Existence and Uniqueness of Solutions to Neutral Stochastic

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 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 1 2

Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China Department of Mechanics, Tianjin University, Tianjin 300072, China

Correspondence should be addressed to Jianguo Tan, [email protected] Received 9 March 2012; Revised 12 May 2012; Accepted 15 May 2012 Academic Editor: Márcia Federson Copyright q 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. A class of neutral stochastic functional differential equations with Poisson jumps NSFDEwPJs, dxt − Gxt fxt , tdt gxt , tdWt hxt , tdNt, t ∈ t0 , T , with initial value xt0 ξ {ξθ : −τ ≤ θ ≤ 0}, 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

2

Abstract and Applied Analysis

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 Ω, F, P be a complete probability space with a filtration {Ft }t≥0 , which satisfies the usual conditions, that is, the filtration is continuous on the right and F0 contains all Pnull sets. Moreover, Ca, b; Rn denotes the family of functions ϕ from a, b to Rn that are right-continuous and have limits on the left, Ca, b; Rn is equipped√with the norm ψ supa n 2 t0 ≤t≤T

≤ C4δn .

3.51

n Since ∞ n0 C4δ ≤ ∞, the Borel-Cantelli lemma yields that, for almost all ω ∈ Ω, there exist a positive integer n0 n0 ω such that 2 1 sup xn1 t − xn t ≤ n , 2 t0 ≤t≤T

n ≥ n0 .

3.52

It follows that, with probability 1, the partial sum

x0 t

n−1 xi1 t − xi t xn t

3.53

i0

is the partial sum of function series x0 t x1 t − x0 t · · · xn t − xn−1 t · · · .

3.54

By the second item of series 3.54, the absolute value of every item 3.54 is less than the corresponding item of positive 1

1 1 1 2 ··· n ··· . 2 2 2

3.55

Moreover, the positive series is convergent; further, by the Weierstrass criterion, series 3.54 is convergent on t0 , T . Furthermore, it is uniformly on t0 , T . Let the sum function be xt. Therefore, the approximate sequence {xn t} uniformly converges to xt on t0 , T and is Ft -adapted, hence xt is also continuous and Ft -adapted. On the other hand, 3.50 implies that, for each t, sequence {xn t} is a Cauchy sequence in M2 t0 − τ, T ; Rn as well. Hence, we also have xn t → xt in M2 t0 − τ, T ; Rn . Letting n → ∞ in 3.46 gives E|xt|2 ≤ c1 ec2 T −t0 ,

t0 ≤ t ≤ T.

3.56

14


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

t0 −τ

|xs|2 ds E

≤E

t0 t0 −τ

t0 t0 −τ

T

|xs|2 ds E

|xs|2 ds

t0

|ξs|2 ds E

3.57

T t0 −τ

c1 ec2 T −t0 ds < ∞.

That is, xt ∈ M2 t0 − τ, T ; Rn . Now we prove that xt satisfies 2.1: 2 t t 2 n n E fxs , s − fxs , s ds E gxs , s − gxs , s dWs t0 t0 2 t n E hxs , s − hxs , sdNs t0 ≤ t − t0 E

t t0

E

t t0

fxsn , s − fxs , s2 ds

gxsn , s − gxs , s2 ds

t 2λ 2λ t − t0 E |hxsn , s − hxs , s|2 ds

2

t0

≤ t − t0 KE

t t0

xsn − xs 2 ds KE

2λ 2λ2 t − t0 KE

t t0

t t0

xsn − xs 2 ds

xsn − xs ds

t ≤ K T − t0 1 2λ 2λ T − t0 E xsn − xs ds

2

t0

t E sup |xn r − xr|2 ds ≤ K T − t0 1 2λ 2λ2 T − t0 t0

t0 ≤r≤s

T E|xn s − xs|2 ds −→ 0 ≤ K T − t0 1 2λ 2λ T − t0

2

t0

n −→ ∞. 3.58


15

Hence, we can let n → ∞ in 3.34 to obtain that xt − Gxt ξ0 − Gξ

t

fxs , sds

t

t0

gxs , sdWs t0

t

3.59

hxs , sdNs. t0

So xt is the solution to 2.1. Step 2. We need to remove the additional condition 3.33. Let σ > 0 be sufficiently small for 3K σ 4 8λ 2λ2 σ σ < 1. δ : κ 1−κ

