5.1 The Construction of Stochastic Integrals
5.1.1 Stochastic Integrals for Martingales Bounded in L 2
We writeH2 for the space of all continuous martingalesM which are bounded in L2 and such that M0 D 0, with the usual convention that two indistinguishable processes are identified. Equivalently,M 2 H2 if and only ifM is a continuous local martingale such thatM0 D 0andEŒhM;Mi1 <1(Proposition4.13). By Proposition3.21, ifM 2 H2, we haveMt D EŒM1 j FtwhereM1 2 L2is the almost sure limit ofMtast! 1.
Proposition4.15(v) shows that, ifM;N 2H2, the random variablehM;Ni1is well defined, and we haveEŒjhM;Ni1j <1. This allows us to define a symmetric bilinear form onH2via the formula
.M;N/H2 DEŒhM;Ni1DEŒM1N1;
where the second equality comes from Proposition4.15(v). Clearly.M;M/H2 D0 if and only ifMD 0. The scalar product.M;N/H2 thus yields a norm onH2given by
kMkH2 D.M;M/1=2H2 DEŒhM;Mi11=2DEŒ.M1/21=2:
Proposition 5.1 The space H2 equipped with the scalar product .M;N/H2 is a Hilbert space.
Proof We need to verify that the vector spaceH2is complete for the normk kH2. Let.Mn/n1be a sequence inH2which is Cauchy for that norm. We have then
m;limn!1EŒ.M1n M1m/2D lim
m;n!1.MnMm;MnMm/H2D0:
Consequently, the sequence.M1n / converges in L2 to a limit, which we denote by Z. On the other hand, Doob’s inequality in L2 (Proposition 3.15 (ii)) and a straightforward passage to the limit show that, for everym;n,
E h
sup
t0.MtnMmt /2i
4EŒ.Mn1M1m/2: We thus obtain that
m;limn!1E h
sup
t0.MtnMtm/2i
D0: (5.1)
Hence, for everyt 0,Mnt converges inL2, and we want to argue that the limit yields a process with continuous sample paths. To this end, we use (5.1) to find an
5.1 The Construction of Stochastic Integrals 99
increasing sequencenk" 1such that E
hX1
kD1
supt0jMntkMntkC1ji
X1 kD1
E h
supt0.MntkMntkC1/2i1=2
<1:
The last display implies that, a.s., X1 kD1
sup
t0jMtnkMtnkC1j<1;
and thus the sequence.Mtnk/t0converges uniformly onRC, a.s., to a limit denoted by.Mt/t0. On the zero probability set where the uniform convergence does not hold, we takeMtD0for everyt0. Clearly the limiting processMhas continuous sample paths and is adapted (here we use the fact that the filtration is complete).
Furthermore, from theL2-convergence of.M1n /toZ, we immediately get by passing to the limit in the identityMntk DEŒMn1k jFtthatMtD EŒZ jFt. Hence.Mt/t0
is a continuous martingale and is bounded inL2, so thatM 2H2. The a.s. uniform convergence of.Mtnk/t0 to.Mt/t0 then ensures that M1 D limM1nk D Z a.s.
Finally, theL2-convergence of.M1n /toZ D M1 shows that the sequence.Mn/
converges toMinH2. ut
We denote the progressive-field on˝ RCbyP (see the end of Sect.3.1), and, ifM2H2, we letL2.M/be the set of all progressive processesHsuch that
E h Z 1
0 Hs2dhM;Mis
i<1;
with the convention that two progressive processes H and H0 satisfying this integrability condition are identified ifHs D Hs0 D 0, dhM;Mis a.e., a.s. We can viewL2.M/as an ordinaryL2space, namely
L2.M/DL2.˝RC;P;dPdhM;Mis/
where dPdhM;Mis refers to the finite measure on.˝RC;P/that assigns the mass
E h Z 1
0 1A.!;s/dhM;Mis
i
to a setA2P(the total mass of this measure isEŒhM;Mi1D kMk2H2) . Just like anyL2space, the spaceL2.M/is a Hilbert space for the scalar product
.H;K/L2.M/DE h Z 1
0 HsKsdhM;Mis
i;
100 5 Stochastic Integration
and the associated norm is
kHkL2.M/D E
h Z 1
0 Hs2dhM;Mis
i1=2 :
Definition 5.2 Anelementary processis a progressive process of the form
Hs.!/D
p1
X
iD0
H.i/.!/1.ti;tiC1.s/;
where0 D t0 < t1 < t2 < < tp and for everyi 2 f0; 1; : : : ;p1g,H.i/is a boundedFti-measurable random variable.
