Hilbert space-valued integral of operator-valued functions

c World Scientific Publishing Company DOI:10.1142/S1793744210000144

HILBERT SPACE-VALUED INTEGRAL OF OPERATOR-VALUED FUNCTIONS

VOLODYMYR TESKO

Institute of Mathematics, National Academy of Sciences of Ukraine 3 Tereshchenkivska Street, Kiev, 01601, Ukraine

tesko@imath.kiev.ua Received 6 December 2009

In this paper we construct and study an integral of operator-valued functions with respect to Hilbert space-valued measures generated by a resolution of identity. Our integral generalizes the Itˆo stochastic integral with respect to normal martingales and the Itˆo integral on a Fock space.

Keywords: Itˆo integral; normal martingale; resolution of identity; Fock space.

AMS Subject Classification: 46G99, 47B15, 60H05

1. Introduction

It is well known that the Itˆo integral of adapted square integrable functions plays a fundamental role in the classical stochastic calculus. In the case of complex- valued integrands such integral can be interpreted (roughly speaking) as an ordinary spectral integral applied to a certain vector from anL²-space. The reason for this is that a square integrable martingale (integrator) can be regarded as a resolution of identity applied to the above-mentioned vector. At the same time, it is considerably more difficult to establish a relation between the Itˆo and spectral integrals forL²- valued integrands, since there is a problem in finding an explicit expression for the corresponding spectral integral (see, for example, [5] and reference therein). In this context a natural problem arises, — to give a suitable definition of “spectral integral” which will generalize the Itˆo stochastic integral.

In this paper we introduce a notion of such “spectral integral” and show that it generalizes the classical Itˆo stochastic integral with respect to normal martingales and the Itˆo integral on a Fock space. Such point of view enables us to treat all these integrals in one framework.

Let us give a short exposition of our constructions. First of all let us recall the definition of an ordinary square integrable martingale and its interpreta- tion using Hilbert space theory, see e.g. [15,5,13]. Throughout this paper, (Ω,A, P) is a complete probability space endowed with a right continuous filtration

135

{A_t}_t∈R+, i.e. with a family{A_t}_t∈R+ ofσ-algebrasA_t⊂ Asuch that A_s⊂ A_t if s≤tand At=

s>tAs for allt∈R+. Furthermore, we assume thatA coincides with the smallestσ-algebra generated by

t∈R+At andA0contains all the P-null sets ofA.

Since At ⊂ A, the Hilbert space L²(Ω,At, P) is a subspace of L²(Ω,A, P).

Denote by E_t the corresponding orthogonal projection in the space L²(Ω,A, P) ontoL²(Ω,At, P). It can be shown that a projection-valued functionR+t→E_t is a resolution of identity in the spaceL²(Ω,A, P) and, for eacht∈R₊,

E_tF =E[F|At], F ∈L²(Ω,A, P),

whereE[·|A_t] denotes a conditional expectation with respect to the σ-algebraA_t. Note that E_t is right continuous (instead of the usual for functional analysis left continuous) since the filtration {At}t∈R+ is right continuous. In what follows, we will consider only right-continuous resolutions of identity.

By definition (see, e.g. [18,19]) a function M:R+ → L²(Ω,A, P) is called a square integrable martingalewith respect to the filtration{A_t}_t∈R+ ifE_sM_t=M_s for all s ∈ [0, t]. Such martingale M is said to be closed by a function M_∞ ∈ L²(Ω,A, P) ifE_tM_∞=M_tfor allt∈R₊.

Assume that a square integrable martingaleM is closed byM_∞∈L²(Ω,A, P).

Then the Itˆo stochastic integral

R+F(t)dM_tof a complex-valued functionF:R+→ Cwith respect toM can be interpreted as a spectral integral

R+F(t)dE_tapplied toM_∞. That is,

R+

F(t)dM_t=

R+

F(t)dE_t

M_∞.

In general case, whenF is a “fine”L²(Ω,A, P)-valued function, the Itˆo integral is well defined but it is impossible to realize the latter equality, because it is not clear what the symbol

R+F(t)dE_tmeans. In view of this it is natural to ask —in what sense the expression

R+F(t)dE_t

M_∞ can be understood in general case?

