relatively to an abstract measure space (X;FX; ), and allows to extend the Skorohod .... for any A 2 FX and any countable partition (Ak)k2IN ..... j i n, then Fj is Fp.
Poisson stochastic integration in Hilbert spaces Nicolas Privault Departement de Mathematiques Universite de La Rochelle 17042 La Rochelle France
Jiang-Lun Wu Institut fur Mathematik Ruhr-Universitat Bochum D-44780 Bochum Germany
Abstract
This paper aims to construct adaptedness and stochastic integration on Poisson space in the abstract setting of Hilbert spaces with minimal hypothesis, in particular without use of any notion of time or ordering on index sets. In this framework, several types of stochastic integrals are considered on simple processes and extended to larger domains. The results obtained generalize the existing constructions in the Wiener case, unify them, and apply to multiparameter time.
Key words: Poisson random measures, Anticipating stochastic integrals, Quantum
spectral stochastic integrals, Fock space. Mathematics Subject Classi cation (1991): 60G60, 60J75, 60H05, 81S25.
1 Introduction The theory of stochastic integration has been developed in an abstract setting, cf. [9], and [19], [20], [21], for the case of abstract Wiener spaces. This construction is closely related to quantum stochastic calculus, cf. [16]. In this paper we present a construction of Poisson stochastic integration in an Hilbert space setting. We consider both the point of views of quantum and anticipating stochastic calculus and since it is based on the Fock space, our approach can be adapted to deal with the Wiener case. Let ?(H ) denote the symmetric Fock space over a given Hilbert space H , and let and respectively denote the ordinary and symmetric tensor product of Hilbert spaces. A notion of adaptedness is introduced on Fock space, relatively to an abstract measure space (X; FX ; ), and allows to extend the Skorohod integral operator via an It^o type isometry to square-integrable adapted vectors (or \processes") in ?(H ) H . Then we de ne the Poisson integral of simple vectors in ?(H ) H , and study its extensions and relationship to the Skorohod integral. No ordering is required on the index set X , moreover we do not make use of a mapping from X to IR or of the real spectrum of a self-adjoint operator to induce 1
an order on X as in [5], resp. [19]. Our integral operator is originally de ned for simple process, and if one removes adaptedness assumptions it still admits several extensions via limiting procedures, although no one is canonical in the sense that they all depend on the particular type of approximation chosen for the integrand. In this sense this type of integral is close to the pathwise integrals of forward, backward and Stratonovich types. Our construction of stochastic integration is also valid in the Wiener interpretation of ?(H ), but in the Poisson case it presents several important di erences with its Brownian counterpart, due to the particular properties of Poisson measures. In classical probability, a -algebra can be identi ed to a space of bounded measurable functions in L1 ( ). In quantum probability, L1 ( ) is replaced with an operator algebra A, -algebras are then identi ed to a sub-algebras of A, and quantum stochastic integrals act on operator processes. The point of view developed in this paper is intermediate between classical and quantum probability. Namely, L2 ( ) is replaced with the Hilbert space ?(H ), stochastic integrals act on vectors in ?(H ) H , and -algebras are identi ed to vector subspaces of ?(H ). Our integral has some similarities with quantum spectral stochastic integrals, but its purpose is di erent since we map vectors in ?(H ) H to vectors in ?(H ). The rst two sections of this paper are only relative to the general symmetric Fock space, not to a particular type of random measure (Gaussian or Poisson). The remaining sections deal with the Poisson case, but each construction is related to its Gaussian counterpart. We proceed as follows. In Sect. 2 we give a de nition of measurability of vectors in the Fock space ?(H ), using a projection system indexed by a -algebra on a measurable space X . In Sect. 3 we introduce a notion of strong adaptedness under which the Skorohod integral operator r+ satis es the It^o isometry
kr+(u)k2?(H) = kuk2?(H) H ; which allows to extend r+ to a larger class of square-integrable processes without hypothesis on the Fock gradient r?u. The Poisson \pathwise" integral is de ned for simple processes in Sect. 4 and its relation to the Skorohod integral is studied in Sect. 5. Under a weak adaptedness condition, the Poisson stochastic integral operator S coincides with the Skorohod integral r+ on simple weakly adapted processes, hence as the Skorohod integral, it can be extended by density to as space of strongly adapted processes. As an application we construct a Fock space-valued multipa2
rameter Poisson process. Other extensions of S without adaptedness conditions are discussed in Sect. 6, in relation with the quantum spectral stochastic integrals of [2]. Sect. 7 recalls the Poisson probabilistic interpretation of the Fock space ?(H ), to be used in Sect. 8, and contains an independent remark on the relation between Guichardet Fock space and Poisson space. In Sect. 8 we work in the Poisson interpretation of ?(H ), using a Poisson random measure on an abstract measure space, and show that our integral coincides with the classical It^o-Poisson stochastic integral under adaptedness conditions. Without adaptedness conditions it becomes a Stratonovich type anticipating integral.