3.60

By Step 1, there is a solution to 2.1 on t0 − τ, t0 σ. Now consider 2.1 on t0 σ, t0 2σ with initial data xt0 σ . By Step 1 again, there is a solution to 2.1 on t0 σ, t0 2σ. Repeating this procedure we see that there is a solution to 2.1 on the entire interval t0 − τ, T . 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 Kn and K such that i (the local Lipschitz condition) for all φ, ϕ ∈ C−τ, 0; Rn with φ ∨ |ϕ ≤ n and t ∈ t0 , T , f φ, t − f ϕ, t 2 ∨ g φ, t − g ϕ, t 2 ∨ h φ, t − h ϕ, t 2 2 ≤ Kn φ − ϕ .

3.61

ii (the linear growth condition) for all φ, t ∈ C−τ, 0; Rn × t0 , T , 2 2 2 f φ, t ∨ g φ, t ∨ h φ, t ≤ K 1 φ2 .

3.62

iii (the constractive mapping) there is a positive constant κ ∈ 0, 1 such that, for all φ, ϕ ∈ C−τ, 0; Rn , G φ − G ϕ ≤ κφ − ϕ.

3.63

Then there exists a unique solution xt to 2.1 with initial data 2.3. Moreover, the solution belongs to M2 t0 − τ, T ; Rn .

16


Proof. For each n ≥ 1, define truncation functions fn , gn , and hn as follows: ⎧ ⎨fxt , t fn xt , t nxt ⎩f ,t xt

xt ≤ n, xt > n,

⎧ ⎨gxt , t ≤ n, xt gn xt , t nxt ⎩g ,t xt > n, xn

3.64

⎧ ⎨hxt , t ≤ n, xt hn xt , t nxt ⎩h ,t xt > n, xn Then fn , gn , and hn satisfy conditions 3.2 and 3.3. By Theorem 3.2, there is a unique solution xn t and xn t ∈ M2 t0 − τ, T ; Rn to the equation xn t − G xtn ξ0 − Gξ

t t0

t t0

t t0

fn xsn , sds 3.65

gn xsn , sdWs hn xsn , sdNs,

t ∈ t0 , T .

Certainly xn1 t is a unique solution to the equation

x

n1

t n1 ξ0 − Gξ fn1 xsn1 , s ds t − G xt t0

t t0

t t0

gn1 xsn1 , s dWs hn1 xsn1 , s dNs,

3.66

t ∈ t0 , T

and xn1 t ∈ M2 t0 − τ, T ; Rn to the equation. Now define the stopping time τn T ∧ inf t ∈ t0 , T : xtn ≥ n .

3.67


17

We can show that xn t xn1 t for t ∈ t0 , T . Then by Lemma 3.4, basic inequality, we have 2 1 2 n1 x t − xn t ≤ Gxtn1 − Gxtn κ 2 3 t n1 n fn1 xs , s − fn xs , s ds 1 − κ t0 2 3 t n1 n gn1 xs , s − gn xs , s dWs 1 − κ t0

3.68

2 3 t n1 n hn1 xs , s − hn xs , s dNs . 1 − κ t0

Taking the expectation, by the Holder inequality and Lemma 3.3, we transform the ¨ Poisson process to the compensated Poisson process Nt Nt − λt, and, by martingale isometry, we have 2 Exn1 t − xn t 2 ≤ κExtn1 − xtn

