# Numerical Solutions to Neutral Stochastic Delay Differential Equations with Poisson Jumps under Local Lipschitz Condition

Authors: Jianguo Tan; Hongli Wang; Yongfeng Guo; Zhiwen Zhu
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

Recently, Liu et al. (2011) studied the stability of a class of neutral stochastic delay differential equations with Poisson jumps (NSDDEwPJs) by fixed points theory. To the best of our knowledge to date, there are not any numerical methods that have been established for NSDDEwPJs yet. In this paper, we will develop the Euler-Maruyama method for NSDDEwPJs, and the main aim is to prove the convergence of the numerical method. It is proved that the proposed method is convergent with strong order 1/2 under the local Lipschitz condition. Finally, some numerical examples are simulated to verify the results obtained from theory.


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## Body

## 1. Introduction

Neutral stochastic delay differential equations (NSDDEs) have recently been studied intensively by, for instance, Kolmanovskii et al. [1, 2], Mao et al. [3–6], Luo et al. [7], Zhou and Hu [8], and Luo [9]. However, explicit solutions can hardly be obtained for NSDDEs; as a result, several numerical schemes have been developed to produce approximate solutions for NSDDEs. For example, Wu and Mao [10] studied the numerical solutions of NSDDEs. Zhang and Gan [11] considered the mean square convergence of one-step methods for NSDDEs. Zhou and Wu [12] studied the convergence of numerical solutions to neutral stochastic delay differential equations with Markovian switching. 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. [13, 14] studied the semi-implicit Euler method for stochastic differential delay equation with jumps (SDDEwJs) and the convergence of numerical solutions to stochastic differential delay equations with Poisson jump and Markovian switching (SDDEwPJMSs). Li et al. [15, 16] discussed the convergence of the numerical solutions for SDDEwJs and SDDEwPJMSs. Luo [17] considered the comparison principle and stability of SDDEwPJMSs. Therefore it is natural and necessary to incorporate jumps in the neutral stochastic delay differential equations. However, the study of neutral stochastic delay differential equations with Poisson jumps is less by far. Cen and Zhou [18] investigated convergence of numerical solutions to neutral stochastic delay differential equation with Poisson jump and Markovian switching. Liu et al. [19] studied the stability of NSDDEwPJs by using fixed points theory. Luo and Taniguchi [20] proved the existence and uniqueness for non-lipschitz stochastic neutral delay evolution equations driven by Poisson jumps. However, there are not any numerical methods that have been established for NSDDEwPJs yet. Therefore, in this paper, we first prove the Euler-Maruyama method applied to NSDDEwPJs converges to the true solution under local Lipschitz condition.The outline of the paper is as follows. In Section2 we will introduce some necessary notations and assumptions, and then the Euler-Maruyama method is used to define the numerical solutions for NSDDEwPJs. Section 3 will present several useful lemmas. In Section 4, we state our main result; that is, the numerical solutions will converge to the true solutions of NSDDEwPJs under the local Lipschitz condition. At last, some numerical examples are given to verify the results obtained from the theory.


