STOCHASTIC INTEGRALS
H. P. McKEAN, JR. THE ROCKEFELLER UNIVERSITY NEW YORK, NEW YORK
1969
A CAD
E M I C P R E S S New York and London
C OPYRIGHT © 1969,
BY
ACADEMIC PRESS, INC
.
ALL RIGHTS RESERVED
NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.
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LIBRARY
OF
CONGRESS CATALOG CARD N UMBER:
PRINTED IN THE UNITED STATES OF AMERICA
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Dedicated to K.
ITO
PREFACE
This book deals with a special topic in the field of diffusion processes: differential and integral calculus based upon the Brownian motion. Roughly speaking, it is the same as the customary calculus of smooth functions, except that in taking the differential of a smooth function/ of the ! -dimensional Brownian path t b(t), it is necessary to keep two terms in the power series expansion and to replace (db) 2 by dt : df(b) = f(b) db + if"(b)(db) 2 = f'(b) db + if"(b) dt, ---+
or, what is the same,
J f'(b) db= f(b) t
0
t -
0
1
J
t
f"(b) ds.
0
This kind of calculus exhibits a number of novel features; for example, the appropriate exponential is eb - t/2 instead of the customary eb. The main advantage of this apparatus stems from the fact that any smooth diffusion t x(t) can be viewed as a nonanticipating functional of the Brownian path in such a way that x is a solution of a stochastic differ ential equation dx= e(x) db + f(x) dt ---+
..
VII
Vlll
PREFACE
with smooth coefficients e and f This represents a very complicated nonlinear transformation in path space, so it can hardly be called explicit. But it is concrete and flexible enough to make it possible to read off many important properties of I. Although the book is addressed primarily to mathematicians, it is hoped that people employing probabilistic models in applied problems will find something useful in it too. Chandrasekhar [I] , Uhlenbeck Ornstein [I] , and Uhlenbeck-Wang [I] can be consulted for appli cations to statistical mechanics. A level of mathematical knowledge comparable to Volume I of Courant-Hilbert [I] is expected. Y osida [2] would be even better. Also, some knowledge of integration, fields, independence, conditional probabilities and expectations, the Borel Cantelli lemmas, and the like is necessary ; the first half of I to ' s notes [9 ] would be an ideal preparation. Dynkin [3] can be consulted for additional general information ; for information about the Brownian motion, Ito-McKean [I] is suggested. Chapter I and about one third of Section 4.6 are adapted from Ito-McKean ; otherwise there is no overlap. I to [9] and Skorohod [2] include about half of Chapters 2 and 3, and Section 4.3, but most of the proofs are new. Problems with solutions are placed at the end of most sections. The reader should re gard them as an integral part of the text. I want to thank K. I to for conversations over a space of ten years. Most of this book has been discussed with him, and it is dedicated to him as a token of gratitude and affection. I must also thank H. Conner, F. A. Gri.inbaum, G.-C. Rota, I. Singer, D. Strook, S. Varadhan, and the audience of 1 8. 54/MITI 1 965, especially P. O 'Neil, for information, corrections, and/or helpful comments. The support of the National Science Foundation (NSF/GP/ 4364) for part of I965 is gratefully acknowledged. Finally, I wish to thank Virginia Early for an excellent typing job. H. P. M c KEA N , JR. South Landaff, New Hampshire 1968
CONTENTS
Vll
Preface
. .
Xl
List of Notations
1.
.
Brownian Motion
Introduction 1 . 1 Gaussian Families 1 .2 Construction of the Brownian Motion 1 .3 Simplest Properties of the Brownian Motion 1 .4 A Martingale Inequality 1 .5 The Law of the Iterated Logarithm 1 .6 Levy's Modulus 1 . 7 Several-Dimensional Brownian Motion
1 3 5
9
11 12 14 17
2 . Stochastic Integrals and Differentials 2. 1 Wiener's Definition of the Stochastic Integral 2.2 Ito's Definition of the Stochastic Integral IX .
20 21
X
CONTENTS
2.3 2.4 2.5 2.6 2. 7 2.8 2.9
3.
Simplest Properties of the Stochastic Integral Computation of a Stochastic Integral A Time Substitution Stochastic Differentials and Ito's Lemma Solution of the Simplest Stochastic Differential Equation Stochastic Differentials under a Time Substitution Stochastic Integrals and Differentials for Several-Dimensional Brownian Motion
Stochastic Integral Equations
24 28 29 32 35 41 43
(d = 1)
3.1 Diffusions 3. 2 Solution of di e (I) db + f(I) dt for Coefficients with Bounded Slope 3.3 Solution of di e (I) db + f(X) dt for General Coefficients Belonging to C1(R1) 3.4 Lamperti's Method 3.5 Forward Equation 3.6 Feller's Test for Explosions 3. 7 Cameron-Martin's Formula 3. 8 Brownian Local Time 3.9 Reflecting Barriers 3.10 Some Singular Equations
50
=
52
=
4.
Stochastic Integral Equations
4.1 4.2 4.3 4.4 4.5 4.6 4. 7 4.8 4.9 4.10
54 60 61 65 67 68 71 77
(d ;;:: 2)
Manifolds and Elliptic Operators Weyl's Lemma Diffusions on a Manifold Explosions and Harmonic Functions Hasminskii's Test for Explosions Covering Brownian Motions Brownian Motions on a Lie Group Injection Brownian Motion of Symmetric Matrices Brownian Motion with Oblique Reflection
82 85 90 98 102 108 115 117 123 126
References
133
Subject Index
135
LIST OF NOTATIONS USAGE: Positive means > 0, while nonnegative means � 0 ; it is the same with negative and nonpositive. A field is understood to be closed under countable unions and intersections of events. The phrase with n probability 1 is suppressed most of the time. c (M) stands for the class of n times continuously differentiable functions f from the (open) manifold M to R 1 ; no implication about the boundedness of the function or of its partials is intended. f is said to be compact if it vanishes off a compact part of M.
a A A b B B
c
an extra Brownian motion the Lie algebra of G (Section 4. 7) a field including the corresponding Brownian field B (Section 1.3) a Brownian motion (Section 1 .2) an event a Brownian field (Section 1.3) a constant XI .
..
Xll
d nn D(G) D a � e e
E(f) f
f
g
G
G G*
H
.
1.0.
n
0
O(d) p P(B)
Q
LIST OF N OTATIONS
the dimension, a differential (Section 2.6) a class of formal trigonometrical sums (Section 4.2) the enveloping algebra of G (Section 4. 7) a 1 -field (Section 4. 1 ), a Lie or enveloping element (Section 4. 7) a partial, the boundary operator a Brownian increment b(k2-n) - b((k - 1 )2- n) (Section 2. 5), an interval a Laplacian, e.g., o 2 jox 1 2 + + o 2 joxd 2 a nonanticipating Brownian functio n al (Section 2.2), the coefficients of 8 2 in G (Sections 3. 1 , 4. 1 ) an exit or explosion time (Sections 3.3, 4.3) the expectation based on P(B) of the function f a function, the coefficients of o in G (Sections 3. 1 , 4. 1) a local time (Section 3.9) the coefficients of o 0 in G (Section 4. 1 ) a group of fractional linear substitutions (Section 4.6), a Lie group (Section 4. 7) an elliptic operator (Sections 3. 1 , 4. 1) the dual of G (Section 4.2) a Hermite polynomial (Section 2 . 7) infinitely often a compact coo function, a patch map (Section 4. 1 ) the Jacobian ox' fox (Section 4. 1) logarithm lg(lg) the space of functions f with 11/ 111 = J Ill < oo the space of function s / with 11/11 2 = (f l/1 2 ) 1 1 2 < oo a manifold (Section 4. 1) an integer an orthogonal transformation (rotation) the orthogonal group an elementary solution of oujot = G * u (Sections 3. 1 , 4. 1 ) the probability of the event B, usually Wiener measure (Section 1 . 2) an elliptic operator on a torus (Section 4.2) a Bessel process (Section 1 . 7) a Riemann surface (Section 4.6) d-dimensional number space ·
·
·
Xlli
LIST OF NOTATIONS
Rn ® Rm
SO(d) sp t t
T
u
u
w
to
X
X
z
3
the applications of Rm into Rn the special orthogonal group [det o + 1 ] (Section 4.7) spur or trace time a stopping time (Section 1 . 3), an intrinsic time or clock (Section 2. 5) a torus [0, 2 n ] d (Section 4.2) a solution of oujot Gu a patch of a manifold (Sectio n 4. 1 ) a point of a covering surface (Section 4.6) a covering Brownian motion (Section 4.6) local coordinates on a patch (Section 4. 1 ) a stochastic integral (Section 2.6), a diffusion expressed in local coordinates (Section 4.3) a point of a manifold M (Section 4. 1 ) a martingale (Section 1 .4), a diffusion on a manifold (Section 4.3), a complex Brownian motion (Section 4.6) the rational integers 0, + 1 , etc. the lattice of integral points of Rd maximum mtntmum the inner product of Rd multiplication, cross product of Rd outer product transpose the norm on Rd, the bound of an application of Rd (y- x)-1[l(y)-l (x)] (x # y),l '(x) (x y) (Section 3.5) J Ill except in Section 4.2 (J 1112 ) 1 1 2 except in Sect ion 4.2 the upper bound of Il l the integral part of intersection . union set inclusion point inclusion increases to decreases to infinity, the compactifying point of a noncompact manifold. =
=
.
X
@
*
I I I.
11 1 111
Ill liz llllloo [ ] n u
c
E
i !
00
=
STOCHASTIC INTEGRALS
1
BROWNIAN MOTION
INTRODUCTION
N. Wiener and K. Ito are the principal names associated with the subject of this book. Wiener [ 1 , 2] put the Brownian motion on a solid mathematical foundation by proving the existence of a completely additive mass distribution P(B ), of total mass + 1, defined on the class of all continuous paths 0 � t --+ b(t) E R1 by the rule � s] = P [b(t) E A I b(r) ·. r ""'
J
A
exp [ - (x - y) 2/2(t - s)] [2n (t - s)] 1/2
dy
for t > s, x = b(s) , and A c R1 • Wiener also proved that the Brownian path is nowhere differentiable. Because of this, integrals such as e(t) db cannot be defined in the ordinary way. Paley e t al. [1] over came this difficulty by putting
J:
J
1
0
e(t) db = e(l)b(l) - e(O)b(O) 1
1
J e'b dt 0
2
1
BROWNIAN MOTION
for sure functions e e ( t) from C 1 [0 , 1] and by extending this integral to L2 [0, 1] by means of the isometry =
Cameron-Martin's [1] formula for the Jacobian of a translation in path space, Wiener's [4] solution of the prediction problem, and Levy's white noise integrals for Gaussian processest should be cited as the deepest applications of this integral. Ito [1] extended this integral to a wide class of (nonanticipating) functionals e e ( t) of the Brownian path with P 1 and e 2 dt < developed the associated differentials into a powerful tool. t Peculiarities of the Brownian integral, such as the formula
oo]
[f:
=
2
fo1 b (t) db
=
b (W
=
- 1,
find a simple explanation in Ito's formula for the Brownian differential of a function f e C 2 (R1) : df( b)
=
f'( b) db +
( 1/ 2)/" (b) dt.
Ito used his integral to construct the diffusion associated with an elliptic differential operator G on a differentiable manifold M. § For M R 1 and Gu (e 2 /2)u" + fu' with e( # 0) and f belonging to C 1 {R1), the associated diffusion is the (nonanticipating) solution of the integral equation =
x(t)
=
x(O) +
t
t
e (x) db +
J/(x) ds t
x
=
(t � O) . � "
Bernstein made an earlier attempt in this direction. tt Gihman [1] carried out Bernstein's program independently of Ito. t See Hida [1 ]. This admirable account of white noise integrals, filtering, prediction, Hardy functions, etc. encouraged me to leave that whole subject out of this book. � See Ito [7] . § See Ito [2, 3, 7, 8]. � See Ito [2, 6]. tt See Bernstein [3] ; see also [2].
1.1
3
GAUSSIAN FAMILIES
The purpose of this little book is to explain Ito's ideas in a concise but (hopefully) readable way. The principal topics are listed in the table of contents. A novel point is the use of the exponential martingale.
to obtain the powerful bound
This bound is used continually below and leads to best possible estimates in my experience, though often it is not a simple task to prove them so. Another novel point (for probabilists) will be the use of Weyl's lemma to check the smoothness of solutions of parabolic equations such as oufot G * u. =
1.1
GAUSSIAN FAMILIES
Consider a field B of events A, B, etc. with probabilities P(A) attached. A class of functions f measurable over B is a Gaussian family if, for each choice of d � 1, 0 # y (y 1 , . , yd) E Rd, and f (/1 , , fd), the form y · f y 1 /1 + · · · + yd fd has a nonsingular Gaussian distri bution : 22 exp / Q)dc ( c . (Q > 0), P[a � y f < b] (2:n:Q)l/2 =
=
. .
•
.
.
=
=
or, what is the same, if
fb
a
E[exp (J - 1 y f )] ·
Q
=
=
2 e-Q/ • t
2 E[(y f) ] is a nonsingular quadratic form in y E R d, and the ·
t E(f) is the expectation based upon P(B ) .
1
4 density function p transform: p
=
=
BROWNIAN MOTION
p(x) (x
( 2n)-d
I
Rd) of f can be expressed as a Fourier
e
exp ( -J - 1 y x) e- Q/2 dy. ·
Rd
Q can be brought into diagonal form Q' = o - 1 Qo by a rotation o of Rd, and since the Jacobian of o is simply ldet o l = + 1 , p can be evalua
ted as
p
=
=
(2n)-d
I
exp c-J=i y. ox) e- Q'/2 dy
Rd
( 2 n) - d12( det Q) - 1 12 exp ( - Q - 1/2) ,
Q- 1 being the inverse quadratic form applied to x E Rd; especially, the distribution is completely specified by the inner products E(/1/ ), etc. This 2
fact will be used without comment below. Because of the above, p splits into factors p 1 p 2 under a perpendicular splitting R1 EB R of Rd if and 2 only if Q splits into a sum Q1 EB Q under the dual splitting, i.e. , 2 statistical independence is the same as being perpendicular relative to the inner product E(/1 ! ). 2
Pro b l e m 1
Check the bounds (a + ( 1/a)) - 1 exp ( - a2/2) <
I
00
a
exp ( - b 2/2) db < a - 1 exp ( - a2/2) .
So l ution
I
00
a
exp ( - b 2/2)
00
a
=
I
b exp ( - b 2/2)- = a - 1 exp ( - a2J2) a
00
a
( 1 + b -2) exp ( - b 2/2)
< ( 1 + a- 2)
I
00
a
exp ( - b 2/2) .
1 .2 1.2
5
CONSTRUCTION OF THE BROWNIAN MOTION
CONSTRUCTION OF THE BROWNIAN MOTION
Consider the space of continuous paths t--+ b(t) E R 1 with b(O) and impose the probabilities:
P
=
0
[Dn (a k :::::; b( t k) < bk) ] ( (a 2 ("exp ( - c 112 /22 t 1 ) exp [ - (c2 - c 1 ) 2/2(1t22 - t 1)] [2n(t 2 - t 1 ]) 1 a1 2 a" (2n tt) 1 =
'
•
•
•
where
n � 1. Because exp [ - ( b - a ) 2 /2 t] (2n t) 1 1 2
=
J
Rl
exp [ - (b - c ) 2 /2 (t - s)] [2n(t - s) ]1 1 2
exp [ - ( c - a) 2 /2s] d X C 2 1 1 (2ns)
( t > s),
this is a permissible definition. The family [b(t): t > 0] is a Gaussian family with
Wiener [1 , 2] proved that P(B ) can be extended so as to be completely additive on the smallest field B 00 including all the events B = (a � b(t) < b) (a < b, t � 0). The space of continuous paths with these extended probabilities imposed is the so-called Brownian motion . Levy's [2] elegant proof of this fact, as simplified by Ciesielski [ 1 ] , will now be explained. t Usage:
a 1\ b
means the smaller of a and
b.
1
6
BROWNIAN MOTION
Ciesielski's nice idea is to use the Haar functions : + (n-1)/2 2
(k- )2-n � t < k2-n 1 k2-n � t < (k + 1)2-n
)/2 ( Jk 2-n{ t) = - 2 n-1
otherwise
0
odd k < 2n, and 0 � t � 1, augmented by f0(t) = 1 (0 � t � 1). These functions provide a perpendicular basis of L2[0, 1]; in fact, defin ed for
n
�
1,
according as i2-m = j2-n or not, and if f E L2[0, 1] is perpendicular to them all, then f
k2-n
(k-1)2-n j
J:
J:
is independent of k � 2n, so that f= (b - a) f= 0 for each choice of 0 �a= i2-n < b =j2-n � 1 and n � 1. f= 0 is immediate from this. Now compute the (formal) Haar coefficients of the fictitious white noise h·t: 1
g o = J/o b• = b(l)
I
1
I' b • = - 2(n-1)/2 gk2-n = 0Jk2-n
Note that [gk2-n: odd k < 2n, n � 1], augmented by g0, is a Gaussian family with and note also that these coefficients are independent since
t The stands for differentiation with respect to time. •
1 .2
CONSTRUCTION OF THE BROWNIAN MOTION
7
Levy's idea is to use the formal Haar series : 00
b·=go fo + L n==
1
L Bk2 -"fk 2 -" kodd< 2 "
to define the Brownian motion for t � 1 . Consider, for this purpose, a Gaussian family [gk 2 -n : k = 0 k odd < 2", n � 1] with the properties listed above and define 00
t
I Bk2 -n f fk2 -n . n==l kodd <2 "
b(t) = 9o f fo + L t
0
or
It is to be proved that this sum converges uniformly for 0 � t � 1 to a continuous path with the correct distribution. 2-(n+ ll/2
FIG. 1.
(k-1)2-n
·
k2-n
(k+l)2-n
J�
Because the so-called Schauder functions fk 2 -n are little tents of height 2-
= 2-(n+l)/2 X max 1Bk 2 -nl, kodd< 2 " t 11/lloo
=max,�
1l/(t)j.
1
8
BROWNIAN MOTION
permitting us to estimate as follows :
P[en > 0(2-n lg 2 )1 12 ] "
[
P max l g k2 -nl > OJ2n lg 2 k < 2" 2 {2 ) exp c ( · t dc �2 2" J 2 6(2 n lg 2)11 J2n < constant n - 1 12 2" exp ( - 02n lg 2)t
=
]
odd
x
=
But if 0 > 1 ,
constant
n -1122n(t-o2).
x
is the general term of a convergent sum, so by the first Borel-Cantelli lemma, P[en � 0(2-n lg 2")112, n j oo] 1, =
proving the desired convergence. As to the distribution, [b(t) : � t � 0 1] is a Gaussian family, so it suffices to check, that
E[b(t 1 )?(t2 )]
tt o t2 o tt t2 Jo f t f + .I1 Jo fk2-n Jo fk2-n 00
=
=
1 J0 jlj2
=
t1
A t
2,
using Parseval's relation for the Haar functions, applied to the indicator functions j1 and j2 of s � t1 and s � t2• Ciesielski-Levy's construction is now extended from 0 � t � 1 to t � 0 by piecing together independent copies bn(t): t �1, n � 0 of the defining Haar series according to the recipe :
b(t) b 0 (t) b 0 (1) + b 1 (t - 1) b 0 (1) + b 1 (1) + b 2 (t - 2) =
= =
etc.
(0 �t < 1) (1 �t < 2) (2 �t < 3)
Because£ [b(t 1 )b(t 2 )] t1 " t2 still holds, this e xtended Brownian motion has the correct distribution. =
t See Problem 1 , Section 1 . 1 .
1 .3 1.3
SIMPLEST PROPERTIES
9
SIMPLEST PROPERTIES OF THE BROWNIAN MOTION
Using the formula E [b(t1 )b(t 2 )] = t1 "t 2 , the reader can easily check the following facts :
(1) b(t + s) - b(s) : t � 0 is a Brownian motion, independent of b(t) : t � s, for any s � 0. This is the so-called differential property of the Brownian motion. (2) cb(t/c2) : t � 0 is a Brownian motion for any constant c > 0. This is the so-called Brownian scaling. (3) tb(1/t): t > 0 is a Brownian motion. This leads at once from P[b(O+) = OJ = 1 to the strong law of large numbers
1 b(t) o] lim t [ p =
=
1.
ttoo
(4)
- b(t) : t � 0 is a Brownian motion.
Dvoretsky et al. [1] found a very simple proof of Wiener's result that the Brownian path is nowhere differentiable. Suppose b(t) : t � 1 is differentiable at some point 0 � s � 1 . Then lb(t) - b(s) l � l(t - s) for t ! s and some integral I � 1 . But this means that lb(jfn) - b((j - 1)/n) I < 11/n
fori = [ns] + 1 , i < j � i + 3, and sufficiently large n, so that the event under consideration is included in
( ( j 71) (j <- , B= u u n 1 u n . b b l�1 m�1 n �m O
and, by the Brownian scaling, it is easy to see that P(B ) = 0, as follows :
p [.0mO
j oo
=
j oo
0.
x
10
1
BROWNIAN MOTION
An important feature of the Brownian motion is the fact that it begins afresh at stopping times, as discovered by Dynkin [3] and Hunt [1]. Define Bt to be the smallest subfield of the universal field Boo , including all the events B= (a � b(s) < b) (a < b, s � t), and call a functional 0 � t � oo of the Brownian path a stopping time if ( t < t) E Bt for each t � 0. A constant time t = t is a stopping time, as is a passage time such as t = min (t : b(t) I)t; but a last-leaving time such as max (t � 1: b(t) = 0) is not. Bt + is now defined as the field of events B E Boo such that B n ( t < t) E Bt for each t � 0 . Bt + = n s> Bs if t = t, as the notation suggests. Roughly speaking, Bt + is the field of b(s) : s � t +. Also, t itself is measurable over Bt + , and with the understanding that b( oo) = oo, so is b(t). t Dynkin-Hunt's statement is as follows : if t is a stopping time, then, conditional on t < oo , b + (t) = b(t + t) - b( t) : t � 0 is a Brownian motion, independent ofb(t) : t � t+ , i. e. , independent ofBt + . This is just a statement of the differential property of Brownian motion [see ( 1) at the beginning of this section ] in the special case of a constant stopping time. =
t
Proof
Define tn= k2- n if (k- 1) 2- n � t < k2- n, take B E Bt + , d � 1, a bounded function f E C(Rd), 0 < t 1 < < t d , and put ·
·
·
e( t) = f [b( t1 + t) - b ( t), .. . , b(td + t) - b ( t)] .
Because t n t t as n i oo , e(t n) tends to e(t ) as n j oo if t < oo, and since B (\(in= k2- n) = B (\ ( ( k - 1 ) 2- n � i < k2- n) E Bk 2 -n ,
an application of (1) gives E[B
n
(t <
00
)
,
e(t) ]= lim E[B n
t oo
= lim
00
n
(t
n
<
oo ),
e(t ) J n
I E[B (\ (t n= k2 - n),
nf oo k =l
e(k2- n)]
n t(t�t)=nm�tUk2-n�t(b(k2- )>1-I/m)EBt and so (t
1.4
11
A MARTINGALE INEQUALITY
00
=
=
lim L P[B k
njoo =l
n
(tn
=
k2 - n)]E [e(O)]
P[B n (t < oo )]E[e(O)].
The proof can now be completed by the reader. A useful extension is as follows. Consider fields A t :::::> Bt (t � 0) such that A t is independent of the field Bt + of b + (s) = b(s + t) - b(t) : s � 0. t � oo is a stopping time if (t < t) E A t for each t � 0, A t + is defined as the field of events A E A 00 such that A n (t < t) E A t for t � 0, and the result is that, conditional on t < oo , b + (t) = b(t + t) b(t) : t � 0 is a Brownian motion independent of b(t) : t � t + , in fact, it is independent of the whole field A t + The proof is identical. -
.
Problem 1
Use Dynkin-Hunt's result to prove Blumenthal's 01 lawt : P(B) or 1 if B E B0 + .
=
0
Sol ution
b + = b if t = 0. B is then measurable both over b + and B0 + , and as such, it is independent of itself.
1 .4
A MARTINGALE INEQUALITY
A chain 3 = [3k: k � n] is a (sub) martingale relative to the increasing fields Zk (k � n) if (a) 3k is measurable over Zk, (b) E(l3kl) < oo , and (c) E(3k I Zk_1) ( � ) = 3k-l for each k � n. Doob's submartingale in equality t is an extension of C ebysev's and Kolmogorov's inequalities : it states that, for a submartin gale, (1 > 0). § t See Blumenthal [1 ]. t See Doob [1 ]. § Usage : x+ is the bigger of x and 0.
1
12
BROWNIAN MOTION
Proof
The event B that 3k � I for some k � lapping events
n
is the sum of the nonover
( k � n) , so E(3 n + ) � E[3 n + , B] = L E[E(3 n + I Zk ), Bk] k�n � L E(3k ' Bk ) � L lP( Bk ) = lP( B ). k�n k�n
Doob's inequality is easily extended to submartingales with con tinuous sample paths. A process 3 = [3t: t � 1] is a (sub)martingale relative to the increasing fields Zt (t � 1) if the obvious analogs of (a), (b), and (c) hold. Under the extra condition of continuous sample paths, Doob's inequality for chains supplies us with the bound :
[
]
(l > 0). P max 3t � z � z -1E(31 + ) t� 1 Note that if 3 is a martingale and if £(32) < oo, then £(32 I Z ) � £(31 Z)2 = 32, so that 32 is a submartingale. THE LAW 01:? THE ITERATED LOGARITHM
l.S
Hincin's law of the iterated logarithmt : �
[. P hm tt0
b(t)
(2t lg2 1ft)112
] =1 =1t
will now be proved using the martingale inequality of Section 1 .4 and the fact that 3(t) = exp [ab(t) - a 2 t/2] is a martingale for each choice of a E R1. This method is used over and over below, so the reader should understand this simplest case completely before proceeding. Because and tb(l /t ) are likewise Brownian motions, Hincin's law implies
-b
t See Hincin [1] . t lg2 stands for lg(lg).
1. 5
13
THE LAW OF THE ITERATED LOGARITHM
p
[hm.
]
b(t) = - 1 =1 1 2 1 1 (2 ) g t �0 t 2 1 I
and p
[-hm. tt
00
t
b(t) (2t 1g2 t )1/2
=
]1 = 1 .
Proof of Iimr + 0 b(t)/(2t lg2 1 /t)112 � 1
3(t) =exp [ etb(t)- et2t/2] (t � 0) is a martingale over the Brownian fields Br (t � 0) . To begin with, E [3(t)]
=
J exp[ac- et2t/2] (2nt) -112 exp(-c2f2t) de J(2nt)-1'2 exp[-(c- ett)2/2t] de
=
=1.
