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

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 proposedmethod 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.


Introduction
Neutral stochastic delay differential equations (NSDDEs) have recently been studied intensively by, for instance, Kolmanovskii et al. [1,2], Mao et al. [3][4][5][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 Section 2 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.
The stochastic integral is defined in the Itô sense, and the integral version of (1) is frequently expressed as We can now define the Euler-Maruyama approximate solution for NSDDEwPJs.
In this paper, the following hypotheses are imposed on (1).

(H1) The Local Lipschitz Condition.
There is a positive constant   such that, for all where From ( 6), we obtain where 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 () on  ≥ −.

Lemmas and Corollaries
In this section, we first establish a few lemmas under local Lipschitz condition.For each  > 0, define the stopping times (As usual we set inf 0 = ∞.) Lemma 2. Under (H1)-(H2), for any  ≥ 2, there exists a constant  > 0 independent of Δ, such that Proof.First, we prove the -moment of the exact solution of (1) is finite.For any  1 ∈ [0, ], we obtain From (2) and inequality we get By condition (H2) and inequality ( + )  ≤ 2 −1 (||  + ||  ), then for any  ∈ [0, ], we have where Similarly By the Hölder inequality and (8), then for any  1 ∈ [0, ], we get Now, we use  to denote that a generic constant may change between occurrences.Using the Burkholder-Davis-Gundy inequality for the two martingale integral terms, we have Similarly, for the jump integral term, we have By the above inequalities, we can obtain that is, where The Gronwall inequality shows that Then we can prove in the same way that the Euler-Maruyama approximate solution to (1) has the property that So, we can obtain by letting  =  1 ∨  2 .

𝐸 ( sup
where  1 is a positive constant independent of Δ.
For the jump integral, we can transform to the compensated Poisson process which is a martingale, and use the isometry The Gronwall inequality shows that This is the desired result.

Main Result
where   is the indicator function of set .
Recall the Young inequality, for 1/ + 1/ = 1 (,  > 0); we have Thus for any  > 0, we have By Lemma 2, then Similarly, the result can be derived for ]  as so that Using these bounds along with in (50) gives Now we bound the first term on the right-hand side of (48).By the definition of () and (), we have For ease of exposition, we abbreviate Thus, for any  ∈ [0, ],  sup By the process of Lemma Taking ( 60)-( 63) into (58), we obtain where Taking ( 55) and ( 66) into (48), we have Given any  > 0, we can choose  sufficiently small for and then choose  sufficient large for and finally choose Δ so that Thus, [sup 0≤≤ |()| 2 ] < .The proof is completed.

Numerical Examples
In this section, we present some numerical examples in support of our previous theoretical results.First, we illustrate the strong convergence of the Euler-Maruyama method for NSDDEs with Poisson jumps.We choose  = 10 and  = 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   : 1 ≤  ≤ 1000;   = 1/1000 ∑ 1000 =1 |  (  )| 2 ;   denotes the mesh point.From the figure, we can see that the exact solutions and numerical solutions match well.
To show the strong convergence order of the Euler-Maruyama method, we apply the Euler-Maruyama method to Example 1. Then simulating the numerical solutions with   We fix  = 20 and  = 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 4 also illustrates that the Euler-Maruyama method for Example 2 is convergent with order 1/2.

Example 1 .Figure 1 :
Figure 1: The exact solution and the numerical method approximations for Example 1.

Figure 2 :
Figure 2: The convergence rate of the Euler-Maruyama method for Example 1.

Figure 4 :
Figure 4: The convergence rate of the Euler-Maruyama method for Example 2.