## 2. Preliminaries and the Euler-Maruyama Approximation

Let(Ω,F,P) be a complete probability space with filtration {Ft}t≥0, which satisfies the usual conditions, that is, the filtration is continuous on the right and F0 contains all P-null sets. Let C([a,b];Rn) denote the family of functions φ from [a,b] to Rn that are right-continuous and have limits on the left; C([a,b];Rn) is equipped with the norm ∥ϕ∥=sup⁡a<t<b|ϕ(t)|, where |·| is the Euclidean norm in Rn, that is, |x|=xTx(x∈Rn). If A is a vector or matrix, its trace norm is denoted by |A|=trace(ATA), while its operator norm is denoted by ∥A∥=sup⁡{|Ax|:|x|=1}. Let p>0 denote by LF0p([-τ,0];Rn) the family of all F0 measurable, C([a,b];Rn)-valued random variables ξ such that E∥ξ∥p<∞. For simplicity, we also denote by a∧b=min⁡{a,b} that a∨b=max⁡{a,b}.In this paper, we consider then-dimensional neutral stochastic delay differential equations with Poisson jumps
(1)d[x(t)-G(x(t-τ))]=f(x(t),x(t-τ))dt+g(x(t),x(t-τ))dW(t)+h(x(t),x(t-τ))dN(t),t∈[0,T],x(t)=ξ(t),t∈[-τ,0],
where τ is a positive fixed delay, and ξ(t)∈LF0p([-τ,0];Rn), W(t) is a d-dimensional standard Wiener process which is Ft-adapted, and N(t) is a scalar Poisson process with intensity λ. Assume that W(t) and N(t) are independent of F0. Moreover the functions G:Rn→Rn, f:Rn×Rn→Rn, g:Rn×Rn→Rn×d, h:Rn×Rn→Rn.The stochastic integral is defined in the Itô sense, and the integral version of (1) is frequently expressed as
(2)x(t)-G(x(t-τ))=ξ(0)-G(x(-τ))+∫0t‍f(x(s),x(s-τ))ds+∫0t‍g(x(s),x(s-τ))dW(s)+∫0t‍h(x(s),x(s-τ))dN(s).We can now define the Euler-Maruyama approximate solution for NSDDEwPJs.Given a step-sizeΔ∈(0,1) which satisfies Δ=τ/m for an integer m, let tk=kΔ for k>0. Compute the discrete approximation yk≈x(tk) by setting y0=x(t0) and performing
(3)yk+1-G(yk+1-m)=yk-G(yk-m)+f(yk,yk-m)Δ+g(yk,yk-m)ΔWk+h(yk,yk-m)ΔNk;
if tk<0, we have yk=ξ(tk). Moreover, the increments ΔWk∶=W(tk+1)-W(tk) are independent Gaussian random variables with mean 0 and variance Δ; ΔNk∶=N(tk+1)-N(tk) are independent Poisson distributed random variables with mean λΔ and variance λΔ.Lety~(t)=yk, y~(t-τ)=yk-m, t∈[tk,tk+1), with initial values y~(t)=ξ(t) on [-τ,0]. The continuous Euler-Maruyama approximate solution y(t) is to be interpreted as the stochastic integral
(4)y(t)={G(y~(t-τ))+ξ(0)-G(ξ(-τ))+∫0t‍f(y~(s),y~(s-τ))ds+∫0t‍g(y~(s),y~(s-τ))dW(t)+∫0t‍h(y~(s),y~(s-τ))dN(t),t∈[0,T],ξ(t),t∈[-τ,0].It is not difficult to see thaty(tk)=y~(tk)=yk; that is, y(t) and y~(t) coincide with the discrete solution at the grid points.In this paper, the following hypotheses are imposed on (1).(H1)The Local Lipschitz Condition. There is a positive constant Cd such that, for all x1,x2,y1,y2∈Rn with |x1|∨|x2|∨|y1|∨|y2|≤d,
(5)|f(x1,y1)-f(x2,y2)|2∨|g(x1,y1)-g(x2,y2)|2∨|h(x1,y1)-h(x2,y2)|2≤Cd(|x1-x2|2+|y1-y2|2).(H2)The Constractive Mapping. There is a positive constant K∈(0,1) such that, for all x,y∈Rn(6)|G(x)-G(y)|<K|x-y|.(H3)The Linear Growth Condition. There is a L>0 such that for all x,y∈Rn(7)|f(x,y)|2+|g(x,y)|2+|h(x,y)|2≤L(1+|x2|+|y2|).Remark 1.
Condition (H3) can be replaced by condition (H1).In fact, from (5) and inequality |a2+b2|≤2(|a2|+|b2|), it is easy to obtain
(8)|f(x,y)|2≤2|f(x,y)-f(0,0)|2+2|f(0,0)|2≤2Cd(|x|2+|y|2)+2|f(0,0)|2≤Ld(1+|x|2+|y|2);
similarly,
(9)|g(x,y)|2≤Ld(1+|x|2+|y|2),|h(x,y)|2≤Ld(1+|x|2+|y|2),
where
(10)Ld=2(Cd∨|f(0,0)|2∨|g(0,0)|2∨|h(0,0)|2).
From (6), we obtain
(11)|G(x)|2≤2|G(x)-G(0)|2+2|G(0)|2≤2K2|x|2+2|G(0)|2≤K1(1+|x|2),
where K1=2(K2∨|G(0)|2).Recently we have studied the existence and uniqueness of solutions to neutral stochastic functional differential equations with Poisson jumps [21]. In an analogous way, we may establish the following existence and uniqueness conclusion that under assumptions (H1)-(H2), (1) has a unique solution x(t) on t≥-τ.


## 3. Lemmas and Corollaries

In this section, we first establish a few lemmas under local Lipschitz condition. For eachd>0, define the stopping times
(12)σd∶=inf⁡{t≥0,|y(t)|≥d},νd∶=inf⁡{t≥0,|x(t)|≥d},ρd=νd∧σd.
(As usual we set inf⁡∅=∞.)Lemma 2.
Under (H1)-(H2), for anyp≥2, there exists a constant A>0 independent of Δ, such that
(13)E(sup⁡-τ≤t≤T|x(t)|p)∨E(sup⁡-τ≤t≤T|y(t)|p)<A.Proof.
First, we prove thep-moment of the exact solution of (1) is finite. For any t1∈[0,T], we obtain
(14)E(sup⁡-τ≤t≤t1|x(t)|p)≤E∥ξ∥p+E[sup⁡0≤t≤t1|x(t)|p].