2 Measurability in Hilbert space with respect to an abstract index set In this section we give a de nition of measurability for vectors belonging to the n 0 symmetric Fock space ?(H ) = L1 n=0 H (with H = IR) over H , by using the notion of projection system. If h1; : : : ; hn 2 H we denote by h1 hn 2 H n the symmetrization of h1 hn 2 H n, n 1, and by S the vector space generated by fh1 hn : h1; : : : ; hn 2 H; n 2 INg : Let r? : ?(H ) ! ?(H ) H and r+ : ?(H ) H ! ?(H ) denote the gradient and Skorohod integral operators, densely de ned on S as
r? (h1 hn) =
iX =n i=1
h1 h^ i hn hi ;
where h^ i means that hi is omitted in the product, and
r+(h1 hn h) = h1 hn h; extended by linearity and density as closed operators with domains Dom(r? ) and Dom(r+ ). Let (f ), f 2 H , denote the exponential vector de ned as 1 X (f ) = n1! f n: n=0 De nition 1 Let (X; FX ) be a measurable space, and let (pA)A2FX be a projection system indexed by (X; FX ) on an abstract Hilbert space H , i.e. a family of self-adjoint operators pA : H ! H , A 2 FX , that satis es
pA pB = pB pA = pA\B ; 8A; B 2 FX ; 3
(2.1)
and
1 X k=0
pAk = pA ;
for any A 2 FX and any countable partition (Ak )k2IN FX of A.
The above assumptions imply that p; = 0. We may further assume as in Sect. 8 that pX is the identity of H , but this is not necessary in general. The following de nition is apparent in [9], [16], [19].
De nition 2 A vector F 2 ?(H ) is said to be FAp -measurable, A 2 FX , if F 2 Dom(r? ) and (Id pAc )r? F = 0; where Ac stands for the complement of A and Id denotes the identity on ?(H ).
Each projection system (pA )A2FX determines a family of vector subspaces of ?(H ) which play the role of -algebras F p = (FAp )A2FX . The conditional expectation of a vector 1 X F = fn 2 ?(H ) with respect to
FAp
n=0
is naturally de ned as
E [F jFAp ] =
1 X p nf
n=0
A n:
Condition (2.1) ensures the relation
E [E [F j FAp ] j FBp ] = E [E [F j FBp ] j FAp ] = E [F j FAp \B ]; F 2 ?(H ); A; B 2 FX : After xing ' 2 H , the projection system (pA )A2FX de nes a nite measure on (X; FX ) as (A) = ('; pA ')H ; A 2 FX : If H is unitarily equivalent to L2 (X; ) via the Hahn-Hellinger theorem (cf. [8], [16]), i.e. there exists a unitary operator U : H ?! L2(X; ), then pA can be identi ed to 1A with pA = U ?11AU , however such an identi cation may not be always possible. In particular, when working with the probabilistic interpretation of ?(H ) under a Poisson random measure on a set M , i.e. H = L2(M; ), we allow the index set X to be di erent from M .
4
3 Adaptedness and It^o isometry in Hilbert space The de nitions introduced in this section are not speci c to the Poisson case.
Proposition 1 The Skorohod integral operator r+ satis es the isometry
kr u)k +(
2 ?(H )
= kuk
2 ?(H ) H
+
iX =n
((ui ; r? Fj )H ; (uj ; r?Fi )H )?(H ) ;
i;j =1
(3.1)
for a simple process of the form u = Pii==1n Fi ui , where ui 2 H and Fi 2 Dom(r? ), i = 1; : : : ; n. Proof. This result is well-known, however it is generally stated in the case where H is a L2 space. By bilinearity, orthogonality and density it suces to choose u = f n g 2 ?(H ) H , f; g 2 H , and to note that
kr u)k +(
2 ?(H )
=n
iX
2 1
i
( n ? i )
f g f
?(H ) = 2
(n + 1) i=0 ?(H )
= kf n gk2
1 ((n + 1)!(n + 1)kf k2nkgk2 + (n + 1)!n(n + 1)kf k2n?2(f; g)2 ) H H H H (n + 1)2 = n!kf k2Hnkgk2H + (n ? 1)!n2kf k2Hn?2(f; g)2H = kuk2?(H ) H + ((g; r? f n)H ; (g; r? f n)H )?(H ) : 2:
=
This isometry has been related to the quantum It^o formula in [13]. From this formula, a bound on the norm of r+ (u) can be deduced as
kr+(u)k2?(H) kuk2?(H) H + kr?uk2?(H) H H : This bound allows to extend r+ to a domain in ?(H ) H , closed for the k k1;2 norm de ned as kuk21;2 = kuk2?(H) H + kr?uk2?(H) H H ; however it imposes some regularity on the process u via its gradient r? u.
De nition 3 Let U ?(H ) H denote the space of simple processes of the form u=
iX =n i=1
Fi (pAi 'i); F1; : : : ; Fn 2 Dom(r? ); A1; : : : ; An 2 FX ;
'1; : : : ; 'n 2 H , n 2 IN.
