that of regularization by means of white-noise type perturbations of the equation (this last is possible in the general situation). Moreover, from what was said it's clear that the limiting process yt does not depend on the diffusion matrix a(x) . This means that the stochasticity of the limiting slow motion is an intrinsic property of the unperturbed system (1.1): a result of its instability near saddle points; the characteristics of the noise that we added for regularization do not influence the result. Some degenerate (but not completely degenerate) stochastic perturbations of Hamiltonian systems that are natural in certain problems are considered in [7]' [3]. A system of the form (2.8) with a small delay and multiwell potential F(q) is studied in [10]. Results concerning weak convergence of the slow component in the multiwell case are of special interest when motion of a large number of particles , or motion of a fluid is considered. For example, motion of particles in an incompressible flow that is close to a planar motion can be described by the equations (2.7). Stochasticity of such a motion and the characteristics of the corresponding stochastic process ar e studied in [5], [6] . In the case of equat ions (2.13) in a tube De with br anching, the limiting slow motion is also a diffusion process on the corresponding graph [11] . Finally, we'll mention in this section perturbations of a diffusion process with conservation law H(x) (equation (2.11)) . Let r be the graph homeomorphic to the set of connected components of level sets of H(x) . Under the non-degeneracy assumptions mentioned at the end of Section 1, a unique invariant measure is concentrated on each connected component C (z, k) . One can prove (see [8], [9]) that the slow component of the perturbed motion, which is again the projection Y(Xf) of Xi onto the graph, converges weakly as E: 1 a to a diffusion process on r. It is worth mentioning that in the multidimensional case the branching of the graph occurs at vertices corresponding to critical points of the Hamiltonian (which is assumed to be generic) at which the quadratic main part of H(x) has only one plus or only one minus; to all other vertices are attached only one or two edges . 4. Systems with many degrees of freedom. Consider a Hamiltonian system with n > 1 degrees of freedom . Suppose it has 1 smooth first integrals HI, ..., Hl (X). If the system is ergodic on each (almost each) connect ed component of the level sets C( z) = {x E lR 2n : HI( x) = l ZI, . .. , Hl(x) = Zl}, Z = (ZI' ... , Zl) E JR. , one can expe ct that a version of the aver aging principle holds for perturbations of this system, and the space (an open book) obtained by identifying all points of each connected component of C(z) is the phase space of the limiting slow motion for the perturbed system (see [13]).
SOME RECENT RESULTS ON AVERAGING PRINCIPLE
15
We have seen that in the case of one degree of freedom , in a region where the Hamiltonian has no critical points, averaging is relatively simple. At the same time, considering such parts of the phase space (parts corresponding to the interior corresponding to the interior of one page of the open book) is a necessary first step to understanding the general situation. In a region without critical points, under some additional assumptions, one can introduce the angle-action coordinates (see [2]' Section 50). So we'll start with a completely integrable system with n degrees of freedom, which can be written in the action-angle coordinates. In the case of many degrees of freedom the situation with the averaging principle is not so simple even in the case of a region without critical points. After an appropriate time rescaling, the perturbed system in this case has the form:
if
= {3I (If,
(4.1)
n
where = (If(t) , ... ,I~(t)) E IR n , and
n.
This seems natural since the normalized Lebesgue measure is the unique invariant measure for most of the tori. Results in this direction can be found in [1]' [18], [19] that can be interpreted as concerning the regularization of the original problem using stochastic perturbations of the initial point. The convergence of maxO~t~T In - Itl to zero as e 1 0 (the averaged trajectory being taken with the same initial condition 10 as 10) is proved to take place in the sense of convergence in Lebesgue measure on the space of initial points (which is the same as convergence in probability).
16
MARK FREIDLIN A ND ALEXANDER WENT ZELL
As we have already mentioned in the pr evious sect ion , reg ularizat ion by a stochastic perturbation of t he initial point does not work, in general , for sys te ms with multi well Hamiltonian s, so we should study regul ariz at ion by stochastic perturbati ons of t he equ ation. As t he first step in this direction, such a regularization shou ld be considered in a region where t he act ion-angle coordinates can be introduced. This problem (in a slightl y mor e general setting) was studied in [14]: Suppose t hat stochastic perturbations are added t o equat ion (4.1): s , X) w:' I'e,x t = f3 1 (I e,t x , cp s,t X) + Vc.y. 0'1 (Ie,x t , cP t t,
.
I~ '
1
(4.2)
.
x = _ n(I~ ' X) + f32(I~ ' " , cP~' X) + .jY.0'2 (I~ ' " , cP~' X) w, e
Here »c 2: 0 is a paramet er, W t a Wiener process in 1R2n , and 0'1 (I, cp), 0'1 (I, cp) are matrices of the appropriat e dimensions with smooth entr ies that ar e 27r-periodic in CPj. Take a(I ,cp) = (aij(I, cp)) =
0'1(I, cp)O'i(I ,cp), aij(I)
/3 1 (1)
=
(21r) -n
=
r f3 (I, cp)
jyn
(21r) - nl d ip,
Let
~
aij (I , cp) dcp, a(1) = (aij(I)) , c IRn be the reson ance set for
t he frequencies Wj(1). It was proved in [14] t hat if the matrix a(I, cp) is non-degener at e and t he reso nance set A has zero Lebesgue measure, t hen , whatever t he initi al conditi ons I~' x = 10 , cpeox = CPo for t he equati on (4.2) are, t he slow compon ent I~ ' X converges weakly as e 1 0 t o the diffusion process in IR n described by the equation
I;
(4.3) where the matrix fun ction O'(I ) satisfies the equ ali ty 0'(1) 0'*(1) = a(I). This means t hat und er the condit ions mentioned above t he averaging is per formed in the natural way. It is easy to understand that t his is not true, in general, if the set A contains a non-empty ope n set. After this, we can consid er the limiting behavior of the processes defined by (4.3) as »c 1 o. It can be proved that they converge i~ probability uniformly on every finit e time interval [0, T] to the solution It of the equa t ion
I;
10
= 10 .
This mean s that t he averaging principle holds for equa tio n (4.1) in the sense of weak convergence after regularizatio n by mean s of a non- degenerate white-noise -type perturbati on of t he equat ion , if t he reson an ce set A has zero Leb esgu e measure. In Hamilt on ian syste ms , it seems t o be na tural to consider white-noisetype pert urbati ons added only to the "velocity" coo rdinates. This lea ds to
SOME RECENT RESULTS ON AVERAGING PRINCIPLE
17
a degenerat e matrix a(I , ep) . Some results in t his case can also be found in [14]. Now let us consider small deterministic perturbations of t he simple complete ly int egr abl e syst em (1.3), possibly with multi well Hamiltonians Hk(xk)(see [15]). After an appropriat e time chan ge, the perturbed syst em has the form
In th e per turbed system, th e oscillators are no longer ind ep endent, t herefore we called them weakly coupled oscillators. The fast component of X i = (Xf(t ), ..., X';( t ») is once again the motion along the non-perturbed traj ect ories (wit h the time change). The phase space of the slow component is the open book II = r 1 x ... x r n (see the notations in Section 1), and th e slow component is the projection Y(Xt) of X i onto II. The averaging principle should describe the limiting behavior of Y(Xt) as c 1 O. In general, limdo Y (Xt) does not exist; and one has to consider regul ari zation of (4.4) by a stochastic perturbation of th e equat ions:
We choose here as ak some non-degener ate (2 x 2)-m atrices with constant ent ries; the two-dimensional Wiener pr ocesses W tk are assumed independent. Inside each n-dimensional page of the open book II (each such page is t he product of n edges l Ii l x 12 i2 X ... X In i n of the graphs r 1 , ... , r n), one can introduce act ion-a ngle coordinates. Thus the limiting as e 1 0 (in t he sense of weak converge nce) slow motion Yt' exists inside the pages, and is a diffusion pr ocess - at least if the corresponding reson an ce set A c jRn has Leb esgue measure O. This diffusion proc ess, in general , hits t he binding of the open book in a finite ti me. It was shown in [15] that if t he pro cess ~x starts at any interior point of a page, then with pr obability 1 it hits the binding at som e point of its (n - 1)-dimensional part: the parts of the binding of smaller dimensions , corresponding to the ends of th e edges h ik being t aken for mor e than one k, are inaccessible. Thus gluing conditions for ~x should be prescribed only at t he (n - 1)-dimensional part of the binding. These gluing conditions, roughly sp eaking, determine the behavior of the process after it hits the bin ding. The differential operato rs determining the pr ocess ~x inside the pages, as well as the gluing conditions, are to be found in [15]. It t urns out t hat , for any Y. > 0, t he gluing conditions ar e indepen dent from th e drift t erm s {3k( Xl, ..., x n ) , and the diffusion coefficient s of ~x in each page ar e pr oporti onal t o »c, Now we t ake »c t o zero, and find t he limiting slow mot ion yo; on II. Since t he diffusion coefficients of Yt' go to 0 as »c 1 0, t he pr ocess yo; will
18
MARK FREID LIN A ND ALEXANDER W E NT ZE LL
be det erminist ic inside t he pages. The sto chasticity of yt is concent rated on th e binding only and can be described by th e prob abili ti es of going from one page t o t his or t hat other page t hat come t oget her at a point be longing to the bind ing; t hese prob abilities may depend on the point of the binding considered. It 's worth menti onin g t hat t he probabili ti es of passing from one page to anot her at t he binding are det ermined by the det erministic par t of t he perturbations and are independent of the matrices G"k responsibl e for the white- noise part of the perturbations. In this sense, we can say that t he sto chasticity of the limiting slow motion for system (4.4) is an intrinsic pr operty of this det erminist ic syst em (caused, actually, by the instabilities in the original non-perturbed syst em). The sto chastic te rms in (4.5) serve just for the purpose of regularization of the problem .
REFERENCES [1] D . ANOSOV, Av eraging in sys tems of differential equations with rapid ly osci llating solutions, Izv . Akad . Nauk SSSR Math., 24 (1960), 721- 742. [21 V . ARNOLD, Mathematical Methods of Classical Me chan ics , Springer , 1978. [31 M . BRIN AND M . FREIDLIN, On stochastic behavior of perturbed Ham ilt on ian system s, Ergodic T h. Dyna m . Sys ., 20 (2000) , 55- 76. [4] M. F REIDLIN, Markov Processes and Differen tial Equation s: Asymptotic Problem s, Birkhauser , 1996. [5] M. F REIDLIN, Deterministic SD-pertur bati on s of planar in com pressible flow lead to st ochasticity, Jou rn . St atistical Phys., 111 (2003), 1209-1218. [6J M. FREIDLlN, Reaction-con vection in in com pressi ble 3D-fluid: a hom ogenization problem, [incomple te]. [7J M . F REIDLIN AND M . WEBER, Ra ndom perturbations of n onlinear oscillators , An n. of Prob ability , 26 (3) (1998) , 925-967. [8] M . FREIDLIN AND M . WEBER, On rando m pert ur bati on s of Ham ilt onian system s with many degrees of freedom, St och . Processes a nd Their Applicat ion s, 94 (200 1) , 199-239. [9J M. FR EIDLIN AND M . WE BER, Ran dom pert urbations of dynamical system s with conservation laws, Probab. Theor y and R elat ed Fi elds , 128 (2004), 441-466. [10] M. F REIDLIN AND M. WE BER, On stochastici ty of solu tions of differential equatio ns with a small delay, St och astics and Dynamics , 5 (3) (2005) , 475-486. [11] M . FREIDLIN AND A . WENTZELL, Diffusi on processes on graphs and the averaging prin cipl e, Ann . of Prob ability , 21 (4) (1993), 2215- 2245. [1 2J M . FREIDLIN AND A . WE NTZELL, Random Perturbati ons of Dyn am i cal System s, Springer, Second ed ition , 1998. [13J M. FREIDLIN AND A. WE NTZELL, Diffu sion processes on an open book an d th e averagi ng principle, Stoch. Process es and Their Applications , 113 (2004 ), 101- 126. [14J M. F REIDLIN AND A . W ENTZELL, A veraging prin ciple for sto chastic pertur bations of multifrequency syst em s, St ochastics an d Dynamics, 3 (3) (2003), 393-408. [151 M . F REIDLIN AND A . WE NTZELL, Long-tim e behavi or of weakly coupled oscillators, J ournal of Statist ical P hysics, 123(6) (2006), 1311-1337. [16] R . K HASMINSKII, Ergodic properties of recurrent diffusion process es and stabiliza tion of solutions of the Cauchy problem f or parabolic equations (R ussian) , Teor. veroyatnost . i p rimen ., 5 (1960) , 179- 196. [17J R. KHASMINSKII, Averaging principle fo r stochastic differen tial equat ion s (R ussian) , Kybernetica, Ac ademi a , Pra ha , 4 (1968), 260-279.
SOME RECENT RESULTS ON AVERAGING PRINCIPLE
19
[18] P . LOCHAK AND S . MEUNIER, Multiphase Averaging for Classical Systems, Springer, 1988. [19] A . NEISTADT, Averaging in multi-frequency systems, Dokl. Akad. Nauk SSSR, 223 (1975) , 314-317; Averaging in multi-frequency systems II, ibid. 226 (1976), 1295-1298.
CRAMER'S THEOREM FOR NONNEGATIVE MULTIVARIATE POINT PROCESSES WITH
INDEPENDENT INCREMENTS FIMA KLEBANER' AND ROBERT LIPTSERt Abstract. We consider a continuous time version of Cramer's theorem with non negative summands St = 2::i:.,., 9 ~i, t - t 00, where (Ti' ~i)i2:1 is a sequence of random variables such that tSt is a random process with independent increments.
t
Key words. Nonnegative summands, boundary effect. AMS(MOS) subject classifications. Primary 60FIO, 60J27.
1. Introduction and main result. The following version of the Cramer theorem, [1], can be extracted from Dembo and Zeitouni, [2]. THEOREM 1.1. Let (~ik~l be a sequence of nonnegative identically distributed and independent random variables with 6 admitting the Laplace transform:
2'(,\) = Ee A6 , ,\ E (-00, A),
:1 A = inf{'\ > 0 : 2'(,\) = oo}.
Then, the family
1 n Sn = ~i , n i=l
L
n -; 00,
obeys the Large Deviation Principle (LDP) in the metric space (JR.+, (}) ( (} is Euclidean's metric) with the rate *- and the rate function
I(u)
SUP AE(-= ,A )
= {
['\u-g('\) ],
-log P(6
= 0) ,
u>O u=O,
where g('\ ) is the log moment generation function , g('\) = log Ee A6 ,
x < A.
In this paper, we study a "continuous time version" of this theorem. For t 2': 0, set
'School of Mathematical Sciences, Building 28M , Monash University, Clayton Campus, Victoria 3800, Australia (fima.klebaner~5ci.monash.edu.au ) . l Depertment of Electrical Engineering Systems, Tel Aviv University, 69978 Tel Aviv, Israel (liptser~eng.tau .ac.il).
21
22
FIMA KLEBANER AND ROBERT LIPTSER
where (~i,Ti)i2:1 is a sequence of random pairs, where random variables: ~i ~
0
and
TO
= 0
~i'S
and
Ti'S
are
< T1 < T2 < ... Ti < .. .
defined on a probability space (D, F , P). Let (~n)n2:0 be the filtration with (0, D) and ~n := O'{h , ~i)isn} ' Random variables ~i are assumed to be identically distributed and independent of ~i-1 with the distribution function ~o =
G(x)
= P(6
The conditional distribution of
Ti
~ x), x ~
given
~i-1
o.
is exponential:
where r is a positive number. Moreover, we assume that
The following theorem is an analogue of Theorem 1.1. THEOREM 1.2 . The family
obeys the LDP in the metric space (lR+, e) with the rate function
[AU - r Jooo(e AZ - l)dG(z)] ,
SU P
I(u) =
t and the u> 0
AE(-oo,A) {
r[1 - G(O+)],
rate
u
= o.
(1.1)
We give two examples illustrating compatibility with Theorems 1.1 and 1.2. For both discrete and continuous time cases, let
P(6
~
x)
= 1-
e- x , x?: 0,
so that, 6 has the Laplace transform with A ating function is
= 1.
The log moment gener-
g(A) = -log(1 - A), A < 1. For G(x) = 1- e- x , x
1
00
~
O. r ).
r(e AZ - l)dG(z) = - - \ ' A < 1. o 1 - /\
CRAMER'S THEOREM FOR MULTIVARIATE POINT PROCESSES
23
8
6
4
o
----------
0.5
1.5
2
2.5
3
FIG. 1. The rate function Id(u)
0.8
o
0.5
1.5
2
2.5
3
FIG . 2. The rate function I C(u) for r = 1
In both cases, rate functions are explicitly computable (see also Figures 1 and 2) , ] d(u ) __
{U - 1 -log(u),
u>0 ( discrete time case) u =0
00,
]c (u)
= {( JT - y'U)2, r,
u> 0 (continuous time case) = O.
u
REMARK 1.1. Related topics to Theorem 1.2 can be found e.g. in Georgii and Zessin , [3], serving a class of marked point random fields. Probably, the proof of Theorem 1.2 can be adapted with arguments from proofs in [3] provided that many details not related to our setting have to be omitted and other ones concerning to the boundary effect have to be added.
24
FIMA KLEBANER AND ROBERT LIPTSER
We prefer to give a complete and direct proof of Theorem 1.2.
2. Counting random measure, its compensator. Laplace transform. We consider ( ~i , Ti) i ~l as a multivariate (m ark ed ) point process (see, e.g. [4], [5]) with the coun ting measure
/-L (dt ,dy) = L)h
wher e O{Ti'~;} is the Dirac delta-function on IR+ x IR+. Parallel to (~n)n~O, we introduce one more filtr ation (~t )t ~O related to (~i,Ti )i ~l : ~t :=
CT(/-L([O , t' l x T) : t' ~ t, r E 88(IR+ )),
where 88(IR+) is the Borel o-algebra on IR+ , and assume that ~o is au gmented by P-zer o set s from :F (not ice that , then, (~t ) t~ O satisfi es the general condit ions). With the help of the counting me asure /-L , one can pr esent t.S; in a form of a stochastic integral with respect to /-L :
is, =
r1
Jo x>o
(2.1)
X/-L( ds, dx ).
Then , the Levy measure v(d s, dx ), related to /-L , is explicitly computed (see, e.g. Theorem III.1.33 , [5])
v(ds,dx) = L I]Ti_l,T;j (t)
dG(x )de-r(S-Ti-tl e-r(S-Ti- t}
= rdsdG (x)
(2.2)
i~ l
and is not random. It is well known (see, e.g. Corollary to Theorem 1 in §4, Ch.4, [6], that und er the deterministic Levy measure the random process t S; has ind ependent increments from non-overlapping time intervals. We recall a useful property: for any nonnegative and (~t ) - predi ct able functi on
f (w, x, t ),
E with
" 00
t'
r
Jo Jx>o
f (w, x ,s)/-L(ds ,dx ) = E . : f (w, x , s)v (ds,dx ) Jo x>o
= 00 " .
LEMMA 2 .1. [La place transform] For any A < A and t > 0,
Proof Though the dir ect computation of Laplace's transform is permissible, we prefer to apply the sto chastic calculus. The pr ocess U, = eA t S, has right-continuous piece-wise constant paths with jumps
L:.Us =
(Us -
Us -
)
= Us -
r
[eAX -l]/-L({s} ,dx ), Jx>o
CRAM ER'S THEOREM FOR MULTIVARIATE POINT PROCESSES
25
so that, for any t > 0,
tj, = 1 +
ri.: r
Jo
Us - [e AX - l ]J.l(ds, dx) .
Since the function f( w, x, s ) := Us - [e AX + 1] is nonnegative and predictabl e, the following equality with A < A holds tr ue:
E
r Us_ [e AX+ l ]J.l (ds, dx) = E r r Us_ [e AX+ l ]v(d s,dx ) « Jo Jx>o Jo Jx>o t
00).
Then, we also have
t
E
t
r r Us_[e AX -l]J.l(ds ,dx ) = E r r Us_[e AX - l ]v(ds,dx ) (E JR.). Jo Jx >o Jo Jx>o
By v(ds ,dx) = rdsdG(x) (see (2.2)) , the later pr ovides the integral equation (EUt ) = 1 + J~ JX>o (EU s) [e AX - l]dG( x)rds which is equi valent to the differe~tial equation d(~~, ) = (EUt) Jx>o [e AX - l ]dG(x) r subject to (EUo) = 1. Thus, the desired result holds. 0 3. The proof of Theorem 1.2. We verify the necessary and sufficient conditions for the LDP t o hold (for mor e det ails , see Puhalskii , [7]): 1) exponential t ightness,
wher e Kj's are compa cts increasing to JR.+; 2) local LDP, defining the rate funct ion I(u ), u E JR.+ 1 t
lim lim -log P(ISt -
8~ O t ~oo
ul :::; J). =
-I(u) .
3.1. The exponential tightness. By choosing
K j = {x E JR.+ : x E [0 , j]} and applying Chern off's inequ ality with par amet er 0.5A , we find that P(St > J.) :::; e - 0 .5Aj + lo g EeO.5AtSt
By Lemma 2.1, we have EeO. 5AtS, = ert !x>o[eO .5Ax - ljdG(z) and, therefore,
~ log P(St > j) t
:::; -0.5Aj
+r
r
Jx>o
[eO. 5Ax - l ]dG(x )
~ J ~OO
- 00.
o
26
FIMA KLEBANER AND ROBERT LIPTSER
3.2. The local LDP. We begin with computati on of 1(0) and prove lim lim ~ log P(St S 8) ~ -r[l - G(O+ )] t-+oo t --1 lim lim -log P(St S 8) S - r [l - G(O+)]. 0-+0 t-+oo t
0 -+0
By (2.2) , {tSt
= O} = {fl((O , t] x {x > O} ) = O.
P(St S 8) ~ P(St = 0) = P(tSt
(3.1)
Consequently for any t > 0,
0) = P(fl( (O, t] , {x > O}) = 0) .
=
The counting process 7l"t := fl((O, t], {x> O}) has ind ep endent increments and the rate Efl( (0, t] , {x > O}) = v( (0, t], {x > O}) = r [l - G(O+ )]t. So, 7l"t is the counting process with the compensator
v((O,t], {x> O})
=
r[l- G(O+) ]t.
Therefor e, by the Watanabe theorem , [8], 7l"t is Poisson process with parameter r [l - G(O+ )]. Hen ce, du e to well known property of the Poisson pr ocess P (7l"t = 0) = e- t r [l - G (O+)], we find that
1
1
= 0) = -r[l -
"tlogP (St S 8) ~ "tlogP (7l"t
G(+)]
and the lower bond from (3.1). The upper bo und from (3.1) is derived with t he help of Laplace's t ransform with 0 < ).. < A. To this end, we use identity 1 = E exp ()..tSt - tr
r
i.:
[eAX -l]dG(x ))
implyin g the inequality 1 ~ EI{s,9} exp (t[M - r Jx>o[eAX 1
-
l ]dG(x )])
being equivalent to "t log P(St S 8) S -)..8 + r Jx>o [e- - l ]dG(x ). Now, passing t ----> 00 , we obtain the following upper bound depending on 0 and X: limt-+oo log P(St S 8) S -)..8 + r Jx>o [eAX - l]dG(x) and , then, passing 0 ----> 0 and)" to -00 we find that AX
i
r
lim lim ~ log P(St S o) S -r dG(x ) = -r[l - G(O +)]. t-+oo t } x>o
0-+0
We continue the proof by checking t he validity of the I (u), u > 0, from (1.1). In ot her words, we sh ow that --1 lim lim -log P(ISt - u l S 8) S -I(u)
0.....0 t .....oo
t 1
lim lim -log P(!St t-+oo t
0 .....0
ul S 0) ~ -I(u ),
CRAMER'S THEOREM FOR MULTIVARIATE POINT PROCESSES
with I(u)
=
sup
27
[AU - r JoOO(e AX -l)dG(x)] .
AE(-oo ,A)
The Laplace transform
1 = Eexp (AtSt - tr ( [e AX -ljdG(x)) , A < A, }x>o implies the inequality 1 ::::
EI{ISt-UI~o}
exp ( - tJu
+ AtU - tr (
i.: [e
AX - l]dG(x))
and, in turn , the upper bound, depending on A: lim lim
0->0 t-ucc
~t log P(ISt - ul ~ J) ~
- (AU - r
r
}x>o
[e AX - l]dG(x)).
The desired upper bound is obtained by further maximization in A over (-00 , A) of (AU - r Jx>o [e AX -l]dG(x)) . The lower bound proof uses a standard approach of changing "probability measure". Denote by A* = argmaxA
o [e AX - l]dG(x)). Since A* solves the equation (with u > 0) (3.2)
A* is a proper positive number strictly less than A. Set
Zt(A*) = exp (A*tSt - i t
1>0 r[eA*X - l ]dG(X)dS) .
(3.3)
First of all we notice that EZt(A*) = 1. Moreover, taking into account (2.1) and applying the Ito formula to Zt(A*) one can see that (Zt(A*), qt)t>o is a positive local martingale with paths from the Skorokhod space JI)[O ,oo ) ' Then, a measure i\, defined by di\ = Zt(A*)dP t, wher e P t is a restriction of P on qt, is the probability measure. We introduce the probability space ([2, F, Pt) . Since Zt(A*) > 0, P-a.s., not only P t « P t but also P t « P t with dPt = Z;-l (A*)P t. This property and (3.3) provide a lower bound P( jSt - u]
~ J) = Pt( ISt - ul ~ J) = (
}{ ISt-UI~6} :::: e- A*to- tI(u)Pt ( ISt - ul ~ J).
Therefore, we find that
Z;-l(A*)dP t
28
FIMA KLEBANER AND ROBERT LIPTSER
It is clear, the desired lower bound to obtain it is left to prove that
lim Pt(ISt -
ul ::; 8) = 1
lim i\(ISt t-+oo
ul > 8) = O.
t-+oo or , equivalently,
(3.4)
Thus the last step of proof deals with (3.4) . To this end, we show that
(Et denotes the expectation relative to Pt) .
-
lim EtlSt -
t-+oo Since Eet (A) 2Ee
ul 2 = o.
= 1, it holds
t(A) 0- 8 8A2
I
A=A*
= t 2 E(St - r
r
xeA*XdG(x)) 2 "ct(A*) _ tr
J{x>O}
.
,
r
x 2e A*xdG(x)
J{x>O} ~
yo
=ti
Hence, -E(St
1 - u)2 = -rE t
1
x 2 e A* xdG(x)
{x>O}
-----+
t-+oo
(see (3.2))
O.
o
REFERENCES [1] CRAMER H. (1938), Sur un nouveau theoreme-limite de la theoriedes probabilites. Actualites Santifiques et Industrielles . Colloque cosacrea la throrie des probabilitrs, 3(736) : 2-23. Hermann, Paris. [2] DEMBO A. AND ZEITOUNI O. (1993), Large Deviations Techniques and Applications, Jones and Bartlet, 1993. [3J GEORGlI HANS-OTTO AND ZESSIN HANS (1993), Large deviations and the maximum entropy principle of marked point random fields . Probability Theory and Related Fields, 96: 177-204. [4] JACOD J ., Multivariate point processes: predictable projection, Radon-Nikodyrn derivatives, representation martingales. Z. Wahrsch. Verw. Gebiete (1975) , 31 : 235-53. [5] JACOD J. AND SHIRYAEV A.N ., Limit Theorems for Stochastic Processes . SpringerVerlag , New York, Heidelberg, Berlin (1987 ). [6] LIPTSER R.S. AND SHIRYAEV A.N. (1989) , Theory of Martingales. Kluwer , Dordrecht (Russian edition 1986) . [7] PUHALSKlI A. (2001), Large Deviations and Idempotent Probability, Chapman & Hall /CRC Press . [8] WATANABE S. (1964 ), On discontinuous additive functionals and Levy measures of Markov process. Japan J. Math., 34: 53-70.
ON BOUNDED SOLUTIONS OF THE BALANCED GENERALIZED PANTOGRAPH EQUATION LEONID BOGACHEV· , GREGORY DERFELt , STANISLAV MOLCHANOV+ , AND JOHN OCKENDON§ Abstract. The question about the existence and characterization of bounded solutions to linear functional-differential equations with both advanced and delayed arguments was posed in the early 1970s by T . Kato in connection with the analysis of the pantograph equation, y '(x) = ay(qx) + by(x) . In the present paper, we answer this question for the balanced generalized pantograph equation of the form oo -a2Y"(x) + alY'(x) + y(x) = fo y(ax ) p(da), where al ;::: 0, a2 ;::: 0, ar + a~ > 0, oo and p is a probability measure. By setting K := fo Ina p(da) , we prove that if K ::; then the equation does not have nontrivial (i.e. , nonconstant) bounded solutions, while if K > then such a solution exists. The result in the critical case, K = 0, settles a long-standing problem. The proof exploits the link with the theory of Markov processes, in that any solution of the balanced pantograph equation is an L:-harmonic function relative to the generator L: of a certain diffusion process with "multiplication" jumps. The paper also includes three "element ary" proofs for the simple prototype equation y' (x) +y(x) = ~ y(qx) + ~ y(x /q), based on perturbation, analytical, and probabilistic techniques, respectively, which may appear useful in other situations as efficient exploratory tools.
°
°
Key words. Pantograph equation, functional-differential equations, integrad ifferential equations, balance condition, bounded solutions, WKB expansion, qdifference equations, ruin problem, Markov processes, jump diffusions, L:-harmonic functions , martingales. AMS(MOS) subject classifications. ondary 34K12.
Primary 34K06, 45J05, 60Jxx; sec-
1. Introduction. The classical pantograph equation is the linear firstorder functional-differential equation (with rescaled argument) of the form
y'(x) = ay(qx)
+ by(x) ,
(1.1)
where a , b are constant coefficients (real or complex) and q > 0 is a rescaling parameter. Historically;' the term "pantograph" dates back to the seminal paper of 1971 by Ockendon and Tayler [24], where such equations/ emerged • Department of Statistics, University of Leeds, Woodhouse Lane, Leeds LS2 9JT, UK (oog a ch evemat hs . l eeds . ac. uk) . tDepartment of Mathematics, Ben Gurion University of the Negev, Beer Sheva, Israel (derfel~math. bgu . ac . ill. >Department of Mathematics, University of North Carolina at Charlotte, Charlotte NC 28223, USA (smo l.chaneuncc . edu) . §University of Oxford, Centre for Industrial and Applied Mathematics, Mathematical Institute, 24-29 St Giles, Oxford OX1 3LB, UK (ock~maths.ox.ac.uk). lThe name "pa nt ograph equation" was not in wide use until it was coined by Iserl es [14] for a more general class of functional-differential equations . 2To be more precise, a certain vect or analog of Eq. (1.1). 29
30
L. BOGACHEV, G. DERFEL, S. MOLCHANOV, AND J. OCKENDON
in a mathematical model for the dynamics of an overhead current collection system on an electric locomotive (with the physically relevant value q < 1) . At about the same time, a systematic analysis of solutions to the pantograph equation was started by Fox et at. [11], where various analytical, perturbation, and numerical techniques were discussed at length (for both q < 1 and q > 1). It is interesting to note that an equation of the form (1.1) (with q > 1) was derived more than 25 years earlier by Ambartsumian [2] to describe the absorption of light by the interstellar matter. Some particular cases of Eq. (1.1) are also found in early work by Mahler [19] on a certain partition problem in number theory (where Eq. (1.1) , with a = 1, b = 0, q < 1, appears as a limit of a similar functional-difference equation) and by Gaver [12] on a special ruin problem (with a = 1, b = -1 , q > 1). Subsequently, the pantograph equation has appeared in numerous applications ranging from the problem of coherent states in quantum theory [26] to cell-growth modeling in biology [28] (see further references in Refs. [7,8,14,20]). These and other examples suggest that, typically, the pantograph equation and similar functional-differential equations with rescaling are relevant as long as the systems in question possess some kind of self-similarity. Since its introduction into the mathematical literature in the early 1970s , the theory of the pantograph equation (and some of its natural generalizations) has been the subject of persistent attention and research effort, yielding over years a number of significant developments. In particular, the classification of Eq . (1.1) with regard to various domains of the parameters.i' including existence and uniqueness theorems , and an extensive asymptotic analysis of the corresponding solutions have been given by Kato and McLeod [17] and Kato [16] . The investigation of such equ ations in the complex domain was initiated by Morris et at. [22] and Oberg [23] and continued by Derfel and Iserles [5] and Marshall et at. [20]. A systematic treatment of the generalized first-order pantograph equation (with matrix coefficients and also allowing for a term with rescaled derivative) is contained in the influential paper by Iserles [14], where in particular a fine geometric structure of almost-periodic solutions has been described. Asymptotics for equations with variable coefficients have been studied by Derfel and Vogl [6] . Higher-order generalizations of the pantograph equation (1.1) lead to the class of linear functional-differential equations with rescaling , e
m
L L ajky(k)(ajx + ,6j) = 0
(1.2)
j=l k=O
3Dep end ing on wh ether q iRb = O.
< 1 or
q
> 1 and also on the cases iRb < 0, iRb > 0, and
BO UNDED SOLUTIO NS OF THE BALANCED PA NTO GRAPH EQUATIO N
31
(see Ref. [4] and references therein)." Kato [16] posed a problem of asym ptoti c ana lysis of Eq . (1.2), including the question of existe nce and characte rizatio n of bounded solutions. Some pa rtial answers to t he latt er questi on have been given by Derfel [7, 4] and Derfel and Molchanov [8]. In par ticular, Derfel [7] considered the "balanced" genera lized firstorder pantogr aph equation of the form l
y' (x ) + y(x ) = I>jy(a j x ),
(1.3)
j=l
subject to the condition
e 2::Pj = 1,
Pj
> 0 (j = 1, .. . .z),
(1.4)
j=l
so that the weights Pj of the rescaled y-terms on the right-hand side of Eq. (1.3) match the unit coefficient of the y(x ) on the left. Not e that, owing to th e bal ance condition (1.4), Eq. (1.3) always has a t rivial solution y = const . The question of existence of nontrivial (Le., nonc onstant) bounded solutions is most interesti ng (and most difficult ) in the case where the righthand sid e of Eq. (1.3) involves both "advanced" (aj > 1) and "delayed" (0 < aj < 1) arg ument s. It turns out that the answer depend s crucially on t he quantity
e K := 2::Pj In oj .
(1.5)
j= l
Derfel [7] has proved that if K < 0 then Eq . (1.3) has no nontrivial bo unded soluti ons , whereas if K > 0 t hen such a solution always exists. In the "critical" case K = 0, this question has remained op en as yet . In t he present pap er , we consider a more general integra-differential equation'' of the pantograph type, namely,
1
00
- a2Y"(x)
+ al Y'(x) + y(x) =
y(a x ) J1 (da ),
(1.6)
where a l ~ 0, a2 ~ 0, at + a~ > 0 (so that al , a2 do not vanish simultaneously), and J1 is a probability measure on (0, 00),
1
00
J1 (0, 00) =
J1 (da ) = 1.
(1.7)
4 Note t hat t he theory of such equations is close ly related t o th e t heory of q-difIer en ce equat ions develop ed by BirkhofI [3] and Ad ams [1] (see also Sect ion 3 be low). 5 In fact, t he result s of the pap er [7J mentioned a bove include first- ord er eq uations of the form (1.6 ), Le., with a2 = O. Let us also remark that mor e gener al first-ord er int egr o-d ifIerential eq uations (b ut wit h delayed arguments only , i.e., o E (0, 1)) were considered by Iserl es an d Liu [15].
32
L. BOGACHEV, G. DERFEL, S. MOLCHANOV , AND J . OCKENDON
The parameter 0: in Eq. (1.6) can be viewed as a random variable, with values in (0 ,00) and the probability distribution given by the me asure JL, i.e. , P{o: E A} = JL(A), A C (0, (0). Note that Eq. (1.6) is balanced in the same sense as Eq. (1.3) , since the mean contribution of the distributed rescaled term y(o:x) is matched by that of y(x) . Moreover, Eq. (1.6) reduces to Eq. (1.3) when al = 1, a2 = 0, and the measure JL is discrete, with atoms Pj = JL(O:j), j = 1, . .. ,f (i.e., 0: is a discrete random variable, with the distribution P{ 0: = O:j} = Pj, j = 1, ... , f). As already mentioned, due to the balance condition (1.7) any constant satisfies Eq. (1.6) , and by linearity of the equation one can assume, without loss of generality, that y(O) = O. Moreover, if x > 0 (x < 0) then the righthand side of Eq. (1.6) is determined solely by the values of the function y(u) with u > 0 (respectively, u < 0) . Therefore, the two-sided equation (1.6) is decoupled at x = 0 into two one-sided boundary value problems,
1
00
- a2yl/(x) y(O)
+ alY' (x ) + y(x) =
y(o:x) JL(do:),
x
~
0,
(1.8)
= O.
For Eq. (1.6), the analog of Eq. (1.5) is given by
1
00
K :=
in 0: JL(do:)
=
E[lno: ].
(1.9)
Our main result is the following theorem, which resolves the problem of nontrivial bounded solutions in the critical case, K = 0 (and also recovers and extends the result of Ref. [7] for the case K < 0) . THEOREM 1.1. A ssume that 0 =f:. Ejln o ] < 00 , so that Kin Eq. (1.9) is well defined and the m easure JL is not concentrated at the point 0: = 1, i.e ., the random variable 0: does not degenerate to the constant 1. Under these hypotheses, the condition K ::; 0 implies that any bounded solut ion of equation (1.6) is trivial, i.e., y(x) == const, x E JR . The apparent probabilistic structure of Eq. (1.6) is crucial for our proof of this result. The main idea is to construct a certain diffusion process X t , with negative drift and "mult iplicat ion" jumps (i.e., of the form x f-t o:x) , such that Eq. (1.6) can be rewritten as .cy = 0, where .c is the infinitesimal generator of the Markov process X t . That is to say, the class of bounded solutions of Eq. (1.6) coincides with the set of bounded .c-harmonic functions . This link brings in the powerful tool kit of Markov processes; particularly instrumental is the well-known fact (see , e.g. , Ref. [9]) that for any L-harmonic function f (x), the random process f(X t ) is a martingale, and hence , for any t 2: 0,
f(x) = E[f(Xt) IXo = x ],
xER
(1.10)
On the other hand, due to the multiplication structure of independent cons ecutive jumps of the process X t , its position aft er n jumps is expressed
BOUNDED SOLUTIONS OF THE BALANCED PANTOGRAPH EQUATION
33
in terms of a background random walk Sk = 6 +.. '+~k (0 ::; k ::; n) , where ~i 's are independent random variables with the same distribution as In a. The hypothesis K ::; 0 of Theorem 1.1 implies that, almost surely (a .s.), the random walk Sn travels arbitrarily far to the left. Using an optional stopping theorem (whereby the boundedness of f(x) is important), we can apply the martingale identity (1.10) at the suitably chosen stopping (firstpassage) times, which eventually leads to the conclusion that f'(x) = 0 and hence f (x) = const. This approach also allows us to give an example of a nontrivial bounded solution to equation (1.6) in the case K > 0 (thus extending the result by Derfel [7] to the second-order pantograph equation). THEOREM 1.2. Suppose that K > 0, and set foo(x) := P{liminf X t = +oolXo = x}, t-+oo
x E lR,
(1.11)
where X t is the random process constructed in the proof of Theorem 1.1. Then the function foo(x) is L-harmonic and such that foo(x) --; 0 as x --; -00 and foo(x) --; 1 as x --; +00. When a2 = 0, Eq. (1.6) becomes
1
00
alY'(x)
+ y(x) =
y(ax) f.l(da),
x E lR.
(1.12)
In this case, the diffusion component of the random process X; is switched off, and it follows, due to the negative drift and multiplication jumps (see details in Section 4), that if X o = x ::; 0, then X t ::; 0 for all t 2: O. That is to say, the negative semi-axis (-00,0] is an absorbing set for the process X t , and hence the function foo(x), defined by Eq. (1.11) as the probability to escape to +00 starting from x, vanishes for all x ::; O. This leads to the following interesting specification of the example in Theorem 1.2. COROLLARY 1.1. If a2 = 0 in Eq. (1.6) then foo(x) 0 for all x ::; O. Moreover, Eq . (1.6) implies that all derivatives of the junction joo(x) vanish at zero, j~)(O) = 0 (k = 1,2, . . . ). Before elaborating on the ideas outlined above, we would like to make a short digression in order to consider the simple prototype example of Eq. (1.3), namely,
=
y'(x)
1
1
+ y(x) = "2 y(qx) + "2 y(q-1x)
(q =I- 1),
(1.13)
and to give several different "sketch" proofs of Theorem 1.1 in this case. Note that, according to Eq. (1.5), we have
K
1
1
1
= "2 In q + "2 In q = 0,
so Eq. (1.13) falls in the (most interesting) critical case. In fact, this example was the starting point of our work and a kind of mathematical test-tube
34
L. BOGACHEV, G. DERFEL, S. MOLCHANOV, AND J . OCKENDON
to try various approaches and ideas. Although not strictly necessary for the exposition, after some deliberation we have cautiously decided to include our early proofs (based on perturbation, analytical, and probabilistic argumerits," respectively), partly because this will hopefully equip the reader with some insight into validity of the result , and also because these methods may appear useful as exploratory tools in other situations. The rest of the paper is laid out as follows. In Sections 2, 3, and 4, we discuss the three approaches to equation (1.13) as just mentioned. In Section 5 we start a more systematic treatment by describing the construction of a suitable diffusion process with multiplication jumps. In Section 6, we discuss the corresponding L-harmonic functions and obtain an a priori bound for the derivative of a solution. Finally, in Section 7 we prove our main Theorems 1.1 and 1.2.
2. Perturbative proof. Following the ideas used by Ockendon and Tayler [24] and Fox et al. [11] in the case of the original pantograph equation (1.1), we start by observing that if q = 1 then Eq. (1.13) is reduced to the equation y' = 0, which has constant solutions only. Therefore, when the parameter q is close to 1, it is reasonable to seek solutions of Eq. (1.13) in a form that involves "superposition" of (exponentially) small oscillations (fast variation) on top of an almost constant (polynomial) function (slow variation). This leads to a WKB-type asymptotic expansion ofthe solution in terms of perturbation parameter e :::::: 0 (see Ref. [25]), which in the first-order approximation yields two first-order differential equations: a nonlinear equation (called the eikonal equation) for the fast variation and a linear equation (called the transport equation) for the slow variation. For simplicity of presentation, we will restrict ourselves to the first-order approximation, but in principle one can go on to the analysis of higherorder terms, which are described by linear equations and therefore can be determined without much trouble. To implement this approach, set q = 1 ± e (s > 0) , x = e-1u, and y(x) = y(c1u) =: f(u). Then
x y '()
= df du
. du dx
= e] '( u ) ,
and Eq. (1.13) takes the form
ef'(u)
1
1
+ f(u) = '2 f((1 ± e)u) + '2 f((1 ± e)-lu).
(2.1)
As explained in the Introduction, without loss of generality we may assume that f(O) = o. 6 It is amusing that these three methods represent nicely the traditional organization of British mathematics into Applied Mathematics, Pure Mathematics, and Statistics , which is reflected in the names of mathematical departments in most universities in the UK.
BOUNDED SOLUTIONS OF THE BALANCED PANTOGRAPH EQUATION
35
Now, suppose that, for small E:, the function f(u) admits a WKB-type expansion,
f(u) "" (Ao(u) + E:A1(u)
+ ... ) exp(E:-1V(u)) ,
(2.2)
which in principle should be valid uniformly for all u , including the limiting values u --t and u --t 00 . Differentiation of Eq . (2.2) yields
°
E:j'(u) "" (Ao(u)V'(u)+E:(A~(u)+Al(U)V'(u))+·,,) exp(E:-1V(u)) . (2.3) From Eq. (2.2) we also obtain
f ((1 ± E:) u) "" (A o(u) + E: (AI (u) ± uA~ (u)) V(u) x exp ( - - ± u V' (u) E:
+ E:U
+ ... )
2VI I ( U ) )
+ ... 2 '
(2.4)
and
f((l ± E:)-Iu) "" (Ao(u) + E:(AI(u) =F uA~(u)) x exp
(V~U)
=F uV'(u)
+ ... )
+ E: (UV'(U) + u
2
V;'(u))
+ ... ). (2.5)
Substituting the expansions (2.3), (2.4) and (2.5) into Eq. (2.1), canceling out the common factor exp (V (u)/ E:), and collecting the terms that remain after setting E: = 0, we get
Ao(u)V'(u)
+ Ao(u) = ~ Ao(u)exp(±uV'(u)) + ~ Ao(u) exp(=FuV'(u)).
Assuming that Ao(u) i= 0, this gives the equation 1 + V'(u) = cosh(uV'(u)),
(2.6)
uV'(u) u - --..,.--'--'--:-- cosh(uV'(u)) - l '
(2.7)
or equivalently
Similarly, equating the terms of order of E: and noting that Al (u) cancels out owing to Eq. (2.6), we obtain
A~(u) = A~( u) u sinh( iN' (u)) + ~ Ao(u) u2 V" (u) cosh( uV' (u)) 2
+ ~ Ao(u) uV'(u) exp(=FuV'(u))
(2.8)
We can now check that the formal expansion (2.2) is compatible with the zero initial condition, f(O) = O. Equation (2.7) implies that if u --t 0 then uV'(u) --t 00, and moreover
uV' (u) "" -In u + In In ~ u
+ ... ,
(2.9)
36
L.BOGACHEV,G.DERFEL,S.MOLCHANOV,ANDJ.OCKENDON
whence In2u 1 V (u) '" - -2- + In u . In In ;: + .. .
(2.10)
Furthermore, differentiation of Eq. (2.9) gives 1 .. · u 2 V "( u ) ",lnu-Inln-+ u
(2.11)
Inserting formulas (2.9) and (2.11) into Eq. (2.8), we obtain for Ao(u) the asymptotic differential equation
1(lnu-Inln-+ 1) .. ·
A~(u) -"'=fAo(u)
2u
u'
which solves to
1)
lnu ( In u - 2lnln;: + .. . In Ao(u) '" =fT
.
(2.12)
Finally, substituting the expansions (2.10) and (2.12) into Eq. (2.2) we obtain that f(u) ......, 0 as u......, 0, as required. Let us now explore the behavior of the solution as u ......, 00 . In this limit, Eq. (2.7) gives uV'(u) ......, 0, and moreover 2
uV'(u) ---+ ... ' 6
u'" - - - UVI(U) whenc e
uV' (u) '"
~ - ~3 + .. . ,
U
3u
4 u 2V"() u '" - -
(2.13)
+ -3u8 3 + ...
(2.14)
2 2 V(u )rvC--+-+ ... , 3
(2.15)
U
From (2.13), we also find
U
9u
where C = const. Inserting the expansions (2.13) and (2.14) into Eq. (2.8), we obtain A~ (u) 1 2 - - r v - ± -2+...
Ao(u)
U
u
'
and hence Ao(u) '" Couexp ( =f~
+ ...) .
(2.16)
BOUNDED SOLUTIONS OF THE BALANCED PANTOGRAPH EQUATION
37
°
The case Co -# is unsuitable, since Eq. (2.16) would imply that Ao(u) ----> 00 as u ----> 00 and , in view of formulas (2.15) and (2.2), the solution f(u) appears to be unbounded, which contradicts our assumption. Therefore, Co = and hence Ao(u) = 0, thus reducing the expansion (2.2) to
°
Arguing as above , we successively obtain A1(u) = 0, A 2(u) = 0, etc. This indicates that f(u) = 0, which was our aim. 3. Analytical proof. In this section, we demonstrate how the theory of q-difference equations (see Refs. [1, 3]) can be used to show that Eq. (1.13) has no nontrivial bounded solutions. In what follows, we assume that q -# 1. As explained in the Introduction (see Eq, (1.8)), Eq. (1.13) splits into two (similar) one-sided equations, so it suffices to consider the boundaryvalue problem
y'(x)
+ y(x)
=
1
1
"2 y(qx) + "2 y(q-1x),
x :2: 0,
(3.1)
y(o) = 0. Assume that y(x) is a bounded solution of Eq. (3.1) , ly(x) 1::; B (x :2: 0) , and let y(s) be the Laplace transform of y(x),
then y(s) is an alytic in the right half-plane,
lY(s)1
<
:s'
~s
~s
> 0, and
> 0.
(3.2)
On account of the boundary condition y(o) = 0, Eq. (3.1) transforms into
1 (1 + s)y(s) = 2 q y(q-l s) + ~ y(qs) ,
(3.3)
or, after the substitution ",( s) := s y( s), (3.4) Note that the estimate (3.2) implies ~s
> 0,
and in particular ",(s) is bounded in the vicinity of the origin.
(3.5)
38
L. BOGACHEV, G. DERFE L, S. MOLCHANOV, AND J. OCK ENDON
Let us rew rite Eq. (3.4) in t he form
cp(q2 S) - 2(1 + qs)cp(qs) + cp(s) =
o.