From (2) and inequality
(15)(a+b+c+d+e+f)p≤6p-1(|a|p+|b|p+|c|p+|d|p+|e|p+|f|p),
we get
(16)E[sup⁡0≤t≤t1|x(t)|p]≤6p-1E[sup⁡0≤t≤t1|G(x(t-τ))|p]+6p-1E|ξ(0)|p+6p-1E|G(ξ(-τ))|p+6p-1E[sup⁡0≤t≤t1|∫0t‍f(x(s),x(s-τ))ds|p]+6p-1E[sup⁡0≤t≤t1|∫0t‍g(x(s),x(s-τ))dW(s)|p]+6p-1E[sup⁡0≤t≤t1|∫0t‍h(x(s),x(s-τ))dN(s)|p].
By condition (H2) and inequality (a+b)p≤2p-1(|a|p+|b|p), then for any t∈[0,T], we have
(17)|G(x(t-τ))|p≤2p-1|G(x(t-τ))-G(0)|p+2p-1|G(0)|p≤2p-1Kp|x(t-τ)|p+2p-1|G(0)|p≤Lp(1+|x(t-τ)|p),
where Lp=2p-1(Kp∨|G(0)|p). Thus for any t1∈[0,T],
(18)E[sup⁡0≤t≤t1|G(x(t-τ))|p]≤Lp(1+E[sup⁡0≤t≤t1|x(t-τ)|p])≤Lp(1+E[sup⁡0≤t≤T|x(t-τ)|p]).
Similarly
(19)E[sup⁡0≤t≤t1|G(ξ(-τ))|p]≤Lp(1+E∥ξ∥p).
By the Hölder inequality and (8), then for any t1∈[0,T], we get
(20)E[sup⁡0≤t≤t1|∫0t‍f(x(s),x(s-τ))ds|p]≤Tp-1∫0t‍E|f(x(s),x(s-τ))|pds≤Tp-1Ldp/2∫0t‍(1+E|x(s)|2+E|x(s-τ)|2)p/2ds≤Tp-1Ldp/2∫0t‍(1+2Esup⁡-τ≤ν≤s|x(ν)|2)p/2ds≤Tp-1Ldp/22p/2∫0t‍(1+Esup⁡-τ≤ν≤s|x(ν)|2)p/2ds≤Tp-1(2Ld)p/2(T+∫0t1‍Esup⁡-τ≤ν≤s|x(ν)|pds).
Now, we use H to denote that a generic constant may change between occurrences. Using the Burkholder-Davis-Gundy inequality for the two martingale integral terms, we have
(21)E[sup⁡0≤t≤t1|∫0t‍g(x(s),x(s-τ))dW(s)|p]≤H∫0t1‍E|g(x(s),x(s-τ))|pds≤(2Ld)p/2H(T+∫0t1‍Esup⁡-τ≤ν≤s|x(ν)|pds).
Similarly, for the jump integral term, we have
(22)E[sup⁡0≤t≤t1|∫0t‍h(x(s),x(s-τ))dN(s)|p]=E[sup⁡0≤t≤t1|∫0t‍h(x(s),x(s-τ))dN~(s)iiiiiiiiiiiiiiiiiiiiiii+λ∫0t‍h(x(s),x(s-τ))ds|p]≤2p-1E[sup⁡0≤t≤t1|∫0t‍h(x(s),x(s-τ))dN~(s)|p]+2p-1λpE[sup⁡0≤t≤t1|∫0t‍h(x(s),x(s-τ))ds|p]≤2p-1H∫0t1‍E|h(x(s),x(s-τ))|pds+(2T)p-1λp∫0t1‍E|h(x(s),x(s-τ))|pds≤(2Ld)p/2[2p-1H+(2T)p-1λp]×(T+∫0t1‍Esup⁡-τ≤ν≤s|x(ν)|pds).