5
(3.2)
If H = L2 (X; ), X = IR and pAh = 1Ah, A 2 FX , a simple process u = Pi=n p i=1 Fi 1]ti;ti+1 ] is usually said to be adapted if Fi is F[0;ti] -measurable, i = 1; : : : ; n. n ((u ; r? F ) ; (u ; r? F ) ) It is well-known that in this case, the term Pii;j==1 i j H j i H ?(H ) in (3.1) vanishes and that the It^o isometry
kr+(u)k2?(H) = kuk2?(H) H holds in place of the Skorohod isometry, thus allowing to extend the Skorohod integral to square-integrable processes indexed by IR. In the general case, due to the lack of ordering on the index set X , there is no analog for the \past" of a set Ai . We introduce in our abstract setting a de nition of adaptedness under which the It^o isometry will hold.
De nition 4 A simple process u 2 U ?(H ) H is said to be strongly adapted if
it has a representation of the form
u=
iX =n i=1
Fi (pAi 'i) ;
A1 ; : : : ; An 2 FX , F1 ; : : : ; Fn 2 S , '1; : : : ; 'n 2 H , such that (Id pAi )r? Fj = 0; or (Id pAj )r? Fi = 0; i; j = 1; : : : ; n; i.e. Fj is FAp ci -measurable or Fi is FAp cj -measurable, i; j = 1; : : : ; n. We denote by UAd the set of strongly adapted simple processes.
The set UAd of simple strongly adapted processes is not a vector space: F (pA ') + G (pB ') may not be strongly adapted even if F (pA ') and G (pB ') are. Moreover the strong adaptedness property is not independent of the representation chosen for u 2 UAd . If X = IR+ , the usual adapted processes u = Pii==1n Fi p[0;ti] ' , 0 t1 tn , F1 ; : : : ; Fn 2 S form a vector space which is contained in the set of strongly adapted processes.
De nition 5 The set of strongly adapted square-integrable processes is de ned to be the completion in ?(H ) H of the set of simple strongly adapted processes. The It^o isometry holds on strongly adapted processes because the second term of the Skorohod isometry vanishes for such processes.
6
Proposition 2 The It^o isometry
kr+(u)k?(H) = kuk?(H) H
(3.3)
holds for all square-integrable strongly adapted process u. Proof. If u 2 UAd is of the form
u=
iX =n i=1
Fi (pAi 'i) ;
A1 ; : : : ; An 2 FX , F1; : : : ; Fn 2 S , then the condition (Id pAi )r? Fj = 0; or (Id pAj )r? Fi = 0;
i; j = 1; : : : ; n;
shows that ((pAi 'i ; r? Fj )H ; (pAj 'j ; r? Fi )H )?(H ) = 0; i; j = 1; : : : ; n: Hence (3.3) holds from the Skorohod isometry (3.1), and a density argument concludes the proof.
2
Although the sum u + v of two square-integrable strongly adapted processes may not be strongly adapted itself, we have u + v 2 Dom(r+ ) ?(H ) H , and r+ (u + v) satis es r+(u + v) = r+ (u) + r+ (v), however the It^o isometry (3.3) may not hold for u + v.
4 Poisson integral We introduce an integral that will be the analog of a pathwise Poisson integral, and will coincide with the Skorohod integral on strongly adapted processes.
De nition 6 Let ' 2 H be such that pA ' = 0 ) pA = 0; 8A 2 FX : A vector u 2 ?(H ) H is said to be a simple '-process if it can be written as
u= We denote
iX =n
Fi (pAi ') ; A1; : : : ; An 2 FX ; F1; : : : ; Fn 2 S ; n 2 IN:
i=1 by U '
the vector space of simple '-processes.
7
(4.1)
The '-simple processes are relative to a single ' 2 H . In the Gaussian case, we could work with U itself, i.e. a simple process could depend on n vectors '1; : : : ; 'n instead of a single ' 2 H , cf. [19] for the case X = IR.
De nition 7 Let u 2 U ' be a simple '-process written as u=
iX =n i=1
Fi (pAi ') ; F1 ; : : : ; Fn 2 S ; A1; : : : ; An 2 FX ; n 1:
(4.2)
We de ne the Poisson stochastic integral of u as
S (u) =
iX =n i=1
r+(Fi (pAi ')) + r+((Id pAi )r?Fi) + (pAi '; r?Fi)H :
(4.3)
This de nition is formulated on the Fock space ?(H ) and it will have a concrete meaning under the Poisson probabilistic interpretation of ?(H ) in Sect. 8. The operator S is in fact analog to a pathwise Stratonovich type integral.