A key point for the answer is the observation that the conditional expectation E[·|At] is a resolution of identity in the Hilbert spaceL²(Ω,A, P) and that a function F:R₊→L²(Ω,A, P) can be naturally viewed as an operator-valued functionR₊ t→A_F(t) whose values are operatorsA_F(t) of multiplication by the functionF(t) in the spaceL²(Ω,A, P). This suggests us to consider the above-mentioned question in a more comprehensive sense —in what sense the expression

R+

A(t)dE_t

M_∞ or,equivalently,

R+

A(t)dM_t,

can be understood? HereE:R+ → H is a resolution of identity in a Hilbert space H,R₊t→A(t) is an operator-valued function whose values are linear operators inH,M_t:=E_tM_∞is an abstract martingale andM_∞∈ H.

In the next section we give the answer to this question for a certain class of operator-valued functionsR+t→A(t). The corresponding integral

R+

A(t)dM_t (1.1)

we define as an element of the Hilbert space Hand call it a Hilbert space-valued stochastic integral (or simply H-stochastic integral). We stress that formally the construction ofH-stochastic integral is similar to the one of the Itˆo integral. Since theH-stochastic integral integrates with respect to a Hilbert space-valued measure and the integrand is, generally speaking, non-bounded operator-valued function one has to analyze the mutual dependencies of the integrand and the integrator.

Therefore, by analogy with the classical Itˆo integration theory in order to give a correct definition of integral (1.1) we introduce a suitable notion of adaptedness between the integrandAand the integratorM.

We illustrate our abstract constructions with a few examples. Thus, in Sec. 3, we show that the Itˆo stochastic integral is the particular case of the H-stochastic integral. Namely, we assume that H := L²(Ω,A, P) and take a normal martin- gale N:R+ →L²(Ω,A, P) as an L²(Ω,A, P)-valued martingale (by definition the process N ={N_t}t∈R+ is a normal martingale if N and t→N_t²−t are both mar- tingales). For a given adapted square integrable functionF ∈L²(R₊×Ω, dt×P), let R+ t → A_F(t) be the corresponding operator-valued function whose values are operatorsA_F(t) of multiplication by the functionF(t) =F(t,·)∈L²(Ω,A, P) in the spaceL²(Ω,A, P), i.e.

L²(Ω,A, P)⊃Dom(A_F(t))G→A_F(t)G:=F(t)G∈L²(Ω,A, P).

Then the H-stochastic integral ofA_F coincides with the Itˆo integral

R+F(t)dN_t ofF. That is,

R+

A_F(t)dN_t=

R+

F(t)dN_t.

In Secs. 4–6 using the notion of theH-stochastic integral, we give the definition and establish properties of an Itˆo integral on a symmetric Fock space F(H) over a real Hilbert spaceH. In the special important case whenH =L²(R₊, dt) we obtain the well-known Itˆo integral on the Fock spaceF=F(L²(R+, dt)). This integral is the pre-image (under the corresponding probabilistic interpretation of F) of the classical Itˆo stochastic integral with respect to normal martingales which possess the Chaotic Representation Property. Observe that the Itˆo integral on the Fock spaceFis the useful tool in the quantum stochastic calculus, see e.g. [4,2] for more details.

Most of the basic results of this paper were announced and partially proved in brief preliminary note [17].

2. A Construction of H-Stochastic Integral

LetHbe a complex Hilbert space with the inner product (·,·)_Hand the norm·_H, L(H) be a space of all bounded linear operators inH. Suppose thatE:R+→ L(H) is a right-continuous resolution of identity inH, i.e.{E_t}_t∈R+ is a right-continuous and increasing family of projections, such thatE_∞:= lim_t→∞E_t= 1.

A slight generalization of the notion of square integrable martingale is the fol- lowing.

Definition 2.1. A functionM:R+→ H, t→M_t,is called anH-valued martingale with respect toE ifE_sM_t=M_s for allt∈R₊ and alls∈[0, t].

It is clear that an ordinary square integrable martingaleM:R+→L²(Ω,A, P) with respect to the filtration {At}t∈R+ is an L²(Ω,A, P)-valued martingale with respect to the resolution of identity E_t = E[·|At]. By analogy with the classical probability, theH-valued martingaleM will be calledclosed by a vectorM_∞∈ H ifM_t:=E_tM_∞,t∈R₊.

Sometimes it will be convenient for us to regard the H-valued martingale M:R₊ → H as an H-valued measure B(R₊) α → M(α) ∈ H on the Borel σ-algebraB(R+). To this end, for any interval (s, t]⊂R+, we set

M((s, t]) :=M_t−M_s, M({0}) :=M₀, M(∅) := 0,

and then extend this definition to all Borel subsets of R+. Using such H-valued measure, we construct a non-negative Borel measure

B(R₊)α→µ(α) :=M(α)²_H∈R₊.