By the above inequalities, we can obtain
(23)E(sup⁡-τ≤t≤t1|x(t)|p)≤E∥ξ∥p+E[sup⁡0≤t≤t1|x(t)|p]≤E∥ξ∥p+6p-1E∥ξ∥p+6p-1Lp(1+E∥ξ∥p)+6p-1Lp(1+E[sup⁡-τ≤t≤t1|x(t)|p])+6p-1(Tp-1(2Ld)p/2+(2Ld)p/2Hiiiiiiiiiiiiiiiiiii+(2Ld)p/22p-1H+(2Ld)p/2(2T)p-1λp)×(T+∫0t1‍Esup⁡-τ≤ν≤s|x(ν)|pds);
that is,
(24)E(sup⁡-τ≤t≤t1|x(t)|p)≤(2Lp6p-1+(1+6p-1(1+Lp))E∥ξ∥piiiiiiiiii+6p-1(2Ld)p/2(Tp+(2p-1+1)HT+Tp2p-1λp))×(1-6p-1Lp)-1+(6p-1(2Ld)p/2(Tp-1+(2p-1+1)H+(2T)p-1λp))×(1-6p-1Lp)-1×∫0t1‍Esup⁡-τ≤ν≤s|x(ν)|pds≤c1+c2∫0t1‍Esup⁡-τ≤ν≤s|x(ν)|pds,
where
(25)c1=(2Lp6p-1+(1+6p-1(1+Lp))E∥ξ∥piii+6p-1(2Ld)p/2(Tp+(2p-1+1)HT+Tp2p-1λp))×(1-6p-1Lp)-1,c2=(6p-1(2Ld)p/2(Tp-1+(2p-1+1)H+(2T)p-1λp))×(1-6p-1Lp)-1.
The Gronwall inequality shows that
(26)E(sup⁡-τ≤t≤T|x(t)|p)≤c1ec2T≡A1.
Then we can prove in the same way that the Euler-Maruyama approximate solution to (1) has the property that
(27)E(sup⁡-τ≤t≤T|y(t)|p)≤A2.
So, we can obtain
(28)E(sup⁡-τ≤t≤T|x(t)|p)∨E(sup⁡-τ≤t≤T|y(t)|p)≤A,
by letting A=A1∨A2.Lemma 3.
Under (H1)-(H2),(29)E(sup⁡-τ≤t≤T|y(t∧ρd)|2)<C1,∀t∈[-τ,T],
where C1 is a positive constant independent of Δ.Proof.
For anyt1∈[0,T], we have
(30)E(sup⁡-τ≤t≤t1|y(t∧ρd)|2)≤E∥ξ∥2+E(sup⁡0≤t≤t1|y(t∧ρd)|2).
By (4) and inequality
(31)(a+b+c+d+e+f)2≤6(|a|2+|b|2+|c|2+|d|2+|e|2+|f|2)
and (11) then we have
(32)E(sup⁡0≤t≤t1|y(t∧ρd)|2)≤6E[sup⁡0≤t≤t1|G(y~(t∧ρd-τ))|2]+6E|ξ(0)|2+6E|G(ξ(-τ))|2+6E[sup⁡0≤t≤t1|∫0t∧ρd‍f(y~(s),y~(s-τ))ds|2]+6E[sup⁡0≤t≤t1|∫0t∧ρd‍g(y~(s),y~(s-τ))dW(s)|2]+6E[sup⁡0≤t≤t1|∫0t∧ρd‍h(y~(s),y~(s-τ))dN(s)|2]≤6K1E[1+sup⁡0≤t≤t1|y~(t∧ρd-τ)|2]+6E∥ξ∥2+6K1(1+E∥ξ∥2)+6E[sup⁡0≤t≤t1|∫0t∧ρd‍f(y~(s),y~(s-τ))ds|2]+6E[sup⁡0≤t≤t1|∫0t∧ρd‍g(y~(s),y~(s-τ))dW(s)|2]+6E[sup⁡0≤t≤t1|∫0t∧ρd‍h(y~(s),y~(s-τ))dN(s)|2].
Using the Cauchy-Schwarz inequality, Fubini’s Theorem, and (8),
(33)E[sup⁡0≤t≤t1|∫0t∧ρd‍f(y~(s),y~(s-τ))ds|2]≤TE[sup⁡0≤t≤t1∫0t∧ρd‍|f(y~(s),y~(s-τ))|2ds]≤TLdE[∫0t1∧ρd‍(1+|y~(s)|2+|y~(s-τ)|2)ds]≤TLd(T+2∫0t1‍E(sup⁡-τ≤ν≤s∧ρd|y(ν)|2)ds).
Then by the Doob martingale inequality and (8), we have
(34)E[sup⁡0≤t≤t1|∫0t∧ρd‍g(y~(s),y~(s-τ))dW(s)|2]≤4E[∫0t1∧ρd‍|g(y~(s),y~(s-τ))|2ds]≤4LdE[∫0t1∧ρd‍(1+|y~(s)|2+|y~(s-τ)|2)ds]≤4Ld(T+2∫0t1‍E(sup⁡-τ≤ν≤s∧ρd|y(ν)|2)ds).
For the jump integral, we can transform to the compensated Poisson process
(35)N~(t)=N(t)-λt,
which is a martingale, and use the isometry
(36)E[|∫ab‍h(y~(s),y~(s-τ))dN~(s)|2]=λ[∫ab‍E|h(y~(s),y~(s-τ))|2ds]
to obtain
(37)E[sup⁡0≤t≤t1|∫0t∧ρd‍h(y~(s),y~(s-τ))dN(s)|2]=E[sup⁡0≤t≤t1|∫0t∧ρd‍h(y~(s),y~(s-τ))dN~(s)iiiiiiiiiiiiiiiiiiiiii+λ∫0t∧ρd‍h(y~(s),y~(s-τ))ds|2]≤2E[sup⁡0≤t≤t1|∫0t∧ρd‍h(y~(s),y~(s-τ))dN~(s)|2]+2λ2E[sup⁡0≤t≤t1|∫0t∧ρd‍h(y~(s),y~(s-τ))ds|2]≤8E[|∫0t1∧ρd‍h(y~(s),y~(s-τ))dN~(s)|2]+2λ2TE[∫0t1∧ρd‍|h(y~(s),y~(s-τ))|2ds]≤(8λ+2λ2T)E[∫0t1∧ρd‍|h(y~(s),y~(s-τ))|2ds]≤(8λ+2λ2T)Ld(T+2∫0t1‍E(sup⁡-τ≤ν≤s∧ρd|y(ν)|2)ds).
Inserting (33)–(37) into (32) gives
(38)E(sup⁡0≤t≤t1|y(t∧ρd)|2)≤6K1E[1+sup⁡0≤t≤t1|y~(t∧ρd-τ)|2]+6E∥ξ∥2+6K1(1+E∥ξ∥2)+6(T+4+8λ+2λ2T)Ld×(T+2∫0t1‍E(sup⁡-τ≤ν≤s∧ρd|y(ν)|2)ds).
Substitute the above inequality into (30), the result is
(39)E(sup⁡-τ≤t≤t1|y(t∧ρd)|2)≤E∥ξ∥2+E(sup⁡0≤t≤t1|y(t∧ρd)|2)≤12K1+(6K1+7)E∥ξ∥2+6K1Esup⁡0≤t≤t1|y~(t∧ρd-τ)|2+6(T+4+8λ+2λ2T)Ld×(T+2∫0t1‍E(sup⁡-τ≤ν≤s∧ρd|y(ν)|2)ds)≤12K1+(6K1+7)E∥ξ∥2+6K1Esup⁡-τ≤t≤t1|y(t∧ρd)|2+6(T+4+8λ+2λ2T)LdT+12(T+4+8λ+2λ2T)Ld×∫0t1‍E(sup⁡-τ≤ν≤s|y(ν∧ρd)|2)ds.
The Gronwall inequality shows that
(40)E(sup⁡-τ≤t≤T|y(t∧ρd)|2)≤(12K1+(6K1+7)E∥ξ∥2iiiiiiiiii+6(T+4+8λ+2λ2T)LdT)×(1-6K1)-1×e12(T+4+8λ+2λ2T)LdT/(1-6K1)≡C1.
This is the desired result.Corollary 4.
Under (H1)-(H2),(41)E|y(t)|2≤C2,∀t∈[-τ,T∧ρd],
where C2 is a positive constant independent of Δ.Lemma 5.
Under (H1)-(H2), for anyt∈[0,T], there exists a constant C3>0 independent of Δ, such that
(42)∫0t∧ρd‍E|y(s)-y~(s)|2ds≤C3Δ.Proof.
For anyt∈[0,T∧ρd], there exists an integer k such that t∈[tk,tk+1). By the definition of y(t) and y~(t), we have
(43)y(t)=G(y~(t-τ))+ξ(0)-G(ξ(-τ))+∫0t‍f(y~(s),y~(s-τ))ds+∫0t‍g(y~(s),y~(s-τ))dW(t)+∫0t‍h(y~(s),y~(s-τ))dN(t),y~(t)=G(yk-m)+ξ(0)-G(ξ(-τ))+∫0kΔ‍f(y~(s),y~(s-τ))ds+∫0kΔ‍g(y~(s),y~(s-τ))dW(t)+∫0kΔ‍h(y~(s),y~(s-τ))dN(t).
It is clear that y~(t)=yk and y~(t-τ)=yk-m, and thus
(44)y(t)-y~(t)=∫kΔt‍f(y~(s),y~(s-τ))ds+∫kΔt‍g(y~(s),y~(s-τ))dW(t)+∫kΔt‍h(y~(s),y~(s-τ))dN(t)=f(y~(s),y~(s-τ))(t-tk)+g(y~(s),y~(s-τ))(W(t)-W(tk))+h(y~(s),y~(s-τ))(N(t)-N(tk)).
Then applying the inequality (a+b+c)2≤3(|a|2+|b|2+|c|2) and (8), we have
(45)|y(t)-y~(t)|2≤3|f(y~(s),y~(s-τ))|2Δ2+3|g(y~(s),y~(s-τ))|2|W(t)-W(tk)|2+3|h(y~(s),y~(s-τ))|2|N(t)-N(tk)|2≤3Ld(1+|y~(t)|2+|y~(t-τ)|2)×(Δ2+|W(t)-W(tk)|2+|N(t)-N(tk)|2).