Proposition 3 The de nition (4.3) of S (u) is independent of the particular representation (4.2) chosen for u. Moreover, S is linear on the vector space U ' of simple '-processes:
u; v 2 U ':
S (u + v) = S (u) + S (v);
Proof. This proof uses the assumptions of Def. 1 and Def. 6. Let n 1 and F1 ; : : : ; Fn ; G1 ; : : : ; Gn 2 S , A1; : : : ; An; B1 ; : : : ; Bn 2 FX , with
Ai \ Aj = ;; Bi \ Bj = ;; ; i 6= j; i; j = 1; : : : ; n:
(4.4)
Assume that u has two representations
u=
iX =n i=1
Fi (pAi ') =
iX =n i=1
Gi (pBi ') :
Applying successively Id (pBj ) and Id (pAi ) on both sides we obtain iX =n i=1
Fi pBj \Ai ' = Gj (pBj '); and Fi (pAi ') =
jX =n j =1
By scalar product with pAi \Bj ', this implies from (4.4):
Fi = Gj if pAi\Bj ' 6= 0; i; j = 1; : : : ; n; and
pAi ' =
jX =n j =1
pAi\Bj ';
pBj ' = 8
iX =n i=1
pBj \Ai ';
Gj pAi \Bj ' :
i.e. from the assumption (4.1) on ':
pAi =
jX =n j =1
pAi\Bj ; and pBj =
iX =n i=1
pBj \Ai :
Consequently, iX =n i=1
r+(Fi (pAi ')) + r+((Id pAi )r?Fi) + (pAi '; r?Fi)H
= = =
iX =n i;j =1 iX =n i;j =1 iX =n j =1
r+(Fi pAi\Bj ') + r+((Id pAi\Bj )r?Fi) + (pAi\Bj '; r?Fi)H r+(Gj pAi\Bj ') + r+((Id pAi\Bj )r?Gj ) + (pAi\Bj '; r?Gj )H
r+(Gj (pBj ')) + r+((Id pBj )r?Gj ) + (pBj '; r?Gj )H :
If A1; : : : ; An are not disjoint sets we choose a partition (Ci;j )1i;j n of A1 [ [ An such that for all i = 1; : : : ; n, the set Ai is written as Ai = Sjj ==1n Ci;j . We have iX =n
u=
i=1
Fi (pAi ') =
iX =n i;j =1
Fi pCi;j ' ;
p Ai =
jX =n j =1
Ci;j ;
(4.5)
and iX =n i=1
r+(Fi (pAi ')) + r+((Id pAi )r?Fi) + (pAi '; r?Fi)H
= =
iX =n i;j =1 iX =n j =1
r+(Fi (pCi;j ')) + r+((Id pCi;j )r?Fi) + (pCi;j '; r?Fi)H
r+(Gj (pBj ')) + r+((Id pBj )r?Gj ) + (pBj '; r?Gj )H :
In order to show that S is linear on the simple '-processes U ' we choose two simple '-processes
u= and with
v=
iX =n i=1 iX =n i=1
Fi (pAi ') ; F1 ; : : : ; Fn 2 S ; A1; : : : ; An 2 FX ; n 1;
Gi (pBi ') ; G1; : : : ; Gn 2 S ; B1; : : : ; Bn 2 FX ; n 1;
Ai \ Aj = ;; Bi \ Bj = ;; ; i 6= j; i; j = 1; : : : ; n: 9
We have
u+v =
iX =n i=1
Fi (pAi ') +
iX =n i=1
iX =m
Gi (pBi ') =
i=1
Hi (pDi ') ;
(4.6)
where D1 ; : : : ; Dm 2 FX are disjoint sets with 8 > if Dj Ai n Bi ; < Fi G if Dj Bi n Ai ; Hj = > i : F + G if D A \ B ; i i j i i
i = 1; : : : ; n, j = 1; : : : ; m. Moreover, jX =m
jX =n
jX =n
j =1
i=1
i=1
(Id pDj )r? Hj =
(Id pAi )r? Fi +
(Id pBi )r? Gi ;
hence S (u + v) = S (u) + S (v).
2
In the Gaussian case, the above proposition is obvious since the analog of S is de ned as
T (u) =
iX =n i=1
r+(Fi (pAi 'i))+(pAi 'i; r?Fi)H = r+(u)+trace (r?u); u 2 U : (4.7)
Remark 1 Given '; 2 H satisfying (4.1), the operator S is linear on U ' and on U , but not on the linear space spanned by U ' and U . More precisely, let u 2 U ', v 2 U , be simple '-process, resp. -process, with
u=
iX =n i=1
Fi (pAi ') ;
v=
iX =n i=1
Fi (pBi ) ;
then u + v may be a simple ' + -process (if Ai = Bi , i = 1; : : : ; n) but even in this case we may not have S (u + v) = S (u) + S (v):
S (u + v) =
iX =n i=1
r+(Fi pAi (' + ) + (Id pAi )r?Fi) + (pAi (' + ); r? Fi)H
= S (u) + S (v) ?
iX =n i=1
r+((Id pAi )r?Fi):
The linearity property of S holds on a given class of processes which is determined by '. This corresponds to the fact that Poisson random measures with same jump sizes are stable by addition but not by linearity. The Gaussian setting, however, is fully linear because in this case the term r+ ((Id pAi )r? Fi ) in (4.3) is absent of the de nition (4.7) of T (u). 10
5 Skorohod and Poisson integrals In this section we deal with the relationship between S and the Skorohod integral operator r+ . First we note that S coincides with r+ on a vector space of weakly adapted simple '-processes which is larger than the set of strongly adapted simple '-processes. Under the strong adaptedness condition, S can (as r+) be extended to a class of square-integrable processes via the It^o isometry (3.3).
