560538.pdf

Chaotic Kabanov formula for the AzeÂma
martingales
NICOLAS PRIVAULT1, JOSEP LLUI
ÂSSOLE Â2and JOSEP VIVES2
1De
Âpartement de Mathe Âmatiques, Universite Âde la Rochelle, Avenue Marillac, 17042 La
Rochelle, Cedex 1, France. E-mail: [e-mail]
2Departament de Mathema
Átiques, Universitat Auto Ánoma de Barcelona, 08193 Bellaterra, Spain.
E-mail: [email protected].es; [email protected].es
We derive the chaotic expansion of the product of nth- and ®rst-order multiple stochastic integrals
with respect to certain normal martingales. This is done by application of the classical and quantum
product formulae for multiple stochastic integrals. Our approach extends existing results on chaotic
calculus for normal martingales and exhibits properties, relative to multiple stochastic integrals,
polynomials and Wick products, that characterize the Wiener and Poisson processes.
Keywords: AzeÂma martingales; multiple stochastic integrals; product formulae
1. Introduction
The Wiener±Itoà and Poisson±Itoà chaotic decompositions give an isometric isomorphism
between the Fock space Ã(L2(R)) and the space of square-integrable functionals of the
process. This isomorphism is constructed by association of a symmetric function
fn2L2(R)nto its multiple stochastic integral. The isometry property comes from the
fact that the angle brackets hBitand h~
Nitof both the Wiener and compensated Poisson
processes (Bt)t2R,(
~
Nt)t2Rare equal to t. Therefore such isometric isomorphisms may exist
for more general processes, provided their angle bracket equals t.
A martingale is said to be normal (Dellacherie et al. 1992) if its angle bracket is equal
to tand if the range of the multiple stochastic integrals isometry de®ned on Ã(L2(R)) is
equal to the space of square-integrable functionals of this martingale. This last property is
called the chaotic representation property.
The quadratic variation ([M,M]t)t2Rof a normal martingale (Mt)t2Rin L4(Ù)with
the chaotic representation property (or less restrictively, with the predictable representation
property) satis®es the equation
[M,M]ttt
0
ösdMs,t2R, (1)
called a structure equation (Emery 1989), where (ös)s2Ris a predictable process. On the
other hand, a martingale satisfying (1) has angle bracket equal to t, but does not necessarily
possess the chaotic representation property.
In this paper we work with solutions of such equations, which include the Wiener
Bernoulli 6(4), 2000, 633±651
1350±7265 #2000 ISI/BS

process, ö0; the compensated Poisson processes, öconstant and non-zero; and the
AzeÂma martingales, ötâMt,â2[ÿ2, 0[.
Note that in the probabilistic framework of the structure equation one has to use a
predictable version of ö, which is ötâMtÿin the case of the AzeÂma martingale.
However, since we are working in L2with stochastic integrals with respect to a normal
martingale, we do not need to distinguish between the adapted and predictable versions of ö
(Dellacherie et al. 1992, p. 199). Thus, in what follows we will use ötâMtin the case of
the AzeÂma martingale.
In the stochastic analysis on both the Wiener and Poisson spaces, multiplication formulae
for multiple stochastic integrals proved to be useful tools as they give the chaotic expansion
of the product of two multiple stochastic integrals. Recently (Russo and Vallois 1998),
multiplication formulae have been proposed for multiple stochastic integrals with respect to
normal martingales. However, these formulae do not give the explicit chaotic expansion of
the considered products.
Our ®rst goal in this paper is to relate the product formulae of Kabanov (1975), Russo
and Vallois (1998) and Surgailis (1984) to their counterparts in quantum probability; see
Section 3. With these tools we compute the chaotic expansion of the product of a multiple
stochastic integral with a single stochastic integral, and refer to this formula as the chaotic
Kabanov formula. This formula is proved for a class of martingales that includes the AzeÂma
martingales; see Section 4. From the Kabanov formula, we have not obtained a general
formula for the chaotic expansion of the product of two multiple stochastic integrals of
orders nand mfor this general class of martingales.
As an application of this Kabanov formula we show that a number of properties in
stochastic analysis such as the possibility of expressing multiple stochastic integrals with
polynomials and some properties of the Wick product are speci®c to the Wiener and
Poisson cases; see Section 5. In Section 6 we consider formulae giving the derivation of a
product.
2. Notation and preliminaries
We denote by
Ã(L2(R)) 
n2NL2(R)n
the symmetric Fock space over L2(R), where L2(R)nis endowed with the norm
k:k2
L2(R)nn!k:k2 L2(R)n, and h:,:idenotes the scalar product in L2(R).
In this paper we work with a normal martingale (Mt)t2R, with M00, that satis®es the
structure equation (1).
The multiple stochastic integral of a symmetric function fn2L2(R)nwith respect to
(Mt)t2Ris denoted by
In(fn)n!1
0tÿ
n
0tÿ
2
0
fn(t1,...,tn)dMt1 dMtn,
634 N. Privault, J.L. Sole
Âand J. Vives