The purpose of this section is to define and study aHilbert space-valued stochas- tic integral(or H-stochastic integral)

R+

A(t)dM_t (2.1)

for a certain class of functionsR+t→A(t) whose values are linear operators in H. We give a step-by-step construction of this integral, beginning with the simplest class of operator-valued functions. Since in the special case (see above) we want to obtain a generalization of the classical Itˆo integral, we have to introduce a suitable notion of adaptedness betweenA(t) andM_tsuch that on one hand gives a natural generalization of the usual notion of adaptedness in the classical stochastic calculus, and on the other hand, allows us to obtain an analogue of the Itˆo isometry property and as a result to extend integral (2.1) from simple to a wider class of operator- valued functions.

Let us introduce the required class of simple functions. For each pointt∈R+, we denote by

HM(t) := span{M_s2−M_s1|(s₁, s₂]⊂(t,∞)} ⊂ H

the linear span of the set {M_s2−M_s1|(s₁, s₂]⊂(t,∞)}, and LM(t) =L(HM(t),H)

will represent the set of linear operators in A:H → H such that continuously act from H_M(t) toH. We stress that the set L_M(t) consists of all linear (bounded or non-bounded) operatorsA:H → Hsuch that

A_LM(t):= sup AgH

g_H

g∈ HM(t), g= 0

<∞. Definition 2.2. A linear operatorAinHwill be calledM_t-measurable if

(i) A∈ LM(t) andA_LM(t)=A_LM(s)for alls∈[t,∞).

(ii) Ais partially commuting with the resolution of identityE. More precisely, AE_sg=E_sAg, g∈ H_M(t), s∈[t,∞).

Evidently, if an operatorA in His M_t-measurable for some t∈ R+ then Ais M_s-measurable for alls∈[t,∞).

Definition 2.3. We say that a family {A(t)}_t∈R+ of linear operators in His an M-adapted operator-valued function if, for every t ∈ R+, the operator A(t) is M_t-measurable.

AnM-adapted operator-valued functionR₊t→A(t) will be called simple if there exists a partition 0 =t₀< t₁<· · ·< t_n<∞ofR+ such that

A(t) =

n−1

k=0

A_kκ_(tk,tk+1](t), t∈R+, (2.2) where κα(·) is the characteristic function of a Borel set α ∈ B(R+). It is clear that each operatorA_k in (2.2) isM_tk-measurable, sinceA(t) is theM_t-measurable operator for everyt∈R+.

Denote byS =S(M) the space of all simpleM-adapted operator-valued func- tions on R₊. For eachA∈S of kind (2.2), we introduce a quasinorm

A_S2 :=

R+

A(t)²_LM(t)dµ(t) ¹₂

:=

_n−1

k=0

A_k²_LM(tk)µ((t_k, t_k+1]) ¹₂

, (2.3) where, as above,µis a Borel measure onR+defined byµ(α) :=M(α)²_H. Since a simple operator-valued functionAisM-adapted, we conclude that, for eacht∈R+,

A(t)_LM(t)=A(t)_LM(s), s∈[t,∞).

Due to the latter equality and finite additivity of the measureµ, definition (2.3) is correct, i.e. it does not depend on the choice of representationAin S.

Let us give the definition of anH-stochastic integral.

Definition 2.4. ForA∈S of kind (2.2), theH-stochastic integralwith respect to M is defined as an element ofHgiven by

R+

A(t)dM_t:=

n−1

k=0

A_k(M_tk+1−M_tk). (2.4) Of course, definition (2.4) does not depend on the choice of representation of the simple functionAin the spaceS. Further, let 0≤s≤ ∞be fixed. Clearly, ifA belongs toS thenAκ_[0,s] also belongs toS. Hence, for every A∈S, we can define an indefiniteH-stochastic integral by

_s

0

A(t)dM_t:=

R+

A(t)κ_[0,s](t)dM_t.

The following statement gives a description of the properties of our integral.