De nition 8 A simple process u 2 U ?(H ) H is said to be weakly adapted if it has a representation
u= such that
iX =n i=1
Fi (pAi 'i) ; F1; : : : ; Fn 2 S ; '1; : : : ; 'n 2 H; n 2 IN; (Id pAi )r? Fi = 0;
i = 1; : : : ; n:
(5.1) (5.2)
We denote by Uad' the set of '-processes in U ' that are weakly adapted.
This de nition states that a simple process u = Pii==1n Fi (pAi 'i) is weakly adapted if Fi is FAp ci -measurable, i = 1; : : : ; n. The weak adaptedness condition is in general dependent on a particular representation (5.1) of u. In the case of weakly adapted simple '-processes the situation is di erent.
Proposition 4 All representations (5.1) of a given simple weakly adapted '-process u 2 Uad' satisfy condition (5.2). Moreover, the set Uad' of weakly adapted simple '-processes forms a vector space.
Proof. Let n 1 and F1; : : : ; Fn ; G1 ; : : : ; Gn 2 S , A1; : : : ; An ; B1; : : : ; Bn 2 FX with Ai \ Aj = ; and Bi \ Bj = ;, i 6= j; i; j = 1; : : : ; n. Assume that
u=
iX =n i=1
Fi (pAi ') =
and that the representation u = Prop. 3
pAi =
jX =n j =1
pAi \Bj ; pBj =
iX =n i=1
Pj =n
j =1 Gj
jX =n j =1
G j p Bj ' ;
pBj ' satis es (5.2). We have as in
pBj \Ai ; and Fi = Gj if pAi\Bj 6= 0; i; j = 1; : : : ; n;
11
hence (Id pAi
)r? F
i
= =
jX =n
(Id pAi \Bj
j =1 jX =n j =1
)r? F
i
=
jX =n j =1
(Id pAi\Bj )r? Gj
(Id pAi )(Id pBj )r? Gj = 0;
i.e. (5.2) is also valid for all representations u = Pii==1n Fi (pAi ') of u, with Ai \ Aj = ;, i 6= j , i; j = 1; : : : ; n. If A1; : : : ; An are not disjoint sets we have the expression (4.5): jX =n =m iX iX =n =n jX u = Fi (pAi ') = Gj pBj ' = Fi pCi;j ' : i=1
j =1
j =1 i=1
From the above discussion we have (Id pCi;j )r? Fi = 0, i = 1; : : : ; n, j = 1; : : : ; m, hence jX =m (Id pAi )r? Fi = (Id pCi;j )r? Fi = 0; i = 1; : : : ; n: j =1
In order to show that the weakly adapted processes form a vector space we proceed as in the proof of the linearity of S on the simple '-processes in U ' (Prop. 3): we choose two '-processes
u= and with
v=
iX =n i=1 iX =n i=1
Fi (pAi ') ; F1 ; : : : ; Fn 2 S ; A1; : : : ; An 2 FX ; n 1;
Gi (pBi ') ; G1; : : : ; Gn 2 S ; B1; : : : ; Bn 2 FX ; n 1;
Ai \ Aj = ;; Bi \ Bj = ;; i 6= j; i; j = 1; : : : ; n:
Writing as in (4.6) of Prop. 3
u+v =
iX =m i=1
Hi (pDi ')
we check that if u and v are weakly adapted, then (Id pDi )r? Hi = 0, i.e. u + v is weakly adapted.
2
If X = IR+ and (E (t))t2IR+ is an adapted operator process in the sense of quantum stochastic calculus, cf. [10], then (E (t)1)t2IR+ is weakly adapted in our sense, where 1 denotes the vacuum vector in ?(H ). The operators S and r+ coincide on simple weakly adapted '-processes but the It^o isometry (and hence the strong adaptedness 12
condition) is needed to extend this identity to a closed space of processes. The space of square-integrable strongly adapted '-processes is the completion in ?(H ) H of the space of strongly adapted simple '-processes.
Proposition 5 Let u 2 Uad' be a simple weakly adapted '-process. We have S (u) = r+(u); i.e. the Poisson and Skorohod integrals of u coincide. This relation extends to the square-integrable strongly adapted '-processes via the It^o isometry
kr+(u)k?(H) = kS (u)k?(H) = kuk?(H) H : Proof. We apply the de nition (4.3) of S , the fact that (Id pAi )r? Fi = 0, since Fi is FAp ci -measurable, i = 1; : : : ; n, and use the self-adjointness of pAi which implies (pAi '; r?Fi )H = ('; (Id pAi )r? Fi )H = 0 in ?(H ). Since strongly adapted processes are also weakly adapted and satisfy the It^o isometry for r+ , the operators S and r+ coincide on the set of square-integrable strongly adapted '-processes.