Then by Corollary 4 and the Lyapunov inequality [22]
(46)E∫0t∧ρd‍|y(s)-y~(s)|2ds≤3LdE∫0t∧ρd‍(1+|y~(t)|2+|y~(t-τ)|2)×(Δ2+|W(t)-W(tk)|2+|N(t)-N(tk)|2)ds≤3Ld∫0t∧ρd‍E(1+|y~(t)|2+|y~(t-τ)|2)E×(Δ2+|W(t)-W(tk)|2+|N(t)-N(tk)|2)ds≤3LdE∫0t∧ρd‍(1+|y~(t)|2+|y~(t-τ)|2)×(Δ2+mΔ+λΔ)ds≤3LdT(1+2C2)(Δ+m+λ)Δ≤C3Δ.
Let C3=3LdT(1+2C2)(1+m+λ); the lemma is therefore complete.


## 4. Main Result

Theorem 6.
Under assumptions (H1)-(H2), the Euler-Maruyama approximate solution (4) converges to the exact solution of the NSDDEwPJs (1) in the sense that
(47)lim⁡Δ→0E[sup⁡0≤t≤T|x(t)-y(t)|2]=0.Proof.
Lete(t)=x(t)-y(t); it is easy to see that
(48)E[sup⁡0≤t≤T|e(t)|2]=E[sup⁡0≤t≤T|e(t)|2I{σd>T,νd>T}]+E[sup⁡0≤t≤T|e(t)|2I{σd≤Torνd≤T}]=E[sup⁡0≤t≤T|e(t)|2I{ρd>T}]+E[sup⁡0≤t≤T|e(t)|2I{σd≤Torνd≤T}]≤E[sup⁡0≤t≤T|e(t∧ρd)|2]+E[sup⁡0≤t≤T|e(t)|2I{σd≤Torνd≤T}],
where IA is the indicator function of set A.
Recall the Young inequality, for1/p+1/q=1(p,q>0); we have
(49)ab≤aδ1/pbδ1/p≤(aδ1/p)pp+bqqδq/p=apδp+bqqδq/p,∀a,b,δ>0.
Thus for any δ>0, we have
(50)E[sup⁡0≤t≤T|e(t)|2I{σd≤Torνd≤T}]≤2δpE[sup⁡0≤t≤T|e(t)|p]+1-2/pδ2/(P-2)P{σd≤Torνd≤T}.
By Lemma 2, then
(51)P{σd≤T}=E[I{σd≤T}|y(σd)|pdp]≤1dPE[sup⁡0≤t≤T|y(t)|p]≤Adp.
Similarly, the result can be derived for νd as
(52)P{νd≤T}≤Adp,
so that
(53)P{σd≤Torνd≤T}≤P{σd≤T}+P{νd≤T}≤2Adp.
Using these bounds along with
(54)E[sup⁡0≤t≤T|e(t)|p]≤2p-1(E[sup⁡0≤t≤T|x(t)|p]+E[sup⁡0≤t≤T|y(t)|p])≤2pA
in (50) gives
(55)E[sup⁡0≤t≤T|e(t)|2I{σd≤Torνd≤T}]≤2p+1δAp+2(P-2)APδ2/(P-2)dp.
Now we bound the first term on the right-hand side of (48). By the definition of x(t) and y(t), we have
(56)x(t∧ρd)-y(t∧ρd)=G(x(t∧ρd-τ))-G(y~(t∧ρd-τ))+∫0t∧ρd‍f(x(s),x(s-τ))-f(y~(s),y~(s-τ))ds+∫0t∧ρd‍g(x(s),x(s-τ))-g(y~(s),y~(s-τ))dW(s)+∫0t∧ρd‍h(x(s),x(s-τ))-h(y~(s),y~(s-τ))dN(s).
For ease of exposition, we abbreviate
(57)G~(s)=G(x(s-τ))-G(y~(s-τ)),f~(s)=f(x(s),x(s-τ))-f(y~(s),y~(s-τ)),g~(s)=g(x(s),x(s-τ))-g(y~(s),y~(s-τ)),h~(s)=h(x(s),x(s-τ))-h(y~(s),y~(s-τ)).
Thus, for any t∈[0,T],
(58)Esup⁡0≤t≤t1|x(t∧ρd)-y(t∧ρd)|2≤4Esup⁡0≤t≤t1|G~(t)|2+4Esup⁡0≤t≤t1|∫0t∧ρd‍f~(s)ds|2+4Esup⁡0≤t≤t1|∫0t∧ρd‍g~(s)dW(s)|2+4Esup⁡0≤t≤t1|∫0t∧ρd‍h~(s)dN(s)|2.