3 t − t0 E 1−κ

t 2 fn1 xsn1 , s − fn xsn , s ds t0

t 2 3 E gn1 xsn1 , s − gn xsn , s ds 1−κ t0 2 t 3 2 2λ 2λ t − t0 E hn1 xsn1 , s − hn xsn , s ds 1−κ t0 t 2 2 6 ≤ κExtn1 − xtn − t t 0 E fn1 xsn1 , s − fn1 xsn , s ds 1−κ t0 t 6 fn1 xsn , s − fn xsn , s2 ds t − t0 E 1−κ t0 t 2 6 E gn1 xsn1 , s − gn1 xsn , s ds 1−κ t0 t 6 gn1 xsn , s − gn xsn , s2 ds E 1−κ t0 2 t 6 2 2λ 2λ t − t0 E hn1 xsn1 , s − hn1 xsn , s ds 1−κ t0 t 6 2λ 2λ2 t − t0 E |hn1 xsn , s − hn xsn , s|2 ds. 1−κ t0

3.69

18


For t0 < t < tn , we have know that fn1 xsn , s fn xsn , s fxsn , s, gn1 xsn , s gn xsn , s gxsn , s,

3.70

hn1 xsn , s hn xsn , s hxsn , s. Again by xtn1 xtn0 ξ, we get that 0 2 Exn1 s − xn s 2 ≤ κExtn1 − xtn

6 t − t0 E 1−κ

t 2 fn1 xsn1 , s − fn1 xsn , s ds t0

t 2 6 E gn1 xsn1 , s − gn1 xsn , s ds 1−κ t0 2 t 6 2λ 2λ2 t − t0 E hn1 xsn1 , s − hn1 xsn , s ds, 1−κ t0 t 2 2 6 κE sup xn1 s − xn s t − t0 E f xsn1 , s − fxsn , s ds 1−κ t0 −τ≤s≤t t0

6 E 1−κ

3.71

t 2 g xsn1 , s − gxsn , s ds t0

6 2λ 2λ2 t − t0 E 1−κ

t 2 h xsn1 , s − hxsn , s ds. t0

Noting the fact that xn1 s xn s ξ, for s ∈ t0 − τ, t0 , we get 2 E sup xn1 s − xn s t0 ≤s≤t

t 2 6 t − t0 1 2λ 2λ2 t − t0 n1 n ≤ E K − x x ds n s s 1 − κ2 t0 t 2 6 t − t0 1 2λ 2λ2 t − t0 n1 n ≤ K E sup − x s s ds. x n 2 1 − κ t0 t0 ≤s≤t

3.72

From the Gronwall inequality one sees that 2 E sup xn1 s − xn s , t0 ≤s≤t

t0 < t < τn .

3.73


19

This means that, for t0 < t < τn , we always have 3.74

xn t xn1 t.

It is then deduced that τn is increasing, that is as n → ∞, τn ↑ T a.s. By the linear growth condition, for all almost all ω ∈ Ω, there exists an integer n0 n0 ω such that τn T as n > n0 . Now define xt by xt xn0 t, t ∈ t0 , T . Next to verify that xt is the solution to 2.1, by 3.74 xt ∧ τn xn t ∧ τn , it follows that t∧τn n xt ∧ τn − G xt∧τn ξ0 − Gξ fn xs , sds t0

t∧τn

gn xs , sdWs

t∧τn

t0

3.75 hn xs , sdNs,

t0

that is, t∧τn n xt ∧ τn − G xt∧τ ξ0 − Gξ fxs , sds n t0

t∧τn

gxs , sdWs

3.76

t∧τn

t0

hxs , sdNs. t0

Letting n → ∞ xt ∧ T − G xtn ∧ T ξ0 − Gξ

t∧T

t∧T fxs , sds t0

gxs , sdWs

t0

3.77

t∧T hxs , sdNs, t0

that is xt − Gxt ξ0 − Gξ

t fxs , sds t0

t t0

gxs , sdWs

3.78

t hxs , sdNs. t0

We can see that xt is the solution to 2.1 and xt ∈ M2 t0 −τ, T ; Rn . 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

20


for their very valuable comments and helpful suggestions which improve this paper significantly.

References 1 X. Mao, “Razumikhin-type theorems on exponential stability of neutral stochastic functionaldifferential equations,” SIAM Journal on Mathematical Analysis, vol. 28, no. 2, pp. 389–401, 1997. 2 V. Kolmanovskii, N. Koroleva, T. Maizenberg, X. Mao, and A. Matasov, “Neutral stochastic differential delay equations with Markovian switching,” Stochastic Analysis and Applications, vol. 21, no. 4, pp. 819–847, 2003. 3 X. Mao, “Razumikhin-type theorems on exponential stability of neutral stochastic functionaldifferential equations,” SIAM Journal on Mathematical Analysis, vol. 28, no. 2, pp. 389–401, 1997. 4 X. Mao, Y. Shen, and C. Yuan, “Almost surely asymptotic stability of neutral stochastic differential delay equations with Markovian switching,” Stochastic Processes and their Applications, vol. 118, no. 8, pp. 1385–1406, 2008. 5 X. Mao, “Exponential stability in mean square of neutral stochastic differential-functional equations,” Systems & Control Letters, vol. 26, no. 4, pp. 245–251, 1995. 6 X. R. Mao, Stochastic Differential Equations and Their Applications, Horwood, Chichester, UK, 1997. 7 Q. Luo, X. Mao, and Y. Shen, “New criteria on exponential stability of neutral stochastic differential delay equations,” Systems & Control Letters, vol. 55, no. 10, pp. 826–834, 2006. 8 S. Zhou and S. Hu, “Razumikhin-type theorems of neutral stochastic functional differential equations,” Acta Mathematica Scientia. Series B, vol. 29, no. 1, pp. 181–190, 2009. 9 J. Luo, “Fixed points and stability of neutral stochastic delay differential equations,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 431–440, 2007. 10 L. S. Wang and H. Xue, “Convergence of numerical solutions to stochastic differential delay equations with Poisson jump and Markovian switching,” Applied Mathematics and Computation, vol. 188, no. 2, pp. 1161–1172, 2007. 11 L. S. Wang, C. Mei, and H. Xue, “The semi-implicit Euler method for stochastic differential delay equations with jumps,” Applied Mathematics and Computation, vol. 192, no. 2, pp. 567–578, 2007. 12 L. Ronghua, M. Hongbing, and D. Yonghong, “Convergence of numerical solutions to stochastic delay differential equations with jumps,” Applied Mathematics and Computation, vol. 172, no. 1, pp. 584–602, 2006. 13 L. Ronghua and C. Zhaoguang, “Convergence of numerical solution to stochastic delay differential equation with Poisson jump and Markovian switching,” Applied Mathematics and Computation, vol. 184, no. 2, pp. 451–463, 2007. 14 J. W. Luo, “Comparison principle and stability of Ito stochastic differential delay equations with Poisson jump and Markovian switching,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, vol. 64, no. 2, pp. 253–262, 2006. 15 J. Tan and H. Wang, “Mean-square stability of the Euler-Maruyama method for stochastic differential delay equations with jumps,” International Journal of Computer Mathematics, vol. 88, no. 2, pp. 421–429, 2011. 16 D. Z. Liu, G. Yang, and W. Zhang, “The stability of neutral stochastic delay differential equations with Poisson jumps by fixed points,” Journal of Computational and Applied Mathematics, vol. 235, no. 10, pp. 3115–3120, 2011. 17 J. Luo and T. Taniguchi, “The existence and uniqueness for non-Lipschitz stochastic neutral delay evolution equations driven by Poisson jumps,” Stochastics and Dynamics, vol. 9, no. 1, pp. 135–152, 2009.

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