(3.6)
Equation (3.6) is a linear q-difference equation of orde r 2. According to the general t heory of such equations (see Ref. [1]), the characteristic equation for Eq. (3.6) (in t he vicinity of s = 0) reads
q2 p
_
2qP + 1 = 0,
and P1 .2 = 0 is its mult iple root. The corresponding fundamental set of solutions to Eq. (3.6) is given by
CP1(S) = P1 (s), In s
CP2(S) = P2(s) + -1- P3 (s), nq
where P1 (s), P2(s), and P3 (s) are generic power series converge nt in some neighborh ood of zero, an d In s denot es t he principal branch of the logarithm . Note t hat the solution CP2(S) is unsui t able because it is unbounded near s = 0 (see Eq. (3.5)). On the ot her han d , t he funct ion CP1(S ) is analytic in the vicinity of zero and, moreover , it can be analytically continued , step by step, into t he whole comp lex plane 1 t hen t he analytic continuation from a disk lsi::; a to the bigger disk lsi ::; qa is furnished by the formula cp(qs) = 2 (1 + s )cp(s ) - cp(q- 1s) (see Eq. (3.4)), and so on . T hat is to say, CP1 (s) can be extended to an ent ire function cp( s), which by construction satisfies Eq . (3.6) for all s E 0 cont radicts the est imate (3.5). Hence, cp(s ) = const, so t hat y( s) = const . s- l, and by t he uniqueness t heorem for t he Laplace transform this imp lies t hat y(x) == const, i.e, y(x) == y(O) = 0, as claimed. 4. Probabilistic proof. In t his sectio n, we give a pr obabilistic inte rpretation of Eq . (1.13) via a certain ruin pr oblem , and pr ove t hat the corresponding solution is constant using elementary probabilist ic considerations. Alt ho ugh our argument does not cover t he whole class of bounde d solutions , it contains some ideas t hat we will use in the second half of the paper to give a complete proof of our general result. Let us consider t he following "double-or-half" ga mbli ng model (in conti nuous time). Suppose t hat a player spends his initial capital, x , at rate v per unit time, so that afte r time t he is left wit h capital x - vt . However , at a random ti me T (wit h exponential distrib ution), he gambles by putting th e remaining capital at st ake , whereby he can either double his mone y or lose half of it , both with probability 1/2. Afte r that , the process cont inues
BOUNDED SOLUTIONS OF THE BALANCED PANTOGRAPH EQUATION
39
in a similar fashion, independently of the past history. If the capital reaches zero and then moves down to become negative , this is interpreted as borrowing, so the process proceeds in the same way without termination. In that case, gambling will either double or halve the debt , and in particular the capital will remain negative forever . More generally, if X t denotes the player's capital at time t ;::: 0, starting with the initial amount X o = x, then the random process X, moves with constant negative drift (-v), interrupted at random time instants (Ji by random multiplication jumps from its location Xi = X cr i - O (i.e., immediately before the jump) to either qx, or q-1 X i (q =1= 1), both with probability 1/2 . We assume that the jumps occur at the arrival times (J1, (J2, .. . of an auxiliary Poisson process with parameter A > 0, so that the waiting times until the next jump, 7i = a, - (Ji-1 ((Jo := 0), are independent identically distributed (Li.d .) random variables, each with the exponential distribution
t > O. According to this description, (X t ) is a Markov process, in that the probability law of its future development is completely determined by its curr ent state, but not by the past history ( "lack of memory") (see , e.g., Refs.
[9, 10]). We are concerned with the ruin problem for this model." Namely, consider the probability fo(x) of becoming bankrupt starting with the initial capital X,
fo(x)
:= P{liminf t-oo
x, :S 0 jXo =
x} == Px{To < oo},
X
E
JR,
where To := min{t ;::: 0 : X, :S O} is the random time to bankruptcy and P; denotes the probability measure conditioned on the initial state X o = x. From the definition of the process X t , it is clear that if X :S 0 then To = 0 and so fo(x) = 1. For x > 0, we note that if the first jump does not occur prior to time x ]» , then the process will simply drift down to 0, in which case To = x ]» < 00 . Otherwise (i.e., if a jump does happen before time x / v) , the ruin problem may be reformulated by treating the landing point after the jump as a new starting point (thanks to the Markov property). More precisely, by conditioning on the first jump instant (Jl (= 71) and using the (strong) Markov property, we obtain
7 A similar ruin problem for the pro cess with deterministic multiplication jumps of the form Xi f--> qx ; (q > 1), was first considered by Gaver [12], leading to the equ ation y' (x )+y(x) = y(qx) (i.e. , with advanced argument, cr. Eq. (1. 3»). The systematic theory of general processes with multiplication jumps was developed by Lev [18J.
40
L.BOGACHEV,G. DERFEL,S .MOLCHANOV ,AND J .OCKENDON
(4.1)
where in the last line we have made the substit ution u = X-V8. The rep resentation (4.1) implies that t he fun cti on f o(x ) is continuo us and, mor eover , (infinitely) differe ntiable , and by differentiation of Eq, (4.1) wit h respect to x, it follows t hat t he fun ction y = fo(x) satisfies t he generalized pantograph eq uation (cf. Eq. (1.13))
*yl(X) +y(x) =
~y(qx) + ~y(q-l X) .
(4.2)
It is easy to see t hat, in fact , t his equation is satisfied on t he whole axis, x ERAs we have ment ioned , f o(x ) = 1 for all x ::; 0, and it is now our aim to show t hat t he same is true for all x > 0, which wou ld mean t hat t he solution y = fo(x) to equation (1.13) is a constant , fo (x) == 1, x E R To this end, note t hat t he position of the process after n jumps is given by
n
=
1,2 , . . . ,
where ~n 's are i.i.d, random variables taking t he values ± 1 with probabilit ies 1/2 . By it er at ions (using that X o = x), we obtain''
X Un = (( (x - VTl) q6 - VT2 ) q~2 _ .. . - VTn ) q~n
= (x - VTl) q~lH2+ " 'Hn = «":
(x - vt
- v T2 q ~2 + " ' Hn - .. . - V Tn q ~n
(4.3)
Tiq -S' -l) ,
. =1
where Sn := 6 + 6 + ... + ~n , So := O. Not e t hat Sn can be inte rpreted as a (simple) r an dom walk, which in our case is symmetric (i.e., P { ~i = I} = P { ~i = -I} = 1/ 2) and t herefore recurrent (see, e.g ., Ref. [10]). In particular , t he eve nts An := {Sn-l = O} (n = 1, 2, ... ) occur infinit ely often, with probability 1. Furt hermore, setting B n := { Tn > I} , we note t hat the events An n B n (n = 1, 2, . . . ) are conditio nally independent, given the realization of t he random walk {Sk' k ;::: I }. Since t he ra ndom variables T n (and t herefo re t he events B n ) are inde pendent of {Sk} , we have, with probability 1, 8Similar random sums as in Eq . (4.3) ar ise in products of certain random ma trices in relati on to random walks on t he group of affine transform ati ons of the line (see Ref. [13]).
BOUNDED SOLUT IONS OF THE BALANCED PANTOGRAPH EQUATIO N ex>
L
41
ex>
P(A n n Bn l{Sk} ) =
n= l
L
P(B n) IA n
n= 1
= e->"#{n : A n occurs} =
00 ,
where 1{... } denot es t he indicator of an event . Hence, Borel-Can celli's lemma (see, e.g., Ref. [10]) implies
P (An n e; occur infinitely often I{ S k}) = 1
a.s.,
(4.4)
and by t aking the exp ect a tion in Eq. (4.4) (wit h respect t o the distribution of the sequence {Sd ), the sa me is true in the uncondition al form,
P(A n n B n occur infinitely oft en) = 1. As a consequence, the terms in the random seri es
will infinit ely ofte n exceed the value 1, all other t erms being nonnegati ve. Therefore, the series diver ges to +00 a.s. , and from (4.3) it follows that lim inf X U n ~ n - ex>
In t urn, this implies that To < as claimed.
00
°
a .s.
a.s., and so fo(x ) = 1 for all x > 0,
5 . Jump diffusions. We now pur sue a more general (and more systematic) approach. Equations of the form (1.2) ar e linked in a natural way wit h certain continuous-time Markov pr ocesses (more speci fically, diffusions with multiplication j umps) . To describe this class of pr ocesses , let us consider a Brownian mot ion B~ ' v , starting at the origin, with diffusion coefficient", 2': and nonpositi ve (constant) drift - v ~ 0,
°
or equivalently
t 2': 0, where B t = B; 'o is a st andard Br owni an mot ion (wit h continuo us sample paths). We assume t hat ",2 + v 2 > 0, so t hat B ; 'v do es not degener at e to a (zero) constant. The random pr ocess B~ 'v determines t he underlyin g diffusion dynamics for a pro cess with jumps, (XI, t 2': 0), which is defined as follows. Sup pose t hat t he jump instants are given by t he arrival ti mes 0'1, 0'2, . . . of
42
L. BOGACHEV , G. DERFEL, S. MOLCHANOV , AND J. OCKENDON
an auxiliary Poisson process with parameter A > 0, so that Ti = a, - (J'i -1 (i = 1,2, . . . ) are i.i.d. random variables with exponential distribution ,
(we set formally (J'o := 0). Furthermore, suppose that the successive jumps are determined by the rescaling coefficients Cti of the form Cti = e~i, where ~i'S are i.i.d. random variables. Then, the (right-continuous) sample paths of the process X, are defined inductively by
0=
(J'o ~
t<
(J'l ,
(J'i~t<(J'i+1,
i=1 ,2, ...
(5.1)
That is to say, the process (X t , t 2: 0) starts at point x and moves as X; = x + B;'v until a random time (J'1, when it jumps to a random point
where (1 := B~~v = B~~v - B~~v. Thereafter , the process proceeds in a diffusive way as X; = XO"l + B;'v - B~~v until a random time (J'2, when it makes the next jump to X 0"2
= e 6X =
= e 6(X + B':" e 6 +6x + e 6 H 1(1 + e 6(2, 0"2 -
0
CT1
0"2
B""V) 0"1
where (2 := B~;V - B~~v, and so on. Iterating, we obtain that the n-th jump, occurring at time (J'n , lands at the point
(5.2) where
(n=1,2 ,oo.)
(5.3)
is a sequence of i.i.d . random variables.
6. L:-harmonic functions. Let us study the properties of the process X t in gre ater detail. DEFINITION 6.1. The infinitesimal operator (generator) .c oj a Markov random process (X t , t 2: 0) is defined by (.cj)(x) := lim Ex[j(Xh)] - j(x) h-->O+ h '
(6.1)
with the domain V(.c) cons isting oj junctions j jor which the limit in
Eq. (6.1) exists.
BOUNDED SOLUTIONS OF THE BALANCED PANTOGRAPH EQUATION
43
In a standard way, by considering possible scenarios for the process (X t ) up to an infinitesimal time h (see Ref. [9]), one obtains the following . PROPOSITION 6 .1. For the random diffusion with jumps (X t ) defined
above, its generator E acts on bounded C 2 -smooth functions as ",2
(£f)(x) = where
~
2
+ A(E [f(ee x)] -
j"(x) - vf'(x)
f(x)) ,
(6.2)
is a random variable with the same distribution as anyone of the
i. i. d. random variables 6, 6
,... .
function f is called £-harmonic if E] = 0. Note that the expectation in Eq. (6.2) can be written as a Stieltjes integral, DEFINITION 6.2. A
1
00
E[f(eex)]
=
f(zx) dF(z),
wher e F(z) is the cumulative distribution function of the random variable Z = ee, i.e., F(z) := P{ee :::; z} = P{~ :::; lnz} (0 < z < 00) . Hence, the equation E] = 0, with £ given by Eq. (6.2), is equivalent to ~
-2 j"(x) + vf'(x) + Af(x) =
roo
A Jo f( zx) dF(z) ,
which is a balanced generalized pantograph equation of the form (1.6) . Our aim is to study the class of bounded £-harmonic functions . Let us denote by II . II the standard sup-norm on R
Ilfl\
:= sup xER
If(x)l·
As a first step, we estimate the derivative of an £-harmonic function. PROPOSITION 6.2. Suppose that v > 0. If Ilfll < 00 and £f = then
°
11f' 1 < 00 .
Proof. If r: = 0 then, according to Eq, (6.2), the equation E] = 0 takes the form vf'(x) = A (E [f(ee x)] - f(x)), which gives
\11' 11 s
~v~ Ilfll <
00.
For x > 0, the condition L] = 0 is equivalent to
j"(x) - 'Y f' (x ) = -g(x) , where 2v > 0, r:
'Y := '2'
2A g(x) := '2' (E[f(eex)] - f(x)) .
'"
(6.3)
44
L. BOGACHEV, G. DERFEL, S. MOLCHANOV, AND J . OCKENDON
Solving equation (6.3), we obtain
1 1
00
f'(x) =
00
= where C = const. Since C =I- 0 then
=
11f'11
(6.4)
g(u + x) e-"Y du + cc-, U
Ilgll S 4>.,..-21Ifll <
f'(x) = 0(1) so that lim x->+oo f(x) Therefore, C = 0 and
g(u) e-"Y(U-X) du + Ce"Y X
00 ,
+ Ce"Y x
---.
00, Eq, (6.4) implies that if (x ---. +00),
00
which contradicts the assumption
Ilfll < 00 .
s Ilgll roo e-"YU du = M < 00. lo
I
o
The proof is completed.
7. Proof of the main results. Let (Ft , t 2: 0) be the natural filtration generated by the process (X t ) , i.e., F t = er{X., sSt} is the minimal rr-algebra containing all "level" events {Xs S c} (c E JR, sSt). Intuitively, F t is interpreted as the collection of all the information that can be obtained by observation of the random process (X s ) up to time t. As is well known (see, e.g., Ref. [9, Ch. 4]), if a function f is .c-harmonic then the random process f(X t ) is a martingale relative to (Ft ) , i.e., for any o S « <», with probability 1, (7.1) In words , this means that if we are trying to predict the mean value of the martingale f (Xd at some future time t using its past history up to time «
t 2:
o.
(7.2)
In addition , if f is bounded then, by Doob's optional stopping theorem (see, e.g., Ref. [29, §8]), Eq . (7.2) extends to
Ex[f(XT) ] = f(x) ,
(7.3)
where T is a random stopping time, i.e., such that {T ::; t} E F t for each t 2: o. In words , one should be able to decide whether the random time T has occurred by observing the process up to a given time. The martingale
BO UNDED SOLUT IONS OF THE BAL A NCED PAN TOGRAPH EQUATION
45
pro perty (7.3) is crucial in t he proof of our main theor em below, wher e we will apply it to t he special sequence of stopping t imes. THEORE M 7.1 (d. Theorem 1.1 ) . A ssume that E [~ ] ~ 0 and P{~ =FO} > O. If £f = 0 and Il f l < 00 , then f (x ) = const, x E JR . Proof For r > 0, set N; := min{n : Sn ~ -r} , where Sn = 6 + .. .+ ~n , and define 00
t; := aN r = L an l n=l
(7.4)
{ N r =n },
where (ai) are the time instants of successive j umps (see Section 3). The assumption E [~] ~ 0 impli es (see, e.g., Ref. [10]) that liminfn -+ oo Sn = -00 a .s., so that N; < 00 a .s. and therefore the random vari able T; is well defined . Not e that T; is a st opping time for the random pro cess (X t ) , i.e., {Tr ~ t} E F t for each t ~ O. Indeed, suppose that we are given a sample path of the pr ocess X s up to time t , and in particular we know t he t ime inst ants a, and th e magnitudes of all the j umps prior to t. Then , using Eq, (5.2) , we can reconstruct the corresponding value s? ~i = In(XujXu;-o). In turn , this allows us t o det ermine if the threshold (-r ) has been re ached by t he associat ed random walk Sn and, t herefore, whether or not the condition T; ~ t holds , as required . Now, applying the optional sto pping t heorem in t he form (7.3), we obtain
f (x ) = Ex[f (XTr )] = E[J(eSNrx + te rms independent of x )).
(7.5)
First , suppose that v > O. Differen t iation of Eq. (7.5) wit h resp ect t o x gives (7.6) whence, using Propositi on 6.2, we get
111'11~ e- r 111'11< 00 . Letting here r -+ + 00 , we conclude that 111'11 = 0 and so If v = 0 then , differenti ating Eq . (7.6) we get
1" (x ) = E[e2SNr 1" (eSNr x
f = const.
+ .. .)),
so t hat
111"11~ e- 2 r ll1"11< 00 . 9There is a slight pr oblem if X ,,; probability , so m ay be igno red .
0
= X,, ; = 0,
(7.7)
but t his only ha pp ens wit h zero
46
L. BOGACHEV, G. DERFEL, S. MOLCHANOV, AND J . OCKENDON
Equation (6.3) (with v = 0) implies II!"II :S Ilgll < 00, so taking r -> +00 in Eq. (7.7) yields !,,(x) == O. Therefore, f(x) = C1X+eo, but since Ilfll < 00, we must have C1 = 0, so that f(x) == Co = const. 0 PROPOSITION 7.1. Denote A oo := {lim inf t --+ oo X, = +oo} , and consider the probability of the event A oo as a function of the initial point of the process X t ,
xER
Then the fun ction foo(x) is £-harmonic, i.e ., (£foo)(x) = 0,
x E JR,
where £ is the generator of the random process (X t ) given by Eq. (6.2). Proof Conditioning on X h « +(0), by the Markov property we obtain foo(x) = Ex [Px(Aoo IX h ) ] = Ex[P XI> (A oo )] = Ex[foo(X h ) ],
when ce, by the definition (6.1) , it readily follows that £foo = O. 0 THEOREM 7.2 (cf. Theorem 1.2). Suppose that E[~ ] > O. Then the fun ction f 00 (x) is a nontrivial bounded £-harmonic function; in particular, f oo(x) -> 0 as x -> -00 and foo(x) -> 1 as x -> +00. Proof The function foo(x) is £-harmonic by Proposition 7.1. In order to obtain the limits of foo(x) as x -> ±oo, note that 1- foo(x) = Px{liminf x, t--+oo
< +oo}
= Px {lim inf X(J"n < +oo} n--+oo = Mlim PX{X(J"n:S M infinitely
(7.8) often} .
--+00
According to Eq. (5.2), the condition X(J"n :S M can be rewritten as
Since E[~] > 0, the strong Law of Large Numbers implies that , with probability 1,
Sn
rv
nE [~] ->
+00
(n
->
(0 ).
(7.10)
It follows that if the inequality (7.9) holds for infinitely many n, then x
+ liminf ((1 + (2e-S1 + . .. + (ne-Sn-1) :S n- oo
lim M e- sn = O. n_~
(7.11)
BO UNDED SOL UTIO NS OF THE BALANCED PANT OGR APH EQUATION
47
Moreover , using the esti mate (7.10) and recalling t hat (i are LLd. random variables (see Eq. (5.3)), it is easy to show (e.g., using Kolmogorov's "t hree series" theorem , see Ref. [10]) t hat the random series 00
TJ :=
L (n e-s" - I
(7.12)
n =l
converges wit h prob abil ity 1. Therefore, from Eqs . (7.9), (7.11) and (7.12) it follows t ha t for any M > 0,
PX{XcT" ::; M infinit ely often} ::; P{TJ::; - x}. Returning to Eq. (7.8), we deduce that
1 2: foo (x ) 2: 1 - P{TJ ::; -x} = P{TJ > -x}->1
(x ----- + 00).
On th e other hand, writing t he left-hand side ofEq. (7.9) as x+ TJ+8 n , where 8n ----- 0 a.s ., we have, for any e > 0, M > 0,
+ TJ + 8n ::; 0 for all n lar ge enough} ::; P {x + TJ + 8n < M e- s; infinit ely ofte n},
P {x + TJ ::; - €} ::; P {x
which in view of Eq. (7.8) implies
0 ::; foo(x) ::; 1 --- P{TJ ::; -x - s} =P {TJ > - X - €}-----O
(x ----- - 00).
o
Thus, the proof is complete d.
Acknowledgments. P art of this research was done when t he second aut hor (G.D.) was visiting t he University of Cambridge in May-J une 2005, and his thanks are due t o Arieh Iserles for stimulating discu ssions and useful remarks. The t hird aut hor (S.M.) grat efully acknowledges t he support from the Center of Advanced Studies in Mathematics of the Ben Gurion University during his visit in May-June 2006, and he would like to thank Michael Lin for kind hospitality.
REFERENCES [1J C.R . ADAMS, L in ear q-difference equations, Bull. Amer . Mat h. Soc. 37 (1931), 361-400. [2J V.A . AMBARTSUMJAN , On the theory of brightness fl u ctu a tion s in the M ilky Way, (Russian) Doklady Akad. Nauk SSSR 44 (1944), 244-247; (E nglish t ra nslatio n) Compt . Rend. (Doklady) Acad . Sci. URSS 44 (1944), 223-226. [3] G.D . BIRKHOFF , The genemlized R iemann problem for lin ear differentia l equations and the allied probl ems for linear difference and q-difference equ ations, P roc. Amer. Acad . Arts Sci. 49 (1913), 521-568.
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L. BOGACHEV, G . DERFEL,S. MOLCHANOV, AND J . OCKENDON
[4] G . DERFEL, Functional-differential and functional equations with rescaling, in Operator T heory and Boundary Eigenvalue P roblems (International Workshop, Vienna, July 27-30, 1993) , Operator Theory: Advances and Applications, Vol. 80, I. Gohberg an d H. Langer (eds.), Birkhauser, Basel, 1995, pp. 100Ill. [5J - - - AND A. ISERLES, The pantograph equation in the complex plane, J. Math. Anal. Appl. 21 3 (1997) , 117-132. [6] - - - AND F . VOGL, On the asymptotics of solutions of a class of linear functiona ldifferential equations, European J . Appl. Math. 7 (1996), 511-518 . [7J G.A . DERFEL, Probabilistic method for a class of functional-differential equa tions, (R ussian) Ukrain. Mat . Zh. 4 1 (1989), 1322-1327; (English translation) Ukrainian Math. J . 4 1 (1990), 1137-114l. [8] - -- AND S.A . MOLCHANOV, Spectral methods in the theory of functionaldifferential equations, (R ussian) Mat. Zametki 4 7(3) (1990), 42-5 1; (E nglish translation) Math. Notes 4 7 (1990) , 254-260. [9J S.N . ETHIER AND T .G. KURTZ, Markov Processes: Characterization and Conve rgence, Wiley Series in P robability and Mathematical Statistics, John Wiley & Sons, New York , 1986. [10] W. FELLER, An I n trodu cti on to Probability Theory and I ts Applications, Vol. II , Wil ey Series in Probability and Mathematical Statistics, 2nd ed ., Jo hn Wiley & Sons, New York, 1971. [11] L. Fox , D .F . MAYERS , J .R. OCKENDON, AND A .B . TAYLER, On a functional differential equation, J . Inst . Math. Appl. 8 (1971), 271-307. [12] D .P. GAVER, JR ., An absorption probabi lity problem, J . Math . An al. Appl. 9 (1964 ) , 384-393 . [13] A .K . GRINTSEVICHYUS, On the continuity of the distribution of a sum of dep endent variables con n ected with independent walks on lines, (Russian) Teor. Vero yatn. i Primenen. 1 9 (1974) , 163-168; (English translation) Theory P roba b. Appl. 19 (1974) , 163-168. [14J A . ISERLES, On th e generalized pantograph fun ctional-differential equation, European J . Appl. Math. 4 (1993) , 1- 38. [15] - - - AND Y .K. LIU, On pantograph integro-differential equations, J. Int egr al Equations Appl. 6 (1994), 213-237. [16] T . KATO , Asymptotic behavior of solutions of the funct ional differential equation y'( x) = aY(Ax ) + by(x), in Delay and Functional Differential Equations and Their Applications (Proc. Conf., Park City, Utah, March 6-11 , 1972) , K. Schmitt (ed .) , Academic P ress, New York , 1972, pp. 197- 217. AND J.B. McLEOD, The functional -differential equation y'(x) = aY(Ax) + [17] - by(x) , Bu ll. Amer. Math. Soc . 7 7 (1971), 891-937. [18J G.SH . LEV, Semi-Markov processes of multiplication with drift , (R ussian ) Teor. Veroyatn. i Primenen. 17 (1972) , 160-166; (English translation) Theory P robab. Appl. 17 (1972), 159-164. [19J K. MAHLER, On a special fu n ct ional equation, J. London Math. Soc. 15 (1940) , 115-123. [20] J .C . MARSHALL, B . VAN-BRUNT, AND G.C . WAKE, A natural boundary for so lutions to the second order pantograph equation, J . Math. Anal. Appl. 2 99 (2004) , 314- 32l. [21J T .E . MASON , On properties of the so lutions of linear q-difference equations with entire function coefficients, Amer. J . Math. 3 7 (1915), 439-444. [22J G .R. MORRIS, A. FELDSTEIN, AND E.W . BOWEN, The Phraqm en -Lin delof principle and a class of fun ctional differential equations, in Ordinary Differential Equations (Proc. Conf., Math . Res . Center , Naval Res. Lab., Washington, D.C ., 1971) , L. Weiss (ed.), Academic Press, New York , 1972, pp. 513-540. [23] R. J. OBERG, Local theory of complex functional differential equations , Trans. Am er. Math. Soc . 1 6 1 (1971) , 302-327.
BOUNDED SOLUTIO NS OF THE BALANCED PANTOGRAPH EQUATION
49
[24] J .R . OCKENDON AND A .B. TAYLER, The dynami cs of a current collection system for an electric locomotive, Proc. Royal Soc . London A 322 (1971), 447-468. [25] B .K. SHIVAMOGGI, P ert urbation Methods for Differential Equations, Birkhauser, Boston, 2003. [26] V . SPIRIDONOV, Universal superpositions of coherent states and self-similar potentia ls, Phys. Rev . A 52 (1995) , 1909-1935. [27] E .C . T ITCHMARSH , The Theory of Fun ction s, 2nd ed., Oxford Un iversity Press, Oxford, 1939. [28] G.C. WAKE, S. COOPER, H .K . KIM, AND B . VAN-BRUNT, Functional differential equations for cell-growth models with dispersion, Commun. Appl. Anal. 4 (2000), 561-573. [29] J . YEH, Martingales and Sto chastic Analysis, Series on Multivariate Analysis, Vol. 1, World Scientific, Singapore, 1995.
NUMERICAL METHODS FOR NON-ZERO-SUM STOCHASTIC DIFFERENTIAL GAMES: CONVERGENCE OF THE MARKOV CHAIN APPROXIMATION METHOD* HAROLD J . KUSHNERt Abstract. The Markov chain approximation method is an efficient and popular collection of methods for the numerical solution of stochastic control problems in continuous time, for reflected-jump-diffusion-type models and the convergence proofs have b een extended to zero-sum stochastic differential games. We apply it to a class of nonzero -sum stochastic differential games with a diffusion system model where the controls for the two players are separated, It is shown that equilibrium values for the approximating chain converge to equilibrium values for the original process and that any equilibrium value for the original process can be approximated by an eo-equilibrium for the chain for arbitrarily small eo > O. The actual numerical algorithm is that for a stochastic game for a finite-state Markov chain.
1. Introduction. The paper is concerned with the convergence proofs of the Markov chain approximation method to non-zero-sum stochastic differential games. The method is widely used for the numerical solution of stochastic control and optimal control problems in continuous time, for controlled reflected-jump-diffusion type models. The method was extended to zero-sum stochastic differential games in [15, 16, 17]. The Markov chain approximation method has been used to solve non-zero-sum differential games [11, 12], but there have not been results concerning convergence of the numerical equilibrium values to the equilibrium values for the original diffusion model. Papers such as [4] deal with approximations to non-zerosum games in normal form , and do not apply to the diffusion-type system models or to the type of approximations that we use. We will work with a discounted cost problem for a diffusion model in a compact set G, with absorption on the boundary. This particular state space and cost function are chosen to simplify the development , allowing us to concentrate on the issues that are particular to the non-zero-sum case. One can replace the boundary absorption by boundary reflection, with the boundary and reflection directions satisfying the conditions in [18, 15]. We will work with two-player games for notational simplicity, but the methods allow the use of any number. The non-zero-sum game is difficult because, as opposed to the zero-sum case , the players are not strictly competitive and have their own value functions. Weak convergence methods as in [18] are still employed, but many of the key techniques that were used previously cannot be carried over and new methods must be applied. The proof for the two-person zero-sum game in [15] has the advantage that the controls are *This work was partially supported by NSF grant DMS-0506928 and ARO contract W911NF-05-10928. t Applied Mathematics Department, Brown University, Providence, RI 02912 (h j k lQdam. brovn . edu).
51
52
HAROLD J . KUSHNER
determined by a minmax operation and that there is a single cost function, so that one player's gain is another's loss, properties that the non-zero-sum game does not have. This difference creates difficulties for the non-zerosum case that require considerable modification of the proofs. The methods that are employed require the use of strong-sense, rather than with weaksense solutions as in [18]. Unlike the single player problem. one must work with strategies and not simply controls. The Markov chain approximation technique first approximates the controlled diffusion by a Markov chain on a finite state space with a discretization parameter h. The cost function is then approximated to apply to the chain. The numerical procedure is to solve the game problem for the chain model. Then one proves convergence, namely, that the equilibrium or e-equilibrium values for the two players for the approximating chain model converge to equilibrium or e-equilibrium values for the diffusion model as the approximating parameter h goes to zero . The methods of proof are purely probabilistic, no PDE techniques are required. The key condition that the approximating chain must satisfy is a simple "local consistency" condition. Getting such approximations is straightforward and many methods are in [18]. The approximating chains are similarly obtained for the game problem. The numerical approximations are processes which are close to the original, which allows great flexibility. In Section 2, the model and the cost functions for the players are defined, the boundary conditions discussed and a review of some background material is given. The convergence proofs use the fact that the player's strategies for the original diffusion process can be simplified (uniformly in the controls), with various approximations to the controls. The first of such results is given in Section 3, which shows that the diffusion model can be uniformly approximated by a discrete-time model, and the relation between randomized and relaxed controls is developed. Section 4 contains the main results concerning approximation of the policies of the players. It is shown the general relaxed controls be discretized in time, approximated by finite-valued ordinary controls, and delayed slightly, without changing the values very much. The definition of equilibrium is in Section 5, where it is also shown that strategies for the players can be well-approximated by a "smoot h" conditional probability, depending only on selected samples of the driving Wiener process (and not on the entire Wiener process). This representation will be of great help in the proofs. The Markov chain approximation is introduced in Section 6 The proofs depend on certain representations of approximations to the approximating chains that allow us to show that the costs for the chain itself change little if the controls are approximated. Some such results are in Section 7. These results are new and should be useful in dealing with numerical approximations. The approximations in Section 7 are applied in Section 8 to show that the 'approximate" equilibrium (values or strategies) for the diffusion are approximate equilibrium (values or strategies) for the
53
NUMERICAL APPROXIMATIONS FOR GAMES
chain for small h. If the e-equilibrium values for the chain are unique for small E > 0, then the convergence proof is complete since an "approximate" equilibrium value for the chain is also one for the diffusion. If the value is not unique then the proof of this last fact is more difficult, and we restrict attention to the case where the diffusion coefficient is constant. The proof uses a representation of the chain that is close to a discrete-time system, and the proof of this representation exploits a strong approximation theorem. This is also new and is developed in Section 9. The final part of the convergence proof is in Section 10. Part of the proof uses a weak convergence analysis as in [18], and the paper is organized to take advantage of those results to the extent possible so that we can avoid duplicating details. For S a topological space, let D[S; 0, 00) denote the S-valued functions on [0,00) that are right continuous and have left-hand limits, and with the Skorokhod topology [6, 18] used. If S = JRv , then we write D[S; 0, 00) = DV [0,00).
2. The model. The system model is x(t)
t
= x(O) + Jo o
2
I)i(X(S), ui(s))ds i=l
+
1 t
O'(x(s))dw(s),
(2.1)
0
where x(t) E JRv, Euclidean v-space, Player i, i = 1 uses control Ui(') ' and w(·) is a standard vector-valued Wiener process. Let (3 > 0 and define E~ as the expectation conditioned on the use of control u(·) and initial condition x. The cost function for Player i is
=
(U1(-),U2('))
where 7 is the first time that the boundary BG of a compact set G is hit (7 = 00 if the boundary is never reached). The set G satisfies (A2.1) below. Define b(·) = b10 + b20 and k(·) = k 10 + k 2( ·). Let Ui denote the set of admissible controls for Player i: Ui( ') E Ui if it is measurable, non-anticipative with respect to w(·), and Ui-valued. The following condition on the regularity of the functions is assumed. A2.1. bi 0, and 0'(.) ere bounded and continuous and Lipschitz continuous in x J uniformly in u. The controls Ui(.) fOT Plauer i take values in Ui , a compact set in some Euclidean space, and k i(·) and giO are bounded and continuous.
The first boundary hitting time T. The proof of convergence, whether here or for the problems in [13, 18], generates a sequence of process approximations (continuous-time interpolations of the approximating chain) and the exit or boundary hitting times of this sequence has to converge to the exit time of the original diffusion process (2.1) . For the convergence of any numerical procedure, one must assure that something analogous takes
54
HAROLD J. KUSHNER
place. If the costs for the approximating problem are to converge to the costs for (2.1) , (2.2), then we need to assure (at least with probability one) that the paths of the process x(·) are not "tangent" to 8G at the moment T of first hitting the boundary. For r/>(.) in DV[O, 00) (with the Skorokhod topology used), define the function f(r/» with values in JR+ = [0,00] by: f(r/» = 00, if r/>(t) E GO, the interior of G, for all t < 00, and otherwise use
f(r/»
= inf{t :
r/>(t) rt. GO}.
A2.2. For a continuous real-valued function ( .) on JRv, define G = {x : (x) s:; O}, and suppose that it is the closure of its interior {x : (x) < O} . For each initial condition and control, the function f(.) is continuous (as a map from DV [0,00) to the compactified interval [0,00]) with probability one relative to the measure induced by the solution to (2.1). The tangency problem would be a concern with any numerical method, since they all depend on some sort of approximation. For example, the convergence theorems for the classical finite difference methods for elliptic and parabolic equations generally use a nondegeneracy condition on a(x) in order to (implicitly) guarantee (A2.2). This issue is discussed in [13, Section 4.4] and in [18, Section 10.2]. In [18, Section 10.2] a randomized stopping criterion is described, which assures that (A2.2) holds and has minimal effect on the costs . See also [18, p 280, sec ed.] where it is shown that the Girsanov transformation method can playa useful role in the verification of (A2.2). Note that typical methods for the solution of parabolic or elliptic equations with Dirichlet boundary conditions require nondegeneracy and smooth boundaries, which assures (A2.2). Relaxed controls: Review. When proving the convergence of sequences or approximations in control theory, there is a great advantage in working with "relaxed" controls in lieu of ordinary controls. They are used only for the purposes of the convergence proofs and have no role in practice. Let {Ft , t < oo} be the filtration on the probability space and let w(·) be a standard vector-valued FrWiener process. Let riC),i = 1,2, be measures on the Borel sets of Ui x [0,00) such that ri(Ui x [0,tJ) = t and the process ri(A x [O, .J) is measurable and non-anticipative for each Borel set A C Ui. Then ri(') is an admissible relaxed control for Player i [7, 18]. We will use Ui for the set of admissible relaxed controls as well as for the admissible ordinary controls. For Borel sets A CUi, write ri(A x [to, tl]) = ri(A, [to, t!l) , and use ri(A, tl) if to = 0. Define U = U I X U2 and U = U I X U2 . For almost all (w, t) and each Borel A CUi, one can define the left derivative (w.p.1) '(A ) _ li ri(A, t) - ri(A, t - 0) r i , t - 1m s: • 8-+0
U
We can suppose that the limit exists for all (w, t), with no loss of generality.
NUMERICAL APPROXIMATIONS FOR GAMES
55
Then for all (w, t), r~(-, t) is a probability measure on the Borel sets of U, and for any Borel set B in U, x [0,00),
r f{("'i ,t)EB}r~(dai, t)dt.
roo
ri(B) =
io i;
If Ui (.) is an ordinary control, then it has the relaxed control representation riC) define by r~(A , t) = h(Ui(t)) , where fA is the indicator function of the set A. The weak topology [18] will be used on the space of admissible relaxed controls. In this topology, any sequence of relaxed controls is compact; hence it has a weakly convergent subsequence. The use of relaxed controls does not change the range of values of the cost functions. The system and cost functions with relaxed controls. Define the "product" relaxed control r(·) by its derivative r' (-,t) = r~ (-,t )r2(" t) . Thus r(.) is a product measure, with marginals ri(')' i = 1,2. We will usually write r(·) = (rlC), r2(')) without ambiguity. The pair (w(·), rC)) is called an admissible pair if each of the riC) is admissible with respect to w(·) . In relaxed control terminology, (2.1) and (2.2) are written as
x(t) = x(O)
+
r1
io
o.
b(x(s), ai)r'(dai, s)ds
+
r
io
a-(x(s))dw(s).
(2.1r)
3. A discrete time approximation and randomized controls.
A relaxed control can be approximated by a randomized ordinary control. This is best illustrated for a discrete time system, for which we will also have need. Let 6. > 0 and suppose that r(t,·) is adapted to F nCl - for t E [n6.,n6. + 6.). Then define x Cl(n6.) recursively by xCl(O) = x(O) and for n ~ 0,
x Cl(n6.
+ 6.) = x Cl(n6.) +
t::
b(x Cl(n6.), a)r'(da, s)ds nCl U +a-(xCl (n6.)) [w(n6. + 6.) - w(6.)].
(3.1)
For t E [n6.,n6. + 6.), the continuous time interpolation can be defined either by x 6(t) = x 6(n6.) or by
x 6(t) = x 6(n6.)
1 t
+
+
it 1
b(xCl(n6.), a)r'(da, s)ds
nCl U a-(xCl(n6.))dw(t),
(3.2)
n6
The cost functions W i6 (X, r) for the interpolated processes are given by (2.2r) , with x 6 C) replacing x(·) and the relaxed control in (3.1) or (3.2) used.
56
HAROLD J . KUSHNER
A note on convergence. Note the following fact concerning the approximation. Let {r Cl (.)} and r (·) be admissible relaxed controls with r Cl (.) --> r (·) w.p.1 (in the weak topology) as!::l --> 0, and r Cl( ., n!::l ) adapted t o F nCl - , as in (3.1). Then it is easy to see that , as !::l --> 0, t he sequence of solutions {x Cl (.)} converges w.p.1, uniformly on any bounded time interval and the limit (x( ') ,r (·),w(·)) solves (2.1r) . By the boundary continuity (A2.2), concerning the w.p.1 continuity of the hitting times, the sequence of first hitting times of the boundary for xCl O converges w.p.1. to that of the limit. These facts imply that the costs converge to those for the limit processes . Randomized controls. We now show that a relaxed control can be viewed as a limit of randomized controls, a fact that will be needed in the convergence proofs. Recall that ri ( ', [n!::l, n!::l +!::lJ) / !::l is a probability distribution on the Borel sets of Ui . Define the randomized control version of this distribution, as follows. Let r( ·) be an admissible relaxed control and let E nCl denote the conditional expect at ion given F nCl - . Let uf"n be a random variable with the conditional distribution r f"n (' ) = EnCl~i (· ,[n!::l ,n!::l+!::lJ) /!::l. Define u~ = (U~n, U2\)' and let u6. 0 denote the continuous time interpolation, with interpolation intervals !::l. Finally, define x Cl (.) by xCl (O ) = xCl (O ) = x(O) and
= xCl(n!::l) + !::lb (xCl (n!::l ), u~ )
xCl (n!::l +!::l )
+a(x Cl (n!::l ))[w(n!::l + !::l) - w(n!::l) ].
(3.3)
wit h continuous time interpolation xCl (t) . Define r~ 0 = r~n O r~\ ('), and let r Cl (.) be the rel axed control with derivative r~ O on [ni , n!::l -+- !::l ). Then we have the following result, where rCl O is used for xClO in (3.2). The theorem shows that relaxed cont rols are approximat ions of randomized cont rols. THEOREM 3 .1. Under (A2.1) and using r~ O f or r' (·,s ),s E [n!::l , n!::l + !::l ), in (3.1) and (3.2), for any T < 00, lim
sup
2
Cl--.O x (O)EG rEU
lim
sup
= 0,
supEsup IxCl(t) - x(t)1 t~T
supEsup jxCl (t) - x Cl(t )1
Cl--.O x (O)EG rE U
t ~T
2
= O.
(3.4a) (3.4b)
If we add the condit i on (A2.2) then the costs for (3.1) and (3.3) converge (uniformly in x(O ), rO) to those for (2.1r) . Proof. Define ox~ = xCl(n!::l) - xCl (n!::l ). Then
8X~+1 = ox~ + !::l
l
[b(x Cl( n!::l), a ) - b(xCl (n!::l),a )]
r~ (da)
+ [a(xCl(n!::l )) - a(x Cl (n!::l ))] [w(n!::l + !::l ) - w(n!::l) ] + M~ ,
(3.5)
57
NUMERICAL APPROXIMATIONS FOR GAMES
where
By the definition ofii~O via a conditioning on FnA- , the random variable M~ is an FnA-- martingale difference and its variance is 0(6 2 ) , uniformly in the controls. With these facts in hand, the Lipschitz condition can be used in the usual way to show (3.4b) and the fact that lim
sup
sup sup
A->O x(O )EG rEU t5.T
E
lit
t
e- f3t k(xA(s), fjA(S))dS-i
l
e- f3t k(x A (s), o:))rA,I (do:, S)dSI =0.
The proof of (3.4a) is similar and is omitted. Now (3.4) and (A2.2) imply that the sequence of first hitting times also converges, uniformly in the controls Then the last expression and (3.4) imply that the costs converge, also uniformly in the controls. 0 4. Approximating the controls. In the convergence proofs we exploit the fact that, without loss of generality in the proofs, the controls can be restricted to be finite-valued , piecewise constant and slightly delayed. The (uniform) approximation facts are developed in this section. Theorem 4.1 shows that we can represent the strategies in a particular "conditional probability" form, which will be convenient in the proofs. We start with a canonical form for the control approximation. For each admissible relaxed control r(·) and each E > 0, let riO be an admissible relaxed control such that, for each T < 00 and bounded and continuous real-val ued function cPi 0 , lim sup Esup
€->O riEUi
t5.T
r1
IJo u, cPi(O:i)
[«do: i , s) -
r~,1(dO:i, s)] -
= 0,
(4.1)
i = 1,2,
Let x( ·) and x€( ·) denote the solutions to (2.1r) corresponding to rO and r€(·), respectively. The Wiener processes are the same, but the initial conditions might be different. In particular, define x€(.) by!
x €(t) = x €(O)
+
it l
b(x€(s), o:)r€,1(do:, s)ds
+
it
o-(x€(s))dw(s). (4.2)
The next theorem shows that the solution x( ·) process and costs are (uniformly) continuous in the controls in the sense of (4.3) and (4.4) . lThe processes x C) and x€(· ) depend on r(· ) and r€(.), but this dependence is suppressed in the notation.
58
HAROLD J. K USHNER
4 .1. Assume (A2.1) and let (r( ·),r' (·)) satisfy (4.1). Then
THEOREM
for each t lim
2
sup
supEsuP lx' (t )- x (t )1 =0.
,-+0 x (O),x' (O) :lx ' (O) -x (O)I-+o r EU
.~t
(4.3)
If we add the condition (A2.2), then lim
sup IWi( x , r ) - Wi (x , r ' )1= 0,
sup
i
= 1,2 .
(4.4)
, -+ 0 x(O) ,x' (O):lx ' (O)-x (O)I-+o rEU
Proof We comm ent only on the use of (4.1), since the proof is close to that of Theorem 3.1. Define <5x'( t ) = x' (t ) - x (t ). Then <5x' (t) = <5x'(O) +
1°t II t
+ +
t f [b (x'(s), a) Jo Ju
b(x(s) , a)] r'Ldcx, s)ds
[a(x'(s )) - a(x(s))] dw(s)
(4.5)
b(x' (s), a) [r, ,t(da, s) - r' (da , s)]ds.
We need to show that the last term is small, uniformly in the control for sma ll E. By (4.1), t his would be the case if b(.) was a continuous functi on that depend ed only on time. The fact that x' (-) is equic ontinuous, with a high prob ability will imply t he result; in parti cular , we will use t he fact t hat lim sup sup sup E sup Ix'(L\ >" -+0
r
'
1>"9
.~>..
+ s) - x' (l>' )12 =
O.
(4.6)
Write t he last term of (4.5) as 2
1 Juf L 1
[t />..] - l
L
1= 0
[t/ >.J-l
+
1>'+ >'
b(x' (l>. ), a ) [r"' (da ,s )-r' (da , s)]ds+O(>' )
I>.
l>..+ >'
(4.7)
[b (x' (s), a) - b(x'(l>'), a )] [r"'(da, s)-r' (da, s)] ds.
1=0
L>.
(4.6) implies that, as >. ----> 0, the expectat ion of the squ ar e of the sup of t he last term over any finite t ime interval goes to zero, uniformly in r(-),r' (·), x (O ), x'(O) , as E ----> O. Now assumption (4.1) impli es that first t erm of (4.7) goes to zero as E ----> 0, uniformly in >. and the controls and init ial conditio ns. (4.3) is a consequence of these facts and the Lipschitz condition. The convergence of the costs follows from t he convergence of t he paths and cont rols, which impli es the conver gence of the exit t imes (via (A2.2)) . 0 2 [tl
A] denotes t he integer part of tl A.
59
NUMER ICAL AP P ROXIMAT IONS FOR GAMES
Finite-valued and piecewise constant approximations. Next, we choose t he approximations in (4.1) t hat will be of interest . F irst we discretize t he Ui . Su ppose t hat U, E IRc;, Euclidean ci-space.3 For each p, > 0, part it ion IRc; into disjoi nt hypercubes {Rr ,/} with diameters u: The bou ndar ies ca n be assigned to t he adjoining sets in any way at all. Define Uf ,1 = U, n Rr ,1 and let Uf ,l, i = 1, . .. , pr , denote t he finite number of non- em pty int ersecti ons. Fix a point ar,1 E Now, for relaxed cont rols (r IO , r2(.)), define approximating relax ed controls r1 0, on t he control value space Uf = {a1,l, l S; pn, via t heir derivatives, rr " (ar ,l,t) = r~ ( Uf'l , t) . The set of su ch finit e-valued a nd piecewise constant cont rols is denot ed by Ui (p,). The next theor em follows from Theorem 4.1.
tr;'.
THE OREM 4.2. A ssume (A2.1)-(A2.2), and the above approximation E Ui(p, ), i = 1,2. Then (4.1)and of either one or both of the ri(' ) by Theorem 4.1 hold for p, replacing e, and the approximation is uniform in the choice of {a1,l}.
rro
Delayed controls. Theorem 4.2 holds if t he actions of t he discretized controls are delayed slightly and approximated by piecew ise constant orand r10 E Ui (p, ). Define dinary controls To do t his, let t:" > t:,,1~ = r1(a1,l, kt:,, ) - r1(ar ,l, kt:" - t:,, ), l S; pr , k = 1, .... The t:"r~ is t he total ti me t hat t he measure defined by the derivative of t he relaxed control is concentrated on on t he time interval [kt:" - t:", kt:,, ). Now define the piecewise constant ordinary controls ur'~ (.) as follows. On t he int erval [kt:" , kt:" + t:,, ), they have values
°
ar
(4.8) On [kt:", kt:" + t:,, ), ur '~ o takes t he valu e a1 ,1 on a ti me interval of length t:"t,{ Note also t hat t he ur'~o are "delayed ," in t hat the values of riO on [k t:" - t:" , k t:,,) det ermine t he values of ur '~ ( , ) on [kt:", kt:" + t:,, ). Thus ur '~(t) , t E [kt:" , kt:" + t:,, ), is .rk~ _ -measurable . Let denot e the relaxed control representation of u1'~(')' with time deri vative rr' ~ " Let Ui(p" o) de note the subset of Ui(p, ) that are ordinary contro ls and constant on [lo, lo + 0), l = 0, 1, . ... It will be requir ed t hat t he approximating controls be constant on intervals t hat are multiples of some small 0 > 0, where t:,, /o is a n integer. To modify t he above construction to acco mplish t his , first divide [kt:" , kt:" + t:,,) int o t:,, / o subi ntervals of length O. To eac h value a1 ,1, assign (in order l = 1,2 . . . ,) [t:"t,~ /o] (t he integer par t ) successive subintervals of length
rr.A(·)
(.).
3The U; could be any compact set in a complete and separable metric sp ace .
60
HAROLD J. KUSHNER
O. On any time interval, the total unassigned time goes to zero as 0 --+ 0, and whatever assignments are made to them is asymptotically (as 0 --+ 0) unimportant. To eliminate any ambiguity, assign the remaining subintervals as follows. On the interval, [k6., k6. + 6.) , the unassigned length of time for value 0'.,:,,1 is £,:, ,8,1 = 6.,:,,1- [6.,:, ,1 / 0]0. The sum 5,:,,8 = 2: £,:, ,8,1 , i .k ', k ' ,k ' ,k I ' ,k is an integral (zero or positive) multiple of o. Then assign each unassigned subinterval at random, with value at~ chosen with probability £t{l / Sf:. Let u. (J.l , 0, 6.) denote the set of such controls." Let rf'll.,8,I(.) d~note the time derivative of rr,ll. ,8(.). The next theorem is a consequence of Theorem 4.1. It states that for fixed J.l and small 0 the constructed controls ut,8,ll.(.) yield good approximations, uniformly in ri( ') and {at ,l}. Equation (4.1) holds in that for each J.l > 0, 6. > 0, and bounded and continuous cPi(')'
lim sup Esup 8- 0 riEUi
t~T
I10tju,cPi(ai) [rf,8,ll.,1(dai' s)-rf,ll.,1(dai, s)] dsl =0.
THEOREM 4 .3.
(J.l, 0, 6.)