The setE of all elementary processes forms a linear subspace ofL2.M/. To be precise, we should here say “equivalence classes of elementary processes” (recall thatHandH0are identified inL2.M/ifkHH0kL2.M/D0).
Proposition 5.3 For every M2H2,E is dense in L2.M/.
Proof By elementary Hilbert space theory, it is enough to verify that, ifK2L2.M/ is orthogonal toE, thenKD0. Assume thatK2L2.M/is orthogonal toE, and set, for everyt0,
XtD Z t
0 KudhM;Miu:
To see that the integral in the right-hand side makes sense, and defines a finite variation process.Xt/t0, we use the Cauchy–Schwarz inequality to observe that
E h Z t
0
jKujdhM;Miu
i E
h Z t
0.Ku/2dhM;Miu
i1=2
.EŒhM;Mi1/1=2: The right-hand side is finite sinceM 2 H2 andK 2 L2.M/, and thus we have in particular
a.s. 8t0;
Z t 0
jKujdhM;Miu<1:
By Proposition 4.5 (and Remark (i) following this proposition), .Xt/t0 is well defined as a finite variation process. The preceding bound also shows thatXt 2 L1 for everyt0.
Let0s<t, letFbe a boundedFs-measurable random-variable, and letH2E be the elementary process defined byHr.!/DF.!/1.s;t.r/. Writing.H;K/L2.M/D 0, we get
E h
F Z t
s
KudhM;Miu
iD0:
5.1 The Construction of Stochastic Integrals 101 It follows thatEŒF.XtXs/D0for everys<tand every boundedFs-measurable variableF. Since the processXis adapted and we know thatXr2L1for everyr0, this implies thatXis a (continuous) martingale. On the other hand,Xis also a finite variation process and, by Theorem4.8, this is only possible ifXD0. We have thus proved that
Z t
0 KudhM;MiuD0 8t0; a:s:
which implies that, a.s., the signed measure having density Ku with respect to dhM;Miuis the zero measure, which is only possible if
KuD0; dhM;Miua:e:; a:s:
or equivalentlyKD0inL2.M/. ut
Recall our notationXT for the processXstopped at the stopping timeT:XtT D Xt^T. IfM 2 H2, the fact thathMT;MTi1 D hM;MiT immediately implies that MT also belongs toH2. Furthermore, if H 2 L2.M/, the process1Œ0;THdefined by .1Œ0;TH/s.!/ D 1f0sT.!/gHs.!/ also belongs to L2.M/ (note that 1Œ0;T is progressive since it is adapted with left-continuous sample paths).
Theorem 5.4 Let M2H2. For every H2E of the form
Hs.!/D
p1
X
iD0
H.i/.!/1.ti;tiC1.s/;
the formula
.HM/tD
p1
X
iD0
H.i/.MtiC1^tMti^t/
defines a process HM2H2. The mapping H7!HM extends to an isometry from L2.M/intoH2. Furthermore, HM is the unique martingale ofH2that satisfies the property
hHM;Ni DH hM;Ni; 8N2H2: (5.2)
If T is a stopping time, we have
.1Œ0;TH/M D.HM/T DHMT: (5.3)
102 5 Stochastic Integration
We often use the notation
.HM/tD Z t
0 HsdMs
and call HM the stochastic integral of H with respect to M.
