22 2 Brownian Motion
Proposition 2.5 Let B be a pre-Brownian motion. Then, (i) B is also a pre-Brownian motion (symmetry property);
(ii) for every > 0, the process Bt D 1B2t is also a pre-Brownian motion (invariance under scaling);
(iii) for every s0, the process B.ts/DBsCtBsis a pre-Brownian motion and is independent of.Br;rs/(simple Markov property).
Proof (i) and (ii) are very easy. Let us prove (iii). With the notation of the proof of Proposition2.3, the-field generated byB.s/is.HQs/, which is independent of .Hs/ D .Br;r s/. To see thatB.s/is a pre-Brownian motion, it is enough to verify property (iv) of Proposition 2.3, which is immediate since B.tis/B.tis1/ D
BsCtiBsCti1. ut
LetBbe a pre-Brownian motion and letGbe the associated Gaussian white noise.
Note thatGis determined byB: Iff is a step function there is an explicit formula forG.f/in terms ofB, and one then uses a density argument. One often writes for f 2L2.RC;B.RC/;dt/,
G.f/D Z 1
0 f.s/dBs
and similarly
G.f1Œ0;t/D Z t
0 f.s/dBs ; G.f1.s;t/D Z t
s
f.r/dBr: This notation is justified by the fact that, ifu< v,
Z v
u
dBsDG..u; v/DG.Œ0; v/G.Œ0;u/DBvBu: The mappingf 7!R1
0 f.s/dBs(that is, the Gaussian white noiseG) is then called theWiener integralwith respect toB. Recall thatR1
0 f.s/dBsis distributed according toN .0;R1
0 f.s/2ds/.
Since a Gaussian white noise is not a “real” measure depending on!,R1 0 f.s/dBs
is not a “real” integral depending on!. Much of what follows in this book is devoted to extending the definition ofR1
0 f.s/dBsto functionsf that may depend on!.
2.2 The Continuity of Sample Paths 23 Definition 2.6 Let.Xt/t2Tbe a random process with values inE. Thesample paths ofXare the mappingsT 3 t 7! Xt.!/obtained when fixing! 2 ˝. The sample paths ofXthus form a collection of mappings fromTintoEindexed by!2˝.
LetB D .Bt/t0 be a pre-Brownian motion. At the present stage, we have no information about the sample paths ofB. We cannot even assert that these sample paths are measurable functions. In this section, we will show that, at the cost of
“slightly” modifyingB, we can ensure that sample paths are continuous.
Definition 2.7 Let .Xt/t2T and.XQt/t2T be two random processes indexed by the same index setT and with values in the same metric spaceE. We say thatXQ is a modificationofXif
8t2T; P.XQtDXt/D1:
This implies in particular thatXQ has the same finite-dimensional marginals asX.
Thus, ifXis a pre-Brownian motion,XQ is also a pre-Brownian motion. On the other hand, sample paths ofXQ may have very different properties from those ofX. For instance, considering the case whereT D RC andE D R, it is easy to construct examples where all sample paths ofXQ are continuous whereas all sample paths ofX are discontinuous.
Definition 2.8 The processXQ is said to beindistinguishablefromXif there exists a negligible subsetNof˝ such that
8!2˝nN; 8t2T; XQt.!/DXt.!/:
Put in a different way,XQ is indistinguishable fromXif P.8t2T; XtD QXt/D1:
(This formulation is slightly incorrect since the setf8t2 T; Xt D QXtgneed not be measurable.)
If XQ is indistinguishable from X then XQ is a modification of X. The notion of indistinguishability is however much stronger: Two indistinguishable processes have a.s. the same sample paths. In what follows, we will always identify two indistinguishable processes. An assertion such as “there exists a unique process such that . . . ” should always be understood “up to indistinguishability”, even if this is not stated explicitly.
The following observation will play an important role. Suppose thatT DIis an interval ofR. If the sample paths of bothXandXQ are continuous (except possibly on a negligible subset of ˝), then XQ is a modification ofX if and only if XQ is indistinguishable fromX. Indeed, ifXQ is a modification ofXwe have a.s.Xt D QXt
for everyt 2 I\Q(we throw out a countable union of negligible sets) hence a.s.
XtD QXtfor everyt2I, by a continuity argument. We get the same result if we only assume that the sample paths are right-continuous, or left-continuous.
24 2 Brownian Motion Theorem 2.9 (Kolmogorov’s lemma) Let X D .Xt/t2I be a random process indexed by a bounded interval I ofR, and taking values in a complete metric space .E;d/. Assume that there exist three reals q; ";C> 0such that, for every s;t2I,
EŒd.Xs;Xt/qCjtsj1C":
Then, there is a modificationX of X whose sample paths are Hölder continuousQ with exponent˛for every˛2.0;q"/: This means that, for every! 2˝ and every
˛2.0;"q/, there exists a finite constant C˛.!/such that, for every s;t2I, d.XQs.!/;XQt.!//C˛.!/jtsj˛:
In particular, X is a modification of X with continuous sample paths (by theQ preceding observations such a modification is unique up to indistinguishability).
Remarks
(i) If I is unbounded, for instance if I D RC, we may still apply Theorem 2.9 successively with I D Œ0; 1; Œ1; 2; Œ2; 3; etc. and we get that X has a modification whose sample paths arelocallyHölder with exponent˛for every
˛2.0; "=q/.
(ii) It is enough to prove that, for every fixed˛ 2 .0; "=q/,Xhas a modification whose sample paths are Hölder with exponent˛. Indeed, we can then apply this result to every choice of˛in a sequence˛k""=q, noting that the resulting modifications are indistinguishable, by the observations preceding the theorem.
Proof To simplify the presentation, we takeID Œ0; 1, but the proof would be the same for any bounded interval (closed or not). We fix˛2.0;q"/.
The assumption of the theorem implies that, fora> 0ands;t2I, P.d.Xs;Xt/a/aqEŒd.Xs;Xt/qC aqjtsj1C":
We apply this inequality tos D .i1/2n,t D i2n (fori 2 f1; : : : ; 2ng) and aD2n˛:
P
d.X.i1/2n;Xi2n/2n˛
C2nq˛2.1C"/n: By summing overiwe get
P
2n
[
iD1
fd.X.i1/2n;Xi2n/2n˛g
!
2nC2nq˛.1C"/nDC2n."q˛/:
2.2 The Continuity of Sample Paths 25
By assumption,"q˛ > 0. Summing now overn, we obtain X1
nD1
P
2n
[
iD1
fd.X.i1/2n;Xi2n/2n˛g
!
<1;
and the Borel–Cantelli lemma implies that, with probability one, we can find a finite integern0.!/such that
8nn0.!/; 8i2 f1; : : : ; 2ng; d.X.i1/2n;Xi2n/2n˛: Consequently the constantK˛.!/defined by
K˛.!/Dsup
n1 sup
1i2n
d.X.i1/2n;Xi2n/ 2n˛
!
is finite a.s. (Ifn n0.!/, the supremum inside the parentheses is bounded above by1, and, on the other hand, there are only finitely many terms beforen0.!/.)
At this point, we use an elementary analytic lemma, whose proof is postponed until after the end of the proof of Theorem2.9. We writeDfor the set of all reals of Œ0; 1/of the formi2nfor some integern1and somei2 f0; 1; : : : ; 2n1g.
Lemma 2.10 Let f be a mapping defined on D and with values in the metric space .E;d/. Assume that there exists a real˛ > 0and a constant K <1such that, for every integer n1and every i2 f1; 2; : : : ; 2n1g,
d.f..i1/2n/;f.i2n//K2n˛: Then we have, for every s;t2D,
d.f.s/;f.t// 2K
12˛jtsj˛:
We immediately get from the lemma and the definition ofK˛.!/that, on the eventfK˛.!/ <1g(which has probability1), we have, for everys;t2D,
d.Xs;Xt/C˛.!/jtsj˛;
whereC˛.!/ D 2.12˛/1K˛.!/. Consequently, on the eventfK˛.!/ < 1g, the mappingt 7! Xt.!/is Hölder continuous onD, hence uniformly continuous onD. Since.E;d/is complete, this mapping has a unique continuous extension to IDŒ0; 1, which is also Hölder with exponent˛. We can thus set, for everyt2Œ0; 1
XQt.!/D ( lim
s!t;s2DXs.!/ifK˛.!/ <1; x0 ifK˛.!/D 1;
26 2 Brownian Motion
wherex0 is a point of E which can be fixed arbitrarily. Clearly XQt is a random variable.
By the previous remarks, the sample paths of the processXQ are Hölder with exponent˛onŒ0; 1. We still need to verify thatXQ is a modification ofX. To this end, fixt2Œ0; 1. The assumption of the theorem implies that
lims!t XsDXt
in probability. Since by definitionXQt is the almost sure limit ofXs whens ! t,
s2D, we conclude thatXtD QXta.s. ut
Proof of Lemma2.10 Fixs;t 2 Dwiths < t. Letp 1 be the smallest integer such that2pts, and letk0be the smallest integer such thatk2p s. Then, we may write
sDk2p"12p1: : :"l2pl
tDk2pC"002pC"012p1C: : :C"0m2pm;
wherel;mare nonnegative integers and"i; "0j D 0or1 for every1 i l and 0jm. Set
siDk2p"12p1: : :"i2pi for every0il;
tjDk2pC"002pC"012p1C: : :C"0j2pjfor every0jm:
Then, noting thatsDsl;tDtmand that we can apply the assumption of the lemma to each of the pairs.s0;t0/,.si1;si/(for1 i l) and.tj1;tj/(for1 j m), we get
d.f.s/;f.t//Dd.f.sl/;f.tm//
d.f.s0/;f.t0//C Xl
iD1
d.f.si1/;f.si//C Xm
jD1
d.f.tj1/;f.tj//
K2p˛C Xl
iD1
K2.pCi/˛C Xm
jD1
K2.pCj/˛
2K.12˛/12p˛ 2K.12˛/1.ts/˛
since2pts. This completes the proof of Lemma2.10. ut We now apply Theorem2.9to pre-Brownian motion.