A ssume (A2.1)-(A2.2). and let ri(')
E
Ui, i
(4.9)
= 1,2. For
> 0, construct rf,8,ll. (.) E u.(J.l , 0, 6.) as above. Then (4.1) holds
for r; ,8,ll.(.) and (J.l, 0, 6.) replacing riO and E, respectively. Also, (4.9) holds: I.e ., for any E > 0, there are J.l, > 0,0, > 0,6., > 0 and ""' > 0, such that for J.l ::; J.l" 0 ::; 0,,6. ::; 6., and 0/6.::; """ sup sup sup IWi(x,r 1 ,r2 ) x
rl
-
r2
Wi (x,rl,u~,8,ll.)I::;
E.
(4.10)
(4.10) holds with the indices 1 and 2 interchanged or if both controls are approximated. Now consider the discrete-time system (3.1) . Then the J.l, > 0,0, > 0,6., > 0 and ""' > 0 can be defined so that sup sup sup IWi(x,r 1,r2 ) x
rl
r2
-
Will.(x,rl,u~,8,ll.)I::;
E.
(4.11)
(4.11) holds with the indices 1 and 2 interchanged or if both controls are approximated or delayed by 6.. 5. Approximations to e-equilibrla. Studies in differential games for models such as (2.1) are usu ally based on the Elliott and Kalton definition of strategy [5, 8], which are generalizations of feedb ack controls. An Elliott-Kalton strategy CIO for Player 1 is a mapping from U2 to U1 with the following property. If two admissible controls for Player 2 are identical up to time t, then the response of CIO is the same to both until at least time t. It will not otherwise depend on what Player 2 might do in the 4The u ; ,8,Ll. (-) are funct ions of r i (') ' but this dependence will be om it t ed in the notation.
NUMERICAL APPROXIMATIO NS FOR GAMES
61
future.f The definition for Player 2 is analogous. Let Ci denot e the set of such strat egies or mappings for Player i . The above definiti on of st r ategy does not account for the po ssibility of randomized controls, where the sample response to a given control pro cess of the other player might depend on the choice of the r andomization. We ext end t he definition t o allow randomized strategies that have the form of the second line in (5.2) for either one or both of the players. Theor em 4.1 shows t he connect ion between relaxed and randomized controls.P The pair Ci(') E Ci , i = 1,2, is an e-equilibrium strategy pair if
W 1(X, Cl , C2)
~
W 1(x , rl , c2) - c,
W 2(X,Cl ,C2)
~
W 2(x , cl , r2) - c
(5.1)
for all admissible controls ri( '),i = 1,2. The notation W2(x , cl , r2) implies that Player 1 uses its strategy Cl ( .) and Player 2 uses the relaxed control 1"2( ' ).
We will require the following assumption.
°
A5.I. For each small c > there is an e-equilibrium Elli ott-Kalton str at egy (c1(·), tH· )) un der which the solution to (2.1) or (2.1r) is well defined . That t he processes are well defined is unrestrict ive, since Theor em 4.3 sh ows that , for the purposes of constr uct ing e-op timal strategies , it is sufficient to restrict attention to strategies whose contro l fun cti ons are piecewise const ant, finite-valu ed and need dep end only on slig ht ly delayed values of th e ot her players cont rol realizati ons. The control pr ocesses tha t a strategy might yield ca n be eit her ordinary or relax ed . The following t heo re m gives a representation of an e-equilibrium strategy as a condit ional distribution, which plays an important ro le in the converge nce proofs. It is not a pr actical control and , like all t he other a pproxi mat ions to the cont ro ls, is used onl y for the pr oofs . THEOREM 5.1. A ssume (A2.1) and (A2.2) . Given
Cl
> 0, there are
(p" <5, .6.) > 0, where .6./<5 is an in teger, su ch that the values for any strategy pair (C l(') , C2 (') ), i = 1,2, under which the solution to (2.1r) is well defin ed can be approximated within Cl by the follo winj simpler strategy pairs (ci ,8, 6 (.), c~, 8, 6 ( . ) ) for which th e realizations of cr ' ,6 (-) are ordinary controls in Ui (p" <5, .6.), and den oted by ur ,8,6(.). Th ey are defin ed as follows. 5T he definit ion in [5] requires that the cont rols r i (A , ·) be pr ogr essi vely m easurable, and not simply measurable and adapte d, for each Borel set A . But due to the approx imation results of Theor em s 4.1-4.3, this added requirem ent is unnecessary in our case. 6 As seen in t he previous section, we need onl y cons ider strat egies t hat give cont ro l processes t ha t are piecewise constant, finite-valued an d slight ly de layed, (AS.1) is t he weakest possible assump tion concerning equilibr ia . If it does not hold , t he n t here is no num erical prob lem , since t here is no solution. Cr ite r ia for t he existence of eq uilibria are in [1, 3, 10]. See also [9] and t he su rvey and references in [19].
HAROLD J . KUSHNER
62
Let ko E [n6., n6.
+ 6.)
for integers n, k. Then for
r
P {u ,<5,c>'(kO) = ailw(s), s =
ai E
ur,
~ ko;u 1j'<5'C>.(10),j = 1,2, 1< k}
p{ ur ,<5,c>'(ko) = aiIW(I6.),l ~ n;u'j,<5,c>'(IO),j = 1,2,10 < n6.}
(5.2)
==Pi,k (a i; w(I6.),1 ~ n;u'j,<5,c>'(lo),j = 1,2,10 < n6.). The functions Pi,kO can be assumed to be continuous in the {w(lo)}arguments, for each value of the other arguments. Proof.
Theorem 4.3 says that we need only work with strategies
cr,<5,C>.(·) whose control process realizations are in Ui(p"o,6.), and that in the time interval [n6., n6. + 6.) depend only on the control process values of the other player up to time n6. - 6.. We will also need to define the response of such a strategy if the other player uses controls that are not discretized. We interpret (5.2), as applied to a non-discretized strategy of the other player, by applying it to a discretization of the realizations of the other players actions. For small p" 0, 6., Theorem 3.3 implies that this interpretation would have negligible effect on the values. Let ur,<5,C>.(·), i = 1,2, denote the control realizations under cr,<5,C>.(·), i
= 1, 2. The probability law of (u't,<5,C>. (.), u~,<5,C>. (.); w(.)) determines the
probability law of (x(.), u't,<5 A (.) , u~,<5,C>. 0; wO) where x( ·) is the solution to (2.1) that corresponds to these controls. Next, write the probability law of the controls as a conditional probability
P {u r ,<5,C>.(kO) = ai!w(s),S
~ n6. ;u'j,<5,c>'(lo),j =
1,2,10 < n6.},
(5.3)
where ko E [n6., n6. + 6.). The rule (5.3) , for i = 1,2, derived from the rules cr,<5,C>.(·), can be considered to be "randomized" Elliott-Kalton strategy pair. The next step is to apply (5.3) to the piecewise-constant continuoustime interpolation of the discrete-time system xC>. 0 defined by (3.1) . Since only discrete-time samples of the Wiener process are used in (3.1), the probability law of the solution to (3.1) on [0, t] is determined by the law of
(u't,<5'C>.(lO),u~,<5'C>.(lo),IO < t;w(n6.) ,n6.
< t).
This implies that the probability law of the controls and paths for xC>. 0 can be written as
P {ur ,<5,c>'(kO)
= aijw(I6.), I ~ n; u'j'<5, C>. (10), j = 1,2 ,10 < n6.} ,
(5.4)
for k5 E [n6., n6. + 6.) . By Theorem 3.1, for small enough 5,6. the paths xC>.O and xO and the associated costs are arbitrarily close, uniformly in
NUMERICAL APPROXIMATIONS FOR GAMES
63
the controls. This argument implies that, with minimal change in the costs, we can replace the rule (5.2) by the rule (5.4), in which the conditioning on w(·) is restricted to the samples w(I6.). Given the form (5.4), the assertion concerning continuity is proved by a "smoothing" procedure." Let p > 0 and define the smoothed functions
P~,k
(O:i; w(I6.),1 ~ n;u'j,8,f:l.(18),j =
= N(p)
J
1,2,18 < n6.)
e-lz-wI2j2PPi,k (O:i;z;u'j ,8,f:l.(18),j = 1,2,18 < n6.) dz
(5.5)
where N(p) is a normalizing constant and w = {w(I6.), I ~ n} . Owing to the smoothing, the functions are continuous in the w-variables, uniformly in the values of the control variables (since there are only a finite number of values for the control variables). As p -; 0, (5.5) converges to
v.»
(O:i; w(I6.),I:=:; n;u'j'8,f:l.(15) ,j = 1,2,15 < n6.)
for almost all {w(I6.), I ~ n} values. Hence, outside of a set of arbitrarily small measure, it converges uniformly in {w(I6.), I :=:; n}. This argument implies that the smoothed conditional probability rule (5.5) and the original conditional probability rule (5.4) will choose the same control values with a probability that goes to unity as p -; O. Hence we can suppose that the Pi,k(.) are smooth in the w-variables. 0
A representation of the rule (5.2). The following method of realizing the random choices given by the rule (5.2) will be useful. Let {Bk } be random variables that are i.i.d. and uniformly distributed on [0,1] with {Bk, k 2: I} being independent of all data on the system before time 18. For each Player i, and each nand kin (5.2), divide the unit interval [0,1] into subintervals whose lengths are proportional to the conditional probability of the o:r,l, I :=:; pr, given by (5.2). Then choose the value ur,8,f:l.(k8) = o:r,l for Player i if the random selection of Bk on [0, 1] falls into that subinterval. The same random variables {Bk } are used for both players. 6. The Markov chain approximation: Brief review and approximations. First we will briefly describe the Markov chain approximation method [13, 14, 18] for a cost minimization problem, mainly to establish notation. Let h > be an approximation parameter. First one determines a finite-state controlled Markov chain ~~ that has a continuoustime interpolation that is an "approximation" of the process x(·). The numerical procedure is the solution of the optimization problem for the chain
°
7See also (18, Theorem 10.3.1J for a similar smoothing. The procedure for showing that it is sufficient for the control rule to depend only on the samples of the Wiener process is different here than in the reference. In the reference it was sufficient to approximate only an e-optirnal control. But here we have two players, and we need an approximation for each player that is valid no matter what the control of the other player.
64
HAROLD J . KUSHNER
and an approximating cost function. Under a simple (and minimal) local consistency condition, the minimal cost function for the chain converges to the minimal cost function for the original problem. The approximating chain and the local consistency conditions are the same for the game problem, except that the game problem for the approximating chain must be solved. The book [18] contains a complete discussion of many simple methods for getting the transition probabilities of the chain. The approximations have the basic properties of the original physical model and this connection can be exploited to improve the algorithms. Since we are concerned mainly with the issues that are new to the game problem, we will use the simplest state space for the chain, one based on the regular h-grid Sh in lRv . Define Gh = Sh n G and G~ = Sh n GO. The numerical boundary points, called 8G h, are those points in Sh - G~ that can be reached in one step from G~ under some control. The process stops on first reaching them. Only G~ U 8Gh is of interest. On G~ the chain "approximates" the diffusion part of (2.1) or (2.1r). Next we define local consistency. Let u~ = (u? n' u~ n) denote the controls at step n, define .6.~~ = ~~+1 -~~, and let E~:i den~te the expectation given the data to step n 8 with ~~ = x and control value 0: = u~ to be used. For the game problem of this paper, 0: = (0:1,0:2) with O:i E Ui. Define the covariance matrix a(x) = O'(x)O"(x) . The chain is locally consistent if (this defines the functions bh (.) and a h (.)) E~:;:.6.~~ == bh(x, o:).6.t h(x, 0:) = b(x,o:).6.th(x ,o:) +o(.6.t h(x,o:)) , covh,Ot [.6. ~n t h _ E h,Ot .6. t h] x,n x ,n ~n
== ah(x ,o:).6.th(x,o:) = a(x).6.t h(x, o:) + o(.6.t h(x, 0:)), lim
sup
h-O xEG,OtEU
h(x
.6.t
(6.1)
,o:) = O.
The function .6.th ( . ) , called an interpolation interval, is obtained automatically when the transition probabilities are calculated; see [18]. Equation (6.1) shows that the approximating chain has the "local properties" (conditional mean change and conditional covariance) of the diffusion process. Let ph(x, ylo:) denote the probability that the next state is y given that the current state is x and control pair 0: = (a1' 0:2) is used. Under our condition that the controls for the two layers are separated in that b(x,o:) = b1(x, 0:1) + b2(x, 0:2), one can construct the (still locally consistent) chain so that the controls are "separated" in that the one-step transition probability has the form (6.2) This might simplify the coding, but will not be necessary here. 8This is {~~ ,l S n ,u~ ,l
< n} .
65
NUMERICAL APPROXIMATIONS FOR GAMES
A one-dimensional example. Let h be small enough so that hlb(o:, x)1 ::; (J"2(x). Then
h(
p x,x±
hi ) _ (J"2(x) ± hb(x, 0:) 0: 2(J"2(x) ,
h
t::.t (x, 0:)
=
h2
(6.3)
(J"2(x)
is locally consistent [18, Chapter 5] Admissible controls. Let F~ denote the minimal o-algebra that measures the control and state data to step n (this does not include u~ , which is determined at step n), with E~ denoting the associated conditional expectation. An admissible control for Player i at step n is a Ui-valued and F~-measurable random variable. Let Uf denote the set of the admissible control processes for Player i . Next, define a relaxed control for the chain. Let r7n 0 be a measure-valued random variable that is a probability distribution on the Borel sets of U, such that n(A) is F~ measurable for each Borel set A E Ui. Then the r7nO are called relaxed controls for Player i at step n. As done for the diffusion model (2.1r) , an ordinary control can be represented by the relaxed control defined by r7,n(A) = I{'U? "EA} for Borel sets A CUi. Define the product relaxed con-
r7
trol r~O by ~~(Al x A 2) = r~,n(Al)r~,n(A2) . The transition probability associated with a relaxed control is
Iph(x,Y la)r~(do:).
Let
Cf
denote
the set of control strategies for the chain E~. Given an approximating cost function, the numerical problem is to solve the game problem for the approximating chain. The approximating cost function for the chain. The cost functions are the analogs of (2.2) or (2.2r). The cost rate for Player i is ki(x ,O:i)t::.th(X, 0:). The stopping costs are gi( '), and T h denotes the first time that the set G~ is exited. Let Wih(x,uI,u~) denote the expected cost for Player i under the control sequencestz/' = {u?,n,n 2 O},i = 1,2. A representation of the transition probabilities. For the convergence proofs, it is useful to have the chains for each h and all possible controls defined on the same probability space." This is done as follows. Let {Xn} be a sequence of i.i.d random variables, uniformly distributed on [0, 1] with {Xl, 12 n} being independent of {E?, u7 , Is: n}. For each value of the current state x = E~ and control 0: = u~ , order the finite number of possible next states y , and divide [0,1] into subintervals whose lengths are ph(x, ylo:). Then with x = E~, 0: = u~ given , the next state value is selected according to where the random choice for Xn falls in [0,1] . The same random variables {Xn} will be used for all h. 9This representation of the chain is not needed and was not used for the single player problem nor for the zero-sum game. For the non-zero-sum game, it helps with the difficulties in the convergence proof that are unique to the problem of this paper.
66
HAROLD J . KUSHNER
e
The continuous-time interpolation h (.) of the chain. The discretetime chain ~~ is used for the num erical computations. The proofs of convergence are based on continuous-time interpolation of the {~~}. These interpolations will approximate the controlled diffusion process x( ·). Two types of interpolations were used in [18]. The one that was of most use in [18] is called 1j;h(.) and will be bri efly described below , so that the results of the reference can be appealed to where needed. In the present paper it is simpler to work mostly with the other interpolation ~h (. ), which is constructed as follows. Define .6.t~ = .6.t h (~~, u~) , and t~ = 2::~:01 .6.t? Define ~h(t) = ~~ and u?(.) = u~ on [t~ , t~+l)' Let the (continuous time) relaxed control representation be denoted by r?(.). Define r h(.) = (r? (.), rq(') ), with time derivative rh,l(0: , t) = r~" (0:1, t)r~ " (0:2, t). We use uI' for the set of continuous time interpolations of the control (ordinary or relaxed) for Player i. By (6.1), we can write
e.. = ~~ + bh(~~, u~).6.t~ + ,B~
(6.4)
where ,B~ is a martingale difference with E~[,B~][,B~]' ah(~~, u~).6.t~. There are martingale differences-? 8w~ with conditional (given .F~) covariance .6.t~I such that (see [18, Section 10.4.1] and [13, Section 6.6]) ,B~ = o-h(~~ , u~ )ow~ . Let whO denote the continuous time interpolation 1 of 2::::-0 8w~ with intervals .6.t~ . Then we can write-!
~h(t) =
I
t
x(O )+ l\h(~h(s) ,Uh(S))dS+ lto-h(~h (S))dWh(S)+Eh(t) , o-h (~h ( s), u h(s) )dwh(s) =
where, for each T <
h(.) 00 , E
I o-(~h t
(s) )dwh (s) + Eh(t) ,
satisfies
lim supEsup [th(s)12 = O. h-> O
(6.5)
uh
s~T
(6.6)
The Eh(t) is due to the O(.6.t h) approximation of ah(x ,o:) by o-(x)o-(x)' .
The interpolation 'lj.Jh (.). In [18] an alternative cont inuous-time interpolation 1j;h(.) was used primarily since it simplified the proofs there . For each h, let l/~ , n = 0, 1, . . . , be mutually ind ependent and exponentially distributed random variables with unit mean, and that are independent of {~~ ,u~ ,n 2: O}. Define .6.T~ = l/~.6.t~ and T~ = 2::~:01.6.TJ'. The OT~ is a new interpolation interval, and T~ provides a new time scale. Define the interpolations 1j;h(.) and u~O by: 1j;h(t ) = ~~ and u~(t) = u~ on [T~ , T~+1)' l Opossibly requiring the augmentat ion of the probability sp ace by the a d di t ion of an independent Wiener process. lIThe equat ion is only valid at the times t~, but the error is as ymptotically negligible, in that it satisfies (6.6).
NUMERICAL APPROXIMATIONS FOR GAMES
67
The advantage of 7jJh(.) is that 'ljJhO is a point process. If the controls u~ are feedback, then 'ljJhO is a continuous-time Markov chain and when in
state x and with control a used the transition rate is 1/ .6.th(x, a) and the probability is just ph(x, Yla) that y is the next state. Given a jump, the distribution of the next state is given by the ph(x, Yla), and the conditional mean change is bh(x, a).6.t h(x, a) . In general, 'ljJhO can be decomposed in terms of a compensator and martingale as
where the martingale Mh(t) has quadratic variation process
It can be shown that ([18, Sections 5.7.3 and 10.4.1]) there is a martingale whO (with respect to the filtration generated by the state and control processes, possibly augmented by an "independent" Wiener process) such that
where ah(.) [ah(.)]' = ah (-) . The martingale whO has quadratic variation It and converges weakly to a standard Wiener process. The martingale ch (-) arises due to the difference between a(x) and ah(x) (recall the o(.6.t h) terms in (6.1)) and satisfies (6.6). Thus,
where r~O is the relaxed control representation of u~O. The time scales based on T~ and T~ are asymptotically equivalent, since the random variations due to the I/~ variables averages out, as seen in the following theorem. Consequently, the interpolations ~h(.) and 7jJ hO are asymptotically equivalent, so that any asymptotic results for one are also asymptotic results for the other. THEOREM 6.1. Assume the last line of (6.1). Then the time scales with intervals .6.t~ and .6.T~ are asymptotically equivalent. Proof. Write .6.T~ - .6.t~ = (I/~ - l).6.t~, a martingale difference, and define fh(t) = min{n : t~ ;::: t}. By the martingale property
68
HAROLD J . KUSHNER n
E sup
'L 6.t?(V~ -
1)
n
< 4E
'L [6.t?12 E(v? -
1)2.
i=O
This goes to zero as h - 4 0 by the last line of (6.1). The result is the same 0 if we define fh(t) = min{ n : 7~ 2 t}.
A comment on weak convergence. Weak convergence methods play an important role in the proofs in [18]. They will playa central role in Theorem 8.1, which shows that a near-equilibrium strategy for the diffusion is also one for the chain For the continuous-time processes we use the path spaces of functions that are right-continuous and with left-hand limits, and with the Skorokhod topology. The sequence (~h(.),r?(.) ,r~O, WhO,7h) is tight. For any subsequence of (~h(.), r?O, r~O , WhO,7h), there is a further subsequence which converges weakly to random processes (x(·), rd·), r2('), w(·), 7), where ri(') is a relaxed control for Player i, (x(·), rl ('), r20, wO,I{ rh ~ .}) is nonanticipative with respect to the standard vector-valued Wiener process w(·) and, writing r(·) = (rIO, r2('), the set satisfies
l1 t
x(t) = x(O)
+
b(x(s), a)r' ido; s)ds
+
l
t
er(x(s)dw(s) .
Also, along the convergent subsequence, Wih(X, r?, r~) - 4 Wi(x, rl, r2)' The proofs are the same as for the one-player control case in [18, Chapter 10]. Example of the construction of 8w h ( . ) . We will give a simple example of the construction of the random variable 5w~ defined below (6.4). More complete details for the general case are in [18, Section 10.4.1], [13, Section 6.6]. Suppose that the matrix er(·) = a is a constant with the form
er
=
[~l ~],
where er l is a square invertible matrix of dimension d l .
Partition the components of x as x = (Xl, X2), where the dimension of Xl is di , and write the analogous partition w h 0 = (w? 0, w~ 0). Partition the ahO in the second line of (6.1) as
h _ [ a?(x , a) a (x, a) h a2 ,1 (x, a)
aJ,2(x, a) ] h a2 (x, a)
.
As h -4 0, aJ(.) -4 erl[erl]' and all other components of ahO go to zero, all uniformly in (x, a). For any Wiener process W2(') that is independent of the other random variables, we can let w~O = W20 and ow~ n = w2(t~+d - w(t~) . The only component of whO that contributes to the limit processes is w? 0 and the increments are defined by
NUMERICAL APPROXIMATIONS FOR GAMES
69
(6.9)
where 6E;,h is due to the approximation of a?O by O"dO"l]' in (6.9) and its continuous time interpolation satisfies (6.6). If an ordinary control is used, then the double integral is just b1 (~~ , u~),6.t~. 7. Approximating the chain: I. The proof of convergence requires the use of a series of approximations to the chain that are analogous to what was done Theorems 4.1-4.3 . One such approximation will be given in this section. It will be used for the proof that a near-equilibrium for the diffusion is also one for the chain, which is one half of the convergence proof and will be given in the next section. These approximations, and the ones in Section 9, have an independent interest and should be useful for ot her convergence analyses for numerical approximations. Theorem 7.1 concerns an approximation to (6.5) that is based on the same whO process, and will be used in Theorem 8.1. Consider the interpolation of the chain as represented by (6.5) . For the control r h ( . ) = (r?( .), r~O) used in (6.5) and the approximation parameters f.J" 6,,6. as used in Theorem 4.3, define the approximation ur,O,£l.,h(.), i = 1,2, analogously to what was done above Theorem 4.3 . For the process whO that appears in (6.5) under the original control rh(-), define the process
Let rr ,O,£l.,h(-) denote the relaxed control representation of ur ,O,£l.,h(-). Keep in mind that the process ~J1-,o,£l. ,h( .) defined by (7.1) is not a Markov chain even if the controls are feedback , since the "driving" process whO is obtained from the process (6.5) under controls rhO and not under the revised controls rr,O,£l.,hO , i = 1,2 that are used in (7.1). Let wt,O,£l.,h(x, rt,O,£l.,h, r~,O,£l.,h) denote the cost for the process (7.1) . Continuing with the approximations, next define the discrete-time system EJ1-,O,£l.,h (n,6. ) with initial condition x(O) and, for n ~ 0,
70
HAROLD J . KUSHNER
t lJo,8,C>. ,h(nl::. + 1::.) = t lJo ,8,C>. ,h(nl::.) +
.:
b(t lJo ,8,C>. ,h(nl::.) , u lJo ,8,C>. ,h(s) )ds
(7.2)
nC>.
+a(tlJo ,8,C>. ,h(nl::.)) [wh(nl::.
+ 1::.) -
wh(nl::.)] .
Denote the piecewise-constant continuous-time interpolation by t lJo,8,C>. ,h(.). r lJo,8 ,C>. ,h, r 2lJo,8 ,C>.,h) denote the associated cost . The next theLet WIJo,8,C>.,h(x t '1 orem is a partial analog of Theorem 4.3. THEOREM 7.1. Assume (A2.1). For (IL, 8, 1::.) > OJ let rf ,8,C>. ,h(.) approximater?O . For each e > 0 andt < OOJ there are u; > 0,8, > 0,1::., > and r: > 0, such that [or IL :s: IL" 8 :s: 8£, I::. :s: 1::., and 8/1::.:S: K",
°
Under the additional condition (A2.2), we can suppose that IWr ,8,C>.,h(x, ri,8 ,C>.,h , r~,8,C>.,h) - Wih(X, r~, r~) I:s: c. (7.4)
lim sUPh-+O sup x ,r~ ,r~
- ,8,C>. ,h(.) (7.3) and (7.4) hold iJ ~1Jo " 8C>.h , 0 and Wr- 8C>.h ' , (.) are replaced by ~1Jo 8 and Wr- ,C>. ,h(') J resp., or iJ only one oj the controls is approximated. Proof. Define 8~1Jo , 8,C>.,h(t) = t lJo ,8,C>. ,h(t) - ~h(t). Then, proceeding as done in Theorem 4.1, write 8e,8,C>. ,h(t) = + +
it°t
r1
[b(e ,8,C>. ,h(s), a) -
JO
U
[a(e ,8,C>. ,h(s)) -
ii
bh(~h(s), a)J rh,l(da , s)ds
a(~h(s))] dwh(s)
b(e,8,C>.,h(s) , a) [rlJo ,8,C>. ,h"(da , s) - rh"(da , s)J ds + c~(t) .
Both of the processes whO , c?O are martingales with respect to the filtration induced by the data (~h(.),rh(·),wh(.)) and whO has quadratic variation v' It and c?O satisfies (6.6). Partition the last integral analogously to what was done in (4.7) , with intervals -\. It can be shown that the process ~1Jo,8,C>.,hO satisfies the following estimate. sup sup sup Esup 1~1Jo,8,C>.,h(l-\ + s) - ~1Jo,8,C>.,h(I-\)12 1Jo,8,c>' r
h
lA:;St
S:;S A
= 0(-\) + sup I::.th(x, a)). 12 Actually, they are martingales only when evaluated at the time points t~ , but the difference is unimportant, since they are constant between such times.
71
NUMERICAL APPROXIMATIONS FOR GAMES
The martingale property and the Lipschitz condition and standard estimates imply that there is a constant K, depending only on t, such that
Esup 18~il,8,L!.,h(s)1 2 ~ K s
L
1
1=0
LA
[t /AJ-1
+KE
(L+l)A
it a
E 18e,8,L!.,h(s)! 2 ds + K,A,h(t)
+ O(A) 2
b(~J1.,8,L!.,h(lA), a) [r J1. ,c5,L!., h,l(da , s)-rh"(da, s)] ds
The error term K,A,h(t ) is due to the E~O and to the use of ~J1.,c5,6,h(lA) in lieu of ~J1.,c5,6,h( s) in the second line, and it satisfies
For each small A, the method of approximation of the controls implies that the second line goes to zero uniformly in h as (IL, 8, 6.) -; O. Then (7.3) follows from the resulting inequality and the Bellman-Gronwall Lemma. The inequality (7.4) follows from (7.3) and (A2.2) . 0 8. Convergence, Part I: An approximate equilibrium for the diffusion is also one for the chain. The convergence proof will be in two parts. In this section, we show that an e-equilibriurn for (2.1) is an Eo-equilibrium for the approximating chain, where EO -; 0 as E -; O. The converse result will be proved in Section 10, under a stronger condition, where it is shown that an approximate equilibria for the chain is also one for the diffusion. THEOREM 8 .1. Assume (A2.1) , (A2.2), and (A5.1). An E-equilibrium value for (2.1) or (2.1r) is an Eo-equilibrium value for the approximating Markov chain , where EO -; 0 as E-; O. Proof. Let E > 0 be given. By (A5.1) , there is an e-equilibrium strategy pair for (2.1r). Theorem 5.1 says that for small enough IL,8 and 6., it can be represented as in (5.2), where 6./8 is an integer, and the Pi,kO are continuous in the w variables. We can suppose, without loss of generality, that for each n, k and i, the rule (5.2) is defined for the finite number of possible conditioning control values, Let (cf"(.), c~O) denote this strategy pair. 13 Recall that when a strategy defined by (5.2) is applied to an arbitrary relaxed control, (5.2) is actually applied to the space-time discretization of that relaxed control, as defined above Theorem 4.3. We will need to adapt the strategies c~ (-) for use on the chain ~~ . This is done by simpl y replacing the w(l6.) in (5.2) by the w h(l6.) process that was used in the representation (6.5), and whose construction was illustrated in (6.9) . These
cf' (-)
13The strategies depend on il and 8 as well as on ~ , but for simplicity we suppress that dependence in the notation.
72
HAROLD J. K USHNER
controls are constructed for purposes of the proof only. For each intege r k, t he control value uf,8,A,h(k8) t hat is obtained from the rule (5.2) with t he w h (.) sam ples used will be applied t o the chain for all steps m for which t~ E [H , k8 + 8). The resulting strat egies are well-defined for all steps of the chain. They will be denoted by c~ ,h (. ) and are in To prove the theorem we will need to show t hat for small enough ( p , ~, 8), there ar e EO > 0 and ho > 0, where EO -+ 0 as E -+ 0 such that for h ::; h o and any sequence T~ O of admissible rel axed (or ordinar y) controls for t he chain,
Cr.
_A,h) > -A,h) W 1h(X, c_A,h ,Cz _ Wh1 (X , T hl ,~ 1 _A ,h) _ > Whz (X, C-A,h,TZ h) W Zh(X, C_A,h ,Cz 1
1
EO ,
(8.1)
EO·
When writing Wf (x, c~ , h, Tq ), we mean that Player 1 uses strategy c~ ,h( .) and Player 2 uses relaxed cont rol TqO (in continuous time interpolation not ation , but for the chain model), or an ordinar y control with this relax ed cont rol representation , with an analogous int erpretation for the other forms. If the strategy pair c~,h( .) , i = 1,2 , is used for the chain , then we write rf,8,A,hO , i = 1, 2, for t he relax ed control rep resentati on of the control actions , in its cont inuous-time interpolation. Let ~h( .), whO, and h T denote t he corresp onding cont inuous-time interpolation of t he chain, t he "pre-W iener" process, and t he first exit t ime, resp. The sequence (~h( .), ri ,8,A,h(.), r~ , 8,~ ,h ( . ) , whO , T h ), par am etrized by h, is tig ht. Select a weak ly convergent subsequence (h -+ 0) with limit denot ed by (x( ') , Tl ( ' ), T20,W(·),T). The set (XO ,Tl( ') , Tz (·), I{r::; .}) is non-an ticipat ive with res pect to the standard vecto r-valued Wiener pr ocess w(' ), and t he set (XO ,Tl( ·), T2(·),W(·) ) solves (2.1r) . The limit T is t he first hittin g t ime of t he boundar y of G by the limit pro cess xO. All of the det ails concerning t he tightness, t he char acterization of the limit pr ocesses and boundar y hitting t imes, the fact t hat the limit pro cesses solve (2.1r), and the convergence of the costs , are t he sa me as for the control pr oblem in [18, Chap ters 10, 11]. From this point on , when weak convergent sequ en ces are dealt with we will suppose (without loss of generality) that the Skorokhod represent ation is used so that all pro cesses are defined on the sa me probability space and the weak convergence is equivalent to converge nce wit h probability one in the t op ology of the path spaces [6, Theorem 1.8 , Chap ter 3]. Under Skor okhod repr esentation t he rule (5.2), with t he whO-samples used , converges w.p.1 to t he sa me rul e with t he w(·)-sam ples used , du e to t he convergence whO -+ w(·) and t he cont inuity of t he pr ob abilities in (5.2) in t he w-variables. Hence t he limits TiO , i = 1,2 , are j ust t he cont rol realizati ons of t he original e-equilibrium strat egies cf'(.), i = 1,2 . Since t he solut ion to (2.1) or (2.1r ) is uniqu e for each admissible (control, Wie ner pr ocess) pair , t he probability law of any limit set (X('),Tl( '), TZ( '),WO) does
73
NUMERICAL APPROXIMATIONS FOR GAMES
not depend on the particular convergent subsequence. Hence the original set of processes converges weakly to this unique limit set, where the control is determined by the rules cf'(.), i = 1,2. By the weak convergence
-c.) W 1 (X, C-c. 1 , C2
~
Wh( -C.,h 'C2 -C.,h) 1 X, C 1
Wh( -C.,h) , 1 x,rAh, c
(8.2)
,h,r2 ) = Wh( --c.,h ,r2 Ah) ' < _ max W 2h(x, C_C. 1 X, C1 1
(8.3)
< _ max W 1h(X,rl,C-C.,h) 2 T1EUi'
-c.) W 2 (X'C-c. 1 ,c2
~
=
1
2
Wh( -C.,h 'C-C.,h) 2 X'C 1
2
T2EU;
Getting fN·), with the strategy cf',hO fixed, involves solving an optimal control for a non-Markov system. But, via a weak convergence argument using a maximizing sequence of controls, it can be shown that the maximizing controls ftO exist. In fact, we need only work with control processes whose cost values approximate the maximum values arbitrarily well, and there always is such a control. We will show that
limsuPh->owNx,f?,c~,h)::;Wl(X,C~,c~) +E+p(,u,<5,~),
°
(8.4)
where the error term p(,u, 8,~) -> as (,u, <5,~) -- 0, with the analogous result for the indices reversed. The three inequalities (8.2), (8.3), and (8.4) imply that if Player 2 uses c~,hO, then as h -- 0, the best policy for Player 1 is to use c~,h(.) (modulo p(,u, 8,~) + E). The analogous result holds if the indices are reversed. Since (,u, 8,~) is arbitrarily small, the theorem is proved. It remains to prove (8.4). Let {ui,8,C.,h(l8)} denote the control values that are obtained by the space and time discretization of f?O, using the method described above Theorem 4.3. These are the values that the strategy c~,hO uses. Let {u~,8,C.,h(l<5)} denote the control choices for Player 2, based on the strategy rule c~ ,hO and the control choices of Player 1. Let rr,8,c. ,h(-) denote the (continuous time interpolation) relaxed control representation of {ur,8,c.,h(l8)}. The processes ~hO and whO now denote the interpolation of the chain and the pre-Wiener process, resp., under the strategy c~,hO and control f?O . The discretized control sequence {ui',8,C. ,h(l<5)} is used by Player 2 to get its controls, but Player 1 uses f? ('), and w h (.) depends on f? 0 as well as on {u~ ,8,C.,h (l<5)}, the actual control sequence that is used by Player 2. This whO process will be fixed for each h and used in the rest of the proof. Define the process ~JL,8,C.,h(-) by (7.1), driven by the {ur,S,c. ,h(l8)}, i = 1,2, and whO. Theorem 7.1 implies that (8.5)
74
HAROLDJ.KUSHNER
where pl(p"rS,fl) --+ a uniformly in fh(.) , as (p"rS,fl,h) construction of r~, o,tl ,h( .) , we have the equality
--+
a.
,h,rIJ. ,o,tl,h) -_ WIJ. ,o,tl,h(x, r u.s.s:» ,C-tl,h) W iIJ. ,o,tl ,h(x, r IJ.,o,tl i 1 z 1 2·
By the
(8.6)
Let TIJ.,o,tl,h denote the first hitting time of the boundary for ~IJ. , o,tl,h (.). The set (~IJ. ,o,tl ,h(.), f?O, ri ,o,tl ,h( .),r~,o,tl,h( .), whO, TIJ.,o,tl,h), parameterized by h, is tight. Extract a weakly conver~ent subsequence, index it by h also, and denote the limit by (x(·), flO, ri'o, (.), r~,o,tl 0, w('), T). Then (x( ·), f 1 (') , ri,o,tl ('), r~,o,tl (') , wO'!{.,.:::;.}) is non-anticipative with respect to the standard Wiener process w(·) , the set (x( '), ri,o,tl(') , r~ ,o,tl (.), w(·)) satisfies (2.lr), and T is the first hitting time of the boundary. The r;,o,tl (-) is just the relaxed control that is defined by the weak-sense limit {ur,o,tl(lrS)} of {u;,o,tl,h(lrS)} . Thus, on any finite interval [a , T], it is determined by a finite set of random variables {ur ,o,tl(lrS), lrS ~ T}. It must be shown that the limits u~,o,tl(lrS) are chosen by the conditional probability law that defines c€'O; i.e., that PZ ,k
((l(Z; wh(lfl), l ~ n ;uj ,o,tl,h(lrS) ,j = 1,2, is < nfl) --+ PZ,k ((l(Z; w(lfl ), l ~ n ;uj ,o,tl(lrS), j = 1,2, is < nfl )
(8.7)
for krS E [nfl, nfl + fl) . Since there are only a finite number of values for the control, for any t < 00 the limit {ui,o,tl(lrS),u~,o,tl(lrS),lrS ~ t} will be achieved after a finite number of steps through the convergent subsequence h --+ a, w .p.L, Using this and the continuity of PZk(') in the w variables, we see that the wh(lrS) in the left side of (8.7) can be replaced by the limit w(lrS) as h --+ a. [We will comment on this point at the end of the proof.] Thus the policy c€'O acting on any relaxed control with discretization {ui ,o,tl(lrS)} will yield the sequence {u~,o,tl(lrS)} and it follows that
Wi,o,tl,h(x, ri,o,tl,h(.), c~ ,h)
--+
W1(x, ri,o,tl(. ), c€,) .
Furthermore, by (8.5) and (8.6) , mod pl(p"rS,fl),
'h ' C-tl,h) W 1h(x, rl z
--+
-tl) ' W 1 (x, "iIJ.,o,tl , C2
We can conclude that
where the € is due to the fact that (cf'(.), c€'O) is an e-equilibrium, The arbitrariness of the subsequence implies (8.4) . The same argument is used when the indices 1,2 are reversed.
75
NUMERICAL APPROXIMATIONS FOR GAMES
Now, we comment briefly on the limit in (8.7). By Theorem 4.3, for small £:1 > 0 the controls that are used on the interval [0, £:1) have a negligible effect on the paths and costs. So let us use ui,8,Cl'\I<5) = U1 (l<5), u~,8 ,Cl,h(I<5) = u2(1<5) for arbitrary ui(I<5) and 1<5 < £:1. For k<5 E [£:1,2£:1), we have the rule (8.9) and the probability of selecting any (l2 E Uf converges as w h(£:1) ~ w(£:1). Then the limit in (8.9) must be the law of u~,8,Cl(I<5) for £:1 :::; 1<5 < 2£:1. Using the method of selecting the control values in terms of the {ez} that was described after the proof of Theorem 5.1, we can suppose that the convergence u~,8,Cl,h(lc5) ~ u~,8,Cl(I<5), £:1 :::; l<5 < 2£:1, occurs in a finite number of steps w.p.1, as h ~ 0 through the convergent subsequence, with the rule (8.9) used. Next, on [2£:1,2£:1 + £:1), we have the rule
The ui,8,Cl,h(lc5), £:1 :::; 1<5 < 2£:1, can be assumed to converge in a finite number of steps. Hence, proceeding as above , so do the selected values of u~ ,8,Cl ,h(I<5), £:1 :::; 1<5 < £:1 + 2£:1. Continuing in this way yields the form (8.7) . 0 9. Approximating the chain: II. Representations and approximations of the chain with control-independent driving noise. We are not able to prove the converse result to Theorem 8.1 for arbitrary 0'(.) due to the fact that the whO process that is used in (6.5) depends on the control. When 0'(.) is constant, this control dependence is small and can be factored out, and (uniform in the control) approximations in terms of an i.i.d. driving sequence can be developed. Once this control dependence is factored out, more convenient approximations to the chain can be obtained. This is done in this section and the results applied to the completion of the convergence proof in the next section. We will work with the model used in (6.9) where
0'
=
[~l ~],
the dimension of
Xl
is d 1 , and
0'1
is
a square and invertible matrix of dimension d 1 . If 0'(.) is constant, then the results of this section will hold for any locally consistent approximation under (A2.1) and (A2.2). But to avoid annoying minor details, we will focus on two main classes of models, each of which is typical of a large class of models and numerical algorithms. Case 1 below arises when one uses the so-called central-difference approximation to get the transition probabilities. Case 2 arises when one uses a central-difference approximation for the non-degenerate part and a one-sided or "upwind" approximation for the degenerate part [18, Chapter 5]. Both forms are locally consistent. Let bi (-) denote the ith component of b(·).
76
HAROLD J. KUSHNER
We will show that whO can be factored as whO = wh(.)+(h(.) where whO does not depend on the control or state and (h(.) is "asymptotically negligible." Case 1. Let d l = V , so that (Y is invertible and define the matrix a = (Y(Y' . Suppose that ai,i - ~j :#i lai,j I ;::: O. The condition can be weakened if the approximation intervals can depend on the coordinate direction. [18, Chapter 5]. We can also apply a linear transformation to diagonalize the matrix a. Then the condition always holds. Note that such a diagonalization might make the coding of the numerical algorithm near the boundary more complicated. Let e, denote the unit vector in the ith coordinate direction. A "central-difference" version of the canonical form of the transition probabilities and interpolation interval in [18, (4.15), Chapter 5] is 2 h( .hl ) _ qi,i ± hbi(x , a) /2 A h( )_ A h_ h P X , x ± e, a Q , u t x, a - ut - Q ' h h a·+ . p (x , x + eih + ej hla ) = p (X,x-eih-ejhla) = 2Q '
h
p (x, x
+ eih -
Q = Lai,i -
ejhla)
=p
h
(x , x - e.b. + ejhla)
L lai,jl /2, i,j:i#}
=
a-:- ·
(9.1)
2Q'
qii = ai,;/2 - L lai,jl/2. j:#i
We also assume that h is small enough so that qi,i-hlbi(x, a)1 ;::: O. A simple computation using (9.1) shows that bh(x , a) = b(x , a) and ah(x , a) = (Y(Y' + O(,0"t h) . By (9.1) we can write ,0"t~ = ,0"t h . In one dimension, (9.1) reduces to (6.3) , where = (Y2 /2.
ql,l
Case 2. Now, use (6.9) where (Y =
[~l ~]
where the dimension of
is d l , and (YI is a square and invertible matrix of dimension d l . The problem concerns the effect of the degenerate part. We will use a canonical model that is motivated by the general model of [18, Chapter 5]. Define v h . b = sUPx ,a~ i=dl+llbi(x,a)1 and Q = Q + hb. For this case, red efine ,0"t h = ,0"t h(x, a) = h 2/Qh . Use the form (9.1) for i :S d l, with Q replaced by Qh. For i = d l + 1, .. . ,V, use Xl
p
h(x , x ± e,.hla ) -_ hb;(x, Qh a)
In this case , we might have transitions from state x to itself, with transition probability p
h(
I) _ hb - h ~~=dl+llbi(X, a)1 Qh
x,x a -
It is easy to show that ah(x, a) = (Y(Y'+O(,0"t h ) and bh(x , a) =b(x ,a). Let E~ denote the expectation given all the system data up to step n.
NUMERICAL APPROXIMATIONS FOR GAMES
77
9.1. With either of the models Case 1 or Case 2, l5w~ = where the components are martingale differences. The 8w~ are i.i. d., {8wr ,l ;::: n } is independent of {~r , u? ,l S n }, and 8w~ = O(h ). For either case, E~ 8w~ [8w~ ]' = t::.t h , and E~ l5(~ [8(~ ]' = O(ht::.t h ), E~ l5 (~ [l5w~ ] ' = O(ht::.t h ). THEOREM
8w~
+ 8(~ ,
Proof. The proof is just a construction of the l5w~ . First we define l5w~ as t ho ugh bO = 0 and ah(x, 0:) = ao', The result will be l5w~ . Then l5(~ is defined to make up the difference, and shown t o have no effect asym pt ot icall y, as h ........ O. The fact that the dominant t erms in the t rans it ion probabilities in (9.1) do not depend on h, and that the contribut ions du e to the drift term b(·) (which contains the control and state information ) are proportional to h makes t his possible. In order t o avoid a complicated notation and to con centrate on the essential points, we start with Case 1 in on e dimension. The treatment of the higher-dimensional model follows the same pattern and this is illu strated via a two -dimen sional case . Then the minor modifications that are required for Case 2 are discussed. These three exam ples illustrate t he gen er al pro cedure, which will hop efull y then be appare nt . The point of the proof is to show that the dw~ and l5(~ can be const ruct ed so that the theorem is satisfied. We need only kn ow t hat t he decompositi on is possible. Define l5~~ = ~~+1 - ~~ . Case 1 , one dimension. Sin ce bh ( . ) = b(.), the do ub le integr al t erm in (6.9) is just b (~~ , u~ )t::.th . To construct t he state tran sit ion s for t he chain , we will use t he represent ation described in Secti on 6, t hat defines t he state tran sit ions in terms of i.i.d r an dom variables {Xn} , un iformly distributed on [0, 1]. In one dimen sion (9.1) is (6.3) and ph(x , x ± hlo: ) = 0.5 ± hb(x , 0: )/ [2a 2] and t::.t h = h 21a 2. Let 8~~ = h if Xn E [0, .5 + hb(~~ , u~ ) 12a2], and set it equal t o -h ot herw ise. The "conditio nal mean" change is 2 h [ hb (~~ , u~) /2a2 ] = b (~~ , u~ )t::.t h , which partially confirms t hat the const ructio n satisfies the local con sisten cy condit ion (6.1) . It also holds t hat t he conditi onal variance of l5~~ is a 2 t::.t h + o( t::.t h ), which completes the requirem ents for local consist ency Next define the martingale difference term l5w~ . Divide [0, 1] into the two segments [0, .5]' (.5, 1], If Xn E [0, .5]' then set 8w~ = hla , otherwise set it equal to -hla. It is what 8w~ would be if bO = 0 and ah(x , o: ) = o a' , Nex t , we de fine l5(~ = w~ - l5w~ to make up for the error s. Ther e are two compon ent s to l5(~ . One com pone nt is due t o the use of a in lieu of [ah ( ~~ , u~ W / 2 as in the last line of (6.9) . Sin ce ah(x , 0:) - a 2 = O(h 2 ) , we have [ah(x, 0:)]1/2 - a = O(h 2) and t he corresponding error compo nent is O(h3 ) . The associated int erpolat ed error pr ocess clearly satisfies (6.6). The second com ponent of the error 8(~ is due to t he neglect of t he term b(·) in const ructi ng 8w~ . T his is handled as follows. Su pp ose t hat
78
HAROLD J . KUSHNER
°
b(~~, U~) ~ (the computation is analogous if b (~~ , u~) < 0). Then the second component of the error is14
8(~ = (2h - b(~~, u~)~th) /o-,
if Xn E [.5, .5 + hb(~~ , u~) /20-2]
and it equals -b(~~ , u~)~th / 0- otherwise. 8(~ is
The conditional variance of
e:n [2h _ b(t<'nh' uh)~th/0-]2 hb(~~,u~) n 20-2 +E~ [b(~~, u~)~th / 0-]2
(1 _hb(;:'2U~)) = O(h)~th .
Clearly the term 8(~ depends on the control u~ and state ~~ . But 8w~ does not. It is just a Bernoulli sequence with values ±h, with {8w?, l ~ n} independent of the data up to step n . Also, E~[8w~]2 = ~th, E~8w~c5(~ = O(h)~th and E~ [c5(~]2 = O(h)~th, all uniformly in the controls. Define the continuous-time martingales wh(t) and (h(t) by interpo1 lating 2::0 c5w? and 2:~:Ol c5(t , resp. , with intervals ~th15 and define wh(t) = wh(t) + (h(t) . The martingale wh(-) has quadratic variation It , and wh (s), s ~ t, is independent of ~h (s),u h (s), s ::; t. The quadratic variation of (h( .) (and its quadratic covariation with wh(.)) is O(h), uniformly in the controls and initial condition. The two-dimensional model for Case 1. The method in higher dimensions follows that in one dimension, except that there are more sub intervals to deal with. The general approach will be illustrated via a twodimensional problem, where 0- is a 2 x 2 non-singular matrix. Let al ,2 ~ 0 , bi(~~ ' u~) ~ O. The method for the other cases is analogous. As for the one-dimensional case, the goal is to realize the choices determined by the transition probabilities (9.1). Divide the unit interval into successive subintervals of lengths ql ,I/Q, q1 .I/Q , Q2,2 /Q , Q2,z/Q ,al ,2/2Q , al ,2/2Q. If Xn E [0, (Q1 ,1 + hb1(~~ ,U~)) /Q j, then set c5~~.n = h, and c5~g.n = O. If Xn E (Q1.1 + hbl(~~ , u~)) /Q, 2Ql ,I/Q j, then set c5~~,n = -h, and c5~q.n = O. Using the two intervals of length Q2,2 /Q, do the analogous computation with the indices 1, 2 reversed. If Xn falls in the next to last of the six subintervals, then set c5~~ = (h, h), and set it equal to equal to (-h, -h) if Xn falls in the last of the six subintervals. Define c5w~ by repeating the above with b(x, a) = 0 and premultiplying by 0-- 1 , as follows: For Xn falling in the six successive subintervals, define
c5w~ = 0--1 { ( ~
) ,(
~h
) ,(
~)
,(
~h
) ,(
~
) ,(
=~
)l
14The 2h in the formula is due to the fact that on the interval Xn E [.5, .5 + hb (~~, u~) /20"2 J, the difference ~~+1 - ~~ was implicitly assigned a value -h when ow~ was constructed , when it should have been assigned the value +h. 15They ar e martingales only when sampled at the times nb.t h = b.t~ , n = 0, 1.2, . . . , but this difference is unimportant , asymptotically.
