On solutions of neutral stochastic delay Volterra equations with singular kernels ∗

In this paper, existence, uniqueness and continuity of the a dapted solutions for neutral stochastic delay Volterra equations with singular kernels are discussed. In addition , continuous dependence on the initial date is also investigated. Finally, stochastic Volterra equation with the kernel of fractional Brownian motion is studied to illustrate the effectiveness of our results.


Introduction
This paper is concerned with solutions of neutral stochastic delay Volterra equations (NSDVE) driven by Poisson random measure as follows: Stochastic Volterra equation(SVE) was first studied by Berger and Mizel ( [1], [2]) for equations: f (t, s, X(s))ds + t 0 g(t, s, X(s))dB(s).
(1.2) Such equations arise in many applications such as mathematical finance, biology.etc.During the past 30 years, the theory of SVE has been developed in a variety directions.Lots of the well-known results are concerned with Eq.(1.2) with regular kernels.In particular, Protter [3] studied SVE driven by a general semimartingale and resolved a conjecture of Berger and Mizel.Using the Skorohod integral, Pardoux and Protter [4] investigated SVE with anticipating coefficients.Recently, Some results of backward stochastic volterra equations were obtained (see e.g.[5], [6], [7], [8]), which can be used for discussing mathematical finance and stochastic optimal control.
On the other hand, there are also some papers which consider Eq.(1.2) with the singular kernel.One can see Cochran et al. [9], Decreusefond [10], Wang [11], Zhang [12], [13] and the references therein.Wang [11] proved that there exists a unique continuous adapted solution to SVE with singular kernels.Zhang [12] established the existence-uniqueness and large deviation estimate for SVE in 2-smooth Banach spaces, and Zhang in [13] studied the numerical solutions and the large deviation principles of Freidlin-Wentzell's type for SVE with singular kernels.
Stochastic differential equations with delay have been widely used in many branches of science and industry (see e.g.[14], [15]), and neutral type stochastic delay differential equations have been intensively studied in recent years(see e.g.[15], [16]).However, few work has been done on the NSDVE with singular kernels.In this paper, we prove the existence, uniqueness and continuity of the adapted solutions to NSDVE with singular kernels.The continuous dependence of solutions on the initial data is also investigated.Moreover, NSDVE with the kernel of fractional Brownian motion is given to illustrate the effectiveness of our results, where the kernel of fractional Brownian motion is a singular kernel, for it may take the infinity at points s = 0 and s = t.
The paper is organized as follows.In Section 2, we give the preliminaries, and devote Section 3 to deal with the existence and uniqueness result.The path continuity of the solution is obtained in Section 4. The continuous dependence of solution on the initial data is presented in section 5. Finally, NSDVE with the kernel of fractional Brownian motion is studied to illustrate the obtained results in section 6.

Preliminaries
Throughout this paper, we let (Ω, F, P ) be a complete probability space with some filtration {F t } t 0 satisfying the usual conditions (i.e., the filtration is increasing and right continuous while F 0 contains all P -null sets).In this paper, we make the following assumptions:

Let
(A.1)For some p > 2, there exist two functions G(.) and K(t, s), such that for s, t ∈ [0, T ], where G(.) is a concave continuous and nondecreasing function from (A.3) Assuming that there exists a positive number κ < 1, such that for and D(0) = 0.
Lemma 2.1.( [17]) For p ≥ 1, x 1 , x 2 ∈ R d and κ ∈ (0, 1), we have Lemma 2.2.(Bihari inequality) Let G : R + → R + be a concave continuous and nondecreasing function such that G(r) > 0 for r > 0. If u(t), v(t) are strictly positive functions on R + such that where ρ(r) = Proof.Let X 0 (t) = ξ(0), X 0 t = ξ, for t ∈ [0, T ].Define the following Picard sequence: where H(t, s, 0, y) N (ds, dy) According to (A.1), By (A.1),(A.2) and Hölder's inequlity, we obtain EJQTDE, 2012 No. 74, p. 4 Substituting the above inequalities of I 1 , I 2 and where . For G(u) is a positive concave function, there exists a positive constant a, such that By Lemma 2.1, we derive Substituting (3.3) into (3.4),we get So we can apply the Gronwall inequality to get the inequality Since k is arbitrary, this leads to the inequality EJQTDE, 2012 No. 74, p. 5 Consequently, we know that X n (t) ∈ L p T for each n ∈ N .In the following, we will prove the existence and uniqueness of the solution to Eq.(1.1), we first study the existence.

Existence. Let
Using Lemma 2.1 and (3.1), we have According to (A.1) and Hölder's inequality, Combining this with (3.6), we see that where . Consequently, According to Lemma 2.2, we have h(t) ≡ 0, for any t ∈ [0, T ].This means that {X (n) , n ∈ N } is a cauchy sequence in L p T , hence there is an X ∈ L p T , such that EJQTDE, 2012 No. 74, p. 6 Moreover, Letting n → ∞, we can get for all t ∈ [0, T ], Similarly, as n → ∞, we can obtain Taking limits on both sides of (3.1) gives the existence.
Uniqueness.Let X(t) and X(t) be two solutions of Eq.(1.1).Set By virtue of Lemma 2.1, we get By a similar argument as (3.7), we derive Substituting this into (3.8)gives Combining (3.9) with the Bihari inequality leads to The uniqueness has been proved.This completes the proof.

Path continuity of the solution
In this section, in addition to the assumptions (A.1) and (A.3), we also assume that: and for γ > 0, there exists a positive constant K 2 , such that (A.5)There exists a positive constant K 3 , such that where 1 < u ≤ p.
If we denote by X(t) = X(t) − D(X t ).Then from Eq.(1.1) we get where and In what follows, we will study the path continuity of the solutions.Firstly, we give an useful Lemma.
EJQTDE, 2012 No. 74, p. 10 To get the path continuity of the solution for Eq.(1.1), we need an additional assumption: where K 4 is a positive constant.
Proof.Let ∆ = t − t ′ .By Lemma 2.1 and 4.1, we obtain Obviously, Substituting (4.12) into (4.11),we get Note that where . By a similar argument as (4.11), we can get