Throughout this chapter, we argue on a probability space.˝;F;P/. Most of the time, but not always, random processes will be indexed byT DRCand take values inR.
Definition 2.1 LetGbe a Gaussian white noise onRCwhose intensity is Lebesgue measure. The random process.Bt/t2RCdefined by
BtDG.1Œ0;t/ is calledpre-Brownian motion.
Proposition 2.2 Pre-Brownian motion is a centered Gaussian process with covari- ance
K.s;t/Dminfs;tg.notD:/s^t:
© Springer International Publishing Switzerland 2016
J.-F. Le Gall,Brownian Motion, Martingales, and Stochastic Calculus, Graduate Texts in Mathematics 274, DOI 10.1007/978-3-319-31089-3_2
19
20 2 Brownian Motion Proof By the definition of a Gaussian white noise, the variablesBt belong to a common Gaussian space, and therefore.Bt/t0is a Gaussian process. Moreover, for everys;t0,
EŒBsBtDEŒG.Œ0;s/G.Œ0;t/D Z 1
0 dr1Œ0;s.r/1Œ0;t.r/Ds^t:
u t The next proposition gives different ways of characterizing pre-Brownian motion.
Proposition 2.3 Let .Xt/t0 be a (real-valued) random process. The following properties are equivalent:
(i) .Xt/t0is a pre-Brownian motion;
(ii) .Xt/t0is a centered Gaussian process with covariance K.s;t/Ds^t;
(iii) X0 D 0 a.s., and, for every 0 s < t, the random variable Xt Xs is independent of.Xr;rs/and distributed according toN.0;ts/;
(iv) X0 D 0a.s., and, for every choice of 0 D t0 < t1 < < tp, the variables Xti Xti1,1ip are independent, and, for every1ip, the variable Xti Xti1is distributed according toN .0;titi1/.
Proof The fact that (i))(ii) is Proposition2.2. Let us show that (ii))(iii). We assume that.Xt/t0is a centered Gaussian process with covarianceK.s;t/Ds^t, and we letH be the Gaussian space generated by.Xt/t0. ThenX0 is distributed according toN .0; 0/and thereforeX0D0a.s. Then, fixs> 0and writeHsfor the vector space spanned by.Xr; 0 r s/, andHQsfor the vector space spanned by .XsCuXs;u0/. ThenHsandHQsare orthogonal since, forr2Œ0;sandu0,
EŒXr.XsCuXs/Dr^.sCu/r^sDrrD0:
Noting thatHsandHQsare subspaces ofH, we deduce from Theorem1.9that.Hs/ and.HQs/are independent. In particular, if we fix t > s, the variableXtXs is independent of.Hs/ D .Xr;r s/. Finally, using the form of the covariance function, we immediately get thatXtXsis distributed according toN .0;ts/.
The implication (iii))(iv) is straightforward. Takings D tp1 andt D tp we obtain thatXtpXtp1 is independent of.Xt1; : : : ;Xtp1/. Similarly,Xtp1Xtp2is independent of.Xt1; : : : ;Xtp2/, and so on. This implies that the variablesXtiXti1, 1ip, are independent.
Let us show that (iv))(i). It easily follows from (iv) thatXis a centered Gaussian process. Then, iff is a step function onRCof the formf DPn
iD1i1.ti1;ti, where 0Dt0<t1<t2< <tp, we set
G.f/D Xn
iD1
i.XtiXti1/
2.1 Pre-Brownian Motion 21 (note that this definition ofG.f/depends only onf and not on the particular way we have written f D Pn
iD1i1.ti1;ti). Suppose then that f and g are two step functions. We can writef D Pn
iD1i1.ti1;ti andg D Pn
iD1i1.ti1;ti with the samesubdivision0Dt0<t1<t2 < <tpforf and forg(just take the union of the subdivisions arising in the expressions off andg). It then follows from a simple calculation that
EŒG.f/G.g/D Z
RC
f.t/g.t/dt;
so thatG is an isometry from the vector space of step functions onRC into the Gaussian space H generated by X. Using the fact that step functions are dense inL2.RC;B.RC/;dt/, we get that the mappingf 7! G.f/can be extended to an isometry fromL2.RC;B.RC/;dt/into the Gaussian spaceH. Finally, we have
G.Œ0;t/DXtX0DXtby construction. ut
Remark The variant of (iii) where the law ofXtXsis not specified but required to only depend ontsis called the property of stationarity (or homogeneity) and independence of increments. Pre-Brownian motion is thus a special case of the class of processes with stationary independent increments (under an additional regularity assumption, these processes are also called Lévy processes, see Sect.6.5.2).
Corollary 2.4 Let.Bt/t0be a pre-Brownian motion. Then, for every choice of0D t0<t1< <tn, the law of the vector.Bt1;Bt2; : : : ;Btn/has density
p.x1; : : : ;xn/D 1 .2/n=2p
t1.t2t1/ : : : .tntn1/ exp
Xn iD1
.xixi1/2 2.titi1/
;
where by convention x0D0.
Proof The random variablesBt1;Bt2 Bt1; : : : ;Btn Btn1 are independent with respective distributions N.0;t1/;N .0;t2 t1/; : : : ;N .0;tn tn1/. Hence the vector.Bt1;Bt2Bt1; : : : ;BtnBtn1/has density
q.y1; : : : ;yn/D 1 .2/n=2p
t1.t2t1/ : : : .tntn1/ exp
Xn iD1
y2i 2.titi1/
;
and the change of variablesxi D y1C Cyi fori 2 f1; : : : ;ngcompletes the argument. Alternatively we could have used Theorem1.3(ii). ut Remark Corollary2.4, together with the propertyB0 D 0, determines the collec- tion offinite-dimensional marginal distributionsof pre-Brownian motion. Property (iv) of Proposition 2.3shows that a process having the same finite-dimensional marginal distributions as pre-Brownian motion must also be a pre-Brownian motion.
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!.