Theorem 2.1. Let 0 ≤ s ≤ ∞ be fixed. For all constants a, b ∈ C and for all simple operator-valued functions A, B∈S, we have

_s

0

(aA(t) +bB(t))dM_t=a _s

0

A(t)dM_t+b _s

0

B(t)dM_t, (2.5) _s

0

A(t)dM_t ²

H≤ _s

0 A(t)²_LM(t)dµ(t). (2.6) Moreover, the indefinite integral _s

0 A(t)dM_t of A ∈ S is an H-valued martingale with respect to resolution of identityE. That is,for any u∈[0, s],

E_{u s}

0

A(t)dM_t

= _u

0

A(t)dM_t. (2.7)

Proof. Equalities (2.5) and (2.7) are obvious. Let us check inequality (2.6).

Taking into account the definition of _s

0 A(t)dM_t, it will be enough to prove (2.6) replacingswith ∞. LetA∈S be of the form

A(t) =

n−1

k=0

A_kκ_∆k(t), ∆_k := (t_k, t_k+1].

Since M is the H-valued martingale with respect to E, we see that M(∆_k) = E(∆_k)M_tk+1,whereE(∆_k) :=E_tk+1−E_tk. Using Definition2.2and the properties of resolution of identityE we get

R+

A(t)dM_t

2 H

=

R+

A(t)dM_t,

R+

A(t)dM_t

H

=

n−1

k,m=0

(A_kM(∆_k), A_mM(∆_m))_H

=

n−1

k,m=0

(A_kE(∆_k)M_tk+1, A_mE(∆_m)M_tm+1)_H

=

n−1

k,m=0

(E(∆_k)A_kE(∆_k)M_tk+1, E(∆_m)A_mE(∆_m)M_tm+1)_H

=

n−1

k=0

(A_kE(∆_k)M_tk+1, A_kE(∆_k)M_tk+1)_H=

n−1

k=0

A_kM(∆_k)²_H

≤

n−1

k=0

A_k²_LM(tk)M(∆_k)²_H=

n−1

k=0

A_k²_LM(tk)µ(∆_k)

=

R+

A(t)²_LM(t)dµ(t).

Inequality (2.6) enables us to extend the H-stochastic integral to operator- valued functions R₊ t → A(t) which are not necessarily simple. Namely, denote by S₂ = S₂(M) a Banach space associated with the quasinorm · S2. For its construction, at first it is necessary to pass from S to the factor space S˙ :=S/{A∈S|A_S2= 0}and then to take the completion of ˙S. It is not difficult to understand that elements of the space S₂ are equivalence classes of operator- valued functions on R₊ whose values are linear operators in the space H. These classes are completely characterized by their following properties:

• An operator-valued function R₊ t →A(t) belongs to some equivalence class fromS₂if and only if, forµ-almost allt∈R+,A(t) is anM_t-measurable function and there exists a sequence (A_n)^∞_n=0 of simple functions A_n∈S such that

R+

A(t)−A_n(t)²_LM(t)dµ(t)→0 as n→ ∞. (2.8)

• Operator-valued functions R+ t → A(t) and R+ t → B(t) belong to the same equivalence class fromS₂ if and only if, forµ-almost allt∈R+,

A(t)g=B(t)g, g∈ H_M(t).

By convention, in what follows we will not make the distinctions between the equiv- alence class and operator-valued function from this class.

Definition 2.5. An operator-valued function R+ t → A(t) is said to be H-stochastic integrablewith respect toM ifAbelongs to the spaceS₂.

LetAbe anH-stochastic integrable function with respect toM and (A_n)^∞_n=0be a sequence of the simple functions A_n∈S such that (2.8) holds. Due to inequality (2.6) a limit

n→∞lim

R+

A_n(t)dM_t

exists in H and does not depend on the choice of the sequence (A_n)^∞_n=0 ⊂ S satisfying (2.8). We denote such limit by

R+

A(t)dM_t:= lim

n→∞

R+

A_n(t)dM_t∈ H

and call it the H-stochastic integral of A with respect to M. It is clear that if A andB belong to the same equivalence class fromS₂,then

R+

A(t)dM_t=

R+

B(t)dM_t.

Note that for all H-stochastic integrable functions the assertions of Theo- rem2.1hold. To prove this fact, one should write (2.6) and other expression from Theorem 2.1 for approximating functions fromS and then pass to the limit. We omit the details, note only that due to (2.6) the limit transition is always possible.

To clarify slightly the definition of H-stochastic integral, we consider a simple example which shows that an ordinary spectral integral applied to a certain vector fromHcan be thought of as theH-stochastic integral.