NUMERICAL APPROXIMATIONS FOR GAMES
79
The error terms c5(~ are computed analogously to wh at was done for the one-dimensional model. The same pattern is followed in higher dimensions. Case 2. The difference between Cases 1 and 2 is t hat for Case 2 the matrix (J is singular. The main complication is notational, and the general procedure will be illustrated via a two-d imensional model, where only the first component of x( ·) has a Wiener process driving term. In this case , b = max X,Q Ib 2(x, a )l, Q = 2q1 ,1 = [(J1]2 and Qh = Q+hb . The pro cedure differs from that of Case 1 mainly in that we need to account for the ph(x , z ]o) , which depends on b. Divid e the unit interval into three subintervals of lengths
and divide the third subinterval into two further subintervals of lengths
resp . There are now four subinte rvals. The values of c5~~ are determined as follows. Analogously t o what was done in the one-dimens ional exa mple of Case 1, if Xn E (O, Q / Qh], then define c5~~ n = o. If Xn E (Q/Qh, 1]' set c5~?,n = 0 and c5w?,n = o. If th e value of Xn falls into the fourth subinterval, th en define c5~g n = o. If Xn falls into the third subinterval , then define c5~g,n = h sign(b2 (~~ ' u~)). If Xn E (0, (qll +hb1 (~~ ,U~))/Qh], set c5~?,n = h, and set it equal t o -h if Xn E [( qll + hb1 (~~ ,U~ ) ) /Qh ,2qll /Q h]. These constructions yield (9.1) with 6.t h = h 2/ Qh. To complete the computation of c5w? n ' repeat the procedure wit h b(·) = 0 and divide by (J1 , as follows. Set c5ti?,n = h/O"l if Xn E (0, q1 ,tfQh]. It is -h/(J1 if Xn E (q1,tfQh, 2q1 ,tfQh], and it is zero otherwise. The variance is h 2/ Qh , which is the current value of 6.t h . The second component of c5w~, namely c5w~ n' is unimportant since it is event ua lly multiplied by zero. So, let us tak~ its values from independent Bernouilli sequ ence with valu es ±h/ y7Jh, each taken with probability 1/2. The error terms (~ ar e computed using a procedure that is analogous t o th at in the one-dimensional Cas e 1 example. In all cases , the qu adratic variat ion process of wh ( .) is It. 0 In the next theorem, where (J(-) = (J , a constant, it is shown that the conti nuous-t ime interpolation ~h (. ) can be writt en in the form
where E~ (-) equals E?(-) plus a st ochastic integral wit h resp ect to t he error martingale (h (. ), and it sat isfies (6.6) . Since wh (-) does not dep end on the
80
HAROLD J . KUSHNER
control and is the sum of i.i.d. zero-mean random variables of size O(h), we can exploit the form (9.2) to get approximation results analogous to those in Theorems 4.1-4.3. In other words, the controls can be space and time discretized with arbitrarily small change in the costs for the chain, just as in Theorems 4.1-4.3. The proof of the various assertions follows the lines of arguments used in Theorem 7.1, exploiting the martingale properties and the Lips chitz condition. The details are very similar and are omitted. THEOREM 9 .2. Assume (A2.1) and any of the models that were used in Theorem 9.1. Define the process (h( .):
Then, for each t
> 0, (9.4)
If (A2.2) is assumed as well , then the costs for ~h(.) and (h(.) are arbitrarily close , uniformly in the control and initial condition. Now, given u , 5, 6., let ut,cl,A,h( .) be th e delayed and discretized approximation of r?O that is constructed by the procedure described above Theorem 4.3, and let rIJ-,cl,A,h(.) denote the relaxed control representation of the pair of approximations. Define the process (IJ-,cl,A,h(.) by (IJ- ,cl,A ,h(t)
= x(O) + +
it
it i
b((IJ- ,cl,A,h(s),a)rIJ-,cl,A ,h,l(da, s)ds
(9.5)
O"dwh(s).
Then for t > 0 and I > 0 there are positive numbers f.L -y , 8-y , 6.-y l h-y , K-y, such that for f.L :'S f.L-y, 8 :'S 6. :'S 6. -y, h :'S h-y , 8/6. :'S K-y,
s.;
(9.6) If th e condition (A2.2) is also assumed, th en for small (f.L , 8, 6. , h) the costs for (IJ-,cl,A,hO and (h (.) are arbitrarily close , uniformly in the control and initi al condition.
Approximating iijh (.) by a Wiener process. By the method of construction, the terms [wh(n6.+6.)-wh(n6.)J, n = 0, 1, . . . are LLd. and have orthogonal components. The covariance is 6.1 , where I is the identity matrix, and the set converge to (i.i.d) normally distributed random variables as h -> O. It will be useful to quantify this closeness to normally distributed random variables for use in Theorems 9.4 and 10.1 via the following strong approximation theorem for i.i.d . random variables.
NUMERICAL AP PROXIMAT IONS FOR GAMES
81
LEMMA 9.3 . [2, Theorem 3.]Let {¢n} be a sequence of JRd-valued i.i. d. rando m variables wit h zero mean an d bound ed (2 + 8) th moment,
whe re 0 < 8 ::; 1. S uppose th at the covariance matrix r is non-singular. Then without changing the distribution, one can redefine the sequence on a richer probability space together with a Wiener process B (· ) with covariance matrix r suc h th at
IL ¢i - B(n)1
=
0(nO.5 -
c
)
(9.7)
':S;n
w.p. l f or large n , f or some 0 < c < 0.5. The next t heorem exploits Lemma 9.3 t o pr ove t hat t he pr ocess defined by (9.5) can be well approximated by the discrete t ime system (3.1), whose solu tion we now write as xJ1.,6, t:., h(. ), since the discretized controls r ;,6,t:.,h(.), i = 1,2, ar e used . The controls that ar e used in (9.5) are obt ain ed from the discretization of the relaxed control represent ati on of the int erpolation of { u~} , the origin al cont rol sequence for the chain. THEOREM 9.4. Assum e (A2.1) and (A2.2) an d any of the mode ls th at were used in Th eorem 9.1 Th en th e probability space can be defined suc h that wh(t) = w(t) + ph(t) , where w( ·) is a vector-valued standard Wiener process . For each t > 0, limh_OE sUPs 0,
lim h -O
su p E sup I x J1. ,6,t:. ,h(s ) - ~J1. ,6.t:. ,h (s)12 = O. r h
,x(O)
(9.8)
s:S;t
Proof. The claim t hat we can app ly t he cont rols for t he chai n to (3.1) requ ires some expla nation and a careful choice of t he probability space . Consider Case 1 and define ran dom variables ¢n = 8w~ / Jh 2 / Q = 8w~ / t::.t h . (For Case 2, t he development is t he same, except t hat Q h replaces Q.) Then {¢n,Xn} satisfies t he conditions of Lemma 9.3 and , wit hout loss of generality, we can suppose that the pr obabilit y space is constructed such that (9.7) holds for a standard vect or-valu ed Wien er pr ocess B (·), which will b e used only to approximate d the sums of t he ¢n . We will not have use for the component of the Wiener pr ocess that approximates t he sums of the Xn' Then , on this probability space define ow~ in t erms of t he ¢n , inverting t he above definition . We ca n suppose that ~~ is also defined on t he same space . Now construct u~, which is just a fu nction of ~~. T hen construc t the rand om variabl e ~~ , which is j ust a funct ion of (Xo, ~~, u~), as done for Case 1 in Theorem 9.1. Conti nuing in t his way, we can construct {~~ ,u~ ,ow~} on t he probab ilit y space , wit h t he sa me law as originally. Given the controls u~ , t hey can t hen be time and space discret ized an d delayed. 16 as in Theorem 9.2. 16 Actually,
it is only required that the controls be approximated and delayed such
82
HAROLD J . KUSHNER
From Lemma 9.3, we for large nand w.p .1, n
LhrPi - hB (n ) = ho(no.s- C ) .
(9.9)
i =O
and w (·) = hB(-jt::.th )/-!Q is a standard vect or-valued Wiener pro cess . It follows that t here is a c > 0 and a th -+ 0 as h -+ 0 such t hat t .]D.t h
8w? -
L
w(t ) = o(t [t::.th n
(9.10)
i= O
w.p .1 for t ~ th and small h. We can now write wh(t) = w(t ) + ph(t) , wher e ph(.) has ind ependent increments, and limh-+oEsuPs
+
= x (O)
itl b(~J.1. ,8,D.,h(s) ,
CK.)r!-,,8, D.,h"( dCK., s)ds + crdw(t ) + crph (t ),
and write (3.1) wit h t he cont rols r!-,,8,D.,h(.) as (in int erpolat ed form)
itl
X!-,,8,D.,h(t) =x(O )+
b(x!-,,8,D. ,h(s), CK.)r!-" 8,D.,h,,(dCK. , s)ds+crdw(t). (9.11)
From here on the pr oof is completed by using the Lipschit z condition, the mar tingale properties, and st andard est imates and the rest of t he det ails are omitted. 0 10. Convergence: II. An approximate equilibrium for the chain is an approximate equilibrium for the diffusion. Theorem 8.1 showed t hat an approximate equilibrium for the diffusion is also one for the approximating chain . It follows from this that if the e-equilibrium value for the chain is unique for arbit rarily small E, then the converse result is true; nam ely that e-equilibrium values for the chain ar e E1-equilibrium values for (2.1r), where E1 -+ 0 as E -+ 0, and we are done , sinc e it would then be implied by Theorem 8.1 that the e-equilibrium valu es for the diffusion are also unique for small E. If the e-equilibrium valu e for the chai n is not un iqu e for small E, th en we will show that this "converse" assert ion is true for the model used in Theorem 9.1. We are not able to show the converse result when cr (·) depends on x, so cr(· ) is constant in t he next theor em. Condition (AS.1) is not need ed for th e pro of, although the problem makes lit tle sen se if (AS.1) does not hold . that the contro l a pplied on [n Ll ,n Ll the dis cretization are not needed.
+ Ll)
is F~Ll._ -rneas ur a ble. T he other as pects of
NUME R ICAL AP P ROXIMAT IONS FOR GAMES
83
T HEOREM 10.1. Assume (A2.1) and (A2.2) and the model used in Theorem 9.1, where 0-(-) is constant. Then for any 10 > 0 there is lOt > 0 which goes to zero as 10 -4 0 such that an e-equilibrium value for the chain ~~ f or small h is an lOt-equilibrium value for (2.1r) . Proof T heorem 9.2 says t hat t he paths and cost functions for (9.2) (which is ~h (-) unde r an arbitrary cont rol ), (9.3) (where t he cont rol is as in (9.2) but t he driving process is wh ( . ) ) , and (9.5) (which is (9.3) with discretized controls) are arbit rarily close, uniformly in t he cont rols, for small (p., 0, 6., h). Theor em 9.4 gives t he same resu lt for (9.5) and x!-, ,
REFERENCES [lJ A . BENSOUSSAN AND A . F RIEDMAN. Nonzero-s um stochastic d ifferenti al games wit h [2J [3]
[4] [5J
[6] [7J
[8J
[9J [10J
[l1J [12]
[13] [14J [15]
stopping times and free boundaries. Tran s. Amer. Math Soc. , 23 1 : 275-327, 1977. 1. BERKES AND W . P HILIPP. Approximation t heorems for ind ependent and weakly dependent random vectors. Ann. Probab., 7: 2!}-54, 1979. R . B UCKDAHN, P . CARDALIAGUET, AND C . R AINIER. Nash equilibrium payoffs for nonzero-sum stochastic differential garnes. SIAM J. Control and Optimization, 43 : 624-642, 2002 . E.ALTMAN, O .PO URTALLIER, A . HAURIE, AND F . MORESINO. Approximating Nash equilibria in nonzero-sum games. International Game The ory R eview, 2 : 155172, 2000. R .J . ELLIOTT AND N.J . KALTON. Existen ce of value in differen tial gam es, Mem. AMS , 126. Amer. Math. Soc, Providence, R I, 1974. S. N. ETHIER AND T .G. K URTZ. Markov Processes: Chara cterization and Convergen ce. Wiley, New York , 1986. W .F. FLEMING. Generaliz ed so lutions in optimal stochastic control. In P.T. Liu , E . Roxin , and R. Sternberg, ed itors, Differential Games and Control Theory: III, pp . 147-165. Marcel Dekke r, 1977. W .H . F LEMING AND P .E . SOUGANlDIS. O n the existence of value fun ctions for two-player zero-sum differential games. In dian a Univ . Math . J., 38: 293-3 14, 1989. A . FRIEDMAN. Stochastic differential games. J. Differen tial Equations, 11 : 79- 108, 1972. S . HAMADENE. Point s d'equilibre dans les j eux stochastiques de somme no n null e. C. R. Acad . Sci . Paris S er . I Math, 318: 25 1-256, 1994. A .B . HAURIE, J.B . KRAWCZYK, AND M . ROCHE. Mo nitoring cooperative eq uilib ria in a st ochast ic differential game. J. of Optimiz . Theory and Applic ., 8 1 : 73-95, 1994. A .B . HAURIE AND F . !vloRESENO. Computing equilibria in stoc has tic games of int ergener a t ional equity. In In tern ation al Conferen ce on Dynamic Games. Intern a t ional Society of Dyn a mic Garnes , 2004 . H .J. K USHNER. Probability M ethod s for Approximations in St ochast ic Control and f or Ell ipt ic Equ ations . Acad emic Press, New York, 1977. H .J. K USHNER. Numerical methods for stochastic control problems in continuous time. SIAM J. Cont rol Optim ., 28 : 999-1048, 1990 . H .J . K USHNER. Numeric al methods for stochasti c d ifferent ial games. SIA M J. Con trol Opt im., pp . 457-486, 2002 .
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[16] H .J . K USHNER. Numerical approximations for stochastic different ial games: The ergodic case. SIA M J. Control Optim., 4 2 : 1911-1933, 2004. [17] H.J. K USHNER. Numerical methods for stochastic differential games: The ergodic cost criterion. In S. Jorgensen , M. Quincampoix , and T. Vincent , editors, A nnals of the International Society of Dynami c Games, 2005. To appear . [18] H .J . K USHNER AND P . D UPUIS. Num eri cal Me th ods f or St ochastic Control Problem s in Contin uous T ime. Springer-Verlag, Berlin and New York, 1992. Second edition, 200l. [19J K .M. RAMACHANDRAN. Stochastic differential games and a pplicat ions. In Handbook of Stochastic Analysis and Applications, pp . 473-532. Marcel Dekker , New York , 2002.
PART
II:
NONPARAMETRIC ESTIMATION
ON THE ESTIMATION OF AN ANALYTIC SPECTRAL DENSITY OUTSIDE OF THE OBSERVATION BAND* ILDAR A . IBRAGIMovt Abstract. We consider a Gaussian stationary process X(t) with an integer analytic spectral density f(A) and study a problem of its estimation. The process X(t ) is nonobservable. Instead of it we observe a linear transformation Yet) , 0 ~ t ~ T, of X (t) with a transfer function a(A), la(A)1 = 1 if A belongs to an interval I . We study how far from I the consistent estimation of f(A) is possible, T -> 00 . Key words. estimation.
Stationary processes, band limited processes, spectral density
1. Introduction. The paper continues author's investigations devoted to the statistical approach to the unicity theorem for analytic functions. The unicity theorem says that if two functions f and 9 are holomorphic in a region G and f(z) = g(z) for all z in some sequence of distinct points with limit point in G, then f(z) = g(z) everywhere in G. The theorem means in particular that if an integer (or entire, " an entire function is an integral function in British usage" [3]) analytic function is observed on an interval I, it can be restored immediately in the whole complex plane. Of course, this problem of restoration is an ill posed one and small perturbations of the observations may drastically change the solution on large distance from the region of observation. We are interested in knowing how far from the region of observation a consistent restoration is possible under small stochastic perturbations. There are a few possible statements of the problem. One of them which has been considered in [8] is the following on e. Problem 1. We are observing an integer analytic function f on an interval [x, y] in the white noise of a small intensity e, i.e, the observation X (t ), x t y , satisfies the relation
:s :s
dX(t) = f(t)
+ cdw(t) ,
(1.1)
w(t) is a standard Wiener process. Denote F(M, 17, p) the class of integer analytic functions which satisfy the following restrictions on their growth sup If(z)1 IzisR
:s M exp{(jRP}.
(1.2)
These are well known classes of integer functions of order p and type 17, see [3], [12]. It has been shown in [8] that if the function f in (1.1) belongs *Supported by the RFBR (05-01-00911), RFBR-DFG (04-01-04000), NSh4222 .2006.1 , INTAS (03-51-5018 ), CRC-701. t p etersbur g Department of Steklov Institute of Mathematics, St . Petersburg 191023, Russia. 85
86
ILDAR A. IBRAGIMOV
to the class F(M ,(T, p), the consistent estimation of j (z ) when e ---> 0 is possibl e approximate ly in (In 1/ c) vicinities of [x, y] (see details in [8]). An other scheme of observation has been considered in [9], [10] . Call it P roble m 2. Let
*
(1.3) be a sa mple from a distribution with an integer ana lytic densit y fun ction j (x ). It is supposed that the sa mple (1.3) is censored by an inter val [x, y]: only observat ions X j E [x, y] can be used to construct estimat ors for j , all other observations dis appear. It is shown in [9], [10] that if the density function j belon gs to a class F (M, (T, p), the consist ent est imat ion is possible approximate ly in (In n) ~ vicinities of [x, y] (see details in [9] ,[10] ). In this paper we consider one mor e variant of the last pr ob lem . Let X (t ) be a Gaussian stationary process with an integer analytic spectral density j (A). The pro cess X (t ) is the Fourier transform of a Ga ussian process Z (A) with indep end ent increments and E ldZ(AW = j (A)dA,
X(t) =
I:
(1.4)
eitA dZ (A),
see, for example, [5], ch. 11. T he process X (t ) is unobservable. Inst ead of it we are obse rvi ng for o ~ t T a linear t ransformation Y (t ) of X (t ),
s
Y (t ) =
I:
eitAa(A)dZ (A)
(1.5)
wit h a transfer fun ct ion a(A). The process Y (t ) is also a station ar y Gaussia n process wit h t he spect ral density g(A) = la(AW j (A). If ja(A)1 = 1 for A E [x, y ], then j (A) = g(A) for x ~ A ~ y . Ther e are man y ways t o constru ct a consistent estimator when T ---> 00 estimators g(A) for g(A). Any such est imat or will esti ma te consiste nt ly j (A) inside [x,y]. How sh all we set pr oblems on the est imat ion j (A ) out side [x , y]? On t he first glance a direct ana logue of the pr oblem 2 looks as follows: t he pro cess Y (t ) is ba ndlimite d to [x , y], i.e. the trans fer function a(A) = 0 for A tf. [x , y]. But t his approac h will not work. Indeed , in the case the obse rvation pro cess
is analyt ic in t he whole complex plan e. Thus t he observation Y (t ) ad mits t he ana lytic continuation on lR 1 . By Ma ruyama's t heorem the Gaussian process Y (t ) wit h conti nuo us spec t ru m is ergodic (see [13] or [6]) and hence by ergo dic t heorem the correlation function R y (t )
-
= EY (t )Y (O) =
liS
lim -8
s ...... OO
0
Y (t
-
+ s)Y(s )ds
SPECTRAL DENSITY ESTIMATION OF THE OBSERVATION BAND
87
can be determined by the observation Y(t), 0 ::; t ::; T. The correlation function Ry (t) determines the spectral density g(A), A E 1~.1, and hence j(A) for x ::; A ::; y. But then the analytic function j(A) is determined for all A. (Note that linear filters with a(A) = 0, A t/:. [x, y] are physically non- realizable.) Therefore we take an engineering conception of bandlimited processes. Namely, we consider linear transformations Y(t) with transfer functions a(A) such that
ja(A)\ = 1, x::; A ::; y,
la(A)1 < 1, A t/:. [x, y].
We denote the class of such transfer functions A. Of course it is supposed that the transfer function a(A) is unknown, otherwise we could estimate j(A) as g(A)la(A) I- z . Thus we consider below the following estimation problem. Problem 3. A stationary Gaussian process X(t ) with the spectral representation (1.4) has the spectral density j E F(M, 0' , p). Suppose also that
I:
(1.6)
j(A)dA ::; A .
Denote the class of spectral densities j which belongs F(M, 0', (!) and satisfies (1.6) G(M, 0', p, A) = G . The observation process Y(t) , 0 ::; t ::; T , is determined by the relation (1.4). The unknown transfer function a E A. The problem is to estimate the spectral density j(A), j E G , of X(t) in and outside of the observation interval [a, b]. Let us formulate the basic results. THEOREM 1.1. There exists an estimate ](A) of j(A) such that
E
rl f (A) - ] (A) j2dA ::; 0
InT T lnlnT
l;
(1.7)
where the constant C depends on M , 0', p, A, x , y only. Denote GT(o: ) the (lnT )", jP-vicinity of the interval [x,y]. THEOREM 1.2 . There exists an estimate ](A) of f(A) such that for all fixed 0: < 1 and f3 < (¥ the jollowing inequality holds
E!{
sup !EG,aEA
sup If(A) G T ({3 )
](A)I} ::; 0",,{3T
12
'" .
(1. 8)
The constant 0",,/3 depends also on M , 0', p, A , x, y . The next theorem shows that the length of consistency intervals can not be essentially increased. THEOREM 1.3. Let L be an exterior point of the interval GT(O:) , 0: > 1. Then for all sufficiently large T (1.9) inf sup E!I] - j(L)l z > Co > O. !
!EG,aEA
The constant Co depends on M, 0', p, x, y , 0: .
ILDAR A. IBRAGIMOV
88
2. Construction of estimators. Proof of Theorems 1, 2. The method of t he proof does not depend on t he observation's interva l and for t he sake of sim plicity we su ppose below that [x ,y] = [- 1, 1]. We consider t he following gene ral scheme of estimator 's construction . Let {'Pn} be a complete orthonormal syst em in L 2 ( -1 , 1). Then we can expand t he res trict ion of f (>.. ) on [- 1, 1] int o t he Fourier series with res pect to {'Pn},
n
If an are estimators for an, esti mate f (>..) by a sum I:n") , the choice of N dep ends on t he class G . This method has been init iated by N.N. Chent sov [4]. To realize the scheme t ake as a complet e ort honormal syste m in L 2 ( - 1, 1) the Legendre polynomials Pn(>.. ), n = 0,1 , . .. : Pn is a polynomi al of degree n and J~ l Pn(>.. )Pm(>..)d>.. = omn' See, for example, [14]. LEMMA 2.1. Let f E F (M , a ,p). Let
f(>..) = L anPn(>" ),
an =
[11Pn(>.. )f (>..)d>.. ,
(2.1)
n
be its expansion into the Fourier series with respect to {P n }. Then
~p ln ~ } epa
(2.2)
lakl :S M exp[ - kln (k / ea )} .
(2.3)
lakl :S cMex p {
-
where c depends on a , p. If p = 1,
T he result sho uld be well known. A proof of it see in [10]. LEMMA 2.2 (see [2], p. 74, or [15], p. 17) . Let Q( z) be a polynomial of degree n. Then for all complex z
IQ(z)! ::; m axQ (x ) .! z+ ~In. Ixl~ l
(2.4)
LEMMA 2.3 (see [14]). Th e f ollowing inequaliti es hold max IPn (x) ! :S IPn(I) 1= J (2n + 1)/ 2. Ixl~ l
(2.5 )
It follows from t he lemm as above that t he series (2.1) converges in t he whole complex plan e and determines t here an analytic function. Hence t he equality
(2.6) n
89
SPECTRAL DENSITY ESTIMATION OF THE OBSERVATION BAND
holds in the whole complex plane. Take as a basic statistics the periodogram Ir(A) of the observation, (2.7) and consider as estimators of linear functionals
(ep,g) =
I:
ep(A)g(A)dA
the statistics J~oo ep(A)IT(A)dA. In particular we estimate the coefficients
by the statistics (2.8) LEMMA
2.4. The mathematical expectation
(2.9) where
DT(A) = sin(~A/2). The covariance fun ction
EIT(Al)Ir(A2) =
{I: (I: +(1:
1r2~2 +
f(l)D?(l-Al)dl
I:
j(l)D?(l-A2)dl
j(l)D?(l- AdDT(l - A2)dl) 2 j(l)D?(l- Al)DT(l- A2)dl)
(2.10)
2}.
Proof for processes with the discrete time see , for example, in [1], Theorems 8.2.7,8.2.8 . The formulas (2.8) , (2.9) can be proved in the same way. LEMMA 2.5. The bias Ea n - an satisfies the inequality (2.11) The constant C depends on the class G.
90
ILDAR A. IBRAGIMOV
Proof Since the equality
J OO D} (>.. -
-1 1r
f.L )df.L = 1
-00
an d (2.8) , we can write
lEan-
ani =
~1[11 Pn(>..) [ _~j1
~~ 1r
r
Pn(>" )d>"
-1
1r
r
J
1p.I'5,T
D~(f.L)g(>.. + 2f.L/T )df.L
JOO D~(f.L)9(>")df.L1 -7'
f.L- 2df.L j
J,p.I >T/2
+~
:
/2
-1
(2.12)
IPn(>")llg(>..)-g(>..+2f.L/T)ld>..
D~ (f.L)df.Lj1
-1
IPn(>")llg(>..)-g (>..+2f.L/T)ld>.. .
The first summand on the right side of (2.11) does not exceed
T- 1[11 IPn (>..)g(>..)d>..
+ T- l [1
1
IPn (>")d>" [ : g(f.L )dj.L
~ CT- l .
If 1>" 1 ~ 1 and 1>.. + 2j.L/ T I ~ 1, then Ig(>..) -g (>.. +2f.L/T )1~ C If.LIT- 1. Hence the second summand in (2.11) does not exceed
1
C j T/2 j T/2 C 1 j.LI D~ (f.L) df.L + C D~ (f.L)df.L IPn(>..)ld>" ~ ;rr.' T - T/2 -T/2 {1-21 p.I / T99} vT
o
The lemma is proved. LEMMA 2.6. Th e vari an ce of an
(2.13)
Proof It follows from (2.9) that
Elan - Ean l2 =
1r2~2 [11[11Pn(>..)Pn(j.L)d>..df.L( [ +
= J,
1r2~2 [11 [11P
n
+h .
: DT(>..-Z )DT (f.L-Z )g(Z )dZr
(>" )Pn(f.L)d>..df.L ( [ : DT(>..-Z )DT (f.L +Z)9(Z )dZ) 2 (2.14)
SPE CTRAL DENSITY ESTI MATION OF THE OBSERVATION BAND
91
We bound the first summand on t he right , the second one can be bounded in t he same way. We have
r
J 1 ::; 1r
28 {j1 j 1 Pn(>')Pn(Jl )d>.dJl( DT(>.-l )DT(Jl-l )9(l)dl)2 T 2 - 1 -1 J{ JlI ~ 2}
+(
r
J { llI$2}
DT(>' -l )DT(Jl-Z )9(l)dl)2} = J 11 + J 12.
Fur ther
and J 12 ::;
~i
22
22 i
g(ll )g(l2)(ill Pn(>' )Pn(Jl )DT (>' - ll) DT(>' -l2)d>.) 2
<
C(m ax lll$2 f (l))2
-
T2 X
ill ill Pn(>.)Pn(Jl )d>.dJl(i: DT (>. - l)DT(Jl- l )dl) 2.
But
i : DT(X - l)DT(Y -l )dl = 1rDT(X - y) . Thus J 12 ::; CT -
1i11IPn(>.)ld>' i~ IPn(Jl) jT- 1 Df (>' - Jl )dJl
::; CT- 111Pn 112 = CT- 1 and the lemma follows. Prove now theorem 1.1 . Consider estimators N
f N(>' ) = ~ anPn (>. ), o
N = 0,1 , .. ..
(2.15)
The sequence {Pn } is orthonormal in £ 2[-1 ,1 ] and N
E { llf - f N112} = ~E l an o
ex>
- an l 2 + L lanl2 N+I
N
::; 2
L lEan o
N
an l
2
ex>
+ 2 LE jan - Eanl2 + ~ lanl2 . 0
N+l
92
ILDAR A. IBRAGIMOV
Lemma 2.2 implies that
f
N+l
lanl 2 :::; Cexp {- ~NlnN}. p
It follows from this inequality and Lemmas 2.5 and 2.6 that
Take N
rv
{:,lc,'!;.
and put]
= It« . We find that
E{II] _ f 11 2 }
:::;
C InT . T lnlnT
The theorem is proved. 0 Remark. Suppose that the observation process is not band limited.We may conjecture (again because of the unicity theorem) that then the bound can be ameliorated. Indeed this is the case. For example suppose that p = 1 and a- > 0 is known. For ItI :::; a- consider the following estimator R(t) for the correlation function R(t) of the observation process
1
r
T
R(t) = T-t Jo If
-
t
X(t+s)X(s)ds,O:::;t:::;a-,
ItI > a , we set R(t) = O.
R(t) = R(-t) , -a-:::;t:::;O.
Consider as an estimator for f()...)
]()...) = 21 7f
ja e-
i t >. R(t)dt.
-a
It is rather easy to show that
Ell] - fIIL2 (-1,1) ;:::: El l] - flloo ;:::: CT- 1 . Comparing the last formula with (1.7) we see how large are losses due to band limitation. Proof of Theorem 1.2. Consider at first the estimators f n defined in (2.14). Let L ~ 1. Because of Lemmas 2.1, 2.2 and 2.6 we will have that
6.(L) = E sup IfN()...) - f()...)! I>'~L
: ; 5r L vnl N
L
a
+
f
N+l
exp { -
+ viP -lll
n
~pepaIn ~}IL + VlL2 -lll
(2.16) n .
93
SPECTRAL DE NSITY ESTI MATIO N OF THE OBSERVATION BAND
It follows t hat
t::. (L ) ::; (CL )N Take N
rv
I'
(T-
1 2 /
+ exp {
- ;
In N}).
~ l~~nTT and set j = I» . Then if L ::; C (ln T ) ~, 0 < a <
1,
I
T ( 1- Pln In lnLC)} 1- a E I~~~ f (>. )-f (>. ) ::;Cexp {In --2T ::;CT -2 -. (2.17) The t heorem is proved.
0
3. Lower bounds. Proof of Theorem 1.3. Below we restrict ourse lf t o the obs ervations (1.1) with the transfer fun cti ons of the following form I, if 1>'1 ::; 1, a(A) = { e ( -AsinA) , 0 < e < 1, if 1>'1> 1.
(3.1)
Denote .40 the set of t hese transfer fun ctions. Take a number L , ILl > 1. Our initial probl em is t he pro blem of estimation of the value f (L ) base d on t he observations (1.5) wit h a E .40. Consider together with the initial pr oblem an additional one-parametric esti mation pr oblem of t he following typ e. Define t he following one-p arametric famil y {io} of spe ctral densities
fo(>') = cpL(A)(l
+ B'l/JL(>')),
B E [0, 1],
(3.2)
wher e CP L E G and 'l/JL E G are two given functions such t hat CP L(L)'l/JL(L) :::: 1/2 . The final choice of t hese funct ions will b e made lat er. Below we usually omit t he index L and write cP, 'I/J inst ead of CPL, 'l/JL. Suppose that t he initi al pr ocess X (t ) has a spe ctral density f o and cons ider t he pro blem of estimation of t he par amet er () on the b ase of obse rvations (1.5) wit h a known tran sfer fun cti on a E A o. Let j be an est imato r of the valu e f (L ), f E G , in t he ini ti al problem. Becau se of
B = fo(L ) - cp(L) cp( L)'I/J (L) we may consider as an estimator for B in the addition al pr ob lem
, j - cp(L) B = cp(L )'I/J(L )' Evidentl y (3.3)
94
ILDAR A. IBRAGIMOV
and hence inf
sup
! !EG,aEA
Eelf - 1e(L)1 2 ~ ep(L)'l/J(L) inf sup EIB - e1 2 .
(3.4)
e e ,aEAo
It follows that we prove the theorem 1.3 if we can establish the inequality sup supEelB aEAo
e
el 2
~ co> O.
(3.5)
We see later that the additional estimation problem corresponds to a regular statistical experiment in the sense of [11], sect. 1.7. Hence the Fisher information I(e) of the additional problem is well defined and by Cramer - Rao 's inequality (see [11], Sect. 1.7): for any estimator Bwith SUPe EelB - el2 < 00 E
e
IB_ el2 > -
(1 + d'(e))2
I(e)
+ d2(e)
.
(3.6)
Here d(e) = EeB - e is the bias of B. It follows from (3.6) that if
supI(e) ::; 1, e
(3.7)
then sup EelB -
e
el 2 ~ ci > 1/16.
(3.8)
In the next part of the section we prove the inequality (3.7) . Because of (3.4), (3.5), (3.8) it will prove the theorem. Because the interval GT(a) of theorem 1.3 depends on p only and to simplify the proof we will not try to construct spectral densities with given (J', p, M , A but satisfy ourselves by constructing 1 E G with given p and some (J', M, A . In other words we prove the following weakened version of Theorem 1.3: for a given p one can find (J', M , A such that for 1 E G(M, (J', p, A) the inequality (1.9) will be satisfied (of course these (J', M, A will not depend on the interval GT(a)). Return to the definition of the functions .p, 'l/J determining the family (3.2). Let epO(A) = (sin (A + i)(A + i)-l)(sin(A - i )(A - i)-i) . Set
epL(A) = ep(A) = epO(A - L) + epO(A + L) .
(3.9)
The function ep(z) is an integer function. It is easy to see that
wher e the constant C does not depend on L . Let now 'l/Jo(z) be a function from F((J',M,p) with some (J',M and let the function 'l/Jo satisfy the following conditions:
SPEC TRAL DENSITY EST IMATION OF THE OBSERVATION BAND
95
on the real line
0::; 1/;0(>") ::; e- 1W , 1/;0(>") = 1/;0(- >.. ), 1/;0(0) = 1;
(3.10)
for all L E R l
l1/;o(z - L )I ::; Cexp{cl zIP }
(3.11)
where the constants C, c do not depend on L. The existence of such functions will be proved at t he end of the section. We define now 7/J L as
1/;d z) = 1/;0 (z
+ L ) + 7/Jo (z -
L) .
(3.12)
Under such choice of 'P,1/; all the functions f e will belong to G (wit h a given p and some a, M, A determi ned by 'PO, 7/Jo) . T he observation process Y(t) of t he additional prob lem determines on the Hilbert space L 2 (0,T) = LT t he family of Gaussian measu res Pe. We show th at all t hese measures are absolute ly continuous one wit h respect to other. To do t his we apply some gene ral results on absolute continuity of Gaussian measures in Hilbert spaces. LEMMA 3. 1 (see [7], Th. 2, Sect. 4, Ch. VII). Let P l and P 2 be two Gaussian measures with m ean values zero and correlation operators R l an d R 2 correspondingly defined in a Hilbert space H . In order that thes e m easures be equivalent, it is n ecessary and suffic ien t that the operator D = R ;-l/ 2R lR ;-1/ 2 - I, I be th e identity operato r , be a Hilbert S chm idt operator and its eigenvalues Jk satisfy the in equality Jk > -1. If th e m easures Pl , P2 are equivalent, the R adon - Nykodim derivative (3.13) where ek are the eigenfunctions of the operator D corresponding to the eigenv alues Jk . In t he case when P is a Gaussian measure on LT = L 2 (0, T) generat ed by a stationary process X (t ) wit h t he correlation function B (t ) and t he sp ectral density b(>.. ) the correlation operator is defined on L T by t he formu la (B u)(t)
=
iT B (t - s )u (s)ds .
To the operat or B corresponds the bilinear form (B U,V)T
= iT iT B (t
- s )u (t )u (s )dtds .
(3.14)
96
ILDAR A. IBRAGIMOV
Continue a function u E LT on the whole real line setting u(t) = 0, t and denote u(>.) the Fourier transform of the continued function ,
~
[0, T]
We can rewrite then the form (Bu,v)r in the terms of b(>.) , namely
(Bu, v)r
=
i:
b(>.)u(>.)v(>.)d>..
(3.15)
Denote ET the class of integer analytic functions cp(z) square summable on the real line and satisfying the inequality
(integer functions which satisfy the last inequality are called integer functions of exponential type T, see their theory in [3] or [12].) The famous Paley-Wiener theorem (see, for example, [3], Th. 6.8.1 ) asserts that the class ET coincides with the class of functions u which can be represented as
u(>.)
= JT eitAv(t)dt,
v E L 2 ( -T, T) .
-T
In particular, the unit ball determined by the form (Bu, u)r : {u E LT : (Bu,u) :::; 1} corresponds to the set {u E ET: J~oo b(>')lu(>')1 2d>':::; 1}. LEMMA 3 .2 (see [7], Th. 3, Sect . 5, Ch . VII). Let PI and P2 be two Gaussian measures generated by two stationary Gaussian processes 6 ,6 possessing spectral densities it (>'),12 (>') correspondingly. Suppose that the spectral densities it, 12 satisfy the following conditions: 1. there exists a function CPo E E(j and positive constants CI, C2 such that
2.
Then the restrictions PIT , P2T of the measures PI, P2 on LT are equivalent for any positive T. The observation process of the additional problem has as the possible spectral densities the functions
and it follows from the last lemma that for all equivalent for all T > O.
e the measures Po, Pe are
97
SPECTRAL DENSITY ESTIMATION OF THE OBSERVATION BAND
Denote Ro(t) the correlation function corresponding to the spectral density gO(A) and let Ro denote the corresponding correlation op erator. Let h(A) = rp(A)la(AW"p(A). The Fourier transform of this function
H(t) =
1:
eitAh(A)dA
is a correlation function with the corresponding correlation operator H on
LT, Hu(t) =
l
T
H(t - s)u(s)ds.
Thus the operators
Ro =Ro+BH and the operators Do corresponding to the operator D of Lemma 6 are
In particular, the eigenvalues of D g are equal to BOk where Ok are the eigenvalues of D . It follows then from Lemma 3.1 that the density fun ction
(3.16)
where
ek
are the eigenfunctions of the operator D.
LEMMA 3 .3 . The
Fisher information (3.17)
Proof. Apply (3.16). We get that
~ln (x)=~", dB
pg
2
x,e k)2 -I}
L.: 1 +OkBOk {(R;;-11 +/2BOk
and hence
To compute the last expectation we apply the following evident result:
98
ILDAR A. IBRAGIMOV
if Z is a Gaussian stationary process with the correlation operator B , then for all v E LT random variables (Z, V)T are Gaussian and
E(Z,vh = EZ(O)· (1,vh, E(Z,uh(Z,vh = (Bu, vh·
It follows that under Pe the random variables ~k = (R01 / 2y , ekh = (Y, R
o 2ekh are Gaussian with means zero and correlations 1
/
where the last 6kl is the Kroneker symbol. Thus the Gaussian random variables ~k are independent with means zero and variances Et;~ = 1 + B6k. The equality (3.18) gives then that
The lemma is proved. It follows from the lemma that
o Below we study two last factors . LEMMA 3.4. The eigenvalues 6k of the operator D satisfy the inequality (3.19) where E: is defined by (3.1). The constant C does not depend on Land T . Proof We have
max Ok
=
sup (DU,U)T = sup (ROl/2HRol/2u,uh IlullT=l
= sup(Hv,vh
IlullT=l
v
where the last upper bound is taken over all v E LT such that (Rav, vh = 1. Taking into account the relation (3.15) we find that max d, =
s~p
=
s~p
i: i:
h('x')lv('x'Wd'x' (3.20) 2
la(,X,)1 1/J('x' )
where the upper bound is taken over all v E E T such that (3.21)
SPECTRAL DENSITY ESTIMATION OF THE OBSERVATION BAND
99
Further a non-negative function u E Err can be represented as IV(A)1 2, v E Err/ 2 (see [12]). Thus max5k :::;
s~p
1:
'IjJ(A)la(AWlv(AWdA
and the upper bound is taken over all v E ET+l such that
The properties (3.10) (3.11) of the function 'l/J imply that max5 k
:::; C(exp{-(L/2)P})
+c 2sup v
max
3L/22:IAI2:L/2
ISi~AV(A)12 /\
(3.23)
To estimate the last summand on the right hand side, take a function v satisfying (3.22) and set Vl(Z) = v(z)Si~ Z . Then VI E ET+2 and satisfies the inequalities
The second inequality together with Paley - Wiener theorem implies that (3.25)
Represent the function Vl(A) by its expansion with respect to the Legendre polynomials (see Sect. 2). We get that Vl(A) = ~n anPn(A) and (see Lemma 2.1)
L
lan l2 :::; 1, lanl:::; Cc- 1 exp{ -n In n /eT} .
n
Hence (see Lemma 2.2) for L/2
IV(A) I < (
t
s IAI :::; 3L/2
IPn(A)12) 1/2 + Cc- 1
jan llPn(A) 12
N+l
o
< (9L)N +
f
CC 1(9L)N e-N1nN/eT.
Take here N ::::: 18TL. We find that E
max
L/2::;IA I::; 3L/2 LEMMA
Iv(A) (sinA)>,-II:::;C(cexp{18TLln9L}+e- TL ) .
(3.26)
o
3 .5. The following inequality holds
trD =
L s, s C(T + ln k
2
c- 1 ) .
(3.27)
100
ILDAR A. IBRAGIMOV
Proof. The trace of the operator D = R~1/2HR~1/2 is
trD = 2:)DUk, Uk)r k
where {Uk} is an orthonormal basis in LT. If we consider such basis for which Vk = R6 /2uk are defined, we find that
trD
= ~(HVk,Vk)T k
where {vd are orthonormal with respect to the bilinear form corresponding to the operator R o, (ROVk, VI)T = bkl . Arguing as above, rewrite the forms (Hv , V)T, (Rov, v)r in the terms of the corresponding spectral densities. We find that (3.28) where the functions Vk E ET+l and satisfy the conditions (3.29) Setting a(>,)vk(>')
= rk(>') we find that trD =
I:
7j;(>.)
~ r~(>\)d>'
(3.30)
and the set of the functions {rd is an orthonormal system in the subspace
aET+l of L 2 ( -00,00) We estimate now the sum
2:k r~(>.) .
Let
Then
and the functions {y'21frk} constitute an orthonormal system in £2(-00,00). It follows that
~ Tk(>') :::; ~ supllOO eiAXr(X)dXI2 k
21T
r
-00
(3.31 )
SPECTRAL DE NSITY ESTIMATIO N OF THE OBSERVATIO N BAND
101
and t he upper bound is taken over all r, llr ll = 1, which are t he Fourier t ransforms of fun cti ons represent ed as av , v E ET+l ' Hence
(3.32) where upp er bound is taken over all v E
ET+l
under the conditi on (3.33)
Consider now separat ely two cases: 1>'1> 1 and Let 1>'1> 1. It follows from (3.33) that
I:
1>'1~
1.
Si~: >'lv(>.) 12d>' ~ g-2 .
Hence
and
L r~ (>.) ~ (l a(>')1 sup Iv(>')1)2 ~ T + 2. v
k
Let now
1>'1
~ 1. Funct ions
vi(>')
st\
sat isfy t he inequalities
Hen ce expanding t he function Vl (>') with respect t o the Legendre polynomials (see Sect .2) we find that for N > eT
Take here N
rv
In 1/ e. We get that (3.35)
o Now we are ready t o pr ove the inequality (3.7). LEMMA 3.6 . Let the numbers e, L in the addition al problem satisfy the relation In g- 1 = 18LT ln9L - LT.
(3.36)
102
ILDAR A. IBRAGIMOV
Then f or sufficiently large T and L > (In T) <>/ P , a > 1, the Fisher inform ation of the additional problem satisfies the in equality
1(8)
s C (Te (-L /W + e- LT/ 2 )
< 1.
(3.37)
The lemma is an immediate corollar y of the inequalities (3.19) and (3.27) . We have noticed above that the inequality (3.7) proves t he theorem. Thus to finish the proof we have only to construct the function 'l/Jo satisfying the condit ions (3.10) , (3.11). T he construction coincides in pr in ciple with an alogous constructions in pa pers [9], [10] and we omit the details. We will distinguish t hree cases. The first and the simplest case is when p is an even integer, p = 2k. We set in t his case 'l/Jo(z) = exp{ _ z2k} . Evidentl y t his function satisfies (3.10) . The function 'l/Jd z) = 'l/Jo(z - L ) is an integral fun cti on and for Izi < L / 4k 3t(z - L )2k > 0 and hence I'l/Jd z)I < 1. If Izi > L /4k, then Iz - L I2k :::; 22k- l(lz I2k + L 2k) :::; (C1zJ)2k. If p = 2k + 1, k ? 0, is an odd int eger , we set 00
'l/Jo(z) =
(
IIn=l
Sin(Zpn - (1+6l ) ) 2
zPn -
(l +6 l
'
<5
> O.
Applying the Stirling formu la we get that
Further
ISi: z I:: ; Iz l-1e P z1 a nd henc e for
Izi :::; a lLj with a bei ng a small posit ive number ,
I'l/Jd z) I :::; C exp{ -cl lz - L IP+ c21~( z - L )PI} :::; C . For
Izi > a lL I evident ly I'l/Jd z)I :::; Ce clziP.
The case of non integer p is more com plicated. Ro ughly speaking t he construction is t he following one. Define the integer p from the relations p < P < p + 1. Consider at first the function
where
{
u2
uP}
G(u,p ) = (l - u )exp u +"2 + . . . + P- .
SPECTRAL DENSITY ESTIMATION OF THE OBSERVATION BAND
Take a = ~; and set V",(z) as follows
= V(zc i "' ) .
103
The function 'lj;o is then defined
For the proof that this function satisfies all the necessary conditions, see [10] . This work has been started in summer of 2006 when I visited the Bielefeld University and finished January of 2007 again in Bielefeld . I am very thankful to Prof. F. Goetze for his hospitality during my visit. I would like to thank the secretaries Mrs . A.L . Cole and Mrs. N. Epp for their help during the visits. REFERENCES [1] ANDERSON T.W., The statistical analysis of time series, J . Wiley, 1971. [2] BERNSTEIN S.N. , Extremal properties of p olynomials, ONTI, Moscow , 1937 (in Russian) . [3] BOAS R ., Entire functions , Academic Press, 1954. [4] CHENTSOV N.N ., Estimation of unknown probability density based on observations, Dokl. Akad. Nauk SSSR (1962), 147: 45-48 (in Russian). [5] Dooa J .L ., St ochastic processes, J . Wiley, 1953. [6] GRENANDER D., Abstract inference, J . Wiley, 1981. [7] GIHMAN 1. AND SKOROHOD A ., The theory of stochastic processes I, Springer, 1974. [8] IBRAGIMOV 1., On the extrapolation of entire functions observed in a Gaussian white noise, Ukraini an Math. J. (2000), 52: 1383-1395. [9J IBRAGIMOV 1., An est imat e for the analytic density of distribution, based on a censored sample, J. Math . Sci. (2006 ), 133: 1290-1297. [10] IBRAGIMOV 1., On cens ored sample estimation of a multivariate analytic probability density, Theory Probab. Appl. (2007 ), 51 : 1-13. [11] IBRAGIMOV LA. AND KHASMINSKII RZ ., Statistical estimation, Asymptotic theory, Springer, 1981. [12J LEVIN B ., Distribution of zeros of entire functions, Amer. Math. Soc. , Providence, RI, 1964. [13] MARUYAMA G ., The harmonic analysis of stationary stochastic processes, M em . Fac. Sci. Kyusyn Univ. A , 4: 45-106. [14] SZEGO G , Orthogonal polynomials, Amer. Math. Soc . Colloq. Pub!. , N.Y. , 1967. [15] WALSH J .L ., Approximation by bounded analytic funct ions, Gauthier-Villars, Paris, 1960.
ON ORACLE INEQUALITIES RELATED TO HIGH DIMENSIONAL LINEAR MODELS YURI GOLUBEV' Abstract . We consider the pr obl em of estimat ing a n unknown vector e fr om the noisy data Y = Ae + E, wher e A is a known m x n matrix and E is a white Gaussian noise . It is assumed that n is large and A is ill-posed. Therefore in or der to esti mate e, a spectral regularization method is used and our goal is to cho ose a spectral regularization parameter with the help of the data Y. We study data-driven regularization m ethods based on t he empirical risk minimization principle and provide some new oracle ine qu alities related to this approach .
Key words. Spectral re gularization, excess risk, ordered smoothers, empirical risk minimization principle, oracle inequality. AMS(MOS) subject classifications. Primary 62G05, 62G20.