and as a convention we let
In(fn)In(~
fn)In(fn(t1,...,tn)), (2)
if fn2L2(R)nis not symmetric, where ~
fndenotes the symmetrization of fnin its n
variables.
We denote by Ftthe ó-algebra generated by fMs:0<s<tg,t2R.
We let =ÿ:Ã(L2(R)) !Ã(L2(R)) L2(R) and =:Ã(L2(R)) L2(R)!
Ã(L2(R)) denote the annihilation and creation operators on Ã(L2(R)), de®ned as
=ÿ
tIn(fn)nI nÿ1(fn(:,t)), t2R,=(In(fn1)) In1(fn1),
where fn1is symmetric in its n®rst variables, n2N.
Let Af]a,b]: a,b2R,a,bg. For ]a,b], ]c,d]2Awe use the notation ]a,b]<
]c,d]ifb<c. Let Sn,n2N, denote the vector space
SnX
m
i1
áiIn(1Ai
11Ai
n): Ai
k\Ai lÆ,1<k,l<n,Ai 1,...,Ai n2A,
(
ái2R,i1, ...,m,m>1,
and let Sdenote the vector space generated by [n2NSn, which is dense in L2(Ù).
Let Vndenote the vector space
VnX
m
i1
ái1Ai
1In(1Ai
1...1Ai
n): Ai
k\Ai lÆ,1<k,l<n,
(
Ak
0\Al 0Æ,1<k,l<m,Ai 0,...,Ai n2A,ái2R,i1, ...,m,m>1:
We denote by Vthe vector space generated by [n>0Vn, which is dense in L2(dë) and in
L1(dë), where dëdt3dP.
Let Undenote the vector space
UnX
m
i1
ái1Ai
0In(1Ai 11Ai
n): Ai
k\Ai lÆ,0<k,l<n,
(
Ak
0\Al 0Æ,1<k,l<m,Ai 0,...,Ai n2A,ái2R,i1, ...,m,m>1:
Note that Unis made of processes in Vnthat do not depend on the `present'. We denote
by Uthe vector space generated by [n>0Un, and by 
Uthe completion of Uin L2(dë).
The space  Uis strictly smaller than L2(dë) but it contains the adapted square-integrable
processes and the range of =ÿ, since F2Simplies =ÿF2U.
We will use the following lemma of Ma et al. (1998).
Lemma 1. Let In(fn), I1(g1)2S.We have
In(fn)I1(g1)In1(fng1)n1
0
Inÿ1(fn(:,t))g1(t)d[M,M]t:(3)
Chaotic Kabanov formula for the Aze
Âma martingales 635

