Research Article | Open Access
The Neutral Stochastic Integrodifferential Equations with Jumps
We study the existence and uniqueness of mild solutions for neutral stochastic integrodifferential equations with Poisson jumps under global and local Carathéodory conditions on the coefficients by means of the successive approximation. Furthermore, we give the continuous dependence of solutions on the initial value. Finally, an example is provided to illustrate the effectiveness of the obtained results.
Stochastic evolution equations (SEEs) are well known to model problems from many areas of science and engineering, wherein quite often the future state of such systems depends not only on the present state but also on its past history (delay) leading to stochastic functional differential equations and it has played an important role in many ways such as the model of the systems in physics, chemistry, biology, economics, and finance from various points of the view (see, e.g., [1, 2]).
Recently, SEEs in infinite dimensional spaces have been extensively studied by many authors (see, e.g., [3, 4] and the references therein). There is much interest in studying qualitative properties: existence and uniqueness, stability, invariant measure, and so forth for SEEs with Wiener process (see, e.g., [3, 5, 6]). Particularly, the existence and stability results of solution to SEEs and integrodifferential systems have also been considered in the literature (see, e.g., [7, 8]). Furthermore, the problem of the existence and uniqueness of solution for neutral stochastic partial functional differential equation in the case where the coefficients do not satisfy the global Lipschitz condition was investigated by Cao et al. , Bao and Hou , and recently Govindan  and Diop et al. .
On the other hand, there have not been many studies of SEEs driven by jumps processes while these have begun to gain attention recently. To be more precise, Röckner and Zhang  showed by successive approximations the existence, uniqueness, and large deviation principle of SEEs with jumps. Luo and Taniguchi  considered the existence and uniqueness of mild solutions to SEEs with finite delay and Poisson jumps by the Banach fixed point theorem. For SEEs with jumps one can see recent monograph  as well as papers ([9, 13, 14, 16] and the references therein). Motivated by the previously mentioned problems, we will extend some such results for the following neutral stochastic integrodifferential equations with Poisson jumps: with an initial function ; that is, is an -measurable, -value random variable such that , and , are linear, closed, and densely defined operators in a Hilbert space ; for . Let the functions , , and be Borel measurable and let be continuous.
The aim of our paper is to establish existence, uniqueness, and stability results for mild solution of (1) under global and local Carathéodory conditions in the Hilbert space based on successive approximation method. Our main results concerning (1) rely essentially on techniques using strongly continuous family of operators , defined on the Hilbert space and called their resolvent (for the precise definition we can refer to Grimmer ).
The rest of this paper is organized as follows: In Section 2, we will give some necessary notations, concepts, and basic results about the Wiener process, Poisson jumps process, and deterministic integrodifferential equations. Section 3 is devoted to prove the existence and uniqueness of the solution. In Section 4, we study stability through the continuous dependence on the initial values. An example is given in Section 5 to illustrate the theory.
2. Preliminaries Results
This section is concerned with some basic concepts, notations, definitions, lemmas, and preliminary facts which are used through this paper. For more details on this section, we refer the reader to [3, 17–19].
Let be a complete probability space equipped with some filtration satisfying the usual conditions (i.e., it is right continuous and contains all -null sets). Let and denote two real separable Hilbert spaces, with their vectors norms and their inner products, respectively. We denote by the set of all linear bounded operators from into , which is equipped with the usual operator norm .
Let and denotes the family of all right continuous functions with left-hand limits (càdlàg) from to . The space is assumed to be equipped with the norm ,
We also assume that denotes the family of all almost surely bounded, -measurable, -valued random variables. Further, we consider the Banach space of all -valued -adapted càdlàg process defined on , with , such that
Let be a -valued -Wiener process defined on the probability space with covariance operator , where is a positive, self-adjoint, trace class operator on . Let denote the space of all Hilbert-Schmidt operators from into with the inner product
Let , (the domain of ), be a stationary -Poisson point process taking its value in a measurable space with -finite intensity measure by , the Poisson counting measure associated with ; that is, for any measurable set , which denotes the Borel -field of . Let be the compensated Poisson measure that is independent of . Denote by the space of all predictable mappings for which We may then define the -valued stochastic integral , which is a centered square-integrable martingale. For the construction of this kind of integral, we can refer to Peszat and Zabczyk .
Next, to be able to access existence, uniqueness, and stability of mild solutions for (1) we need to introduce partial integrodifferential equations and resolvent operators.
Let be two Banach spaces such that for all ; and are closed linear operators on and satisfy the following assumptions:(H1)The operator is the infinitesimal generator of a strongly continuous semigroup on .(H2)For all , is a closed linear operator, , and are the set of all bounded linear operators from into . For any , the map is bounded, differentiable and the derivative is bounded uniformly continuous on
By Theorem 2.3 in , we can see that () and () imply the integrodifferential abstract Cauchy problem has an associated resolvent operator of bounded linear operators , , on Hence, we can give the mild solution for the integrodifferential equation where is a continuous function.
Let us give the definition of mild solution for (1).