2
In the Gaussian case we also have T (u) = r+(u) if u 2 U is a simple weakly adapted process, and T can be extended as in Prop. 5 to the completion of the set UAd of simple strongly adapted processes. A rst application of the operator S is the construction of a multiparameter Poisson process in Fock space. Taking X = IRd with its canonical (partial) ordering \ " we let t = (t1 ; : : : ; td ), [s; t] = [s1; t1 ] [sd ; td ]; s; t 2 IRd ; s t; and we construct the d-parameter compensated Poisson '-process in ?(H ) as
n~ t = p[0;t ] '; t 2 IRd+: This process has intensity ([0; t ]) = kp[0;t ] 'k2H , t 2 IRd+ , and ' controls the \size" of jumps as noted in Remark 1. The Hilbert space H needs not be equal to L2 (IRd+). In this case we can adopt the notation Z IRd+
udn~ t := S (u)
13
for the compensated Poisson stochastic integral of a simple '-process u 2 U ' with respect to (~nt)t2IRd+ . Let u be a simple '-process of the form
u=
iX =n i=1
Fi (p[si;si+1 ] '):
If si si+1, i = 1; : : : ; n ? 1, and u is adapted in the usual sense, i.e. Fj is F[0p ;si] measurable, 1 j i n, then Fj is F[psi;si+1 ]c -measurable, 1 j i n, and u is strongly adapted (in particular it is also weakly adapted in our sense). In this case, Z IRd+
udn~ := S (u) = r+(u)
and this relation extends to square-integrable strongly adapted processes by density from Prop. 5.
6 Other extensions of S as a linear operator on U' In this section we deal with the possibility to extend the operator S without the strong adaptedness condition. Clearly, the operator S is not closable because it involves traces of the gradient r? and there is no canonical way to extend it to a larger domain. We mention two possibilities of extension, one of them being based on the relation of S with quantum spectral stochastic integrals. a) Let u 2 Dom(r? ) ?(H ) H , and let be a set of partitions = fA1; : : : ; An g 2 FX of X with (Ai) > 0, i = 1; : : : ; n, and jj = sup0in?1 (Ai), 2 , such that (i) we have
(u; p ')
2 1 A H
lim sup
( p ' ) ? (I = 0; A d pA )u
jj!0 A2 (A) ( A ) 1 ; 2 2
(ii) (u; pA ')H 2 Dom(r? ), 8A 2 , and there exists an element of ?(H ) H denoted trace r? u, such that
1
2 1
? ?
(I pA )r (u; pA ')H ? (Id pA )trace r u
= 0: lim sup jj!0 A2 (A) (A) d 1;2 2
(This requires the existence of at least one sequence (n)n2IN such that limn!1 jn j = 0). We let
u =
(u; pA ')H (p ') 2 U '; A A2 (A) X
14
(6.1)
and
1 (I p )r? (u; p ') : d A A H A2 (A) Under the above assumptions, u and (trace r?u ; ')H converge respectively to u and (trace r?u ; ')H in ?(H ), and trace r? u converges to trace r?u for the kk1;2 norm as jj goes to 0, due to the bounds trace r? u =
ku ? uk21;2
X
(u; p ')
2 1 A H
( p ' ) ? (I ; (X ) sup A d pA )u
(A) A2 (A) 1;2
ktrace r?u ? trace r?uk21;2
2
1
1
? ?
; (I
p ) r ( u; p ' ) ? (I
p )trace r u (X ) sup d A A H d A
( A ) ( A ) A2 1;2
and
k(trace r?u ; ')H ?(trace r?u; ')H k2?(H) ktrace r?u ?trace r?uk2?(H) H k'k2H : Under the above assumptions we can de ne S (u) as the limit in ?(H )
S (u) = jlim S (u ) = r+(u) + r+ (trace r?u) + (trace r?u; ')H : j!0 2
(6.2)
An example of a vector u 2 ?(H ) H satisfying the above assumptions is provided + by letting u = P1 n=0 Fi pBi ' 2 Dom(r ) ?(H ) H , where (Bi )i2IN FX is ? 2 a disjoint family and (Fi )i2IN Dom(r? ), with P1 i=0 k(Id pBi )r Fi k1;2 < 1. In ? this case, trace r? u is de ned to be P1 i=0(Id pBi )r Fi , and (6.2) de nes S (u). Conditions (i)-(ii) are naturally satis ed since (u;p(AA'))H (pA ') = (Id pA)u and 1 ? ? (A) (Id pA )r (u; pA ')H = (Id pA)trace r u, A 2 , 2 . Here, can be de ned as the set of partitions = fA1; : : : ; Ang 2 FX of X with (Ai ) > 0, i = 1; : : : ; n, such that all A 2 is contained in a Bj for some j 2 IN. b) The integral S (u) can also be linked to the quantum spectral stochastic integrals of [2], [3], which provide a di erent extension of S . Given a simple operator process (E (x))x2X de ned as E (x) = Pii==1n Ei 1Ai (x), x 2 X , where E1 ; : : : ; En : ?(H ) ?! ?(H ) are bounded operators, the quantum spectral stochastic integral of (E (x))x2X is the operator denoted by Z
X
E (x)(a+' (p(dx)) + a?' (p(dx)) + a' (p(dx)) 15
1 n and de ned on exponential vectors (f ) := P1 n=0 n! f , f 2 H , as
Z
X
=
E (x)(a'
p(dx)) + a?(p(dx)) + a (p(dx))
+(
iX =n i=1
'
'
(f )
r+(Ei (f ) (pAi ')) + (r?Ei (f ); pAi ')H + r+((Id pAi )r?Ei (f )):
Given u a simple process de ned as u = Pii==1n(Ei (f )) (pAi '), we have the relation
S (u) =
Z
X
E (x)(a'
p(dx)) + a? (p(dx)) + a (p(dx)))
+(
'
'
(f ); f 2 H:
If X = [0; 1] and (E (t))t2[0;1] is an adapted operator process such that t 7! E (t) (f ) is continuous from [0; 1] into ?(H ), f 2 H , then as a consequence of Lemma 3.2 of [2] the quantum spectral stochastic integral of (E (x))x2[0;1] is well-de ned on the exponential vector (f ) in the sense that lim jj!0
iX =n i=1
S (E (ti ) (f ) (p[ti;ti+1 ] '))
exists in ?(H ), with = f0 < t1 < < tn < 1g. However, X should be an interval of IR and this requires to consider processes such as (E (t) (f ))t2[0;1] , which then satisfy a condition similar to (ii) above, due to the particular form of exponential vectors which implies r? (f ) = (f ) f .
7 Isomorphism of Guichardet space and Poisson space This section recalls the Poisson probabilistic interpretation of the Fock space ?(H ), which will be used in the next section, and presents a remark on the relations between Poisson space, Fock space and Guichardet Fock space. The Guichardet interpretation of ?(H ), cf. [7], has been used in connection with quantum stochastic integrals, cf. [4], [5], usually in the particular case M = IR+. We assume that H = L2(M; G ; ), where (M; G ; ) is a measure space with nite di use measure . In this case, H n is the space L^ 2(M n ) of symmetric square-integrable functions on M n, n 1. The gradient r? F 2 ?(H ) H of F 2 Dom(r? ) ?(H ) is denoted as (rx F )x2M and u 2 ?(H ) H is denoted by (u(x))x2M . Let P0 = f;g,
Pn = f(x1; : : : ; xn) 2 M n : xi 6= xj ; i 6= j; 1 i; j ng; n 1 16
and P = S1 n=0 Pn. Let A denote the collection of subsets of B of P such that
B \ Pn 2 G n, n 1. The measure de nes a measure n on Pn, n 1, and a measure Q on (P ; A) as 1 X Q(B ) = n1! n(B \ Pn); n=0 i.e. 1 X Q = n1! n n=0
0 with the convention (f;g) = 1. In the Guichardet interpretation of ?(H ), each element F = P1 n=0 fn 2 ?(H ) is isometrically identi ed to a square-integrable function in L2(P ; Q). The Guichardet interpretation of ?(H ) is not a probabilistic interpretation because it does not induce a product on ?(H ). On the other hand, the underlying probability space of a Poisson random measure (N (A))A2G on M can be taken equal to P . We identify each element ! 2 P = S1 n=0 Pn to a ( nite) sum of Dirac measures
!
X
x2!
x ;
and for A 2 G we de ne N (A) : P ?! IR+ as
N (A)(!) = !(A) = card(fx 2 ! : x 2 Ag): The compensated random measure N~ is de ned by N~ (A) = N (A) ? (A). Let P denote the Poisson probability measure with intensity on (P ; A), such that for disjoint sets A1; : : : ; An 2 G , the applications N (A1); : : : ; N (An ) are independent Poisson random variables with respective intensities (A1 ); : : : ; (An ), n 1. This measure can be written as 1 X P (B ) = e?(B) n1! n(B \ Pn); B 2 A: n=0 In this setting, ! 2 Pn corresponds to a Poisson sample of n points in M , we refer to [14] for this construction of the Poisson measure. The n-th order multiple stochastic integral of a symmetric function fn 2 L^ 2(M n ) is the functional in L2 (P ; P ) de ned as Z In(fn) = fnd(! ? ) d(! ? ); ! 2 P ; with the isometry
Pn
kIn(fn)k2L2(P ;P ) = n!kfnk2L2 (M n; n); 17
cf. e.g. [11], [18]. The isometric isomorphisms between the Fock space ?(L2 (M; )), the Guichardet Fock space L2 (P ; Q) and the Poisson space L2(P ; P ) can be described as follows: J L2(P ; P ) L2 (P ; Q) I? ?(H ) ?! n!fn - fn ,! In(fn ) The isometric isomorphism between ?(H ) and the Poisson space L2(P ; P ) will be denoted by J , i.e. J associates fn 2 L^ 2 (M n) to its multiple stochastic integral In(fn ), cf. [12] for a recent account of this isomorphism via the Charlier polynomials. The isometric isomorphism between ?(H ) and the Guichardet space L2 (P ; Q) is denoted by I , it maps fn 2 L^ 2(M n) to the function n!fn in L2 (P ; Q). The Guichardet interpretation of the operators r? and r+ is
r?x F ( ) = F ( [ x); 2 P ; F 2 S ;
(7.1)
and for u 2 U a simple process,
r+(u)( ) =
X
x2
u(x; n x); 2 P ;
(7.2)
cf. e.g. [5], where we identi ed respectively F; r? F; u, r+(u) to J F , J r?F , J u, J r+(u). On the other hand we know from e.g. [6], [11], [15], [17], that the probabilistic interpretation of r? is given by
r?x F (!) = F (! [ x) ? F (!); x 2 M; ! 2 P ; F 2 S ; and
r u)(!) = +(
X
x2!
u(x; ! n x) ?
Z
X
u(x; !)d(x); ! 2 P ; u 2 U ;
with respectively F; r? F; u, r+(u) in place of I F , I r? F , I u, I r+ (u).
8 Stochastic integration with respect to a Poisson random measure In this section the Hilbert space H is H = L2(M; ), is a nite measure, ?(H ) is identi ed via J to the Poisson space L2(P ; P ), ?(H ) H is identi ed to L2(P M; P ), and (X; FX ) remains an abstract index set with projection system (pA )A2FX in the next proposition. We assume that pX is the identity of H .
18
Proposition 6 In the Poisson stochastic interpretation of ?(H ), the stochastic integral S (u) of a simple 1M -process u = Pii==1n Fi (pAi 1M ) 2 U 1M satis es S (u) =
iX =n i=1
Z
Fi
M
(pAi 1M ) (x)(N (dx) ? (dx)):
Proof. This result follows from the expression on Fock space of the multiplication R operator by M (pAi 1M ) (x)(N (dx) ? (dx)), cf. e.g. [16]: Z
Fi (pAi 1M ) (x)(N (dx) ? (dx)) M = r+(Fi (pAi 1M )) + r+ ((Id pAi )r? Fi ) + (pAi 1M ; r? Fi )H : 2 The Poisson stochastic integral S (u) can be extended to square-integrable strongly adapted processes via the It^o isometry (3.3) with the relation
S (u) =
Z
M
u(x)(N (dx) ? (dx)):
The Poisson integral S (u) de ned above is for simple processes. It can be extended to a given process u by taking limits from a sequence of simple processes approaching u, however the result depends in general on the choice of an approximating sequence. Considering u de ned in (6.1), we have
ku kL2(M;) = 2
=
X (u; pA 1M ) 2 L (M )
(A) A2
X
(u; pA 1M )L2 (M; )
(A)
A2
2
(pA 1M )
L2 (M;)
2
(pA 1M )
L2 (M;) 1 X
= (1A) (u; pA 1M )2L2 (M;) = (A) (u; pA pA 1M )2L2 (M;) A2 A2 X X 1 2 2 = (A) A2(pA u; pA 1M )L2 (M;) A2(pA u; pA u)L2 (M;) X (u; pAu)2L2(M;) (u; u)2L2 (M;); X
A2
i.e. ku k2L2 (M;) kuk2L2 (M;) ; P-a.s. On the last line we used the assumption that pX is the identity of H . Hence in assumption (i) of Sect. 6, L2 convergence can be replaced by almost sure convergence of the approximation:
2
(u; pA 1M ) 2
1 L (M;)
( p 1 ) ? (I
p ) u
= 0; a:s:; lim sup A M d A
2 jj!0 A2 (A) (A) L (M;) and this suces to de ne
S (u) = jlim S (u ); j!0 19
provided assumption (ii) is also satis ed. As a particular case we can take X = M and the projection pA, A 2 FX , can be taken equal to the multiplication operator by 1A . If X = M = IRd with the canonical partial ordering \ " and the projection system fpA gA2B(IRd ) is de ned as pA f = 1Af , f 2 H = L2(M; ) = L2 (X; ), = , then our de nition of adaptedness extends the usual de nition and for strongly adapted processes the Poisson and Skorohod integrals coincide. A similar result in the Gaussian case can be obtained by replacing the compensated Poisson random measure N~ with a Gaussian random measure with variance .
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[13] J. M. Lindsay. Quantum and non-causal stochastic calculus. Probab. Theory Related Fields, 97:65{80, 1993. [14] J. Neveu. Processus ponctuels. In Ecole d'ete de Probabilites de Saint-Flour VI, volume 598 of Lecture Notes in Mathematics, pages 249{445. Springer-Verlag, 1977. [15] D. Nualart and J. Vives. A duality formula on the Poisson space and some applications. In E. Bolthausen, M. Dozzi, and F. Russo, editors, Seminar on Stochastic Analysis, Random Fields and Applications, Progress in Probability, pages 205{213, Ascona, 1993. Birkauser. [16] K.R. Parthasarathy. An Introduction to Quantum Stochastic Calculus. Birkauser, 1992. [17] J. Picard. Formules de dualite sur l'espace de Poisson. Ann. Inst. H. Poincare Probab. Statist., 32(4):509{548, 1996. [18] D. Surgailis. On multiple Poisson stochastic integrals and associated Markov semigroups. Probability and Mathematical Statistics, 3:217{239, 1984. [19] A.S. U stunel. Construction du calcul stochastique sur un espace de Wiener abstrait. C. R. Acad. Sci. Paris Ser. I Math., 305:279{282, 1987. [20] A.S. U stunel and M. Zakai. The construction of ltrations on abstract Wiener space. J. Funct. Anal., 143(1):10{32, 1997. [21] L. Wu. Un traitement uni e de la representation des fonctionnelles de Wiener. In Seminaire de Probabilites, XXIV, 1988/89, volume 1426 of Lecture Notes in Mathematics, pages 166{187. Springer-Verlag, 1990.
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