Now if t > s and 3 + = exp[et[b(t) - b(s)] - et2(t- s)/2] , then 3(1) = 3(s)3 + , and by the first step,
E [3(t) I BsJ =3(s)E [3 + I BsJ =3(s)E [3(t - s)] = 3(s). This completes the proof that 3 is a martingale and permits the applica tion of the martingale inequality of Section 1.4 to prove
[
] [
]
P max [b(s) - ets/2] > P = P max 3(s) > eaP � e-ap E [3(t)] e-ap. s�t s�t Define h(t) =(2t lg 2 1/t)112 and choose 0 < () < 1, t =en-t , 0 < fJ < 1, et (1 + fJ)e-nh(On), and p =h(On)/2, so that ap =(1 + fJ) lg 2 en and e-ap =constant x n -1 -o is the general term of a convergent sum. An application of the bound just proved gives =
=
]
[
P max [b(s) - ets/2] > P � constant s�t
x
n -1 -0 ,
so that, by the first Borel-Cantelli lemma,
[
P max [b(s)- ets/2] � p, n i s�t
oo] = 1 ,
1
14
BROWNIAN MOTION I,
especially, for n i 00 and (}" < t � enrxfJ"-1 1 + () 1 () 1 + () 1 b(t) � .���1b(s) � 2 + P = 2() + 2 h( ) < 2() + 2 h(t) , since h E j for small t. Making (} i 1 and () ! 0 completes the proof of limr-t, oh/h � 1 . Proof of lim t -1- 0 b(t ) /(2 t lg 2 1/t)1 1 2 � 1
[
]
n
]
[
Define independent events (0 < (} < 1 , n By Problem 1 , Section 1 . 1 ,
J
P( B ) = 1 ,JOlg2 o- ")t/2 ( 2 0 1n
:t
exp (- c /2)
;?:;
1) .
de
( 2n) I n - r< 2 -JO+ O) / (1 - O)J �constant ( lg n )1 / 2 is the general term of a divergent sum [1 - 2Je + (} < 1 - fJ], and an x
1-
application of the second Borel-Cantelli lemma permits us to conclude that b(fJ") ;;::; ( 1 - j{J)h(fJ") + b(fJ"+ 1 ) i.o. , as n i oo. But also, b(fJ"+ 1 ) < 2h((}" + 1) as n j oo by the first part of the proof, so
b(fJ") > (1 - J O)h(O") - 2h(fJ"+ 1 ) > [ 1 - JO - 3J fJ]h(O"), i.o., as n i oo; i.e. , lim t -1- 0 bfh � 1 - 4JfJ, and to complete the proof,
it suffices to make (} ! 0. 1 .6
LEVY 'S MO DULUS
Levy proved that h(t) =(2t lg 1/t)1 1 2 is the exact modulus of continuity
of the
Brownian sample path :
This will now be verified using Levy ' s [1] own elegant method.
LEVY ' S MODULUS
1.6
15
Proof of lim �1
Define h(t)
[
=
(2t lg 1/t)112 as above and take 0 < fJ < 1. Then
P max [b(k2-n )- b((k- 1)2-n )] � (1 - fJ)h(2-n ) k � 2n
=
[
exp (- c2 /2) de 112 1 ( 1 - �) ( 2 lg 2n)l/2 (2n )
J
]
2n
]
= ( 1 - /) 2 n < exp(-2n/).
By Problem 1 of Section 1.1,
exp (-c2 /2) de 2 n 1 = 2n 2 1 ( 1 - �)( 2 Ig 2 n )l/2 (2n) 1
J
x
> constant for
n
2n
Jn exp [-(1- fJ) 2 lg 2n ] > 2n �
j oo. An application of the first Borel-Cantelli lemma now gives
]
[
P lim max [b(k2-n )- b((k- 1)2-n )] /h(2-n ) �1 n foo k � 2 n completing the first half of the proof. Proof of lim
=
1,
�1
Given 0 < fJ < 1 and e > [(1 + fJ)/(1 - fJ)] - 1, p
max
0 < k =j- i� 2 n o
O � i<j� 2n
�
" �
o < k � 2n o
exp (- c2 /2) dc 2J 2 1 ( 1 + e)( 2 Ig 11k2 - n) t I 2 (21t ) I
O � i < j � 2n
< const ant
x
1�
J
+ 2n (l H)2-n(l-6)(l •)2
is the general term of a convergent sum [(1 - fJ)(1 + e ) 2 > 1 + fJ], so the first Borel-Cantelli lemma implies lb(j2-n )- b(i2-n )l < (1 + e)h(k2-n ) ( 0 � i < j � 2n ' k
=
j - i � 2n �' n i 00).
1
16
BROWNIAN MOTION
Now pick 0 � 1 1 < t 2 � 1 so close together that t = t 2 - t1 < 2-m( 1 -�> with m so large that the last estimate holds for all n �m. Pick n so that 2-(n + l )(l -� ) � t < 2-n(l-�>, and expand t1 and t 2 as follows : Pt
P2
(n < Pt < P2 <etc.) t 1 - ·2 - n - 2 - - 2 - - etc. t2 = j 2- n + 2 - q1 + 2- q2 + etc. (n < q 1 < q 2 < etc.), verifying that t 1 � i2- n < j2 - n � t 2 and 0 < k = j- i � t2n < 2n�. Because b(t) is continuous, _
z
l b(t2 ) - b(t l ) l � 'l b(i2- n) - b(tl) l + l b(j2 - n) - b(i2- n) l + lb(t2 ) - b(j2 n)l � L (1 + e) h (2- p) + (1 + e) h ( k2- n ) + L (1 + e) h(2- q) . p>n q>n But also, for n j oo , L h(2- P) � constant h(2- n) < eh[2- (n + l ) (l - � )] -
x
p>n
and since h E j for small t, Because e> 0 can be selected at pleasure by choosing fJ> 0 sufficiently small, P[lim � 1] = 1 , and the proof is complete. Problem 1
Give a proof of Kolmogorov's lemmat : a process x E R 1 3(x) : which satisfies E[ l3(x) - 3(Y) I rl] � constant x l x - yiP for some a> 0 and P> 1 has continuous sample paths. More precisely, if 3 * (x) = lim 3(y ) as --+
y = k2- n ! x, then
P [l3 * ( x) - 3 * ( y) l < l x - y jl'
locally] = 1
for any
and P [3 * ( x) = 3(x)] = 1 t See Slutsky [1].
for any x E R1 •
y
< (p - 1 ) fa,
1 .7
17
SEVERAL-DIMENSIONAL BROWNIAN MOTION
Use the proof of Levy's modulus as a m odel, but notice that the present problem is not so delicate. Check that Kolmogorov's lemma also holds for processes x E Rn 3(x) for n � 2. This will be used in Chapter 3. �
Solution for
n =
1
Given y <([3 - 1 )/a and <5 > 0 so small that ( 1 - <5)(/3 - ay) > 1 , P[l3(j 2-n) - 3(i2-n)l > (k2- n)Y
for some 0 � i2 -n <j 2 - n with k
�
L
O � i2-"<j2-"� k < 2"0 1
� constant
x
� COnstant
X
=
j
-
i < 2n«5]
�
1
( k2- n) - aY£ [ 13(j 2- n) - 3( i2 - n) ja]
L ( k2 - n)p - ay
2n[ 1 - ( 1 - «5)(p - ay)]
is the general term of a convergent sum. The rest is plain sailing over the course laid out for the proof of Levy's modulus. 1 .7
SEVERAL-DIMENSIONAL BROWNIAN MOTION
A d-dimensional Brownian motion is just the joint motion b(t) [b 1 ( t ) . .., b d ( t )] (t � 0) of dindependent ! -dimensional Brownian particles. Boo is now the obvious product field, P(B ) is the correspond ing product distribution on Boo, and t is a stopping time if (t < t) is measurable over the field B1 of b(s) : s � t for each t � 0. As before, the Brownian traveler begins afresh at stopping times, i.e. , if t is a stopping + time, then, conditional on t < oo , b ( t ) = b(t + t) - b(t) : t � 0 is a d-dimensional Brownian motion, independent of the field Bt+ of b(t) : t � t + , especially , for t < oo , =
,
P[b( t + t) E db I B1+]
=
(2 n t) - d 12 exp ( - l b - al2/2 t) db t
depends upon t > 0 and a = b(t) alone. A projection of the d-dimensional Brownian motion onto a lower dimensional subspace is likewise a Brownian motion. By projecting onto
18
1
BROWNIAN MOTION
'V
sufficiently many !-dimensional su bspaces, the laws of Hincin and Levy :
follow from Sections 1 .5 and 1 .6. Brownian motion is invariant under a d-dimensional rotation o, i.e. , b * = ob is likewise a d-dimensional Brownian motion. Because of this, the radial motion r = lb I = (b 1 2 + + bd2) 1 1 2 begins afresh at its stopping times. In fact, a stopping time t of r is also a Brownian stopping time, so for t < oo , a = b( t) , and t > 0, ·
P [r (t + t) <
R
I Bt+J
=
(2n t) - d1 2
·
·
J
l b i< R
exp ( - l b - a l 2 /2t) db ,
and since this expression is insensitive to rotations of a , it must be a function of Ia I = r(t) alone. Because the field Rt+ of r(t) : t � t + is part of B t + , the proof is complete. + o 2 fobd 2 ) is associated with the d-dimen A/2 = i(o 2/ob 1 2 + sional Brownian motion via the double role of (2 nt) - d 1 2 exp ( - l b - a l 2 /2t) ·
·
·
as (a) the Green function (elementary solution) of the heat flow problem oufot = Au/2 and (b) the transition /unction of the Brownian motion. The radial part of A/2 : +; _ 1 o 2 d - 1 A 2 -2 2 + r or or
(-
a)
is associated with radial motion r = lb I in just the same way, so it is apt to call r the Bessel process. 1 Use Levy ' s modulus for the !-dimensional Brownian motion to check that P[r > 0, t =1= 0] = 1 for d � 3 (see Problem 7, Section 2.9, for the proof in case d = 2). Problem
1.7
SEVERAL-DIMENSIONAL BROWNIAN MOTION
19
Solution
Because of Levy ' s modulus of continuity for the Brownian path, the existence of a root of r( t) = 0 between 0 < f) < 1 and 1 implies the occurrence of the event for some k between ()2" an d 2" for all sufficiently large
n.
But, for a !-dimensional Brownian motion,
P[ lb( k2-") l < (3 2-n lg 2") 1 1 2 ] � constant 2-n/ 2 Jn if k2-n �f) > 0, and so P(Bn) � constant 2"[2-"1 2 Jn]d x
·
x
is the general term of a convergent sum if d � 3. An application of the first Borel-Cantelli lemma completes the proof. Problem 2
P[r = 0 i.o., t ! OJ = 1 for d = 1 .
Solution
Use the law of the iterated logarithm of Section 1 . 5 in the form : 1.
1m
t �0
+ b(t) (2t lg 2 1/t) 1 1 2
==
1.
2
2.1
STOCHASTIC INTEGRALS AND DIFFERENTIALS
WIENER 'S DEFINITION OF THE STOCHASTIC INTEGRAL
Because
ln = L l b(k2- n ) - b((k - 1)2- n )l k�2n
increases as n i oo , while
! 0,
the length /00 of the Brownian path b(t) : t � 1 is infinite, so that it is impossible to define the integral J: e db by any of the customary recipes. t Use the estimate e-x
< 1
-
x + x2 /2 for x > 0. 20
2.2
21
ITO ' S DEFINITION
Paley et a/. [ 1 ] overcame this obstacle by defining
1
f0 e(t) db -J0 e'bdt =
1
for (sure) functions e = e(t) (t � 1) of class C 1 [0, 1 ] with e(1) then making use of the isometry
[( 1 1 1 ) 2] fo Jot1 E Jo e db
=
0, and
fo
1 2 e dt
t2 e'(t 1 )e'(t2 ) dt 1 dt2 = to extend the integral to all (sure) functions e E L 2 [0, 1 ] . t Ito [ 1 ] extended =
1\
this integral to a wide class of Brownian functionals e = e(t) depending upon the path t � b(t) in a nonanticipating way , as will now be explained. "
2.2
ITO 'S DEFINITION OF THE STOCHASTIC INTEGRAL
Consider the field C of Borel subsets of [0, oo) and an increasing family of fields At => Bt ( t � 0) such that As is independent of the field Bs + of b + (t) = b (t + s)- b(s) : t � 0. A function e = e(t) depending upon t � 0 and the Brownian path t � b(t), plus possible extra stochas tic coordinates measurable over A00 , is a nonanticipating Brownian functional if (1) e is measurable over C x A00 , and (2) e(t) is measurable. ?ver A, for any t � 0. The program is to define e db, simultaneously for all t � 0, for almost every Brownian path , under the condition
P
[{e2 ds < oo , t � 0J
J:
=
1.
Problem 1 , Section 2.5, shows that this condition cannot be dispensed with. To make things clear, it will be enough to discuss J� e db (t � 1) under the condition
[ P ( e 2 dt < oo J
=
1.
The estimates are based upon the martingale trick of Section 1 . 5. The discussion differs from that of Ito [ 1 ] in this point only. t Problem 1, Section 2.3, contains additional information about this isometry.
22
2
STOCHASTIC INTEGRALS AND DIFFERENTIALS
Step 1
A nonantictpating Brownian functional e is called simple if e( t ) = e((k - 1)2 - n) for (k - 1)2- n � t < k2 - n (k � 2n) and some n � 1 . Given such e, define
t f e db = L e((k - 1)2-m)[b(k2-m) - b((k - 1)2-m)] 0
k�l
+ e(l2-m)[b(t) - b(l2 - m)] for t � 1 , m � n , and I = [2mtJ,t and note the following points : (a) the integral is independent of m � n , (b) (e1 + e 2 ) db = e 1 db + e 2 db, (c) k e db = k db for any constant k, and (d) the integral is a continuous function of t � 1.
I� I�
I>
I�
I�
Step 2
I�
To define e db (t � 1) for the general nonanticipating functional, a powerful bound for the integral of a simple functional is needed :
Proof
f�
I�
]
For simple e, 3(t) = exp [ e db - t e 2 ds is a (continuous) martingale over the fields At (t � 1), and E[3(1) ] = 1 . In fact, if e is constant ( =c) for s � t, then cis measurable over As and so is indepen dent of b(t) - b(s), with the result that
E [3(t) I As] = 3(s) E [exp(c[b(t) - b(s)] - c 2 (t - s)/2) I As] = 3(s), as in Section 1 . 5. A simple induction completes the proof of this point, and the stated bound follows upon replacing e by rxe and using the martingale inequality of Section 1 .4 : t [x] means the biggest integer <x.
2.2
P[ Step 3
ITO ' S DEFINITION
23
P]
1 max e db - J e 2 ds > J 2 0 t� 1 0 = P max 3(t) > erxP � e-rxP£ [3(1) ] = e - rxp . t� 1 t
r:t
[
J
P [max1,;; 1 s; en db < 0(2 - n + 1 lg n)1 ' 2 , n j
oo
]=1
for simple en with P[f: e/ dt � rn, n j ] = 1 , and any 0 oo
>
1.
Proof
Choose (2"+ 1 lg n) 1 1 2 and f3 0(2_"_1 lg n) 1 1 2 in the bound of Step 2. e - rxP = n - 0 is the general term of a convergent sum, so the first Borel-Cantelli lemma justifies the estimate 1 1 1 0) - n + 1 n 1 1 2 n / f3 ( lg P [max ) , j ] = 1. en db� f e ds � (2 f 2 2 2 t� 1 Now repeat with - en in place of en. r:t
0
=
=
oc
0
+
+
oo
Step 4
Given a nonanticipating Brownian functional e with J: e 2 dt < it is possible to find simple nonanticipating functionals en (n � 1) so that oo,
P [J: (e - en) 2 dt � rn, n j Proof
oo
]
=
1.
Define e=O (t�O), e' = 21f,� 2 _,e ds, and e" = e'(Tm[2mt]). Because J: (e - e") 2 dt tends to 0 as i and I j (in that order), it is possible for each n � 1 , to pick I and so as to make P [J: (e - e") 2 dt Tn] � Tn. en= e" is nonanticipating and simple, and the desired estimate m
oo
oo
m
>
P [J: (e - en) 2 dt � rn, n j oo] = 1
is immediate from the first Borel-Cantelli lemma.
2
24
STOCHASTIC INTEGRALS AND DIFFERENTIALS
Step 5
J; e db (t � I) can now be defined. Choose simple en (n � I) so that n ] I as in Step 4. According to Step 3, P u: (e - en) 2 dt � 2 , n j max1.;1 J� (en - en _ 1 ) db tends to 0 geometrically fast as n i so it is permissible to put J� e db = limntoo J� en db (t � I). The estimate of oo
-
=
oo,
Step 3 shows that the integral does not depend on the particular choice of simple approximations en (n � 1). Because the convergence is uniform, J� e db (t � I) is a continuous function, especially, it is defined simultaneously for all t � 1 , for almost every Brownian path. Problem
1
Prove that under the condition P [f0"' e 2 dt < ] I, Jo"' e db can be defined in such a way as to make P [lim11"' J� e db Jo"' e db] I. oo
=
=
=
Solution
such that J0"' (en - e)2 dt � 2-n Choose simple en = 0 near t (n � 1). The estimates used above can easily be extended to show that max1;;,o Jo"' (en - en _ 1) db tends to 0 geometrically fast as n i oo. Because J� en db is a continuous function of t � so is J� e db. = oo,
oo,
2.3 SIMPLEST PROPERTIES OF THE STOCHASTIC INTEGRAL
Ito's integral is now defined, and the next job is to note some of its simplest properties for future use; e is a nonanticipating Brownian functional with P [f� e 2 dt < t � o] I. oo,
=
J� (e1 + e 2 ) db J� e1 db + J� e 2 db. (2) J� ke db k J� e db for any constant k. (3) J� e db is a continuous function of t < (4) J� e db Jo "' ef db if t < is a Brownian stopping time and
( I)
=
=
oo.
=
oo
is the (nonanticipating) indicator function of (t � t).
2.3
25
SIMPLEST PROPERTIES
00 00 2 2 2 (5) E [(f0 e dbrJ � ll e ll = E [f0 e dt ] ( � oo) if P [f0 e dt < oo ] = I; if H e ll .::: oo , then E[ Uooo e db rl = ll e ll 2 and E[fooo e db ] = 0. (6) 3(1) = exp [J; e db - ! s; e 2 ds] is a supermartingale, i. e., -3 is 00
a submartingale over the fields At ( t � 0), P
[
£(3) � 1 , t and
[maX1;;,0 J0 e db - � ( e2 ds f3J � e -aP. 1
>
J;
(7) P max,,0 en db < 8(2 - n + 1 lg n) 1 1 2 , n j oo ] = Ifor any 8 00 P f e/ dt � rn, n j oo ] = 1 . 0 e db + ! e 2 dt )] = I if ( 8) exp J - I
[
E[ (
>
I if
Jooo E [exp(t Jooo e 2 dt)] < oo .
Jooo
The proofs of ( 1 ), (2), and (3) are trivial. Proof of (4 )
Clearly, s; e db = fooo ef db is trivial if e is simple and 0 far out; and if the general e is approximated by such simple en (n � 1) as in Problem I, Section 2.2, then max,,0 J; (e - en) db will tend to 0 as n i oo while Jooo e n f db will tend to Jooo ef db , since Jooo (e - en )2f 2 � 2 -n and (7) is applicable. =
Proof of (5)
E[(fooo e db r] = ll ell 2 if e is simple and
far out, as a direct computation shows. As to the general e, it is possible to find simple en = 0 far out, so closely approximating the nonanticipating functional e X the indicator function of s; e 2 � n that P
=
0
[ J0 (en - e)2 dt � 2 - n, n j 00] = 1 00
t £(3) < 1 is possible, as will be verified in Section 3. 7.
2
26
and limnt
00
II en I
STOCHASTIC INTEGRALS AND DIFFERENTIALS
= II e 11. For this choice of en ,
2 =llel l 2 , e =lim l nl l ntoo and if l el < it is possible to make limntoo lien-el =0, so that 2 ' lim l i en - e ll 2 =0. E [ ({ (e. - e) db) ] �lim ntoo ntoo The reader will easily supply the rest of the proof. oo,
0
Proof of (6)
Approximate e by simple en (n � 1) as in Problem 1 Section 2.2, and use Step 2 of Section 2.2. Proof of (7)
Use (6) as in Step 3 of Section 2.2. Proof of (8)
Prove this (a) for simple e vanishing far out, (b) for the producten of a general e and the indicator function of J� e 2 �n, and (c) for the general oo oo 2 r r e, using�the domination ) o en � ) o e2• A
Pro b l e m 1
Deduce from (5) the result of Akutowicz-Wiener [1] that an orthog onal transformation o of L2 [0, ) induces a measure-preserving auto morphism of the space of Brownian paths t b(t) via the mapping b ( t) Joaa oe 1 db (t � 0), e 1 =e 1 (s) being the indicator function of s � t. oo
�
�
Sol ution
E
[J
00
0
oe. db J
00
0
] oe1 db =J
00
0
oe.oe1 =
J
00
0
eset =sA
t.
2.3
27
SIMPLEST PROPERTIES
Problem 2
Use the fact that 3(t) =exp [yb(t) - y 2 t/2] is a martingale to prove the formulas: (a) E [e - yt] =(cosh (2y) 1 1 2a) - 1 for t =min (t : lbl = a) (b) E[e-yt] =exp ( - (2y) 1 1 2a) for t= min (t : b = a) for y and a 0. Deduce from (b) the distributions: (c) P[t edt] = (2nt3) - 1 1 2 a exp ( - a 2f2t) dt (d) P[b(t) E dx, maxs� b(s) E dy] =(2fnt3) 1 1 2 (2y - x) exp [ - (2y - x) 2/2t ] dx dy (0 � y > x). >
t
x
Sol ution
as the smaller of t � 0 and tn + =min (k2 -n > t). b(tn) is the integral J� e db of the (simple nonanticipating) indicator function function e of (s � tn + ) . E[exp (yb(tn) - y 2tn/2) ] = 1 follows, and since b(tn) � maxs� t b(s), the martingale bound tn is defined
P [max b(s) > c] � P [max b(s) - s > P] 2 � � s t
s t
r:t
< e -rxP
=exp (- c 2f2t) (a = eft, P = c/2) permits us to make n i oo under the expectation sign, obtaining (! 00 =t t). Because b(t 00) � a, 1 � eYaE[exp ( - yt/2)] , as follows upon making t j oo, and P(t < oo ) = 1 is deduced by making y ! 0. Now it is permis sible to make t i oo under the expectation sign in (e), and (a) and (b) follow upon substituting (2y) 1 1 2 for y and noting that P[b(t) = - a] = P[b(t) = :+-a] = 1 in the first case, and P [b(t) = a] = 1 in the second. (c) follows upon inverting the transform (b), and (d) is deduced from (c) and the elementary formula 1 P [b(t) dx , max b(s) > y] =J P[t E ds]P[b(t- s) + y E dx] ( x < y), � 1\
e
s t
0
in which tis now min (t : b =y).
28
2.4
2
STOCHASTIC INTEGRALS AND DIFFERENTIALS
COMPUTATION OF A STOCHASTIC INTEGRAL
At this point, it is instructive to compute a stochastic integral from scratch. The simplest interesting example is
Section 2.6 contains an explanation of the unexpected - t; the multiple integral
(db (t1 ) ( db (t2 ) ( • • •
"-
d b (tn)
is evaluated for n � 3 in Section 2.7. Define the simple nonanticipating functional en = b(2-"[2"t ] ). Because (e - en) 2 dt tends to 0 as n j oo for any t � 0, it is enough
s;
s;
en db = t(b 2 - t) . Besides, for A = b(k 2 - n) to prove that limnt - b((k - 1)2-"), I = [2"t ] , and n i oo , t
ao
[
]
2 J en db = 2 L b((k - 1)2 - ") A + 2b( l2- ")[b(t) - b(l 2- ")] 0
k�l
= L [b(k2- ") 2 - b((k - 1) 2- ") 2 ] - L A2 + o(l) k�l
k�l
= b(t) 2 - L A2 + o(l) , k�l
so it is actually enough to prove the following lemma, stated in a sharper form than is actually needed. Lemma
Define 3 n{t) = L A2 + [b(t) - b(l2-")] 2 - t k�l
for I = [2"t] and t � 1. Then
[
]
P max l3n(t) l < 2- "1 2 n, n i oo = 1. t� 1
2.5
29
A TIME SUBSTITUTION
Proof
3n(t ) (t � 1) is a continuous martingale over the Brownian fields Bt (t � 1), so 3n2 is a continuous submartihgale, and the submartingale inequality of Section 1 .4 supplies the bound
]
[
P max l3n(t)l > 2-"1 2 n t� 1 � 2"n-2 E[3n{l)2 ] n n = 22 "n -2 E[(b( 2- )2 - 2- )2 ] = constant
x
n-2 ,
using the Brownian scaling b(2 - ") --+ 2- n / 2 b(1) in the last step. But n - 2 is the general term of a convergent sum, so an application of the first Borel-Cantelli lemma completes the proof. Problem 1
The Brownian differentials under a stochastic integral should always stick out into the future. For instance, the backward integral:
J0 b db =lim L b( k2-")[b( k2-" ) - b(( k - l)T")] ntoo k�2 " 1
has the value }[b(1) 2 + 1] instead of �[b(1) 2 - 1 ] . Prove this. Problem 1 , Section 2.6, contains additional information on this backward integral. 2.5
A TIME SUBSTITUTION
Consider a stochastic integral t(t) = s; e db based upon a non
s;
anticipating Brownian functional e with t(t) = e 2 < co (t � 0) , let t - 1 be the left-continuous inverse function t- 1 (t) = min (s : t(s) = t) defined for t < t( oo ), and let us check that a = x(t- 1 ) is a Brownian motion for times t < t( oo ). Because � is constant if t is flat, this is the same as saying that x(t) = a(t) (t � 0) with a new Brownian motion a. t is called intrinsic time (clock) for x. Section 2.8 contains additional information about such time substitutions. Problem 1 , Section 2.9, can be used for an alternative proof.
30
2
Proof
Define t - 1
STOCHASTIC INTEGRALS AND DIFFERENTIALS
(t) = oo for t � t( oo) , let a(t) = x (t- 1 ) = x( oo) + c(t - t( oo))
(t
t( 00))
with an independent Brownian motion c, and, for n � 1 , 0 � It is A 1 < < and '}' = 1 , , E R", put Q = enough to prove that is a Brownian motion, and for this, it suffices to check
t
·
·
tn ,
·
a
(y
•
•
Yn)
•
L Yi Yi ti ti .
Integrate the extra Brownian motion c out of I = E (exp
[
= E e xp this gives
[
l = E exp A
Because
(j - 1 L yi a(tJ + Q /2)] J 1 j Yi L [ 0 (
t - l (t · )
yd L l j0 (
' e db +
t - l (t i )
t i � t( oo )
e db - t
(tj - t( 00)) + Q j2) ]
L yic(ti - t(oo))] + Q/2)] ; L YiY/ ti - t(oo))
t i , tJ � t( oo )
t-1( t ) is a stopping time,t this can be expressed as
t (t- 1(1) � s) = (t � t(s)) E As (s � 0).
2.5
A TIME SUBSTITUTION
31
with the nonanticipating Brownian functional f = L Y i the indicator of t � t - 1 ( ti) , and since x
e
follows from (8), Section 2.3, and the proof is finished. From the formula x(t ) = a(t) and the results of Sections 1.5 and 1.6, it is possible to read off the analogs of the strong laws of Hincin and
I= 1
Levy :
[. P hm
J
x( t)
t-1,0 ( 2t lg2
=1 =1 ! 2 1 1 1/ )
and p
in which A = [t1, ) and t(A) = JA with the understanding that 0/0 = 1 . Additional applications of time substitutions will be made below. 2 e ,
t2
Problem 1
Prove that if P [f� ds < oo , t < I ] = I and if P [f: dt = oo ] = I , then t [t P lim J db = - lim J db = oo = 1 . t t1 0 tt1 0 J This shows that the condition P [f: dt < oo ] = I is indispensible for the existence of J: db. e
2
e
e
2
e
e
2
e
So l ution
J�
with a new Brownian motion a. Now use the fact that lim ttoo a = - lim ttoo a = oo . e
db = a(t) for t < I
32
2
STOCHASTIC INTEGRALS AND DIFFERENTIALS
"
2.6
STO CHASTIC DIFFERENTIALS AND ITO 'S LEMMA
A stochastic integral is an expression
x(O) + J e db + J ! ds 0 o number x(O) independent
x(t)
t
t
(t �0)
=
based upon (a) a of the Brownian field Boo , (b) a nonanticpating Brownian functional with p u�e 2 ds < t � 0] 1 ' and (c),a nonanticipating Brownian functional/with e
00 '
=
P[f�ifl ds < oo, t �o]
=
1.