By the process of Lemma 5 and (45), we can get
(59)Esup⁡0≤t≤t1|y(t∧ρd)-y~(t∧ρd)|2≤3Ld(1+2C2)(Δ2+mΔ+λΔ).
By condition (H2) and (59), we have
(60)Esup⁡0≤t≤t1|G(x(t∧ρd-τ))-G(y~(t∧ρd-τ))|2≤2Esup⁡0≤t≤t1|G(x(t∧ρd-τ))-G(y(t∧ρd-τ))|2+2Esup⁡0≤t≤t1|G(y(t∧ρd-τ))-G(y~(t∧ρd-τ))|2≤2K2Esup⁡0≤t≤t1|x(t∧ρd-τ)-y(t∧ρd-τ)|2+2K2Esup⁡0≤t≤t1|y(t∧ρd-τ)-y~(t∧ρd-τ)|2≤2K2Esup⁡0≤t≤t1|x(t∧ρd-τ)-y(t∧ρd-τ)|2+6K2Ld(1+2C2)(Δ2+mΔ+λΔ)≤2K2Esup⁡0≤t≤t1|x(t∧ρd)-y(t∧ρd)|2+6K2Ld(1+2C2)(Δ2+mΔ+λΔ).
By Cauchy-Schwarz inequality and condition (H1) and Lemma 5, we get
(61)Esup⁡0≤t≤t1|∫0t∧ρd‍f(x(s),x(s-τ))iiiiiiiiiiiiiiiiiiii∫0t∧ρd‍-f(y~(s),y~(s-τ))ds|2≤TE∫0t1∧ρd‍|f(x(s),x(s-τ))-f(y~(s),y~(s-τ))|2ds≤TCdE×∫0t1∧ρd‍(|x(s)-y~(s)|2+|x(s-τ)-y~(s-τ)|2)ds≤4TCdE∫0t1∧ρd‍(|x(s)-y(s)|2+|y(s)-y~(s)|2)ds≤4TCd∫0t1‍E[sup⁡0≤ν≤s|x(ν∧ρd)-y(ν∧ρd)|2]ds+4TCdC3Δ.
Similarly, by Burkholder-Davis-Gundy inequality and (8), we have
(62)Esup⁡0≤t≤t1|∫0t∧ρd‍g(x(s),x(s-τ))iiiiiiiiiiiiiiiiiii∫0t∧ρd‍-g(y~(s),y~(s-τ))dW(s)|2≤4E∫0t1∧ρd‍|g(x(s),x(s-τ))-g(y~(s),y~(s-τ))|2ds≤4CdE×∫0t1∧ρd‍(|x(s)-y~(s)|2+|x(s-τ)-y~(s-τ)|2)ds≤16CdE∫0t1∧ρd‍(|x(s)-y(s)|2+|y(s)-y~(s)|2)ds≤16Cd∫0t1‍E[sup⁡0≤ν≤s|x(ν∧ρd)-y(ν∧ρd)|2]ds+16CdC3Δ.
Then by (37) and condition (H1) and Lemma 5, we get
(63)Esup⁡0≤t≤t1|∫0t∧ρd‍h~(x(s),x(s-τ))dN(s)|2≤(8λ+2λ2T)E[∫0t1∧ρd‍|h~(y~(s),y~(s-τ))|2ds]=(8λ+2λ2T)E×[∫0t1∧ρd‍|h(x(s),x(s-τ))-h(y~(s),y~(s-τ))|2ds]≤(8λ+2λ2T)CdE×∫0t1∧ρd‍(|x(s)-y~(s)|2+|x(s-τ)-y~(s-τ)|2)ds≤4(8λ+2λ2T)CdE×∫0t1∧ρd‍(|x(s)-y(s)|2+|y(s)-y~(s)|2)ds≤4(8λ+2λ2T)Cd×∫0t1‍E[sup⁡0≤ν≤s|x(ν∧ρd)-y(ν∧ρd)|2]ds+4(8λ+2λ2T)CdC3Δ.
Taking (60)–(63) into (58), we obtain
(64)Esup⁡0≤t≤t1|x(t∧ρd)-y(t∧ρd)|2≤8K2Esup⁡0≤t≤t1|x(t∧ρd)-y(t∧ρd)|2+24K2Ld(1+2C2)(Δ2+mΔ+λΔ)+16TCd∫0t1‍E[sup⁡0≤ν≤s|x(ν∧ρd)-y(ν∧ρd)|2]ds+16TCdC3Δ+64Cd∫0t1‍E[sup⁡0≤ν≤s|x(ν∧ρd)-y(ν∧ρd)|2]ds+64CdC3Δ+16(8λ+2λ2T)Cd×∫0t1‍E[sup⁡0≤ν≤s|x(ν∧ρd)-y(ν∧ρd)|2]ds+16(8λ+2λ2T)CdC3Δ.