2.2 The Continuity of Sample Paths 27 Corollary 2.11 Let B D .Bt/t0 be a pre-Brownian motion. The process B has a modification whose sample paths are continuous, and even locally Hölder continuous with exponent12ıfor everyı2.0;12/.
Proof Ifs<t, the random variableBtBsis distributed according toN .0;ts/, and thusBtBshas the same law asp
ts U, whereUisN.0; 1/. Consequently, for everyq> 0,
EŒjBtBsjqD.ts/q=2EŒjUjqDCq.ts/q=2
whereCq D EŒjUjq < 1. Takingq > 2, we can apply Theorem2.9with" D
q
2 1. It follows thatBhas a modification whose sample paths are locally Hölder continuous with exponent˛for every˛ < .q2/=.2q/. Ifqis large we can take˛
arbitrarily close to12. ut
Definition 2.12 A process.Bt/t0is aBrownian motionif:
(i) .Bt/t0is a pre-Brownian motion.
(ii) All sample paths ofBare continuous.
This is in fact the definition of areal(orlinear) Brownian motionstarted from 0. Extensions to arbitrary starting points and to higher dimensions will be discussed later.
The existence of Brownian motion in the sense of the preceding definition follows from Corollary 2.11. Indeed, starting from a pre-Brownian motion, this corollary provides a modification with continuous sample paths, which is still a pre-Brownian motion. In what follows we no longer consider pre-Brownian motion, as we will be interested only in Brownian motion.
It is important to note that the statement of Proposition2.5holds without change if pre-Brownian motion is replaced everywhere by Brownian motion. Indeed, with the notation of this proposition, it is immediate to verify that B;B;B.s/ have continuous sample paths ifBdoes.
The Wiener measure. LetC.RC;R/be the space of all continuous functions from RCintoR. We equipC.RC;R/with the-fieldC defined as the smallest-field on C.RC;R/for which the coordinate mappings w 7!w.t/are measurable for every t0(alternatively, one checks thatC coincides with the Borel-field onC.RC;R/
associated with the topology of uniform convergence on every compact set). Given a Brownian motionB, we can consider the mapping
˝!C.RC;R/
!7!.t7!Bt.!//
and one verifies that this mapping is measurable (if we take its composition with a coordinate map w7! w.t/we get the random variableBt, and a simple argument shows that this suffices for the desired measurability).
28 2 Brownian Motion TheWiener measure(or law of Brownian motion) is by definition the image of the probability measureP.d!/under this mapping. The Wiener measure, which we denote byW.dw/, is thus a probability measure onC.RC;R/, and, for every measurable subsetAofC.RC;R/, we have
W.A/DP.B2A/;
where in the right-hand side B: stands for the random continuous functiont 7!
Bt.!/.
We can specialize the last equality to a “cylinder set” of the form AD fw2C.RC;R/Ww.t0/2A0;w.t1/2A1; : : : ;w.tn/2Ang;
where0Dt0 <t1 < <tn, andA0;A1; : : : ;An 2B.R/(recall thatB.R/stands for the Borel-field onR). Corollary2.4then gives
W.fwIw.t0/2A0;w.t1/2A1; : : : ;w.tn/2Ang/ DP.Bt0 2A0;Bt1 2A1; : : : ;Btn2An/ D1A0.0/
Z
A1An
dx1: : :dxn
.2/n=2p
t1.t2t1/ : : : .tntn1/ exp
Xn iD1
.xixi1/2 2.titi1/
;
wherex0D0by convention.
This formula for theW-measure of cylinder sets characterizes the probability measureW. Indeed, the class of cylinder sets is stable under finite intersections and generates the -fieldC, which by a standard monotone class argument (see AppendixA1) implies that a probability measure onC is characterized by its values on this class. A consequence of the preceding formula for theW-measure of cylinder sets is the (fortunate) fact that the definition of the Wiener measure does not depend on the choice of the Brownian motionB: The law of Brownian motion is uniquely defined!
Suppose thatB0is another Brownian motion. Then, for everyA2C, P.B02A/DW.A/DP.B2A/:
This means that the probability that a given property (corresponding to a measurable subset A ofC.RC;R/) holds is the same for the sample paths of B and for the sample paths ofB0. We will use this observation many times in what follows (see in particular the second part of the proof of Proposition2.14below).
Consider now the special choice of a probability space,
˝ DC.RC;R/; F DC; P.dw/DW.dw/: