128 5 Stochastic Integration for any choice of0 Dt0 <t1< <tnand1; : : : ; n 2R, is dense in the space L2C.˝;F1;P/of all square-integrable complex-valuedF1-measurable random variables.
Proof It is enough to prove that, ifZ 2L2C.˝;F1;P/is such that E
h Z exp
i
Xn jD1
j.BtjBtj1/i
D0 (5.17)
for any choice of0Dt0 <t1< <tnand1; : : : ; n 2R, thenZD0.
Fix0Dt0 <t1 < <tn, and consider the complex measureonRndefined by
.F/DE h
Z1F.Bt1;Bt2Bt1; : : : ;BtnBtn1/i
for any Borel subsetFofRn. Then (5.17) exactly shows that the Fourier transform of is identically zero. By the injectivity of the Fourier transform on complex measures onRd, it follows that D 0. We have thusEŒZ1A D 0for everyA 2 .Bt1; : : : ;Btn/.
A monotone class argument then shows that the identityEŒZ1A D 0 remains valid for anyA2.Bt;t0/, and then by completion for anyA2F1. It follows
thatZD0. ut
Proof of Theorem5.18 We start with the first assertion. We first observe that the uniqueness ofhis easy since, if the representation of a given variableZholds with two processeshandh0inL2.B/, we have
E h Z 1
0 .hsh0s/2ds iDE
h Z 1
0 hsdBs Z 1
0 h0sdBs
2i D0;
hencehDh0inL2.B/.
Let us turn to the existence part. LetH stand for the vector space of all variables Z 2 L2.˝;F1;P/for which the property of the statement holds. We note that if Z2H andhis the associated process inL2.B/, we have
EŒZ2D.EŒZ/2CE h Z 1
0 .hs/2ds i:
It follows that H is a closed subspace of L2.˝;F1;P/. Indeed, if .Zn/ is a sequence in H that converges to Z in L2.˝;F1;P/, the processes h.n/ corre- sponding respectively to the variablesZnform a Cauchy sequence inL2.B/, hence converge inL2.B/to a certain processh 2 L2.B/– here we use the Hilbert space structure ofL2.B/– and it immediately follows thatZDEŒZCR1
0 hsdBs. SinceH is closed, in order to prove thatH DL2.˝;F1;P/, we just have to verify thatH contains a dense subset ofL2.˝;F1;P/. Let0Dt0<t1 < <tn
5.4 The Representation of Martingales as Stochastic Integrals 129
and1; : : : ; n2R, and setf.s/DPn
jD1j1.tj1;tj.s/. WriteEtf for the exponential martingaleE.iR
0f.s/dBs/(cf. Proposition5.11). Proposition5.11shows that exp
i
Xn jD1
j.Btj Btj1/C 1 2
Xn jD1
2j.tjtj1/
DE1f D1Ci Z 1
0 Esff.s/dBs
and it follows that both the real part and the imaginary part of variables of the form exp
iPn
jD1j.BtjBtj1/
are inH. By Lemma5.19, linear combinations of such random variables are dense inL2.˝;F1;P/. This completes the proof of the first assertion of the theorem.
Let us turn to the second assertion. IfM is a martingale that is bounded inL2, thenM12L2.˝;F1;P/, and thus can be written in the form
M1DEŒM1C Z 1
0 hsdBs; whereh2L2.B/. Thanks to (5.9), it follows that
MtDEŒM1jFtDEŒM1C Z t
0 hsdBs
and the uniqueness ofhis also immediate from the uniqueness in the first assertion.
Finally, ifMis a continuous local martingale, we have firstM0DC2Rbecause the-fieldF0contains only events of probability zero or one. IfTn D infft 0W jMtj ngwe can apply the case of martingales bounded inL2toMTnand we get a processh.n/2L2.B/such that
MtTn DCC Z t
0 h.sn/dBs:
Using the uniqueness of the progressive process in the representation, we get that h.sm/ D 1Œ0;Tm.s/h.sn/ ifm < n, dsa.e., a.s. It is now easy to construct a process h2L2loc.B/such that, for everym,h.sm/D1Œ0;Tm.s/hs, dsa.e., a.s. The representation formula of the theorem follows, and the uniqueness ofhis also straightforward. ut Consequences Let us give two important consequences of the representation theorem. Under the assumptions of the theorem:
(1) The filtration.Ft/t0is right-continuous. Indeed, lett0and letZbeFtC- measurable and bounded. We can findh2L2.B/such that
ZDEŒZC Z 1
0 hsdBs:
130 5 Stochastic Integration
If" > 0,ZisFtC"-measurable, and thus, using (5.9), ZDEŒZjFtC"DEŒZC
Z tC"
0 hsdBs: When"!0the right-hand side converges inL2to
EŒZC Z t
0 hsdBs:
Thus Z is equal a.s. to an Ft-measurable random variable, and, since the filtration is complete,ZisFt-measurable.
A similar argument shows that the filtration.Ft/t0is also left-continuous:
If, fort> 0, we let
Ft D _
s2Œ0;t/
Fs
be the smallest -field that contains all-fields Fs for s 2 Œ0;t/, we have FtDFt.
(2) All martingales of the filtration.Ft/t0 have a modification with continu- ous sample paths. For a martingale that is bounded inL2, this follows from the representation formula (see the remark after the statement of the theorem). Then consider a uniformly integrable martingaleM(ifMis not uniformly integrable, we just replaceM byMt^a for everya 0). In that case, we have, for every t0,
MtDEŒM1jFt:
By Theorem3.18(whose application is justified as we know that the filtration is right-continuous), the processMthas a modification with càdlàg sample paths, and we consider this modification. LetM1.n/be a sequence of bounded random variables such thatM1.n/!M1inL1asn! 1. Introduce the martingales
M.tn/DEŒM1.n/jFt;
which are bounded inL2. By the beginning of the argument, we can assume that, for everyn, the sample paths ofM.n/are continuous. On the other hand, Doob’s maximal inequality (Proposition3.15) implies that, for every > 0,
P h
sup
t0jM.tn/Mtj> i 3
EŒjM1.n/M1j:
5.4 The Representation of Martingales as Stochastic Integrals 131
It follows that we can find a sequencenk" 1such that, for everyk1, P
h
supt0jM.tnk/Mtj> 2ki 2k:
An application of the Borel–Cantelli lemma now shows that supt0jM.tnk/Mtj !a:s:
k!10
and we get that the sample paths of M are continuous as uniform limits of continuous functions.
Multidimensional extension Let us briefly describe the multidimensional exten- sion of the preceding results. We now assume that the filtration .Ft/ on ˝ is the completed canonical filtration of a d-dimensional Brownian motion B D .B1; : : : ;Bd/started from0. Then, for every random variableZ 2 L2.˝;F1;P/, there exists a uniqued-tuple.h1; : : : ;hd/of progressive processes, satisfying
E h Z 1
0 .his/2ds
i<1; 8i2 f1; : : : ;dg;
such that
ZDEŒZC Xd
iD1
Z 1
0 hisdBis:
Similarly, ifMis a continuous local martingale, there exist a constantCand a unique d-tuple.h1; : : : ;hd/of progressive processes, satisfying
Z t
0.his/2ds<1; a.s. 8t0; 8i2 f1; : : : ;dg; such that
MtDCC Xd
iD1
Z t 0 hisdBis:
The proofs are exactly the same as in the casedD1(Theorem5.18). Consequences (1) and (2) above remain valid.
132 5 Stochastic Integration