Abstract and Applied Analysis

Volume 2012 (2012), Article ID 371239, 20 pages

doi:10.1155/2012/371239

Research Article

Existence and Uniqueness of Solutions to Neutral Stochastic Functional Differential Equations with Poisson Jumps

Jianguo Tan,¹ Hongli Wang,² and Yongfeng Guo¹

¹Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China

²Department of Mechanics, Tianjin University, Tianjin 300072, China

Received 9 March 2012; Revised 12 May 2012; Accepted 15 May 2012

Academic Editor: Márcia Federson

Copyright © 2012 Jianguo Tan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

A class of neutral stochastic functional differential equations with Poisson jumps (NSFDEwPJs), , with initial value , is investigated. First, we consider the existence and uniqueness of solutions to NSFDEwPJs under the uniform Lipschitz condition, the linear growth condition, and the contractive mapping. Then, the uniform Lipschitz condition is replaced by the local Lipschitz condition, and the existence and uniqueness theorem for NSFDEwPJs is also derived.

1. Introduction

Neutral stochastic functional differential equations (NSFDEs) have recently been studied intensively, for instance, Mao [1] and Kolmanovskii [2], Mao et al. [3–6], Luo et al. [7], Zhou and Hu [8], and Luo [9]. Poisson jumps are becoming increasingly used to model real-world phenomena in different fields such as economics, finance, biology, and physics. There is an extensive literature concerned with Poisson jumps, for example, Wang et al. [10, 11], Ronghua et al. [12, 13], Luo [14], and Tan and Wang [15]. Therefore, it is natural and necessary to incorporate jumps in the neutral stochastic functional differential equations. However, the study of NSFDEwPJs is limited by far. Liu et al. [16] studied the stability of NSFDEwPJs by using fixed point theory, Luo and Taniguchi [17] proved the existence and uniqueness of non-Lipschitz stochastic neutral delay evolution equations driven by Poisson jumps. However, no theory for the existence and uniqueness of solutions to NSFDEwPJs has been established yet. Therefore, in this paper, we first prove the existence and uniqueness of solutions to NSFDEwPJs.

The outline of the paper is as follows. In Section 2 we will introduce some necessary notations and assumptions. In Section 3, we will present several useful lemmas, and then we prove the existence and uniqueness of NSFDEwPJs under the uniform Lipschitz condition, the linear growth condition, and the contractive mapping. Furthermore, the uniform Lipschitz condition is replaced by the local Lipschitz condition, and the existence and uniqueness theorem is also derived.

2. Preliminaries

Let be a complete probability space with a filtration , which satisfies the usual conditions, that is, the filtration is continuous on the right and contains all P-null sets. Moreover, denotes the family of functions from to that are right-continuous and have limits on the left, is equipped with the norm , where is the Euclidean norm in , that is, . Let denote the family of all -valued -adapted process , , such that . We denote by the family of the processes in such that . means the maximum of a and b, and means the minimum of a and b.

In this paper, we consider the -dimensional neutral stochastic functional differential equations with Poisson jumps

where can be regarded as a -valued stochastic process, is an -dimensional standard Wiener process that is -adapted, and is a scalar Poisson process with intensity . Assume that and are independent of . Moreover, the functions , , , .

The stochastic integral is defined in the Itô sense, and the integral version of (2.1) is frequently expressed as

The initial value is followed:

3. The Existence and Uniqueness Theorem

Definition 3.1. An -valued stochastic process on is called a solution to (2.1) with initial data (2.3) if it has the following properties:

(i) is continuous and is -adapted;(ii) and ;(iii) and (2.2) hold for each .

A solution is said to be unique if any other solution is indistinguishable from it, that is,

Let us now begin to establish the theory of the existence and uniqueness of the solution to (2.1) with initial data (2.3).

Theorem 3.2. Assume that there exist two positive constants and such that

(H1) (the Lipschitz condition) For all and ,

(H2) (the linear growth condition) for all ,

(H3) (the constractive mapping) there is a positive constant such that, for all , ,

Then, there exists a unique solution to (2.1) with initial data (2.3). Moreover, the solution belongs to .

To prove Theorem 3.2, we give the following useful Lemmas 3.3 and 3.4 first.

Lemma 3.3. If , such that

then .

In particular, for , there is equality.

The proof of Lemma 3.3 can be found in [6].

Lemma 3.4. For any , and one has

The proof of Lemma 3.4 can also be found in [6].

To prove the uniqueness of the solution for (2.1), we will establish Lemma 3.5.

Lemma 3.5. Let (3.3) and (3.4) hold. If is a solution to (2.1) with initial data (2.3), then

where is the contraction constant in (3.4). In particular, belongs to .