1. Introduction and main results. In this paper, we deal with a classical problem of recovering an unknown vector () = (() (1), ... , () (n) )T E lR n from the noisy data Y
= A() + E,
(1.1)
where A is a known m x n - matrix and E E lRn is a white Gaussian noise with a known variance (T2 = EE 2 (k), k = 1, . .. , m. The standard way to estimate () is based on the maximum likelihood estimator eo = argmin IIY -
A()1I 2 ,
I1ElRn
where IIxl1 2 = 2::;;'=1 x 2 (k). It is easy to see that eo = (A T A )-l ATy and the mean square risk of this estimator is comput ed as follows
where Ak and ePk E lRn ar e the eigenvalues and the eigenfunctions of (AT A )-l :
In what follows, it is assumed that A is ill-posed i.e., A1 :S A2 :S . .. :S An. The equation (1.2) reveals the principal difficulty in eo: its risk may be ' CNRS and Institute for Probl ems of Information Tran smission CMI, 39 rue F. J oliotC urie , 13453 Marseille, France. 105
YURIGOLUBEV
106
very large when n is large or when A has a large condition number. In this paper , we suppose that n is large (it may be infinity), so the risk of is also large. The basic method to improve is to make t he variance 0'2 2::~=1 Ak smaller by suppressing larg e Ak. The simplest way to implement this idea is to smooth with the help of a properly chosen n x n - matrix H i.e., using a new estimator He o. In this paper, we focus on with the following famil y of linear estimators
eo
eo
eo
ea = Haeo = u; [(AT A )-I] (AT A )-1 ATy, where Ha(z) is an analytic function Ho:(z) = 2::%"=0ho: (k )zk such that lima_o Ho:(z) = 1, lim z _ oo Ho: (z) = O. This method is called spectral regularization (see [3]). The regularization parameter 0 controls the quality Indeed, with an elementary algebra we get the standard bias-variance of decomposition
s:
n
Eileo: - el12 = :2)1 -
n
Ho: (Ak)]2 (e,?/Jk)2 + 0'2
k=1
L
AkH~ (Ak), (1.3)
k=1
wher e ?/Jk = AcPn / IIAcPkll and (e, ?/Jk) = 2::;:'1 e(l)?/Jk(l) . It is clear that the sp ectral regularization may substantially improve when (e, ?/Jk)2 is small for large k. In practice, a goo d choice of HaO is a delicate pr oblem related to the num eri cal complexity eo:. For instance , to make use of the sp ectral cut-off regul arization with Ha(z ) = l{ o z ~ I} , one has to compute the singular valu e decomposition (SVD) of A. For large n this numerical problem may be difficult or even infeasible. The very popular Tikhonov 's [7] regularization is defin ed by
eo
eo: = arg:nin{ IIY - A e ll 2 + 01 IeI12 }
.
For this method we have Ha(z) = 1/ (1 + o z ), Notice here that this regularization technique is good if A is really ill-posed. Indeed in view of (1.3), if a may improve the banal estimator
e
eo
This means for instance that for inverse problems with Ak ~ 1 Tikhonov's regul ari zation makes no sense. Another widespread regularization technique is due to Landweber. This method is bas ed on a very simple idea: t o find recursively a root of equation
107
ON ORACLE INEQUALITIES
Notice that for positive a we can write ATy = [ATA-aI]B+aB, or equivalently B = [I - a-I ATAJ e+ a-I ATy. This formula motivates Landweber 's iterations defined by
It is easy to see that these iterations converge if aAl < 1. It is also easy to check that (1.4)
In spite of its iterative character, the numerical complexity of Landweber 's iterations may be very hight. Indeed, when the noise is small, Hi(z) should be 1, and (1.4) results in . con d(A) clef An 2> = ~ So, if A is severely ill-posed, the number of iterations may be very large, thus making the method infeasible. A substantial improvement of Landweber 's iterations is provided with the v-method (see e.g., [3]). Whatever an inversion method is used , the principal question is how to choose its regularization parameter. Intuitively, (see (1.3)), this parameter should minimize in some sense the risk n
L [a, B] =
n
2:)1 -
Hr:x(Ak)J2 (B, 1/Jk )2 + CT 2 I>kH;(Ak).
k=l
k=l
In statistics, there are two main ideas to formalize this optimization problem: • to assume that
e belongs to
a known set
e E JRn
and to take
a* = argminsupL [a,B] IiE9
Q
• to construct based on the data an "estimate" i ra, Y j of L [a,B] and to compute & = arg min j.jo, Y] Q
Statistical literature related to these approaches is so vast that it would impractical to cite it here. We refer interested reader to [6] and [2] as typical representatives of its. Notice that the first approach is related to the theory of minimax estimation [4]. This paper focuses on the second approach, namely, it deals with datadriven regularization parameters computed with help of the empirical risk
108
YURI GOLUBEV
minimization principle. This method says that the regularization parameter should be computed as follows & = argmin Rpen[Y, a], a
where
and Pen(·) is a given function IR+ --.., IR+. A heuristic motivation behind this method is rather transparent. Indeed, the best regularization parameter is obviously given by a* = argmin
110 - 19,,11 2 .
(1.5)
" Evidently, a* cannot be used since it depends on 0 which is unknown. So, the first idea is replace 0 in (1.5) by 190 . It is clear , that directly this idea doesn 't work because min " 11190 - Ba l1 2 = O. Therefore we need to corr ect 11190 - 19,,112 by an additional term, thus arriving at Rpen[Y, a]. Intuitively, this idea assumes that the best Pen(a) should be a minimal function such that uniformly in 0 E IR n (1.6) Unfortunately, the mathematical formalization of "the best penalty" and (1.6) is a very delicate problem. We refer interested readers to [1], which provides a reasonable approach to this formalization . In this paper, we assume that the penalty is given and our goal is to bound from above Eo IIB", - 011 2 . The simplest way to analyze this risk is to use SVD . Let Ak and (/Jk be the eigenfunctions and eigenvectors of (AT A)-l . Denoting Wk = A
N(O ,l).
Notice that
Ba
admits the following
and n
11 190 - 19,, 11 2
= L)l- H a (.Ak)]2y 2(k)
(1.8)
k=l
n
110 -
B,, 11 2
= :2)O(k) - Ha (Ak)y (k )f k= l
(1.9)
109
ON ORACLE INEQUALITIES
We have already mentioned that the main idea in the empirical risk minimization is to control Eolleo, - 811 2 with the help of EoRPen[Y, 0]. Mathematically this idea can be expressed as follows. For any jJ. > 0 and a given penalty Pen( .) define the excess risk by
6.Pen(jJ.)
= sup OER n
{Eo118 - eo, 11 2 -
(1 + jJ.)EoRpen[Y, Oi]}
and we will say that the penalty is admissible if 6.Pen(jJ.) < jJ. > O. If Pen (ex) is admissible, then we obtain immediately
(1.10) 00
for any
(1.11) where
n
+ (Y2
L AkH~(>\k) + (Y2Pen(ex) -
n
2(Y2
k=l
L AkHex(Ak) . k=l
This equation can be interpreted as a bias-variance decomposition of Bex related to the empirical risk. The empirical bias term is given by I:~=d1 Hex(Ad1 2(8,1/Jk )2 and it coincides with the standard one. However the variance term differs from (Y2 I:~=l AkH~(Ak) and it is computed as follows n
l:Pen(ex)
=
(Y2
L AkH~(Ak) + (Y2 Pen(ex) -
n
2(Y2
k=l
L
AkHex(Ak)'
k=l
The inequality (1.11) can be rewritten in the form of an oracle inequality
Eo lle -
Bo,I12 :::; rpen[e] + i~f{jJ.rpen[8] + 6. pen(jJ.)} ,
where rpen[8] = inf., Rpen [8, ex] is the oracle risk. Thus, to control the risk of our data-driven method we need to compute the excess risk . Notice that when n is large the exact computation of the excess risk is infeasible: indeed, for given 8 with the Monte-Carlo method we can compute
D(8, jJ.) = Eol18 -
Bo,J12 -
(1 + jJ.)EoRpen[Y, oj
but we cannot maximize this function numerically over JRn for large n. In order to overcome this difficulty, let us introduce
6.~en(jJ.) ~f EoS~P{2(1 + jJ.) 'j;e(k)AkHex(Ak) -jJ.
t AkH~(Ak)e(k) k=l
(1.12)
-1
+ Cmaxk AkH~(Ak) jJ.
(l+jJ.)pen(ex)} .
110
Y URIGOLUBEV
Notice that in cont rast to t he excess risk , 6. ~en (11) can be compute d by t he Monte-Carlo method , and we will show that for a sufficient ly lar ge class of spectral regul ariz ation methods 6Pen(ll) ~ 0"2 6.~en(Il). These regul arizat ion methods (smoot hers) are called ordered sm ooth ers. They were firstly introduced in [5]. D EFI NITIO N 1.1. Th e f amily of smooth ers {Ha (-) , a ;:::: O} is called ordered if: 1. f or all a ;:::: 0 arul ). ;:::: 0, 0 ~ H a ( '\ ) ~ 1 2. Hal (,\) ;:::: H a2(,\), for all a l s a2 and all x > O. T ypi cal examples of ordered smoothers are provided t he Tikhonov regul arization, the spe ctral cut-off method, the Landweber it erations. The main result of this paper is given by the following theorem THEOREM 1.1. Let {HaO , a ;:::: O} be a family of ordered smoothers. Th en for some C > 0 and f or all 11 > 0
Let us illustrat e how this theorem works. Suppose n
P en (a ) = P en(a ) =
22: Ha('\k)'\k. k= 1
It is well known that this pen alty is related to t he un biased risk est imation, since for given a
Assume also t ha t A is not severely ill-pos ed . More pr ecisely, suppose there exists K < 1 such t hat for all a ;:::: 0 (1.13)
(1.14) It is easy to see that t hese conditions allow onl y a polyn omi al growth of '\ k . Indeed , if '\k m '\ 1 = k for some m E [0, 00),
t hen for t he spect ral cut-off with H a ( '\ ) = l { a '\ t h at K = m / (m + 1).
~
1} it is easy to check
111
ON ORACLE INEQUALITIES
THEOREM 1.2. Let {Ha (-) , a 2: O} be a family of ordered smoothers and (1.13-1 .14) hold true, then for some C > 1 and any J.L E (0,1)
f:::.P en (J.L)
~
C 1 / ( 1-
(1 _
K) A
K; )1 /(1-1<)
uZ
J.L(~+K)/(l-I<) .
This theorem improves oracle inequalities from [2] over the classes of ordered smoothers. It says that the upper bound for the excess risk doesn 't depend on n. Moreover, it allows the minimization of the empirical risk over all possible regularization parameters, thus showing that a priory restrictions on a are not essential. On the other h and, it reve als some serious difficulties related to the penalty Pen(a) : the upper bound for excess risk explodes as K; -> 1. It means that this penalty is not good for severely ill-posed inverse problems. More details about this effect along with an improved penalty can be found in [1]. 2. Proofs. 2.1. Ordered processes and their properties. Our method of deriving oracle inequalities is related to a spe cial class of random processes. Let ~(t), t 2: 0 be a random process with E~(t) = 0 and a finite variance Ee(t) = UZ(t) which is assumed to be monotone
a Z(t 2) 2: uZ(h), The process
~(t) ,
tz
2: t 1·
t 2: 0 , is called ordered if it is separable and for all t 2 2: t1
(2.1) Obviously, this condition can be rewritten as
So, an ordered process can be viewed as a natural generalization of the Wiener process W(t) for which EW(t 1)W(tz) = min{EW Z(t1), EWZ(tz)}. Denote for brevity
The main property of ordered processes is given by the following lemma. LEMMA 2 .1. Suppose there exists A > 0 such that
00.
(2.2)
tl ,t2
Then there exists a constant C depending on A such that for all T > 0 and all p 2: 1 [E
sup t ,sE [O ,T J
I~(t)
-
~(s)IP ]
l/p
~ Cpu(T),
(2.3)
112
YURIGOLUBEV
where C is a generic constant. Proof We provide the proof of (2.3) only for reader 's convenience. In fact, its proof is standard and it is based on the classical chaining arguments [8]. For simplicity, we assume that (j2 (t) is a continuous fun ction. Then for a given integer s ~ we can find points t'k on [0, T] such that
°
(j2(t'k)
= 2- sk(j2(T) ,
k
= 0, . . . ,2 s -
1
and denote by T'" the set of these points. Let U be an arbitrary point in TS. Then we can find a chain, i.e. the points 'Tj (u) E Tj , j = 0, . .. , s such that 1. U = 'Ts(u), 0= 'To(u)
2. j(j2('Tj(u)) - (j2('Tj_1(U))1 :s; 2- H 1(j2(T).
To verify that such points exist, one can imagine the standard binary tree with the nodes at the level j associated with the points t{ , I = 0, ... ,2 j -1. It is clear that there exists a unique way connecting U E T" and (top of the tree). This way passes via nodes, which are denoted by 'Tk(U) . So, we can write
°
s-l
U = i)'Tk+1(U) - 'Tk(U)], k=O and for arbitrary points u, v in T" we get s-l
s-l
U - V = L['Tk+1(U) - 'Tk(U)] - L['Tk+1(V) - 'Tk(V)], k=O k=O Therefore in view of (2.2), we have
E 11p sup I~(u) - ~(v)IP 1J., vETs
s-l
:s; 2
L [E k=O
sup uETk +l
s-l
= 2
X
L [E k=O
sup
1~ ('Tk+1 (U) ) - ~('Tk(U))IP
I6.E('Tk+1 (u ),'Tk (u)) I
r
I p
P
(2.4)
uETk+l
[E[~('Tk+1(U)) - ~('Tk(U))]2] s-l
:s; 2(j(T) LTkI2[E sup k=O
Let L( x) = log" (x + eP convex on (0,00), since
uETk +l 1
) •
P12] l ip
I6. E(Tk+1(U),Tk(U)W ]
l ip
.
Notice that for any p ~ 1 this function is
113
ON ORACLE INEQUALITIES
Therefore, using convexity of L(x) and (2.2), we obtain El /p
sup 16..;(Tk+l (u), Tk( u)) I P uETk +l
~ ~L [
L
EeAIA~ (Tk+l (U) 'Tdtl))I]
uETk+l
1
[k+2
~ :\ 1og 2
(\ )
+e
p-l]
=
(k+2)10g(2)+p-1
>.
+
log[') ]
>.
.
Substituting this in (2.4) , we arrive at the inequality [E
sup I~(u) - ~(v)IP ]
l ip
~
which proves the lemma, since by separability of [E
I~(u)
sup tl ,vE[O ,T]
-
CO"(T)
1J.,vETs
~(v)IP ]
l/P
= lim sup [ E 8-+00
~(t)
sup I~(u) - ~(v)IP
]l lP
u, vETs
.
n
Lemma 2.1 almost immediately results in the following fact: LEMMA 2 .2. Let ~(t) be an ordered process satisfying (2.1, 2.2) and such that ~ (O) = O. Then there exists a constant C depending on >. such that for all "( > 0
Esup[~(t) _ "(O"q(t)]P ~ C[2q(p + 2) - 4]q(P+2 )-2, t2:0
+
(2.5)
"(p l(q-l )
where [xl+ = max(O, x ). Proof We will use the following form of the Markov inequality
(2.6) which immediately results from the banal inequality
Without loss of generality, we may assume that 0"2(t) is continuous and such that limt-+oo 0"2 (t) = 00 . Then for any integer k ~ 0 we can find tk b) such that
YURIGOLUBEV
114
Using that f (x ) = x Pl {x > xo} is monotone in x > 0, we have E sup [~ (t) - I'aq(t)]: t ~O
00
s 2:E k=O
e (t) l{~ (t) ~ I'aq(t) }
sup t E [t k(-r ),t k+l (-Y)]
00
:::; 2: E
k=O
e (t )l{ ~(t) ~ I'aq(tk(,,)) }
sup t E [t k(-y) ,tk+ l(-y)]
(2.7)
00
:::; 2:E
k=O
:::; E
e (t )l{
sup tE [tk (-y) ,t k+l (-Y )]
sup tE [t k(-y) ,tk+l (-Y)]
«o ~ I'aq(t k(,,)) }
1~ (t)IP
sup °991 (-Y) 00
+ 2: E k =l
sup
0::; t::; tk+ 1(-y)
e(t) l{
sup
O::;t::;tk+l (-y)
~(t) ~ I'aq(tk(,, ))} '
By Lemma 2.1, the first term at t he right-hand side of the ab ove inequality is bounded as follows
whereas the second one , in view of (2.6), is controlled by 00
2: E
e (t) l{
su p O::;t::;tk+ l(-Y)
k= l
C
:::;
sup O::;t::;t k+ l (-y)
~(t) ~ I'aq(tk(,,)) }
d p+d 00 aP+d(tk+l(,,)) _ C(p + d)p+d 00 (k + l )p+d (p +) 2: baq (tk (,,) )]d I'p/(q- l) 2: . kqd/(q-l) k= l
:::;
k=l
C[2 (p + d)] p+d 00 1 I'p/(q- l) 2: kd/ (q -l )-p ' k=l
Setting d = (q -l )(p + 2) in the above inequ ality and using (2.7) toget her 0 with (2.8), we prove (2.5).
2.2. Some examples of ordered processes . The simplest example of an ordered process is ~ (t) = ~t , where ~ is a zero mean random variable with a finite exponential moment E cosh ( A~ ) < 00 for som e A > O. As we have already mentioned, the Wiener process W (t) is an ordered process. At the first glance, ~t and W (t ) are qu ite different, but from the viewpoint of Lemma 2.2 they ar e equivalent. Of cours e, t he distribution of maxt ~o [W (t ) - I't ] is well-known
P {max[W(t ) t~O
I' tl ~ x} = exp ( - 2I'x ).
115
ON ORACLE INEQUALITIES
The next two examples play an essential role in adaptive estimation. Let H, (.) be a family of ordered smoothers (see Definition 1.1). Consider the following Gaussian processes n
e+(t) = 2:)Hto(Ak) - Hto+t(Ak)]bke(k) , k=1
t
>: 0
n
e-(t) = LlHto(Ak) - Hto-t(Ak)]bke(k), k=1
0 ~ t ~ to,
where e(k) are i.i.d. N(o , 1) and 2::1 b; < 00. It is easy to see that e+(t) and e-(t) are ordered processes. Indeed, in view of (1.12) we have for
t2 :::: t1 n
Ee~(h) = L lHto(Ak) - HtO+t1(Ak)][Hto(Ak) - HtO+tl (Ak)]b~ k=l n
~ L [Hto(Ak) - HtO+tl (Ak) ][Hto(Ak) - HtO+t 2(Ak)]b~ k=1 and similarly, n
Ee~(t1) = LlHto- t1(Ak) - Hto(Ak)][Hto-tl (Ak) - Hto(Ak)]b~ k=l n
~ L[Hto- t1(Ak) -
u.; (Ak)][Hto- t2(Ak) - n.;(Ak) ]b~
k=l
Therefore with Lemma 2.2 we get
E sup [e+(a) - , i)H"o(Ak) "~" O
E sup [e-(a) - , f)H" 0 (Ak) "~"o
k=1
H"(AkWb~] p ~ C(~), +
k=1
"[
H"(AkWb~]P ~ C(~) , +
,
thus arriving at LEMMA 2.3 .
Let {H,,(-) , a :::: O} be a fam ily of ordered smoothers,
th en for any , > 0
E~~~[~lH"o(Ak) -, 't[H"o(Ak) k=l
H,,(Ak)]he(k)
H"(Ak)fb~] P ~ C(~) . +
,
(2.9)
YURIGOLUBEV
116
The next imp ort ant orde red pro cess is defined by n
T/(t ) = I: H 1 / t (>\k)(e (k) -1 ), k= 1
where e(k) are i.i.d. N (O, l) and {HtO , t 2: O} is a family of ordered smoothers. It is easy to check t hat
So, in order to apply Lemma 2.2, it remains to check (2.2). Denoting for brevity n
IIH"'2 - H"'l 11 2= I:lH"'2(>"k ) -
u.; (>"k)]2,
k= 1
we have
(2.10)
Since obviously
t hen using the Taylor expansion for 10g(1 - .) at the right-hand side of (2.10) , we get for>" ::; 1/2 E exp[>".6. e(02 ' 01)]
s exp (C>..2),
t hus pr oving (2.2). Therefore using Lemma 2.2, we obt ain t he following fact . LEMMA 2.4. Let {H", O , 0 2: O} be a f amily of ordered smoothe rs, then for all , > 0
117
ON ORACLE INEQUALITIES
2.3. Proof of Theorem 1.1. Denote for brevity ek = (e, ¢k ). We begin with a simple auxiliary lemma that is cornerstone for the proof. LEMMA 2.5. Let {H,A ·), 0:: 2': O} be a family of ordered smoothers. Then there exists a constant C such that for any data-driven smoothing parameter 6
(2.12)
00
::; CEo m:xAkHl(>\k)Eo 2:)1
-
He,(Ak)]2e~.
k=l
and
(2.13)
00
::; CEo m:XAkHl(Ak)Eo
2:[1 - He,(Ak)]2e~ . k=l
Proof Let
0::0
be a given smoothing parameter. We obviou sly have
00
Eo
2:[1 - He,(Ak)h!\~ek~(k)
(2.14)
k=l 00
= Eo 2:[Hc>o(Ak) - He,(Ak)]Aek~(k) . k=l
It follows immediately from (2.9) that
lEo ~[Hc>o(Ak) - He,(Ak)]Aek~(k)1 ::; l Eo f)Hc>o (Ak) - He, (Ak)]2 Ake~
+ C.
k=l
I
Therefore minimizing the right-hand side in I ' we obtain
To bound from above the right-hand side at the above display, we use once again that Hc>(-) are ordered smoothers. So, when 6::; 0::0 we obtain
YURI GOLUBEV
118
00
~ m:XAkH~o()\k)L [l-H& (AkWB~ k=l and simil arly for &
~ 0:0
00
00
L[Hao(Ak) - H& (Ak)]2AkB~ ~ m:XAkH&(Ak ) L[l - Hao (Ak) ]2B~. k=l
k= l
Therefore combining these inequalities with (2.14) and (2.15), we get
Using the element ary inequality 2ab ~ ab ove display as follows
f.W2
+ b2 / u ; we
can conti nue the
i- ~[1- H& (Ak)]ABk~ (k)1 ~[
::; fL L..t 1 -
k=l +fLEo
Hao (I\k \ )] 2112 CmaXk AkH~o (Ak ) k + ----::::-".-.:..--'-(l
I:[l -H& (Ak)]2B~ +
fL
CEo maXk AkHl (Ak) .
k= l Therefore minimizing the right-hand side in
fL 0:0 ,
we get
lEo ~[1 - H& (Ak)]ABk~(k)1 . f{ fL Z:: ~ [ 1 - H aD (\I\k )J2 Bk2 + ----'--"--CmaXk AkH~o(Ak )} ::; m ~
~l
I:[l -H& (AkWB~ + I:[l- H&(Ak )] 2B~ +
+EO{fL
k=l
::; 2EO{fL
~l
fL
CmaXk AkHl(Ak)} fL CmaXk AkH l (Ak)}. fL
119
ON ORACLE INEQUALITIES
To finish the proof of (2.12) it suffices to minimize the right-hand side in f-L . Inequality (2.13) follows from (2.12) since Hcx(A) = 2Hcx(A) - H;(A) are ordered smoothers and we can apply (2.12) with HcxO = HcxO. 0 In view of the definition of the empirical risk and (1.9) , we have n
Rpen [Y, &] =
,
'2
IIBo - B&II + a
2
2"'"
Pen(&) - o Z:: Ak
k=l n
= L[l - H&(Ak)]2B~
+ (]'2 Pen(&)
k=l n
+ L[H~(Ak) - 2H&(Ak)]Ake(k ) k=l n
+2(]' L[l - H&(Ak)]2.j:\;Bk~(k)
k=l and n
liB - 8&11 2 = L[B(k) - H&(Ak)y(kW k=l n
00
= L[l - H&(Ak)]2B~
k=l
+ (]'2 L AkH~(Ak)e(k) k=l
n
-2(]' L [l - H&(Ak)]Bk.j:\;H&(Ak)~(k) .
k=l Therefore for the excess risk we have
n
-2(]' L[l - H&(Ak)lBk.j:\;~(k)
k=l
(2.16)
n
-2f-L(]' L[l - H&(Ak)]2Bk.j:\;~(k) k=l
+ Cmaxk (]'2AkH~(Ak) + 2(1+f-L)(]'2 f-L - (1
+ f-L)(]'2 Pen(&)
t
AkH&(Ak)e (k)
k=l - f-L(]'2
~ AkH~(A k)e(k)}.
YURIGOLUBEV
120
The last two lines can be bounded by cr 2 !:l~en (p,). Indeed,
E{ Ccr 2
maXk
AkH~()'k) + 2(1 + J-L)cr 2 t
AkH&(Ak)e(k)
~l
J-L
-(1 + J-L)cr 2Pen(&) - J-Lcr 2 t AkH~(Ak)e(k) } k=l Ccr2 maxi, AkH~(Ak)
::; E sup {
'"
+2(1 + J-L)cr 2
J-L
n
(2.17)
L Ak H", (Ak)e(k) k=l
-(1
+ J-L)cr 2Pen (ex) -
2
J-Lcr 2 t AkH;(Ak)~2(k) } k=l
= cr !:l~en (J-L) .
Finally, with Lemma 2.5 we obtain Eo {-J-L t[l _
k=l
H&(Ak)]2B~
_ Ccr
2 maxk
AkH~(Ak)
J-L
-2cr ~[l-H&(>\k)]Bk~~(k) - 2J-Lcr ~[1-H&(Ak)]2Bk~~(k)} ::; O. This inequality together with (2.16) and (2.17) completes the proof of the theorem. 2.3.1. Proof of Theorem 1.2. In view of Theorem 1.1, it suffices to check that
(2.18) where
121
ON ORACLE INEQUALITIES
We begin the proof of (2.18) with the deterministic term. By (1.13) we get su p { CmaXk
)..kH~(>'k) -
!!:-
fl,
,,>0
~ sup {
1
CA
-
K
_1_
,,>0
fl,
[Ln AkH~(Ak) ]K k=1
~ su p { C)..~-K xK _ x:2:0
t AkH~(Ak)}
J.L
(2.19)
2 k=1
!!:-x} =
~
n AkH~(Ak) } L k=1
C 1/(1- K) A1fl,(K+1) /(K-1).
2
Denote for brevity
H (A) = 2(1 + J.L)H,,(Ak) - J.LH~(Ak) " 2 + J.L . Our next step is to show that
E
~~~{ (2 + J.L) ~ AkH,,(Ak)[e(k) - 1] _ .~ ~ AkH~(Ak)}
(2.20)
:S C 1/(1-K)A1 (1- ,,")-1/(1-K)J.L(K+l) /(K-l) . It is easy to see that in view of (1.14)
Next notice that if {H",( ·), a?: O} is a family of ordered smoothers, then {H,,(-), a?: O} is also a family of ordered smoothers. Therefore by Lemma 2.4, for any J.L > 0 we obtain E
s~p{ (2 + J.L) ~ AkH,, (Ak)[e(k) -
1] -
~ ~ AkH~(Ak)}
~ (2 + J.L)ES~P{~ AkH",(Ak)[e(k) -1 ] _ ~
K J.L A1 [0"2(a) (1 - ,,")(2 + J.L)2] 1/ (1+ )} 2(2 + J.L) 2Ai(2 + 2J.L)2
C(l
+ J.L )2/ (1-1< )2(4+K) / (1-K)A1(1- ,,")-1/ (1-K)J.L (K+1 )/ (K-1).
thus proving (2.20) . Thus (2.18) follows obviously from (2.19) and (2.20) .
122
YUR IGO LUBEV
REFERENCES [1J L . CAVALIER AND Y U. GO LUBEV, R isk hull m et hod and regularization by projections of ill-posed inverse problem s, A nn . of St at . (2006), 34: 1653-1677. [21 L . C AVALIER, G .K. GOLUBEV, D. PICARD , AND A .B . T SYBAKOV, Oracle in equalities fo r inverse problems, Ann . of St a t . (2002), 30: 843-874. [3J H .W . E NGL, M . H ANKE, AND A . N EUBAUER, R egulari zat ion of Inverse Pro blem s, Kl uwer Academic Publishers , 1996. [4J L A . IBRAGIMOV AND R.Z. KH ASMINSKII, Statisti cal Estimation. A symptoti c T heory , Springer-Verlag , NY, 198!. [5] A . K NEIP, Ordered lin ear smo others, Ann. Statist. (1994), 22 : 835-866. [6J B . M AIR AND F .H . R UYMGAART, Statistical estimation in Hilbert scale. SI AM J . Appl. M a th. (1996) , 56 : 1424-1444. [7J A .N . TIKHONOV AND V .A. ARSENIN, Solution of Ill-pos ed Problems , Winston & Sons, Washington, 1977 . [8] A . VAN DER VAART AND J . W ELLNER Weak convergenc e and em pirica l process es. Springer-Verlag, NY , 1996.
HYPOTHESIS TESTING UNDER COMPOSITE FUNCTIONS ALTERNATIVE OLEC V. LEPSKI* AND CHRISTOPHE F . POUET*t Abstract . In this paper, we consider the problem of the minimax hypothesis testing in the multivariate white gaussian noise model. We want to test the hypothesis about the absence of the signal against the alternative belonging to the set of smooth composite functions separated away from zero in sup-norm . We propose the test procedure and show that it is optimal in view of the minimax criterion if the smoothness parameters of the composition obey some special assumption. In this case we also present the explicit formula for minimax rate of testing. If this assumption does not hold, we give the explicit upper and lower bounds for minimax rate of testing which differ each other only by some logarithmic factor. In particular, it implies that the proposed test procedure is " almost " minimax. In both cases the minimax rate of testing as well as its upper and lower bounds are completely determined by the smoothness parameters of the composition. Key words. nonparametric hypothesis testing, separation rate, minimax rate of testing, composite functions , structural models, metric entropy, gaussian random function, Implicit Function Theorem. AMS(MOS) subject classifications. 62ClO.
1. Introduction. In this paper, we study the problem of minimax hypothesis testing in the multidimensional gaussian white noise model
dXe(t)
= g(t)dt + cdW(t),
t
= (tI, . . . , td) E'D o
(1.1)
where 'Do is an open interval in lR d , d 2: 1, W is the standard Brownian sheet in lR d and 0 < e < 1 is the noise level. Our goal is to test the hypothesis on the absence of the signal g, i.e.
H:
g=O,
against the alternative written in the following form
G(1Pe) = {g
E
A:
9 E
G(1Pe) ,
G:
Ilglloo
~ sup Ig(t)12:1Pe} . tED
Here G is a compact subset of 1L 2 ('Do) endowed by the metric generated by 11 ·1100 , 'D c 'Do is an open interval in lR d and 1Pe -> 0, E -> 0, is the separation sequence. We consider the observation set 'Do which is larger than 'D in order to avoid the discussion of the boundary effects. Without loss of generality we will assume that 'Do = [-1, l]d and 'D = [- 1/2, 1/2]d. We define a decision rule to be any measurable function of the observation {Xe(t), t E 'Do} taking the values 0 and 1. *Laboratoire d'Analyse, Topologie et Probabilites, UMR CNRS 6632, Universit e d 'Aix-Marseille 1, 39, rue F . Joliot-Curie, 13453 Marseille cedex 13, France (lepski~cmi.univ-mrs. fr ). t (p ouet ccmt . univ-mrs . fr ). 123
124
OLEG V. LEPSKI AND CHRISTOPHE F. POUET
1.1. M in imax approach. To measure the performance of any decision rule we will use so-called minimax criterion. Let J!l'9 be the probability measure on the Borel a-algebra of C (D o) generated by the observation (1.1). For any decision rule ii we int roduce the risk function
1<.(ii ,1fe)=J!l'o{ii=1}+
sup J!l'g{ii=o} ,
(1.2)
gEG C1/J. )
which is the sum of the first and t he second errors probabilities. We say that the separation sequence 'Pe is the minimax rate of testing and ii is t he minimax decision rule if 1. limc~oo limsuPe->o 1<.(ii, Cepe) = 0; 2. limc~oliminfe~oinf}. 1<.(ii ,C'Pe) = 1, whe re inf is taken over all possib le decision rules. It is worthy of mentioning t hat t he minimax rate of testing as well as t he minimax decision rule usu ally dep end on the parameter set G t hat makes its choice very import ant. In t he next section we discuss t he choice of G for a model described by a mult ivar iate function .
1.2. Choice of parameter set. Structural assu m ptio ns. It is well known t hat the main difficulty in testing under alternatives being multivariate functions is the curse of dimensionality: the best attainable rate of t esting decreases very fast , as the dimension grows . To illustrate this effect , suppose, for example, that the underlying function 9 belongs to G = lHId( V, L ), u :» O,L > 0, where lHId(V, L) is an isotropic Holder class of fun ctions. We give the exact definition of this functional class later . Here we only mention that lHId(V, L) consists, in particular, of functions 9 with bounded partial derivatives of order less or equal LvJ and such that , for all x, y E Do,
where Pg(x ,y - x) is the Taylor polynomial of order LvJ obtained by expansion of 9 around the point x , and II . II is the Euclidean norm in JRd. P arameter u characterizes the isotropic (i.e., t he same in each direction) smoothness of function g . If we use the risk (1.2) , t hen ( [11], [16]) the r at e of testing is given by
It is clear that if v is fixed and d is large enough this asymptotics of the rate is too pessimistic to be used for real data: the valu e 'Pe ,d(V) is small only if the noise level e is unreasonably small. On the other hand, if the noise level e is realistically small the above asymptotics might be of no use alr ead y in dime nsion 2 or 3. This problem arises because the d-dimensional Holder class lHId(V, L ) is t oo massive . A way t o overcome the curse of dimensionality is to consider
HYPOTHESIS TESTI NG UNDER COMP OSIT E FUNCTI ONS
125
models wit h poorer functional classes G. Clearly, if the class of candidate functions g is smaller , t he rat e of testing is fast er. Note t hat t he "poverty" of a funct ional class can be describ ed in t erms of rest ricti ons on its met ric ent ropy and there are several ways t o do it. The way we will follow in the pres ent paper consists in imposing a structural assumption on the function g . The classical exa mples are pr ovided by the single ind ex additive and projection pursuit structures ([3], [4], [6], [7], [8], [9], [20] and [21] among others). (i) [Sin gle-index m odel.) Let e be a dire ction vecto r in]Rd, and assume that g(x) = f (eT x) for some unknown univari ate functi on f. (ii) [Additive m odel.). Assum e that g(x ) = L~=l f i (Xi ) , where fi are unknown univariat e functions. (iii) [P rojec ti on pu rsuit regressi on.) Let e 1 , .. . ,ed be dir ect ion vectors in ]Rd , and assume that g(x) = L~=lfi(eTx) , wh ere I, ar e as in (ii). (iv) [Multi-in dex model.) Let e1, . . . , em, m < d are dir ect ion vectors and assume that g(x ) = f eeT x , . . . , e;;'x ) for some unknown rndimension al function f. In genera l, und er structural assumpt ions the rate of testi ng improves, as compared t o the slow d-dim ensional rat e 'Pe.,d(l/). On t he other hand, the assumption that the underlying fun ct ion g belongs to a poor functional class can lead t o inad equ ate model. In general, it is qu it e restrict ive to ass ume t hat g has some parametric str ucture, i.e. the str uct ure described by a finit e dimension al par amet er and remained un chan geable in the whole doma in of observati on. Thus, we seek a rather general (nonparametric) str uct ural restriction which would ad mit powerful testing pro cedures wit hout sacrificing flexibility of t he mod elin g. We argue t hat t his program can be realized if the un derlying function g is a composition of two smooth funct ions . 1.3. Composite functions. We now define our nonparametric struc ture imp osed on the model. We will assume that g is a com posite fun ction, i.e., t hat get) = f (G (t )), where f : [0, 1] -; ]R and G : V o -; [0,1 ]. All our results remain valid if we replace [0, 1] in the definition of f and G by some other bounded interval in R We will further suppose that f and G are smooth functi ons such that f E lHhb, L 1 ) and G E lHId(,8 , L2) where "( , L 1 ,,8, L 2 are positive const ants . Here and in wh at follows JH[1 b, L 1 ) and lHId (,8, L 2 ) ar e t he Holder class on [0, 1] and t he isotropic Holder class on V o respectively (see Definiti on 1.1 below). The class of composite funct ions g with su ch f and G will be denoted by lHI(a, £), where a = b,,8) E lR~ and £ = (L 1, L 2 ) E ]R~. The performan ce of a t est ing pro cedure will be measured by the risk function (1.2) where we set G = lHI(a, £). We will see t hat t he val ue of a det ermines th e quality of testing associated to our mo del, i.e. , t he rate of testing.
126
OLEG V. LEPSKI AND CHRISTOPHE F . POUET
We now give the definitions of anisotropic Holder ball and discuss some trivial cases of testing under alternatives being composite functions. DEFINITION 1.1. Fix v > 0 and L > 0 and an interval V' in ]Rd. Let lv j be the largest integer which is strictly less than v and for f = (k l , .. . , kd) E N d set IfI = k l + .. .+ kd. The ISOTROPIC HOLDER CLASS lHId(V, L) ON Viis the set of all functions G : V' ~ ]R having on V' all partial derivatives of order lv J and such that J
-
"" I fJlkIG(y)~ kd L..J ~ kl
Ikl=Lvj
uX'" uX d 1
I
fJ1kIG(z) < Lily - zllv-Lv j I ~ kl !:l kd , V Y, z E V . UX I ' " uXd
where Xj and Yj are the jth components of x and y . REMARK 1.1. It is easily to see if G E lHId(V, L) then
rr d
G(Y) - ""
I
L..J
O~lk'~ Lv j
fJ 1kIG(x) fJxk1 ... fJx kd I
d
(Yj_Xj)k k .!
j=1
j
I -< Lily-xliV ,
V X,Y E V'.
J
where Xj and Yj are the jth components of x and y. REMARK 1.2 . It is also evident that if G E lHId(V, L) then fJG
~ E
lHId(V - 1, L) ,
UXI
Trivial cases. Zone of super-slow rate: 0 < "(, (3 < 1. Clearly, in this zone JHI(a, £) c lHI d ('y(3 , L 3 ) , where L 3 is a positive constant depending only on "(, j3 and £. Due to this inclusion, a standard testing procedure [11] converges with the rate (3, "( > 1. In this zone we easily get the inclusions lHId((3, L 4 ) c lHI(A, L) c lHI d((3 , £5), where £4 and £5 are positive constants depending only on (3 and £ . To show the left inclusion it suffices to fix a linear function f. Therefore, the rate of testing corresponding to lHI(a, £) is the same as for any isotropic Holder class lHId((3 , ') i.e, coincide with
HYPOTHESIS TESTING UNDER COMPOSITE FUNCTIONS
127
We finally remark that if 13 ~ 1 the composite function 9 is rather non-smooth. The minimax rate of testing on lHI(a, £) is the same as on the Holder class lHId«l/\ ,)13, .). This is a very slow rate 'Pe,d«l/\ ,)13). Therefore, only for 13 > 1 we can expect to find decision rules with powerful statistical properties and later on only the case 13 > 1 will be studied. Moreover, all results in the paper are proved under assumption d = 2. Below we discuss in detail this restriction.
2. Test procedure and main results. 2.1. Test procedure. Our test procedure uses two different constructions of the decision rules. Both of them are based on the kernel estimators of the underlying function 9 = f(G). Let K : JR --. JR be a function satisfying the following assumption. Assumption (K). 1. K(u) = 0, Vu ef. [-1/2,1 /2].
JK(u)du=1.
2.
3N E N* such that JujK(u)du = 0, Vj = 1,N. 4. KElHI 1(1, 1) on JR. Put K(v) = K(Vl)K(V2) , v E JR2 . We will say that K is a kernel. First construction. Put 3.
213
h*
=
e 2-Y13 + 13 + 1
if
13>2,-1;
if
f3~2,-1;
'213
{
[eJln (l /e))
where a =
13 - lf3J,
(13+1)2 ,
1, a,
if if
e-: 2; 13>2;
and define VA > 0
h[A] We set for any r
X= {
= (q, A, x),
K-13
= h*A7J+f.
where q : JR --. JR, A> 0, x E
1),
Later on we will see that ge,i(r)(IIKI12/vIXh),i = 1,2, with the special choice of the function q and parameter A can be viewed as the kernel estimator of the function 9 at the point x. Set _ { e2-yS+J+1 / Jln (l/e), c; -
if
13 > 2, - 1;
if
13 ~ 2,- 1,
13--y
[eJln (l/e)] :rwm ,
128
OLEG V. LEPSKI AND CHRISTOPHE F. POUET
A=
{Aj =
(~2jr,
1/2 < r;2 J
j = O,l},
~ 1,
where K, = 1/({3 - 1) VI,. Put 'I' = lHh ({3, S) ® A ® 1), where S is the constant from Lemma D.I. Introduce the random events
{~~~ Ig""i(T) (h[A]/Axf/2131 ~ 3eL({3,L2)} , i = 1,2,
A =
where £({3, L 2 ) = yC(8S) 21 [1 - II {31 and the constant C is defined in Lemma A.I. The first decision rule is defined as RA1 V RA2 . Second construction. Put 13+1
f-l
=
e~ Jln(~/e),
if
{3 > 2, - 1;
[eJln (lie))
if
{3
{
and define for any x E
- ( )-
g",
X
-
i3+T,
~
2, - 1,
1)
1
f-lII K ii2
J (t K
1 - Xl
f-l'
t2 -f-l X2) dX""tl" ( '\
Clearly, that g",(x)[IIKlldf-l] is the standard kernel estimator with the bandwidth f-l . Introduce the random event
The second decision rule is defined as RB . Our final decision rule is simply the maximum between the first and second one, i.e .6.; = RA1 V RA2 V RB .
2.2. Main results. For any separation sequence ¢'" us denote
-t
0, e
-t
0 let
and consider the separation sequence
=
e2")~~+l
if {3 > 2, - 1; '13
{ [eJln (l ie))
i3+T,
if {3 ~ 2, - I.
THEOREM 2.1. Let 11 E JR~ and £ E JR~ be fixed. There exists C* = C(ll, £) > 0 such that VC > C*
lim sup ",--0
[lP'o{A IUA2 U B } +
sup 9ElHI(C'P
lP'g{;hn,,42nB}]
=0 .
HYPOTHESIS TESTING UNDER COMPOS ITE FUNCT IONS
129
The explicit expression of the constant C* can be found in the proof of t he theorem. THEOREM 2 .2. For any given n E JR~ and i: E JR~ lim inf lim inf il}f [JPlo{Li e = C-+O
e-+O
6.
•
1} +
sup ( g EH C
( l/ e ) l ~
)
IP'g{tS e = O}]
=
1,
where inf is taken over all possible decision rules. Here T/ = 0 if (3 ::; 2'Y - 1 and T/ > 0 is an arbitrary number if (3 > 2'Y - 1. R EM A RK 2.1. (Upper bound) 1. The assertions proved in Th eorem 2.1 and Theorem 2.2 allow us to state that 6; is minimax decision rule and 'Pe(n) is the minimax rate of testing in the case (3 ::; 2, - 1. 2. Moreover, if (3 ::; 2'Y - 1 the rate of testing 'Pe( n) does not depend on , and is equal to the minim ax rate 'Pe,2((3) associated to the isotropic Holder class lHId((3, .) . This is rather surprising: the quality of the test under alternatives written in the form the 9 = f (G) is the same as for the altern atives described by function G independently of function f. Such a behavior cannot be explained in the terms of smoothness: f (G ) E lHI2 b ,.) and does n ot belong to lHI2((3, .) 3. Also , it uiorihs to m ention that the method of obtaining of lower bound in the case (3 ::; 2, - 1 can be simply generalized for an arbitrary dimension d > 2 . The corresponding domain would be (3 ::; db - 1) + 1. REM ARK 2.2 . (Lower bound) 1. In the case (3 > 2'Y -1 the separation sequences in Theorem 2.1 an d in Theorem 2.2 do not coincide. However, they differ from each other only by the fa ctor [In (1/ gj'7 , where T/ is an arbitrary positive number. It allows us to say that the decision rule 6; is "alm ost " minimax decision rule . We are definitely sure that the separation sequence 'Pe(n) from Theorem 2.1 is the minimax rate of testing and, therefore, the lower bound result should be improved. 2. Let us also note that the lower bound construction used in the case (3 > 2, - 1 is heavily based on the assumption d = 2. In t he context of t he las t remark it is interesting to compare our results with the resul ts recently obtained in [13] for t he estimation of composite function. In [13] t he authors study t he minimax rate of convergence of estim ators, i.e. t he asymptotics of minimax risk whic h is defin ed as
where inf is taken over all possible estimators . The lower bo und result proved in [13] states t hat for any dimension d 2: 2 minimax rat e of conver gence ca n not be fast er than
130
OLEG V. LEPSKI AND CHRISTOPHE F. POUET 213-r
"l/J,,(a) =
{
[eJln(1/e)]2 "Y1'+i3+d 21' [eJln (l ie)] 2 i3 +d,
1 ,
if (3
> db - 1) + 1;
if (3::; db
-
1) + 1.
Comparing this result with the results given by Theorem 2.1 we can conclude that the minimax rate of convergence of estimators and minimax rate of testing differ if (3 > 2,-1 and d = 2. This result is very unusual. As far as we know it is the first problem described in the literature where the minimax rate of convergence of estimators and the minimax rate of testing are different in the case of sup-norm losses. In particular, the upper bound result proved in [13] under additional assumption, E (0,2) , (3 E (1,2] shows that "l/Jr;;(a) is the minimax rate of convergence of estimators if j3 ::; db - 1) + 1. As we see both rates, CPr;; (a) and "l/Jr;; (a) coincide if (3 ::; 2, - 1 and d = 2. 2.3. Open problems. OTHER STATISTICAL MODELS AND PROBLEMS .
1. We considered the gaussian white noise model because it is the simplest and idealized object to study, its analysis requires a minimum of technicalities. Composition structures can be studied in the same spirit for more realistic models, such as nonparametric regression with random design , nonparametric density estimation and classification. Note that our theorems can be directly transposed to gaussian nonparametric regression model with fixed equidistant design using the equivalence of experiments argument
[2], [18]. 2. We restrict our study to the sup-norm loss and to the Holder smoothness classes . A natural extension would be to consider models where the risk junction is described by other norms and other smoothness classes, such as Sobolev and Besov classes, or, for instance, by the classes of monotone or convex functions. The case of functional classes with anisotropic smoothness is of interest as well. 3. We consider only the simplest composition f(G), where f : JR -+ JR and G : JRd -+ R A more general description could be , G k ) , where! : JRk -+ JR and G s : JRd. -+ JR, s = K" and ! (G 1, d1 + + dk = d. 4. A related more complex modeling can be based on Kolmogorov's theorem of representation of a continuous function of several variables by compositions and sums of functions of one variable addition [14], [19] . REFINEMENT OF ASSUMPTIONS.
1. In the present paper we are able to treat only the case d = 2. We believe that our upper bound result (Theorem 2.1), including t esting procedure, with minor changes can be directly applied in any dim ension in the case (3 ::; d(, - 1) + 1. It will require onl y
HYPOTHESIS TESTING UNDER COMPOSITE FUNCTIONS
131
to prove smoothness properties of the implicit function of d - 1 variables analogously to result obtained in Lemma D.l. This conjecture is partially confirmed by the results from [13] for the case "( E (0,2),,6 E (1,2]. To construct the decision rule one can apply the estimation procedure proposed in [13] which is quite different from the constructions used in the present paper. In view of Remark 2.2 this approach would bring a minimax decision rule. 2. However, the lower bound result given by Theorem 2.2 if ,6 > 2"(- 1 uses rather sophisticated construction of random walk on JR recently proposed in [5]. As far as we know such result does not exist in the dimension larger than 1, that restricts the use of our lower bound construction in the dimension larger that 2. Moreover, as it has been already mentioned in Remark 2.2, even if d = 2 the lower bound is not exact. As a consequence, in the case ,6 > d("t - 1) + 1 the extension to an arbitrary dimension as well as the exact lower bound for d = 2 for the risk function (1.2) remain open problems. 3. Proofs. 3.1. Proof of Theorem 1. I. Let us find the upper bound for the first error probability, i.e. for lPo{AI U A 2 U B}.
For given and put
x E A let
us denote 'I.x
=
{T = (q,.\, x),
q E lHl(,6, S),
XED}
Note that
lPO{Al
U
A2 U B} :S lPo{B} + 2
L
lPo{A.x,t},
(3.1)
.xEA
since the distributions of the gaussian random functions cide under lPo. Put also
First we find the upper bounds for IE sUPrE'I".>. IE sUPxE'D ((x). I. La. Upper boundjorIEsuPrE'I"€(T).
90:,1
«T),.\
and
90:,2
coin-
E A, and for
132
OLEG V. LEPSKI AND CHRISTOPHE F. POUET
Let us fix A E A, T = (q,A,x) E'I A, T' = (q',A,X') E'I A and consider the intrinsic semi-metric generated by gaussian random function €(-) :
Putting 'it E JR
we have
and, therefore , h(T, T')
~ IIK~x - K~\xIl2 + IIK~\x - K~\x'112 A
(3.2)
yIJ; + yIf;.
IIK;,xIl
Taking into account that = 1, 'iT = (q, A, x) E 'I and the definition 2 of the kernel K we obtain by direct calculations
It =
2[1- (K; ,,,, K;,,,, )] = -2(K;,,,,K;,,,,-K;,,, )
= -I I;II~
I x::
2V e 2)K(v l) [X::(Vl+eA"'1h[A])[QeV2)-Q'eV2) J)-K(Vl)] dv 1dv2 .
Applying the assumption (Kl), (K2) and (K4) we get
h
~2 [::~::~] [~:]] Ilq-q'lloo ~2 [~:]] Ilq-q'lloo A Q(A)llq-q'lloo. (3.3)
Putting q~ , x'(-) = q'(-) -
q'(. +(X2 - X2)/A)
we have similarly to (3 .3)
Taking into account that q' E IHI(.8, S) and, therefore , 2 llq'x ,x ' II < sIx -A x21 (X)
-
(3.5)
133
HYPOTHESIS TESTING UNDER COMPOSITE FUNCTIONS
applying assumptions (Kl), (K2) and (K4) we obtain from (3.4) and (3.5)
where
II· Ih
is lLl-norm on lR 2 . Thus, we have
2V2[h* A~/2rl [1 + ~ CEllx - x'/h.
12 :::;
diam(D) + [S VI] diam(D)] Ilx
- x'iiI
(3.6)
Finally, we obtain from (3.2), (3.3) and (3.6)
fJ>..(T,r ') :::; VQ(A)llq - ql/loo + /CEllx - x'iiI.
(3.7)
For any 8> 0 let us denote by E~oo)(lHh(,8,S)), Ey)(D) and E~ji).)('I,,),A E A, the 8-entropy OfJH[l (,8, S) w.r.t uniform norm, 8-entropy oi D w.r.t. 11·111norm, and 8-entropy of 'I" w.r.t. intrinsic semi-metric p" respectively. Clearly, that \/8 > 0
E~l) (D) :::; 2ln (
dia;(D))
(3.8)
Note also that (3.7) implies Yu > 0 and YA E A
E~ji",)('I,,) :::; ECc::} (JH[(,8,S))
+ E~l
4Q("')
(D).
(3.9)
4C<
Therefore, we deduce from (3.8), Lemma A.l and (3.9) that \/u > 0 and YA E A
E~ii>. )('I,,) :::;C[4SQ(A]*U-~ +4ln(l /u) +2ln(4C diam(D)) E
Noting that IEt 2(T) = 1, Yr E 'I, and applying Lemma B.l with obtain from (3.10) YA E A
V' J-k
IE 7"s:~K t(r) :::; £(,8 , L 2 ) [ h[A] where £(,8 , L 2 )
+
(J"
(3.10)
= 1 we
~ + £,
= VC(8S) -k [1 - 1/,8] and £ = J2ln (4 diam(D)) + 2.
134
OLEG V. LEPSKI AND CHRISTOPHE F. POUET
Simple calculations show that In Cc; :=: In (1/ c) and (3.11) where a > 0 depends on (3 only. Thus, we have for all and VA E A
E
> 0 small enough
(3.12) Ll.b. Upper bound for IE SUPxE'D ~(x). Putting Vx, x' E V
we obtain similarly to (3.6) Vx, x' E V
p(x ,x /) :::; f.l-1J diam(V) llx - x'iiIThis together with 1E~2(x) = 1, Vx E V, (3.8) and Lemma B.1 gives (3.13) Note that VA E A in view of (3.12) and (3.11) we have
, 1/2(3} lP'O{AA,I} = lP' { sup 1~(r)1 ? 3£((3 , £2) ( AX) / h[AJ rE'!A ) 1/2(3} , = 2lP' { sup ~(r) ? 3£((3 , £2) ( AX / h[AJ rE'!;,
,
,
( ) 1/2(3 }
:::; 2lP' { sup ~(r) - IE sup ~(r) ? £((3, £2) AX / h[AJ rE'!A rE'!;, :::; 2lP' { sup rE'!A
~(r) -
Applying Lemma B.2 with
(J
IE sup €(r) ?
a«,£2)C- a} .
rE'! A
= 1 we get VA E A
and , therefore
L lP'o{ AA,I} :::; 2log2 (1/() exp { _~£2(fJ, £2)C- 2a} . AEA
(3.14)
HYP OTHESIS T ESTI NG UNDER COMPOSIT E FUNCT IONS
13 5
Using t he same arguments we have from (3 .13)
Jl!'o{B} ::;
2J1!'{SUP~ (X) - lEsup~(x) 2: V ln( I //1) } xE D x ED
::; 2.Jii. (3.15)
Taking together the bounds found in (3.14) and (3.15) we get from (3.1)
Jl!'o { A l U A2 U B}
--+
0, as
e --+ O.
(3 .16)
II. Let us find t he upper bound for t he second err or probabilit y, i.e. for
Jl!'g {A l n A2n B}.
sup gE IHl (C cp c (a ))
For any g E lHI(a, ..c) , r E 'I' and x E 'D let us denote by ,
1
Bg(T) = ,\h[,\J
/
K
Bg(x ) = : 2/ K
(tI
-Xl - '\"'Q((t2- X2)/'\ ) h-X2) h['\]
, -,\-
f(G (t ))dt - f(G (x )) ;
Cl ~ Xl ,t2 ~ X2 ) f (G(t ))dt - f (G(x )).
It is clear that
[%&r] -
Bg(r ) =
s,
Bg(x ) =
z, [ge (X)~KI I2]
f (C(x)) , (3.17)
- f (C(x)) .
Let M ~ M ({3 , L 2 ) be t he constant from Lemm a D.l. Put j = 1, J
Sjo(x) =
{gE lHI(a, ..c) :
Sj j (x) = {
g E
IC 1 ,o(x)! V !CO, l (X)! ::;
A' 1]1/'"::; lHI( a, ..c) : [ ~
+ 1,
M~/"' } ;
IC1,o(x)1V IC ,l (X)! ::; O
[A.] ~ l /"'} ,
where, AJ+l ~ M (L 2) "', C l ,o(x) and CO,l(X) ar e [8C/8t l ] (x ) and [8C/8 t2] (x) respe ctively. First, for given x . E 'D let us find the upper bound for sup g Eii o(x )
IBg(x) l,
sup
!Bg(r )l , r = (q,Aj,X ), j = I ,J + 1.
gEiij (x )
vVe denote by 3 1 , 3 2 , . . . , the functions uniformly bounded by 1 on 'D and by Gl , G2 , ... , t he absolute constants and wit hout loss of genera lity we will assume that (3 .18)
136
OLEC V. LEPSKI AND CHRISTOPHE F. P OUET
II.l.a. Upper boundforsuP gEYJ o(x) IBg(x )!. In view of R em ark 1.1 we have \:It E V
where
Since f E lHI 1 (r, L l ) we have 'Vt E V (t he summation with respect t o em pty set of indexes is supposed to be 0)
bJ
L jCl )(G(x)+Px(t- x )) [5 1(t)L21It- xlli3f
f( G(t )) = f(G(x)+ Px(t - x )) +
1=1
= f (G(x ) + Px(t - x ))
bJ
= f (G(x ))
+L
f (l )(G(x )) [Px(t - x )]l + L 154 (t) [Px (t - x)f
1= 1
>
wher e Xl = 0 if , :5 1 and Xl = 1 if , Tak ing into account that 'Vt E (1/ 2)/L, X2 + (1/ 2)/L] and 'Vg E 5)o(x ),
l.
[X l -
2
IPx(t - x )I :5 C2C;/L + X2 C3L2/L,
X2
(1/ 2)/L, X l
=
{
+ (1/ 2)/L]
X
[X2 -
0, if 1 < f3 :5 2; 1, if f3 > 2,
and using the assumptions (Kl), (K2) and (K3) with N ~ f3 2 we obtain from (3.19) : sup
!B g(x) l :5 Cl (a, £ ) [(C;/L )"
+ X2 /L2'Y + Xl/L13 + /L'Y I3] ,
(3.20)
gE.l'io (x )
where Cl (a, £ ) is the constant depe nding only on " f3, L l and L 2. P ut ting 5 = 2, /\ f3 /\ ,f3 we note t hat (3 .20) implies Vu E 2t sup
gEJ'J o(x)
!B g (x)!:5 3C 1 (a, £ ) [(C;/L )"
+ /L s ]
:5 6C l (a, £ )
(3.21)
HYPOTHESIS TESTING UNDER COMPOSITE FUNCTIONS
ILl.b. Upper bound for SUP9Ej'Jj(X) 1139 (T) I, j = 1, J
137
+ 1.
Let us denote by qxO the implicit function being the solution of the equation G(t) = G(x). In view of Lemma D.1 VG E 5'j j(x) the function
qx(.) is uniquely defined on the interval t, and I J + 1 = [x2 -
=
[x2 - Aj , X2
+ Aj] , j = 1, J
1,X2 + 1] .
In view of Remark 1.1, taking into account that G(qx(t2) ,t2)
G(x), Vt2 E Ij we have "It = (tl, t2) E V such that t2 E I j
(3.22) Therefore, repeating all lines in (3.19) we obtain "It E I j 0 I j bJ
f(G(t))
=
f(G(x)) +
L
f(l )(G(x)) ['-Px(tl - qx(t2) ,t2) ]1
1=1
+X 1 C4 S 7 (t)L l [L2 V 1rltl
- qx(t 2)1 13
(3.23)
+LlSS (t)['-Px(tl - Qx(t2), t2)r + S6(t)L lL;ltr - qx(t2)1 , 13, Set (3.24) and let Tx ,j
= (qj,x,Aj,x). Note that
Here we used , in particular , the definition of the kernel K . Let us also remark that the kernel K is vanished on lR \ [- 1/2, 1/2] and, therefore, the integration w.r.t t2 in (3.25) is done on the domain of definition of qxO (see Lemma D.1 and the definition of Aj). Changing the variables in (3.25) u = (tl - Qx(t2)) /h [Aj ], v = (t2 X2) /Aj , noting that the functions Zk ,x( ' ) in (3.22) are independent of u and using the assumptions (Kl), (K2) and (K3) with N ;::: (32, we obtain from (3.23) and (3.25) similarly to (3.20)
138
OLEC V. LEPSKI AND CHRISTOPHE F. PO UET
sup
IBg (Tj,x)1
gEnj(x)
~ 02(a,.£){ ([Ajf /l
(3.26)
~ 302(a,.£) {([Aj]l/l
and,t herefore , "It E 'D such that It 1
-
qx(t2)1 ~ (1/ 2)h[Aj] and t2 E [X2 -
Aj , X2+ Aj]
Now let us return to (3.26). First, we note that
Next,
if {3 > 2, - 1; if {3 ~ 2, - 1, where
!!Z
() - { u -
[1 _K,({3 - x )({3 - 1)]
,
~
{3 + 1
[1-
2{3({3
if {3 > 2, - 1;
+ 1)
K({3 -X)({3- , )] 2, {3
if (3
~
2, - 1.
and y = y(a) > 0 is the given number. 1. Case , ~ 1 , 1<{3 ~ 2. {3 - 1
z( a) = (3 [ 1 - 2{3({3 + 1)
]
=
2{32 + (3 + 1 2({3 + 1)
> 1.
(3.28)
139
HYPOTHESIS TESTING UNDER COMP OSIT E F UNCTIONS
2. Case "I :::; 1, {3 > 2.
z(a) = 2 [1 _ ({3 - a )({3 - 1)] > 2 _ /3 - 1 > 1. /3+ 1 2/3({3 +1 ) Note also that in t hese cases necessaril y /3 > 2"1 - 1. 3. Case 1 < "I < /3 :::; 2, /3 > 2"1 - 1. z(a)
f3 -1 ] =;;f3 [ 1- 2f3(f3+1) ~
1]=
2f3 [ f3 f3+1 1- 2f3(f3+ 1)
2f32 + f3 + (f3+ 1)2
1>
l.
4. Case 1 < "I < /3 ::; 2, /3 :::; 2"1 - 1.
/32 ~/3"1 ] ?
z(a) = / : 1 [1 -
2~/3-: 1)] > 1.
/ : 1 [1 -
5. Case "I> 1, (3 > 2, /3 > 2"1 - 1.
=
z(a)
[2 1\ ~]
(/3~~~~ D 1)] ? /32: 1 [1- 2~~\)]
[1-
/32 + 3/3
= ({3 + 1)2 > 1. 6. Case "I > 1, /3 > 2, /3 :::; 2"1-1.
z(a) =
2+3{3 [1 - ({3 -a)({3 - 'Y) ] > ~ [~_ P.-] > /3 > 1. /3+ 1 2"1/3 - /3+ 1 2 2"1 - ({3+1)2
~
T hus, we have from (3.26), (3.27) and (3.28) t hat sup
sup
!B (7j ,x)l ::; 6C (a, £ )epe(a).
(3.29)
2
g
j =I,J+ I gESij(x )
°
11.2. Let C > be t he constant the choice of which will be done lat er. Since 9 E lHI(Cepe(a)) there exists x = x(g) E'D such that (3.30) Let us introduce the following notations:
) = ) (g) = {j = 0, J + 1: q(v) = q3,x(v ) = [qx(X2
9 E lHI (Cepe( a))
+ VA3 ) - Xl]/
n Sjj (x )} ;
(A3(;
T = 73,x'
where qj,x is defined in (3.24) . Note also that ) is correctly defined becau se "Ix E 'D J+ I
lHI(a, £ ) =
USj j (x ),
j =O
Sj j (x ) nSjj/(x) = 0, j
=1=
1',
j ,j'
= 0, J + 1.
140
OLEG V. LEPSKI AND CHRISTOPHE F. POUET
In view of Lemma D.1 (1) and (3) QEJH[I(,8,S) , on [-1 ,1] .
(3.31)
Put
and note that due to (3.31)
nA"j ,i
J+l
Ai =
t,
S;; ...
13 S;; B.
j=1
if if
3=f: 0, 3= O.
(3.32)
Without loss of generality we will assume that (3.18) is true for x and, therefore it is sufficient to find the upper bounds for JP9 {AI} and JP 9 {B} . II.2.a. Upper bound for JPg{AI}. First let us note that V).. > 0
Next ,
9",1(1')
=
(3.33)
=
[f(G(x)) +.8g (1')] +s€(1').
Thus, we have from (3.29), (3.11) , (3.33) and the definition of l'
Al =
{
<;;; {
(h[>-] />-x) kl§",l(f) 1:S 3C£((3,£2)}
EII~llf:) [I f (G(f)) 1- IB
g(
<;;;
{It (f) [ ~ (>-X /h[>-]) $
<;;;
{ ltcr)1 ~ l~~ (AX /h[A]) k
<;;;
{ It(f)1 ~ E- a [(0/2 -
f) I] -E (h[>-J/>-x)
kIt(
f) I:s 3C£((3, £2) }
[(0/2 - 602(a,,C))/ IIKI12 - 3£((3, £2)] } [(0/2 - 602(a, 'c)) /IIKI12 - 3£((3, £2)] }
602(a, ,C)) /IIKI12 - 3£((3,£2)] } ,
141
HYPOTHESIS TESTING UNDER COMPOSITE FUNCTION S
where 0 is chosen such that 0 > 1202(a, £) + 31IKI 12.c(.8 , L 2) and a is defined in (3.11). Taking into account that is the standard normal random variable and denoting by its distribution function we obtain vs such that 1= 0
ten
IP'9 {AI}
3
:::; (e- a [ ( 0 /2 - 6C2(a, £)) /IIK II 2 -
II.2.b. Upper bound for First, let us note that
3e.c (,B, L 2)]) .
(3.34)
IP'g{ B}. In (1/ p,) ~ Z (a) In (l/e),
p, = e Jln (1/e)
(3.35)
where /3+ 1
2~,8+,8+1
Z( a) = {
,8+1
if
,B> 2, - 1,
if
,B:::; 2, - 1.
Next,
ge(X)
= 11:1 12
[f(G (X)) + Bg(x)] +e~(x) .
Similarly to previous calc ulation we get from (3.21) and (3.35)
B = { ge(X) <
~ { 11:
[3Jln (1/p,) + 2] e }
112 [If (G(x))I-
IBg(x)l] -
el~(x)1 :::; [3Jln (1 /p, ) + 2] e }
{
~ :Y:~I%~; [If(G(X))1-I Bg (x)l] - el~(x)1 :::; [3Jln (1 /p,) + 2]e } ~ { 1~(x) l 2: J ln (1/e ) [(0 /2-601 (a, £)) / IIKI12-3 JZ (a)] } , where 0 is chosen such that 0 > 120 1(a, £) + 31IKI12JZ (a). Taking int o acco unt that ~(x) is t he standar d nor mal random variable we obtain Yg such t hat = 0
3
IP' g{ B} :::; ( J ln (l/e) [( 0 /2 - 601(a, £ )) /II KI12 - 3JZ (a)J) .
(3.36)
It remains to note that upper bounds in (3.34) and (3.36) ar e independent of 9 and therefore, we finally obtain from (3.32) lim sup
sup
e ~O
9E IlI( C 'P, (a l )
IP'g{A 1 n A2 n B} =
0,
142
OLEC V. LEPSKI AND CHRISTOPHE F. POUET
3.2. Proof of Theorem 2. There are two different constructions for the lower bound. The first one is used in the case 13 > 2"( - 1 and it is based on a rather sophisticated random walk. As to the second construction, used in the case 13 :S 2"( - 1, it is absolutely standard. Putting f(t) = t the problem is reduced to the testing under alternative described by the function G and, after that, one can use the result from [11] . Thus, only the case 13 > 2"( - 1 will be considered. In the proof of the theorem C 1 , C2 , . . • denote absolute constants. Each time they appear, we explain the link between these constants and the constants a and 'c. I General remarks As usual (see, for example, [12]), we will construct a lower bound for inf},. R(fS., C
where JPl 1r (A) =
Ie JPl
fe
1
2: 1 - "2!JPlo - JPl1r 12 ,
(A) dat (8) and !JPlo- JPl1r
I; =
!Epo
(~~
_
1)
2.
The proof is done in the following way. First, the parametric subset in the alternative is defined. Next , we do some calculation on the L 2-distance. Finally, the probability tt is chosen such that it leads to the expected rate of testing up to a logarithmic factor. 1.1 Parametric subset of alternatives. Let us choose a function F : lR. - + lR.+ such that 1. F(~»O.
2. F(1 /2+u)=F(1/2-u). F(u) = 0, u't [0, 1]. 4. FE1Hh("(,1) .
3.
Let us also choose a function r : lR. -+ lR. such that (i) . The function r belongs to the Holder space lHI 1 (13, L 3 ) with L 3 a constant compatible with property (iii). (ii) . The function t7 is non-decreasing. (iii) . t7 (0) = 0, r (1) = 1 and Vl < m = Lj3J : t7(I) (0) = t7(I) (1) = O. Introduce the parameters ,x, hand T such that
1h L L"Y "Y F (1/ 2) = C 3
C{Je(a)
(In (1/c)2+1J) 2"113+13+1 2 -y{3
h=,X(3,
Let~=(6" .. ,~T)E3T={0, l}T . Define k
s, (0 =
L ~i , i=l
So = 0.
T=[l /'x].
HYP OTH ESIS T ESTING UNDER COMPOSITE FUNCTIONS
Set D.. k = [- 1 + (k - 1»., -1
+ k>.] , k = 1, T
and define
14 3
v» E [-
1, 0]
~[Sk - l + ~k t ( y- (k-1 x ».)] lI6.k (Y)·
g~ (y) = ~
Finally, we put g~ (y ) = [g ~(y) + g ~ (-y ) ] /2 , y E [- 1, 1]. The par ametric subset of alte rnatives is defined as follows.
r~ (x) = L 1h"l F Clearly that
r~ =
( Xl -
>.:g~ (X2) ), ~ E 2T.
f (G), where
L 1 h"lF (Lh3 +2~) . f( u ) = £"I U
3
f
By const ruction G E lHI 2( ,B, L 2 ) ,
E
lHI 1('y , L 1 ) . and, moreover ,
1.2 Calculation of the B ayesian likelihood ratio. Let 1r be a probability measu re on 2T which will be chosen later. According t o Gir sanov formula , t he Bayesian likelihood rat io unde r lP'o is
t; = z, ( ex p
( ~J r~ (x)
dW (x ) -
J
2~2 r~ (x) 2 dX) )
.
Put
where the absolute constant C 1 depends only on L 1 , L 2 and L 3. Let €be an independent version of~ . Then,
lEo (l;)=lE>T x>TlEo exp(~J[t~ (X)+tf(X)] dW(x) - 2~2J[t~ (X) +t~(x)]
=lE", x >T exp{c-2/t~ ~ lE>T X>TexP { [172/c2] ;, \ ,ti)} , L.. u\Sk(
dX)
_\}
O - Sk (O $1
k= l
~ tA;E>TX>T ( 19:1<.L.. . =0
where A" = exp (6 2/0"2)
I lsk1-1W....Skl-1(f )19 · . . I ISk' _lW-SkS-1(f)19) '
< k.$1'
- 1.
144
OLEC V. LEPSKI AND CHRISTOPHE F. POUET
Note that up to this point, all formulas hold for any prior distribution on 2T. 1.3 Prior distribution and final calculation. In order to get the rate of testing, we choose the prior distribution 1f as the distribution of the random walk defined in Lemma C.l. Putting
1f
we have
L
lE"x" (
lllskl_ICO-skl_l(e)!$l'"
l$k , < ...
L
L
lE"x" (
l:5kl < .. .
L
llISkS_'(~)-Sk,-1(e)19)
L s,xllls k1_1(e)-xlI9 " ' ][ISkS- l (O- XS!9 )
Xl ,""X S
L
lE"x" (
1:5k 1 < ...
L s ,x][ lskl _l (e)- xd 9 ' "
X l, ' '' 'X s
][ISks_ 1- l (t) - XS- 119
- k s - 1 + l)l+ry k, - k s - l
3c1n (k s
3cln (k s -k 1 +1)1+'7 k s-k 1
3cln(kl)1+7 3cln(k 2 - k1+1)1-h7 l$k , < .. .
k 1-1
k2-kl
S (3cln(T)2+'7)' S (cRwln(T))s (2+'7 ). Thus, the following upper bound holds T
lEo (l;) ~
LA: (cRwInT)(2+
7J )s
s=o
<
- 1- (cRwInT)
2+
1 '7J(exp[O' 2/c: 2] -1)
.
(3.37)
The right-hand side of (3.37) is of order 1 + (CRW In T) (2+'7J ) 0'2 / c: 2 . Recall that
h "Y =
C
L~
3
( / [I
L 1P (1/2) c:
n
1/] 1+'7J/ 2) e
2~
2..ri3+13+1
.
HYPOTHESIS TESTING UNDER COMPOSITE FUNCTIONS
145
Therefore, 1
l-(cRwlnT)
(2+ ) T)
(exp[0'2/c2] -1)
-
1. (3.38)
Thus,
liminfiI}fR(.6. ,Ccpe) e-+O
fl.
~ 1- -2
1
C
2JI3+/+1
1- C
-y
C3
2JI3+/+1 -y
C3
,
(3.39)
where the absolute constant C 3 depends only on L 1 , L 2 , L 3 , (3, I , F (1/2) and CRW. It remains to note that the left-hand side of (3.39) goes to 1 if C -> O. 0
APPENDIX
A. Metric entropy. Let ('1', p) be metric space and let T c '1' be a compact set. For any 5 > 0 we denote by Ef(T) 5-entropy of T W.r.t p. The result below can be found in [10]. LEMMA A.1. If P = 11 ·1/00 (uniform norm on [-1 ,1]) then V5 > 0
C is the constant independent on 0 and L. The upper bound given in the lemma is sharp. We do not present the lower bound because it is not used in the paper.
B. Gaussian random functions. In this section we present some results concerning the large deviation probability of the extrema of gaussian random function . These results can be found, for example, in [1] and [17]. LEMMA B.1. Let T)(t), t E T be a centered gaussian random function and p(s, t) = jEiTJ(t) - T)(s)1 2 its intrinsic semi-metrics. Then EsupT)(t)S;4V2 tE'[
r VE[;(T)du,
l;
where 0' = VSUPtE'[ Var(T)(t)) . Moreover, if Dudley integral is finite then T)(-) is a.s . bounded on T . LEMMA B .2. Let ry(t), t E T be a centered gaussian random function a.s. bounded on T. Then Vx > 0
lP'{SUPT)(t) > ESUP17(t)+x} S; exp{ -x 2/ 20'2} . tE'[
tE'[
146
OLEG V. LEPSKI AND CHRISTOPHE F. POUET
C. Random walk. T he following lemma deals with the existence of a random walk whic h is more unpredictable than the simple random walk . It is a dir ect consequence from [5] (T heorem 1.4). Let ( = {(j , j E N} denote a sequence of random variables taking values in {O, 1} and Let §n (() denote the partial sum =1 ( j . LEMMA C.l. For any 1] > 0, there exists a probability tt on [{O}, {l} ]OO and a constant c > 0 independent of n such that for any k > 0 and any x ~0 :
I:7
D . Implicit functions. In t his sect ion we establish some smoothness properties of implicit functi ons. In spite of the fact t hat the main argume nt here is Implicit Funct ion Theorem [15], we were unable to find req uired results in t he existing lit erature and we give below t heir direct proof. Let G E JHlz (,8, L ) on 'Do ~ IR 2 , Xo E 'Do be the fixed point , and let us consider the equation (D .1)
G(t) = G (xo) Without loss of generality we will suppose t hat
LEMMA D .l. There exist the positive constant M = M (,8, L ) and 5 = 5 (,8, L ) such that VG E 1HI2 (,8, L ) the implicit function qO satisfying (D.1) 1. exists and is uniquely defined on the set [YO - Ma t<; ,Yo + Mat<;] ;
2.
• takes its values in [xo - M (4a)t<; , Xo
+M
(4a) t<;] ;
• Iq' (y )l :::; tc; Vy E [Yo -Mat<;,Yo + Mat<;] , where the constant K 1 depend only on ,8; 3. belongs to lHI l (,8, 5 a""-I3 ) on [yO- M at<; ,Yo + Mat<; ] , where, remind, '" = [1/(,8 - 1)] V 1. Proof. We will use the following not at ions: K t , K 2 , . .. denote absolute constants depending only on ,8, 8 t O, 8 2 0 ,.. . de note bounded functions and F' denotes the derivative of the function F . Also wit ho ut loss of generality we put (x o, yo) = (0, 0) and ass ume that Gl ,o(O, 0) = a. Since Gl ,o E lHI 2 (,8 - 1, L ) the following inequality holds: Gl ,o(x , y) ~ a/2 ,
Vjx l, Iyl :::; (a/ 2Lm)'\ m = ,8 - a .
This follows from Gl ,o(x , y) ~ G t,o(O, 0) - Lm llx _ Yllt / t<; .
(D.2)
HYPOTHESIS TESTING UNDER COMPOSITE FUNCTIONS
147
Put 5 = (a/4{3Lmt. We have
G(5,0 ) = G (0 ,0 ) +G l ,o ( 0 ,0 ) 5 + S1 ( 5) 513f\ 2~ a5_L513 f\ 2 ~ al<+!( I/ LtA{3), where p({3 ) is constant depending on {3 only. In the same way as above, we have
As the function G E IHb ({3, L ), the following exp ansion at point (5, 0) holds
This implies
G(5,y) ~ p({3)al<+l(1 /Lt G( -5, y ) ::; -p({3)al<+l (l / Lt Thus, G(5,y) > 0, G(-5 ,y ) < 0,
'lilyl ::;
alyl- Lmlyl13 f\ 2; + alyl + Lmlyl13f\ 2 . (a/16{3mLt.
Let us fix y E [-(a/ 16{3m Lt, (a/16{3m Lt ] and consid er the function R(-) = G(·,Y). We have
R (5) > 0, R (- 5) < 0, R' (x ) ~a/ 2,
'lixE [- (a/4{3mL)1/13-1, (a/ 4{3mLt].
Then , there exists a unique point x such that R(x) = 0. The implici t function q(-) is defined by
q(y) = x(y). Clearly it exists on the set [ - (a/16{3m L)I< , (aj I6{3m L)I<] and t akes its value in [ - (aj 4{3mL)I< , (aj 4{3mL )1<]. The implicit functi on theorem [15] asserts that the fun ct ion q(.) is m tim es differentiable and we can calculate its derivatives using formula G(q(y ), y) == 0. In parti cular ,
q'(y) = _ GO,l (q(y), y) . G1 ,o(q(y) , y) and in view of (D.2) Iq' (y )1 ::; K 1, 'liy E [ - (aj I6{3m L) I<,( aj 16{3m L)I<] . where K 1 depends only on (3. Thus, th e assertions (1) and (2) of the lemma are proved . Let us now study the regulari ty of the implicit funct ion q(.). Put ting Gk r = Xl X k, r = 1, m , and different iating the identity G(q(y ), y ) = 2 we first prove that for n = 1, .. . , m : I
ez:
°
q(n)(y)G1,o(q(y) , y) + P n (q ( l ) (y), ... ,q(n- l )(y) , y)+ GO,n(q(y ), y) = 0, (D.3)
148
OLEG V. LEPSKI AND CHRISTOPHE F. POUET
where Pn (Zl , . . . , zn- l , y) is a polynomial such that each term is ofthe form n-l
n-l
i=l
i=l
Gk,r(q(y), y)
IT z~ ;, I: ili :::; n,
k+r:::; n.
The relation is true for n = 1 and n = 2. Assume that the relation is true up to n . Then the derivative of each term in the polynomial Pn is also a polynomial such that each term is of the form
Gk,r(q(y), y)
n
n
i=l
i=l
IT z~; , I: u. s n + 1,
k+r:::;n+1.
Thus, we have
where Pn+1 (Zl , ... ,Zn-l ,zn,y) is a polynomial such that each term is of the form
Gk,r(q(y), y)
n
n
i=l
i=l
IT z~; , 2:= il, :::; n,
k + r :::; n .
Therefore the relation holds for any n = 1, . . . , m. In the same way as above we prove that Iq(n)(y)1 :::; K (L /at- 1 . Indeed , this relation is true for n = 1 and n = 2. Let us assume now that the relation is true up to n. Then, each term in the polynomial Pn + 1 is bounded
Gk,r(q(y), y)
n
n
i=l
i=l
IT q(i)(y)l; :::; K 4L IT (L /a)(i-l )l; :::; K 4L (L /at .
It leads to Iq (n+l )(y)! :::; K s (L /at . Let us now study the regularity of points y and z and proceed as follows.
q(m).
We use equality for n = m at
m-1
IGk,r(q(y) ,y)
II q(i)(vd; -
m- 1
Gk,r(q(y),y)
m-1
::; L
II (Lla)(i- 1)1; (L iar jy -
II q(i)(Wi)l; I i=l
i=l
z lK 6 (L la) (r-1 )(lr- 1) ::; L (L la)m-1
Iy -
z];
i=1 i¥r
n
,.",...1
!Gk,r(q(y), y)
i= l
,.",...1
q(i)(z)I ;-Gk,r(q(Z) ,z)
II q(i)(z)I ;I::; K 7Lly- z i=l
IGo,m(q(y) , y) - GO,m(q(Z) ,
z)1 ::; KsLly -
z l,B-m.
l
(L la )""""1 i
HYPOTHES IS TESTING UNDER COMPOSITE FUNCTIONS
149
Therefore summing all the differences and dividing by a we get
where Kg is a constant depending only on {3. These results entail t hat the implicit function is in H; ({3, Kg (L/a)m). 0 The authors are grateful to Fabienne Castell for suggesting to use the resu lts from [5] .
REFERENCES [lJ R .J . ADLER, An Introduction to Continuity, Extrema and Related Topics for Gen[2J [3J
[4] [5J
[6] [7J [8J [9] [10J
[11]
[12J [13] [14J
[15]
[16J [17] [18] [19]
eral Gaussian Processes, IMS Lecture, Notes-Mongraph Series, 12 Institute of Mathematical Statistics, Hayward CA, 1990. L. BROWN AND M . Low, Asymptotic equivalence of nonparametric regression and white noise, Ann. Statist., 2 4 (1996), pp. 2384-2398. H . CHEN, Estimation of a projection-pursuit type regression model, Ann. Statist., 19 (199 1), pp. 142-157. G.K. GOLUBEV, Asymptotically minimax estimation of a regression function in an additive model, Problems Infor m . Transmission, 28 (1992), pp. 101-112. a. HAGGSTROM AND E. MOSSEL, Nearest-neighbor walks with low predictability profile and percolation in 2 + E dimensions, Ann. Probab., 26 (1998), pp. 1212-1231. P . HALL, On projection-pursuit regression, An n. Statist., 1 7 (1989), pp . 573-588. M. HRISTACHE, A. JUDITSKY , AND V . SPOKOINY, Direct estimation of the index coefficient in a single-index model, Ann. Statist. , 29 (2001), pp. 595-623. M . HRISTACHE, A. JUDITSKY, J. POLZEHL, AND V. SPOKOINY, Structure adaptive approach for dimension reduction, Ann. Statist., 29 (2001), pp . 1537- 1566 . P. HUBER, Projection pursuit. With discussion., Ann. Statist., 1 3 (1985), pp . 435-525 . LA . IBRAGIMOV AND R.Z. HASMINSKII, Statistical Estimation. Asymptotic Theory, Springer-Verlag, 1981 (Originally published in Russian in 1979) . Yu.1. INGSTER Asymptotically minimax testing of nonparametric hypotheses, Probability theory and mathematical statistics, Vol. I (Vilnius, 1985) , pp. 553-574, VNU Sci. P ress, Utrecht, 1987 . Yu .1. INGSTER AND LA . SUSLlNA, Nonparametric Goodness-of-Fit Testing Under Gaussian Models, Lecture Notes in Statistics, Springer-Verlag, New York, 2003. A. IOUDITSKI, a.v. LEPSKI, AND A .B. TSYBAKOV, Statistical estimation of com posite functions, Manuscript, 2006. A .N . KOLMOGOROV, On representation of continuous functions of several variables as superposition of continuous functions of one variable and addition, Dokl. Akad. Nauk SSSR, 11 4 (1957) , pp. 369-373. S . LANG, Analysis II, Addison-Wesley, Reading, 1969. O .V. LEPSKI AND A .B. TSYBAKOV, Asymptotically exact nonparametric hypothesis testing in sup-norm and at a fixed point, Probab. Theory and Related Fields, 117 (2000), pp . 17-48. M. LIFSHITS, Gaussian Random Functions, Kluwer Academic Publishers, Dordrecht , 1995 . M . REISS , Asymptotic equiva lence for nonparametric regression with multivariate and random design (2006), Submitted. D .A . SPRECHER, An improvement in the superposition theorem of Kolmogorov,. J . Math. Anal. Appl. , 38 (1972), p p. 208-213.
150
OLEG V. LEPSKI AND CHRISTOPHE F. PO UET
[20] C. J . STONE, Optimal global rates of convergence for nonparametric regress ion, Ann. Statist., 10 (1982) , pp. 1040--1053. [21] C .J . STONE, Additive regression and other nonparametric models, Ann. Statist., 13 (1985) , pp. 689-705.
PART
III:
STOCHASTIC
PARTIAL DIFFERENTIAL EQUATIONS
ON PARABOLIC PDES AND SPDES IN SOBOLEV SPACES W~ WITHOUT AND WITH WEIGHTS NICOLAI V. KRYLOV " Abstract. 'We present a "st reamlined" t heory of solvabili ty of parabolic P DEs and SPDEs in half sp aces in Sobolev spaces wit h weights . T he approach is based on inte rio r estimates for equatio ns in the whole space and is easier t han and quite di fferent from t he standard one . Key words. St och ast ic partial differential eq uat ions , Sobolev spaces wit h weights, interi or estimates . AMS(MOS) subject classifications. Primary 60H1 5, 35K 20.
1. Introduction. Let jRd be a d-dimensional Euclidean sp ace consisting of points x = (xl, ..., x d ) . We denote
8 x'
D , = -8"
Dij =
u.o;
Introduce D u as the gra dient of a function u wit h resp ect to x and D 2u as its Hessian matrix. In domains G C jRd we are considering t he following par abolic equation (1.1) where T E (0, 00] and L tu = ( 1 /2)a~j Dij u
+ b~DiU + CtU,
an d t he stochastic par t ial different ial equat ion
where r is a stopping t ime
Here and everywhere below we are using t he summation convention. The precise framework in which th ese equat ions ar e treated is given late r in the ar t icle. We present t hree sets of results : Set 1 . Int erior Wi-esti mates for solut ions of (1.1) (Sect ion 3) and (1.2) (Sect ion 2) when G = jRd . "127 Vi nce nt Hall, University of Minnesota, Min nea polis, MN, 55455 . The work was partially suppo rted by the NSF Grants DMS-0140405 and DMS-0653121. 151
152
NICOLAI V. KRYLOV
Set 2 . Boundar y estimates for solutions of (1.1) (Section 5) and (1.2) (Sect ion 4) in spaces like with weights when G = IRt, where
Wi
IRt
= {x = (x l , x' ) E IRd
:
x l> O,x' E IR d -
I
}.
Set 3 . Exist ence and uniqueness t heorems for (1.1) (Sect ion 7) and (1.2) (Section 8) in spaces like with weights when G = IRt. In case of equatio n (1.1) we take p E (1,00) and in case of (1.2) we restrict p to [2, 00).
Wi
This is, of course , a quite nar row subclass of t he class of pro blems treated in t he modern t heory of SP DEs and we do not try to overview t his very wide ly develop ed theory referring the reader t o t he collection of articles [4]' to [6]' [16], [20]' and many references t herei n. Our main resul ts collected in Set 3 are known from [7] and [9] (apart from Theorem 8.1) even for spaces of Bessel pot enti als HJ, which coincide with W ;r if 'Y is a nonnegative int eger . However , t he proofs here are qui t e differe nt and mu ch more eleme ntary albeit valid only for 'Y = 2 (see however Rem ark 2.1). Conce rn ing earlier res ults on stochastic parti al differ enti al equations in domains t he reader can consult [2]' [1], [3]' [5]' [17]' [18]' an d [19]. Notice that t he ap proach in [2]' [1], [3], and [5] is based on semigroup theory and req uires , sometimes imp licitl y, certain com patibility of the data on the bou ndary. We work in Sobolev spaces wit h weights allowi ng the derivatives of solutions to blow up near t he boundary, which enables us to avoid any com patibility conditions. In a sense the result of Section 3 also belongs to "well-known results" . However , we could not find it in the form we need and therefore give a complet e proof. To the best of our knowledge, all other results of t he paper are new. \Ve derive the results in Set 3 from t hose in Set 2, whic h in t urn are derived from t he resu lts in Set 1. The approach in [7]' [9], [17]' [18]' and [19] requires first developing t he full t heory of solvab ility in 5j;,e(T)-s paces (see t he definition of 5j;,e(r) in Sect ion 4) for equations in IRt wit h coefficients independent of x . In case of (1.2) this leads t o t he fact t hat from t he start one needs t he followin g restriction on t he par amet er e calibrat ing the rat e wit h which t he deri vati ves of solutions can blow up near GIRt:
d - 1 + P [1 -
1_
p( l - <5)
_] < o < d - 1 + p ,
(1.3)
+ <5
where J E (0, 1] is t he constant of "parabolicity" of (1.2) in t he direction of x l axis (see condition (8.6) below) . By the way, observe that (1.3) is obviously satisfied if
d - 2+p
~
e < d - 1 + p.
Also if J = 1, t hen (1.3) becomes a condition necessary and sufficient for solvability theory of equations like (1.1) in domains: d - 1 < < d - 1 + p.
e
ON SPDES IN SOBOLEV SPACES WITH WEIGHTS
153
e
Here from the very beginning we treat variable coefficients, allowing to be any number, and introduce condition (1.3) in the very end. Actually, apart from Section 8 and a few auxiliary results in Section 6 there is no restriction on e whatsoever. We will see that (1.3) is needed for only one purpose: to estimate the zeroth order norm of solutions of equations with variable coefficients. Therefore, if for a particular equation and particular T one can prove such an estimate for 0 beyond the range in (1.3), then automatically, the solvability theory becomes available for this equation on (0,T) (see Theorem 8.1). We will present some examples when it really happens in a subsequent article. There we will show an example in which for a (beyond the range in (1.3)) for the coefficients frozen at some points Xo the equation is not solvable in Sj;,O(T), but the original equation with variable coefficients does admit a unique solution in Sj;,O(T). One of motivations for writing this paper was to prepare necessary tools for extending to the case of variable coefficients the pointwise boundary Holder regularity result from [13] proved there for equations with constant coefficients. For that purpose we prove Theorem 4.1 and say more about it in Remark 4.3. One of advantages of our approach is seen from the proof of the uniqueness of solutions of SPDEs on random time intervals [0, T]. This issue was not properly addressed in quite a few articles. In [8J uniqueness for random time intervals was obtained by proving it first for deterministic times and then using a method of continuation of arbitrary solution on [0, TJ beyond T. We do not use this roundabout here. We have outlined above the contents of all sections in the article apart from Section 6 in which we collect some results needed to justify the way we prove the estimates of the zeroth norm of solutions.
e
2. Interior estimates for SPDEs in the whole space. Let (D,:F, P) be a complete probability space with a given filtration (:Ft , t ~ 0) of a-fields :Ft C :F complete with respect to :F,P. Let do ~ 1 be an integer, possibly taking the value 00 and let w~, defined for finite integers 1 ~ k ~ do, be independent one-dimensional processes each of which is a Wiener process with respect to the filtration {:Ft , t ~ O}. Let P be the predictable a-field on D x lR+, where jR+ = (0,00). Introduce £2 as the Hilbert space of sequences y = (yk, 1 ~ k ~ do , k < 00) of real numbers with norm defined by do
lyl~2
=
L
ly k l2.
k=l
Let d ~ 1 be an integer and assume that for each x E jRd and i, j = 1, ..., d we are given a~j (x), b~(x), Ct(x), which are real-valued processes defined for t > O. We also assume that for each x E jRd and i = 1, ... ,d we are given £2-valued processes a;(x) = (a;k(x) , k = 1,2, ...) and lIf(x) = (lI~(x), k = 1,2, ...) defined for t > O.
154
NICOLAI V. KRYLOV
A SSUMPT ION 2 .1. (i) The functions a,b , c, a , and v are P x B (JRd)m easurable; (ii) Th e functions a i k and v k are continuously differentiable in x; (iii) For som e constants 80,81 , J E (0, 1] for all values of the indices and arguments we have a i j = a j i and for all unit ); E JR d ,
(2.1) where
This assumption is supposed to be satisfied throughout the st ochast ic part of t he ar ticle and in this sect ion we are considering equat ion (1.2). Equation (1.2) is understood in the sense introduced in [11] or [10] and the solutions are looked for in the space 'H.~,o (r) with f E IL p (r ) and 9 = (gk , k = 1,2 , ...) E lHI~ (r). We remind bri efly the definition of these spaces . Everywhere in the stochastic part of the article unless spe cifically ind icat ed otherwise we t ake p ~ 2.
For a sto pping t ime r and integer n W; (JRd) and we set
W; =
(O , rn = {(w,t ):
= 0,1 ,2 we take Sobolev sp aces
°< t::; r (w),t < oo},
In an obvious way we int rod uce t he norms in t hese spa ces and extend these definiti ons for £2- valued funct ions su ch as g = (gk, k = 1, 2, ...). If for any (w, t ) E (0, r n we are given a generalized functi on u = Ut (w) = Ut on JRd such t hat U E n O
(Ut,¢) = ( (fs , ¢ )ds +
Jo
do
t
L ( (g;,¢) dw: k=l
Jo
(2.2)
holds for all finit e t ::; r with pr obability 1. We set
If (2.2) holds we write
(2.3)
ON SPDES IN SOBOLEV SPACES WITH WEIGHTS
155
and this explains the sense in which (1.2) is understood. If G is a domain in ]Rd and (2.2) holds for any 1> E C!)(G), then we say that (2.3) holds in G. To describe the first main result of this section we take an integer d 1 E [1, d] and for x E ]Rd introduce X,
= (1 X ,
... ,xd 1 )
,
Bk =
{x
E]Rd :
Ix'i < R}.
THEOREM 2.1. Take some s , R E (0, 00) and a stopping time rand assume the following: (i) There exists a constant K E (0,00) such that for x E Bk and y E ]Rd, such that Ix - yl ::; eR, and all values of the indices and other arguments
RIDO";(y)le2
+ Rlb~(y)1 + RIVt(y)le2 + R 2!ct(y)1 + R2IDVt(y)le2 ::; K , 10"; (x) - 0"~(y) le2 + la~j(x) - a~j(y)1 ::; 130,
where 130 = 13o(d,P,OO ,Ol) E (0,1] is a (small) constant from Lemma 2.1 below. (ii) We are given a function U E 'H.;,o(r) which is a solution of (1.2) with some f E lLp(r) and g = (gk, k = 1,2, ...) E JH[~(r). Finally, assume that Ut(x) = 0 if x rf. Bk. Then there exist constants N = N(d ,p,oo,od and N* = N*(d,p ,oO ,Ol,C,K).such that
::; N(llf lllLp(r)
IID 2uIILp(r) + R-11IDuIILp(r ) 1 2 + IID gII Lp(r ) + R- IlgIILp(r)) + N* R- lIuliLp(r) '
(2.4)
To prove this theorem we need a lemma. LEMMA 2.1. Let r be a stopping time and assume that we are given an U E 'H~ ,O(T) which satisfies (1.2) with some f E lLp(r) and g = (gk, k = 1,2, ...) E JH[~(r). Assume that for a domain G c]Rd (perhaps, G =]Rd) we haveut(x) = 0 if x rf. G. Also assume that there are constantsc ,K E (0,00) such that for all x E G and y E ]Rd, such that Ix - yl ::; c, and all values of other arguments we have
IDO";(y )le2 + Ib~(y)1 + IVt(y)le2 + ICt(y)1 + IDvt(y) le2 ::; K , 10"; (x) - 0";(y)!e 2 + la~j (x) - a~j (y)1 ::; 130 , where 130 = 130 (d, p, 0o, 81) E (0, 1] is a (small) constant an estimate from below for which can be obtained from the proof. Then there exist constants N such that
= N(d ,p,oO,Ol),
N*
= N*(d,p,oO,Ol, €,K)
156
NICOLAI V. KRYLOV
P roof. Take a nonnegative ( E CO' ( jRd) with support in t he ball of rad ius f: centered at the origin and such that
Introdu ce (Y(x ) = ( (x - y ) and observe that with N = N (d, p)
ID2Ut (x )IP = { I( Y(x )D2Ut(x )IP dy :s; N { ID 2((Y(x)u(x) )IP dy
l ad
+N
l ad
(2.6)
{ ID(Y(x)IPI Dut(x)IP dy + N { ID2(Y(x )IP lut(x)IP dy.
l ad
lad
For each y E jRd the function (Y Ut satisfies
d((YUt) = (L¥((Y ut)
+ in dt + (A¥k((y Ut) + g¥k) dw~,
where
L¥ = ( 1/2) a~j (y) Dij , i f = ( Yi t + ( YLtUt - L¥(( Yut),
A¥k = aik(y )D i ,
g¥k = (Yg~
+ ( YA~ut -
A¥k(( yut ).
By the results for equati ons with coefficients ind ependent of x (see [10] or [11]), for each y
IID2(( Yu )lI i
p
(T)
:s; N( lI i Ylli
p
(T )
+ II DgYll t(T» )'
(2.7)
where here and below N = N( d, p, 50, 51 )' Below we also denote by N* generic constants depending only on d,p,5 0 ,5 1 ,f: , and K . Observe that
(Y (x) LtUt(x) - L¥(( Yut )(x ) = (Y (x) (1 /2 ) [a~j (x) - a~j (y)]Dij Ut(x)
+ (Y(x)[ Ltut(x) -
( 1/2) a~j(x) Dij ut(x)]
+ [(Y(x)( 1 /2)a~j(y) Dij ut(x) - L¥((Yut )(x )]. Here the right-hand side is zero unless x E G and Ix- yl :s; f:
and it follows
from our assumptions that
where
Therefore,
lI i Y lli p(T) s N II(Y i ll i p(T) + N,Bg II (Y D2u tlIC (T) + N* I\77 Y (IDu t l + !Utl)lIi (T) ' p
(2.8)
ON SPDES IN SOBOLEV SPACES WITH WEIGHTS
157
Furthermore,
D((Y Atut - A¥((Yut}) = (YAtDUt - A¥((YDUt) + [D((YAtut) - (YAtDUtl + [A¥((Y DUt) - DA¥((Y ut)], and
A¥((YDUt)I~2 ::; N,B5'I(YIPID2utIP + N*lryYIPIDutIP , ID((YAtut) - (YAtDutI~2 ::; N*lryYIP(IDuti P + IUtIP), IA¥((Y DUt) - DA¥((YUt)I~2 ::; N*lryYIP(IDutIP + IUt IP).
I(Y AtDut -
Hence,
II D gYII[p(r) ::; N(II(Y Dgllt(r)
+ IlryYglI[p(r») + N,Bb'II(Y D 2Utll[p(r) +N*II77Y(I Dutl + lutI)ll[p(r)"
Coming back to (2.7) we conclude that for each y E ]Rd
IID 2(( Yu)ll[p(r)::; N(II(Yfllt(r) + II(YDgll[p(r) + IlryYgll[p (r») +N,B5'II(YD2Utll[p(r) + N*II77 Y(IDutl + IUtJ)ll[p(r)" We now integrate both part of (2.6) with respect to also use simple observations like
r II(YD2Utll[
JIRd
(r ) p
dy =
X,
t, and w . We
Err ( r (P(x-y)ID2Ut(xW dY)dxdt Jo JJRd JIRd
1Ld r
= E
ID 2Ut(xW dxdt
2ulI[p(r)'
= IID
Ld IlryY( IDutl + IUtl) II~p(T) dy ::; N* (1IDullt(r) + Ilull[ p(r»)· Then we find
2ulI[p(r) IID ::; N(llfll[p(r) + Ilg ll~~(r») +N1,Bb'II D 2ullt(r ) + N*l lul l~~(r)'
(2.9)
where N 1 = N 1(d,p , 60, 81 ) . We can now specify the value of ,Bo by taking it such that
N 1 ,B5' ::; 1/2. Then by collecting like terms in (2.9) we come to
158
NICOLAI V. KRYLOV
Here by interpolation inequalit ies, for any I E (0, 1],
Il u ll~~ C"r) ::; IIID2u IIC(T) + N (d,p)t-l llulli p(T)' so t hat by choosing I to satisfy Ni l::; 1/2 we get (2.5) from (2.10). The lemma is pr oved. 0 Proof of Theorem 2.1. If R = 1 the result follows directly from Lemma 2.1. The case of general R we reduc e t o t he par t icular one by using dilations. Introduce
,f-:F k J"t = R2t, T• = R- 2 T , W· tk = R -1 WR2t' (at, i; Ct, O"t , Vt)(x ) = (aR2t, Rb R2t , R 2cR2t, (jR2t, RVR2t )(Rx), Ut(x ) = UR2t(Rx ), f t(x ) = R 2fR 2t(Rx) , gf (x ) = Rg~2 t (Rx) . Also introduce the operators i; and A~ constructing them from the above introduced coefficients. It is easily seen that w~ are indepen dent Ft - Wiener processes, f is an Ft-stopping time, all the above pr ocesses with hats are pr edictable wit h respect to the filtr ation {Ft } , and U E H;,o(-T) , f E Lp(f) , 9 E Jfu~(f) , where the sp aces with hats are defined on t he basis of {Ft} . Ob serve that for t ::; f
£ tUt(X) = (1/2)a i j (x )DijUt(x ) + b~ (x)DiUt (X) + Ct(X)Ut(x) ij i R 2 [( 1/2)aR2t Dij uR2 t + bR2tD iUR2t + CR2tUR2t1 (Rx) = R2LR2tUR2t(Rx ),
=
1 t
[£ sUs(X) + f s(x )]ds =
1
~t
[Lsus( Rx ) + f s(R x )]ds.
Of course , we un derst and t his equality in the sense of distributions:
for any ¢ E CO" (lR d ) . One also knows that if ht is an Ft-predictable process satisfying a natural integrability condition with resp ect to t , then (a.s.). Therefore, (a.s.) for finit e t ::; f
it[A:u + s
it[A~2sUR2S + g~2s] (Rx) 1
g: ](x ) dill: = R
2
R t
=
[A: us+ g:](Rx) dw: .
dw:
ON SPDES IN SOBOLEV SPACES WITH WEIGHTS ~
It follows that (a.s.) for finite t
it
[tsUs(X) + ls(x)] ds +
159
f
it [A~us +
g;](x) dw;
= UR2t(Rx) = Ut(X),
so that u satisfies equation (1.2) with new operators and free terms. It is also easy to see that our objects with hats satisfy the assumptions of the theorem with R = 1. Therefore, by the result for R = 1
Now it only remains to notice that changing variables shows that this inequality is precisely (2.4). The theorem is proved. Here is an interior estimate. THEOREM 2.2 . Take some e.R E (0 ,00) and a stopping time T and suppose that the assumptions (i) and (ii) of Theorem 2.1 are satisfied. Then, for any r E (0, R) , we have
IIIB~D2UlllLp (7")
+ R-IIIIB~DulllLp (7")
~ N(II IBjJ lllL p(7")
+ IIIBnD9IiJLp(7") (2.11)
+ (R -
r)-IIIIBn9 11ILp (7"»)
+ N*(R -
r)-21IIBnulllLp (7" ) ,
where N = N(oo, 01, d, p) and N* = N*(K, e, 00 , 01 , d,p) . Proof. We follow a usual procedure taken from the theory of PDEs. Let x( s) be an infinitely differentiable function on lR such that x( s) = 1 for s ~ 1 and X(s) = for s ~ 2. For m = 0,1,2 , ... introduce, (ro = r)
°
m
rm = r
+ (R -
r) ~ Ti,
~m(x)
= X(2 m+l(R -
r)-l(lx'j- r m) + 1).
i=1
As is easy to check, for
Q(m) = B~ , '"
it holds that (m
= 1 on Q(m) ,
(m
=
°
outside
Q(m + 1).
Also (observe that N2 m+l = N 12 m with N l = 2N)
ID(ml ~ N2 m(R -
r)-I,
ID2( m l ~ N2 2m (R _ r)-2.
Next, the function (mUt is in 'H.~,O(T) and satisfies
where
160
NICOLAI V. KRYLOV
Since (mUt(x) = 0 for x
rt Bk, by Theorem 2.1
Am := IID 2((m u) IIE
(r)
p
+ R-PIID((mu)IIE
1 r
+ N* R- 2pUO + N E
(B mt + c';
(r ) :::;
p
N(P' + R-PCO) (2.12)
+ G;"t + R-PC~t) dt ,
where
P' =
IIIBjJIIEp(r)'
GO =
IIIB R91 It(r)'
UO =
IIIB RuIIEp(r)'
B mt = Ilu(Lt - Ct)(m - a~j DiUtDj(mllt (IRd), C;'t = I ID(UtO"~Di(m)llip (IRd )'
G;"t = IID ((m9t)llt(IRd) ,
C~t = IlutO"~Di(mllip(IRd )" Observe that by the above mentioned properties of (m and the assumption on b, we have
It follows that
e.; :::; N*2 2mp(R -
r)-2 PU? + N*2
mp(R
- r)-PIIDutllip(Q(m+l»'
where
Furthermore, by interpolation inequalities for any "I >
II DUtIlip(Q(m+l»
a
:::; IID( (m+l Ut) Ilip (IRd)
:::; 'Y (R - r)P2-mpIID2((m+1Ut) llip(Q(m+2» + N'Y-12mp(R - r)- PU? , so that for "I E (0,1) (with "I, perhaps, different from the one above)
B mt :::; 'Y II D 2((m+1Ut)lli p(Q(m+2»
+ N*'Y-122mp(R - r)-2 PU?
Simil arly, for "I E (0, 1)
C;'t :::; 'YIID2((m+lUt)llip (Q(m+2»
+ N*'Y-122mp(R - r)-2 PU?
and almost obviously
:::; N2 mp(R - r)-PU? :::; RPN 'Y-122mp(R - r)-2 PU? mp(R G;"t :::; N( IID9t llip(BR ) + 2 - r)-PI19tllip(BR »)·
C~t
Hence (2.12) yields
Am:::; 'YAm+l
+ NP + N2 mp(R - r)-PGO + N*'Y-122mp(R - r)-2 PUO ,
161
ON SPDES IN SOBOLEV SPACES WITH WEIGHTS
where
Now we take "( = 8- P and get "(m Am::; "(m+1 A m+ 1
+ N"(mp + N"(m2mp(R _
r)-PC o
+N*"(m"(-122mp(R _ r)- 2PUO, 00
Ao +
L
(2.13)
00
"(mAm ::;
m=l
L
"(m Am
+ NP + N(R -
r)-PCo
m=l
+N*(R - r)-2 PUO. In order to cancel like terms in (2.13) we need the series to converge. Observe that
and therefore 00
L
"(m A m ::; N(l
+ (R -
r)-2P)llull~~(r)"
m=l
However, the right-hand side may be infinite. In order to circumvent this difficulty we take in the beginning of the proof T 1\ T in place of T , where T E (0,00) . Then by the definition of'H~(T) we have
lIull~~(r/\T) < 00 and we get from (2.13) that A o for modified T is less than the right-hand side of (2.11). Since its left-hand side, with TI\T in place of r , is obviously less than A o, we obtain (2.11) with T on the left replaced with T 1\ T. After this it only remains to let T ~ 00. The theorem is proved. 0 REMARK 2.1. One can prove interior estimates for higher order derivatives of solutions if the coefficients are more regular. This is a routine matter and is achieved by considering equations for derivatives of solutions, which one obtains by differentiating the equation. On this way one would obtain more regular solutions of equations in half spaces. However, for brevity we do not pursue this issue. 3. Interior estimates for solutions of parabolic PDEs in the whole space. Assume that for i , j = 1, ... , d we are given real-valued Bor el measurable functions a~j (x), b~(x), Ct(x) defined for (t, x) E JR+ X JR d , where
JR+ = (0, (0).
ASSUMPTION 3.1. For all values of the indices and arguments a ij j i a and, for a constant 50 E (0,1] and all unit>' E JR d ,
5- 1 > aij>.i>.j > 5 .
° _
t
_
°
=
162
NICOLAI V. KRYLOV
This assumption is supposed to be satisfied throughout the deterministic part of the paper. Here we take a constant T E (0,00] and consider equation (1.1) The solutions of (1.1) are looked for in the space H;,o(T) with f E lLp (T). We remind briefly the definition of these spaces. Everywhere in the deterministic part of the article p>1.
= 0, 1,2 we take Sobolev spaces W; = W;(JRd )
For an integer n we set
and
In an obvious way we introduce the norms in these spaces. Observe that we are using the same notation as in the stochastic case . Deterministic spaces are subspaces of stochastic ones and there is no real danger of confusion. If for any finite t E [0, T] we are given a generalized function U = Ut on ]Rd such that U E no<s
holds for all finite t
~
T. In that case we write
a
at Ut = ft, which explains the sense in which (1.1) is understood and set
We take d 1 , x' , and Bk from Section 2 and state the first main result of this section. THEOREM 3.1. Take some e, R E (0,00) and assume the following : (i) There exists a constant K E (0,00) such that for x E Bk and y E ]Rd, such that Ix - yl ::; eR, and all values of the indices and other arguments Rlb~(y)1
+ R2Ict(y)1 ~ K ,
la~j (x) - a~j (y)1 ~ 130 ,
where 130 = 13o(d, p, 50, 5d > a is a (small) constant from Lemma 3.1 below. (ii) We are given a function U E H; ,o(T) which is a solution of (1.1) with an f E lLp(T). Finally , assume that Ut(x) = a if x ¢ Bk. Then there exist constants N = N(d,p,5 0) and N* = N*(d,p,5 0,e: ,K) such that
IID2U lllLp (T)
+ R-11 IDuliJL p(T) ~
Nllf lllLp (T) + N* R-21 IulIlLp (T)'
ON SPDES IN SOBOLEV SPACES WITH WEIGHTS
163
Similarly to T heore m 2.1 t his t heorem is obtained by using the parabolic dilations from the following lemma in which one can take G = Bk. LEMMA 3 .1. As sum e that we are given a U E 1i~, o (T ) which satisfies (1.1) with an f E Lp(T) . Assume that for a domain G C IR d (perhaps, G = IR d ) we have Ut(x ) = 0 if x rf. G. Also assume that there are constants e, K E (0, 00) such that for all x E G and y E IRd , with Ix - yj :S e, and all values of other arguments we have
I b~ (y) 1
l a~j (x) - a~j (y)1:S (30,
+ ICt(y)1:S K ,
where (30 = (3o(d, p, 00) > 0 is a (small) constant an estimate from below for which can be obtained from the proof. Then there exist constants
= N(d,p,oo),
N
N*
= N*(d, p,oo,c, K)
such that (3 .1 )
Proof. We take a nonnegative ( E Off (lR d ) from the proof of Lemma 2.1 and use t he facts that (2.6) holds and for each y E IR d the function (YUt satisfies
where
By classical results for equations wit h coefficients independent of z , for each y (3.2) wher e here and below N = N(d , p, 00) ' Below we also denote by N* generic constants de pendi ng only on d, p, 00, e, and K . By combining (2.8) wit h (3.2) we see that for eac h y E IR d
IID2(( Yu )llf p(T) :S N(II(Y fl lfp(T) + N(3b'II(Y D 2Ut llf p(T) + N *II T)Y(I DUt l + IUtI) IIC (T)' We now integrate both par t of (2.6) wit h respect to t , x. We also use simple observations like T II ( YD2Ut li r (T) dy = (P(x - y )ID2Ut(x )IP dy) dxdt
r
i;
p
r r(r
Jo J'iR d J'R,d
= iT
ld
ID 2Ut(x)IPdxdt =
II D2ullf p(T)'
164
NICOLAI V. KRYLOV
Then we find
where N 1 = N1(d ,p, bo). We can now spe cify the value of 13o by taking it such that N l13g
:s; 1/2.
Then by collecting like terms in (3.3) we come to (3.4) Here by interpolation inequalities, for any v E (0,1 ]'
so that by choosing 'Y such that Ni, :s; 1/2 we get (3.1) from (3.4). The lemma is proved. Our next result is the following interior estimate. THEOREM 3.2. Take some € , R E (0,00) and suppose that the assumptions (i) and (ii) of Theorem 3.1 are satisfied. Then, for any r E (0, R) , we have
I I IB~ D2 u ll lLp(T)
+ R-IIIIB~Dul llLp (T)
:s; N IIIBjJ IIlLp(T) + N*(R - r)-2 1IIBkU lllLp (T),
(3.5)
where N = N(bo,d,p) and N* = N*(K,€,bo ,d,p) . Proof. We take T'm , (m , and Qm from the proof of Theorem 2.2 and notice that the function (m Ut is in 1{;,0 (T) and satisfies
where
By Theorem 3.1 2 Am := IID (( mu )llt
(T) + R-P IID((mu)ll r p(T)
:s; N F + N* R- 2
pU O
+N
iT e.; dt ,
(3.6)
ON SPDES IN SOBOLEV SPACES WITH WEI GHTS
16 5
where
F = II IBiJlltcT)' Be«
=
Uo = II IBRull[pCT)'
II U( Lt - Ct )(m - a~j DiUtDj (m ll ~p (Rd )"
Observe that by t he propert ies of ( m and its derivatives and t he assumption on b, we have
It follows that
where
Fur thermore, by interpolat ion inequalities for any , >
°
II Dutll ~pCQ(m+l)) 5 II D((m+l Ut)ll ~p(lRd) 5 ,(R - r)pTmp IID2 ( (m+l Ut ) l l ~ p (Q(m+2) ) + N ,-12mp(R - r )- PU?, so t hat for , E (0, 1) (wit h " perhap s, different from the one ab ove)
s.; 5
,II D2((m+l Ut)l l ~p(Q (m+2) )
+ N *, - 122mp(R -
r)-2 PU?'
Hence (3.6) yields
Am 5 , Am + 1 + N F
+ N *, - 122mp(R -
r )-2PUO.
Now we t ake , = 8- P and get
, mAm 5 , m+ 1A m+ 1 + N ,mF 00
+ N *,m, - 122mp(R _ r )-2PUO,
00
A o+ '2::,mA m 5 '2:: ,mAm+NF+N*(R-r )-2 PU O. m=l m=l
(3.7)
In order to ca nc el like terms in (3.7) we need the series to converge . Observe that
and therefore 00
'2:: ,mAm 5 N( l + (R - r)- 2P ) lI u l l ~~ cT)"
m=l
166
NICOLAI V. KRYLOV
However, the right-hand side may be infinite if T = 00. In order to circumvent this difficulty in that case we take in the beginning of the proof S E (0,00) in place ofT = 00. Then by the definition ofH~(oo) we have
Ilull~~(s) < 00 and we get from (3.7) that A o for modified T is less than the right-hand side of (3.5) . Since its left-hand side, with S in place of T, is obviously less than A o, we obtain (3.5) with T on the left replaced with S. After this it only remains to let S -- 00. The theorem is proved. REMARK 3 .1 . One can prove interior estimates for higher order derivatives of solutions if the coefficients are more regular. This is a routine matter and is achieved by considering equations for derivatives of solutions, which one obtains by differentiating the equation. On this way one would obtain more regular solutions of equations in half spaces. However, for brevity we do not treat this issue . 4. Local regularity near the boundary for SPDEs in half spaces. In the setting of Section 2 we will be considering (1.2) in !Ri . We need the stochastic Sobolev spaces JH[;,ii(r), n = 0,1,2, and 5);,ii(r) introduced in [15] in the following way. We take a number e E !R and introduce Lp,ii as the space of functions on !Ri having finite norm given by
Then H~,ii is the set of functions such that u, MDu E Lp,ii, where M is the operator of multiplying by xl, and H;,ii is the set of functions such that u, M Du, 1\1[2 D2u E Lp,ii' The norms in these spaces are introduced in a natural way (cf. Remark 4.2 below) and, as before, we extend these definitions for .e2 -valued functions such as g. Naturally, we write H~,ii =
i.:
These definitions are used for all p E (1, (0) . REMARK 4.1. In [12] the spaces H;,ii are introduce for all real, and it is proved that CQ'(!Ri) is dense in H;,ii' In our case that, = 0,1 ,2 this is a rather simple fact. To prove it , it suffices to show that the subset of H;:'ii ' n = 0,1 ,2, consisting of functions on !R~ with compact support is dense in H;:'ii ' To do that, we take some nonnegative ~ E CQ'(JR) and 7] E CQ'(!Rd-l) such that ~(O) = 1, e(t) = for ItI ~ 1, and 7](0) = 1 and for m = 1,2, ... define
°
(m(x) = ~(m-Ilnxl)7](e-2mx') ,
x E !Ri.
Notice that
Ix l Dl(m(x)\ Ixl Di(m(x)1
= Im-I((m-Ilnxl)7](e-2mx')I, = xle-2ml~(m-llnxl)(Di7])(e-2mx')I,
i ~ 2,
ON SPDES IN SOBOLEV SPACES WITH WEIGHTS
167
and if ~(m-qnxl) i 0, then m-qnx 1 ~ 1 and xl ~ em. It follows that Ix1D(m(x)1 tend to zero as m -+ 00 uniformly on The following formulas in which i ,j ?: 2
Ri.
l(x l)2 DU(m(x)! = Im- 2C(m-1lnx1) - m-1nm-1ln x 1)ITJ(e- 2mx'), I(x l)2 DU(m(x)1 = xle-2mm-Ilnm-lln x) (D i TJ )(e- 2mx')I, l(x l)2D ij(m(x)1 = (xl)2e-4ml~(m-llnxl)(DijTJ)(e-2mx') 1
Ri.
show that l(x 1)2D2(m(x)1 tend to zero as m -+ 00 uniformly on Finally, observe that (m are uniformly bounded and tend to 1 pointwise. Now the dominated convergence theorem and the formulas lu - u(ml = 11 - (mllul, IMDu - MD((mu)1 ~ 11- (ml lDul + luMD(ml, 1M2D 2u - M2D2((mu)1 ~ 11- (m11M2D 2ul + 2IMD(m l tMDul + luM 2D 2(ml easily show that , if n E {O, 1, 2} and u E H;,o, then (mu -+ U in H;,o as m -> 00 . REMARK 4.2. A few times in the future we will be dealing with functions u such that M-Iu E H;,o' In connection with this it is useful to have in mind that M D i(M-Iu) = DiU - JliM-1u , M 2Dij(M-Iu) = M Diju - JUDju - Jlj DiU + 2JUJlj M-Iu implying that
IIM- I u IIHl = IIMD(M-1u)IIL e + IIM - Iu IIL e ~ N(IIDu IILp ,e + IIM - I uIILp, e), IIDuIILp,e::; NIIM-Iu IIH~ ,e ' IIM- I u IIH2 = IIM 2D2(M-1u)IIL e + IIM-IuIlHl ~ N(IIMD2uIILp ,e + IIDuIILp ,e + IIM-IuIILp,e), p,e
p ,e
p,
p,
p,
p ,e
II M D 2u liLp , e + IIDu llLp, e + IIM-1u IIL p, e ~ N IIM- 1u IIH2p,a . It follows that the H;,o norm of M-1u is equivalent to
Next , for n = 0,1 ,2 and stopping times 7 we define JHI;,O(7) as the set of functions f = ft = ft(w) on (0,7] , t < 00 , with values in the set of distributions on which are P-measurable and have finite norm given by
Ri
I lfll~;,e (7") = E 17" 11ft II iI;,e dt.
168
NICOLAI V. KRYLOV
Define Lp,l:I (r) = lHI~, I:I (r) . For a function U = Ut = Ut(w) given on (0, for finite t wit h values in t he set of distribut ions on lRi we write U E 55; ,I:I ,o(r ) if and only if M -Iu E lHI;,I:I(r) and t here exist a real valued f E Lp,l:I (r) and an £2valued 9 = (gk, k = 1, 2, ...) E lHI~,I:I (r) such that for any ¢ E CO" (lRi ) wit h prob ability 1 we have
rn
f
(Ut, ¢) = t (M-Ifs, ¢ )ds+ t (g: , ¢ )dw: k=IJO Jo for all finite t E (0, r]. We set
I l ull jJ~.e (T) = I IM-Iu lI lH~, e (T) + IIfIlLp,e(T) + I l g l llHI~ ,e (T) ' Recall that Assumptions 2.1 is supposed to be satisfied throughout the article. For r > 0 denote QT={ X ElRd :O<x l < r }.
THEOREM 4.1. Take an R E (0, 00] and a stopping time -r and assum e the f ollowing. (i) For som e constants s E (0, 1] and K E (0, 00) we have ' 1 IxI b~. (x) 1 + Ix I DC7;(x )ll2 + Ix Vt(X)!l2
(4.1) + l(x l f DVt(x )ll2 + l(x l ) 2 Ct( x) j ::; K , l a~j (x) - a~j (y)1 + 1C7:(X) - C7;(y )ll2 ::; (30, whenever x ,y E lRi, x l ,yl ::; R , Ix - y l::; s( x l/\ yl ), i, j = 1, ... , d, t > 0, where{3o = (30(d, p, 50, 5r) E (0, 1] is the constant from Theorem 2.1; (ii) We have a fun ction U such that ¢u E 1{~ , o (r ) for any ¢ E CO" (QR ) and u satisfies (1.2) in lRi with some f E Lp,l:I(r) , g = (gk, k = 1,2 , ...) E lHI~,I:I(r) .
Then for any r E (0, R /4 ) II IQrM D 2u llLp ,e(T) + II I QrDu lhLp, e(T) ::; N IIIQR M f liJL p,e(T)
(4.2)
+ NI IIQRM Dg IIJL p,e(T) + N IIIQ RgII JLp ,e (T ) + N *I IIQ RM-l u II JLp.e(T)' where N = N(d,p,5 0,5 l) and N * = N*(d,p,5 0, 51 , s , K ). Proof. We are going to appl y Theorem 2.2 t o shifted B Rwhen d l = 1. For n = -1 ,0,1 , ..., set r n = 2- n / 3r . Observe that if n 2: 0, t hen the half widt h of Qrn_l \ Qrn+2 equals Pn := rn+2/2 and r n-l
+ Pn ::; 2r -1 < 4r < R ,
r n+2 - Pn = Pn·
It follows th at for x E Qrn_l \ Qr n+2 and y , such that Ix - yj ::; SPn, we have
R > rn-l 2: x l 2: r n+2 2: Pn, yl::; Xl + cPn < R , yl 2: x l - SPn 2: Pn, Pn ::; Xl /\ v', Ix _ yl ::; c (X l /\ yl ),
ON SPDES IN SOBOLEV SPACES WITH WEIGHTS
169
so that by our assumptions
Furthermore, if n :::: 0, ( E 00((0 , R)) and «(z) = 1 for rn + 2 ::; z ::; rn-l , then (u satisfies (1.2) in IR d with certain f and g which on Qrn-l \ Qr n+2 coincide with the original ones. Finally, if n :::: 0, then the distance between the boundaries of Qr n \ Qrn+l and Qrn_l \ Qr n+2 is (21 / 3 - l)rn + 2 . It follows by Theorem 2.2 that for n :::: 0
We multiply both parts by r~t~-d and use the facts that rn-l on Qrn_l \ Qr n+2 the ratio xl/r n + 2 satisfies 1 ::; xl / r n + 2
::;
= 2r n+ 2 and
2.
Then we obtain
Upon summing up these inequalities over n :::: 0 we conclude
IIIQ~MD2ull[p,O(T) + IIIQrDull[p ,o(T) ::; N(IIIQ~_l M f ll[p,o(T )
+ I IIQ~_l M Dgll[p,O(T) + Ilh Qr_ gllt ,o(T)) + N*llhQ~_l M-lullt,o(T)' 1
which is somewhat sharper than (4.2). The theorem is proved. 0 By letting r ~ 00 in (4.2) we get the following . COROLLARY 4.1. If the assumptions of Theorem 4.1 are satisfied with R = 00, then
+ IIDulllLp.O(T) ::; NIIM fl llLp,o(T) + NllglllHI;,o (T) + N*IIM-lUlllLp,o(T) '
11M D2u lllLp,O(T)
REMARK 4.3. This corollary for equations with constant coefficient is known from Lemma 3.6 of [14] for d = 1 and Lemma 3.8 of [15] for d :::: 1. For variable coefficients it is Lemma 4.1 of [8] if d = 1. However, there is a
170
NICOLAI V. KRYLOV
very imp ortant distinction betwee n Corollary 4.1 and t he above mentioned references, where from t he st art it is assumed t hat u E .fj ~, II ,O (T ). As we have mentioned in t he Introduction, we intend to use Theorem 4.1 to extend to t he case of variable coefficients t he pointwise boundar y Holder regularity result from [13] proved there for equations with constant coefficients. In a subseq uent article we will show that solutions in .fj ~ , II ,O (T ) spaces wit h rather larg e e admi t estimates of M-I u in lLp ,I'(T) with much smaller J.L if T is chosen in a special way. This, Theorem 4.1, and embedding t heorems will lead to proving t he pointwise Holder continu it y of solutions near t he boundary. REMARK 4.4. We discuss assumption (4.1) in case that R = 00. It impli es that l a~j (x) - a~j (y)1 ::; (3 for a small (3 > 0 and all x, y E JR~ satis fying Ix - yl ::; x l 1\ yl . If Xl = yl = r, then we need l a~j (r, x' ) a~j (r, y' )1 ::; (3 for lx' -y'l ::; r and when r becomes larger we need a~j (r, x' ) to be close to const ants on larg er and larger balls in JRd-l . Basically, we need a~j (r, x ' ) to be independent of x' for large r. On the other hand , the behavior of a~j (x) near aJR~ can be quite irregular. For instance, take d = 1 and introduce the functi on
a(x ) = 2 + cosln x . If x ~ y > 0 and [z - yl ::; e(X 1\ y ), t hen x ::; (1 +e )y, In x ::; In y + lnfl and, for a ~ E (In y, In x), by the mean value theorem we have
+ e)
la(x) - a(y) 1= Isin ~ I (ln x -In y ) ::; In( 1 + e), which can be made arbitra rily small by choosing an appropriat e e. REM ARK 4. 5. It is worth emphasizing that in this section t here is no restrict ions on e, say like (1.3). 5. Local regularity near the boundary for parabolic PDEs in half spaces. In the setting of Secti on 4 we will be considering (1.1) in JR~ . We need the weighted Sobolev spaces lHI;,II (T ), n = 0,1 ,2 , and .fj ~ ,II (T ) introduced for p > 1 in t he following way. We take a number e E JR and t ake L p ,1I and H;,II from Section 4. Next, for n = 0,1,2 and T E (0, 00] we define lHI;,II (T ) as the set of functi ons i = it on (0, T ) wit h values in the set of distributions on JR~ which are measurable and have finite norm given by
I l fl l~;. 8(T) =
I TIIftlliI;.8
dt .
Define lLp ,II (T ) = lHI~, II (T) . For a function u = Ut given on [0, T ] n [0, 00) wit h values in t he set of distrib utions on JR~ we write u E .fj ~, II ,o (T ) if an d only if M -1u E lHI~,II (T)
ON SPDES IN SOBOLEV SPACES WITH WEIGHTS
and there exist a real valued we have
171
f E JLp,e(T) such that for any ¢ E Co(JRt)
for all finite t E (0, T]. In that case we write
and set
Recall that Assumptions 3.1 is supposed to be satisfied throughout the deterministic part of the article. For r > denote
°
Qr={XEJRd :O<X l
(i) For some constants e E (0, 1J and K E (0, (0) we have
whenever x, y E JRt, xl , yl ::; R, Ix - yl :$ c(x l 1\ yl), i, j = 1, .. ., d, t > 0, where f30 = f3o(d,p, (0) E (0,1) is the constant from Theorem 3.1; (ii) We have a function u such that ¢u E H~ ,o(T) for any ¢ E Co(QR) and u satisfies (1.1) in JRt with some f E JLp,e(T) . Then for any r E (0, R /4)
IIIQrM D2ulllLp ,o(T) ::; N IIIQR M flllLp ,o(T )
+ II I Qr D ul iJL p.o(T)
+ N*IIIQRM-lul llLp ,o(T),
(5.1)
where N = N(d,p,oo) and N* = N*(d,p,oo,c ,K) . Proof. As in the proof of Theorem 4.1 for n = -1,0,1 , ... we set r n = 2- n / 3 r and observe that if n 2: 0, then for x E Qrn_l \ Qr n +2 and y , such that Ix - y l ::; cpn, where Pn is the half width of Qrn _l \ Qr n +2' we have
Furthermore, if n 2: 0, ( E Co((O, R)) and ((z) = 1 for rn + 2 ::; z :$ rn-l, th en (u satisfies (1.1) in JRd with certain t, which on Qr n_l \ Qr n+ 2 coincides with the original one. Finally, if n 2: 0, then the distance between the boundaries of Qr n \ Qrn+l and Qrn-l \ Qr n+2 is (2 1/ 3 - 1)rn + 2 '
172
NI COL A I V. KRYLOV
It follows by Theorem 3. 1 that for n 2: 0
We multiply both p art s by r~t~-d and use the facts t hat rn-l = 2r n+ 2 and on Q rn-l \ Qr n+2 the ratio xl /rn + 2 satisfies
1 :::;
xl /rn + 2 :::; 2.
Then we obtain
Upon summing up these inequalities over n 2: 0 we concl ude
IIIQrMD2uIIEp.9(T) + III QrDu ll[ p.9(T) :::; N IIIQr_l Mf ll[p,9(T) + N' lIh~r_l M- l ull [ p,9(T)' which is somewhat sharper than (5.1) . The theor em is pr oved. 0 By letting r -+ 00 in (5.1) we get the following. COROLLARY 5 . 1. If the assumptions of Th eorem 5 . 1 are satisfied with R = 00, then
6 . Auxiliary results. In this section
p>l ,
,:= B - d - p+ 1.
He re we present some auxiliary facts needed for proving existence theorems in Sections 7 and 8. Several times in t he future we use t he following Scheffe 's lemm a , which we prove for com pleteness of presentation. LEMMA 6. 1. Let (E, E , J.1.) be a measure space, r E [1,00) , un ,u E L; (J.1.) 1 and Un -+ U in meas ure . Fin ally, let
Th en
(6.1)
173
ON SPDES IN SOBOLEV SPACES WITH WEIGHTS
Proof We have
Upon integrating through this equation and observing that (Iul r - Iunn+ :::; lulr we conclude by the dominated convergence theorem that
L
Ilul r -Iunn f.L(dx)
Next, if IU n r
(1 /2)lunl
-
ul ~ 31u/, then lunl + lui
~
~ O.
(6.2)
31ul, lui:::; (1/2)lunl, lulr
:::;
,
lunl r -Iul r
~ (1/2)lu n l r
r
,
rlu
IUn - ul ~ 2
ul :::; lun l + lui:::; 2lunl, r n l -Iun , n l :::; 4 IU n
r
-
r(lu
which along with (6.2) imply that
L
r
~ O.
Ie
r
~0
IUn - ul Ilu n-ul2':3Iul J-L(dx)
Furthermore,
IUn - ul Ilu n-ul<3I ul J-L(dx)
by the dominated convergence theorem. By combining the last two relations we come to (6.1) . The lemma is proved. 0 COROLLARY 6.1. Let (E,L"J-L) be a measure space, r,s E (1,00), r- 1 + s-l = 1, Un,U E L; (J-L) , Vn, V E Ls(f.L) , Un ~ u and Vn ~ v in measure. Finally, let
Then
Indeed, it suffices to use Holder 's inequality and the formul a
UnV n - UV = (un - u) v + (v n - v)u + (un - u)(vn - v). Next, we prove a version of Corollary 6.2 of [12]. LEMMA 6.2. If we are given a d x d matrix a, then , for any u we have
E
M H P. 2e,
174
NICOLAI V. KRYLOV
and if ii is nonnegative and symmetric and'Y < 0, then
where (if p < 2) by 0- 10 we mean O. Proof. If U has compact support, then, owing to the assumption that 'Y < 0, the result follows from Corollary 6.2 of [12]' which is proved by using integration by parts and Hardy's inequality. Observe that the case 1 < p < 2 is quite nontrivial. In the general case we take (m from Remark 4.1 apply (6.3) to U rn = (rn U in place of U and pass to the limit as m -> 00. To justify this procedure we claim that
(6.5) are continuous functions on M Lp,fJ and M H;,fJ' respectively. The first one is continuous because it is just the p-th power of the Lp ,o-norm of M- 1u. To prove the continuity of the second one, take Vn such that M-1 vn -> M-1 u in H;,o. Then
in Lp,fJ by Remark 4.2 . By Lemma 6.1 and Corollary 6.1 we also have p IM - 1v n l - 2M- 1vn
->
IM- 1ulp- 2M- 1u
in
L p/ Cp- 1) ,fJ,
{ (xl )1'+1IVnlp-2VnDijVn dx
),fiI·t
= { (Xl )fJ-d( IM- 1v n!p-2 M-1Vn)(MDijVn) dx
JRt
->
{
JRt
(xl )fJ-d(!M- 1u I P - 2M- 1vu)(M Diju) dx
and this proves the continuity of the second function in (6.5). If p ;:::: 2 then a similar argument shows that the last expression in (6.3) is continuous on M H;,o and this proves the lemma if p ;:::: 2. For arbitrary p > 1 we need to proceed differently. Since (6.3) holds with Urn in place of u, we conclude that
ON SPDE S IN SOBOLEV SPACES WITH WEIGHTS
17 5
where
nim .= . IM- I UmIP - 2 (D iUm)D jU m =
(!:.IM -IuIP-2(DiU) Dju + IM- Iu !p-2(M- Iu)( D
iu)(nM o«; + !M-IuIP-2(M-Iu)(Dju)(nM »«;
(6.7)
+IM- Iu IP(M D i( n)M o.c;
Here the fact ors of t he functi ons
are integr abl e against (Xl )iI- d dx, which follows from Holder 's inequ ality, and the functions themselves are un iformly bounded and tend t o zero on Therefore, (6.6) implies that
lRt.
By taking here (iij = rS ij and using Fat ou 's lemma we see t hat
Afte r t hat by (6.7) and t he domi nated converge nce t heore m we obtain
which along with (6.6) lead to (6.3). The above argument also shows that one can pass to the limit in (6.4) written for Urn in place of u . This proves the lemma. 0 C ORO LL A RY 6. 2. Den ote
Then
Q ::; N (d,p) P
+ (p _1 )- lp- I(r2 + 'Y)II M - I u lI ~
P ::; II M- I ul l ~~.~ 11MD u IIL p , B ' 2
p .B
,
(6.9) (6.10)
176
NICOLAI V. KRYLOV
Indeed, estimate (6.10) follows from Holder's inequality. By (6.3) for (iij = oij we have
(p - l)Q = "(b + l)p-IIIM-Iullt
::; "(b + l)p- I IIM- Iu llt
r
l p,9 - JJRd (X p +l lu JP- 2u6.u dx p
,9
+ N(d)P
and (6.9) follows. Next, we need an auxiliary function . Set
Bk =
E ~d-l :
{x'
Ix'i < R}.
LEMMA 6.3. For any 0> 0 there exists an R = R(o,d,p) > 1 and a nonnegative function fJ E CO"(~d) vanishing outside (1, R) x Bk such that In R ::; N(d,p)O-I/2 and
r
~fJP(x) dx =
JJRd X +
(6.11)
1,
Proof. Take a nonnegative ~ E C6"'(~d) with unit norm in L p , vanishing outside (-1 ,0) x B l , and then for some r > 0 set 'T)(x) = r-l /Pe-(d-l)r/P~(r-Ilnxl, e- r x') Then
r
~fJP dx = r- l
JJRd+ xl
=r- l
r
~e(r-Ilnxl, x') dx
JJRd+ Xl
r e(r- Ix l,x')dx=l,
JJRd
r 'T)p- 2IDlfJI 2xl dx = r- JJRdr (e-2IDI~12)(r-Ilnxl, x')~xl dx 3
JJRd+
+
r e-2IDI~ 12 dx,
= r- 2
Ja
d
where the last integral is finite even if p < 2 since ID~12 ::; N(d)~ sup ID2~1 because ~ is nonnegative and smooth. Also , since ~(x) = 0 for xl > 0 and by assumption r > 0, for i 2: 2 we have
r
JJRi
r (e-2IDi~ 12)(r-Ilnxl ,x')x dx JJRi = r- Ie- 2r r (e-2IDi~12)(r-Ixl , x')e 2X1 dx JJRd 2r ::; rr (e-2IDi~12)(r-lxl, x') dx JJRd
'T)p- 2IDifJI 2xl dx = r- Ie- 2r
1e-
= e- 2r
r (e-2IDi~12)(xl, x') dx.
J,ll{d
1
ON SPDES IN SOBOLEV SPACES WITH WEIGHTS
177
By observing that e- 2r :S Nr- 2 , we have both relations in (6.11) satisfied for an r = N(d,p)rS- 1 / 2 . Now we have to take care of the support of ry. The one constructed above has support in
(e- r , 1) x
B~r.
It turns out that conditions (6.11) are dilation invariant. Therefore, the function ry( e- r x) satisfies them as well. This function has support in
(1, er ) x
B~2r
and we see that we can take R = e2r . The lemma is proved. 0 Now we extend (6.4) for variable a by using a localization procedure. LEMMA 6.4 . Assume that, < and let€,(3,K E (0,00) be some constants and let a(x) be a measurable junction given on JR~ with values in the set oj symmetric nonnegative matrices and such that la ij I :S K and
°
(6.12) whenever x , y E JR~ and Ix - yl :S €(x 1 /\ yl) . Then jor any u E M H~,o and rS > 0,11; E (0,1] we have
1:=
r (xlrr+l!uIP-2aij(Diu)Djudx
lJR~
2:: (1 - 1I;)/2 p-2
r (x1p-1allluI
lJR~
P
dx - N(€-lR + 1)(3Q (6.13)
-tite:' R(3 + (3 + lI;-lrS) IIM-1ullt,9' where N = N(d,p,K ,B) , InR = N(d ,p)rS- 1/ 2 and Q is introduced in Corollary 6.2. Proof. Take the function ry from Lemma 6.3 and for any y E JR~ set
ryY(x) = ry(y1x 1, yl(y' _ X'))(yl)(d-2 )/ p, (Y (x) = ryp(y1x1,yl(y' - x'))(yl)d-2 = [ryY(x )]p. Observe that , for any x E JRi and
0:
E
JR
(6.14)
For
0:
°
= this yields 1=
r I(y) dy ,
lJRd+
178
NICOLAI V. KRYLOV
where I(y)
= h(y) + 12(y)
h(y) = aij(y) h(y)
=
r (x }JRt
and with
y = ((yl)-l,y')
r (Xl)'+1(YluIP-2(DiU)Djudx ,
IiRt
Y+I(YluIP-2[a ij(x) - aij(y)](Diu)Djudx.
l)"
Here and below on many occasions we drop the argument x but not y. With R from Lemma 6.3 , if (Y(x) =1= 0, then 1 < ylx l < Rand ylly' _ xii < R implying that
yl < xl < Ryl, Iy' - xii < Ryl = R(x l /\ yl), 0< xl _ yl < Ryl, Ixl _ yll < R(x l/\ yl), Ix _ yl < 2R(x l /\ yl). In case 2R ~ e introduce n := [4Rc l] + 1 and find n points Xi, i = 1, ... , n, evenly spread on the straight segment connecting x and y and such that Xl = x, X n = y. Then n - 1 = [4Rc l] ~ 4Rc l - 1 ~ 2Rc l and
It follows that laij (x) - aij (Y)I ~ (n - 1){3 and
laij (x) - aij (Y)I ~ 4(C I R + 1){3, if 2R ~ c. Estimate (6.15) is also true if 2R 2R(x l /\ yl) ~ c(x l /\ yl) . Hence,
~
(6.15)
e, because then Ix - y! <
and by the construction of ( and the definition of Q
Thus , (6.16) To estimate
h (y)
we introduce
ON SPDES IN SOBOLEV SPACES WITH WEIGHTS
179
so that
where
Jj'(x) = (j,ij(Y)l vYIP-2(DiVY)DjvY,
J¥(x) = (j,ij(Y)lvYIP-2u2(Di1]Y)Dj1]Y .
It follows that
r h(y)dY"2(l-",) JJRir JJRtr (x1rr+1Jj'(x)dxdy _",-1 r r (Xl)1+1J~(x)dxdy. JIRt JIRt
JJRi
(6 .17)
Observe that
r (x 1)1+1J¥ dx s N r (x 1)1+l(7)Y)P-2ID1]YI 2Iu\P dx , JIRd'rJ.~t . r r (x 1rr+1J¥ dxdy s N r (xl)'Y+lluIP( r (1]Y)P- 2ID1]YI 2 dy) dx JIRt JIRt JIRt JIRt and by the construction of 1]
r (1]Y)P- 2ID1]YI 2dy JJRtr [1]P-2ID1]12](ylxl, yl(y' _ x'))(yl)d dy =
JIRi
=
r [1]p-2 ID1]12](ylxl ,y')yldy
JIRt
= (X l)-2
r 1]P- ID1]1 JIRt 2
2 l y
dy S O(xl )- 2,
r r (xlrr+lJ¥ dxdy S No JIRdr (Xl)1-lluI Pdx = NoI IM-1ulli.
JIRd JIRd +
+
+
. p,e
We now conclude from (6.17) that
r
JIRd
h(y) dY"2 (1- "')
+
r r
JIRd JIRd +
(x l)1+lJi(x) dxdy - ",- lNoI IM-lu lli.
. p,e
+
Next, apply Lemma 6.2 for each y to get
r (xl )1+1 Jj' dx "2 ,..2p-2(j,1l(y) r (Xl )1-llvY Pdx
JIRt = ,..2p - 2
r (xlrr-l(j,ll (y)(Ylu IP dx
JIRt
+ ,..2 p-2
JIRi
= ,..2p - 2
I
r (x l rr-l(j,ll(Y lu I dx P
JJRt
x l rr- l [(j,ll (y) _ all(x)](Y luIPdx. r ( JIRt
180
NICOLAI V. KRYLOV
We use again (6.15) on the support of (Y and then integrate with respect to y . Then we conclude
r r (x lp +1Jf dxdy 2: ·l p- 2 la~r (xl p - l a ll (Y lu IPdx
la~ j.iR~
- N (e- l R + l )f3 I1 M- Iu lli,p .9 ,
r h (y ) dy 2: (1 - K)'ip-2 At~r (x l p - l a ll lu IPdx
la~
- N[ (1 - K)(e- 1R + 1)13 + K- lol IIM-1u lli,p,9 ,
which in comb ination with (6.16) leads to (6.13). The lemma is proved. 0 We now generalize (6.3) for variable a. LEMMA 6.5 . Let e, 13 E (0,00) be some constants. Assume that we are given bounded m easurable functions a,ij(x) defi ned on lRi for i , j = 1, ..., d and such that (6.12) holds for all i , j = 1, ..., d and x, y E lRi such that Ix - yl ::; e(x l A yl) . Then for any u EM H; ,IJ we have
1:=
r (Xl P+ 1Iu jP- 2u ai j D ij u dx lad +
::; N f3(IIM D2u Il Lp, 9 1 I M- I u ll i,~.~ + e-2x IIM-1u llt.J
r (x lrr+1lu IP- 2ai j( Di u) Dj u dx lad+ + p-l , (-Y + 1) r (x l)"Y- Ia lll u IP dx ,
(6.18)
- (p -1 )
JR~
where X = (-y2 + 1)(1 + e2) and N = N( d, p) . P roof. As in the pr oof of Lemma 6.2 we may assume that t he support of u is a compact subset of lRi . W ithout losing generality we may also assume that a ij = a ji for all i,j. Next, take a nonnegat ive ( E C8"(lRi) with un it int egr al and such that ( (x) = 0 if xl rf. (1, 1 +e/2) or Ix'i 2: e/2. On ca n construct such functions in a unified way, first choosing such a function, say (1 corresponding to e = 1 and t hen for any e > 0 setting
Below we are also usin g the function 1](x)
=
(x lp +2(( x) .
One shows easily that for the indicator function 7f; of the set (1, 3/ 2) x we have
B~ / 2
ID 21](x)1 ::; Ne - 2- d (-y 2 + 1)(x I P [(x l )2 +e 2]7f;(1 + e- l(x l - 1), e - 1x ' ) ::; N (x 1 p e -
2
-
d
x 7f;(1 + e-l (x l - 1), e- Ix' ),
ON SPDES IN SOBOLE V SPACES WITH WEIGHTS
181
where N = N (d). In particular, (6.19) This estimate will be used later and now for y E ]Rd define
Observe that similarly to (6.14) we have
It follows that
1 = ( I (y) dy,
lad+
where I (y) = h ey) + 12 (y),
h ey) = { ( YluIP-2uaij (y)DijUdx ,
l Ri
12(y) =
y = « yl) - l,y'),
{ (YluIP- 2u[a ij (x ) - aij (Y)]Dijudx.
lai
By t he choice of ( we have t hat if (Y(x ) 1= 0, then 1 < yl x l < 1 + e/ 2 and yl lY' - x' i < c/ 2 implying that
yl < Xl < (1 + c/2)yl, Iy' - x' i < i/c/2 = (c/ 2)(x l /\ yl ), 0 < xl - yl < y le /2, Ix l - yl l < (e/2)(x l /\ ii) , [z - 111< c(x l /\ 11 1 ) , laij (x ) - aij(y)1:=:; (3. Hence,
and by (6.20) and Corollary 6.2
Ild1
2(y) dyl :=:;
+
We conclude
N(3I1MD2uIILp .9 1IM-I ulli~.~ ·
182
NICOLAI V. KRYLOV
To estimate h (y) we integrate by parts which is a very simple matter if p ~ 2 and is also justified for p E (1,2) in Lemma 2.17 of [12] . Then we find that h(y) = J1(y) + J2(y), where
J1(y)=- (p-1) J2(y) = -
r (YlujP-2aij(y)(D iu)Djudx,
JlR +d
r (Dj(Y)luIP-2uaij(y)DiUdx
JlR +d
=
_p-l
=
v:'
r (Dj(Y)a ij (y)DiluIP dx
JlR +d
r JlR
lulPai j (y)Dij(Y(x) dx.
d
+
As above
r
IJ1(y) + (p-1)
I
(YluIP-2aij(Diu)Djudxl:::; N (3
JlRi
r
(Ylu I P- 2IDuI 2dx,
JlRi
kd J1(y) dy + (p - kd 1)
+
(xlyr+1 luIP-2aij (DiU)DjUdxl :::; N(3Q ,
+
wher e Q is taken from Corollary 6.2. It follows that
I
kd J1(y) dy + (p -1) kd (Xlyr+lluIP-2aij(Diu)DjUdX! +
+
:::; N(3I IMD2uIILp ,eIIM-lulli~,~ +N(3(r2
+ li DII M - 1u ll i p, e'
Therefore, (6.21) yields I :::;N(3(IIMD2u IILp ,eI IM-lulli:,~
-(p-1)
+ (·l + 1)IIM-1ullip,e)
r (xlyr+l luIP-2aij(Diu)Djudx+ JlRir J2(y)dy. JlRi
(6.22)
Similarly, to the above estimates
jh(y) -
p-l
kdlu(x )IPaij Dij(Y dxl :::; N(3 kd lu1PjD 2( YI dx. +
+
To estimate ID 2(Y I introduce the function
e(z,y)
= (zl )'Y+2((zl,yly' _ z' )(yl )d-"Y - 3 .
Obviously, (Y(x) = e(y1x, y) . Therefore,
ID 2( Y(x )1 = (yl)2 1(D;e)(y 1x , y)\, I(D;e)(z, y)1 = I(D 21])(zl, yly' _ z') I(yl)d-"Y- 3 , ID 2(Y(x) 1= (yl )d-l-"YI(D 21])(X1yl , yl(y' - x'))I .
(6.23)
183
ON SPDES IN SOBOLEV SPACES WITH WEIGHTS
In particular , owing to (6.19)
where we have used Fubini's theorem which is possib le since the rig ht-hand side of (6.24) is finit e. Finally, owing to (6.20)
r
JRi
D ij(Y(x) dy
= D ij
r
JRi
(Y( x) dy
= D ij (X1)'Y+1 = -y(t + 1)(Xl p -l Jl iJl j ,
so that
By combining this with (6.22) we come to (6.19) and t he lemma is pr oved . 0 7. E x istence and u n iq ueness for parabolic PDEs in h alf s p a ces. We t ake pE (1, oo ),
TE (O , oo ),
d - 1
and prove the following version of Theorem 2.14 of [9]. THEOREM 7.1. Th ere exist constants {3 E (0, 1] and N <
(7.1 )
00,
depend-
ing only on 150 , d, p, and B, su ch that if
lRi,
whenever x, y E Ix-yl ::; X1 l\y l , i, j = 1, oo ., d, t E (O, T ), then for any f E M -1JLp,o(T) there exi sts a unique u E J) ~, o ,o (T ) satisfying (1.1) in Furthermore, fo r any u E J)~,o,o (T) we have
lRi ·
REM ARK 7.1. Part of condition (7.2) could have been repl aced with the requirement that for an e E (0, 1]
l a~j (x ) - a~j (y) j ::; e{3
184
NICOLAI V. KRYLOV
for all x , y satisfying Ix - yl ::; .s(x 1 /\ yl) and all i, i, t . However, this would not make the theorem more general as follows from the argument we used to obtain (6.15) . REMARK 7.2 . Theorem 7.1 is used in [9] to prove the solvability of parabolic equations in Sobolev spaces with weights in bounded Cl-domains. The proof for spaces like 5J;,o(T) with n = 2 is quite standard and is based on partitions of unity and flattening the boundary. Then equations are solved in small neighborhoods of the boundary, where conditions like (7.2) allow one to treat equations with quite irregular a i j and blowing up band c. The reader is sent to [9] for details. . To prove the Theorem 7.1 we need a lemma, in which G is a domain in ]Rd and by Lp ,loc(G) we mean the set of functions u such that u¢ E Lp(G) for any ¢ E CO'(G). LEMMA 7.1. Let Ut and ft be measurable junctions given jar t ::; T with values in Lp, loc(G). Assume that jar a continuous junction 7j; = 7j; (x ) given on G we have 7j; > 0]
and for t ::; T and each ¢ E
CO' (G)
we have
(7.4) Then
Proof. The inequality in (7.5) follows by Holder's inequality:
Next, we reduce the general situation to the one in which Ut has compact support in G . Let ( E CO'(G) and ( :::: O. Then
If (7.5) is true for UtC then iT
fc
(P7j;p-2IUtlp-2Udt dxdt = p-l
fc
7j;p-21(UTIP dx.