Definition 1. A càdlàg stochastic process , , is called a mild solution of (1) on if (i) is -adapted, for all ,(ii)for arbitrary , , and satisfies the following integral equation:(iii),
Throughout this paper, we always assume the following assumptions are satisfied.(H3) (i)The growth condition: there exists a nonnegative real valued function , , which is locally integrable in for any fixed and is continuous monotone nondecreasing in for any fixed . Furthermore, for any fixed and , the following inequality is satisfied: (ii) For arbitrary nonnegative numbers and , the integral equation has a global solution on .(H4) (i)The global condition: there exists a nonnegative real valued function , , which is locally integrable in for any fixed and is continuous monotone nondecreasing in for any fixed . Furthermore and for any fixed and , the following inequality is satisfied: (ii) If there exists a nonnegative continuous function satisfying and then on , where is a positive number.(H5) (i)The local condition: for any integer there exists a nonnegative real valued function , , which is locally integrable in for any fixed and is continuous monotone nondecreasing in for any fixed . Furthermore and for any fixed and with , the following inequality is satisfied: (ii)If there exists a nonnegative continuous function satisfying and then on , where is a positive number.(H6)The contractive mapping: the mapping satisfies that there exists a positive such that, for any and ,
Remark 2. The function , , , where is nonnegative and locally integrable and is a concave, continuous function, satisfies Osgood’s condition; that is, it is a nondecreasing function with and , , such that Then we can show that the function satisfies assumption (H4)-(ii) (cf. ).
To illustrate this remark, we give two examples which satisfy the conditions of in Remark 2. Let . Set where is sufficiently small and , , is the left derivative of , , at the point . Then and are both concave nondecreasing functions definition on satisfying , .
Remark 3. (1) If there exists a positive constant , such that , , then assumption (H4)-(i) implies the Lipschitz condition.
(2) From assumption (H5), for any fixed integer , there exists such that , ; then assumption (H5)-(i) implies the local Lipschitz condition.
We now remark that for the proof of our main results we need the following lemmas.
Lemma 4 (see ). For and , the following inequality is true:
Lemma 5 (see [3, Proposition 7.3]). Suppose that , , is -valued predictable process and let , . Then for any arbitrary there exists a constant such that Moreover, if , then there exists a continuous version of the process . If is a contraction semigroup, then the above result is true for .
Lemma 6 (see [20, Proposition 1.3]). Let be a predictable function satisfying for all almost surely. Let . If is a contraction semigroup, then there exists a constant such that
3. Existence and Uniqueness of Solution
In this section, we will investigate the existence and uniqueness of the mild solution to (1) under the non-Lipschitz condition and a weakened linear growth condition.
We introduce the successive approximations to (5) as follows:and for is defined by
Theorem 7. Assume the assumptions of (H1)–(H4) and (H6) hold. Then, there exists a unique mild solution to (1) in
Proof. The proof is split into the following three steps.
Step 1. We claim that the sequence is bounded. Obviously, . Moreover, we easily show that , for and . In fact, from (18), for , using the basic inequality , we can getBy assumption , with , it follows thatNote that
By using the Hölder inequality and Lemmas 5 and 6, then, associated with assumption for the term , we obtainwhere is a positive constant.
Hence, putting (20) and (21) into (19) yields While, by Lemma 4, it follows that From assumption (H3)-(ii) we show that there is a solution that satisfies where ;
On the other hand, since , we deduce thatHence, , for and . This proves the boundedness of
Step 2. We claim that the sequence is a Cauchy sequence in . For and , from (18), , and Step 1, we can show that there exists a positive constant such that Therefore applying Lemma 4 and assumption again, we obtain Let From (25), condition -(ii), and the Fatou lemma, we have where
By condition -(ii) we get , which implies that This shows that sequence is Cauchy sequence in .
Step 3. We claim the existence and uniqueness of the solution to (1).
Existence. By Step 2, we known that is a Cauchy sequence in ; then the standard Borel-Cantelli lemma argument can be used to show that, as , holds uniformly for . So, taking limits on both sides of (18) we obtain that is a solution to (1). This shows the existence.
Uniqueness. Let both and be two mild solutions of (1) in ; then by the same way as Step 2, we can show that there exists a positive constant such that We can apply -(ii) again and infer that , which further implies almost surely for any This completes the proof of Theorem 7.
Next, we present the existence and uniqueness of mild solutions for (1) with the local Carathéodory conditions.
Theorem 8. Assume the assumptions of (H1)–(H3) and (H5) and (H6) with hold. Then, there exists a unique mild solution to (1) in
Proof. Let be a natural integer and let We define the sequence of the functions , , and as follows: Then, the functions , , and satisfy assumption (H3) and the following inequality:where ,
Thus, by Theorem 7, there exists a unique solution and , respectively, to the following equations:Now define the stopping times We claim that , for all , a.s. .
By (34) and for , and estimated as above we infer that there exist positive constants such that Hence, by assumption (H5)-(i) we have the following inequalities:For all , by assumption (H5)-(ii) we obtain that This means that, for all , we always have For each , there exists , such that . For all , define by Since , it holds that Letting , for all , we infer that The uniqueness is obtained by stopping our process. The proof for Theorem 8 is thus complete.
4. Stability of Solution
In this section, we study the stability through the continuous dependence of mild solutions on the initial value. From now on, we will use to represent the mild solution of (1) to emphasize that the solution depends on the initial value . We need the following assumption:()For all , , there exists a positive constant such that
Theorem 9. Let assumptions (H1), (H2), (H6) with , and (H7) be satisfied. Then the mild solution of (1) is continuous in the initial value (with respect to the strong topology on ).
Proof. Let and be two mild solutions of (1) with initial values and , respectively. Then, for all we can show that there exists a positive constant such thatThus, Applying Gronwall’s inequality, we have which means the mild solution is continuous in the initial value. This completes the proof of Theorem 9.
In this section, an example is provided to illustrate the obtained theory. We consider the following neutral stochastic integrodifferential equations with Poisson jumps of the form: