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.


Introduction
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. [9], Bao and Hou [10], and recently Govindan [11] and Diop et al. [12].
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 [13] showed by successive approximations the existence, uniqueness, and large deviation principle of SEEs with jumps. Luo and Taniguchi [14] 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 [15] 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: 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 { ( ), ≥ 0}, defined on the Hilbert space H and called their resolvent (for the precise definition we can refer to Grimmer [17]).
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.

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][18][19].
Let (Ω, F, (F ) ⩾0 , P) be a complete probability space equipped with some filtration (F ) ⩾0 satisfying the usual conditions (i.e., it is right continuous and F 0 contains all Pnull sets). Let (H, ‖ ⋅ ‖ H , ⟨⋅, ⋅⟩) and (K, ‖ ⋅ ‖ K , ⟨⋅, ⋅⟩) denote two real separable Hilbert spaces, with their vectors norms and their inner products, respectively. We denote by L(K; H) the set of all linear bounded operators from K into H, which is equipped with the usual operator norm ‖ ⋅ ‖. Let Let ( ) be a K-valued (F ) ⩾0 -Wiener process defined on the probability space (Ω, F, P) with covariance operator , where is a positive, self-adjoint, trace class operator on K. Let L 2 fl L 2 ( 1/2 K; H) denote the space of all Hilbert-Schmidt operators from 1/2 K into H with the inner product ⟨ , ⟩ L 2 = tr( * ).
Let = ( ), ∈ (the domain of ( )), be a stationary F -Poisson point process taking its value in a measurable space (U, B(U)) with -finite intensity measure ( V) by ( , V), the Poisson counting measure associated with ; that is, ( , U) = ∑ ∈ , ≤ I U ( ( )) for any measurable set U ∈ B(K − {0}), which denotes the Borel -field of (K − {0}). Let̃( , V) fl ( , V) − ( V) be the compensated Poisson measure that is independent of ( ). Denote by P 2 ([0, ] × U; H) the space of all predictable mappings : We may then define the H-valued stochastic integral ∫ 0 ∫ U ( , V)̃( , V), which is a centered square-integrable martingale. For the construction of this kind of integral, we can refer to Peszat and Zabczyk [15].
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 ‖ ‖ fl ‖ ‖ + ‖ ‖ for all ∈ ; and ( ) are closed linear operators on and satisfy the following assumptions: has an associated resolvent operator of bounded linear operators ( ), ≥ 0, on . Hence, we can give the mild solution for the integrodifferential equation where : [0, +∞) → is a continuous function. Let us give the definition of mild solution for (1).
Chinese Journal of Mathematics 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 ≥ 0 for any fixed ≥ 0 and is continuous monotone nondecreasing in for any fixed ∈ [0, ]. Furthermore, for any fixed ∈ [0, ] and ∈ 2 (Ω, C), the following inequality is satisfied: (ii) For arbitrary nonnegative numbers and 0 , the integral equation has a global solution on [0, ].
We now remark that for the proof of our main results we need the following lemmas.

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 ≥ 1 is defined by Proof. The proof is split into the following three steps.
Step 3. We claim the existence and uniqueness of the solution to (1).

Existence. By
Step 2, we known that { ( )} ≥0 is a Cauchy sequence in B ; then the standard Borel-Cantelli lemma argument can be used to show that, as → ∞, ( ) → ( ) holds uniformly for ∈ [0, ]. 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 B ; then by the same way as Step 2, we can show that there exists a positive constant 6 such that E sup We can apply (H4)-(ii) again and infer that E sup ∈[0, ] ‖ ( ) − ( )‖ 2 H = 0, which further implies ( ) ≡ ( ) almost surely for any 0 ≤ ≤ . This completes the proof of Theorem 7.
and estimated as above we infer that there exist positive Chinese Journal of Mathematics 7 Hence, by assumption (H5)-(i) we have the following inequalities: For all ∈ [0, ], by assumption (H5)-(ii) we obtain that E sup This means that, for all ∈ [0, ∧ ], we always have The uniqueness is obtained by stopping our process. The proof for Theorem 8 is thus complete.

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: Thus, Applying Gronwall's inequality, we have E sup which means the mild solution is continuous in the initial value. This completes the proof of Theorem 9.
To rewrite (48) into the abstract form of (1) Moreover, if is bounded and 1 function, where stand for the space of all continuous functions such that is bounded and uniformly continuous, then (H1) and (H2) are satisfied and hence there exists a resolvent operator ( ( )) ≥0 on H. As a consequence of the continuity of Υ, , it follows that Γ, are continuous on R + ×C with values in H, and from the continuity of it follows that Σ is continuous on R + × C with values in L(K, H). Thus, (48) can be expressed as (1) with , Γ, , , Σ, and as defined above.
By assumption (i), we have ‖Γ( , 1 ) − Γ( , 2 )‖ 2 ([0, ]) ≤ On the other hand, in hypotheses (ii) and (iii) above, if there exists a positive constant , such that ( , ) = , then there exists a positive constant Λ such that assumption (H7) is established. Hence, all the assumptions of Theorems 7 and 9 are fulfilled. Therefore, there exists a unique mild solution of (48) by Theorem 7. Furthermore, this solution depends on the initial value by Theorem 9.