Example 2.1. Let M:R₊ → H be an H-valued martingale closed by a certain vectorM_∞ ∈ H. Suppose f:R+→ Cis a square integrable function with respect to µ, that is f ∈L²(R₊, µ). In the space H we consider a family {A_f(t)}_t∈R+ of the operators of multiplication by the complex numberf(t), i.e.

A_f(t)g:=f(t)g, g∈ H. It is clear thatA_f ∈S₂and

R+

A_f(t)dM_t=

R+

f(t)dM_t=

R+

f(t)dE_t

M_∞.

Note one simple property of the integral introduced above. Let U be some unitary operator acting fromHonto another complex Hilbert spaceK. Then

G:R₊→ K, t→G_t:=U M_t,

is aK-valued martingale with respect to the resolution of identity X:R+→ L(K), t→X_t:=U E_tU⁻¹.

Proposition 2.1. If an operator-valued function R₊ t → A(t) is H-stochastic integrable with respect toM,then the operator-valued functionR+t→U A(t)U⁻¹ isH-stochastic integrable with respect to Gand

U

R+

A(t)dM_t

=

R+

U A(t)U⁻¹dG_t∈ K.

Proof. These properties are immediate if A is a simple adapted function, and a proof of the general case is not difficult.

3. The Itˆo Stochastic Integral as a Particular Case of the H-Stochastic Integral

Before stating the results let us recall the definition of the Itˆo stochastic integral.

In our presentation of stochastic integration we will restrict ourselves to normal martingales. Note that this family of martingales includes Brownian motion, the compensated Poisson process and the Az´ema martingales as particular cases, see e.g. [8,6,13,11,3].

Let (Ω,A, P) be a complete probability space with a right continuous filtra- tion{A_t}_t∈R+. Suppose thatAcoincides with the smallestσ-algebra generated by

t∈R+At andA0 contains all theP-null sets ofA. In addition, assume that A0is trivial, that is everyα∈ A₀has probability 0 or probability 1.

By definition a processN ={N_t}t∈R+is anormal martingaleon (Ω,A, P) with respect to{At}t∈R+if{N_t}t∈R+ and{N_t²−t}t∈R+are martingales with respect to {A_t}_t∈R+. In other words,N is a normal martingale if N_t ∈ L²(Ω,A_t, P) for all t∈R+ and

E[N_t−N_s|As] = 0, E[(N_t−N_s)²|As] =t−s (3.1) for all s, t ∈ R₊ such that s ≤ t. In what follows, without loss of generality we assume thatN₀= 0.

Let us introduce the space of functions for which the Itˆo integral is defined. We will denote byL²(R+×Ω) the space of allB(R+)×A-measurable functions (classes of functions) F:R+×Ω→Csuch that

R+

Ω|F(t, ω)|²dP(ω)dt=

R+

F(t)²_L2(Ω,A,P)dt <∞,

andL²_a(R+×Ω) will present the subspace ofA-adapted functions. Recall, a function F ∈L²(R₊×Ω) is calledA-adapted ifF(t,·) isA_t-measurable for almost allt∈R₊, that is F(t,·) =E[F(t,·)|At] for almost allt∈R+.

Assume thatF(t) = F(t, ω) is a simple function from the space L²_a(R₊×Ω), i.e. there exists a partition 0 =t₀< t₁<· · ·< t_n<∞ofR+ such that

F(·) =

n−1

k=0

F_kκ_(tk,tk+1](·)∈L²_a(R₊×Ω).

TheItˆo integralof such functionF with respect toN is defined by

R+

F(t)dN_t:=

n−1

k=0

F_k(N_tk−N_tk−1)∈L²(Ω,A, P) and has the Itˆo isometry property

R+

F(t)dN_t

2

L²(Ω,A,P)

=

R+

F(t)²_L2(Ω,A,P)dt.

Since the set L²_a,s(R₊ ×Ω) of all simple functions from L²_a(R₊×Ω) is dense in the spaceL²_a(R+×Ω) (with respect to the topology ofL²(R+×Ω)), extending the mapping

L²_a(R+×Ω)⊃L²_a,s(R+×Ω)F →

R+

F(t)dN_t∈L²(Ω,A, P)

by continuity, we obtain a definition of the Itˆo integral on L²_a(R+×Ω). In what follows, we keep the same notation

R+F(t)dN_tfor the extension.

Let us show that the Itˆo integral can be interpreted as theH-stochastic integral.