The stochastic differential dx e db + f dt is a more compact expression of the same state of affairs. For example, the integral formula =
f0 b db t
=
t [b(t) 2 - t]
of Section 2.4 is the same as the differential formula d(b 2 ) 2b db + dt. A stochastic integral is itself a nonanticipating Brownian functional, so the class of stochastic integrals is closed under ordinary integration x -+ J; x ds and under Brownian integration x -+ J; x db ; it is also closed under addition and under multiplication by constants. Ito's lemmat states that it is closed under the application of a wide class of smooth functions. =
Ito 's Lemma
Consider a function u continuous partials Uo
=
=
u[t, x1 , . . . , xn] defined on [0, oo)
x
Rn with
oujot,
and take n stochastic integrals X;(t ) t See Ito [7].
=
x ;(O) + J; e; db + J; !; ds (i � n) .
2.6
STOCHASTIC DIFFERENTIALS AND ITO ' S LEMMA
33
Then the composition x ( t ) u[t, x 1 (t), . . . , xn (t)] is likewise a stochastic integral, and its stochastic differential is =
"" u . - dx . dx . u 0 dt + "' i...J u '· dx.' + � --z i...J i�n i ,j � n 'J ' J ' with the understanding that the products dx i dxi (i, j � n) are to be computed by means of the indicated multiplication table, i.e.,
dx
=
X
db
dt
db
dt
0
dt
0
0
A number of simple examples will illustrate the content of Ito's lemma. Exam ple 1
d(b 2 )
2b db + (db) 2 2b db + dt as noted above. In fact, Ito's lemma states that for u E C2(R1),t the stochastic differential of x(t) u[b(t)] is dx u'(b) db + !u"(b) dt, or, what is the same, =
=
=
=
u [ b ( t )]
=
u(O) +
t
t
J0 u'( b) d b + J0-!- u"(b) ds
( t � 0) .
Exam p l e 2
ItO's lemma applied to 3 exp [J� e db - -!- J� e2 ds] gives d3 3(e d b - !e2 dt) + l3(e db - le2 dt)2 3(e d b - 1e2 dt) + -!-3e2 dt 3e d b , especially, d3 3 db if e = 1 , showing that 3 exp (b - t/2) plays the role of the customary exponential (see Section 2. 7 for additional infor mation on this point). =
=
=
=
=
=
t Warning : C"(R1) denotes the class of n (�oo) times continuously differentiable
functions on R1 ; no implication of boundedness of the functions or of their part ials is intended.
2
34
STOCHASTIC INTEGRALS AND DIFFERENTIALS
Exam p l e 3
Ito 's lemma applied to the product u = It i 2 gives d(Iti 2) = I 2 dit + It di 2 + et e 2 dt, justifying the rule for partial integration: I t i 2 = J I t dx 2 + J I 2 d r.t + J e t e2 ds, 0 0 0 0 especially , this example shows that the class of stochastic integrals is closed under multiplication. t
t
t
t
Proof of Ito ' s Le m m a
Ito's differential formula is short for an integral expression for I = u[t, It , . . . , In] . By the definition of the integrals, it suffices to prove this integral formula for simple e i and fi (i � n), and by the addi tive nature of the integrals, it is enough to prove it for � 1 and constant e i and fi (i � n).t But in that case, I = u[t, et b + ft , . . . , en b + fn] can be expressed as u[t, b(t)] with a new (smooth) function u defined on [0, oo) R t , and a moment's reflection shows that it is enough to prove Ito's lemma for this new function, i.e., for n = 1 , e = 1 , and f = 0; it nis also permissible to take t � 1 . Define A = b(k2 - n) - b((k - 1 )2 - ) and I = [2n t] . For n j sufficiently fast and t � 1 , t
x
oo
u [ t, b(t)] - u [O, OJ = L { u [ k2- n , b(k2- n)J - u [( k - 1 )2 - n , b(k2- n)J } k�l
+ I { u [(k - 1 )2 - n , b (k2 - n )J - u [( k - 1 )2 - n , b((k - 1 )2 - n)J } k�l + u [t, b(t ) ] - u [ l2 - n , b(l2 - n )] = L { u o [ Ck - 1 )2 - n , b(k2 - n )J2 - n + o(2- n ) } k�l
+ I { u t [ { k - 1 )2 - n , b((k - 1 )2 - n )] A k�l
=
J0u 0 [s, b(s)] ds + Jo u 1 [s, b(s)] db + fo!u 1 1 [s, b(s)] ds t
t
t
+ I t u t t [( k - 1)2 - n , b((k - 1 )2 - n ) ] (A 2 - 2 - n ) + o( l ), k�l
t Use the fact that if e is nonanticipating, then e(O) is independent of Boo .
2.7
SIMPLEST STOCHASTIC DIFFERENTIAL EQUATION
35
using the lemma of Section 2.4 in the last step. To finish the proof, it suffices to estimate the maximum modulus of the martingale (l � 2") 3z = L t u 1 1 [(k - 1 )2 - n, b((k - 1 ) 2 - ")](A 2 - 2 -") k� l figuring in the last formula. Under the extra condition ll u1 1 ll oo < oo , the proof of the lemma of Section 2.4 is easily adapted to give P [max 1 31 1 < 2-"12n, n j oo ] = 1 , �� 2 " and Ito's lemma follows. The reader will now check that the condition ll u1 1 l l oo < oo is harmless since P[maxs� t lb(s) l < oo ] = 1 . Problem 1
Define the backward integral 1 J u(b) db = ntlimoo k�L "u [b ( kT n)] [b(kT n) - b(( k - 1 )2-")] 2 for u E C 1 (R1). Prove that : I u(b) db = I: u(b) db + I: u'(b) dt. Problem 1, Section 2.4, contains the simplest instance of this: 1 Ib db = 1-[b( l )2 + 1]. 0
0
So l ution
1 0
I u(b) db = ntlimoo k �L " { u [ b(( k - 1)2- n] ll + u'[b(( k - l)T n ) ] !!2 2
+ o (A 2)}
with L\ a.s before, and the lemma of Section 2.4, adapted as for the proof of Ito's lemma, does the rest. 2.7
SOLUTION OF THE SIMPLEST STOCHASTIC DIFFERENTIAL EQUATION
Given a nonanticipating Brownian functional e with P [ f>2 ds < oo , t � 0 ] = 1,
36
2
STOCHASTIC INTEGRALS AND DIFFERENTIALS
the exponential supermartingale 3(t) =exp [(e db - t {e 2 ds]
is a solution of the stochastic differential equation d3 =3e db with 3(0) = 1 [see Example 2, Section 2.6] . Ito's lemma implies that if 1J is a second solution, then so 3 is the only solution with 3(0) = 1 . The moral is that 3 is the counter part for ItO's integral of the customary exponential exp [f� e db ] . A second expression for 3 will now be obtained: 3o = 1 ,
Proof
Bring in the intrinsic time t(t) =J� e 2 and suppose Jooo e 2 = oo so that t - 1 (t) =min (s : t(s) = t) oo is left continuous and joo as t j oo. Because t - 1 (t) is a stopping time, 3n(t - 1) is a stochastic integral, and recalling (5), Section 2. 3 , we find -• 1 E [3/ ( C ) ] =E [ u: 3n - t e db rJ <
=E u:
]
-• 3: - 1 dt
=E [f;3:- �cc 1 ) as] ,
so that E[3n2 (t - 1 )] � tnfn ! (n � 1), by induction. Now for fixed s � 0
2.7
SIMPLEST STOCHASTIC DIFFERENTIA L EQUATION
37
and f = t- 1(s), 3n(t + f) - 3n(f) is a stochastic integral over the Brownian motion b + (t) = b( t + f) - b(f) since 3n _ 1 e(t + f) is nonanticipating over b+. Because t- 1 (t) - f is a stopping. time of b+ for t � s, 1(5), Section 2.3, implies that E[3n(t- 1 (t)) - 3n(f) I Af + ] = 0, so that 3n(t- ) is a martingale over the fields At - + . An application 1of the martingale inequality of Section 1 .4 to the submartingale 3n 2 (t - ) gives 1
leading, via the first Borel-Cantelli lemma, to the geometrically fast (local) uniform convergence of the sum 3 = 3o + etc. to a solution of a(t) = I + J� a e db (t � 0). This completes the proof except for noticing that the extra condition Jooo e 2 = oo is superfluous. Define the Hermite polynomials: (n ;;?!: 0)
and deduce from the power series for exp ( - x2f2t) that 00
y Hn = exp (y x - y 2 t/2) . L n n
=O
Expanding the solution 3 = exp [Y J� e db - y 2 t(t)/2] of d3 = yae db by Y n3n solution series the with comparing and formula this of means L . gtves proving a special case of a formula of Ito [5] and Wiener [3] :
38
2
STOCHASTIC I NTEGRALS AND DIFFERENTIALS
For e = 1 , this gives the evaluation: (n � l) . t
The moral is that the Hermite polynomials are the counterparts for Ito's integral of the customary powers b(t) n/n ! (n � 1) . Ito-Wiener's general formula is developed in Problems 1-3 below; in these problems e stands for a nonanticipating Brownian functional with P [f000 e2 dt < oo ] 1 . =
Pro b l e m 1
(n + l) Hn + 1 + tHn - 1
=
( n � 1) .
x Hn
Sol ution
Use the generating function exp [yx - y2t/2].
Pro b l e m 2
Ito [5] defines the multiple integral of e1 @ over [0, oo )n as
·
·
·
@ en = e 1 (t 1 )
• •
•
en ( tn)
being the symmetric group of all permutations of n letters. Prove Ito's formula: G
+ L I [e l ® . . . ® ek ® . . . ® en ] k�n
with the help of Example 3, Section 2.6. n=3 Example 3,
Sol uti on fo r
By
Section 2.6, it develops that
t Section 2.4 contains the case n 2, done by hand. � The .... signifies suppress this letter. =
00
f0 eo ek dt t
2.7
SIMPLEST STOCHASTIC DIFFERENTIAL EQUATION
oo h t1 + J e 1 db t e 2 db J e 3 db J e0 db 0 0 0 to
Now permute 1 , 2, 3 and add, obtaining
+ 1 2 similar integrals
39
2
40
STOCHASTIC INTEGRALS AND DIFFERENTIALS
and reduce this to 00 I [e0 ® e 1 ® e ® e3] + t e 0 e 1 dt I[e ® e3 ] + 2 similar integrals, 2 2 using a similar reduction of t e0 e1 dt I [e2 ® e3] . 00
Pro b l em 3 (formu Ia of Ito-Wien e r)
Define e n= e ® i "I= j. Then I[e�• ® ei2 ® etc.]
·
·
·
®e
Jooo e; ei dt = 0 for
(n-fold) and suppose
'
= n1 ! H , n
[I
00
0
e1 2
dt, J e 1 db 0 00
Sol ution
1
X
n !H2 2 n
[too
e 2 2
dt,
too db1 X etc. e
2
Because of Problem 2, I[e1 ] I [e�• ® ei2 ® etc. ] 00 1 1 + 2 = J [e';.' ® e;2 ® etc.] + n 1 t e1 dt I[e�' - ® ei2 ® etc.]. Now use Problem 1 and repeat for e 2 , etc. Problem 4
Sol ution
Because H4 [t, x] = x4 - 6tx2 + 3t 2 , t1
tJ
t2
J0 e db J0 e db J0 e db J0 e db 00
=
(Jo db) 00
e
t See Skorohod [2].
4
-6
Jo dt (t db ) + (Jo dt) . 00
e2
00
e
2
3
00
e2
2
2. 8
41
TIME SUBSTITUTION
But for nice e, the left side has expectation 0, so that
E [ (('e dbrJ � 6E [(('e dbr ('e2 d t] '2 (E [ (fooo e2 dtf]) 112 � 6 (E [ ( fooo e db rJ f
The proof may be completed by an easy approximation. STOCHASTIC DIFFERENTIALS UNDER A TIME
2.8
SUBSTITUTION
Consider nonanticipating functionals e and 0
P [ (e2 ds + {f-2 ds < oo , t � oJ = 1 . = J� e db and t J� (f - 2 ds, and let us prove that =
< t( oo), with a new Brownian motion a, i.e., in the language of differentials, (t < t( )) t To give this a nice sense, e (t - 1 ) and f(t - 1 ) must be nonanticipating functionals of a ; for that a small technical condition must be imposed upon e and/, as explained in Step 2, below.1 To simplify the proof, it is supposed that t( oo) = oo so as to make t - (t) < oo for 0 � < oo.
for
t
00
t
Step 1
The first thing is to define a(t) =
t- 1
Jo db /f ;
this is a Brownian motion, as proved in Section 2. 5. t The means differentiation with respect to time. � K. Ito and S. Watanabe helped me with the proof. •
.
2
42
STOCHASTIC INTEGRALS AND DIFFERENTIALS
Step 2
The next task is to prove that e(t - 1 ) is a nonanticipating functional of a ; for this, it suffices to require that e(t) =
lim 2" J
nt oo
t+2 - n
e(s) ds
t
of e without for t � 0, which can always be achieved by modification changing x. The same proof will show that f(t - 1 ) is a nonanticipating functional of a after a similar modification of f Given t � 0, define f = t - 1 (t). Then it suffices to verify the following facts: (a) a(s) : s � t is measurable over Af + . (b) ,a + (s)1 = a(s + t) - a(t) : s � 0 is independent of Af + . (c) e(t - ) is measurable over Ar + . Both (a) and (c) are easy. As to (b), f + = f( · + f) is a nonantici pating functional of the Brownian motion b + = b(· + f) - b(f), and putting t + (t) = I� (f + ) - 2 ds gives By Section 2.5, it follows that a + , conditional Af + , is a Brownian motion, i.e., (b) holds. Step 3
Because
t- 1 t-1 1 (ef ) 2 ( C ) ds = e2 ds < oo , ( ef ) 2 dt =
Io the integral I� (ef)( t
this integral with
C 1)
da
fo
Io
is now defined, and for the identification of t- 1 1 e db , x( C ) =
fo
it suffices to deal with simple functionals e , as the reader will verify.
2.9
43
STOCHASTIC INTEGRALS AND DIFFERENTIALS
As in
us to suppoe that Section 2.6, this permits e = 1 . Choose simple /, ' with jumps at times k2 - n' so close to f that
f0 l fn (t - 1 ) - j( C 1 W ds = I0
t-
t
1
Un - f) 2 dt
t- 1
= Jo
(/,j - 1 ) 2 ds
<
2- n
(n i oo)
for t � 0. Now for any integral I � 1 , Jo fn (C 1) da t(l 2 - ")
= L !nCCk - 1) 2- n) [a(t(k2 - n )) - a(t( ( k - 1)2 - n))J k�l k2 - n = L fn ((k - 1) 2 - n) f db /J ( k- 1 )2 - n k�l
But, for I = [2nt] and n f oo, this can be read backward to get b (t) = lim b ( l2-" ) = lim n too n too
I0
l2 -
n J:
..!!
f
db
=lim I fn( t- 1 ) da = I f(C 1 ) da , 0 n too 0 and substituting t- 1 (1 ) for t finishes the proof. t(l2 - ")
2.9
t(t)
STOCHASTIC INTEGRALS AND DIFFERENTIALS FOR SEVERAL-DIMENSIONAL BROWNIAN MOTION
Ito's integral is easily extended to the d-dimensional Brownian motion b of Section 1 . 7. A nonanticipating functional e t Rn is defined as before and is automatically a nonanticipating functional of b1 , for example, so that for n = I and P[J; e 2 ds < oo , t � 0] = I , J; e db 1 can be defined as in Section 2.2. More complicated integrals :
�
44
2
STOCHASTIC INTEGRALS AND DIFFERENTIALS
can be built up piece by piece. A few samples for d idea.
R3 ,
=
3
will indicate the
J; l e l 2 ds J; (e/ + e/ + e3 2 ) ds < oo . J; e · db J; e1 db1 + J; e2 db2 + J; e 3 db 3 (b) e : R\ J; l e l 2 ds oo . J; e db (f; e 2 db 3 - J; e3 db 2 , etc. ) (c) e : t R 3 ® R 3 , J; i el 2 ds < oo .t J; e db (f; e11 db 1 + J; e12 db 2 + J>1 3 db 3 , etc. ) (a) e : t
�
=
=
-
t
<
�
x
=
�
=
Ito's lemma is also easily extended. The differential is computed out to terms involving products (dt) 2 , dt db i ( i � d), and dbi dbi (i, j � d) as before, and these are reduced using the multiplication table below.
dbl db 2
dt
dbl
dt
0
0
db 2
0
dt
0
dt
0
0
0
X
Proof of Ito ' s Le m ma
The proof is just the same as in Section 2.6 except that the cross multiplication of Brownian differentials [db 1 db 2 0 , etc.] must be justified. This can be reduced to proving that if e is a bounded non anticipating functional of the 2-dimensional Brownian motion =
t Rn ® Rm always denotes with Rm itself,
is the class of linear applications of Rm into Rn ; for e E Rn ® Rm, le i the bound of this application ; for n = 1, Rn ® Rm can be identified and l e i coincides with the usual norm lei = (e 1 2 + e2 2 + etc.) 1 12•
2.9
45
STOCHASTIC I NTEGRALS AND DIFFERENTIALS
b = (b1 , b 2 ) , then the maximum modulus of the martingale 3, = L e(( k - 1) 2 - n)[b 1 (k2 - n) - b 1 CCk - 1)2- n)J k�l
is bounded by a constant multiple of 2 - n1 2 n as n f oo . But this follows as before from the martingale inequality of Section 1 .4. Additional properties of several-dimensional Brownian integrals and differentials will be explained below in a series of 9 problems with solutions. In these problems e : t � Rn is a nonantici p ating functional with P[f� l e2 1 ds < oo , t � 0] = 1. Problem 1
If e: t
-4
R4 and t(t ) = J; lel 2 ds, then t - 1 (t) a (t) = Jo e · db
is a ! -dimensional Brownian motion for times t < t( oo ) . t- 1 (t ) stands for the customary inverse function min (s : t(s) = t) for t < t( oo ) . Sol uti on for
t( oo ) = oo
An application of Ito's lemma justifies
3 (t) = exp
; s:lel 2 ds] = 1 + J - 1 y (3e . db
[J - 1 y s:
y . e db +
(t � 0)
for y E R1 . The reader will now check that 3 (t - 1) is a martingale over the fields At - 1 + (t � 0), so that, for t � s, E [exp [J - 1 ya (t)] j A t - 1(s) + ]
E [3 (t - 1 (t)) I At- 1(s) + ] exp ( - y 2 t/ 2) = 3 ( t - 1(s)) exp ( - y 2 t/ 2 ) = exp [J - 1 ya ( s)] exp [ - y 2 (t - s)/2] Because a(r) : r � s is measurable over A t - 1 (s) + , the p roof is complete. The proof of Section 2.5 could have been made in the same way. =
2
46
STOCHASTIC INTEGRALS AND DIFFERENTIALS
Problem 2
J:
If e : t --+ Rn ® R4, then a( t ) = e db is an n-dimensional Brownian motion if and only if its differentials have the correct (Brownian) multiplication table : da i dai = dt or 0 according as i = j or not. So l ution
3(1) = e xp [J - 1 y a( t ) + y 2 t/2] is a martingale over the fields A t (t � 0) for each y E Rd if the multiplication table for a is Brownian . But this gives ·
E[exp J'=l y a( t) I A s] = exp [J=l y a(s) ] exp [ - y 2 (t - s)/2] ·
·
for t � s, as in the solution of Problem 1 . Problem 3
If e : t --+ O(d),t then a( t ) = motion.
J� e db
is a d-dimensional Brownian
Sol ution
Use Problem 2. Problem 4
If e : t --+ R4 ® R4 and t ( t ) = strong laws : p
and
[-.
J: i e"i 2, then x(t) = J� e db satisfies the
l x ( t) l ��� (2t lg2 1/t) 1 1 2
J
1=
:::; 1
1
with A = [t 1 , t ), t(A) = 4 i e i 2 , and the understanding that 0/0 = I (see 2 Section 2. 5 for the case d = 1). t O(d) is the group of rotations of Rd.
2.9
STOCHASTIC INTEGRALS AND DIFFERENTIALS
47
Sol ution y
x
is a !-dimensional Brownian motion run with the clock t(t) = le *yl 2 for each direction y E S4 - 1 in accordance with Problem 1 . The rest is obvious. ·
J;
Problem 5
J;
[f;
]
If e : t -+ R4 , then 3(t) = exp e · db - t lel 2 ds is a super martingale over the fields A t ( t � 0), and t � e - aP . P max e · d b - rx e l 2 ds> 2 0 t� 1 0 Sol ution
[ J
P]
Ji
Do this first for simple e as in (6), Section 2.3. The rest is easy. Problem 6
Off ( t : b = 0), the stochastic differential of the Bessel process r = lb l = (b 1 2 + · · · + bd 2 ) 1 1 2 is dr = da + (d - 1)(2r) - 1 dt with a new !-dimensional Brownian motion a(t) = - 1 b · db. Sol ution
J>
Use Ito's lemma and Problem 2. Problem 7
Use the fact that for the 2-dimensional Brownian motion, d lg r = r - 1 da off (t : r = 0) to confirm P[r> 0, t ¥= OJ = 1 . This was proved in Problem 1 , Section 1 . 7, for d � 3 by another method. Give a similar proof for d � 3.
d= 2 P[r(l) > OJ = 1 , so d lg r = r - 1 da for 1 � t < f = min (t � 1 : r = 0), and according to Section 2. 5, if t(t) = > - 2 ds and C 1 (t) = min (s � 1 : t(s) = t), then c(t) = lg r(t - 1 ) will be a ! -dimensional Brown ian motion up to time t(f) � oo . But if f < oo , then c tends to - oo as t i t(f), and that is impossible for a Brownian motion to do, either because t(f) < oo , or because t(f) = oo and c(t) � 0, i.o., as t j oo . Sol ution fo r
J
48
2
Sol ution fo r
d� 3
STOCHASTIC INTEGRALS AND DIFFERENTIALS
Do the same with _ ,d- 2 in place of lg r. Problem 8
Check that the spherical polar coordinates qJ = cos - 1 (b3/r) = colatitude, = (b 1 2 + b 2 2 + b 3 2 ) 1 1 2 , () = tan - 1 (b 2 /b 1 ) = longitude
r
of the 3-dimensional Brownian motion b = (b 1 , b 2 , b3) evolve according to the stochastic differential equations
d r = da 1 + r- 1 dt, d() = (r sin qJ ) - 1 da 3
d qJ = r - 1 da 2 + ! r- 2 cot qJ d t,
with a new 3-dimensional Brownian motion a :
fa a 3 = fo (r sin t
a 2 = (r2 sin cp )- 1 b 3 (b 1 db 1 t
cp
+
fo
t
b 2 db 2 ) - sin cp db 3
)- 1 ( b 1 db 2 - b 2 db 1 ) .
Sol ution
Use Ito's lemma and Problem 2. Pro b l e m 9
Prove by stochastic differentials that for t � 1 , the 2-dimensional Brownian motion b = (bb b 2 ) can be expressed in circular polar coordinates as
[
b(t) = r(t), a
((asjr(si) + 0(1)] .
being the Bessel process b = (b 1 2 + b 2 2 ) 1 1 2 , and !-dimensional Brownian motion.
r
a
an independent
2.9
STOCHASTIC INTEGRALS AND DIFFERENTIALS
49
Sol ution
Take a Bessel process r, an independent ! -dimensional Brownian motion a, and put C 1 (t) = J� r - 2 ds. Because of the independence of a and r, f(t) = r(t) - 1 is a nonanticipating functional of a. Now t = J�! - 2 ds, so by Section 2.8, the differential of b = [r, a(C 1 )] can be expressed as [dr, r - 1 de] with the new Brownian motion t - 1 (t ) r(t) da. c( t) = Jo A moment's reflection shows that c, conditional on r, is still a Brownian motion, i.e. , c is independent of r, and it follows easily that the multi plication table of the reactangular coordinates of db is Brownian, as needed for the identification of b as a 2-dimensional Brownian motion by means of Problem 2.
3
3.1
STOCHASTIC INTEGRAL EQUATIONS (d === I )
DIFFUSIONS
A diffusion on R 1 is a collection of motions with continuous sample paths t I(t) e R 1 , defined up to an explosion time 0 < e � oo, such that I( e - ) - oo or + oo if e < oo . One such motion is attached to each possible starting point I(O) x e R 1 , and these separate motions are knit together according to the rule that if t � oo is a stopping time of I, i.e. , if (t < t) is measurable over I(s) : s � t for each t � 0, then conditional on t < e and I( t) y, the future I + (t) I(t + t) : t < e + = e t �
=
=
=
=
-
is independent of the past I(s) : s � t + and identical in law to the motion starting at y ; in brief, I begins afresh at its stopping times. Given such a diffusion I and a nice function v on R 1 , think of u E [v(I(t)), t < e] as a function of t �0 and I(O) x e R 1 • Because I begins afresh at constant times, =
=
50
3. 1
51
DIFFUSIONS
u(t, x) E [E [v(I ( t)), t < e I I( r) : r � s]] E [ u ( t - s, I(s)), s < e] =
=
for t � s, so that the map exp ( tG ) : v exp (tG )
=
�
u(t, · ) is multiplicative :
exp ((t - s)G) exp (sG ) ,
as the notation suggests. G limt + 0 t - 1 [exp (tG) - 1 ] is a differential operator, expressible in nice cases as Gu (e 2 /2)u" + fu' with e( ¥= 0) and / belongin g to C 00 (R 1 )t ; in such a case, p ( t, x,y) oP [I (t) < y, t < e]joy - can be identified as the (smallest) elementary solution of oujot G *u with pole at x I (O) t (see Section 3.6 for the exact statement) . Because the distribution of I can be computed from p via the rule : =
=
=
=
=
P[.'�nn ( ai � x( tJ < bJ, tn < e] f f . . . f P( t t , x, Y t ) dy 1 p( t2 - t 1 , Y t , Y2 ) dy2 =
b1
a1
b2
a2
bn
an
for it is apt to say that G governs I. Ito [9] contains an excellent introduc tion to this circle of ideas. A more exhaustive (-in g) account can be found in Dynkin [3 ] and/or Ito-McKean [ 1 ] . Brownian motion with a general starting point [ I x + b] is the simplest example of such a diffusion : p ( 2rrt) - 1 1 2 exp [ - (x y) 2 /2t] is the elementary solution of oujot ! o 2 ujox 2 , SO I is governed by Gu u"/2. A slightly more complicated diffusion is I X + eb + ft with constant e ¥= 0 and f Now p (2rre 2 t) - 1 1 2 exp [ - (y - x - ft) 2/2e 2 t] is the elementary solution of oufot G*u with Gu (e 2j2)u" + fu'. The second example already suggests how to make, out of the Brownian sample paths, the sample paths of the diffusion associated with =
=
·
=
=
=
-
=
=
=
t Warning : C"(R 1) denotes the class of n ( � co) times continuously differentiable functions on R 1 ; no implication of boundedness of the function or of its derivatives is intended. t G*
is the dual of G : G*u = (e2u/2)" - (fu)'.
52
3
STOCHASTIC INTEGRAL EQUATIONS
(d = 1 )
Gu = (e 2 /2)u" + fu ' for nonconstant e( # 0) andf from C00 ( R 1 ) : it suffices to make the recipe I = x + eb + ft local, i.e. , to solve the stochastic differential equation di = e db + f dt with e = e[I(t)] , f = f[I(t)], and I(O) = x E R 1 • Sections 3.2-3.4 are devoted to solving this problem and Section 3.5 to the proof that G governs I. 3.2
i = e( I ) db + /(I) dt
SOLUTION OF d
FOR COEFFICIENTS WITH
BOUNDED SLOPE
As the first step of the program outlined in Section 3. 1 , it is proved that for coefficients e andf belonging to C 1 (R 1 ) and of bounded slope,
di = e(I) db + f(I) dt has only one nonanticipating solution I(t) : t � 0 beginning at I(O) = x E R 1 . Ito's [2] original proof is used ; for simplicity, it is assumed that li e ' I I oo � t and II /' II � t, and the proof is spelled out for t � 1 only. oo
t�1 Define the nonanticipating Brownian functionals :
Proof of existen ce for
I 0 ( t) = I x.( t) = x +
J0 e(x. - 1 ) d b + J/ Cx. _ 1 ) ds t
t
( n � 1).
