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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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 UhlenbeckWang [I] can be consulted for appli cations to statistical mechanics. A level of mathematical knowledge comparable to Volume I of CourantHilbert [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, ItoMcKean [I] is suggested. Chapter I and about one third of Section 4.6 are adapted from ItoMcKean ; 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 SeveralDimensional 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 SeveralDimensional 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 CameronMartin'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(k2n)  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) ddimensional 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 =
CameronMartin'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 eQ/ • 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 cJ=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)
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 � eap E [3(t)] eap. s�t s�t Define h(t) =(2t lg 2 1/t)112 and choose 0 < () < 1, t =ent , 0 < fJ < 1, et (1 + fJ)enh(On), and p =h(On)/2, so that ap =(1 + fJ) lg 2 en and eap =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 BorelCantelli 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 limrt, 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 BorelCantelli 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(k2n ) b((k 1)2n )] � (1  fJ)h(2n ) 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 BorelCantelli lemma now gives
]
[
P lim max [b(k2n ) b((k 1)2n )] /h(2n ) �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)2n(l6)(l •)2
is the general term of a convergent sum [(1  fJ)(1 + e ) 2 > 1 + fJ], so the first BorelCantelli lemma implies lb(j2n ) b(i2n )l < (1 + e)h(k2n ) ( 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 < 2m( 1 �> with m so large that the last estimate holds for all n �m. Pick n so that 2(n + l )(l � ) � t < 2n(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
SEVERALDIMENSIONAL 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 (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
SEVERALDIMENSIONAL BROWNIAN MOTION
A ddimensional 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 ddimensional 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 ddimensional 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 ddimensional rotation o, i.e. , b * = ob is likewise a ddimensional 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)
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 BorelCantelli 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 ex
< 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)2m)[b(k2m)  b((k  1)2m)] 0
k�l
+ e(l2m)[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 � erxP£ [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 BorelCantelli 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 BorelCantelli 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 � 2n 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 lienel =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 AkutowiczWiener [1] that an orthog onal transformation o of L2 [0, ) induces a measurepreserving 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[eyt] =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"n2 E[3n{l)2 ] n n = 22 "n 2 E[(b( 2 )2  2 )2 ] = constant
x
n2 ,
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 BorelCantelli 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 leftcontinuous 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 )
t1( 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)
t1,0 ( 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
�
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
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 measuretheoretical 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 t1. 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 BorelCantelli 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 e1] = E(3). CameronMartin [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 ]
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) e1 0 = lim (2e)  1 I ds e10
= lim (2e) 1 e l 0
s�t
z( t  1 (s)) < e
J
s � t  l (t ) z(s)
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
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 = IFI 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 ddimensional 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 ddimensional 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 25. 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 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 ddimensional 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 CameronMartin formulat implies that P[lJ E U" ] is positive also. The reader will notice that Step 3 is simply the socalled 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 etu(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 det 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
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 BorelCantelli 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 24, 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 3dimensional unit ball rolling without slipping on the plane 3 R 2 x  1 c R while its center performs a standard 2dimensional 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 righthanded 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 ddimensional 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 2dimensional 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 SatoUeno [ 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 outwardpointing 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 OJ ; this drift acts to push 3 away from the singular point. Because the standard reflecting Brownian motion [
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
J1 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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AM BSMSP
Ann. Math . Berkeley Symp. Math. Statist. and Prob. DAN Dokl. Akad. Nauk SSSR IJM Illinois J. Math. JMP J. Math. Phys. MA Math . Ann.
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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 2dimensional, 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 severaldimensional, 1 7 covering, 1 08 law of iterated logarithm, 1 8
c
CameronMartin'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 severaldimensional Brownian motion, 43 under time substitution, 41 Diffusion 1 dimensional, 50 backward equation, 63 CameronMartin's formula, 67 explosion 0f, Feller's test, 65 forward equation, 60 generator, 50 reflecting, 71 stochastic integral and differential equations for, 52 on severaldimensional manifold, 90 backward equation, 98 CameronMartin 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