Research Article | Open Access
Some Stochastic Functional Differential Equations with Infinite Delay: A Result on Existence and Uniqueness of Solutions in a Concrete Fading Memory Space
This paper is devoted to existence and uniqueness of solutions for some stochastic functional differential equations with infinite delay in a fading memory phase space.
Let denote the Euclidian norm in . If is a vector or a matrix, its transpose is denoted by and its trace norm is represented by . Let be the minimum (maximum) for .
Let be a complete probability space with a filtration satisfying the usual conditions; that is, it is right continuous and contains all -null sets.
denotes the family of all -measurable valued processes , such that
Assume that is an -dimensional Brownian motion which is defined on ; that is, .
Let denote the family of continuous functions defined on with norm .
Consider the -dimensional stochastic functional differential equationwhere can be regarded as a -value stochastic process, and and are Borel measurable.
The initial data of the stochastic process is defined on . That is, the initial value is a -measurable and -value random variable such that
Our aim, in this paper, is to study existence and uniqueness of solutions to stochastic functional differential equations with infinite delay of type (1) in a fading memory phase space.
The theory of partial functional differential equations with delay has attracted widespread attention. However, when the delay is infinite, one of the fundamental tasks is the choice of a suitable phase space . A large variety of phase spaces could be utilized to build an appropriate theory for any class of functional differential equations with infinite delay. One of the reasons for a best choice is to guarantee that the history function is continuous if is continuous (where ). In general, the selection of the phase space plays an important role in the study of both qualitative and quantitative analysis of solutions. Sometimes, it becomes desirable to approach the problem purely axiomatically. The first axiomatic approach was introduced by Coleman and Mizel in . After this paper, many contributions have been published by various authors until 1978 when Hale and Kato organized the study of functional differential equations with infinite delay in . They assumed that is a normed linear space of functions mapping into a Banach space , endowed with a norm and satisfying the following axioms.
There exist a positive constant and functions , with being continuous and being locally bounded, such that for any and , if , , and is continuous on , then for all in , the following conditions hold:(i),(ii),(iii).
For the function in , is a -valued continuous function for in .
The space is complete.
Later on, the concept of fading and uniform fading memory spaces has been adopted as the best choice.
For and , we define the linear operator by is exactly the solution semigroup associated with the following trivial equation: We define Let be the set of continuous functions with compact support. We recall the following axiom.
If a uniformly bounded sequence in converges to a function compactly on , then and
Definition 1. (1) is called a fading memory space if it satisfies the axioms as , for all (2) is called a uniform fading memory space if it satisfies the axioms as .
Examples. We recall the definitions of some standard examples of phase spaces .
We start first with the phase space of -valued bounded continuous functions defined on , that is, with norm .
(1) Let where is the space of all bounded continuous functions mapping into provided with the uniform norm topology.
(2) Let and provided with the norm
(3) For any continuous function , we define endowed with the norm
Consider the following conditions on : () is locally bounded for , () , ()
Properties of each phase space are summarized in Table 1.
For other examples, properties, and details about phase spaces, we refer to the book by Hino et al. .
Fengying and Ke  discussed existence and uniqueness of solutions to stochastic functional differential equation with infinite delay in the phase space of bounded continuous functions defined on with values in , that is, with norm .
Lemma 2 (page 22 in ). If the phase space satisfies axiom , then is included in .
3. Existence and Uniqueness
Lemma 3 (see ). If such that , then
Lemma 4 (Borel-Cantelli, page 487 in ). If is a sequence of events and then where is an abbreviation for “infinitively often.”
Definitions 1. -value stochastic process defined on is called the solution of (1) with initial data , if has the following properties:(i) is continuous and is -adapted,(ii) and ,(iii), for each , is called unique solution, if any other solution is distinguishable with ; that is, Now, we establish existence and uniqueness of solutions for (1) with initial data . We suppose a uniform Lipschitz condition and a weak linear growth condition.
Theorem 5. Assume that there exist two positive number and such that,
(i) for any and , it follows that(ii) for any , it follows that such thatThen, problem (1), with initial data , has a unique solution . Moreover, .
Proof. For each number , define the stopping time Obviously, as a.s. Let , and then satisfy the following equation: Using the elementary inequality , we get Taking the expectation on both sides and using the Hölder inequality, Lemma 3, and (15) and (16), we get for all in We have also for each in Letting , we get where
By the Gronwall inequality, we inferThat is, Consequently Letting , that implies the following inequalityNow, to prove the second part or the lemma, suppose that . ThenThe demonstration of the lemma is complete.
Proof of Theorem 5. We begin by checking uniqueness of solution. Let and be two solutions of (1), by Lemma 6 and . Note that By the elementary inequality, , one then gets By Hölder inequality, Lemma 3, and (15) and (16), we have From the fact , and We have Applying the Gronwall inequality yields The above expression means that a.s. for . Therefore, for all a.s., the proof of uniqueness is complete.
Next, to check the existence, define and for . Let , and define Picard sequenceObviously . By induction, we can see that .
Hence, for any , one can derive that Note that and then where .
From the Gronwall inequality, we have Since is arbitraryFrom the Hölder inequality, Lemma 3, and (15) and (16), as in a similar earlier inequality, one then has That is,By similar arguments as above, we also have Then where . Similarly