Using the bound (A + B ) 2 � 2A 2 + 2B 2 and (5), Section 2.3, the reader will easily see that for en = e(In) - e(In _ 1 ) and fn = f(In) - f(In - 1 ), D n = E [ l xn + 1 - I n l 2 ]
[( / ds + {1/ ds ] � 2( 11 e' ll � + II ! ' II �) fovn - 1 � foD. _ 1 � constant t"fn ! � 2E
e
t
t
x
(n � 1 ),
3.2
COEFFICIENTS WITH BOUNDED SLOPE
53
J�
by induction. Now 3 . (t) = e. db is a martingale over the Brownian fields B t( t � 0), so the bound of Section 1 .4 applied to the submartingale 3 n 2 provides us with the estimate
[{ e/ dt] � z - 2 ll e' ll � fn.- 1 � constant X l - 2/n !.
= z - 2E
0
- 0
Combining this with a simpler bound for IJ . (t) =
� z - 2 II!' II �
. gtves
[
1
Jo' f. ds :
fo D. - 1 � constant X z - 2/n ! 1
J
P max l xn + - xn l � 21 � constant X z - 2 fn !. t� 1 Pick / - 2 = (n - 2) !. Then / - 2 /n ! is the general term of a convergent sum, and by the first Borel-Cantelli lemma,
[
P max l xn + 1 - xn l � 2 [( n - 2) !] - 1 1 2 , t� 1
n
]
j oo = 1 .
Because of this, I n converges uniformly for t < 1 to a nonanticipating Brownian functional I 00 , and since
J
1
0
1
l e (x«,) - e (xn) l 2 dt � max l x oo - xn l 2 t�
tends to 0 fast for n i oo, (7) of Section 2.3 implies l00 ( t )
=X+
fo e(X 00 ) db + Jo /(X 00 ) ds t
completing the proof of existence.
t
t 1 "'
at
( t � 1),
54
3
STOCHASTIC INTEGRAL EQUATIONS
Proof of u n i q ueness for
(d = 1 )
t�0
Given two nonanticipating solutions x 1 and x 2 , bring in the Brownian stopping time t = min (t : lx 1 1 or lx 2 1 = n), and let x * be the product of x and the (nonanticipating) indicator function of (t � t). Then
x 2 * - x 1 * = J [e (x 2* ) - e(x 1 * )] d b + 0 t
fo [J(x2* ) - f (x 1* )] ds t
t < t,
and D = E [ lx 2 * - x 1 * 1 2 ] � 4n 2 < oo can be bounded by J� D as in the proof of existence. D = 0 follows, and making n i oo , it develops that P[x 1 = x 2 , t � 1] = 1 , as advertised.
3.3
SOLUTION OF dx
=
e( x) db + f( x) dt
COEFFICIENTS BELONGING TO
FOR GENERAL
C 1 (R1)
Using Section 3. 2, it can now be proved that for the general e and f belonging to C 1 (R 1 ) and fixed x E R 1 , there is just one Brownian func tional x defined up to a Brownian stopping time 0 < e � oo (explosion time) such that (a) the product of x(t) and the indicator function of (t < e) is non anticipating, (b) x(t) = x + e(x) db + f(x) ds (t < e), and (c) x(e - ) = - oo or + oo if e < oo .
J�
J�
Besides this, it will be proved that x begins afresh at Brownian stopping times, i.e. , if t is a Brownian stopping time, then, conditional on t < e and x(t) = y, the future x + (t) = x(t + t) : t � 0 is independent of the Brownian field Bt + (over which the past x(s) : s � t + is measurable) and identical in law to the solution of dx = e(x) db + f(x) dt with x(O) = y. Because a stopping time of x is likewise a Brownian stopping time, x is a diffusion as described in Section 3. 1 . Because of (c), it is natural to put x(t) = x( e -) for t � e. P[ e = oo] = 1 for coefficients with bounded slope (see Section 3 .2). A practical test for deciding if P[ e = oo] = 1 is given in Section 3.6 (see also Problems 1 and 2 of this section).
3.3
GENERAL COEFFICIENTS BELONGING TO
C 1 (R 1 )
55
Step 1
Extend e(f) outside [ - n, + n] to en Cfn ) E C 1 (R 1 ) with bounded slope, let xn be the solution of dx = en(x) db + fn(x) dt with x(O) = x, and define the Brownian stopping time en = min (t : lx n l = n) (n � 1 ) . As in the second half o f the proof in Section 3 . 2 , xn _ 1 (t) = xn(t) for t < e n - t ( � e ), and it follows that (a) and (b) hold for the path x(t) = xn(t) n ( t < e n , n � 1 ) and the Brownian stopping time e = lim n t e n � oo. Any other nonanticipating solution agrees with x up to time e. The proof may be adapted from that of Section 3 . 2 . oo
Step 2
x( e - ) = - oo or + oo if e < oo , i.e., (c) holds. Prooft
If (c) did not hold, it would be possible to choose a point of R 1 (such as 0) and a positive number (such as 1 ) such that P(Z) > 0, Z being the event that x returns to 0 from l x l � 1 , i.o. , before time e < oo. Each of the returning times
( � t 2 = min ( t � t 1 : r (t) = 0,
t 1 = min t � 0 : r(t ) = 0, rr;:1x l r (s) l � 1
)
���;1 l r (s ) l � 1
� etc.
)
is a Brownian stopping time, and the loop Xn (t ) = x( t + tn - 1 ) =
J e(rn) dbn + J f(rn) ds t
t
0
0
is the same (nonanticipating) functional of the Brownian , motion
( t � 0) for any n � 2 . The reader will easily see from this that the loops are independent and identically distributed, especially, the passage times t H. Conner showed me this nice proof, improving upon my earlier try.
3
56 t n - tn _ 1
STOCHASTIC INTEGRAL EQUATIONS
(d = 1 )
(n � 2) are such, so by the strong law of large numbers,
But, on Z c (e < oo), L � 2 (tn - t n _ 1 ) � e < oo , which is contradictory unless P(Z) = 0. Step 3
The final job is to check tha t x begins afresh at any Brownian stopping time. Step 2 involved a simple instance of this. The reader will easily amplify the proof indicated below with the proper measure-theoretical flourishes. Proof
Given a Brownian stopping time t, consider b + (t ) = b(t + t) - b(t), x + ( t) = x( t + t), and e + = e - t, conditional on t < e and Bt , b + is a + Brownian motion since (t < e) E Bt + , and (a), (b), and (c) hold with b + , x + , e + , and y = x + (O) in place of b, x, e , and x. This means that for almost every y, x + is identical in law to the solution of l) ( t) = Y + e(tJ) db + ds :
J�
lJ
J�!(lJ)
lJ
P[P[x + E B I t < e , Bt J = P( E B)] = 1 . +
But x(s) : s � t + is measurable over Bt + , and so the proof is complete. Problem 1 P [e = oo] = 1 if e 2 So l ution fo r x(O)
=
+ / 2 � constant
(1 + x 2 ).
x
0
Call the constant k. Define en as in Step 1 and put xn (t)
=
x(t
1\
en) =
J
t
0
A
en
e (x) db
+
J
t
0
A
en
f(x)
ds
(n � 1 ) .
3.3
GENERAL COEFFICIENTS BELONGING TO
C 1 (R 1 )
57
Then for t � m,
= 2km
foo + D) ds . t
m
But this means that D � e 2 k t - 1 , and since k does not depend upon n, the result follows from the bound P[e n � m] � P [x n(m) � n] � D(m)jn 2 ! 0 as n i oo . Problem 2 P[e = oo] = 1 iff = 0. So l ution
Section 2.5 implies that x(t) a(t) with a new Brownian motion a and t(t) = e (x) 2 ds. No explosion can occur since a Brownian motion cannot tend to - oo or to + oo at any time t � oo ; see Problem 7, Section 2.9, for a similar argument. =
s;
Prob lem 3
Prove that
[
��
l x( t ) - x( ) P lim = l e [x (O)] I t-1. 0 ( 2 t lg 1 f t ) I 2
J=
1
and P
[
lim t= tl - tt -1. 0 O �tt < t2 :s:; < e
1
l x( t 2 ) - x( t 1 ) l = max l e[x(s) ] l = 1 . 1 1 2 s�t ( 2t 1 g l /t)
]
Sol ution
Use the strong laws cited at the end of Section 2.5.
3
58
STOCHASTIC INTEGRAL EQUATIONS
(d
=
1)
Problem 4
P[I t � x 2 , t � OJ 1 if I t and I 2 are solutions of di + /(I) dt and I t (0) � I 2 (0). =
=
e(I) db
Sol ution
t min (t : I t I 2 ) is a Brownian stopping time, and since solutions begin afresh at such a time, I t = I 2 (t � t) if t < oo. =
=
Pro b l e m 5
Take a compact and / from C 00 (R 1 ) . Given X < y, let I(lJ ) be the solution of di e(I) db + /(I) dt starting at x(y), put 1J y - x, and notice that x• 1J - 1 (1) - I) solves
e
=
=
=
x•
=
1+
t
t
J0e• x•db + f0J.6. x• ds
with nonanticipating
e•
= =
e( e(
( lJ - I) - t [ lJ ) - I)] (I)
e'
(tJ # I) (lJ I) =
J
and a similar definition of • . Use the formula of Section 2.7 to express I• in the form
[ {e• db - ! J�(e.6.) 2 ds + J�!.._ ds J.t Define e' e'(I), f' / ' (I) , and r ' exp [ f�e' db - ! J�(e') 2 ds + J� !' dsl Prove that E[(x• - x ' ) 2 ] tends to 0 as ! 0. Do the same with x' • - 1 ( lJ ' - I' ) and r" r ' [f�e "r' db - J�e'e " r' ds + J�f "r ' ds] t x•
=
exp
=
=
=
1J
1J
=
in place of x• and I'. Give a similar formula for I "', etc. t Incidently, a new solution of Problem 4, is contained in this formula. t e " = e '' (I) and f " = f"(X).
=
3.3
GENERAL COEFFICIENTS BELONGING TO
C1(R1 )
59
Sol ution
Denote the exponential formulas for x• and x ' by eA and eB, respec· tively. Use the bounds l x' - x• l � I B - A l (eB
+ eA),
l e ' - e• l � £5 eA II e " ll oo ,
I f' - 1•1 � £5e A I I /" I I oo , E(e4A) + E(e4B) � 2 exp ([6 1 1 e ' ll oo 2
+ 4 11 / ' I I oo J t)
to verify that for bounded t, E[(x ' - x•) 2 ] � E[(B - A) 2 (e A
� constant X E
+ eB) 2 ] � constant
[ (f� db r dsr + ( f�(f '
x
E[(B - A) 4] 1 1 2
(e' - e•)
- j•) ds r r ( f�l e' - e• l � constant E [ J�( e ' - e•)4 ds ] 1 /2 + Jo(f' - /•) 4 ds t
+
12
x
t
� con stant � constant
x
£5 2 E(e4A
x
£5 2 •
+ e4 B) 1 1 2
The same line of proof works for x " - x' • , etc. Problem 6
Take e and from C 00 (R 1 ) . Use the result of Problem 5 to show that x can be defined as a function of 0 � t < e and x(O) = x E R 1 in such a way that, for any n � 0, P [ an x is continuous on [0, e) X R 1 ] = 1 t and P
f
[ on r
for each x e R 1 •
=
onx +
f� one(r) db + f� Onf(r) ds, t < e ]
t See Problem 4, Section 2.7.
:t: o
=
of ox .
=
I
3
60
STOCHASTIC INTEGRAL EQUATIONS
(d = I)
Sol ution
Use Kolmogorov's lemma (see Problem I , Section I .6) to show that an i can be modified so as to be continuous on [0 , e) x R 1 for any n � 0. LAMPERTI'S METHODt
3.4
r
Given f E C 1 { R 1 ) with bounded slope, ( t ) = X + b (t) + (t � 0) can be solved much more simply using the sure bound
Dn = max 1In + 1 - In l � I 1/ (In ) - / (In - 1 ) 1 � 11 / ' l l oo I Dn - 1 s�t 0 0 t
t
J;t(r) ds (n � 1)
to ensure the geometrically fast convergence of I n . Dropping the condi tion 1 1/' 11 oo < oo, I can be defined up to its explosion time e � oo as in Section 3.3. Now make a change of scale x -+ x* = j(x) with j E C 2 (R 1 ). Ito's lemma implies that for t < e, di* = j ' (I) [d b + / ( I ) d t ] + ti" (I) d t
= e*(I *) d b + f*(x*) d t
with (a) e*(j) = j', and (b) f*(j) = j'f + j"/2. Lamperti's idea is to construct the solution of di* = e*(I*) db + f * (x * ) dt by solving (a) and (b) for j and f Given 0 < e* from C 1 (R 1 ) and/* from C ( R 1 ), (a) can be solved locally for j E C 2 with j' = e*(j) > 0.
f = (j') - 1 [/*(j) - j"/2]
follows from (b). To keep f differentiable, the extra conditions e* E C2(R 1 ) and f* E C 1 (R 1 ) must be imposed, and for the existence of a global solution, additional conditions are needed. Ito's method applies to a wider class of coefficients, but Lamperti's is simpler, because it eliminates the use of the martingale inequality and the Borel-Cantelli lemma. Unfortunately, Lamperti's method fails in several dimensions not just for technical but topological reasons, as will be pointed out in Section 4. 3. t See Lamperti [1 ].
3.5 3.5
61
FORWARD EQUATION
FORWARD EQUATION
Define G * to be the dual of G : G * u = (e 2 u/2 ) - (fu) ' . Using Section 3.5 and Weyl's lemma (Section 4.2), it is easy to see that for e( =F O) andfbelonging to C 00 (R 1 ) , G governs I in the sense that the density p = p(t, y) = oP[x(t) < y, t < e]joy is the smallest elementary solution of the forward equation oujot = G* u with pole at x (O). This means "
(a) (b) (c) (d) (e)
0 � p, lim t -1- 0 fu p dy = 1 for any neighborhood U of x, p E C 00 [ (0, 00) X R 1 ] , op/8t = G *p, and p is the smallest such function.
Step 1
A special case of Weyl's lemma (Section 4.2) states that if u is the (formal) density of a mass distribution on (0, oo) x R 1 and if
0=
f
( O , oo ) x R t
u [O/ Ot + G] j dt dy
for any compact j E C 00 [(0, oo ) x R 1 ] ,t then u can be modified so as to belong to C 00 [(0, oo) x R 1 ] ; after this modification, u solves ou/ot = G * u in the customary sense. This fact is now applied to the (formal) density p = oP[x(t) < y, t < e]joy as follows. Ito's lemma states that dj (t, x) = j 1 (t, x)e(x)e(x) db + [o / ot + G] j (t, x) dt.t
Because E and so
f[ ; {j1 e) 2 dt ] < oo by the compactness ofj, E [f; j1 e db] = 0, §
0 = E[j(t, x) I� J = =
f
( O , oo ) x R 1
fo dt E [(O/O t + G)j(t, x), 00
t < e]
p[O fO t + G] j dt dy.
t Warning : a compact function defined on an off a subcompact of this figure. § See (5) , Section 2.3. t j1 = ojjox.
open figure is a function vanishing
3
62
STOCHASTIC INTEGRAL EQUATIONS
(d = 1)
Weyl's lemma now provides us with a function q E C 00 [(0, oo ) X R 1 ] such that 8q/8t = G*q and p = q as formal densities on (0, oo) x R 1 • But then for compact j E C 00 (R 1 ), J pj dy = E [j(x), t < e] = J qj dy for any t � 0, since both J pj and J qj are continuous functions of t � 0. This shows that p(t, y) = 8P[x(t) < y, t < e]/8y ( = q) exists and satisfies (c) and (d). The rest is plain except for (e) which occupies the next 2 steps. Step 2
Before proving (e) a little preparation is needed. Take f = min (t : l xl = n) and compact nonnegative j e C00 ( - n, n) and let us borrow from the literature the fact that inside lxl < n, 8u/8t = Gu has a non negative solution u E coo [(0, oo ) x [ - n, n ]] with data u(O + , ) = j and u(t, + n) = O.t By Ito's lemma, ·
du[t - s, x(s)] = u1 [t - s, x(s)]e(x) db for lx(O) I < n and s < t A f, and so A1 0 = E f u1 (t - s, x)e(x) db
J
[:
. I.e. ,
f = E[u(t - s, x) l �t - J = E[j(x), t < f] - u, t
u = E[j(x), t < f]. Step 3
Coming to the proof of (e), take a second elementary solution q with pole at x(O) = x e ( - n, n) . Define
n
Q = f_ q(t - s, x, y)u(s, y) dy n
for s < t and notice that
n
iJQfiJs = f_ [ - (G*q)u + q(Gu)] n = [ - (e 2q/2)'u + (e2q/2)u' + fq u] j � n � 0
t See, for example, Bers et
a/. [1 ].
� To see this, note that if t = f, then limstf t > f , then limstf u(t - s , I) = u[t - s, I(f)] = 0.
u(t - s, I) = j[I(f)] = 0, while if
3.5
63
FORWARD EQUATION
� 0. But then 0 � Ql� = u - f qj, and the desired estimate p � q follows from Jpj = lim E[j( :r) , t < f] = lim u � fqj.
since u( + n) = 0 and + u'( + n)
nt oo
n t oo
Problem 1
Deduce from Weyl's lemma and the results of Step 2 that for compact nonnegative j E C00 (R 1 ) , J pj = E[j(I), t < e] is the smallest nonnegative solution of auj at = Gu which belongs to C00 [(0, oo) X R 1 ] and reduces to j at time t = 0. Sol ution
By Step 2, E[j( I), t < f] = un E coo [(0, oo) x [ - n, n]] satisfies aujot = Gu for l x l < n and any n � 1 . But then U 00 = J pj satisfies
00 00 f0 f u co [iJjiJ t + G*]k dt dx = 0 - 00 for any compact k E C00 [(0, oo) x R 1 ], and an application of Weyl's lemma permits us to deduce that U oo E C00 [(0, oo) X R 1 ] solves au;at = Gu in the usual sense. Now tak� a second nonnegative solution u. By I to's lemma,
du[t - s, x(s)] = u1 [t - s, x(s)]e(x) db for s < t
"
f, so
[ ] E [ lim u( t - s, I),
u � E lim u(t - s, x) sttA f
�
s tt
= E[j(x),
t
]
t < f]
and this increases to E[j( x) , t < e] = u oo as n i oo.
3
64
STOCHASTIC INTEGRAL EQUATIONS
(d
=
1)
Problem 2
Regard p as a function of (t, x, y) E (0, oo ) x R 2 • The problem is to check that p belongs to C 00 [(0 , oo) X R 2 ] and solves the backward equation
using Weyl's lemma (Section 4.2) for 2K
=
! e2 ( x )
::2 + f(x) :x + 1 ::2 e2 (y) - :y f(y)
=
Gx +
G: .
Sol ution
Given compact jl
E
C 00 (0,
00
) , j2 E C 00 (R 1 ),
and
J(O, oo ) x R2p[iJfiJt + K*]j 1j2 j 3 dt dx dy + � =
0,
J
J
[ 8 /o t + Gx * ]j j dt d x pj 3 d y 2 1 { 0 , oo) x R1 R1
the first integral vanishing as in Step 1 , and the second by a small elaboration of Problem 1 . But then
0
=
J
p[iJfiJt + K * ] j dt d x d y
00 ) X R 2 for all compact j E C oo [(0, oo ) {0,
x
and since K is elliptic on R 2 , Weyl's lemma supplies us with a function q E C 00 [ (0, oo ) x R 2 ] such that 8qj8t Kq and p q as formal densities on (0, oo ) x R 2 • But then s qj3 dy E C 00 [ (0, oo ) X R 1 ] coincides with s pj3 dy E [j3(I) , t < e] except on a null set of (0, oo ) x R1 , and the proof is finished by verifying, as for Step 2, that the latter belongs to C [(0, oo ) x R1 ] . =
R 2] ,
=
=
3.6
FELLER ' S TEST FOR EXPLOSIONS
65
(
3.6
FELLER 'S TEST FOR EXPLOSIONS
Given e( =F 0) and f belon gin g to C 1 (R 1 ), think of the solution of dx = e(x) db + f(x) dt in the natural scale : x * = j( x ) =
( exp (- 2 J/fe2 )
so that x * = j(x) satisfies dx* = e*(x * ) db with e * (j) = j' e, and let us establish Feller's testt : either P[ e = oo] = 1 for all x(O) or P[ e = oo] < 1 for all x(O) according as
J
0
[j - j( - oo )]e * (j) - 2 dj =
- oo
J0 [j( oo ) - j] e* (j) - 2 dj = oo 00
or not. A d-dimensional analog of Feller's test, due to Hasminskii, is proved in Section 4.5. Proof of Fe l l e r ' s test
explodes to - oo or + oo at time e < oo if and only if x * = j(x) tends to j(( - oo ) *) or to j( oo *) as t j e. Define u = u (j) to be the solution x
U=
00
l�0 Un , ..
Uo =
1,
Un =
2 fo dj J0 un - l e * - 2 dj X
(n � 1)
of e * 2 u "/2 = ut and use the obvious bounds 1 + u1 � u � exp (u 1 ) § to prove that u tends to oo at both ends of j(R 1 ) precisely in the divergent case :
J
0
- oo
[j - j( - oo)]e * - 2 dj =
By Ito's lemma,
J0 [j( oo) - j]e * - 2 dj = oo . 00
de - 'u (x * ) = - e - 'u (x * ) dt + e - 'u ' e * (x * ) db = e - t u ' e * (x * ) db
+ e - 'u" e * 2 (x *) dt/2
t See Feller [1 ]. Additional information on this subject can be found in Ito-McKean [1] . t The ' stands for differentiation with respect of j. § Un � u 1 n;n ! (n � 0) is easily proved by induction ; the stated bound is immediate from this.
3
66
STOCHASTIC INTEGRAL EQUATIONS
(d = 1)
[f;
for t < e. Because E e - z•(u' e*) 2 (r*) ds] < oo for t = min (t : lrl = n) , it is clear from (5), Section 2.3, that for paths starting at I(O) = x between - n and + n, E [e- tu[I * (t) ]] = u(x * ), and making n j oo, it follows that
[
]
P lim t = e = oo = 1 n t oo
J
in the divergent case. Contrariwise, if 000 [j(oo) - j]e* - 2 dj < oo, then 1 < u( oo * ) < oo , and putting t = min (t : I = 0 or n) , it follows from E [e -t u(x * (t))] = u(x*) that for paths starting at x between 0 and oo,
1 < u(x * ) = nlim E [e- tu(x * (t))] � 1 + E(e- e)u( oo * ) . t oo
But that i s impossible if P [ e = oo] = 1 , so the proof of Feller's test is complete. Pro b l e m 1
Prove that P[e < oo] = 1 if
0
00 2 * f- oo [j - j ( - oo)]e - dj + J0 [j(oo) - j]e* - 2 dj < oo . So l ution
J� u(r* ) ds for u as and t( t ) = J� (u' e*) 2 (r*) ds.
By ItO's lemma and Section 2. 5, u(r * ) = a(t) +
above, t < e, a new Brownian motion a, Because u � 1 , e = oo implies u(I*) � t/2, i.o., as t i oo, so u must be unbounded if P[ e = oo] > 0. Pro b l e m 2
Prove that for e = 1 and/ = l x l 1 + b near + oo, explosion is impossible or sure according as J � 0 or not. Problem 1 , Section 3.3, also covers the case J � 0. So l ution
Use Feller's test and the test of Problem 1 .
3.7
3.7
CAMERON-MARTIN ' S FORMULA
67
CAMERON-MARTIN 'S FORMULA
Given e and f from C 1 (R1), let x be the (nonexploding) solution of dx = e(x) db,t let x.f be the solution of dx = e(x) db + f(x) dt with the same starting point x E R 1 and the explosion time e1 < oo, and let us prove that, for
3(t) = exp
[f�(ffe)(r) db - t (ute) 2(r) ds J
and events B depending upon x(s) : s � t only, P[x1 E B, t < e1] = E [x E B, 3(t)] , especially, P [t > e1] = E(3). Cameron-Martin [1] discovered the proto type of this formula. t P[ef < oo ] = 1 for e = 1 and / = x 2 according to Problem 2, Section 3 . 6 , so E(3) < 1 (t =F 0) in this case. This possibility was mentioned but not substantiated in Section 2.3. For simplicity, the proof is made for e( =F O) and f e C 00 (R 1 ) only. Proof n
B can be approximated by events B ' = B (t � e) with e = min (t : l x l = n) and n i oo, so it suffices to prove the formula for e = 1 and f = 0 far out, especially, it can be supposed that P [ e1 = oo] = 1 . Using 11//e ll < oo, it is easy to see that oo
E
[ f�(ffe) 2(:t)32 ds ] <
oo .
Because d3 = (f/e)(I)3 db, an application of (5), Section 2.3, shows that 3 is a martingale, especially, it is permissible to take E(3) = 1 . But then E[3(t2 )/3(t1 ) I x(s) : s � t1 ] = 1 for any t 2 � t1 , which implies that for B depend.i ng upon x(s) : s � t1 only, Q(B) = E[B, 3(t 2 )] is independent of t 2 � t1 , and it follows that the motion with probabilities Q(B ) begins afresh at constant times. To finish the proof, it is enough to verify E[x(t) E A , 3(t)] = P[x1(t) E A ] for t � 0, x E R 1 , and A c R 1 . Define Gu = e 2 u"/2 + fu ' as usual. Ito's lemma implies that for compact t Problem 2, Section 3.3, gives a proof of this nonexplosion. t See Dynkin [1 ], Tanaka [1 ], and Problem 5 , Section 4. 3, for additional informa tion.
3
68
STOCHASTIC INTEGRAL EQUATIONS
(d = 1)
j e C 00 [ (0, oo) X R 1 ] , dj (t, I)3 = (ojfot + Gj) 3 dt + (it + jffe) 3 db, and integrating this from 0 to oo and taking expectations gives o
= E[i3 IO J = E =
[('(ajJat
+
Gj)3 dt
f( 0 , oo) x Rl dt E [x
e
]
dy , 3] ( ajJa t + Gj) .
Weyl's lemma now implies that p = oE[x < y, 3] /oy belong s to C 00 [(0, oo ) X R 1 ] and satisfies opfot = G*p. Also 0 � p, s p dy = 1 , and lim, -1- 0 p dy = 1 for any neighborhood U of x, so the proof can be completed by appealing to the description of p1 = oP[x1(t ) < y]foy as the smallest such function (see Section 3.5) : namely, p1 � p and J p1 = s p = 1 ' so p1 = p.
fu
3.8
BROWNIAN LOCAL TIME
Levy [2] proved that the Brownian local time :
f(t) = lim ( 2e ) - 1 measure (s � t : 0 � b(s) < e)
e-1. 0
exists and is a continuous function of t � O.t This fact will now be proved, for use in Section 3. 10, with the help of Problem 4, Section 2. 7, and an unpublished formula of Tanaka expressing f - b + as a Brownian integral :
f (t) = b( t) + Step 1
foe0 00 (b) db. t t
Define j(x) = J� ex 00 (b) db and j*(x) = lim j(y) as y = kT n ! x. Then, for any t � 0, P[j* e C{R 1 )] = 1 and P[j*(x) = j (x) ] = 1 for any x e R 1 • t Ito-McKean [1] contains an exhaustive account of local times ; the present proof, together with Problem 1 of this section is adapted, after much simplification, from McKean [3 ]. t x + i s the bigger of x and 0. ex, is the indicator function of the interval [x, y) c R 1 •
3.8
BROWNIAN LOCAL TIME
69
Proof
By Problem 4, Section 2. 7,
[ {e db 4] 2 � 36E [ {e ds ] � constant xy
E [ l i (x) - j ( yWJ = E
xy
x
lx - yl 2 •
Kolmogorov's lemma supplies the rest of the proof.t Step 2
An application of Ito's lemma gives
J
b(t)
dx
J
x
eafJ dy =
- oo Because j * e C(R 1 ) and b(O)
J
t
0
db
J
b(s)
- oo
eafJ dy + t
f f- ooeap dy - a �k2L- n < pek2 - n 00(b)2 - n t
b(s)
0
for n i oo ,
J dx J b(t)
b(O)
X
-
oo
eafJ dy - t
J
t
2
t
J eap (b) ds. 0
ds � const ant X 2 - n
J eap (b) ds t
0
I, ek 2 - n 00 (b)T " db 0 a �k 2 - n < p fJ = lim I, j * (kT")2 - n = j *, a n t oo a �k 2 - " < fJ first for each separate pair ap and then for all af3 simultaneously. Tanaka's formula together with the existence of the local time f(t ) b (t ) + - j*(O) follows for each t � 0, separately. Because b+ - j*(O) is a continuous function of t � 0 while measure (s � t : 0 � b(s) < e) is an increasing function of t � 0, the existence of f and the correctness of Tanaka's formula follow for all t � 0, simul taneously.