To this end, we set H := L²(Ω,A, P) and consider in this space a resolution of identity

E:R+→ L(L²(Ω,A, P)), t→E_t:=E[·|At],

generated by the filtration{A_t}_t∈R+. It is easy to see that the normal martingale N is theL²(Ω,A, P)-valued martingale with respect toE_t:=E[·|At] and

µ([0, t]) =N_t²_L2(Ω,A,P)=E[N_t²] =E[N_t²|A0] =t is the ordinary Lebesgue measure onR₊.

Let us prove that in the context of this section N_t-measurability is equivalent to the usualA_t-measurability. More precisely, the following result holds (the proof being similar to the one in [17] is omitted).

Lemma 3.1. Let t∈R+. For a given function F ∈L²(Ω,A, P) the operator A_F of multiplication byF in the spaceL²(Ω,A, P), i.e.

L²(Ω,A, P)⊃Dom(A)G→A_FG:=F G∈L²(Ω,A, P), Dom(A_F) :={G∈L²(Ω,A, P)|F G∈L²(Ω,A, P)}, isN_t-measurable if and only if the function F isA_t-measurable.

Moreover, ifF ∈L²(Ω,A, P)is aAt-measurable function, then A_FLN(s)=F_L²_(Ω,A,P), s∈[t,∞).

This allows us to establish the following result.

Theorem 3.1. For givenF∈L²(R+×Ω)the family{A_F(t)}t∈R+ of the operators A_F(t)of multiplication by F(t)∈L²(Ω,A, P)in the spaceL²(Ω,A, P),

L²(Ω,A, P)⊃Dom(A_F(t))G→A_F(t)G:=F(t)G∈L²(Ω,A, P), isH-stochastic integrable with respect to the normal martingaleN (i.e. A_F belongs toS₂=S₂(N))if and only ifF belongs to the space L²_a(R₊×Ω).

Proof. This fact is an immediate consequence of Lemma 3.1 and the definitions of the spacesL²_a(R₊×Ω) andS₂.

The next theorem shows that the Itˆo stochastic integral with respect to the normal martingaleN can be interpreted as theH-stochastic integral.

Theorem 3.2. Let F ∈L²_a(R₊×Ω)and{A_F(t)}_t∈R+ be the corresponding family of the operators A_F(t)of multiplication by F(t) in the spaceL²(Ω,A, P). Then

R+

A_F(t)dN_t=

R+

F(t)dN_t.

Proof. In view of Theorem 3.1, Lemma3.1and the definitions of the integrals, it is sufficient to prove the statement only for simple functions F ∈L²_a(R+×Ω). But in this case it can be easily proved directly.

4. An Itˆo Integral on a Fock Space

Let F(H) be a symmetric Fock space over a complex separable Hilbert space H, that is

F(H) :=C⊕ ∞ n=1

Hⁿn!,

wherestands for the symmetric tensor product (⊗is the ordinary tensor product).

Thus, F(H) is a complex Hilbert space of sequencesf = (f_n)^∞_n=0 such that each f_n belongs toHⁿ (H⁰:=C) and

f²_F(H)=|f₀|²+ ∞ n=1

f_n²_Hnn!<∞.

Let us fix a resolution of identityR+t→P_t∈ L(H) in the spaceH. For each t∈R₊, we define the second quantization ofP_tby the formula

ExpP_t:=I⊕^∞

n=1

P_t^⊗n, i.e.

ExpP_tf := (f₀, P_tf₁, . . . , P_t^⊗nf_n, . . .)∈ F(H) (4.1) for all f = (f_n)^∞_n=0 ∈ F(H). It is clear that ExpP:R+ → F(H) is a resolution of identity inF(H).

Suppose that M:R₊ → F(H) is an F(H)-valued martingale with respect to E_t:= ExpP_tof the form

M_t= (0, . . . ,0 k times

, M_k(t),0,0, . . .)∈ F(H), (4.2) where k∈Nis fixed. In what follows, we fix such martingaleM.

In this section, starting with the notion of theH-stochastic integral, we define an Itˆo integral (on the Fock spaceF(H)) with respect to such martingaleM and derive its properties. More exactly, we give a meaning to the expression

R+

f(t)dM_t (4.3)

for a certain class ofF(H)-valued functionsf:R+→ F(H).

First, let us describe a class of functions f:R₊ → F(H) for which expression (4.3) will be defined. In the Fock space F(H), for a fixed vector f ∈ F(H), we consider the operatorA_f of a Wick multiplication byf, i.e.

