STOCHASTIC INTEGRAL OF PROCESSES TAKING VALUES OF GENERALIZED OPERATORS

Abstract

In this paper, we study the stochastic integral of processes taking values of generalized operators based on a triple E ⊂ H ⊂ E^∗, where H is a Hilbert space, E is a countable Hilbert space and E^∗ is the strong dual space of E. For our purpose, we study E-valued Wiener processes and then introduce the stochastic integral of L(E, F^∗)-valued process with respect to an E-valued Wiener process, where F^∗ is the strong dual space of another countable Hilbert space F.

1. Introduction

Since the stochastic integral for a standard Brownian motion initiated by K. Itô in [7], so called the Itô integral, the stochastic calculus has been successfully and extensively developed with wide applications to various fields with randomness. The stochastic integral is one of the main topics in the study of stochastic calculus. In Itô integral, the noise term to explain the diffusion part of a stochastic dynamics is corresponding to the white noise which is formally understood as the time derivative of a standard Brownian motion. Recently, to understand various random phenomena, the stochastic calculus associated with various noise processes has been studied by several authors.

In another direction to study of stochastic calculus, the stochastic integrals of operator-valued (stochastic) processes with respect to vector-valued noise processes have been studied by several authors [1,2,3,5,8,10,4], and the references cited therein. In particular, Applebaum [2], Curtain & Falb [3], Kunita [8] and Da Prato & Zabczyk [4] studied the stochastic integral of operator-valued processes in a Hilbert space, and Albeverio & R¨udiger [1], Mamporia [10] and van Neerven & Weis [12] studied the stochastic integral of operator-valued processes with respect to a Banach space valued Wiener process. Also, Yadava [13] studied the stochastic integrals of operator-valued processes with respect to a Wiener process taking values in a locally convex space.

The main purpose of this paper is to develop the stochastic integral of generalized operator-valued stochastic process with respect to a Wiener process taking values in a countable Hilbert space. For our purpose, we focus on the following twofold:

(1) Wiener process values in a countable Hilbert space;

(2) the stochastic integral of processes taking values of generalized operators.

By taking E = H, from our Wiener process we construct a Wiener process taking values in a Hilbert space (see Remark 3.2). Therefore, our results are generalizations of the results for the case of Hilbert space. Also, our approach can be applied for the study of the Wiener process taking values in a locally convex space and the stochastic integral of processes taking values of generalized operators between locally convex spaces (see Remark 3.2).

This paper is organized as follows. In Section 2, we briefly review a standard construction of a countable Hilbert space E from a Hilbert space H and a triple: E ⊂ H ⊂ E∗, which is necessary for our study. In Section 3, we construct an E-valued Wiener process. In Section 4, we introduce a stochastic integral of processes taking values of generalized operators from a countable Hilbert space into the strong dual space of another countable Hilbert space.

 

2. Countable Hilbert Spaces

Let H be a real separable Hilbert space with the norm ║·║0 induced by the inner product ⟨·, ·⟩. Let A be a positive self-adjont operator with the dense domain Dom(A) ⊂ H such that inf Spec (A) > 0. For each p ≥ 0, the dense subspace Dom(Ap) ⊂ H equipped with the norm

becomes a Hilbert space, which is denoted by Ep. Note that since inf Spec (A) > 0, A−1 is a bounded operator on H of which the operator norm is given by

For each p ≥ 0, we define the norm ║·║−p on H by

Let E−p be the completion of H with respect to the norm ║·║−p. Then we have a chain of Hilbert spaces:

where each inclusion is continuous and has a dense image, and

From a chain (1), we obtain the projective limit space:

equipped with the locally convex topology generated by the family {║·║p}p≥0 of Hilbertian norms. Note that E is sequentially complete. Since the norms are linearly ordered as given in (2), we may choose a countable set of defining norms and so E becomes a countable Hilbert space. The strong dual space E∗ of E is identified with the inductive limit space of {E−p}p≥0, i.e.,

Thus, we have a triple:

Note that if there exists p > 0 such that A−p is a Hilbert-Schmidt operator, then E becomes a countable Hilbert nuclear space, and in this case, the triple given as in (3) is called a Gelfand triple. The canonical real bilinear form in E∗ × E is denoted by ⟨·, ·⟩. For further study of the triple, we refer to [6,11].