= lim n t oo
=
t See Problem 1 , Section 1 . 6.
J
3
70
STOCHASTIC INTEGRAL EQUATIONS
(d = 1)
Problem 1
Step 2 above leads at once to the fact that for each separate t � 0, the Brownian local times f ( x) = lim ( 2e)- 1 measure (s � t : x � b(s) < x + e)
e-1.0 = [b( t) - x] + - [ - x] + - j* ( x)
exist and define a continuous function of x e R1 .t Use (6), Section 2.3 , to deduce the law of Ray [1] : P
�� [� = llim x - y l -1.0 ( 2<5 lg 1/ <5 )
]
\ f ( x ) - f(y l � ( l l fl l ) 1 2 = 1 oo 1
( t � O) .t
Sol ution
Put <5 = y - x. Using (6) , Section 2.3 , and the fact that t J0 exy(b) ds = JY f � (y - x) ll f II oo , you see that for fixed n � 1 , X
I
for some - n � x = i2-n < j2-n = y < n and <5 < 2-( 1 -e) n]
�P
[ (exy(b) db > � (<5 -l fg 1 /<5)1 12 (exy(b) ds + P(<5 lg 1/ <5 )1 1 2 for some
�
L -
-n <
- n < x, etc.
]
2 exp [ - cx(<5 - 1 lg 1 / <5) 1 ' 2P( <5 lg 1 / <5 ) 1 1 2 ]
� constant X n2"[ 1 + e ( 1 + cxP) - cxPJ . t Trotter [ 1 ] was the first to prove this fact. t Ray proved that this bound is the best possible.
3.9
REFLECTING BARRIERS
71
This is the general term of a convergent sum for ap > 1 and 0 < e < (af3 - 1)/(a/3 + 1), so the first Borel-Cantelli lemma implies
[
n n P l f UT ) - f { ir ) l �
G ll f ll "' + P) <<> Ig 1 /<5) 1 1 2
for - n � i2 - n < j 2 - n < n, � = (j i )2- n < 2- ( 1 - e) n , _
] .
and n j oo = l
The proof is completed by making ( a/2) II f II + f3 as small as possible, subject to rt/3 � 1 , and using the method of Section 1 . 6. 3.9
oo
REFLECTING BARRIERS
Skorohod [ 1 ] discovered that for e( :;t= O) and f belonging to
C 00 [0, oo ), and x � 0,
fo
x( t ) = x + e(x) db + J/(r) ds + f ( t) t
t
has just one nonanticipating solution (x, f) defined up to a Brownian stopping time 0 < e � oo (explosion time), such that (a) x(e -) = oo if e < oo , (b) x � 0, and (c) f is continuous, increasing,jlat off 3 = (t : x(t ) = 0), and f(O) = 0. Skorohod identified this solution with the so-called reflecting diffusion governed by G cut down to [0, oo ), subject to u + (0) = 0. t This means that p = oP[x(t ) < y]joy is the smallest elementary solution with pole at x of o ufo t = G*u (y > 0), subject to u + (O) = 0. Skorohod's result is proved below, following McKean [4] . At the same time f is identified as the associated local time : f( t ) =
e(0) 2 lim (2e) - 1 measure (s � t : x(s) < e) . x
eJ,O
Because the problem is local, it is permissible to take/ = 0 near oo . This will simplify the proof. t u + (O) - lim6 ! 0 (e) - 1 [u(e) - u(O)].
3
72
STOCHASTIC INTEGRAL EQUATIONS
(d = 1)
Proof of u n i q u e n ess i n a s pecial case
Consider two solutions x 1 = x + b + f 1 and x 2 = x + b + f 2 for e = 1 and / = 0. If x 2 < x 1 , then x1 > 0 and f1 is flat, so that x 2 - x1 = f2 - f 1 is increasing, while if I 2 > x1 , then x 2 > 0, f 2 is flat, and x 2 - x 1 is decreasing. The moral is that x 1 = I 2 , as stated. This neat proof is from Skorohod [1] . Proof of u n i q u e n ess i n the ge neral case
The difference 3 of two solutions I 1 and x 2 satisfies
, 3(t)
=
fo elt.3 db + J/•3 ds + f2 - f 1 t
t
(t < e 1
A
e2)
with nonanticipating
e(x 2 ) - e(x 1 ) e• = x 2 - Xt and a similar definition off ia.. According to (c), 3 d(f 2 - f 1 ) � 0, so Because e• and jia. are bounded up to time t = min (t : x 1 or x 2 = n), D = E[3(t) 2 , t < t] < oo can be bounded by a constant multiple of s; D. D = 0 follows, and the proof is completed by making n j oo . Proof of existe n ce fo r
x = 0 (th e ge neral case bei ng left to the read e r)
Step 1
x = b + f with f
=
- mins � t b(s) is a solution if e = 1 and/ = 0.
Step 2
Given e =I= 0 and / = 0, if x = b + f as in Step 1 and if
t(t) = J e(x) t
0
-l
ds,
3.9
73
REFLECTING BARRIERS
then the time substitution rule of Section 2.8 implies that x* = I(t - 1 ) is a solution of di* = e(I * ) da + df* with a new Brownian motion t- 1 a (t )
=
fo
dbfe(x)
and f* f(t - 1 ). Because I * (t) is measurable over B t - l ( t ) + , it is independent of a + (s) a(s + t) - a(t) (s � 0). As such, it is a non anticipating functional of a, and since the same holds for f * , (I * , f *) is a solution. =
=
Ste p 3
Given e =I= 0, if x is the solution of di e(x) db + df and if j E C 00 [0, oo) with}' > O,j(O) 0, andj(oo) = oo , then a mild extension of Ito's lemma implies that I* = j(I) is a solution of =
=
d :r *
=
j' e db
+ j" e 2 dt /2 + j'(O) df = e * (x * ) db + f * (I * ) dt + df * ,
and to complete the construction, it is enough to show how to obtain the general e* ( =1= 0) and /*( = 0 near oo ) belongin g to C 00 (R1) from e*(j) j' e and f*(j) j"e 2 /2, by choice of e and j. But 0 =1= e = e * (j)/J ' e C 00[0, oo) if j is as described, so it suffices to solve j"(j ') - 2 = 2f */e* 2 (j) for j E C 00 [0, oo) with j ' > 0, j(O) = 0, and j( oo) oo . This problem can be converted into =
=
=
and it is easy to see that an admissible solution exists iff* = 0 near oo , as is assumed. I d entification of I as the refl ect i ng d iffu sion gove rned by
G
Step 1
For e = 1 and/ = 0, the solution of Skorohod's problem for x � 0 is I = x + b - mins � t (x + b) A 0, and much as in Section 3.5, it follows from the uniqueness of solutions that this motion begins afresh at Brownian stopping times. Now evaluate P[I(t) < y] for x 0 using the joint distribution of b (t ) and ma xs � b(s), stated as (d) of Problem 2, Section 2.3 : =
t
3
74
STOCHASTIC INTEGRAL EQUATIONS
[
P[x ( t) < y] = P rr::� b(s) - b ( t) < y 00
=
J dYJ J 0
=
,-y
J
d � (2/n t 3 ) 1 1 2 (2YJ - �) exp [ - (2YJ - � ) 2 f 2t]
fo d17 (2fn t) 1 12 [exp ( - 1] 2/2 t) - exp ( - (17 + y) 2/2t)] r (2/n t) 1 12 exp ( - 1] 2/2 t) d17 P[ l b( t) l < y] 00
=
,
(d = 1)
=
0
and use this to compute p = oP[I(t) < y]foy for x � 0 with the help of formula (c) of Problem 2, Section 2.3, and the fact that I begins afresh at its passage time to 0. Because lx + b(t ) I (t � 0) begins afresh at its stopping times, the result must be the same as p = oP[ I x + b ( t ) l < y]fo y = (2n t ) - 1 1 2 [exp ( - ( x - y ) 2 /2 t) + exp ( - ( x + y) 2 /2t )] . p
is the elementary solution with pole at x of oufot = (1/2) o 2 ufoy 2 , subject to u+ (0) = 0, S O the proposed identification of I as a reflecting Brownian motion is complete.
But this Ste p 2
Define I = X + b. Given e( ;6 0) from C 00 [0, oo), t(t) = s; e( lxl ) 2 , and I * = x(t 1 ), the time substitution rule of Section 2.8 can be used to verify that dx* = e( II * I) da with a new Brownian motion t- 1 a( t) = dbfe( !x!). -
-
J
0
Because e( lxl) is even, l x* I begins afresh at its stopping times, as the reader will easily verify, and since Gu = e 2 ( l x l ) u /2 governs I*, it is easy to see that l x* I is governed by G cut down to [0, oo ), subject to u+(O) = O . t In fact, if p( t, x, y) is the elementary solution of oufot = G*u on Rl, then for x � 0, p (t, x, + y) + p(t, x, - y) is the transition density for l x* I , and the result is trivial from this. Because the solution produced in Step 2 of the existence proof comes from the reflecting Brownian motion x + b - min s � ( x + b) v 0 via the same recipe as "
t
t e( lx l ) need not be smooth at x = 0, so the statement that G governs I* is not
automatic from Section 3.5. The reader is invited to make a proof which avoids this obstacle.
3.9
75
REFLECTING BARRIERS
leads from the reflecting Brownian motion l x l = lx + b l to lx* l , it must also be governed by G cut down to [0, oo ), subject to u + (0) = 0. Step 3
This is merely the application of the mapping j to the motion of Step 2. The reader will fill in the details. Id entification of f as local t i m e at x =
Step 1
0
For e = 1 , / = 0, and x = b - mins � t b(s), it is enough to prove that - min s t b(s) coincides with the local time f(t) = lim (2e)- 1 measure(s � t : x(s) < e).t �
e J, O
The existence of this local time follows from Section 3.9 and the fact that l b l is a second description of I. Using the joint distribution of b(t) and maxs � t b(s) from (d) of Problem 2, Section 2.3, it develops that
[
2 1 D = E - mi n b (s) - (2e) - measure(s � t : x(s) < e) with
s�t = A - 2B + C
]
[ ] = J (2/nt) 1 1 2x2 exp ( - x2 f2t) dx = t ; B = E [max b(s) (2e)- 1 measure(s � t : b(s) > max b(r) - e) ] s�t r�s = (2e) - 1 J� ds E [ �:; b (s), b(s) > �:; b(r) - e J � (2e)- 1 J� ds E [ (s) + ��x [b (r + s) - b(s)], . b(s) > �:; b(r) - e ]
A = E max b(s) s�t
2
00
0
b
J J dq J t
= ( 2e)- 1 ds 0
x t See Levy [2].
00
0
,
,- e
d� [17 + (2(t - s)fn) 1 1 2 ]
(2/ns3) 1 1 2 (2YJ - �) exp ( - (2YJ - � ) 2 / 2s),
3
76 and
STOCHASTIC INTEGRAL EQUATIONS
t
(d = 1)
lim B � -! J ds J dq [17 + (2(t - s) /n) 1 ' 2 J e -1- 0 0 0 x (2/n s 3) 1 1 2 rJ exp ( - 17 2 /2 s) 00
t
1 t
= ! J ds + - J (t - s) 1 f2 s - 1/ 2 ds o n o 1 = t/2 + (tfn) ( 1 - () ) 1 1 2 () - 1 1 2 d() = t ;
t
C = E[l (2e)- 1 measure(s � t : x(s) < e)l 2 ] s t
t ds t dr P[x(r) < e, x(s) < e] s t = -!e - 2 t ds I dr t (2/nr) 1 ' 2 exp ( - � 2 /2r) d� o
= 2(2e)- 2
00
x
and
e
J (2n(s - r))- 1 1 2 [exp ( - (17 - � ) 2/2(s - r)) 0
+ exp ( - (17 + �) 2 /2(s - r)) ] drJ ,
s
1 t
lim C = 1C
e -1- 0
J ds J [r(s - r)] - 1 1 2 dr = t. 0
0
lim e + 0 D = 0 follows. Because f and - min s � b(s) are both continuous functions of t � 0, the identification holds for every t �0 simultaneously. t
Ste p 2
Putting I = b + f with f = - mins � t b(s) as in Step 1 and x * = x(t- 1 ) with t(t) = e(x) - 2 gives
I:
f = lim (2e) - 1 measure(s � t : x* (s) < e) e-1- 0 = lim (2e) - 1 I ds e-1-0
= lim (2e)- 1 e -l- 0
s�t
z( t - 1 (s)) < e
J
s � t - l (t ) z(s) <
= e(0)- 2 f(t- 1 ),
e
i.e. , f* = f(t- 1 ) is the local time of x * .
e(x)- 2 ds
SOME SINGULAR EQUATIONS
3.10
77
Step 3
This is merely the application of the mapping j, as before, and can be left to the reader. SOME SINGULAR EQUATIONS
3.10
The present section illustrates some of the pathological things that can happen to the solutions of dx = e( x) db + f( x ) dt for singular coefficients e and f 3.10a
Lordan 's Example
Lordant proved that if f = 0 for x � 1 and f = - 1 for x > 1 ,
then
x (t) = a(t) + J /(I ) d s t
has only one nonanticipating solution for Brownian paths a = b, but no solution at all for a = t/2. Proof of nonexiste nce for
a = t /2
I
I · = + t according as x < 1 or x > 1 , so rises from 0 to 1 between time t = 0 and t = 2 and sticks at = 1 from t = 2 on. But for t � 2, x · = t + f( 1) = 0, and that is impossible.
I
Proof of exi ste nce fo r
a=b
Given decreasingf_ � � � f+ from C 1 (R1) with /+ = 0 (x � 0) and = - 1 (x � 2), let x ± be the nonanticipating solution of x = and notice that � x _ since, for I + < x _ , b+
f±
J� f±(I),
I+
(I + - I _ ) · = f+ ( x + ) - f_ (x _ ) � f+ ( x _ ) - f_ ( x _ ) � 0.
Make f_ if and f+ ! ! for x =1= I , f_ ( l ) i - 1 , and f+ (l) ! 0. Then x _ i l) _ , x + ! l) + , and since t Private communication.
78
3
STOCHASTIC INTEGRAL EQUATIONS
(d =
1)
Cameron-Martin's formula (Section 3.7) shows that P[x ± ( t) e A] = E [b ( t) e A , 31 :t: ( t)] tends to the common value P [ lJ ± ( t) e A] = E [ b( t) e A , 31( t)] , in which
But lJ � lJ + and lJ ± is continuous, so P[ lJ P[measure(t � 0 : l) (t) = 1 ) 0]. = 1 , .
_
=
Jo/( l)) t
=
t
= lim x + - b = 1J - b = lim x _ - b
=
3.10b
1 - .J-1
J _ (x_ ) ;?; lim J f- (lJ ) = J f(lJ ) , f 1 - tl 1 - tl
= lim
of x
= lJ + , t � 0] = 1 , and since
lJ ) ;?; lim J f+(x+ ) J f+( I + .J-1 o I + .J-1 o lim
t
I + .J-1
i.e. , 1) = b +
_
t
t
t
0
0
0
s;f(lJ ) . The proof is finished by noticing that any solution b + s:f(x) lies between and X+ . L
Girsanov's Example
The classical problem x(t) = s; l x l " ds has only one solution X = 0 for rx � 1 , but for 0 < rx < 1 and 1 rx = p, x(t) (Pt) 1 1P is also a solution and so is -
=
3. 1 0
SOME SINGULAR EQUATIONS
x(t) = 0 = [p( t - t )J 1 1
79
Ip
for each choice of t 1 � 0. Girsanov [2] discovered a similar phenomenon for the Brownian case :
has only one nonanticipating solution x = Ofor rx � t, but for 0 < rx < !, it has an infinite number of them. Girsanov's result will now be proved. Proof of u n i q u e n ess fo r
rx
�!
Suppose that x ¥= 0 is a nonanticipating solution for small times and define t(t) lxl 2". Section 2. 5 tells us that a(t) = x(C 1 ) is a Brownian
= I�
=
1
= I�
motion near t 0. Because x =/= 0, C (t) Ia I - 2" < oo for small times, and now a contradiction is obtained by introducing the local times f(x) of Problem 2, Section 3.8 : f(O) is identical in law to maxs � b(s), as can be seen from Step 1 of Section 3. 1 0, especially, f(O) > 0:- Besides, f is continuous at x 0, and so the integral la l - 2" 2 J f lxl - 2" dx diverges. C 1 (t) t
= I�
=
=
Proof of non u n i q u en ess fo r 0 <
rx
= I�
ib l - 2". Because E(t) < oo , the integral converges, Define t(t) and using the time substitution rule of Section 2. 8, it develops that
with the new Brownian motion
= I;
a(t) = J
t- 1
0
lbl - " db,
i.e. , besides x = 0, x lx l " db has a second nonanticipating solution x b(t - 1 ). Additional nonanticipating solutions can now be obtained from x = 0 and x b(t - 1 ) either as in the classical case or by (singular)
=
=
80
3
STOCHASTIC INTEGRAL EQUATIONS
(d =
1)
time substitutions t --+ j - 1 (t) with j(t) = t + m f (t - 1 ) and m � 0, f being the local time f( t ) = lim (2 e ) - 1 measure(s � t : 0 � b (s) < e).t e J. O
3.10c
The Bessel Process
Consider the Bessel process r = (b 1 2 + b 2 2 + b3 2) 1 1 2 associated with the 3-dimensional Brownian motion. P[r =I= 0, t > OJ = 1 t and dr = db + dt/r with a new !-dimensional Brownian motion b, § so ,
r(t) = b(t) + I; , - 1 (s) ds.
McKean� proved that r(t) = a(t) +
I� r - 1 (s) ds has
(a) a single nonnegative (nonpositive) solution for any continuous path a with a(O) � 0 ( � 0), (b) no other solutions for Brownian paths a = b, and (c) an infinite number of solutions for some (non-Brownian) paths. Proof of (a)
I;
Because r = - a + ( - r) - 1 , it is enough to deal with nonnegative solutions. Consider the difference D of two nonnegative solutions r1 and r2 • D = D/r1 r2 , and since D/r 1 r2 is summable, it is permissible to differentiate and to conclude from D D = - D 2 /r1 r � 0 that 2 r= D 2 � D2(0) = 0. As to existence, the Bessel process satisfies b + r - 1 , so it is possible to pick translated Brownian paths -
- I�
•
I�
b 1 � b 2 � etc. > a such that bn ! a as n i oo and r = bn + I> - 1 has a positive solution r rn for each n � 1 . D = rn - rn _ 1 cannot change sign since D � - D/rn rn _ 1 , so r 1 � r � etc. ! r and since r; 1 i r 1 2 1 as n i oo , r oo = a + I; r;;:, . =
•
00 ,
00
t Ito-McKean [1 ] is referred to for the proof of this second recipe. Girsanov [2]
gives a description of all the nonanticipating solutions. t Problem 7, Section 2.9. § Problem 6, Section 2.9. � McKean [1 ] ; the present proof is much simplified.
3. 1 0
SOME SINGULAR EQUATIONS
81
Proof of (bj
Use the fact that for the Bessel process, P[r # 0, t > OJ = 1 .
Proof of (c)
Choose a continuous function r � 0 with r(O) = 0 and I� r - 1 < oo (t � 0) such that r(t) = 0 has an infinite number of roots. Define a = r I; r - 1 . Then r satisfies r = a + r - 1 , but at each root of r(t) = 0, a is negative, and it is possible to switch over to the non positive solution, especially, an infinite number of solutions exist. -
I�
4
4.1
STOCHASTIC INTEGRAL EQUATIONS (d � 2)
MANIFOLDS AND ELLIPTIC OPERATORSt
A d-dimensional manifold M is a path-wise connected Hausdorff space covered by a countable number of (open) patches U with patch maps j attached. j is a topological mapping of U onto the open unit ball l x l < 1 of R d , and j2 j1 1 is an infinitely differentiable topological mapping (diffeomorphism) of j1 ( U1 n U2 ) onto j2 ( U1 n U2 ). j permits us to introduce local coordinates x j(z) for z E U, and the overlap conditions permit us to speak of the class C 00 (M) of infinitely differen tiable functions from M to R 1 . t A mapping G : C 00 (M) C 00 (M) is an elliptic differential operator if it can be expressed on a patch U as o
=
�
G
=
1 L eij o2joxi oxj + L fi ojoxi + g i, j �d
i�d
t Singer [1 ] is suggested for general information about manifolds. t Warning : no implication of boundedness of the function or of its partials is intended.
82
4. 1
MANIFOLDS AND ELLIPTIC OPERATORS
83
with coefficients e ii e ii (x) (i, j � d), !'i !'i(x) (i � d), and g belonging to C00( U), e = [e ii] being symmetric [e e *J t and positive [e > 0, i.e. , y ey > 0 (y =I= 0)] . Because the action of G on C00(M) does not depend upon the patch map, a change of local coordinates x --+ x' induces a change of coefficients, expressible in terms of the (nonsingular) Jacobian J ox'/ox as =
=
=
·
=
e --+ e'
= JeJ *
f --+ f ' =
(G g)x ' t -
g --+ g' = g .
Define Je to be the positiv� symmetric root of e and let us verify the following simple facts for future use. (1) J; E C00( U).
( 2) Je '
JJeJ*. A second expression for Je' is JJe o with orthog onal o = J(je) - 1 JJeJ* E C00( U). (3) Define e 1 1 2 to be a root of e if e 1 1 2 (e 1 1 2)* e and e 1 1 2 E C00( U) ; such a root transforming according to the rule (e 1 1 2)' = Je 1 1 2 does not exist in general. (4) Jdet e - 1 dx defines a volume element on M. § =
=
Proof of (I)
Je can be expressed as
l�o c�2) (e
-
l)n
if 0 < e �
1,
and this sum can be differentiated termwise. The general case follows easily. Proof of (2)
e'
=
JeJ*, so the first statement is plain. Now compute oo* = 1 and
deduce from (1) that
0E
C00( U).
t The * means transpose. t f=(ft , • • • , icJ).
§ det means determinant.
4
84
STOCHASTIC INTEGRAL EQUATIONS
(d � 2)
P roof of (3)
A mapping D : C00(M) --+ C00(M) is a nonsingular 1 -field if D(uv) (Du)v + u(Dv) for any u and v from C00(M) and if D =1= 0 at any point of M; such a map can be expressed on a patch U as D I i � d fi a; ax i = f grad, with f =I= 0 belonging to C00( U) and transforming by the rule f' = Jf Because [det e 1 1 2 ] 2 det e =I= 0, the rule (e 1 1 2 )' Je 1 1 2 states that the columns of e 1 1 2 define d independent nonsingular 1 -fields : =
=
·
=
=
(j � d). But this is not possible in general ; for instance, the spherical surface S 2 : lx I = 1 c R 3 does not admit any nonsingular 1 -field. A simple proof of this classical fact can be made as follows. A nonsingular 1 -field on S2 can be regarded as an actual tangent direction y =1= 0 attached to each point of the spherical surface. Consider a longitudinal circle
C : cp
=
colatitude = constant =1= 0 or
rc,
0 � (}
=
longitude < 2n ,
let t/1 be the inclination of y to the eastward direction at a point of C, and let n be the winding number of t/J during an eastwards passage around C. n is a continuous function of 0 < cp < rc. But near the north pole [small cp ] , n = - 1 , while near the south pole [big cp ] , n + 1 , and this con tradiction finishes the proof. =
Proof of (4)
jdet (e') - 1 dx' = jdet J * - 1 e - 1 J- 1 j det J l dx = jdet e- 1 dx .
Pro b l em 1
Q is an application into R 1 of the class C00(0) of germs of infinitely differentiable functions at 0 e R d . Prove that Qf � 0 for any non negative f with f(O) = 0 if and only if Q = t L eij o 2 fox i axj + I fi o f oxi + g i �d i , j� d with e = [e ii] � 0. Problem 6, Section 4.3, contains an application of this fact. Sol ution
Given f E C00(0) with f(O) 0 � f, define o 2f to be the Hessian [ o 2f/ ox i oxi] evaluated at x 0 . This is a nonnegative form since =
=
4.2
WEYL ' S LEMMA
85
y · o 2fy c < 0 implies f(ey) e 2 c/2 + O(e 3 ) < 0 for e ! 0, so for Q as stated, 2Qf sp [ e o 2f]t is nonnegative. Now suppose Qf � 0 for any fE C 00 (0) with f(O) 0 � � Given a general fE C 00 (0), f - [f(O) + grad f(O) · x + tx · o 2f x] + clxl 2 vanishes at 0 and is > 0 or < 0 for small x ¥= 0 according to the sign of c ¥= 0. Thus Qf = Q[f(O) + grad /(0) · x + !x · o 2fx] , so that Qf is of the desired form : Qf f(O) Q l + grad f(O) · Q x + t i C o 2f) ii Q x i xi . Put f = (x · y) 2 for fixed y E R d . Then 0 � 2Qj' = L Yi Yi Qxi xi , and this shows that [ e ii] = [Qx i xi] is nonnegative. =
=
=
=
=
4.2
WEYL'S LEMMA
Weyl's lemma, already used in Section 3.6, will now be proved. The reader can just note the statement and skip to Section 4.3 if he likes. Consider an elliptic operator G defined on a manifold M as in Sec tion 4. 1 and let G* be its dual relative to the volume element dz (det e - 1 ) 1 1 2 dxt : =
for compact j1 and j2 E C 00 (M). Weyl's lemma states that if u is th e (formal) density of a mass distribution on (0, oo) x M and if
J
u( - 8/ot - G * )j dt dz =
J
vj dt dz
( 0 , 00 ) X M 00 ) X M for some v E C 00 [(0, oo) X M] and any compact j E C00 [ (0, oo) X M] , then u can be modified so as to belong to C 00 [(0, 00 ) X M] ; after this modification, (ojot - G)u v in the usual sense. Because the proof is particularly simple for u and v not depending upon t � 0 [Gu = - v], it will be easiest to begin with this special case. The proof is adapted from Nirenberg. § (0,
=
t sp means spur or trace. t See (4), Section 4. 1 . § Nirenberg [1 ] ; see Bers et parabolic problems.
a/.
[1 ] for general information about elliptic and
4
86
STOCHASTIC INTEGRAL EQUATIONS
Step 1
Bring in the space Dn (n >
(d � 2)
- oo) of formal trigonometrical sums :
f = I j(l) exp (j - l l · x)t l e zd
with J = conjugate ]( - ·), and
viewing/as a (formal) function on the d-dimensional torus T = [0, 2n)d, and let us prepare some simple facts for future use. ( 1 ) nn - 1
Dn ,
and n n > Dn = C00 (T) is dense in Dn . (2) a is a bounded application of nn into nn - 1 with bound 11 8 11 � l . t (3) f jf is a bounded application of Dn into nn for any j E C00(T), and ll jf l l n � ll j ll oo ll / ll n + c1 l l / ll n - 1 � c2 ll f ll n with constan ts depending upon j and n but not upon f § ==>
_
oo
�
Proof of (1)
This can be left to the reader. Proof of (2) o is defined first on C00 {T)
Dn
and then closed up. The bound is plain from the formal sum for of if f E C00 {T). c:
Proof of (3)
f jf is defined first on C00 (T) and then closed up as before, so it is enough to prove the bound for f E C00(T). But for such f, �
IIi! II / =
f) if 1 2 � IIi II
oo
2 11 / II o 2 ,
and since 111 11; + 1 = La l l (d - 1 12 + o)f ll n 2 , the bound for some n � 0 implies the bound for n + 1 : t Zd is the lattice of integral points of Rd. t o stands for any one of oj ox1 (i � d) . § IJjoo II is the upper bound of Ul on T.