As a consequence of this lemma, we have
In(1A11An)I1(1A1)I1(1An)
whenever Ai\AjÆ,1<i,j<n.
We assume that there exists a set Pof functionals dense in L2(Ù) and included both in
L1(Ù) and in Dom(=ÿ), and such that Pis stable by =ÿ
t,t2R. This assumption is satis®ed
in the cases of interest to us, that is in the Wiener and Poisson cases, and also in the case of
the AzeÂma martingales since the latter is bounded (in this case it suf®ces to take PS).
We recall the following identity, satis®ed in general on Fock space:
E[=(u)2]kuk2
L2(Ù)L2(R)E1
01 0
=ÿ
sut=ÿ tusdsdt
:(4)
De®nition 1. Let Dom1,2(=)denote the set of processes u 2L1(dë)such that there exists a
sequence (un)n2NVconverging in L1(dë)to u and such that (=(un))n2Nconverges in
L1(Ù)to an element of L2(Ù)denoted as =(u).
Remark 1. The domain Dom1,2(=)is well de®ned. We can show this as follows. Assume
that (un)n2NVconverges in L1(dë) to 0 and that (=(un))n2Nconverges in L1(Ù)to
G2L2(Ù). From the duality relation
E[=(un)F]E1
0
un(t)=ÿ
tFdt
,F2P,
we obtain E[GF]0, F2P, hence G0 since G2L2(Ù).
This also means that if u2Dom1,2(=) and there exists a sequence (un)n2Nconverging
in L1(Ù)touand such that (=(un))n2Nconverges in L1(Ù)toG2L2(Ù), then
G=(u). Naturally, Dom1,2(=) contains the usual L2domain of =which is denoted as
Dom2(=).
Finally, let
TnX
m
i1
Fi1]ai,bi]In(1Ai
11Ai
n) : [0, bi]\Ai
jÆ,j1, ...,n,
(
Ai
k\Ai lÆ,1<k,l<n,Ai 1,...,Ai n2A,Fi2L2(Fai), i1, ...,m,m>1,
and Tbe the vector space generated by [nTn. The space T\L2(dë) is not dense in L2(dë),
but it contains U, and it is dense in  U.
Remark 2. Let u,v2U, where vis an adapted process. Then uv2T. To show this, we
write u1AIn(1A11An)withA1,...,Ak<Band
v1BIm(1An11Anm)
with An1,...,Anm<B, since vis adapted. Then
636 N. Privault, J.L. Sole
Âand J. Vives

uv1A\BIkm(1A11Ak1An11Anm)Inÿk(1Ak11An),
hence uv2T.
Proposition 1. Fo r u 2Tsuch that 1
0utdMt2L2(Ù), we have u 2Dom1,2(=)and
=(u)1
0
utdMt:(5)
Proof. We start by choosing uof the form uF1]a,b]In(1A11An), with F
Im(1B11Bm), Ai\[0, b]Æ,i1, ...,n, and B1,...,Bm[0, a]. Then, from
Lemma 1:
=(u)Inm1(1B11Bm1]a,b]1An)
I1(1]a,b])Im(1B11Bm)In(1A11An)
FI1(1]a,b])In(1A11An):
Hence, by linearity, for any u2Tand F2S,
=(u)FI1(1[a,b])In(1A11An)1
0
utdMt:
If u2T, then choosing a sequence (Fn)n2NSthat converges in L2(Ù)toF, and
letting umFm1]a,b]In(1A11An), we have the convergence of (um)m2Nin L1(dë)tou
and of (=(um))m2Nin L1(Ù)to1
0utdMtwhich belongs to L2(Ù) by hypothesis. h
3. Classical and quantum product formulae
The aim of this section is to link two versions of the product formula for a multiple stochastic
integral and a single stochastic integral with respect to normal martingales.
The ®rst one is called the Kabanov formula in reference to Kabanov (1975), which
treated the Poisson case, ö1, and can be stated as follows. For fn2L2(R)nand
g2L2([0, T]), in particular gwith compact support, Lemma 1 was extended by Ma et al.
(1998) and Russo and Vallois (1998) as
In(fn)I1(g)In1(fng)n1
0
I
nÿ1(fn(:,t))g(t)d[M,M]t:(6)
The second term in this formula is an integration over a diagonal, due to the Itoà formula, and
the notation `I
nÿ1(fn(:,t))' will be made precise in De®nition 2.
The second product formula uses only chaos expansions and Fock space, and can be
found in the work of quantum probabilists. It is often stated in the formalism of quantum
stochastic integrals:
Chaotic Kabanov formula for the Aze
Âma martingales 637