Therefore
(65)Esup⁡0≤t≤t1|e(t∧ρd)|2≤((24K2Ld(1+2C2)(Δ+m+λ)(1-8K2)-1iiiiiiiiiii+[16T+64+16(8λ+2λ2T)]CdC3)iiiiiiiii×(1-8K2)-1)Δ+[16T+64+16(8λ+2λ2T)]Cd1-8K2×∫0t1‍E[sup⁡0≤ν≤s|x(ν∧ρd)-y(ν∧ρd)|2]ds.
That is, for K2<1/8(66)Esup⁡0≤t≤T|e(t∧ρd)|2≤C4e[16T+64+16(8λ+2λ2T)]TCd/(1-8K2)Δ,
where
(67)C4=(24K2Ld(1+2C2)(Δ+m+λ)ii+[16T+64+16(8λ+2λ2T)]CdC3)×(1-8K2)-1.
Taking (55) and (66) into (48), we have
(68)E[sup⁡0≤t≤T|e(t)|2]≤E[sup⁡0≤t≤T|e(t∧ρd)|2]+E[sup⁡0≤t≤T|e(t)|2I{σd<Torνd<T}]≤C4e[16T+64+16(8λ+2λ2T)]TCd/(1-8K2)Δ+2p+1δAp+2(P-2)APδ2/(P-2)dp.
Given any ɛ>0, we can choose δ sufficiently small for
(69)2p+1δAp<ɛ3,
and then choose d sufficient large for
(70)2(P-2)APδ2/(P-2)dp<ɛ3,
and finally choose Δ so that
(71)C4e[16T+64+16(8λ+2λ2T)]TCd/(1-8K2)Δ<ɛ3.
Thus, E[sup⁡0≤t≤T⁡|e(t)|2]<ɛ. The proof is completed.


## 5. Numerical Examples

In this section, we present some numerical examples in support of our previous theoretical results.Example 1.
First we consider the following linear NSDDEwPJs:(72)d[x(t)-0.5x(t-1)]=[-9x(t)+7x(t-1)]dt+x(t-1)dW(t)+0.5x(t-1)dN(t),t∈[0,T],x(t)=t+1,t∈[-1,0].First, we illustrate the strong convergence of the Euler-Maruyama method for NSDDEs with Poisson jumps. We chooseT=10 and λ=10. To the best of our knowledge, there are not any analytical solutions available for Example 1. Therefore, we use the Euler-Maruyama method to compute an “explicit solution” with step-size Δ=2-10 in our experiments. We draw the numerical solution obtained from the Euler-Maruyama method with step-size Δ=2-6 in Figure 1. The data used in the figure is obtained by the mean square of data by 1000 trajectories; that is ωi:1≤i≤1000; yn=1/1000∑i=11000‍|yn(ωi)|2; tn denotes the mesh point. From the figure, we can see that the exact solutions and numerical solutions match well.Figure 1
The exact solution and the numerical method approximations for Example1.To show the strong convergence order of the Euler-Maruyama method, we apply the Euler-Maruyama method to Example1. Then simulating the numerical solutions with 5 different step-sizes h=2p-1Δ for 1≤p≤5, Δ=2-14. The mean-square errors ɛ=1/1000∑i=11000‍|yn(ωi)-x(T)|2 all measured at time T=10 are estimated by trajectory averaging. We plot our approximation to ε against Δ on a log-log scale in Figure 2. For reference a dashed line of slope one-half is added. It clearly shows that the Euler-Maruyama method for Example 1 is convergent with order 1/2.Figure 2
The convergence rate of the Euler-Maruyama method for Example1.Example 2.
Consider the following nonlinear NSDDEwPJs:(73)d[x(t)-0.5x(t-τ)]=[-12x3(t)+10x(t-1)]dt+x(t-1)dW(t)+0.1x(t-1)dN(t),t∈[0,T],x(t)=t+1,t∈[-τ,0].We fixT=20 and λ=1 for Example 2. The same as Figure 1, the data used in Figure 3 is obtained by the mean square of data by 1000 trajectories. In Figure 3, we show the numerical simulation of Example 2 by Euler-Maruyama method at step-sizes Δ=2-6 (upper), Δ=2-8 (middle), Δ=2-10 (lower).Figure 3
The Euler-Maruyama method approximations for Example2 with step-sizes Δ=2-6 (upper), Δ=2-8 (middle), and Δ=2-10 (lower).Figure4 also illustrates that the Euler-Maruyama method for Example 2 is convergent with order 1/2.Figure 4
The convergence rate of the Euler-Maruyama method for Example2.



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*Source: 976183-2014-06-04.xml*