4.2
WEYL ' S LEMMA
87
I i! I I ;+ 1 Ia I Cd- 1 1 2 + a)ifl l n2 � La [l l j ( d - 1 1 2 + o )f lin + I I ( oj)f 11nJ 2 � Ia [I I i I oo I I Cd - 1 1 2 + o)f l i n + c 1 I Cd - 1 1 2 + o)f l n- 1 + l oil l oo l f l n + c 2 l f l n- 1 J 2 � Ia [l l il l oo I Cd - 1 1 2 + o)f l n + c3 1 1 !11nJ 2 � [l l i l oo II! l i n + 1 + c4 1 1 f llnJ 2 , completing the proof for n � 0. As to the case - n < 0, Dn and n - n have a natural pairing under which the dual of the multiplication Dn }Dn is just the multiplication n- n JD - n , so that l i/ 1 1 - n � c 5 11!1 1 - n . Now }(1 - �) - (1 - �)it is a differential operator of degree � 1 with coefficients from C00(T) ; as such, it is a bounded application of nm into nm - 1 (m > - 00 ) by (2). ( 1 - L\) - 1 is an isometry of nm onto nm + 2, so =
�
�
n+ 1 L=O( 1 - L\) - k [j( 1 - �) - (1 - L\)j] ( 1 - L\) - n + k - 1 k is a bounded application of n - n - 1 int o n - n- 1 + l( n+ l) - 1 nn . (3 ) n ow follows for - < 0 : l j/1 1 - n 1 (1 - �) - njfl l n � II j(1 - L\) - 'J I n + II [j(1 - L\) -n - (1 - L\)- nj]J I n � II i l l oo 1 (1 - �) - 'JI I n + c6 1 1 (1 - �)-'JI I n 1 + c7 1 !1 1 -n - 1 � l jl l oo 11/1 1 - n + Cs l / 1 1 - n - 1 · =
=
n
=
Step 2
Because of (2) and ( 3) of Step 1 , an elliptic operator Q on T can be regarded as a bounded application of nn + 2 into nn ; the purpose of this step is to prove an a priori bound:
1 1 ! 1 n + 2 � c 1 11 Qflln + c2 11!1 1 n+ 1
4
88
STOCHASTIC INTEGRAL EQUATIONS
(d � 2)
with constants depending upon Q and n but not upon f Q can be expressed using the global coordinates 0 � x i < 2n (i � d) of T. Because the part of Q of degree � 1 is a bounded application of D" + 1 into D", it is permissible for the proof to suppose that Q has no such part : Q = 1 L e ij a 2 I ax i axj . As usual, it is enough to prove the bound for f E C00(T). Proof for con stant coeffi cients
Define y to be the smallest eigenvalue of the quadratic form based on the (top) coefficients e/2 of Q . Then
[I I Qf l n + j2 y I J I n + t] 2 � 11 Qf lln 2 + 2y2 IIJ I ; + 1 L 1]( 1)1 2 ( 1 + 1 11 2 ) " [l ( 1 · e1 ) 2 + 2y2 (1 + 1 11 2 )] � Y 2 II! 11; + , 2 since ( / · e/) 2 � 4y2l/l 4 • This establishes the required bound y - 1 and c2 j2. =
c1 =
with
=
Proof for n on con stant coeffi cients
Define y > 0 to be the minimum of the lowest eigenvalue of the (top) coefficients of Q and take b > 0 so small that on a ball of diameter < b, Q can be replaced by Q' with constant coefficients and lowest eigenvalue � y , keeping the moduli of the (top) coefficients of Q - Q' smaller than yj2d2• By the bound for constant coefficients,
I jj I n + 2 � }' 1 I Q jj I n + Ji II jj I n + -
1
1
for j E C00(T), 0 � j � 1 ' and j = 0 outside a ball of diameter < b. But also
II Q'jfl l n � II ( Qj - j Q )f l n + l j QJ I n + II ( Q' - Q )jflln � Ct l f l l n + tt + c 2 11 Q fl l n§ + L li e 0 2 jf l nt � ct l fl l n + l + c 2 1 Q f l n + I: [ll e l oo 1 a 2 jflln + c3 1 a 2 jfl l n- 1 J § � c4 11f lln + + c 2 I Qf l n + ( y/ 2) l iflln + 2 , 1
t Qj - jQ is of degree � 1 . t L e 82 stands for Q - Q ' . § (3) of Step 1 .
4.2
WEYL ' S LEMMA
89
so
ll jfl l n + 2 � Cs !I Qflln + c6 llf lln + l· Now express the function 1 as a finite sum of such functions j and conclude that l flln + 2 � Ill iflln + 2 � c7 II Qflln + Cg l fl l n + l· Step 3
Weyl's lemma for Gu = - v can now be proved with the help of the a priori bound of Step 2. The statement is that, if u is the (formal) density of a mass distribution on M, and if
J
M
uG *j d z = - J
vj dz M
for some v E C00(M) and any compact j E C00(M), then u can be modified so as to belong to C00(M) ; after this modification, Gu - v in the usual sense. =
Proof
Because the statement is local, it suffices to prove it on a patch U. Modify the local coordinates x on U so that the torus T = [ - n, n) d sits inside U, pick cotnpact jl and j E C00 (M) such that
2
i t = 1 on [ - n/4, n/4] d j 2 = 1 on [ - n/3, n/3] d = 0 off [ - n/3, n/3] d = 0 off [ - n/2, n/2] d , and let Q be an elliptic operator on T coinciding with G on [ - n/2, n/2] d . Regardj1 u as belonging to n - n for n > d/2.t Q j1 u + j1 v can be expressed
as a differential operator of degree � 1 with coefficients from C00(T) acting on j u, so the a priori bound of Step 2 implies
2 l itu l - n + l � ct i ! Qjtu l - n - 1 + c 2 ll j tu 1 - n � ct l i tvl l - n - 1 + C3 ll j 2 u l -n + C 2 i l jtu ll - n
i.e. , jlu E
n -n + 1 •
< 00 , Repeating the estimation, we find that
The rest is plain. t (jt u)" is bounded.
jlu E n D" = C00( T) .
n> -
oo
90
4
STOCHASTIC INTEGRAL EQUATIONS
(d � 2)
Step 4
Weyl's lemma for (8/8t - G)u = v can now be proved in much the same way. Bring in the space n m f n of formal sums
f(k , 1) exp (j - l kt) exp (J=l 1 · x ) f = kL e Z1 l
e zd
with J = conjugate ]( - ·) and II ! 11; 1 " = L l]( k , 1) 1 2 ( 1 + k2)m( l + 1 1 1 2 )" < oo , viewing f as a (formal) function on T = [n, n)d + 1 . The map a;at - Q is a bounded application nm ln into nm - 1 /n - 2 for Q as in Step 2, and the a priori bound
ll f ll m + 1 / n + ll f ll m ;n + 2 � c 1 1 1 ( 8/ 8 t Q ) f ll m;n + c 2 11 f ll m;n + 1 -
is proved much as before. Q can be supposed to have no part of degree � 1 . Then 1 (8/8t - Q) exp ( j�Ikt + J- 1 l · x) l 2 = IF-I k + ! l · ell 2 � constant X (k 2 + 1 1 1 4), S O that there is no interference between 8j8t and Q ! The rest of the proof is similar to the elliptic case. Warning : from this point on, G stands for an elliptic operator with G 1 = 0. G * denotes its dual relative to the volume element ( det e - t ) 1 1 2 dx. 4.3
DIFFUSIONS ON A MANIFOLD
Ito [3, 8] proved that if G is an elliptic operator on a manifold with G 1 = 0, then the local solutions of
x ( t) = x + J je (x) db + J !(x) ds t
t
0
0
M
on the patches U of M can be pieced together into a diffusion 3 governed by G. This means that (a) the path 3 : t M is defined up to an explosion time 0 < c � oo , (b) c = oo if M is compact, while 3(e - ) = oo if c < oo and M is noncompact,t (c) 3 begins afresh at its stopping times, i.e. , if t is a stopping time of 3, then , conditional on t < c and 3 (t) = z, the future 3 + (t) = 3(t + t) : t < c + = c - t is independent of the past 3(s) : s � t + and identical in law to the motion starting at z, �
t oo is the compactifying point of M in the noncompact case.
4.3
DIFFUSIONS ON A MANIFOLD
(d) if t < c is a stopping time of 3 and if 3 (t) belongs to a patch with patch map j, then
91 U
x(t ) = j(3 + ) = x(0) + J Je (x ) db + j f(x ) ds t
t
0 0 up to the exit time of 3 + from U, for a suitable Brownian motion b depending upon the patch map j. (e) the density of the distribution of 3 ( t) relative to the volume element (det e - 1 ) 1 1 2 dx is the smallest elementary solution of 8u/8t = G * u with pole at 3 (0) = z E M, i.e., it is the smallest function p � 0 belonging to c oo [ (0, 00 ) X M] such that 8pf a t = G *p and limt.).O f p (de t e - 1 ) 1 1 2 dx = u 1 for each patch U containing z. Step 1
G can be expressed on a patch u as 1 L e ij 8 2 /8x i axj + L fi a;axi ' and thinking of U as part of R d , Je and f can be extended from the closed ball B : lxl � 1 /2 to the whole of R d so as to make them compact and belong to C00(R d). Given a d-dimensional Brownian motion b ,
x ( t) = x + J Je (x) db + J/ (x ) ds t
t
0 can be solved as in Section 3.2, and for lxl � 1 /2 and e = min (t : l x l = 1 /2), it is easy to see that the (nonanticipating) local diffusion : x 1 (t ) = x(t ) ( t < c) = x( e ) (t � c) begins afresh at Brownian stopping times and does not depend upon the mode of extension of the coefficients. Step 2
Define a path 3 on the union of two overlapping balls B1 : lx 1 l � 1 /2 c U1 and B2 : lx 2 1 � 1/2 c U2 as follows : (1) Begin at 3 (0) = z E Bb say, take a d-dimensional Brownian motion b 1 , base upon it a copy x 1 of the local diffusion for B1 starting at x1 = j1 (z) , t define 3 = j1 1 (x 1 ) up to the exit time c 1 = min (t : lx 1 1 = 1 /2), and if either c 1 = oo or c 1 < oo and 3 ( c 1 ) E 8(B1 u B2 ), stop and put en = 0 (n � 2). t j is the patch map of
U.
92
4
STOCHASTIC INTEGRAL EQUATIONS
(d � 2)
e 1 < oo and 3( e 1 ) E B2 t take the Brownian motion b 2 = b 1 (t + e 1 ) - b 1 ( e 1 ), base upon it a copy x 2 of the local diffusion for B2 starting at x 2 = j2 [3( e 1 ) ] , define 3 = }2 1 [x 2 (t - e 1 )] up to the sum of e 1 and the exit time e 2 = min ( t : I x 2 1 = I/2) , and if either e 2 = oo or e 2 < oo and 3( e 1 + e 2 ) E 8(B1 u B2 ), stop and put e n = 0 (n � 3). (3) But if e 2 < oo and 3( e 1 + e 2 ) E B1 take the Brownian motion b3 = b 2 (t + e 2 ) - b 2 ( e 2 ), base upon it a copy x3 of the local diffusion for B1 starting at x3 = j1 [ 3( e 1 + e 2 )] , define 3 = Ji"" 1 [x3 ( t e 1 - e 2 )] up to the sum of e 1 + e 2 and the exit time e 3 = min (t : lx3 1 = 1 /2), and if either e 3 = oo or e 3 < oo and 3( e 1 + e 2 + e3 ) E 8(B1 u B2 ), stop and put e n = 0 (n � 4), etc. o,
(2) But if
o,
-
3 is now defined up to the explosion time e = lim n t oo e 1
-+-
and the product of 3(t) and the indicator function of (t nonanticipating functional of the Brownian motion b 1 •
•
+ en ,
�) is a
•
•
Step 3
The next step is to prove that the path 3 pieced together in Step 2 is a diffusion compatible with the local diffusions dx = J e db + f dt : namely, 3 begins afresh at stopping times t < e, and if 3( t) belongs to a patch U c B1 u B 2 with patch map j, then
x (t) = j(3 + ) = x (O) + J J ( x ) db + J f( x ) ds, t
0
t
e
0
up to the exit time of 3 + from U, for a suitable Brownian motion b depending upon j. Proof
On a patch U contained in the overlap B1 n B2 c U1 n U2 , 3 can be expressed either as j � 1 ( x 1 ) or as j2 1 (x 2 ) . The point of this step is that this ambiguity does not get us into trouble. Ito's lemma states that under a change of local coordinates x � x' on a patch U, the differential dx = Je (x) db + f(x) dt is changed into
dx ' = J(x )Je ( x) db + (Gx')(x ) dt = Jj db + f' dt.t e
t Bo is the inside of B. � J = ox'fox.
4.3
93
DIFFUSIONS ON A MANIFOLD
(2) of Section 4. 1 states that JJ� Je' 0 with orthogonal 0 E C 00 ( U), so Jj� db = Je' db' with the new Brownian motion b'(t) I� db.t Because of this, the motions }1 1 (x 1 ) and }2 1 (x 2 ) are identical in law on =
=
the overlap B1
n
o
B2 The rest of the proof is left to the reader. •
Step 4
Before Step 5 can be made, an a priori bound is needed. This states that for x ( t ) Je (x) db + J(x) ds and t ! 0, =
I�
l
P �:� l x ( s ) l
I�
�RJ � exp ( - R 2 /2 dy t),
y being the biggest eigenvalue of e(x) for lxl � R.t Proof
Define f3 to be the upper bound of If I for lxl � R. Given a direction (J E sd - l ' Problem 1 Section 2.9, tells us that up to the exit time ' min (t : l x l R) , (J x can be expressed as a !-dimensional Brownian motion a run with the clock t(t) e e( x) e ds � yt, plus an error of magnitude � {J t. Because of this, max s � t 1 8 x (s) l � max s � y t la(s) l + {J t up to the exit time, so that for (J running over the d coordinate directions of space, =
•
=
l
]
�I
·
·
J-]d � dP l max la(s) l � J - {Jt ] s � yt d l
P max j x ( s ) l � R � I P max e . X � s�t s�t (J
� 2d
I(ooyt) -
l f 2 [( R /v'd) - P t ]
exp ( - x 2 /2) d x §
( 2n ) 1/ 2
for t ! 0. t See Problem 3, Section 2.9. t Problem 3 of this section gives a bound in the opposite direction. § See Problem 2, Section 2.3. � See Problem 1 , Section 1 . 1 .
4
94
STOCHASTIC INTEGRAL EQUATIONS
(d � 2)
Step 5
3( e - ) exists and belongs to 8(B1
u
B2 ) if e < oo .
Proof
Using the a priori bound of Step 4, it is easy to see that 3( e - ) exists if e < oo ; for if not, then it is possible to find a pair of nested surfaces contained in a single patch U inside B1 u B2 and separated by a distance R > 0, such that P(Z) > 0, Z being the event that 3 passes from the inner to the outer surface and back, i.o. , before time c . But if t 1 < t 2 < etc. < e are the successive times of returning to the inner via the outer surface, then the a priori bound implies
(n i oo )
with a suitable constant y, and an application of the first Borel Cantelli lemma gives the absurd result : oo > c � the tail of L l /n = oo on Z. Step 6
Define Bn (n � 1 ) so that Bn overlaps U i < n B i and U n � l Bn = Mt and let 3 2 : t < e 2 denote the motion of Steps 2-5. Using the same recipe with 3 2 and the local diffusion x3 for B 3 in place of x 1 and I 2 gives a motion 33 : t < e 3 on B1 u B 2 u B 3 with the same properties as those elicited for 3 2 in Steps 3 and 5 : namely, 33 is defined up to time c 3 , it begins afresh at its stopping times, it agrees with the appropriate local diffusions on patches of B1 u B2 u B3 , and 33( e3 ) E 8(B1 u B2 u B3 ) if e 3 < oo . Continuing in this way, it is easy to define such motions 3 n : t < en on u i � n B so as to have 3n 1 = 3n up to time en 1 < en ( n � 3) . But then the path 3 = 3n( t < en ) is defined up the explosion time e = l imn t oo en and satisfies (a), (b) , (c), and (d) , as the reader can easily verify. The only tricky point comes in connection with (b ) if M is compact. Then M can be covered by a finite number of balls B so that a U i
-
Step 7
G governs 3, i.e. , (e) holds. t M is connected.
-
4.3
DIFFUSIONS ON A MANIFOLD
95
Proof
As in Step 1, Section 3.5, it follows easily from the lemmas of Ito and Weyl that the (formal ) density p of the distribution of 3(t ) belongs to C00 [(0, 00 ) X M] and is an elementary solution of oujot = G * u with pole at 3(0) = z ; it remains only to show that it is the smallest such. The proof is divided into 2 cases according as M is compact or not. For compact M, we borrow from the literature the fact that for any j E C00 (M) , oujot Gu has a solution u E C 00 [(0, 00 ) X M] with data u(O + , · ) = }. t This solution is easily identified as £[}(3 )] , as in Step 2, Section 3.5, and it follows by the line of reasoning of Step 3, Section 3.5, that p is the only elementary solution of oujot = G * u. For non compact M, the proof is only a little more elaborate. Take a smoothly bounded region B c M with compact closure B and borrow from the literature the fact that for compact nonnegative j E C00 (B), ouj ot = G u has a nonnegative solution u E C00 [(0, oo) x B] with data u(O + , · ) = j and u = 0 on oB. t Define f = min (t : 3 E oB) . Then u can be identified as £[}(3) , t < f] inside B, as in Step 2, Section 3.5, and then the argu ment of Step 3, Section 3 . 5, can be carried out with the aid of Green's formula. The conclusion is that p is smaller than any other elementary solution of oujot = G * u, i.e. , (e) holds. =
Step 3 should be amplified by noting that it is not always possible to define the local diffusions I using a single Brownian motion b in such a way that under a change of coordinates x � x ' , I changes into I ' = x' (I) . By the computation of Step 3, this would mean that di ' = fl db + f' dt = Jj e db + f' dt. But then Je' = Jje, and this is not possible for M = S 2 for instance, even permitting nonpositive roots of e. t Lamperti's method for solving di + J� (I) db + /(I) dt§ meets the same geometrical obstruction , as the reader can easily check. Ikeda [ 1 ] has adapted Ito's method to the case of manifolds with boundary. Unfortunately, it would be too long a job to explain this beautiful development, but see Section 4. 1 0 for a discussion of the Brownian motion in a disk with oblique reflection. For the most up to date information, see Motoo [2] and Sato-Ueno [ 1 ] . Ito's method can t See, for example, Bers et a/. [1 ]. t See (3), Section 4. 1 . § See Section 3 .4.
4
96
STOCHASTIC INTEGRAL EQUATIONS
(d � 2)
also be used to solve 8u/8t = Au/2 for differential forms on a manifold.t Problems 1-5 below concern the case M = Rd. G is expressed using the global coordinates of Rd, and x denotes the solution of t -t x (t) = x(O) + Je (x) d b + (x ) ds .
t
Pro b l e m 1
J/
Define y = y(x) to be the biggest eigenvalue of e (x) . Prove that for x(O) = 0, p
and
[-.
l x ( t) l = Jy ( O) 1/� (2 t lg2 1/ t) 11 2
J
=
1
p
Sol ution
Proceed as in Problem 4, Section 2.9. Problem 2
P[ e
=
oo]
=
1 if l e l 2
+
1/1 2 � constant
x
(1 + lxl 2 ).
So l ution
Proceed as in Problem 1 , Section 3 . 3 . Pro b l e m 3
[
]
P max lx (s)l � R � constant x J t exp ( - R 2/2dyt) s�t
for t ! 0 and x (O)
=
0, y being the smallest eigenvalue of e (x) for lxl � R.
So l ution
Recall Step 4. Up to the exit time min(t : l x l
=
R),
max lx(s )l � max la (s) l - {3 t , s�t
s � yt
a being a ! -dimensional Brownian motion and {3 the upper bound of I l l for lxl � R. The reader will finish the proof much as in Step 4. t See Ito [1 0].
4.3
DIFFUSIONS O N A MANIFOLD
97
Problem 4 '
Define }' + (Y ) to be the biggest (smallest) eigenvalue of e(x) for lxl � R. Prove that for f = 0, x(O) = 0, and tR = min (t : II I = R), }' - E(tR) � R 2 � }' + E(tR). _
So l ution
Up to time tR ,
max lx( s )l � max la(s)l s�t
s�y_t
with a !-dimensional Brownian motion a, as in Problem 3 . Accordingly, P[tR < oo] = I. By Ito's lemma, dlxl2 = 2x Je db + sp e(x) dt,t so
·
t R2 = lim E[lxl2(tR A n)] = E[ J a sp e(x) dt ] , n
0
i oo
and the result follows. Problem St
Define I to be the solution of di = Je (x) db, x1 the solution of di = Je (I) db + ef( x) dt with the same starting point x'(O) = x(O), e the explosion time for I, e1 the explosion for x1, and 3 the functional Je (x) db - t ! ef( x) ds . The problem is to prove that exp P[e = oo] = 1 and that the Cameron-Martin formula,
[ f�J·
J�
·
]
P[I1 E B, t < e1] = E[x E B, 3(t)] , holds for events B depending upon I(s ) : s � t alone. §
So l ution
Proceed as in Section 3.7. Problem 6
Given a (nonexplosive) diffusion 3 on a manifold M, defined as in (a), (b), and (c) at the beginning of this section, put u = E[v(3)] for compact v E C00 (M) and assume that Gv = t lim t -l- 0 t - 1 [u - v] exists pointwise for all such v. Deduce from Problem 1 , Section 4. 1 , that G is an elliptic partial differential operator. t sp means spur. � Section 3. 7 contains the ! -dimensional case of this. § Dynkin [1 ] ; see also Girsanov [1 ] or Motoo [1 ].
4
98
STOCHASTIC INTEGRAL EQUATIONS
(d � 2)
So l ution
Deduce first, from Step 4, that G is an application of germs of c oo functions. By Problem 1 , Section 4. 1 , it suffices to verify that Gv � 0 at a (local) minimum of v E C 00 (M ). Pro b l e m 7
Use the results of Step 7 and Weyl's lemma to prove, as in Problem 1 , Section 3 . 5 , that for compact j � 0 from C00(M), J pj = E [j(3), t < e] is the smallest nonnegative solution of 8u/8t = Gu which belongs to C 00 [(0, 00 ) X M] and reduces to j at time t = 0. Pro b l e m 8
Deduce, as in Problem 2, Section 3 . 5, that p, as a function of t � 0, its pole, and its argument, belongs to C00 [(0, 00) X M2] and satisfies the back ward equation 8p/8t = Gp as a function of t > 0 and the pole. 4.4
EXPLOSIONS AND HARMONIC FUNCTIONS
Regard the chance of explosion P[ e < co ] as a function p of the starting point 3(0) = z E M and let us verify that p belongs to C 00 ( M) and is a solution of Gp = 0. Proof
E [v(3), t < e] E C 00 [(0, co) X M] is a solution of 8u/8t = Gu for compact v E C00 (M),t so
u= 1
-
J uj d z �� J J uG*j d z t
=
M
0
M
for compact j E C 00 ( M ) , and
J pG*j dz = lim M
lim
t j oo O � v j l
J
M
1 = lim lim t t j oo O � v j l
t See Problem 8 of Section 4.3 .
u
G *j d z
J uj dz = 0. M
4.4
EXPLOSIONS AND HARMONIC FUNCTIONS
99
Weyl's lemma now supplies us with a function q E C00 (M) such that Gq = 0 and q = p off a null set of M, and to finish the identification of p with q it suffices to note that 1
- p = P[t < e, no explosion after time t] = E[l - p(3), t < e]
is insensitive to null sets a s regards p(3) and therefore tends to 1 - q as t ! 0. A s imple but useful consequence is that for compact M, the path visits each patch, i.o. , as t j oo . For the proof, it is enough to verify that if U is a small patch with smooth boundary and if e is the entrance time inf (t : 3 E U), then p = P[e < oo] = 1 off U. Step 1
p belongs to
C00(M -
U) and Gp = 0 off U.
Proof
e is the explosion time for the motion governed by G on the open manifold M - U. Step 2
p tends to 1 on a u. Proof
p � Pn = P[3(k2 - n) E U for some k � n2n] . Because the elementary solution of au; at = G * u belongs to C 00 [ (0, 00) X M2J ,t Pn E C00(M), and as a point 0 E au is approached from the outside of U, lim p � Pn(O) . As n j oo , Pn(O) j p(O), so it is enough to prove that p(O) = 1 . Express the path by means of local coordinates x about 0 : x(t) = J Je (x) d b + J f(x) ds t
t
0
0
= J e (O)b + J [j e (x) - Je ( 0) ] d b + J f (x) ds . t
t
0
0
t See Problem 8, Section 4. 3,
1 00
4
STOCHASTIC INTEGRAL EQUATIONS
Because lxi = O(t 113) for t ! 0,
J� Je (x) db - Je (O)b
(d � 2)
J� j je (x) - je (O) jl = O(t 5 13),
so that
= O(t 2 13), t and I = Je (O)b + O(t213) for t ! 0. Je (0) is nonsingular and b is isotropic, so it is enough to prove that the path a = b + an error of magnitude o(t 1 12 ) is sure to enter a cone C : a l � n( a 2 2 + . . . + a d2 ) 1 12, i.o., as t ! 0, however big n may be. But this event (Z) contains the event that b 1 � n(b 2 2 + · · · + bd 2)112 + t 1 1 2 , i.o. , as t ! 0, as the reader will easily verify, so P(Z) � lim P [b 1 ( t ) � n(b 2 2 ( t) + t ,J.. O
·
··
+ b d2 ( t )) 1 1 2 + t 1 1 2 ]
is positive, and since Z belongs to the field B 0 + , an application of Blumenthal's 0 1 law does the rest.t Step 3
Because p tends to 1 on o U, it has a minimum at some point 0 inside M - U, M being compact, and this means that p is constant ( = 1 ) , as will n ow b e proved. Draw a small patch U ' about 0 and modify the local coordinates x so that the closed ball lxl � 1 li e s inside it. E(e ') < oo for paths I starting at 0 and e ' = min (t : I I I = 1). § Define 1) = I ( e'). Because
p(l)) - p( O) =
fo grad p e
'
·
Je db,
p(O) = E[p(l))], and since p(l)) � p(O), the fact that p is constant on lxl = 1 would follow from the lemma : P[l) E U "] is positive for every patch U " of the surface lxl = 1 . This would propagate to the whole of M - U and would show that p = 1. t See Problem 4, Section 2.9. i See Problem 1 , Section 1 . 3 ; the reader will supply the easy extension to the d-dimensional Brownian motion. § See Problem 4, Section 4. 3.
4.4
EXPLOSIONS AND HARMONIC FUNCTIONS
1 01
Proof of the l e m m a t
Consider the motion In governed by Gn = G/n + y grad for a fixed y E U" , up to its exit time e n = min (t : I In l = 1 ) . As n j oo , ·
max li n - ty l t � en
tends to 0 as the reader can easily verify, so P[In(en) E U"] is positive for n i oo , and an application of the Cameron-Martin formulat implies that P[lJ E U" ] is positive also. The reader will notice that Step 3 is simply the so-called maximum principle for the problem Gp = 0 : if Gp = 0 on an open region and if p assumes its maximum (or minimum)
inside this region, then it is constant.
