Local Existence of Solution to a Class of Stochastic Differential Equations with Finite Delay in Hilbert Spaces ()
![](https://www.scirp.org/html/17-7401161\fe4c21e1-80c9-449c-85f9-a575cbb63edb.jpg)
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 ![](https://www.scirp.org/html/17-7401161\905429f0-4823-4b59-bfec-ad2cb2914180.jpg)
(1)
where
is a linear (possibly unbound) operator, and with a constant
we define
by
![](https://www.scirp.org/html/17-7401161\e40406c4-d734-43dd-9442-11db39dea92e.jpg)
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
![](https://www.scirp.org/html/17-7401161\e2800c0d-8fd4-4527-9ca1-07ba3cf310b6.jpg)
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
![](https://www.scirp.org/html/17-7401161\303077f6-9da9-4f37-a059-8a89e6d3531e.jpg)
and we define the space of all Hilbert-Schmidt operators from
into
by
![](https://www.scirp.org/html/17-7401161\b8644960-b835-43dd-974e-bd3352ab4e62.jpg)
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
![](https://www.scirp.org/html/17-7401161\3afa4d66-ee83-43e1-9a9e-573c82e25172.jpg)
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
![](https://www.scirp.org/html/17-7401161\6a3cb3fc-1c37-4739-a33d-3eeeb20dcf5d.jpg)
or
(7)
By the definition of strong solution, we have
(8)
Since
![](https://www.scirp.org/html/17-7401161\e252e6e4-ec31-4671-a8e2-45725ea2d59c.jpg)
We have
![](https://www.scirp.org/html/17-7401161\1920b04c-dbe2-4b85-b542-66a9db33f7c6.jpg)
Substituting equation above for (8), we received
![](https://www.scirp.org/html/17-7401161\2b13567c-c651-4bdf-9502-4638ba5c18e1.jpg)
Hence,
![](https://www.scirp.org/html/17-7401161\5a9e3d90-ba31-4edc-a28c-9a96ca5be5d5.jpg)
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
![](https://www.scirp.org/html/17-7401161\47aa4950-33b4-4a92-95c0-5ecae0adf4f6.jpg)
![](https://www.scirp.org/html/17-7401161\6170856a-d0bb-4405-8463-efc049c2a015.jpg)
where
![](https://www.scirp.org/html/17-7401161\fc5bf89f-0f8c-4670-97a2-153dd4392368.jpg)
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
![](https://www.scirp.org/html/17-7401161\ae8eedd5-a8d6-4d4c-8aba-ca59cb081caa.jpg)
It is easy to verify that
is a closed subspace of
.
Let
be the operator defined by
![](https://www.scirp.org/html/17-7401161\c864837b-96f8-4b0f-91ac-c4bc8e16b5a2.jpg)
We now prove that
. Indeed,
![](https://www.scirp.org/html/17-7401161\e1902a69-514e-4a70-b534-75bb01b07585.jpg)
Since
,
with
, we have
for any
.
Furthermore,
![](https://www.scirp.org/html/17-7401161\59170d3f-1d3c-40f6-9031-5e850ca6195f.jpg)
Hence
![](https://www.scirp.org/html/17-7401161\0aa6e1da-bc7a-4f3d-8a05-95ad58af437f.jpg)
with
.
If we choose
small enough, such that
![](https://www.scirp.org/html/17-7401161\86ebd39e-a4ba-4a0f-8fa2-77c42f42d1fd.jpg)
Then, for any
we have
. In other words, we have
.
For any
,
![](https://www.scirp.org/html/17-7401161\1fdf0818-70f1-4f87-9dbc-6fff90eddfa6.jpg)
In addition, for any
and
, we have:
![](https://www.scirp.org/html/17-7401161\be9b3d60-1c95-46f1-bb48-63b4b48d9ef8.jpg)
Therefore,
![](https://www.scirp.org/html/17-7401161\ab8a6c9b-eb28-4bdf-ade3-d7c8b39bfe3c.jpg)
Finally, if
, we have
is contraction map in
respects to the norm
![](https://www.scirp.org/html/17-7401161\54fcf630-52ba-417a-98d6-d1fee3912f4e.jpg)
Because this norm is equivalent to
, by applying fixed point principle we conclude that (1.1) has only mild solution on
.