(7 .6)
185
ON SPDES IN SOBOLEV SPACES WITH WEIGHTS
The above estimates show that if we take ( = (n i 1, then (7.5) would follow from (7.6). Therefore, in the rest of the proof we assume that there is a compact set reG such that Ut(x) = ft(x) = 0 if x .;. r. In that case the values of'ljJ outside I' become irrelevant and we may assume that G = ]Rd, 'ljJ is bounded away from zero and infinity, Ut, ft are measurable and Lp-valued, and
We now take a nonnegative ( E eff (]Rd) which integrates to one and for an x E ]Rd substitute .s-d((C1(x - .)) in place of ¢ in (7.4) . Then by using the notation V(E) (x) =
s.r v(x-.sy)((y)dy
we find that for any x E ]Rd and t ::; T
U~E)(X) =
it f~€)(x)ds.
Then we have
p-llu~\xW =
iT
IU~E)(x)IP-2u~E)(x)f~E)(X)ds ,
Now we integrate through this relation over x E ]Rd after multiplying it by 'ljJp-2(x). It turns out that we can interchange all integrals since , say u~€)(x) is infinitely differentiable in x, measurable with respect to t, by Minkowskii's inequality (7.7) and the estimates as in the beginning of the proof are valid. conclude that iT
Ie 'ljJP-2Iu~E)IP-2u~E) f~€)
dxds
= p-l
Ie 'ljJP-2Iu~)IP
dx
Then we
(7.8)
We also observe that (for any t :::; T) we have U~E) ....... Ut in L p. Similar relation holds for ft. In particular, U~E) ....... Ut in measure
p,( dtdx) = 'ljJp-2 dxdt on [0, T] x ]Rd. Furthermore, (7.7) and the dominated convergence theorem imply that
186
NICO LAI V. KRYLOV
Simil ar relati on hold s for
f . By Lemma 6.1 and Corollary 6.1
lu(e) IP
lu(e)I P -
1
-t
lul
lulP in p
L 1(f-L),
1
in L pl (p- l )(f-L) , p 2u lu(e)IP-2u (e) - t lul in L pl (p_1) (f-L ), lu(e)IP- 2u (e)f ee) - t lulp- 2uf in L1(f-L ). -t
-
T hus , passing to t he limi t as e - t a in (7.8) lead s to (7.6). T he lemma is proved. 0 Proof of Theorem 7.1 . Rather elementary arg ume nts (cf. t he proof of Lemma 5.7 of [12]) show that if f E Cgo(JR+ x JRi), then the uni que class ical bounded solution of au/at = 6.u + fin JR+ x JRi with zero initi al and boundar y condit ion belongs to f:J ~ , e , o (T) . This gives us a starting point in t he method of cont inuity and , du e to the fact that f:J ~, e , o (T ) is a Ban ach space, shows t ha t t o prove the t heorem we need only prove t he apr iori estimate I luI ISj~ ,o(T) ::; N IIM (L u - au/ at )lllLp,o(T)
for any u E f:J ~ , e , o (T ) , where and below we denote by N generic constants dep ending only on 00, d, p , e, K . We will keep track of (3 up until we will see how to choo se it. By the definition of the norm in f:J ~ , e , o (T) we have
and
II M au/ at lllLp,o(T) ::; IIM (L u - au/at)liJLp ,o(T)
+ 11M Lul llLp,o(T),
wh ere by our ass umptions on b and c
with
F inally, by recalling t hat by Rem ark 4.2 the left-hand side of (7.3) is equivalent t o we see that we need only estimate jI l p in t erms of t he right-hand side of (7.3) . By Corollary 5.1 we have
iv»,
11M D 2u lllLp,o(T) + IIDulllLp,o(T) ::; N IIM fll lLp,o(T) + N II M -1ul llLp ,o(T), where f
= (a/at - L )u.
(7.9)
ON SPDES IN SOBOLEV SPACES WITH W EI GHTS
187
Next, we use Lemma 7.1 with G = lR~ , 1jJ = (Xl )-1, and M ((i - dJ/PUt and M (B-dJ/p ft in place of Ut and ft , resp ectively. We also obs erve that
Also
111jJ-l M (B -dJ/P(Ltut + ft)IILp (GJ = :::; NI IMD 2UtIILp,o + IIMlbtlDutllLp,o+
IIM (Lt ut + ft)IIL p,o IIM 2Ct (.!VI- IUt)liLp,o + II M f tII Lp,o'
so that by our assumptions and Remark 4.2
It follows that Lemma 7.1 is applicable and by noting that
1jJp-2I M( B- dJ/PUt lp- 2(M( B- dJ/PUt )(M (B- dJ/P(LtUt + ft)) = 1jJp- 2MB- dl ut lp-2 (L tut + ft) = M 1'+1Iutl p- 2(L t ut + ft), where I
=
e-
d - p + 1, we obtain
Observe that by (7.2)
2Iutlp- 2Ut(Ltut + it) :::; I Ut I P - 2 ut a~j DijUt + N j3M-Ilut IP-l IDuti +Nj3M-2Iut IP + Nlutlp-1 Ift l, and by Young's inequ ality
M-I IUtlp-lIDUtl = (M-2 (p-I J/Plut IP-1 )(M(P- 2J/ PI Dutl) :::; M-2 1ut1P + Mp- 2I DutIP, IUtlp-Ilft l :::; j3M - 2Iut IP + j3 1-p M 2(p -I JIf t iP. Furthermore,
188
NICOLAI V. KRYLOV
Therefore , coming back to (7.10) we get
By combining this with Lemma 6.5 we obtain
N (3J + N (3I-PIIMfl lr p,o(T)
+ p-I 'Y('r + 1) iT
ld
(xl)'-la:l lutIP dxdt
+
r
- (p-1) {
JJR~
Jo
(xl)'+l lutlP-2a~j(DiUt)DjUtdxdt;::: 0.
Lemma 6.4 with 8 E (0,1 ] and Corollary 6.2 allow us to est imat e the last t erm and we conclude that for any K E (0,1]
N(K - 18 + e NoT +v
l
2 /
(3 )J
+ N (3 -PIIM fl lt
,o(T)
r r (xl )'- la: l lutI dxdt ;::: 0, I; JJR~
(7.12)
P
where
v = p-I 'Y ('r + 1) - (p - 1)(1 - K)"(2p-2 = Kp-2(p - 1)"(2 + p-2(B - d - p + l)(B - (d - 1)) .
°
If we consider u as a function of K , then by virtue of (7.1) we have v < if K = 0. It follows that there exists a K = K(B,p, d) > such that v = - l/N , where N = N(B,p, d) E (0, (0) . Then (7.12) and the assumption that all ;::: 00 imply that
°
By using (7.9) we conclude
2
1 :S:; N I(8 + e N,o- ' / (3)1 + N(8
+ eN,O-, /2 (3 + (3 - P)II M f llt ,o(T)' (7.13)
Now we can specify the choice of 8 = 8(B, d,p, 80 , K), (3 (0, 1). We take them so that
s.s : 1/ 4,
= (3(B, d,p, 80 , K)
E
Nle N,o-, /2 (3 :s:; 1/ 4
and then (7.13) yields
which along with (7.9) lead to (7.3) and the theorem is proved.
D
ON SPDES IN SOBOLEV SPACES WITH WEIGHTS
189
8. Existence and uniqueness for SPDEs in half spaces. In t his section 7 is a fixed stopping t ime, p ~ 2. We start wit h a "conditional" res ult we were talking about in the Introdu ct ion. Here we only have a "nat ur al" ass umption on e. Of course, we are working in the setting of Sect ion 2. T HEOREM 8 .1. Let
d- 1
< e < d - 1 +p.
Assume that for a constan t K E (0,00) we have
Let 13 be the sma llest of the constants called Theorem 7.1 and assume th at
130 in Theorem 4.1 and 13 in
whenever x, y E lR.~, Ix - yl S; xl 1\ y 1, i,j = 1, ... , d, t E (0,7) . Fix an E lLp •Ii (7). A ssum e that there is a constant No < 00 such that f or any
f
(8. 1)
we have the apriori estimate
provided th at
in lR.~ (estimate (8.2) is not supposed to hold if there is no solution u E 5)~.1i ,0(7) of (8.3)). Then for any 9 E lHI~,1i(7) there exis ts a unique u E 5)~,1i,0( 7) satisfying (8.3) with A = 1. Furtherm ore, f or this solution
where N depends only on d,p, e, 60,61, K , and No. Proof. We know that 5)~, 1i , 0 (7 ) is a Banach space. From Theorem 7.1 we also know t hat for A = eq uation (8.3) is uniquely solvab le in 5)2p , e,0(7) . T here fore , by t he method of continuity to prove t he un iqu e solvab ility of (8.3) for A = 1 we only need to show that t here is a constant N such t hat for any objects in (8. 1) we have
°
(8.5)
190
NICOLAI V. KRYLOV
provided that (8.3) holds in ~t. By the definition of the norm in S)~,o ,o(T) we hav e
IlullSjz (r) = IIM- 1uIIIHIZ p, e
p ,o
(r)
+ IIM(Lu + f )IIILp,o(r ) + .xIIAu + gllIHI'
p,()
(r) ,
where as in the proof of Theor em 7.1
Also by our assumptions on a and v
IIAu + gilIHI1 oCr) p,
:::;
IIgII HI1oCr ) + N( IIM- 1u II ILp,0(r) + IIDu II ILp ,o(r) + 11M D2u IIILp,0(r)) ' p,
By combining this with Remark 4.2 we see that to prove (8.5) it suffices to prove (8.4) or else that IIM- 1u II ILp,0(r ) + IIDuI IILp ,o(r) :::; NI IMf II ILp,o(r)
+ II M D 2uIIILp,0(r )
+ N llgIIIHI; ,eCr)'
However, the lLp,o(T)-norms of MD 2u and Du ar e estimated through the lLp,o(T)-norm of M- 1u in Corollary 4.1 and the lat t er admits an est imate by assumption (8.2). The theorem is proved. D Here is a vers ion of one of the results of [7] in which we assume that for a const ant J E (0, 1] and all t and unit .x E ~d we have
ij (a t - C/t j ).xi.xj
>J_l_( ~a1j.xj)2 11 t ,
-
at
~
(8.6)
j
REMARK 8 .1. In light of (2.1) one can always take J = 01 in (8.6) becau se for the positive definite matrix (a~j) and TJ = (1,0, ..., 0) it holds that
(L aij.x j)
2
= (L
J
a~jTJi.xj) 2 :::; (a~jTJiTJj)a~j »» = ai1a~j »» .
J
On the ot her ha nd sometimes J = 1 and 01 may be less t ha n 1. This happens , for instance, if aij == 0 for all j and ai j == 0 for j =I- 1. THEOREM 8.2. Let (1.3) be satisfied and assume that for a constant K E (0,00) we have l
'
1 2
[z Da;( x) IRz + I(x ) DVt(x) IRz :::; K.
W e assert that th ere exist constants (3 > 0 and N < 00 depending only on oo,ol,J,d,p,(), and K , such that if
la~j (x) - a~j (y)1+ la;(x ) - a;(y)I Rz +IX1b~(x) 1 + IX1Vt(x)IRz + l(x 1)2 Ct(x)1 :::; (3,
(8.7)
ON SPDES IN SOBOLEV SPACES WITH WEIGHTS
191
wh en eve r x , y E JR~, Ix - y l :0:; Xl A y l , i , j = 1, ... ,d, t E (0,7) , th en for any f E M - llLp ,0(7), 9 = (gk, k = 1,2, ...) E JH[~,0(7) th ere exists a unique u E SJ ~, 0 ,O(7) satisfying (1.2) in JR~. Furthermore,
R EMARK 8 .2 . For d = do = 1 an exam ple of (J suitable for Theorem 8.2 can be obtained in the following way. Take a smooth fun ction f ~ 1 on (0, 00) such that f(x) = -ln x for small x and f(x) = 1 for x large . Then Ix!, (x )1 is bounded , say by a constant No . Next, for b E (0 ,1) introduce (J(x) = cos f O(x ). Then
1(J/(x)1 = bfO-l( x)lf'(x)llsinfO( x)I :O:; N ob x-l, and if a < y < x and Ix - yl :0:; x A y , then betw een x and y , so that y :0:; t;, we have
Ix -
yl :0:; y and , for a point
t;
which can be made arbitrarily small if b is chos en appropriately. R EMARK 8 .3. One could have introduced the parameter e as in Remark 7.1 and as there this would not make the theor em any more gener al. R EMARK 8.4. Theorem 8.2 is used in [7] to prove the solvabilit y of SPDEs in Sobolev spaces with weights in bounded Cl-domains. The proof for sp aces like 5);,0(7) with n = 2 is quite standard and is based on partitions of unity and flattening the boundary. Then equat ions are solved in sm all neighborhoods of t he boundary, where conditions like (8.7) allow on e to treat equat ions with quite irregular a ij , (Jik and blowing up b, c, and v (cf. Remark 4.4) . The reader is sent to [7] for details. To prove Theorem 8.2 we need the followin g counterpart of Lemma 7.1 in which we use the same notation. LEMMA 8.1. L et Ut = Ut (w), it, and gt = (g[, g; , .. .) be predictable proces ses given for t < 7 with values in Lp ,loc(G) . Assume that for a con ti nuous function 'IjJ = 'IjJ (x ) given on G we have 'IjJ > 0,
and
du;
=
ft dt
+ g~ dw~
in G for t < 7 in the sense that fo r each ¢ E Co(G) (a. s.) for all t E [0, 7) at once w e have
192
NICO LAI V. KRYL OV
Then
E
iTfc
,¢P- 2[2IUtlp- 2Ud t + (p -
1) IUt IP-2 Igtl~2 ] dxdt ~ O.
(8.8)
Proof. We mimic t he proof of Lemma 7.1. Fi rst , we obse rve t hat t he left-h an d side of (8.8) makes sense, since by Holder 's inequality (p ~ 2)
iTfc l'¢Ut IP-2Igtl ~2 (E iT I '¢utll~p(G) 1-2/p(E iT Il gtll~p(G) E
:'S
dt
dt)
dt )2/p
and t he t erm containing f t is taken care of as in the pr oof of Lemma 7.1. Next, as in that proof we redu ce t he general sit ua t ion to t he one in which Ut has compact support in G. More pr ecisely, in the rest of the proof we ass ume that there is a compact set I' c G such t ha t Ut(x) = ft (x ) = g~ ( x ) = 0 if x rf. f . In that case t he valu es of '¢ outside I' become irr elevant and we may ass ume that G = ]Rd, '¢ is bounded away from zero and infinity, Ut , f t ,gt are pr edi ct abl e and Lp-valu ed , and
E
I' (1Iut ll~p+ Ilftll~ p+ Il gtll~ p) dt < 00. Jo
In add it ion one can repl ace here p wit h 2 if one re places 7 wit h 7 /\ T , where T is a constant from (0,00). T hen by using t he not at ion from t he pr oof of Lemma 7.1 we find t hat for any x E ]Rd with probability one for all t < 7
u~"')(x) =
it
f ;"' )(x) ds + i t g~"')k (x) dW:.
By It o's formula for any constant T E (0,00) (here we use again that p ~ 2)
(2/ p) IUS~T(X)JP =
i
T
f\ T [2Iu~"' )(x) IP-2u~"' ) (x) f;"') (x) ds
+(p -1)lu~"')(x) IP-2Ig~"')1 ~21 ds + m tf\T(x ), where mt(x) is a local martingale start ing at zero . Ob serve t hat t he lefthand side is nonnegative and
because of t he sa me reasons as in t he beginning of t he pr oof and t he estimat e
ON SPDES IN SOBOLEV SPACES WITH WEIGHTS
193
that follows from Holder 's inequ ality. Then we find that
Now we integrate through this relation over x E jRd afte r multiplying it by 1jJp- 2(x ). It turns ou t that we can interchang e all integr als since, say u~e)(x) is infinitely differentiabl e in x, measurable with resp ect t o other variables, by Minkowskii's inequ ality (8.9)
and the est imate s as in the beginning of the proof are valid. Aft er int egrating with resp ect to x and set ting T -* 00 we obtain
Now notice that (for any wand t su ch that t < 7) we have u~e) in L p . Similar relations hold for It and g t . In particular, u~e) --> measure
p(dwdtdx)
-* Ut
Ut
in
= 1jJp-2 P(dw)dtdx
on (0,7n x jRd . Furthermore, (8.9) and the dominated convergen ce t heore m imply t hat
Simil ar relations hold for
I
Iu(e) IP- 2
lu(c)
and g. By Lemma 6.1 and Corollary 6.1 -->
IP- 2 Ig(e)
lulp - 2
1£2
L p/ (p-2 )(p) ,
in
2 -* luI P -
Igl£2 in t., (p ).
The t erm with I (c) in (8.10) is t aken care of in the sam e way as in the pr oof of Lemma 7.1 . Thus, passin g to the limit as c -* 0 in (8.10) leads to (8.8). The lemma is proved . 0 Proof of Theorem 8.2 . By Theorem 8.1 we onl y need to find (3 = (3 (00, 01,5, d,p, B, K) E (0 , 1] such that if the condit ions of the theorem are sat isfied with this (3, then (8.2) holds with No dep ending only on 00 ,01,5 , d,p, B, and K , whenever (8.1) and (8.3) ar e sat isfied. Ther efore, we concent rate on estimating 'f l / p .-
IIM- l u II IL
p
,9(T) '
194
NICOLA I V. KRYLO V
lRt,
In Lemma 8.1 we take G = 'ljJ (x ) = (xl) -l , and rep lace ut, lt , gt wit h M (£J-d)/PUt , M (lJ-d)/P(Ltut + I t ), M (£J-d)/P(Atut + gt), respectively. Vie also observe t hat as in t he proof of Theorem 7.1
11'ljJ - I M (O -d)/P (Ltut + I t)IILp(G) ::; NI IM - IUt IIH;,o + 11M I tlILp,o, where and below we denote by N generic constants depending only on 8o,8 1 ,d,p,B, K . We keep t rack of j3 up until we will see how to choose it. Finally,
IIM(O -d)/P(A t Ut
+ gt)IIL p(G)
IIAtut + gtllLp,o ::; N ll Dutll Lp,o + IIMlIt(M-Iut)IILp,o+ Ilgt llLp,o ::; N IIM- Iut IIH2 p ,o + IlgtllL o : =
p.
It follows that Lemma 8.1 is applicable and noting that
'ljJP- 2I M( O- d)/put lp-2 (M( O- d)/pud (M(O - d)/P(LtUt + It ))
= 'ljJp- 2M O-dl utIP-2(L tUt + It) = M ,+1lutlP-2(L tut + I t), 'ljJ p- 2I M (O- d)/putlp- 2IM (O- d)/P(Atut + gt )I;2
= 'ljJ p- 2M O-dl utlp- 2IAtut + gt l;2 = M ,+1lutlP- 2IAtut + gt l;2' where ,
=
B- d - p
+ 1, we obtain
E r r (x l )'+1 [2IUt /P-2Ut( LtUt + I t) Jo lrr{t p+ (p - 1)/utl 2I A tut + gtl;2] dxdt 2: O.
(8.11 )
Next , set
Jl / p := 11M D 2u II Lp,o(T) + II DuIILp,O(T ) + IIM- IuIIlLp,o(T) and note for t he future t hat by Corollar y 4.1 we have
JI / p ::; NIIM I IILp,O(T) + N ll g lllHI~ , o (T)
+ N tv».
(8.12)
We deal with the t erm containing LUt + I t in (8.11) as in t he proof of Theorem 7.1 and conclude that
N j3J + Nj3l- PIIM/II[
(T) p,o
+ E r r (xl ),+l [IUt IP- 2Uta~j o ;»,
Jo JJRd +
+ (p -1 )lutlp- 2IAtUt + gtl;J dxdt 2: O. By combining t his with Lemma 6.5 we obtain
-(p- 1)E
l' r (xl)'+ l lut IP-2[a~j(DiUt)Djut - IAtut+ gt l;2 ] dxdt Jo JJRt +p-l , (/+ l) E r
r (xl ),- lai 1IutJPdxdt
Jo JIRt
+ N j3J
+ N j3I-PIIM/ llt
,o(T ) 2: O.
(8.13)
195
ON SPD ES IN SOBOLEV SPACES WITH W E IGHTS
Her e
2 ij 2 2 2 IAtUt + gtl£2 = a t (DiUt)DjUt + IVtl£21Utl + Igtl£2
+ 2(O"ti ,Vt)£2 UtDiUt
+2(0":' gt)£2DiUt + 2(vt , gt)£2Ut· By our ass umpt ions ij
ij
-
11 -1
Ij
2
(at - at )(DiUt)DjUt ;::: J(at) (at Djud , IVt l;2 !UtIP::; ,BM - 2IUt IP, I U t l p- 2 ( 0"~, Vt) £2UtDiUt
::; N ,B(M- 2IUtIP
< N ,BM-IIUt !p-l IDUtl + M p- 2I DUtI P) ,
IUt IP-2 (O"~ , gt)e2DiUt < NIUt Ip- 2IDUt I!gt 1£2 ::; N ,Bl- pM P- 21gt 1 ~2 + N ,BM (2- p)/(p - l) IUt IP(p- 2) / (p- l ) IDUt IP/(p- l) ::; N ,Bl- pMP- 2 Igtl ~2 + N ,B(M-2IUtIP + Mp- 2I Dutl P) , pIUtlp- 2(Vt ,gt) £2Ut < N ,BM- 1I Utl 1Igtl £2 ::; N ,BM- 2IUtI P + N,BMP-2 Igtl ~2' By usin g the computations in (7.11) , adding that
E r r (x 1p +l MP-2 Igtl~2 dxdt =
Jo JlRt
Ilgll~
OCT)' P,
and con centrating on ,B ::; 1, we infer from (8.13) that
N ,BJ + N,B-P( IIMfll ~p , O (T)
+ IIg ll ~p , o (T))
b + l)E r r (xl)'Y-lail lutlP dxdt
+ p-l i
Jo JIRt
r (xlp+llutIP-2(ail)-I(aij D j Ut)2 dxdt > 0.
- (p -l)JE r
t; JlRt
Lemma 6.4 with J E (0,1 ] and Corollary 6.2 allow us to est imate the last t erm and we conclude that for any K, E (0,1]
N(K,-
IJ
+ eN 8-
1 /
2 ,B)J
+ N,B-P( IIMfl l~p , O (T) + Ilgllt,o(T) )
+vE
t'
r (xlp-laillut lpdxdt;:::O ,
(8.14)
Jo JIRt
wh ere
v = p-l i + p-2 [p(1 - J)
b + 1) - (p - l)J(l - K,)J2 p-2 = K,p-2(p - 1)J2J
+ J](O -
d - p + 1)
(0 - (d - 1) _ p[l _ p(l - 1_J) + J-1).
196
NICOLAI V. KRYLOV
If we consider v as a function of n; then by virtue of (1.3) we have v < 0 if", = O. It follows that there exists a '" = ",(B,J,p,d) > 0 such that u = -l iN , where N = N(B, J,p,d) E (0,00) . Then (8.14) and the assumption that a 11 ~ 50 imply that
By using (8.12) we conclude
I:::; N 1(5 + eNID-I /2 (3 )I
+ N(5 + eNID -I /2 f3 + f3- P )(IIM f llt .e(T) + Ilgll~~.O (T))·
(8.15)
Now we can specify the choice of 5, f3 E (0,1) depending only on
B,d,p,50,51 ,J, and K. We take them so that
and then (8.15) yields
which is (8.2) and the theorem is proved.
o
REFERENCES [1] Z. BRZEZNIAK, Stochastic partial differential equations in M-type 2 Banach spac es , Potential Anal. (1995) , 4(1) : 1-45. [2] Z. BRZEZNIAK, On stochastic convolution in Banach spaces and applications , Stochastics Stochastics Rep . (1997), 61(3-4) : 245-295. [3] G. DA PRATO AND A. LUNARDI, Maximal regularity for stochastic convolutions in LP spaces , Atti Accad. Naz . Lin cei Cl. Sci . Fis. Mat . Natur. Rend . Lincei (9) Mat. Appl. (1998) , 9(1) : 25-29. [4] G. DA PRATO AND L. TUBARO eds , "St ochas t ic partial differential equations and applications-VII" . Papers from the 7th Meeting held in Levico Terme, J anuary 2004 , Lecture Notes in Pure and Applied Mathematics, Vol. 245 (2006), Chapman & Hall/CRC, Boca Raton , FL. [5] F. FLANDOLI, Dirichlet boundary value problem for stochastic parabolic equations: Compatibility relations and regularity of solutions , Stochastics and Stochastics Reports (1990), 29(3) : 331-357. [6] M . HAIRER AND J .C . MATTINGLV, Ergodicity of the 2D Navier-Stok es equations with degenerate sto chastic forcing, Ann. of Math. (2), (2006) , 164(3) : 993-1032. [7J KVEONG-HUN KIM, On stochastic partial differential equations with variable coeffi cients in C1 domains , Stochastic Process. Appl. (2004), 112(2) : 261-283. [8] KVEONG-HuN KIM AND N.V. KRVLOV, On SPDEs with va riable coefficients in one space dimension, Potential Analysis (2004) , 21(3) : 209-239. [9] KVEONG-HuN KIM AND N .V. KRVLOV, On the Sobolev space theory of parabolic and elliptic equations in C 1 domains, SIAM J . Math. Anal. (2004), 36(2) : 618-642.
ON SPDES IN SOBOLEV SPACES W ITH WEIGHTS
197
[10J N .V. KRYLOV, On Lp-theory of stochastic partial differential equations in the whole space , SIAM J. Math. Anal. (1996), 27(2): 313-340. [l1 J N .V . KRYLOV , An analytic approach to SPDEs, pp. 185-242 in Stochastic P artial Differential Equations: Six Perspectives, Mathematical Surveys and Monographs , Vol. 64, AMS, Provide nce , R I, 1999. [12J N .V . KRYLOV , We ighted Sobolev spaces and Lap lace 's equation and the heat equ ations in a ha lf space , Comm. in PDE (1999) , 2 4 (9- 10): 1611-1653. [13J N.V. KRYLOV , Ma ximum principle fo r SPDEs and its app lications , in "St ochastic Differential Eq uations : T heory and Applications, A Volume in Honor of P rofessor Boris L. Rozovskii", P .H . Baxendale, S.V . Lototsky, eds ., Interdi scipli nary Mathematical Scie nces, Vol. 2 , World Scientific, 2007 . http:/ /arxiv .org/math .PR/0604125 . [14] N .V . KRYLOV AND S .V. LOTOTSKY, A Sobo lev space th eory of SPDEs with constant coefficients on a half lin e , SIAM J . Math. Anal. (1999) , 3 0 (2) : 298-325. [15J N.V . KRYLOV AND S .V . LOTOTSKY, A Sobo lev spa ce theory of SPDEs with con stan t coeffi cients in a half space, SIAM J . Mat h. Anal. (1999) , 31 (1) : 19-33. [16J S .B . KUKSIN , R emarks on the balan ce re lations for the two -dimensional NavierStokes equation with random forcing , J . Stat. Phys. (2006), 1 2 2(1) : 101-114. [17] S .V . LOTOTSKY, Di ric hlet problem for stochastic paraboli c equations in smooth domains , Stochastics and Stochastics Reports (1999) , 68 (1- 2): 145- 175. [18J S.V. LOTOTSKY, Sobo lev spaces with weights in domains and boundary value problems for degenerate elliptic equations , Methods a nd Applications of Analysis (2000), 1 (1) : 195- 204. [19] S .V . LOTOTSKY, Linear sto chastic para bolic equations, degenerating on the boundary of a domain, E lectron. J. Probab. (200 1) , 6 (24) : 14. [20J S . LOTOTSKY AND B. ROZOVSKII, Wiener chaos solutions of linear stochastic evolution equ ation s , Ann. Probab. (2006) , 34 (2) : 638-662.
STOCHASTIC PARABOLIC EQUATIONS OF FULL SECOND ORDER* SE RGEY
v.
LOTOTSKyt AND BORIS L. RO ZOVSKII+
A bst r act. A procedure is described for defining a generalized so lution for stochastic differential equations using t he Cameron-Martin version of t he W ien er C haos expansion. Existence and uniqueness of this Wiener Chaos so lution is established for parabolic stochastic PDEs such that both the drift and the d iffusion operators are of the second order.
1. Introduction. Conside r a stochastic evolution equation
du(t)
= (Au(t) + j (t ))dt + (Mu(t) + g(t))dW(t),
(1.1)
where A and M are differential operators , and W is a Wie ner process on a probability space (n, F , JP». Tr aditi on ally, t his equation was stud ied under t he followin g ass umptions : • AI. T he operator A is elliptic, t he order of the ope rator M is less t han t he order of A , and A - ~ M M* is ellipt ic (possibly deg enerate) ope rator , In fact, it is well known t hat unl ess ass um ption Al hold s, E quati on (1.1) has no solut ions in L 2 (n ; X ) for any reason abl e choice of t he state space X . It was shown recentl y (see [4, 5, 6] and t he referen ces t here in) t hat if only t he operator A is elliptic and t he order of M is sm aller than t he order of A , t he n t he re exists a unique generalized solution of E qu ati on (1.1). This solution is oft en referred to as Wie ner Chaos solution . It is given by t he Wi ener chaos expansion u (t) = 2: 01<00 U o (t) ~o , where { ~o }l o l < oo is the Cameron-Martin ortho normal basis in the space L 2 (n ; F W ; X ) of square integr able random elements in X measurable wit h resp ect to t o t he sigmaalgebra F W generated by t he W iener process. The Cameron-Martin basis {~o } is indexed by mul t iind ices a = (aI, a2, ...) . It was show n t hat for certain positive weights Q = {q (a)}lol
I lu l l~,x:=
L
q2 (a)
IluoIIL «o,T);X) < 00 ,
101 <00 *Sergey V . Lototsky acknow ledges support from NSF CAREER award DMS-0237724 . Bo ris 1. R ozovskii acknow ledges support from NSF Grant DMS 0604863, ARO Grant W9 1lNF-07-1-0044, a nd ONR G rant N00014 -07-1-0044 . tDepartment of Mathematics, USC , Los Angel es , CA 90089 (l ot ot s ky0 math. us c . edu) , http : / /www- rcf. usc .ed u /~ lototsky. +Division of A pp lied Mathematics , Brown University, P rovid ence, RI 029 12 (r oz ovs ky0dam. brown . edu ). 199
200
SERGEY V. LOTOTSKY AND BORIS L. ROZOVSKII
where X is the appropria t e Hilbert space characterizing t he "regularity" of the solut ion . Note that without assumpt ion Al
E
L
IluIIL«O,T);X) =
Ilu a
(t)IIL«o ,T);X) = 00.
lal <ex:>
In this pap er , we consider the Cauchy problem for the following stoch astic partial differential equation:
du = (aijDiD ju + biD iu + cu + f)dt + (PijDiD ju + (TiD iU + VU + g)dW,
t E [0 , TJ, x E lR.
(1.2)
In contrast to the pr evious work , t his is a par abolic SPDE of the full second order , in that the drift and diffusion op erators have the same order 2. We const ruct a scale of weighted Wi ener chaos spaces (re late d but not identical to Kondratiev 's spaces) and prove the existence and uniquene ss of t he solution in the spaces from this scale.
2. Constructing a solution: an example. Let IF = (D, F, {Fdo::;t::;T, lP') be a stochastic basis with the usual assumptions and W =
W(t), 0:::; t :::; T , a standard Wiener process on IF. For a Hilbert sp ace X , denote by L 2( W ; X) the collect ion of X-valued random eleme nt s that are square integrable (JEll , IIi < 00) and are measurable with resp ect to the sigm a-algebra generated by W(t) , t E [0, T ]. Consider the Ito equation u(t, x) =e-
x2 2 / +
it
Uxx(s, x)ds +
it
Uxx(s , x)dW(s) ,
(2.1)
t E [O ,T] , x E lR.
If there is a solution , its Fourier t ransform in sp ace , u(t , y) (1/ J27f) fIR e-iXYu(t , x )dx satisfies
y2 u(t , y) =e- / 2 - y2it u (s, y)ds - y2it u( s, y)dW(t) ,
t
E
[0, TJ,
(2 .2)
y E lR.
For each fixed y, (2.2) defines a geom etric Brownian motion:
u(t , y) = e- (l+t)y2_(y4/ 2)t- y2W(t).
(2.3)
Let H'Y(lR ) be the Sobolev space
(2.4) Since
(2.5)
SECOND-ORDER SPDES
201
the solution of (2.1 ) cannot be an element of L 2(W ; L 2 ((0, T) ; H i(JR))) for any I E JR, even though the initial condition is non-random and is an element of H i (JR) for every I E R Let us try another approach. Once again, assuming that the solution exists, we apply the Ito formula to the product u(t , X)£h(t), where
and h = h(t) is a smooth deterministi c function . Since
(2.7) we conclude that the fun ction
(2.8) if defined , must satisfy the heat equation
(2.9) If SUPt Ih(t)1 < 1, then this equ ation has a unique solution in every H i(JR) and
Uh(t, x )
= lEexp( -(X(t , x))2 / 2),
(2.10)
wh ere
1 t
X(t, x ) = x +
J2(1
+ h(s)) dW(s).
(2.11)
In other words , while existe nce of a solution of Equation (2.1) is st ill unclear, we now have a family of functions Uh(t, x ) defined by (2.10) . All we need now is a systematic procedure of relating the family of det erministic fun ctions ui, = Uh(t, x) to a random pro cess U = u(t , x) ; then this process is natural to call a solution of (2.1) . Here is a possible way of constructing a stochastic process from Uh . Let m = {mk ' k ~ 1} be the Fourier cosin e basis in L 2( (0, T) ): (2.12) Then
h(t)
= L hkmk(t). k 2::1
(2.13)
202
SERGEY V. LOTOTSKY AND BORIS 1. ROZOVSKII
For every fixed t E [0, T ] and I E JR, we can now interpret the fun ction (t , .) as a mapping from the set of sequenc es h = (h 1 , h 2 , . . .) to the space H'Y(JR d ) , and , as equalities (2.10) and (2.11) sug gest, this mapping is analyt ic in the region {h : L:k>l h~ < s ] for sufficiently small c. We will now compute the derivatives ofthis mapping. Let :J be the collect ion of multi-indices a = {ak ' k :::: I} . E ach a E :J has non-negative integer element s ak and
Uh
lal = L
ak <
(2.14)
00 .
k
We also use the notation
(2.15) and consider special multi-indices, lal = 1, ai = 1. For each a E :J defin e
a=
(0) with
la[ =
0 and
a = e. , with
(2.16) Then
(2.17) where
(2.18) On the other hand, by direct computation,
(2.19) where
(2.20) and
(2.21)
203
SECOND-ORDER SPDES
is n-th Hermite polynomial. It is a standard fact [1] that the collection {~o: , a E J} is an orthonormal basis in L 2(W;lR). The functions uo:(t, x), a E J, uniquely determine Uh(t, x) according to (2.17) . On the other hand , if
I: Iluo:(t)llk'Y(lR) <
00 ,
(2.22)
I: uo:(t,x)~o:
(2.23)
o:E:J
then the H 'Y(lR)-valued random process
u(t,x) =
o:E:J satisfies IE(u(t ,X)£h(t)) then also
Uh(t,X) ; if, in addition, u
=
u(t,x)
FF-adapted,
IS
I: uo:(t,x)~o:(t) .
=
(2.24)
o:E:J If condition (2.22) fails, then (2.23) is a formal series, which we define to be the stochastic process corresponding to the family Uh . As (2.5) suggests, if Uh is the solution of (2.9), then (2.22) fails for every v, Let us now see how fast the series diverges. Equality (2.9) implies
U(O)(t ,x)=et
x 2/ 2
r 82u(o)(s,x) 8x2 ds ,lal=O;
+io
it 82u~~~s ,x) -! +
UEi(t)=l
82u~~:,x) ds +
Uo: t -
82uo:(s , x) d 8 2 S
()-it o
~
(2.25)
82UO:_ Ek(S ,X) ()d 8 2 mk s s , lal > 1.
LJ yak
k=I
X
mi(s)ds , lal = 1;
0
X
Equations of the type (2.25) have been studied [4, Section 6 and References]. In particular, it is known that
I: Iluo:(t)llk'Y(lR) = lo:l=n
t~ II D;ntuo IIk 'Y(lR),
(2.26)
n.
where D x = 8/8x, t is the heat semigroup, and uo(x) = e- x 2 / 2. To simplify further computation, let us assume that "Y = O. Then, switching to the Fourier transform, II
D 2n x
t
U 11 2 d) 0 L2(lR
=
i rf*. !y I ne4
y2
(t+I )dy
= r (2n + ~) (1 + t)2n
.
(2.27)
Using Stirling's formula for the Gamma function I', (2.28)
204
SERGEY V. LOTOTSKY AND BORIS L. ROZOVSKII
wher e the numbers C(n) ar e uniformly bounded from above and below. Similar result holds in every H'Y(IR) . Thus, (2.22) does not hold , but instead, by (2.28), we have
c:
'" ~ 1 l lua(t)IIW' 2 (IR)
<
00.
(2.29)
«e.r
We denote by (£)o,o(W ; H'Y(IR)) the collection of formal seri es (2.24) satisfying (2.29); the reason for using (£)0,0 in the notation will become clear later . Note that we had equalit ies in all computations for Equation (2.1) that lead to (2.29) , which suggests that (£)o,o(W ; H'Y(IR)) is the natural solution space for Equation (2.1) . For a more gener al stochastic parabolic equat ion of full second order in IR d , the natural solution space turns out to be (£)p ,q(W;L 2((0 ,T); H'Y(lR d ) ) ) for suitable p, q :::; 0. In the next section we address the following questions: 1. How to define the spaces (£)p ,q(W; X) for p, q E IR without relying on an orthonormal basis in L 2((0, T)) ? 2. How to construct a solution of a general stochastic parabolic equations of full second order? 3. General constructions and the main result. As before, let IF = (0" F, {Ft}o:'O t:'OT, ]]D) be a stochastic basis with the usual assumptions and W = W(t), t :::; T , a standard Wiener process on IF. Denote by H' = HS((O , T)), s ~ 0, the Sobolev spaces on (0, T) with norm
°:: ;
(3.1)
wh ere A is the op erator
(3.2) with Neumann boundary conditions. This norm ext ends to functions of several variables via the tensor product of the spaces HS. DEFINITION 3 .1. Given real numbers p, q and a Hilbert space X , (£)p ,q(W; X) is the closure of the set of X -valued random elements
with respect to the norm (3.4)
where each TJk, k ~ 1, is a smooth symmetric function from [0, T]k to X .
205
SECOND-ORDER SPD ES REMARK
3 .1. (a) It is known [2, 7] that, for TJ of the type (3.3) , N 2
",1
2
JEllTJllx = IITJollx + 6
k! 1IIITJklio
2
Ilx·
(3.5)
k =l
(b ) The definition of each individual (£) p,q(W j X) inevitably involves arbitrary choices , such as the norm in Hq((O ,T)) . Further analysis shows that different choices result in shifts of the indices p , q , and the sp ace Up,q(£ )p,q(W ; X) does not dep end on any arbitrary choices. In the white noise setting, where n is the sp ace S' (JR.d) of the Schwartz distributions and IF' is the normalized Gaussian measure on S , the inductive limit Up,q (£) p,q (W ; JR.) is the Kondratiev sp ace (S)-l [2]. R EMARK 3.2 . If X is the Sobol ev sp ace H'Y(JR.d), then we denote t he norm II . IIp,q;x by II ·lIp,q;'Y: (3.6)
PROPOSITIO N 3.1.
Let TJ = f~O!.! where
f
E X and ~O!. is defined by
(2.20). Th en 2 _ 210!.Ip 2qO!. 1ITJ IIp,q; X - ~N Ilf
2
ll x,
(3.7)
where
N 2qO!.
=
II k
2q O!. k .
(3.8)
k21
Proof. Let
c;u = ~ ya!
where
E O!.
10:1 = n.
It is known [3] that
T r t: ... ('2 E U(Sl, "" Jo Jo Jo
Sn)dW(SI) .. . dW( Sn_l)dW(Sn) , (3.9)
is the symmetric function E O!. ( Sl , . .. , Sn)
=
L
mi, (Sa (l )) ... m i n ( Sa(n )) '
(3.10)
aEPn
In (3.10) , the summation is over all permutations of {I , . . . , n}, the functions mi, are defined in (2.12), and the positive integer numbers i 1 :::; i 2 :::; . . . in are such that , for every sequence (bk , k 2: 1) of positive numbers,
II b~k = b s.; .... . i, .
k 21
bi n '
(3.11)
206
SERGEY V. LOTOTSKY AND BORIS L. ROZOVSKII
For example, if a = (1 ,0 ,2,0,0 ,4,0,0, . . .), then 1001 = 7 and i l i 2 = i 3 = 3, i 4 = ... = i 7 = 6. Thus, in the notations of (3.4) , we have
TJk =
~Ee< f,
v a! { 0,
if k
= n,
l,
(3.12)
otherwise.
Not e that (3.13)
By definition (3.2) of the operator A we have
o
The result now follows. COROLLARY
3.1. A formal series (3.15)
with TJe< E X , is an eleme nt of (£ )p,q(W; X) if and only if (3.16)
Proof. This follows from (3 .14) and the equa lity
(3.17)
o
Deno t e by (£) p,q (W) the Hilbert space dual of (£) p,q(W ; IR) relative t o the inner product in L 2( W ; IR) , and by ((' , .)) the corresponding du ality. In the white noise setting, n p, q(£ )p,q (W ) is the space (Sh of the Kondratiev t est functions [2]. If TJ E (£)p,q(W ; X) and ( E (£)p,q(W) , then ((TJ, ( )) is defined and belongs to X. For h E L 2((0, T)) , define
e; =
Eh(T) = exp
(iT
h(s)dW(s) -
~
iT
Ih (SW dS) .
(3.18)
PROPOSITIO N 3.2 . Th e random variabl e Eh is an eleme n t of (£ )p,q (W)
if an only if (3.19)
SECOND-ORDER SPDES
207
Proof. Since
(3.20) it follows that
Eh
= 1 + 2::= CXl
k=l
iT1 ...1 8k
0
0
82
h(Sk) ' " h(sddW(sd '" dW(Sk_ddW (Sk)'
0
(3.21) By (3.4) and (3.5), Eh E (£)p,q(W) if and only if (3.22) that is, Ilh ll~q < 2P .
D
DEFINITIO N 3 .2. W e say that the fun ction h is suffic iently small if (3.19) holds for sufficiently large (positiv e) -p, -q . PROPOSITIO N 3.3. If u E U p,q(£)p,q(W; X) and h is sufficiently
small, th en Uh
= ((u, Eh ))
(3.23)
is an X -valu ed analytic function of h . P roof. For every u E Up,q(£) p,q(W; X) , there exist p , q such that u E (£) p,q(W ;X) ; by Proposition 3.2, Uh will indeed be defined for sufficiently small h . Similar to (2.17) we have (3.24) and this power series in hCY. converg es in som e (infinite-dimensional) neighborhood of zero. D From now on, D; = a/aXi, and the summation convention is in force: c.d, = 2:i Ci di, et c. Consider the linear equat ion in lR d
with initial condit ion u(O, x ) = v(x) , under the following assumptions: BO. All coefficient s are non-random. Bl. The fun ctions aij = aij(t,x) , Pij = Pij(t , X) are me asurable and bounded in (t, x) by a positive number Co , and (i) laij(t, x)- aij(t ,y)I+lpij(t,X)-Pij(t,y) 1::;Col x-yl , x , y E lR d , 0 < t ::; T;
208
SERGEY V. LOTOTSKY AND BORIS 1. ROZOVSKII
(ii) the matrix (ai j) is uniformly positive definite, that is, there exists a 8 > so that, for all vectors Y E Jltd and all (t, x), aijYiYj ~ 81y1 2 . B2. The functions b, = bi(t , x), c = c(t, x), a, = O'i(t, x) , and v = v(t ,x) are measurable and bounded in (t,x) by the number Co . B3.
°
p ,q
(3.26)
p ,q
For simplicity, we introduce the following notations for the differ ential operators in (3.25) :
A = aijDiDj + biDi + C, B = pijDiDj + a.D, + t/.
(3.27)
DEFINITION 3 .3. A solution u of (3.25) is an element of Up,q(£)p,q(W;L 2((0,T); H1(Jltd))) such that, for all sufficiently small hand all t E [0, T], the equality
Uh(t, x)
= Vh(X) +
it
(A
+ h(s)B)uh(s , x)ds
(3.28)
holds in H-l(Jltd) . The following theorem is the main result of this paper. THEOREM 3.1. Assume that, for some p > and q > 1, Uo E (£)p,q(W; L 2 (Jltd)) and I , 9 are elements of the space (£)p,q(W; L 2((0, T) ;H-l (Jltd))). Then there exist r , < such that Equation (3.25) has a unique solution U E (£)r,e(W; L 2((0, T) ; H1(Jltd))) and
°
e
l
T11u (t )II; ,t ;l dt < C·
°
I (llf (t)II;,q;_l + Ilg(t)II;,q;-l) T
(IIvll;,q;o +
°
dt).
(3.29)
e,
The number C > depends only on 8, Co ,p, q, r , and T. Proof. The proof consists of two steps: first, we prove the result for deterministic functions v, i, 9 and then use linearity to extend the result to the general case. Step 1. Assume that the functions v E L 2(Jltd), t, 9 E L 2((0, T); H-l(Jltd)) are deterministic. Then Vh = v, fh = I, gh = g , and classical theory of parabolic equations shows that, for sufficiently small h, Equation (3.28) has a unique solution ui, and the dependence of Uh on h is analytic. As in the previous section, we write U(t,x)
=
L ua(t,x)~a aE:J
(3.30)
SECOND-ORDER SPDES
209
where the coefficients u'" satisfy
U(O)(t, x) = v( x)
+ it (Au(o)(s, x) + f (s, x))ds,
UEk(t ,X) = i t AUEk(S,x)ds + i t (BU(O)(S, X) + g(s,x))mds)ds , (3.31) u",(s, x ) = t Au", (s, x)ds +
Jo
L y(ik Jot Bu"'_Ek (s , x)mk(s)ds , lal > 1. k
Denote by cP = cPs,t, t 2': s 2': 0 the semigroup gen erated by t he op erator AIt follows by ind ucti on on lal t hat
T herefore, using t he usual par abolic estimates,
(3.33)
and t hen (3.29) follows from (3.16). Step 2. As in Step 1, existence and uni quen ess of solution follows from un ique solvability of t he par abolic equation (3.28), and it remains to establish (3.29). Denote by u(t, x; V, F, G, ')') , ')' E J , t he solution of (3.25) with v = V(p f = g = If v = L "'EJ v",f;,,,,, etc., t he n
rc;
ce;
U(t,x) =
L
u(t ,x;v-y,f-y,g-y,')') ·
(3 .34)
-yEJ
It follows from (3.31) t hat u",(t, x; V, F , G, ')')
u",+-y(t, x ; V, F, G, ')') J(a + ')')!
u'"
= 0 if lal < ill and
(t,x;~ , -ftr,-7rr' (0)) va!
(3.35)
210
SERGEY V . LOTOTSKY AND BORIS L . ROZOVSKII
Using the results of Step 1,
1 T
<
~ (1IV'1'IILcJRd) +
Ilu(t,·; v'1',f.y,9'1') 11;,£;1 dt
I
T
(1If'1'(t)II~-lCJRd) + 119'1'(t)II~-lCJRd»)dt).
Now (3.29) follows from (3.34) by the triangle inequality.
(3.36)
o
REFERENCES [1] R.H. CAMERON AND W .T . MARTIN. The orthogonal development of nonlinear functionals in a series of Fourier-Hermite fun ctions. Ann. Math ., 48(2): 385-392, 1947 . [2] H. HOLDEN, B . 0KSENDAL, J. UB0E, AND T . ZHANG . Stochastic Partial Differential Equations. Birkhauser, Boston, 1996. [3] K. ITo. Multiple Wiener integral. J. Math. So c. Japan, 3 : 157-169, 1951. [4J S .V . LOTOTSKY AND B.L. ROZOVSKII. Stochastic differential eq uat ions: a Wiener chaos a p proach . In Yu . Kabanov , R. Liptser, and J . Stoyanov, editors, From stochastic calculus to mathematical finance: the Shiryaev f estschrift, pp. 433507. Springer, 2006 . [5] S .V . LOTOTSKY AND B .L . ROZOVSKII. Wiener chaos solutions of linear stochastic evolut ion equat ions . Ann. Probab ., 34(2) : 638-662, 2006 . [6J R . MIKULEVICIUS AND B .L . ROZOVSKII . Line ar parabolic stochastic PDE's and Wiener Chaos. SIAM J . Math . Anal., 292: 452-480, 1998. [7] D . NUALART. Malliavin Cal culus and Related Topics, 2nd Edition. Springer, New York, 200 6.