 

3. E-valued Wiener Processes

Let (Ω, F, P) be a complete probability space. A function X : Ω → E is said to be Bochner measurable if there exists a sequence of simple functions from Ω to E such that Xn(ω) converges to X(ω) in E for almost all ω ∈ Ω. A function X : Ω → E is said to be Bochner integrable if there exists a sequence of simple functions from Ω to E such that Xn(ω) converges to X(ω) in E for almost all ω ∈ Ω and

for any p ≥ 0, and in this case, we define

A Bochner integrable function X : Ω → E is called a (E-valued) random variable and for which, we denote

which is called the expectation of X with respect to P.

For each 1 ≤ ℓ < ∞, we denote Lℓ(Ω, Ƒ, P;E) the space of all Bochner integrable functions X : Ω → E such that for all p ≥ 0.

For two locally convex spaces E and F, we denote L(E, F) the space of all continuous linear operators from E into F. If E is a nuclear Fréchet space and F is a Fréchet space, then

by the kernel theorem (see [11]), where E∗ ⊗π F∗ is the π-tensor product of E∗ and F∗. Therefore, an operator in L(E, F∗) is sometimes called a generalized operator. It is well-known that for each Φ ∈ L(E, F∗), there exist p, q ≥ 0 such that Φ ∈ L(Eq, F−p). In this case, the operator norm of Φ is denoted by ║Φ║q,−p.

From now on, let Q ∈ L(E∗,E) be given by

for some S ∈ L(E∗,E), and let be a sequence of mutually independent real-valued Brownian motions.

Proposition 3.1. Suppose that for any p ≥ 0,

for a complete orthonormal basis ⊂ E for H. Then there exists an E-valued stochastic process {Wt}t≥0 ⊂ L2(Ω, Ƒ, P;E) such that

Proof. The proof is a modification of the proof of Proposition 4.2 in [4]. Let ⊂ E be a complete orthonormal basis for H such that (5) holds. For each t ≥ 0, we can easily see that the series of E-valued random variables:

is summable in E almost surely on Ω. In fact, for each p ≥ 0 and any m, n ∈ ℕ, we obtain that

which implies that the sequence converges in L2(Ω, Ƒ, P;Ep) for any p ≥ 0, and so converges in L2(Ω, Ƒ, P;E). Therefore, for each t ≥ 0, we define Wt by

Then {Wt}t≥0 is an E-valued stochastic process and it is obvious that E[Wt] = 0 for all t ≥ 0. On the other hand, by applying the dominated convergence theorem, we obtain that

which proves (6). For any ϕ, ψ ∈ E∗, we obtain that

which gives the proof of (iii). Let 0 ≤ s < t ≤ u < v and ϕ, ψ ∈ E∗. Then we obtain that

which means that {Wt}t≥0 has independent increments. □

Remark 3.2. Since every absolutely summable sequence is square summable, the condition that

is stronger than the condition given as in (5). For a special case of the operator S and a locally convex space E, a condition similar to the condition given as in (9) can be found in [13]. In the case of Hilbert space valued Wiener processes studied in [4], the condition given as in (9) implies that Q is a Hilbert-Schmidt operator. But for the study of Hilbert space valued Wiener processes, it is enough to assume that Q is a trace-class operator (see [4]). In this sense, the condition given as in (5) is natural.

Remark 3.3. In Proposition 3.1, the identity given as in (7) is essential. If E is a locally convex space, then as a counterpart of (7), by applying the Cauchy-Schwarz inequality, we obtain that

for which we used only the triangle inequality and the Cauchy-Schwarz inequality, and so it is a sharp iequality. Therefore, for the study of Q-Wiener processes in a locally convex space, the condition given as in (9) is natural.

Definition 3.4. Let Q = S∗S ∈ L(E∗,E) be a nonnegative symmetric operator for an operator S ∈ L(E∗,E) satisfying (5). An E-valued stochastic process {Wt}t≥0 ⊂ L2(Ω, F, P;E) is called a Q-Wiener process if

Theorem 3.5. Let Q = S∗S ∈ L(E∗,E) be a nonnegative symmetric operator for an operator S ∈ L(E∗,E) satisfying (5). Then there exists an E-valued Q-Wiener process {Wt}t≥0.

Proof. The proof is immediate from Proposition 3.1. □

Proposition 3.6. Let {Wt}t≥0 be an E-valued Q-Wiener process. Suppose that there exist a complete orthonormal basis for H and a sequence of nonnegative real numbers such that for any p ≥ 0,

Then there exists a sequence of mutually independent real-valued Wiener processes such that

where the series given as in (11) converges in L2(Ω, Ƒ, P;E).

Proof. The proof is a modification of the proof of Proposition 4.1 in [4]. For each n ∈ ℕ, define

and put Λ = {n ∈ ℕ| ⟨Qen, en⟩ > 0}. Then it is easy to see that {ζn}∈Λ is a sequence of mutually independent Gaussian process such that for any m, n ∈ Λ,

Therefore, we define

where {ηn}n∈ℕ╲Λ is any sequence of mutually independent real-valued Brownian motions. Then it holds that

where the last identity is as in L2(Ω, Ƒ, P;E). In fact, it holds that

Therefore, to complete the proof, it is enough to see the identity:

and then, since the identity holds as in L2(Ω, Ƒ, P;H), it is enough to see that the series converges in L2(Ω, Ƒ, P;E), which follows from that

Hence, the proof is completed. □

 

4. Stochastic Integration of L(E, F∗)-valued Process

Let H and K be Hilbert spaces. Then we have chains such as (1): for p, q ≥ 0,

Let {Wt}t≥0 be an E-valued Q-Wiener process. For each t ≥ 0, we denote Ƒt the σ-algebra generated by {Ws|0 ≤ s ≤ t}. Then Ƒ = {Ƒt}t≥0 is a filtration, i.e., an increasing family of σ-algebras.

A L(E, F∗)-valued stochastic process {Φt}t≥0 is said to be Ƒ-adapted (or simply adapted) if for each t ≥ 0, Φt is Ƒt-measurable.

For a L(E, F∗)-valued (adapted) simple process Φ = {Φt}t≥0 of the form

where 0 = t1 < t2 < · · · < tN = T and Φti (i = 1, · · · ,N −1) is Ƒti-measurable, the stochastic integral of Φ with respect to the E-valued Q-Wiener process {Wt}t≥0 is a F∗-valued stochastic process defined by

Proposition 4.1. For a L(E, F∗)-valued simple process Φ = {Φt}t≥0 of the form given as in (13), we have

Proof. (i) For any ζ ∈ F, we obtain that

Therefore, the proof of (i) is immediate since {Wt}t≥0 is an E-valued Q-Wiener process and

for any ϕ ∈ E∗.

(ii) For some p, q ≥ 0, {Φt}t≥0 ⊂ L(Eq, F−p) and so we obtain that

where ⟨·, ·⟩ is the bilinear form on F∗ × F and is a complete orthonormal basis for the given Hilbert space K. Therefore, we have

where is a complete orthonormal basis for the given Hilbert space K. On the other hand, we obtain that

Hence we have

which gives the proof.

(iii) From (ii), we obtain that

which gives the proof. □

Let L2(E, F∗) be the space of all (adapted) L(E, F∗)-valued process such that

for some p, q ≥ 0, and let the space of all (adapted) simple process Φ ∈ L2(E, F∗) of the form given as in (13). We define a map I : → L2(Ω, Ƒ, P;F∗) by

From (iii) in Proposition 4.1, we see that I is a continuous linear map from into L2(Ω, Ƒ, P;F∗). Since L2(Ω, Ƒ, P;F∗) is complete and is dense in L2(E, F∗), we extend I as a continuous linear map from L2(E, F∗) into L2(Ω, Ƒ, P; F∗). Therefore, for Φ ∈ L2(E, F∗), we define

which is called the stochastic integral of Φ with respect to Q-Wiener process {Wt}t≥0.

Theorem 4.2. For any L(E, F∗)-valued process Φ in L2(E, F∗), we have

Proof. (i) It is obvious from (i) in Proposition 4.1.

(ii) Let be a sequence of simple processes in such that Φn converges to Φ in L2(E, F∗) and I(Φn) converges to I(Φ). Then for some p, q ≥ 0 and any arbitrary ϵ > 0, by applying the Cauchy-Schwarz inequality, we obtain that

from which, by taking limit as n → ∞, we obtain that

Since ϵ > 0 is arbitrary, we prove the desired result. □

The study of stochastic differential equations and stochastic control systems (see [9]) is in progress.