Proof. For every integer , define the stopping time

Clearly, a.s. Set for . Then, for

where is the indicator function of a set A, that is,

Set

Appling Lemma 3.4 twice and (3.4), one derives that

Hence,

Noting that , one sees that

Consequently,

Using the basic inequality and the Hölder inequality, and for the jump integral, we can transform to the compensated Poisson process and we can get

Since both the Wiener process and compensated Poisson process are martingales, using the Doob martingale inequality, we have

Substituting this into (3.15),

Therefore,

Now the Gronwall inequality yields that

Consequently

Letting , it then implies the following inequality:

The proof is complete.

Proof of Theorem 3.2. Uniqueness. Let and be two solutions to (2.1). By Lemma 3.5 both of them belong to . Note that

where

By Lemma 3.4 and condition (3.4), one sees easily that

Therefore,

which implies that

On the other hand,

Noting the fact that , for , we have

Then

Therefore,

The Grownwall inequality then yields that

Therefore the uniqueness has been proved.

Existence. We divide the whole proof of the existence into two steps.

Step  1. We impose an additional condition: is sufficiently small so that

Define and for . For each , set and define, by the Picard iterations

Obviously, , and by induction . In fact,

Let

Then by Lemma 3.4 and (3.4) we have

Then,

Therefore,

By the basic inequality , the Hölder inequality, Lemma 3.3, and martingale isometry, we have

Therefore,

Hence, for any , we have

Note that

Therefore,

where + , .

By the Gronwall inequality, we have

Because is arbitrary, we have

Hence, for all , .

Note that, for ,

In a similar way to that in the proof of the existence, we get that

Note also that, for and ,

In the same way as in the proof of the uniqueness, one derives that

Next we verify that converges to in the sense of and probability 1 on . Moreover, is the solution to (2.1) with initial data (2.3). Hence, by the Chebyshev inequality,

Since , the Borel-Cantelli lemma yields that, for almost all , there exist a positive integer such that

It follows that, with probability 1, the partial sum

is the partial sum of function series

By the second item of series (3.54), the absolute value of every item (3.54) is less than the corresponding item of positive

Moreover, the positive series is convergent; further, by the Weierstrass criterion, series (3.54) is convergent on . Furthermore, it is uniformly on . Let the sum function be . Therefore, the approximate sequence uniformly converges to on and is -adapted, hence is also continuous and -adapted. On the other hand, (3.50) implies that, for each , sequence is a Cauchy sequence in as well. Hence, we also have in . Letting in (3.46) gives

Therefore, by the use of the above result, we get that

That is, .

Now we prove that satisfies (2.1):

Hence, we can let in (3.34) to obtain that

So is the solution to (2.1).

Step  2. We need to remove the additional condition (3.33). Let be sufficiently small for

By Step 1, there is a solution to (2.1) on . Now consider (2.1) on with initial data . By Step 1 again, there is a solution to (2.1) on . Repeating this procedure we see that there is a solution to (2.1) on the entire interval . The proof is complete.

For NSFDEwPJs, we know that the global Lipschitz condition imposed on Theorem 3.2 is a big restriction; now we will replace the global Lipschitz condition by the local Lipschitz condition. Then Theorem 3.6 follows.

Theorem 3.6. Assume that there exist two positive constants and such that

(i)(the local Lipschitz condition) for all with and ,

(ii)(the linear growth condition) for all ,

(iii)(the constractive mapping) there is a positive constant such that, for all ,

Then there exists a unique solution to (2.1) with initial data (2.3). Moreover, the solution belongs to .

Proof. For each , define truncation functions , and as follows:

Then , , and satisfy conditions (3.2) and (3.3). By Theorem 3.2, there is a unique solution and to the equation

Certainly is a unique solution to the equation

and to the equation.

Now define the stopping time

We can show that for . Then by Lemma 3.4, basic inequality, we have

Taking the expectation, by the Hölder inequality and Lemma 3.3, we transform the Poisson process to the compensated Poisson process , and, by martingale isometry, we have

For , we have know that

Again by , we get that

Noting the fact that , for , we get

From the Gronwall inequality one sees that

This means that, for , we always have

It is then deduced that is increasing, that is as , a.s. By the linear growth condition, for all almost all , there exists an integer such that as . Now define by , . Next to verify that is the solution to (2.1), by (3.74) , it follows that

that is,

Letting

that is

We can see that is the solution to (2.1) and . The proof of existence is complete. By stopping our process, uniqueness is obtained. This completes the proof.

Acknowledgments

This research was supported with funds provided by the National Natural Science Foundation of China (no. 11102132 and no. 10732020). The authors thank two anonymous reviewers for their very valuable comments and helpful suggestions which improve this paper significantly.

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