Local Existence of Solution to a Class of Stochastic Differential Equations with Finite Delay in Hilbert Spaces

Abstract

In this paper, we present a local Lipchitz condition for the local existence of solution to a class of stochastic differential equations with finite delay in a real separable Hilbert space which has the following form:

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Minh, L. , Nam, H. and Thuan, N. (2013) Local Existence of Solution to a Class of Stochastic Differential Equations with Finite Delay in Hilbert Spaces. Applied Mathematics, 4, 97-101. doi: 10.4236/am.2013.41017.

1. Introduction

The purpose of this paper focuses on the local existence of mild solution to a class of the following stochastic differential equations with finite delay in a real separable Hilbert space

(1)

where is a linear (possibly unbound) operator, and with a constant we define

by

In which, is the space of all continuous functions from into equipped with the norm

.

(and are continuous functions; is a Weiner process defined in Section 2).

In [1], if A is the generator of a uniformly exponentially stable semi-group in; satisfies Lipchitz and linear growth conditions then the mild solution of Equation (1) is exponentially stable in mean square.

In this paper, we prove the local existence of solution for Equation (1) with the weaker condition on; and.

2. Preliminaries

In this section, we will recall some notions from Bezandry and Diagana [1].

Let H, K be real separable Hilbert spaces, be a filtered probability space; and is a sequence of real-valued standard Brownian motions mutually independent on this space. Furthermore,

.

where are nonnegative real numbers; and

is the complete orthonormal basis in.

In addition, we suppose that is an operator defined by such that

Then, and for all the distribution of is. The K-valued stochastic process is called a -Weiner process.

The subset is a Hilbert space equipped with the norm

and we define the space of all Hilbert-Schmidt operators from into by

Clearly, is a separable Hilbert space with norm

.

Let be all valued predictable processes such that

.

Then, for all the stochastic integral is well-defined by

.

where is the -Weiner process defined above. We have

(2)

In the following, we assume the stochastic integrals are well defined. For stochastic differential equation and stochastic calculus, we refer to [1-8].

2.1. Definition [1]

For, a stochastic process is said to be a strong solution of Equation (1) on if 1) is adapted to for all;

2) is continuous in almost sure;

3) for any almost surely for any, and

(3)

for all with probability one.

4) almost surely.

2.2. Definition [1]

For, a stochastic process is said to be a mild solution of Equation (1) on if 1) is adapted to for all;

2) is continuous in almost sure;

3) is measureable with almost surely for any and

(4)

for all with probability one;

4) almost surely.

3. Main Results

We assume that

(*) The operator generates a strongly semi-group

in.

(**)and satisfy local Lipchitz conditions respects to second argument that means for be a given real number, there exits such that with , and, we have

If condition (*) holds, we will prove that if is a strong solution of Equation (1) then it also is a mild one. It is expressed by Theorem 3.1.

3.1. Theorem

If (*) holds then (3) can be written in the form (4).

Proof: Applying Fubini theorem, we have

(5)

On the other hand

(6)

From (5) and (6), one has

or

(7)

By the definition of strong solution, we have

(8)

Since

We have

Substituting equation above for (8), we received

Hence,

Now, we turn our attention to the local existence of mild solution of Equation (1).

3.2. Theorem

If the condition (*) and (**) are satisfied, then (1) has only mild solution.

Proof: Let be a fixed number in, for each, there exists , such that

where

For any, we chose. Let be a subspace of containing all functions X which adapt with such that and

is continuous. Then is a Banach space with norm

.

Let us consider a set Z which is defined by

It is easy to verify that is a closed subspace of.

Let be the operator defined by

We now prove that. Indeed,

Since, with

, we have for any.

Furthermore,

Hence

with.

If we choose small enough, such that

Then, for any we have . In other words, we have .

For any,

In addition, for any and, we have:

Therefore,

Finally, if, we have is contraction map in respects to the norm

Because this norm is equivalent to, by applying fixed point principle we conclude that (1.1) has only mild solution on.

Conflicts of Interest

The authors declare no conflicts of interest.

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