Remark The quantityH hM;Niin the right-hand side of (5.2) is an integral with respect to a finite variation process, as defined in Sect.4.1. The fact that we use a similar notationHAandHM for the integrals with respect to a finite variation processAand with respect to a martingaleMcreates no ambiguity since these two classes of processes are essentially disjoint.
Proof As a preliminary observation, we note that the definition ofH M when H 2 E does not depend on the decomposition chosen forHin the first display of the theorem. Using this remark, one then checks that the mappingH 7!HM is linear. We next verify that this mapping is an isometry fromE (viewed as a subspace ofL2.M/) intoH2.
FixH2E of the form given in the theorem, and for everyi2 f0; 1; : : : ;p1g, set
MitDH.i/.MtiC1^tMti^t/;
for everyt0. Then a simple verification shows thatMiis a continuous martingale (this was already used in the beginning of the proof of Theorem4.9), and that this martingale belongs toH2. It follows thatHM D Pp1
iD0Mi is also a martingale inH2. Then, we note that the continuous martingalesMiare orthogonal, and their respective quadratic variations are given by
hMi;MiitDH.2i/
hM;MitiC1^t hM;Miti^t
(the orthogonality of the martingalesMias well as the formula of the last display are easily checked, for instance by using the approximations ofhM;Ni). We conclude that
hHM;HMitD
p1
X
iD0
H2.i/
hM;MitiC1^t hM;Miti^t D
Z t
0 Hs2dhM;Mis: Consequently,
kHMk2H2 DEŒhHM;HMi1DE h Z 1
0 H2sdhM;Mis
iD kHk2L2.M/:
By linearity, this implies thatHMDH0MifH0is another elementary process that is identified withHinL2.M/. Therefore the mappingH7!HMmakes sense from
5.1 The Construction of Stochastic Integrals 103 E viewed as a subspace ofL2.M/intoH2. The latter mapping is linear, and, since it preserves the norm, it is an isometry fromE (equipped with the norm ofL2.M/) into H2. Since E is dense inL2.M/(Proposition 5.3) and H2 is a Hilbert space (Proposition5.1), this mapping can be extended in a unique way to an isometry fromL2.M/intoH2.
Let us verify property (5.2). We fixN2H2. We first note that, ifH2L2.M/, the Kunita–Watanabe inequality (Proposition4.18) shows that
E h Z 1
0 jHsj jdhM;Nisji
kHkL2.M/kNkH2 <1
and thus the variableR1
0 HsdhM;Nis D .H hM;Ni/1is well defined and inL1. Consider first the case whereHis an elementary process of the form given in the theorem, and define the continuous martingalesMi,0 i p1, as previously.
Then, for everyi2 f0; 1; : : : ;p1g, hHM;Ni D
p1
X
iD0
hMi;Ni
and we have
hMi;NitDH.i/
hM;NitiC1^t hM;Niti^t : It follows that
hHM;NitD
p1
X
iD0
H.i/
hM;NitiC1^t hM;Niti^t D
Z t
0 HsdhM;Nis
which gives (5.2) whenH 2 E. We then observe that the linear mappingX 7!
hX;Ni1is continuous fromH2intoL1. Indeed, by the Kunita–Watanabe inequality, EŒjhX;Ni1jEŒhX;Xi11=2EŒhN;Ni11=2D kNkH2kXkH2:
If.Hn/n1is a sequence inE, such thatHn!HinL2.M/, we have therefore hHM;Ni1D lim
n!1hHnM;Ni1D lim
n!1.Hn hM;Ni/1D.H hM;Ni/1; where the convergences hold inL1, and the last equality again follows from the Kunita–Watanabe inequality by writing
Ehˇˇˇ Z 1
0 .HsnHs/dhM;Nisˇˇˇi
EŒhN;Ni11=2kHnHkL2.M/:
104 5 Stochastic Integration We have thus obtained the identityhHM;Ni1D.H hM;Ni/1, but replacingN by the stopped martingaleNtin this identity also giveshHM;NitD.H hM;Ni/t, which completes the proof of (5.2).