F(H)⊃Dom(A_f)g→A_fg:=f♦g∈ F(H), Dom(A_f) :={g∈ F(H)|f♦g∈ F(H)}.

Recall, for given f = (f_n)^∞_n=0 and g= (g_n)^∞_n=0 from F(H) the Wick productf♦g is defined by

f♦g:=

_n

m=0

f_mg_{n−m ∞}

n=0

(4.4) provided the latter sequence belongs to F(H). It is easily seen that Dom(A_f) is dense inF(H), sinceFfin(H)⊂Dom(A_f) andFfin(H) is a dense subset ofF(H).

HereFfin(H) denotes the set of all finite sequences (f_n)^∞_n=0 fromF(H).

The following statement underlies the definition of a class of functions for which integral (4.3) will be defined.

Lemma 4.1. For given t∈R+ andf = (f_n)^∞_n=0∈ F(H)an operator A_f of Wick multiplication byf in the spaceF(H)isM_t-measurable if and only if

ExpP_tf =f. (4.5)

Moreover, iff ∈ F(H)has property(4.5),then

A_fLM(s)=f_F(H), s∈[t,∞). (4.6) Proof. Suppose that equality (4.5) holds for somet∈R+andf = (f)^∞_n=0∈ F(H).

Let us prove that the corresponding operatorA_f of Wick multiplication byf isM_t- measurable. Clearly, for this we need to verify hypotheses of Definition2.2.

First, let us check thatA_f ∈ LM(t), i.e. there exists a constantC >0 such that A_fg_F(H)≤Cg_F(H)

for allg∈ HM(t) := span{M_s2−M_s1|(s₁, s₂]⊂(t,∞)} ⊂ F(H).

Fixg∈ H_M(t). From the definition of the spaceH_M(t) and (4.2), we conclude that

g= (0, . . . ,0 k times

, g_k,0,0, . . .). (4.7)

Therefore, according to (4.4) we have A_fg=f♦g= (0, . . . ,0

k times

, f₀g_k, f₁g_k, . . .).

Taking into account that P_t^⊗kg_k = 0 (because g ∈ HM(t) and M is the F(H)- valued martingale with respect to ExpP) andP_t^⊗nf_n=f_n for alln∈N∪ {0}, we obtain

f_ng_k²_Hn+k= n!k!

(n+k)!f_n²_Hng_k²_Hk. Hence

A_fg²_F(H)=f♦g²_F(H)= ∞ n=0

f_ng_k²_Hn+k(n+k)!

= ∞ n=0

f_n²_Hng_k²_Hkn!k! =f²_F(H)g²_F(H). (4.8) So, we can assert that A_f ∈ LM(t) and equality (4.6) holds.

Let us check thatA_f is partially commuting with ExpP, i.e.

A_fExpP_sg= ExpP_sA_fg, g∈ HM(t), s∈[t,∞). (4.9) Using (4.1) and (4.7) we get

A_fExpP_sg=f♦ExpP_sg

= (0, . . . ,0

k times

, f₀P_s^⊗kg_k, . . . , f_nP_s^⊗kg_k, . . .). (4.10) On the other hand, we have

ExpP_sA_fg = ExpP_s(f♦g)

= (0, . . . ,0

k times

, f₀P_s^⊗kg_k, . . . , P_s^⊗nf_nP_s^⊗kg_k, . . .). (4.11) Since f = ExpP_tf and ExpP_s ExpP_t = ExpP_t for all s ∈ [t,∞), we conclude that P_s^⊗nf_n =f_n for alln∈Nand alls∈[t,∞). Hence, from (4.10) and (4.11) it follows (4.9).

So, the first part of the lemma is proved. Let us prove the converse statement:

if for givenf ∈ F(H) the corresponding operatorA_f of Wick multiplication byf in the spaceF(H) isM_t-measurable then (4.5) holds.

SinceA_f is theM_t-measurable, we see that for anys∈[t,∞) AExpP_sg= ExpP_sAg, g∈ HM(t).

Lets∈(t,∞) and (s₁, s₂]⊂(t, s] be fixed. We take g:=M_s2−M_s1 = (0, . . . ,0

k times

, g_k,0,0, . . .)∈ HN(t), where g_k:=M_k(s₂)−M_k(s₁). Then, from the one hand,

AE_sg=Ag=f♦g= (0, . . . ,0

k times

, f₀g_k, f₁g_k, . . .),

and on the other hand, taking into account thatP_s^⊗kg_k =g_k, we get E_sAg=E_s(f♦g) = (0, . . . ,0

k times

, f₀P_s^⊗kg_k, . . . , P_s^⊗nf_nP_s^⊗kg_k, . . .)

= (0, . . . ,0 k times

, f₀g_k, . . . , P_s^⊗nf_ng_k, . . .).

Hence, f_ng_k = P_s^⊗nf_ng_k for each n ∈ N and all s ∈ (t,∞). Since resolution of identity (4.1) is right continuous, the latter equality still holds for s = t, and therefore equality (4.5) takes place.

Now we are ready to introduce a class of F(H)-valued functions for which integral (4.3) will be defined. Denote by L²(R+, µ;F(H)) := L²(R+ → F(H), µ) the Hilbert space ofF(H)-valued functions

f:R₊→ F(H), f²_L2(R+,µ;F(H)):=

R+

f(t)²_F(H)dµ(t)<∞, with the corresponding scalar product, where as aboveµ(α) :=M(α)²_F(H). Definition 4.1. A function f(·) = (f_n(·))^∞_n=0 ∈ L²(R₊, µ;F(H)) is said to be adaptedwith respect to ExpP if forµ-almost allt∈R+

ExpP_tf(t) =f(t).

We will use the notation L²_a(R₊, µ;F(H)) for the corresponding subspace of L²(R+, µ;F(H)) formed by adapted functions.

LetE=E(ExpP) denote the class of simple adapted functions with respect to the resolution of identity ExpP. That is, a functionf belongs toE if it belongs to L²_a(R₊, µ;F(H)) and can be written as

f(t) =

n−1

k=0

f_(k)κ_(tk,tk+1](t)∈ F(H) (4.12) forµ-almost allt∈R₊, where 0 =t₀< t₁<· · ·< t_n<∞is a partition ofR₊. We denote byE2 =E2(ExpP) the closure of the setE in the space L²(R+, µ;F(H)).

Observe that if the structures of the elements of the Hilbert space H and the orthogonal projectionP_tare not known, then it is difficult to characterize the space E2(clear only thatE2⊂L²_a(R+, µ;F(H))). In a special case whenH =L²(R+, dt) and P_t is the resolution of identity of the operator of multiplication by t in the space L²(R+, dt), the spaceE2 can be analyzed in a deeper way, in particular, it can be shown thatE2=L²_a(R+, µ;F(H)), see the next section for more details.

Let us derive the connection between the spacesE₂(ExpP) andS₂(M).

Proposition 4.1. A function f : R₊ → F(H) belongs to the space E₂(ExpP) if and only if the corresponding operator-valued function R+ t → A_f(t) whose

values are operators of Wick multiplication byf(t)in the Fock space F(H)belongs to the space S₂(M).

Proof. This fact is an immediate consequence of Lemma 4.1 and the definitions of the spacesE2(ExpP) andS₂(M).

Now we are ready to give a definition of Itˆo integral on the Fock spaceF(H).

Definition 4.2. An Itˆo integral off ∈ E2 with respect toM is defined by

R+

f(t)dM_t:=

R+

A_f(t)dM_t∈ F(H),

where A_f(t) is the operator of Wick multiplication byf(t) in the spaceF(H).

Clearly, for a simple functionf ∈ E of kind (4.12), we have

R+

f(t)dM_t=

n−1

k=0

f_(k)♦(M_tk+1−M_tk)∈ F(H).

Theorem 4.1. Let f, g∈ E2 anda, b∈Cthen

R+

(af(t) +bg(t))dM_t=a

R+

f(t)dM_t+b

R+

g(t)dM_t

and

R+

f(t)dM_t

2 F(H)

=

R+

f(t)²_F(H)dµ(t). (4.13)

Proof. It is sufficient to prove the statement for simple adapted functions. The first assertion is obvious. Concerning the second one, we use the technique of the proof of Theorem2.1and equality (4.8). Thus, for a simple functionf ∈ E of type (4.12) we obtain

R+

f(t)dM_t

2 F(H)

=

n−1

k=0

f_(k)♦(M_tk+1−M_tk)

2

F(H)

=

n−1

k=0

f_(k)♦(M_tk+1−M_tk)²_F(H)

=

n−1

k=0

f_(k)²_F(H)M_tk+1−M_tk²_F(H)

=

n−1

k=0

f_(k)²_F(H)µ((t_k, t_k+1])

=

R+

f(t)²_F(H)dµ(t).