Bernstein � proved the extraordinary result that if M = R2 and
f = 0, then without any conditions as to the smoothness of e, every solution p E C2(R2) of Gp = 0 is constant, provided only that e1 1 e 22 ei 2 > 0 at each point of R2 and that p is bounded on both sides, e.g. , 0 � p � 1 . This is made still more striking by an example of Hopf [ 1 ] , showing that the dimension 2 cannot be raised : 02 02 02 02 G = (1 + b 2 ) o + 2b o o + o 2 + exp (2a - b 2 ) o c2 a b b a2 p = exp ( - exp (a - b 2 /2)) sin c + 1 .
Bernstein ' s theorem contains a surprising probabilistic fact : for plane diffusions with f = 0, P[ e < oo] is either 0 or 1 , independently of the starting point. Here is the proof. Bernstein's theorem shows that p = P[ e < oo] is constant since p E C00(R 2 ) , Gp = 0, and 0 � p � 1 . But then 1 - p = P[e = oo] = E[e > n , P( e = oo i Zn ) J � = P[e > n](l - p ) ! ( 1 - p)2 (n i oo ),
so p is = 0 or = 1 .
t From S.S.R. Varadhan (private communication). i See Problem 5, Section 4. 3. § See Bernstein [1 ]. Hopf [2] gives a correction to Bernstein's proof. � zn is the field of &(t): t � n.
1 02
4
STOCHASTIC INTEGRAL EQUATIONS
(d � 2)
The fact that P[ e = oo] = 1 does not mean that the path visits each disk, i.o. , as t i oo . Problem 4, Section 4. 5, shows that P[limtt oo 3(t) = oo] 1 is still possible even for Bernstein's case, and the Brownian motion itself provides a counterexample for d = 3 . Bernstein's theorem implies that P[limtf oo 3 = oo] is either = 0 or = 1 for plane diffusions with f = 0. The proof is the same, and one may conjecture that this is always the case for any noncompact M. A rough proof can be made as follows. t Take P[ e = oo] = 1 and define p = P[limtt oo 3 = oo] . Then p E C00 (M) and Gp = 0 is proved as before, and either p = 0 or p is positive on an open region U. A simple adaptation of the lemma of Step 3 shows that P[3 enters U] is positive for any starting point 3(0), and it follows that p > 0 on the whole of M. Now suppose p < 1 for some starting point 3(0). This means that you must hit some fixed compact K, i.o. , as t j oo with a positive chance, and that is not possible because each time you hit K, you have a positive chance (not smaller than the minimum of p on K) of not coming back, i.o., as t i oo . The reader is invited to fill in the details of the proof. =
4.5
HASMINSKII 'S TEST FOR EXPLOSIONS
Hasminskii [1 ] proved a pair of useful tests for explosions of diffusions on M = R d, similar to Feller's test for d = 1 ( Section 3 . 6). Define e and f for G using the global coordinates of R d and introduce B = A - 1 [2! A _ = min A
·
x
+ sp e]
lxi = R
B_ = min B lxl = R
c_ = exp
[(n-]
A + = max A lxi = R
B+ = max B lxi = R
t From H . Kesten (private communication). t Warning : J stands for integration with respect to R dR throughout this section.
4. 5
1 03
HASMINSKII ' S TEST FOR EXPLOSIONS
Hasminskii's first test states that no explosion is possible [P( e = oo ) = 1 ] if e
and his second that explosion is sure [P( < oo ) = 1 ] if R
f1 c = 1 f1 C_jA _ < oo . oo
The idea is to pretend that G is radial, to form the integral for Feller's test at oo for the associated radial motion 13 1 , and then to make it as difficult as possible for this integral to diverge (converge). If the integral still diverges (converges), then the conclusion of Feller's test still holds. Proof of Hasm i n s ki i 's fi rst test
Define u = u(R2 /2) to be the positive increasing solution : 00
u0 = 1 , u = I un , (n � 1 ) n=O of u = 1A + [u" + B + u ' ] = 1 (A + /C + ) ( C+ u ') ' t for R � 1 , and extend it to R < 1 so as to make the extended function belong to C (R d) Under the condition of Hasminskii's first test, u � u1 j oo as R j oo . Because u ' and u" + B + u ' = ufA + are both positive for R � 1 , Gu = -!-A [u" + Bu ' ] + 1 A[u" + B + u ' ] . � 1 A + [u" + B + u ' ] = u (R � 1 ), and Ito's lemma implies that 00
.
de - 'u(x) = e- r grad u · Je d b + e - '(G - 1)u d t � e- r grad u · Je d b for lxl � 1 . But for < oo and paths starting at l x(O) I = 1 say, this can be integrated between the time f = max (t : lxl = 1) < and a time t between f and e , with the result that e
e
e- 'u(lxl 2 /2) - e- 1u( lf2) ::;;; J e-• grad u · Je d b. t
f
t Warning : the ' stands for differentiation with respect to R2 /2 throughout this
section.
1 04
STOCHASTIC INTEGRAL EQUATIONS
4
(d � 2)
Because I� e- • grad u Je db is a !-dimensional Brownian motion run with the clock t( t) = I� e- z s grad u · e grad u ds, t ·
a
e - eu(oo) = lim e-tu(III 2 /2) = lim a( t) - a( f ) + e- fu( 1 /2) < oo. t fe tfe This con_tradicts e < oo since u( oo) = oo, and so P [ e = oo] = 1 . Proof of H asm i n s k i i ' s seco n d test
Define u = u(R 2 /2) as before, but with A _ , B _ , C _ in place of A + , B , C + , and use the sum for u to verify that +
is bounded as R j oo under the condition of Hasminskii's second test. Define tR = in (t : III = R). Gu � (R � 1) so that de-t u(I) � e - t grad u Je db for III � 1 , much as before, and integrating up to t 1 A tR for paths starting at 1 < II(O) I = R 1 < R, it follows that m
u
·
But, for R and R 1 j oo in that order, we find lim {E [e - e , e < t 1 ]u( oo) + E[e- tt , t 1 < e]u( l /2) } � u( oo ) . R t t oo Because u(l/2) < u( oo) and the sum of the coefficients of u( oo ) and u(1 /2) on the left side is � 1 , 1 = lim E[e- e , R t f oo
e <
t 1 ] � lim P[e < oo ] , R t f oo
and since P[tR < oo] = 1 t, P [ e < oo] = 1 follows from the fact that tR j e as R j oo. t See Problem 1 , Section 2.9. t
See Problem 4, Section 4. 3.
4. 5
HASMINSKII ' S TEST FOR EXPLOSIONS
Pro b l e m 1 t
1 05
I
Use Hasminskii's first test to prove that P[ e oo] 1 if d 2, f 0, and e t 1 x 2 2 - 2e 1 2 x 1 x2 + e 2 2 x t 2 R 2 sp e -1 -1 1 � lg R) ( + 2 + 2e 2 e e e
-X •
=
=
=
X
=
+ 2 2 X2
1 2 X 1 X2
1 1X 1
=
for R � 2 . So l ution B+ �
[2 + ( lg R) - 1 ] R - 2 , so C+ �
f2 C+ 1 � t 00
00
R2
dR/R
Ig R, and
lg R
=
oo ,
causing the integral of Hasminskii's first test to diverge. Pro b l e m 2
Take f 0 and define y ( R) to be the biggest eigenvalue of e for l x l � R . The problem is to prove that P[ e oo] 1 if either limRtoo R 2 jy + oo or =
=
=
=
f
00
1
R dRfy +
=
oo .
Hasminskii's first test does not cover this. Sol ution
lx
Because d l 2
=
2I · Je db + sp e dt,
[� ( ir(tW - l x(O)I 2 - { sp e ) { exp [ {r Je ]
3(t) = exp =
ds
a
db - ia 2
·
2 - ia
{r
· er ds
x · er ds
is a supermartingale.t In particular, for paths starting at lx(O)I 1
� E [3 ( tR) ] � exp
t See Hasminskii [1 ].
[�
t Problem 5, Section 2.9.
(R
2
J
]
<
R,
- l r(OW ) E[exp ( - ! ( d + et R 2 )ety + tR)]
4
1 06
STOCHASTIC INTEGRAL EQUATIONS
(d � 2)
with Y + = Y + (R), so that for (d + aR 2 ) r:t.Y + = 1 ,
l - � (R 2 - lx(OW)J l ( d) 2 R 2 ] 1 1 2 ) � co nstant exp - 1 , + 2 ( Y+
E[exp ( - tR /2) ] � exp
x
and
E [ex p ( - e/ 2)] � E [exp ( - tR /2) ] ! 0 as R j oo in case lim R 2 fy + = oo . Similarly, for paths starting at lx(O) I < R, 1
� E [3(tR + �)/3(tR ) I x ( s ) : s � tR ]
� exp ( r:t.b R)E[exp { - ! [ d + r:t. (R + b ) 2 ] r:t.y + (tR + � - tR) } I x( s) : s � tR ] with y + = y + ( R + <5), and for ay + = 1 , it develops that E[exp [ - c(tR + � - t R )] I x ( s ) : s � tR ] � exp ( - bRfy + ) with 2c = d + sup R 2 Jy + . But this means that
[ -J00lx(O) IR dRfy + ] , = oo ] = 1 in the divergent case [J100 R dRfy + = oo ] if c < oo , i .e. , lim E[exp ( - ctR )] � exp
R j oo
so P[ e
if lim R 2 fy + < oo . Pro b l e m 3
Use Hasminskii's second test to prove that P[ e < co ] = 1 for G = y(R)A/2, d � 3, and R dRfy < oo . This shows that the test of Problem 2 cannot be improved.
J100
So l ution
A = R 2 y and B = djR 2 , causing the integral R oo oo 1 f c - t f1 C/A = (d - 2) - f1 R dRfy to converge.
1
4.5
1 07
HASMINSKII ' S TEST FOR EXPLOSIONS
Pro b l e m 4
P[ e = oo ] = 1 for d = 2 and 2G = ( 1 + b 2 ) o 2 foa 2 + 2 b o 2 joa ob + o 2 fob 2 • Hasminskii's first test does not cover this. Prove also that P[lim, t oo III = oo ] = 1 , substantiating the statement at the end of Section 4.4. So l ution
Denote the components of I by a and b. (da - b db) db = O, (da - b db) 2 = d t,
(db ) 2 = dt,
so by Problem 2, Section 2.9, (da - b db, db) is the differential of a 2-dimensional Brownian motion. Because b db = }(0 2 - t) (see Section 2.4), no explosion is possible, but III 2 = a 2 + b 2 tends to oo as t j oo , since, for t j oo , either b 2 � t/2 or b 2 � t/2 and a , whi ch is the sum of a ! -dimensional Brownian motion and f(b 2 - t), is bounded above by (3t lg 2 1 /t)1 1 2 - t/4 � t/5.
J�
-
Pro b l e m 5
Take a function f E C 1 (R1) of the same sign as solution t --+ [I, I. ] E R 2 of
di = I. dt,
x
and regard t he
dx· + j(I) dt = db
as the response of the oscillator I + /(I) = b. with restoring force f to the (formal) white noise b ·, b being a ! -dimensional Brownian motion. The problem is to prove that P[ e = oo ] = 1 and to obtain the bound ..
[
]
P Iim Hjt 1g 2 t � e = 1 t i 00
for the Hamiltonian H = (x Y /2 + oscillator I .. + /(I) = O.t
J: f associated with the
So l ution
Up to the explosion time
e,
dt 2 dH = I di + !(di ) + /(I) di = I d b + 2 , •
t See Potter [1 ].
•
•
•
unforced
4
1 08
STOCHASTIC INTEGRAL EQUATIONS
so
(d � 2)
t
t t t · + ) = H(O) + = + a x H H(O) + f db ( 2 2 0 with a new ! -dimensional Brownian motion a run with the clock
t(t) =
fo(£) 2 ds. t
But if e < oo, then either t( e) < oo and both [x - x(0)] 2 � tt( t ) and (x· ) 2 /2 � H stay bounded as t j e, which is a contradiction, or t( e) = oo and 0 � lim t t e H = - oo , which is also absurd. P[ e = oo] = 1 is now proved. Now use the familiar martingale bound
for e
>
n j oo ,
]
[
P max H - H(O) - ! - rx t > {3 � e - ap 2 2 t � O" 1 ' (X = e - n , and {3 = en + 1 lg n to prove that for t � en and
t e-n t + en + 1 lg n. H � H(O) + - + 2 2 Because (x ) 2 /2 � H, it follows that for t � en , lg n [ l + o( l )] + e - n
H � + en + t
t
f H, 0
so that and Now make e � 1 . 4.6
H � en + 1
lg n[ l + o(l )] exp ( en t )
,
lim Hf2t lg 2 t � 0 2 e. tt 00
COVERING BROWNIAN MOTIONS
Ito's lemma can be used to give a neat proof of Levy' s observation that the 2-dimensional Brownian path is a conformal invariant, t meaning that if 3 = 3(0) + a + J - 1 b is a standard Brownian motion on R 2 t See Levy [2].
4.6
1 09
COVERING BROWNIAN MOTIONS
and iff is a non constant analytic function defined on a domain D c R 2 containing 3(0), then up to the exit time e of 3 from D , /(3) is a standard Brownian motion run with a new clock t(t) = Jf ' (3W, especially, if R is a Riemann surface over D, then the Brownian path 3 : t < e can be lifted up to R via the inverse of the projection R --+ D, t and on a patch of R with local coordinate w, this covering path performs the corresponding standard Brownian motion ro run with the clock t ( t ) = J w' (3W.
J�
J�
Proof
Using the Cauchy-Riemann equations f' = /1 = - � /2 t and the fact that Af = 0, an application of Ito's lemma gives df( 3) = !1 da + !2 db + !f1 1 ( da) 2 + /1 2 da db + lf2 2 (db ) 2 = /' (3) d3 .
Define (J = argj'(3). Then exp (J - 1 0) is a nonanticipating functional of 3, so 3 * ( t ) = exp (� 0) d3 is a standard Brownian motion, § df(3) = 1/'(3) 1 d3 * , and 1/'(3) 1 is a nonanticipating functional of 3 * . A mild extension of the time substitution rule of Section 2.5 now shows that up to time t( e) ,
J�
1
3 * * (t) = / [3(C )] =
f
t- 1
0
J /' (3) 1 d3 *
is likewise a standard Brownian motion, completing the proof. Besides its own importance, this conformal invariance of the Brownian path has entertaining applications to the winding of the 2-dimensional Brownian path as will be explained below.� A know ledge of covering groups, the modular group of second level, and the Jacobi modulus k 2 is now needed. This information can be found in Lehner [ 1 ] and Weyl [ 1 ] . Besides this, it is also necessary to know that the 2-dimensional Brownian path hits each plane disk , i.o., as t j oo . t See Seifert-Threlfall [1 ].
t f1 = of! ox ! , !2 = o[j ox2 , etc.
§ See Problem 3, Section 2.9. � Ito-McKean [1 ] contains the discussion of the Riemann surfaces of lg k 2 given below. The rest is new.
z
an d
4
1 10
STOCHASTIC INTEGRAL EQUATIONS
(d � 2)
Proof
The punctured sphere S 2 - (0, 0, I) is mapped onto the plane R2 via the stereographic projection : x = {x 1 , x 2 , x 3 ) --+- z =
xl
+ Fi x 2 . 1 - x3
Bring in the spherical Brownian motion x governed by
G = A /2 =
l
! (sin
a�
]
a2 2 .t ae
Because the stereographic projection is conformal, 3 z(x) is (locally) a Brownian motion run with a new clock. 3 # 0 for t # 0�, so x cannot hit the south pole (0, 0, - 1 ) for t # 0, and since G commutes with spherical rotations, it cannot hit the north pole (0, 0, 1) either. But this means that for x(O) # (0, 0, 1 ), the projection 3 is well-defined for t � 0, and the fact that x visits each spherical disk i.o. , as t j oo § is mirrored iri the fact that 3 hits each plane disk, i.o. , as t j oo. Consider th e Riemann surface R of w = lg z as a plane divided into horizontal strips of height 2n, with projection z = ew mapping it onto the punctured plane R 2 - 0 as in Fig. 2. R can be viewed as the universal =
•
FIG. 2.
t 0 � cp colatitude � 7T , 0 � 8 longitude < 27T. t See Problem 7, Section 2.9. § See Section 4.4. =
=
4.6
COVERING BROWNIAN MOTIONS
111
covering surface of R 2 - 0 ; as such, its covering group is identified with the fundamental group Z 1 t of R 2 - 0. Because the plane Brownian motion 3 never hits 0 if 3(0) =I= Ot, a Brownian path on R 2 - 0 can be lifted up to R. Levy tells us that this lifted path is a Brownian motion run with a new clock, and it follows that the lifted motion hits each disk of R, i .o. , as t j oo . Regarding R as the universal cover of R 2 - 0, it now follows that the 2-dimensional Brownian path winds both clockwise and counterclockwise about 0, i.o., as t j oo and also unwinds itself, i.o., as t j oo , reflecting the fact that the covering motion makes unbounded vertical excursions but comes back to the strip 0 � b < 2n, i.o. , as
t i 00 .
Define R to be the open upper half-plane for the next application. Given k2 =1= 0, 1 , the inverse function of the elliptic integral
fo
dtf[(I - t 2 )(1 - k 2 t 2 )] 1 / 2
is a Jacobi elliptic function, and Jacobi's modulus k 2 , expressed as a function of the ratio w E R of its fundamental periods, maps R onto the twice-punctured plane R 2 - 0 - I . k 2 is a modular function of the group G of substitutions iw + j O i j modulo 2, i l - kj = + 1 = 0 w --+ 1 k kw + 1 , [modular group of second level ] . G maps R onto R, and dividing R into sheets in accordance with this action as in Fig. 3, k 2 maps each sheet 1 : 1 onto R 2 - 0 - 1 . R can be regarded as the universal covering surface of the twice-punctured plane R 'b - 0 - 1 . G is both the covering group R and the fundamental group of R 2 - 0 1 ; as such, it is iso morphic to the free group on 2 generators. A plane Brownian path 3 cannot meet 0 or 1 if 3(0) =I= 0 or 1 , so such a path can be lifted up from the punctured plane to R and will perform on R a Brownian motion run with a new clock. R is a half-plane, so this covering motion tends to the line R 1 X 0 = o R as t i 00 . Regarding each sheet of R as labeled by an element of the fundamental group G, it follows that, unlike the
l ] - [1 1J
-
case of the once-punctured plane, the winding of the Brownian path about the two punctures 0 and 1 becomes progressively more complicated as t j oo and never gets undone. t Z 1 denotes the rational integers 0, ± 1 , ± 2, etc.
t See Problem 7, Section 2.9.
1 12
4
STOCHASTIC INTEGRAL EQUATIONS
(d � 2) R
-I
I 2
- -
I 3
--
I 3
0
I 2
FIG . 3.
Weyl [1] defines the class surface of the twice-punctured plane to be the biggest surface R 1 between the modular figure R and R 2 - 0 - 1 still having a commutative covering group G1 . R 1 can be regarded as the Riemann surface of w = lg z + J-- 1 lg (z - 1) ; as such, it can be depicted as an infinite number of copies of the Riemann surface w lg z connected by logarithmic ramifications at the points 2n r--i x Z 1 . G1 is just the group G made commutative, i.e. , considered modulo its commutator subgroup. G1 can also be identified with the homology group Z 2 t of R 2 - 0 - 1 . Now lift up the Brownian path from R 2 - 0 - 1 to R 1 • Levy tells us that this lifting is a Brownian motion on R 1 run with a new clock. But such a Brownian motion visits each disk of R 1 , i.o. , as t i oo (the proof is explained below), so the winding of the plane Brownian path about the points 0 and 1 undoes itself, i.o., as t j oo from the point of view of homology with integral coe.fficients. =
t Z2 denotes the lattice of integral points of R2 under addition.
4.6 Proof th at
a
1 13
COVERING BROWNIAN MOTIONS
Brow n ian motion o n R 1 vis its eac h d i sk, i . o. , as
t j oo
Define Q to be the commutator subgroup of G. As n i oo , the image of J--=1 under the commutator
tends to jf l, and, for the first factor running over G (j even and I odd with no common divisors), such fractions are dense on the line o R. Q also maps R onto R, so it is a principal-circle group of the first k ind, so-called, and therefore by a theorem of Poincare,
L ( i 2 + j2 + k2 + 1 2) - 1 = Q
oo . t
Now consider a standard Brownian motion w = w(O) + a + J - 1 b = x + Fl l) on R run with the clock t- 1 inverse to t(t) = : l) - 2 •
J
Clearly t - 1 is defined up to time J: lJ - 2 , e being the exit time min (t : t) = 0) of w from R, and this integral is + oo since - oo = lg
tJ (e - )/tJ(O) = lim t je
rrlJ - 1 d b - 1 rlJ - 2 ds] . 0
0
But then the projection 3 1 of w(t - 1 ) onto R 1 is a Brownian motion run with a new clock defined for 0 � t < oo , and since the expected time spent by w(t - 1 ) in a disk D R with indicator function f is c
J 00E[l)(t)- 2, w(t) t < e] dt 00 = J dt t(2 t ) - 1 { exp ( - lw - w(O) I 2 f 2 t) o =
E D,
0
n:
- exp ( - lw* - w (O)I 2 /2t)}y - 2 dx dy
1 =n
J
n
w* - w(O) Ig y - 2 d d y ,+ w - w(O)
t Lehner [1 , pp. 1 79-1 83 ]. t The * means conjugate in this formula and the next.
X
+
1 14
4
STOCHASTIC INTEGRAL EQUATIONS
(d � 2)
it follows easily from the invariance of the volume element y - 2 dx dy under G that the expected time 3 1 spends in the projected disk is
* - w (O) - 2 d d " 1 J. w Ig y X y l...J O w w( ) Q n [� {] n � L constant Q
x
( i2 + j2
+ k2 + 12) - 1
=
oo .
Because 3 1 is the same as the lifting of the plane Brownian motion from R 2 - 0 - 1 to R 1 up to a change of clock, it is now enough to verify that 3 1 visits each disk of R 1 , i.o. , as t i oo . But this follows easily from the divergence of the expected sojourn time of 3 1 in a disk and the fact that 3 1 begins afresh at its passage times. In fact, if D is a closed sub disk of the open disk B c R 1 , if t 0 � t1 � t 2 � etc. are the successive passage times of 3 1 to D via R 1 B, and if rx P[ t 0 < oo] and =
-
{3
=
E[measure (t : 3 1
E D,
t0 < t < t1 )]
are regarded as functions of 31 (0), then the total expected sojourn time of 3 1 in D is oo
=
=
00
I1 E [measure (t : 3 1 E D : tn _ 1 < t < tn)J
n=
00
L E[tn - 1 < 00 , f3(31(tn - 1))] n= 1
f
[ ]
sup rx � 1X(3 1 (0)) 1 n = oB
n- l
sup p. oD
But for small B, {3 is bounded on o D as is clear upon lifting 3 1 back up to R. Besides, a is harmonic off D, t so the di vergence of the sum implies that rx 1 at some point of oB with the result that a = 1 ,t and now the proof is finished. =
t See Section 4.4. � See Step 3 of Section 4.4.
4. 7 4.7
115
BROWNIAN MOTIONS ON A LIE GROUP
BROWNIAN MOTIONS ON A LIE GROUP
A (connected) Lie group G is a manifold as defined in Section 4. 1 and also a group endowed with a smooth multiplication G x G � G Smooth means that for g1 (g 2 ) contained in a small patch U1 ( U2 ) , the product g = g1 g 2 is confined to a small patch and its local coordinates belong to ceoc ul X U2 ). Define ceo(l ) to be the class of germs of infinitely differentiable functions at the identity 1 of G. A derivation 1 D of ceo(l) is a map ceo( 1 ) � R with D(ft/2 ) = (D.ft)/2 (1) + /1(1)(D/2 ). Such a map can be expressed in terms of local coordinates x on a patch U about 1 as Df = a · gradf( l ) for some a E Rd , and this correspondence D � Rd is an additive isomorphism between Rd and the tangent space A of G at 1 , consisting of all derivations of ceo( 1 ). Define D( G) to be the class of all partial differential operators on G with coefficients from ceo( G) that commute with the left translations g : f � gf = f( g ) D E A can be viewed as a member of D(G) using the recipe Df(g ) = Dgf( l ) For members of A, it turns out that the commutator [D1 , D 2 ] = D1 D 2 D 2 D1 , computed in D(G) and then applied to ceo ( I), belongs again to A . A endowed with this commutator product is the Lie algebra of G. D (G) endowed with the usual product is the enveloping algebra of A , so-named because, up to isomorphism, it is the smallest associative algebra containing A as a Lie subalgebra under the commutator product. A is provided with a mapping into G, the so-called exponential map defined in the neighborhood of 0 E A by the rule x(exp (tD)). = (Dx)( exp (tD)).t exp maps the ! -dimensional subspaces of A onto the ! -dimensional subgroups of G ; it is a local diffeomorphism. A simple example is provided by the group G = S0(3) of proper rotations of R 3 . S0(3) can be identified as 3 x 3 orthogonal matrices of determinant + 1 , A as 3 x 3 skew-symmetric matrices under the com mutator product, and exp as the usual exponential sum : exp (D) = I Dn/n ! . A is spanned by the three infinitessimal rotations : 0 0 0 0 1 0 0 1 0 D3 = 1 0 0 0 , D2 = D1 = 0 0 - 1 , 0 0 . .
.
.
.
-
0
1
0
-1
0
0
[D 1 , D 2 ] = D 3 , [D 2 , D 3 ] = D1 , [D 3 , D1 ]
0
=
0
0
D2 ,
t The stands for differentiation with respect to t, and x for local coordinates on a patch about 1 . •
1 16
4
STOCHASTIC INTEGRAL EQUATIONS
(d � 2)
so that A is isomorphic to R 3 under the outer product. The exponential mapping sends into the rotation about the axis a E R 3 through the angle lal in the sense prescribed by the right-hand screw rule. t Besides the above general facts, only Ado's theorem is needed here (see Step 1 of Section 4.8). Helgason [ 1 ] is recommended for proofs and general information. A (left) Brownian motion on G is a continuous movement 0 � t � 3(t) E G with 3(0) = 1 , beginning afresh at its stopping + times t in the sense that, conditional on t < oo , the future 3 (t) = 3(t) - t 3(t + t) : t � 0 is independent of the past 3(s) : s � t + (i.e., indepen dent of the usual field Bt + ) and identical in law to the original motion 3(t) : t � 0 starting at 1 . This is the analog of the differential property of the !-dimensional Brownian motion. Yosida [ 1 ] proved that such a Brownian motion is governed by a (possibly singular) elliptic differential operator G E D( G) , expressible in terms of a basis D = ( D1 , . . . , Dd) of A as G = -!D eD + f D = t L eii D i Di + L h D i i , j� d i�d with constant 0 � e = e * E Rd ® Rd and f E Rd . For nonsingular e , the statement means that the density p(t, g) of P[3(t) E dg] , relative to the volume element of G, is the elementary solution of oufot = G * u with pole at 1 . A formal proof consists of using the (left) differential property of 3 to check that, for the map G E D(G) defined by Gf(l) = lim t - 1 [E[f(3)] - f( l )], t ..l- 0
E[gf(3)] = exp (tG)f, and then using Problem 1 , Section 4. 1 , and the fact that [A, A] c A to reduce G to the desired form. Ito [4] proved that every such G arises from some (left) Brownian motion by constructing the associated sample paths as in Section 4.3. Y osida also proved this fact by constructing the elementary solution of ouf ot = G * u. A third method is to inject the differentials of a d-dimensional (skew) Brownian motion Je b + ft from A (identified with Rd) t Gelfand-Sapiro [1 ] can be consulted for detailed information about this group.
4.8
1 17
INJECTION
into G via the exponential map and then to put them back together as a so-called product integral:
n exp [Je d b + f ds J
s�t
= lim fl exp [J; [ b (k2- ") - b ((k - 1 ) 2 - ")] + f 2- "] . n t oo
k � 2 "t
This program is carried out in Section 4.8. t 4.8
INJECTION
Given constant 0 � e = e * and f, and a standard d-dimensional Brownian motion b, regard a = Je b + ft as a (skew) Brownian motion in the Lie algebra A of G, i.e. , identify a E R d and a · D E A , and let us verify the following recipe : if 3n( t )
= =
( t = 0) ( t � 0, l = [2" t]) ,
1 3n ( l2 - ") exp [a (t) - a ( l2 - ") ]
then the so-called product integral: 3 oo (t )
=
n exp ( d a ) = lim 3n (t) n t oo
s�t
exists and is a left Brownian motion on G governed by G = DeD/2 + fD. Step 1
Ado's theoremt states that A can be faithfully represented as a Lie subalgebra of Rm ® Rm , under the customary commutator product, for some dimension m. The classical exponential sum exp (a) = L a"fn ! maps this faithful copy of A onto a Lie subgroup of the general linear group GL(m, R 1 ) and this subgroup is locally isomorphic to G but perhaps not globally so. But the injection recipe is local, so it is per missible for the proof to suppose that G is faithfully embedded as a Lie ,
t McKean [2] did this for the special case G S0(3) . See also Gangolli [1 ] for a more general application of the same idea, and Perrin [1 ] for a discussion of Brownian motion on S0(3) from a more classical standpoint. � See Bourbaki [1 ]. =
1 18
4
STOCHASTIC INTEGRAL EQUATIONS
(d � 2)
subgroup of GL(m, R 1 ). Rm ® Rm is provided with the norm Jsp a * a, not to be confused with the bound Ia I of a as an application of R'". The bounds
a l � Jsp a *a � m la l , lsp a c b l � constant
x
Jsp a * a Jsp b* b ,
sp (a + b)* ( a + b) � 2 sp a* a + 2 sp b * b will be of frequent use. Warning : * a stands for the transpose of a E A as an application of R'" ; this will make the formulas come out more neatly. Step 2
be
The product integral for 3oo suggests that in the enveloping algebra D(G) c Rm ® R'", d3oo = 3oo [exp (da) - 1 ] = 3oo [da + -!(da) 2 ] ; it is to proved(that this problem has just one nonanticipating solution 3 : t D( G) with 3(0) = 1 . -+
Proof of u n i q uen ess
Consider the difference 1J of two nonanticipating solutions and let the Brownian stopping time t be the smaller of t � 0 and min (t : 11J I = n). An application of Ito's lemma to lJ * lJ implies lJ* t) (t) =
f t)[dj + *dj + dj *dj] * t) t
0
with dj = da + (da) 2 /2 = Je db + k dt. Using the bounds cited in Step 1 , it develops that the expected spur D of l) * l) (t ) is � (mn) 2 < oo and bounded as in
D
=
E [( sp tJ(k ds + *k ds + � db * Je db)* lJJ
:::;; constant
x
Jo D, t
permitting us to conclude that D = 0. The proof is completed by making n j oo .
4. 8
1 19
INJECTION
Proof of existe n ce
Define 3 to be the sum of t) 0 (t) = 1 and lJ n ( t) = much as in Section 2.7. Then
Dn = E[sp tJ/tJ nJ � constant
x
J0 Dn - 1
J� lJ n - 1 dj for n ;;;: 1 ,
t
(n � 1 )
is proved just as the bound for D above, and using the martingale inequality and the first Borel-Cantelli lemma much as in Section 2. 7, the sum for 3 is found to converge geometrically fast to a solution of 3 = 1 + 3 dj.
J�
Step 3
Define 3 n(O) = 1 and 3 n (t) = 3 n(/2 - n) exp [a ( t) - a(/2 - n)] for I = [ 2"t] , t � 0, and n � 1 ; it is to be proved that
[
P max l3n (t) l � 2et", n j t� 1
oo
]
for any rx > 0.
=1
Proof
The norm of exp (a) is � exp (I Jel lbl + 1/l t). Because of E[exp (yb 1 )] exp (y 2 t/2), E[lexp (a) I P ] is bounded for t � 1 , for each f3 > 0 separately, so E[I31 P] is bounded too, and =
[
]
P max l3n (l2 - ") l > 2et " � constant x 2n [ l - et P J �� 2" is the general term of a convergent sum for rx/3 > 1 . An application of the first Borel-Cantelli lemma, completes the proof. Step 4
[
1 P max 3n - 1 - J 3 n dj � 2 - 8" , n j 0 t� 1
oo
]
= 1
for any (} < 1.
Proof
Define A = [(k - 1)2- " , k2 - " ] and a(d) = a(k2 - ") - a((k - 1)2 - ") for k � 2 ". Using Levy's modulus (Section 1 .6), its counterpart for
4
1 20
STOCHASTIC INTEGRAL EQUATIONS
(d � 2)
Brownian integrals (Section 2.5) , and the bound of Step 3, we find for n i oo and (} < ! , that up to errors of magnitude � constant x 2-6", 1 3n ( t ) - 1 - 3 n dj
J
=
0
[
L 3n (( k - 1) 2-") exp [a(L\) ] - 1 -
k�l
=
L 3 n (( k - 1) 2-" )h ( L\)
JA dj]
k�l
J,�,
with h{L\) = [Je (b(L\)]2 - (J e db )1. Because the final sum ( = 1) 1 ) is a martingale, sp lJ L * lJ L is a submartingale, and using the bound E[ l3 n l 2 ] � constant (t � 1 ) from Step 3 and the independence of 3 n ((k - 1)2 - ") and h (L\), it develops that
[
]
P max sp 1) 1 *lJ z > 2 - 26 " l� " 2 � 226"E[sp l) 2 n * 1J 2 n] =
[
226"E L SP 3n CCk - 1) 2-")h(L\)*h(L\)* 3n CC k - 1 ) 2-" ) k � 2"
� 2 26n m 2 L E[ / 3n ((k - 1) 2- n) j 2 ] E [ /h (L\)j 2 ]
]
k � 2"
� constant
x
226n + n - 2 n .
But for (} < ! , this is the general term of a convergent sum, so an applica tion of the first Borel-Cantelli lemma does the rest. Step 5
[
P max l 3n - 31 � 2 - on , t� 1
n
j
oo
]
=
1
for any 0 < ! .
Proof
s:
s:
3 n - 3 lJn + (3 n - 3) dj with lJ n = 3 n - 1 - 3 n dj. This last expression is of magnitude � 2 - o n for t � 1 , n i oo, and (} < 1 in accord ance with Step 4. Bring in the Brownian stopping time t n defined either as the first time t � 1 such that 1 3 n - 3 1 2cxn or ltJ n l = 2 - 6", or as t 1 if neither of these events occurs before. Because of Steps 2-4, tn = 1 for n n i oo . D = E [ sp ( 3 n - 3) * (3n - 3 )(t n ) J < m 2 2cx < oo can be bounded as in =
=
=
4. 8 D
121
INJECTION
� 2E [sp lJ n * lJ n (tn)] " sp (3. - 3) [dj + * dj + dj * djJ *C3n - 3) + 2E
r(
:::;; 2m 2 2 - 2 9" + constant
x
J D, t
1
0
with the result that D is bounded by a constant multiple of 2 - 2 0" for t � 1 , and now the usual martingale trick applied to the sub martingale tn t tn t sp (3 n - 3) j e db (3n - 3) j e db *
f
J\
J\
0
0
implies
f
P max (3. - 3) je db � 2 - 0", t� 1 0 The analogous bound
n
j
oo
] = 1.
P max (3. - 3)k ds � 2 - 0" , t� 1 0 is even easier to prove, and the result follows.
n
j
oo
]
r
f
t
{
r
= 1
Step 6
3 oo = limn t oo 3 n
exists, and for nonsingular e, it is the left Brownian motion on G governed by G = DeD/2 + fD. Proof
Step 5 leads at once to the existence of the product integral 3oo = 3 for t � 1 , and the reader will easily check that this propagates for t � 1 . It is also plain that 3 oo is a left Brownian motion, and so it suffices to prove the last statement. But for compact u E C 00 (G), n i oo , t � 1 , I = [ 2" t ] , and () < i , it is easy to see that up to errors of magnitude � constant x 2 - 0" , u (3 00 ) - u ( 1 ) = u(3n) - u ( 1 )
I { u [3n ( k2 - ") J - U [3 n (( k - 1 )2 - n) ] } k�l = I I ai (L\) Di u + � I ai (L\ ) a j(L\ )Di Dj u i, j � d k� 1 i�d evaluated at 3n (( k - 1)2 - n), =
r
]
4
1 22
STOCHASTIC INTEGRAL EQUATIONS
and it is easy to see that as
n
(d � 2)
i oo , this expression tends to
But this means that on a patch U with local coordinates x, x = x(3 ocJ is a solution of dx = Je (x) db + f(x) dt, e and f being (just for the moment) the local coefficients of G. This permits us to identify 3oo as the left Brownian motion governed by G and completes the proof. A simple but amusing example of injection is provided by the motion of a 3-dimensional unit ball rolling without slipping on the plane 3 R 2 x - 1 c R while its center performs a standard 2-dimensional 3 Brownian motion b = (b 1 , b 2 ) on the plane R 2 x 0 c R .t G = S0(3), the infinitessimal rotations
0 0 0 D1 = 0 0 - 1 0 1 0
0 0 1 0 0 0 , D2 = -1 0 0
'
-1 0
0 0 0 0
span A, and the exponential maps a1 D 1 + a 2 D2 + a3 D3 E A into the right-handed counterclockwise rotation through the angle lal = 3 1 2 2 2 2 1 (a1 + a 2 + a3 ) about the axis a = (a1 , a 2 , a3) E R , as noted in Section 4.8. As the Brownian particle moves from b((k - 1)2 - ") [point 1 of Fig. 4] to b(k2 - n) [point 2 of Fi g . 4] , the ball suffers the approxi mate rotation exp [e3 x b( �) D] = exp [ - b 2 ( �)D 1 + b 1 ( �)D 2 ]t ·
of angle b(�), counterclockwise about the axis e3 x b( �), as in Fig. 4, so the total rotation suffered up to time t � 0 is just the corresponding product integral : namely, the (left) Brownian motion on S0(3) governed by G = !(Dt 2 + Dz 2 ) .
FIG. 4.
t McKean [2] ; see also Gorman [1 ]. t e3 (0, 0, 1). The x = the outer product. =
4.9
BROWNIAN MOTION OF SYMMETRIC MATRICES
1 23
Pro b l e m 1
Prove that the induced motion 3oo e3 of the north pole on the surface of the rolling ball is the spherical diffusion governed by a2 1 G + = 1 (sin cp ) - a sin cp a + cot 2 cp a e 2
0�
qJ =
[
:
:
]
0 � (} = longitude < 2n.
colatitude � n,
Sol u tion
[ D 3 , G] = 0, so G commutes with the subgroup S0(2) of rotations about the north pole e 3 • Because of this, 3oo e3 is a diffusion on the spheri cal surface M = S0(3)/S0(2), and for the rest, it suffices to compute the 2 2 action G + of G = ! (D1 + Dz ) on u E C00(M) regarding u as a member oo of c ( G) depending only on cosets gS0(2). 4.9
BROWNIAN MOTION OF SYMMETRIC MATRICES
Regard R4 [d = n(n + 1 )/2] as the space of n x n symmetric matrices with coordinates x ii (i � j � d) and define M c R4 to be the submanifold of symmetric matrices with simple eigenvalues. O(n) acts on M by con jugation [x -+ o * xo] . M/O(n) can be identified with the submanifold R of diagonal matrices y with entries y 1 < · · · < Y n down the diagonal, and since the stability group of x E M is the (finite) subgroup D of diagonal rotations ( + 1 down the diagonal), M can be identified with R x O(n) considered modulo D, via the diffeomorphism (y, o) -+ o *yo . G = O(n) x ( + 1) x R4 acts as a motion group on R4 by conjugation [x -+ o * xo] , reflection [x -+ - x] , and translation [x -+ x + y] , and up to constant multiples, the only elliptic operator on C00(R4) commuting with the action of G is 2 G = t L o /ox� + ! I o 2 / oxfj · i�n
i <j
G governs a Brownian motion I lj ·
or
·
I
- I lJ (0) ·
·
on R4 expressible as =
b l} ·
·
(i = j) (i < j),
b ii (i � j) being independent standard ! -dimensional Brownian motions.
4
124
STOCHASTIC INTEGRAL EQUATIONS
(d � 2)
An easy computation shows that
P[I(t 2 ) - I(t 1 ) E dx I I(s) : s � t 1 ] ( 2nt) - nf 2 (n t) - n( n - l ) / 2 exp [ - sp ( x) 2 /2t] dx for t = t2 - t1 > 0, dx being the volume element ni � i dx ii . Using the =
in variance of this formula under the action of G, it is easy to see that the eigenvalues of I begin afresh at stopping times and perform on R the diffusion governed by the action G + of G on C 00 (R) : G + = 1 iLn o 2 f oy i 2 - 1 Ii (yj - Yi) - 1 a ;ayi , j= � up to the exit time e of I from M.t A more picturesque statement is that as I performs the Brownian motion governed by G on M, its eigenvalues perform a standard d-dimensional Brownian motion on R subject to mutual repulsions arising from the potential U : e - 2 = n ( yj - Y i) . t j>i Because of this repulsion, it is natural to conjecture that the exit time e is infinite if I(O) E M, as will now be proved.
u
Step 1
Rd
=
M u o M, oM being the sum of d - 1 submanifolds like
Y 2 < · · · < Yn J ' d - 2 submanifolds like M3 = [x : = y 1 y 2 = y3 < · · · < Y nJ , and so on, plus the single submanifold Mn = [x : y1 = y 2 = · · · = Y n] · It is to be proved in this step that codim o M = 2.
M2
=
[X : Y 1
=
Proof
codim M2 is just 1 plus the dimension of the subgroup of O(d) com muting with the diagonal matrices belonging to M2 • But this subgroup is the product of a copy of 0(2) and the diagonal subgroup of O(d - 2), so the codimension is 2. A similar proof shows that codim M3 = 2 + dim 0(3) = 5, and so on. t Section 1. 7 contains the prototype of the proof. t See Dyson [1 ].
4.9
BROWNIAN MOTION OF SYMMETRIC MATRICES
1 25
Step 2
I cannot hit a submanifold Z of R4 of codimension
2 for t -=1= 0.
Proof
Define
2 2 p (x ) = J sp (x - y) ] - df + t d o , /
do
being the product of the volume element of Z and a positive function belonging to C00(Z) such that p < 00 off z. As X approaches a point of Z from the outside, p is bounded below by a positive multiple of f
rr /2
o
rr/2 (sin 0) 4- 3 d O (sin 8)4- 3 d O = 00 ' f 2 2 1 41 1 ] ] i t z dt 0 b [2 ( [2 ( 1 + £5) ( 1 - cos 0 ) + cos 0)
b being the distance of x from Z. Now suppose I(O) E Z and define e to be the passage time of I to Z. e < oo implies lim t t e p(I) = oo , while for t < e , dp(I) = grad p di + Gp(I) dt is a pure Brownian differential since G[sp (x - y)2] - 412 + 1 = 0 (x -=1= y) . But this means that up to the passage time e , p(I) is a ! -dimensional Brownian motion run with some clock,t and this leads to a contradiction as in the solution of Problem 7, Section 2.9, or Problem 5, Section 4 . 5 . ·
Pro b lem 1
Prove that the eigenvalues of I perform the diffusion governed by G + for n = 2 by direct stochastic differentiation of Y2 = � ( b 1 1 + b22) + Q = b i2/ 2 + ( b 1 1 - b22)2/4
Y1 = } ( b 1 1 + b22) -
j Q,
j Q,
So l ution
in which
b1 1 -
[2 1 + ( - )i I[ 1 - ( - )i +
da i = �
hzzl Y2 - Yt J
2
t See Problem 1, Section 2. 9.
db 1 1 + ( - ) i
b11 - b22 db22 Y2- Y t
J
b 12
Y2 - Y1
db12
(j = 1 ' 2).
4
1 26
STOCHASTIC INTEGRAL EQUATIONS
(d ";3::. 2)
Now use Problem 2, Section 2.9, to prove that a1 and a 2 are independent ! -dimensional Brownian motions. Pro b l em 2
Prove that for n 2, the determinant y 1 y 2 can be expressed as 1 (b 2 - r 2 ), b being ! -dimensional Brownian motion and r an independent 2-dimensional Bessel process. =
So l ution
and d ( y 2 - Y t )fJ2 = db 2 + ! [ ( Y 2 - Yt )fJ2]- 1 with new independent ! -dimensional Brownian motions b 1 and b 2 Now use Section 3. 1 1 c to identify r = (y 2 - y 1 )/ J2 as a Bessel process and express the determinant as i (b 1 2 - r 2 ). •
Prob lem 3
Use the method of Step 2 to prove the topological fact that, for d '?::- 2 , R 4 minus a submanifold of codimension '?::- 2 is still connected. t So l ution
Denote the submanifold by Z, take x and y E R 4 - Z, and draw about y a small ball A not meeting Z. 0 < P[x + x( l ) E A ] , and since, as in Step 2, x + x(t ) : t � 1 cannot meet Z, it is possible to find a continuous path joining x to y in R 4 - Z by going from x to A via a Brownian path x + x(t) : t � 1 and then joining x + x( l) to y by a line segment. 4.10
BROWN IAN MOTION WITH OBLIQUE REFl ECTION
A nice example of a diffusion on a manifold with boundary is the Brownian motion with oblique reflection on the closed unit disk of R 2 • Consider the open unit disk M: l z l < 1 , assign to the point 0 � () < 2n of oM a unit direction I making an angle - n � qJ < n with the outwardpointing normal in such a way that exp (J - 1 (/)) E C 00 (aM), and t See Helgason [ 1 ] .
4. 1 0
BROWNIAN MOTION WITH OBLIQUE REFLECTION
1 27
suppose that l cp l -=1= n/2 except at a finite number of singular points at which cp' -=1= 0. Denote this singular set by Z, and call a singular point attractive if cp' < 0, repulsive if cp' > 0. Brownian motion with oblique reflection along I is the diffusion on M - Z governed by G = A /2, subject to
oufo l = cos
qJ
au;an + sin qJ au;ae = 0
on
oM - Z.t
Dynkin [ 2] and Maliutov [ 1 ] have made a very complete study of this motion. For general information about diffusions on manifolds with b0undary, see Ikeda [ 1 ] , Motoo [2] , and Sato-Ueno [ 1 ] . Co nstructi on fo r cp
=
0 (sta n d a rd reflect i ng B rown i a n motio n )
Using Section 2.8, it is easy to deduce from Problem 9, Section 2.9, that the plane Brownian motion starting at 3(0) = r(O) exp (J - 1 8) -=1= 0 can be expressed as
r being a Bessel process starting at r(O) and a an independent ! -dimen sional Brownian motion.t Replace r by the reflecting Bessel process on (0, 1 ] governed by A + /2 = i [o 2 /or 2 + r - 1 ajar] subject to u - (1) = 0. This motion can be obtained as in Section 3. 10 from a ! -dimensional Brownian motion b by solving dr = db + dtf2r - df for the path 0 < r � 1 and the local time f = lim ( 2a)- 1 measure(s � t : r(s) > 1 - a )
.
£tO
Using this modification of 3 , Ito ' s lemma gives
0 = E (j( t , 3 ) i�J = E
l(' (OjOt + A/2)j(t, 3) dt - (' (Oj/On) (t, 3) dfJ
for compact j E C 00 [ (0, 00) X MJ . Weyl's lemma now implies that the density p of the distribution of 3(t) belongs to C 00 [(0 , 00) X M] and t ojon denotes differentiation along the outward-pointing normal.
t &(0)
-F
0 is assumed only to permit us to use this expression for
3.
4
1 28
STOCHASTIC INTEGRAL EQUATIONS
(d � 2)
solves opjot = Ap/2 inside M. Using Green ' s formula to transform . gives
0
=
Joo ! dt 0
]
[ I
oj op d (} p - j on on E oM
[Joo oj dfJ 0
on
'
granted that p belongs to C00 [(0, oo) X MJ .t But for j = it (t)j2 (r)j3 (0) with COmpact it E C00(0, 00 ) , COmpact J2 E C00(0, 1 ] , }2 (1) = 1 , j2 - {1) = 0, and }3 E C00(oM), this gives
0= so opjon
=
-
Jooi t d t o
op j3 -;- d (} , un oM
I
0 on oM, and the identification of the motion follows.
Construction fo r qJ # n/2
Using the reflecting Bessel process r, its local time f, and the inde pendent ! -dimensional Brownian motion a, solve
1/J ( t ) =
1 J J O 1/J( ) + r- da - tan
0
t
0
This is easily done, since (tan qJ )' is bounded. Define 3 Ito's lemma gives
0 = E[j(t, 3)!0]
=
=
r exp ( J - 1 t/1 ) .
[(' (0/0t + A/2)j(t, 3) d t - (' ( OJ On + tan OjOfJ)j(t, 3) d fJ
E
cp
for compact j E C00 [(0, 00) X M. ] As before, it develops that p belongs to C00 [(0, oo) X M] and solves opjot = Ap/2 inside M, while opfon - o(tan
BROWNIAN MOTION WITH OBLIQUE REFLECTION
4. 10
1 29
,
Co n st r u ction i n t h e p rese n ce of re p u lsive s i ng u laritiecs o n ly
Figure 5 depicts the drift coefficient - tan OJ ; this drift acts to push 3 away from the singular point. Because the standard reflecting Brownian motion [
-
FIG. 5.
- ta n
cp
e
e
FIG. 6.
Co n struction i n t h e p rese n ce of att ractive s i n g u larities
Figure 6 depicts the drift coefficient at an attractive singularity [ arg 3(e - ), but not both, (e) the density of the distribution of 3( t ) is the smallest elementary solution of oufot = Au/2 with pole at 3(0) subject to oujol = 0 on oM - Z.
The proof is carried out only in the simplest case : l = the horizontal direction [
1 30
4
Ste p
STOCHASTIC INTEGRAL EQUATIONS
(d � 2)
1
Z comprises just the two attractive singularities + J--=--1 , and for 3(0) ¢ Z, 3 can easily be defined up to the explosion time e = lim min ( t : I 3 n i oo
J - l l or I 3 + J - l l =
l /n )
so that (a) holds. Ste p 2
By Ito's lemma and Problem 1 , Section 2.9, du (3) = i Au d t +
au or
db +
= 1 Au d t + c(t)
au of)
[ au au ] + tan r - 1 d a - df or
q>
ae
au
- df sec q> af
for u E C 00 (M - Z) and t < e, c being a ! -dimensional Brownian motion and t the clock
Jo I grad u(3W . t
Define 3 = I + J - i 1) and put u = y. Because Au = 0 and oujol = ouj ox = 0 on oM, it follows that 1) is a ! -dimensional Brownian motion run with the clock I grad u(3) l 2 = t up to the explosion time. Because 1) is bounded, e < oo , proving (b), t and the existence of 1)( e - ) is also evident. But then 1)( e - ) = + 1 by the definition of e, and this forces the existence of I( e - ), proving (c).
J�
Ste p 3
Consider the angles a and f3 depicted in Fig. 7 and define u = a - [3. u E C 00 (M - Z), Au = 0 inside M, and oujol = oujox = 0 on oM - Z, so u(3) can be expressed as a ! -dimensional Brownian motion run with the clock t(t) = lgrad u(3) l 2 up to time e. Because u is bounded, t( e - ) < oo, t lim t t e u(3) exists, and it follows that as t j e, 3 approaches
J�
t See Problem 7, Section 2.9, or Problem 5, Section 4.5.
4
. 10
BROWNIAN MOTION WITH OBLIQUE REFLECTION
131
3( e - ) E Z at a definite angle. But as stated before, the standard reflect ing Brownian motion [ cp = OJ does not hit a point of oM named in advance, so r(t) = 1 , i.o. , as t i e, and it is immediate from the picture that as 3 approaches J=-1 , say, a tends to + n /2. This proves (d).
FIG. 7
-J-1 Ste p 4
Using (a)-(d), it is easy to prove (e) as in the nonsingular case, granting that p E C 00 [(0, oo) X M] . Problem
1
Prove that the standard reflecting Brownian motion [ cp = OJ does not hit a point of oM named in advance. So l ution
Define u = 2 lg lz - I I . Au = 0 inside M and oufon = 1 on oM, so, using Ito's lemma and Problem 1 , Section 2.9, as before, we find that u ( 3) is the sum of the local time - f and a ! -dimensional Brownian motion run with the appropriate clock. As such, it cannot tend to - oo at a finite time, so 3 cannot hit 1 .
4
1 32
(d � 2)
STOCHASTIC INTEGRAL EQUATIONS
Problem 2
Prove that for the Brownian motion associated with the boundary condition oufox = 0 starting at z = x + � y,
[
P 3(e - )
]
1
= J - 1 , lim a(3) = n /2 = l( l + y) + ( + x /2) . n t fe
rx
Sol ution
1) is a ! -dimensional Brownian motion up to the explosion time, so
y = E [l) ( e - )] = 2 P[3(e - ) = J - 1 ] - 1 , showing that P[3(e - )
= J-=1] = 1 (1 + y).
Define u = a - x/2. Au = 0 inside M and o uj o l = 0 on oM - Z, so
[ fe J
[ fe ]
n n x a - = E lim u(3) = E lim a(3) = Q - [ 1 ( 1 + y) - Q] , 2 t
t
2
2
Q being the desired probability. Now solve for Q. Pro b l e m 3
Prove that the functions 1 , y, a - x/2 , and f3 - x/2 span the solutions of Au = 0 subject to the conditions : (a) u E C 00 (M - Z) , (b) oufox = 0 on oM - Z, (c) u approaches a finite value as point ofZ.
z
EM -
Z tends tangentially to a
Sol ution
Use Ito ' s lemma and Problem 1 , Section 2.9, as before to prove that any such u can be expressed as u = E[lim, t e u(3)] .
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Math. 2.
SUBJECT INDEX
B
Levy's modulus, 1 8 on Lie group, 1 1 5 skew, injected, 1 1 7 of symmetric matrices, 1 23 winding of 2-dimensional, 1 10 with oblique reflection, 1 27
Backward equation for d = 1 , 63 for d � 2, 98 Bernstein's theorem, 1 10 Bessel process, 1 8 Brownian motion !-dimensional construction of, 5 differential property, 9 distribution of maximum, 27 law of iterated logarithm, 1 3 Levy's modulus, 1 4 local times, 68 nowhere differentiable, 9 passage time distribution, 27 scaling, 9 stopping times, 1 0 several-dimensional, 1 7 covering, 1 08 law of iterated logarithm, 1 8
c
Cameron-Martin's formula for d = 1 , 67 for d � 2, 97 D
Differential, stochastic, definition, 32 Ito's lemma for d = 1 , 32 for d � 2, 44 1 39
see also
Integral
1 40
SUBJECT INDEX
for several-dimensional Brownian motion, 43 under time substitution, 41 Diffusion 1 -dimensional, 50 backward equation, 63 Cameron-Martin's formula, 67 explosion 0f, Feller's test, 65 forward equation, 60 generator, 50 reflecting, 71 stochastic integral and differential equations for, 52 on several-dimensional manifold, 90 backward equation, 98 Cameron-Martin formula, 97 exploeions of harmonic functions and, 97 Hasminskii's test, 1 02 forward equation, 91
iterated, and Hermite polynomials, 37 Ito's definition for d = 1 , 21 for d � 2, 43 simplest properties, 24 under time substitution, 29 Wiener's definition, 20 Integral equation, stochastic general idea, 52 general solution for d = 1 , 52 Lamperti's method, 60 on patch of a manifold, 90 singular examples, 77 solution of simplest, 35 K
Kolmogorov's lemma, 1 6
L
F
Lie algebras and groups, 1 1 5
Feller's test, 65 Forward equation for d = 1 , 61 for d � 2, 91
M
G
Manifolds, 82 Martingales, 1 1
Gaussian families, 3 H
T
Time substitutions, 29, 41
Hasminskii's test, 1 02 I
Integral , stochastic, see also Differential backward, 35 computation of simplest, 28
w
Weyl's lemma application for d = 1 , 61 for d � 2, 95 proof, 85