It is easy to see that (5.2) characterizesHM among the martingales of H2. Indeed, ifXis another martingale ofH2that satisfies the same identity, we get, for everyN2H2,
hHMX;Ni D0:
TakingNDHMXand using Proposition4.12we obtain thatXDHM.
It remains to verify (5.3). Using the properties of the bracket of two continuous local martingales, we observe that, ifN2H2,
h.HM/T;NitD hHM;Nit^T D.H hM;Ni/t^T D.1Œ0;TH hM;Ni/t
which shows that the stopped martingale.HM/Tsatisfies the characteristic property of the stochastic integral.1Œ0;TH/M. The first equality in (5.3) follows. The second one is proved analogously, writing
hHMT;Ni DH hMT;Ni DH hM;NiTD1Œ0;TH hM;Ni:
This completes the proof of the theorem. ut
Remark We could have used the relation (5.2) todefinethe stochastic integralHM, observing that the mappingN7!EŒ.H hM;Ni/1yields a continuous linear form onH2, and thus there exists a unique martingaleHMinH2such that
EŒ.H hM;Ni/1D.HM;N/H2 DEŒhHM;Ni1:
Using the notation introduced at the end of Theorem5.4, we can rewrite (5.2) in the form
h Z
0 HsdMs;NitD Z t
0 HsdhM;Nis:
We interpret this by saying that the stochastic integral “commutes” with the bracket.
Let us immediately mention a very important consequence. IfM 2 H2, andH 2 L2.M/, two successive applications of (5.2) give
hHM;HMi DH.H hM;Mi/DH2 hM;Mi;
5.1 The Construction of Stochastic Integrals 105 using the “associativity property” (4.1) of integrals with respect to finite variation processes. Put differently, the quadratic variation of the continuous martingale HMis
h Z
0 HsdMs; Z
0 HsdMsitD Z t
0 Hs2dhM;Mis: (5.4) More generally, ifNis another martingale ofH2andK2L2.N/, the same argument gives
h Z
0 HsdMs; Z
0 KsdNsitD Z t
0 HsKsdhM;Nis: (5.5) The following “associativity” property of stochastic integrals, which is analogous to property (4.1) for integrals with respect to finite variation processes, is very useful.
Proposition 5.5 Let H 2 L2.M/. If K is a progressive process, we have KH 2 L2.M/if and only if K2L2.HM/. If the latter properties hold,
.KH/MDK.HM/:
Proof Using property (5.4), we have E
h Z 1
0 Ks2H2sdhM;Mis
iDE h Z 1
0 K2sdhHM;HMis
i;
which gives the first assertion. For the second one, we write forN2H2, h.KH/M;Ni DKH hM;Ni DK.H hM;Ni/DK hHM;Ni and, by the uniqueness statement in (5.2), this implies that.KH/M DK.HM/.
u t Moments of stochastic integrals. LetM 2 H2,N 2 H2,H 2 L2.M/andK 2 L2.N/. SinceHMandKNare martingales inH2, we have, for everyt2Œ0;1,
E h Z t
0 HsdMs
iD0 (5.6)
E h Z t
0 HsdMs
Z t
0 KsdNs
iDE h Z t
0 HsKsdhM;Nis
i; (5.7)
using Proposition4.15(v) and (5.5) to derive (5.7). In particular, E
h Z t
0 HsdMs
2i DE
h Z t
0 Hs2dhM;Misi
: (5.8)
106 5 Stochastic Integration Furthermore, sinceHMis a (true) martingale, we also have for every0s<t 1,
E h Z t
0 HrdMrˇˇˇFs
iD Z s
0 HrdMr; (5.9)
or equivalently
E h Z t
s
HrdMrˇˇˇFs
iD0
with an obvious notation forRt
sHrdMr. It is important to observe that these formulas (and particularly (5.6) and (5.8)) may no longer hold for the extensions of stochastic integrals that we will now describe.