Mean-Square Convergence of Drift-Implicit One-Step Methods for Neutral Stochastic Delay Differential Equations with Jump Diffusion

A class of drift-implicit one-step schemes are proposed for the neutral stochastic delay differential equations NSDDEs driven by Poisson processes. A general framework for mean-square convergence of the methods is provided. It is shown that under certain conditions global error estimates for a method can be inferred from estimates on its local error. The applicability of the mean-square convergence theory is illustrated by the stochastic θ-methods and the balanced implicit methods. It is derived from Theorem 3.1 that the order of the mean-square convergence of both of them for NSDDEs with jumps is 1/2. Numerical experiments illustrate the theoretical results. It is worth noting that the results of mean-square convergence of the stochastic θ-methods and the balanced implicit methods are also new.


Introduction
In stochastic numerical analysis, the order of convergence plays a crucial role in the design of numerical algorithms.Unlike in the deterministic modelling situation, there exist in the stochastic environment different types of convergence.Both in the literature and in practice, most attention is focused on two major types of convergence, that is, strong convergence and weak convergence.There is a rich literature on this subject; we here only mention 1-4 and the references therein.
For stochastic differential equations SDEs , Milstein 1 presented a fundamental convergence theorem which established the order of mean-square convergence of explicit one-step methods.The conditions of this theorem use both properties of mean and meansquare deviation of one-step approximation.The theorem showed that under certain conditions global error estimates for a method can be inferred from estimates on its local error.Buckwar 4 extended the convergence theory in 1 to stochastic functional differential equations.Recently, Zhang and Gan 5 extend the theory to neutral stochastic differential delay equations NSDDEs .Therefore, the convergence theory in 1 and its generalization have received some attention in the case of nonjump SDEs.However, in the jump-SDE context, which is becoming increasingly important in mathematical finance 6-8 , to our best knowledge, no corresponding convergence theory of numerical methods for NSDDEs with jumps has been presented in the literature.Motivated by the work of Zhang and Gan 5 , our aim is to establish a relationship between the consistent order and the convergence order of the methods for the NSDDEs with jumps.
In this paper, a class of drift-implicit one-step schemes are proposed for NSDDEs driven by Poisson processes.A general framework for mean-square convergence of the methods is provided.It is shown that under certain conditions global error estimates for a method can be inferred from estimates on its local error.The applicability of the theory about mean-square convergence is illustrated by the stochastic θ-methods and the balanced implicit methods.It is derived from Theorem 3.5 that the order of the mean square convergence of both of them for NSDDEs with jumps is 12.It is worth noting that the results of mean-square convergence of the stochastic θ-methods and the balanced implicit methods are also new.

Neutral Stochastic Delay Differential Equations with Jumps
Let Ω, F, {F t } t≥0 , P be a complete probability space with a filtration {F t } t≥0 satisfying the usual conditions i.e., it is right continuous and F 0 contains all the P-null sets .Let W t : W 1 t , . . ., W b t T be a b-dimensional Wiener process, and N t is a scalar Poisson process with intensity λ λ > 0 , both defined on the space Ω, F, {F t } t 0 , P .And | • | is used to denote both the norm in R d and the trace norm F-norm in R d×b .Also, C t 1 , t 2 ; R d is used to represent the family of continuous mappings ψ from t 1 , t 2 to R d .Finally, L p Consider the neutral stochastic delay differential equations NSDDEs with Poissondriven jumps where By the definition of It ô-interpreted stochastic differential equations, the integral version of 2.1 is expressed as follows: ii iii x t ψ t , −τ ≤ t ≤ 0, and 2.3 holds for every t ∈ 0, T with probability 1, where L p 0, T ; R d denotes the family of Borel measurable functions : 0, In order to guarantee the existence and uniqueness of the solution, we impose the following hypothesis.
Assumption 1 global Lipschitz condition .There exists a positive constant K such that for all

2.5
Assumption 2 linear growth condition .There exists a positive constant L such that for all

2.6
Assumption 3.There is a constant η ∈ 0, 1 such that for all Remark 2.2.In this paper, we always assume that η / 0. Otherwise, the system 2.1 reduces to the stochastic delay differential equations with jumps.
We use J 1 , J 2 , . . . to denote the constants which are independent of the stepsize h.
, where the constant H depends on constants H, T , initial function ψ t , and positive integer s.
Lemma 2.4 is a modified version of 10, Lemma 2.1 .In a similar way, it is not hard to obtain the estimate 2.10 .
In this paper, we will use the following inequality.For any a, c > 0 and 0 < α < 1, we have

Implicit One-Step Schemes
Define a mesh with uniform step h which satisfies τ mh for an integer number m for convenience, we assume that m 2 , and suppose that q is a positive integer with q T/h.Let t n nh, n 0, 1, . . ., q.The drift-implicit one-step methods for the simulation of the solution x t of 2.1 are defined as follows: where Y n is an approximation of the exact solution x t n , and Φ v v f, g, and u are increment functions.ΔW n W t n 1 − W t n , ΔN n N t n 1 − N t n , and ΔW n is independent of ΔN n .Y n−m ψ t n − τ when n − m 0. Remark 3.1.Now, we discuss the solvability of 3.1 .If increment function Φ f does not depend on Y n 1 , it is not difficult to find that the approximations Y n can be computed iteratively.If Φ f depends on Y n 1 , in order to guarantee the existence and uniqueness of a solution, the general approach is to assume Lipschitz continuity of Φ f with respect to Y n 1 with the Lipschitz constant less than 1, and then to apply Banach's contraction mapping principle 4 .
We denote by Y t n 1 the value that is obtained when the exact solution values are inserted into the right-hand side of 3.1 , that is,

3.2
We introduce the following definitions, which are presented in the literature, see 1, 3 , for example.Definition 3.2.The local error of method 3.1 is the sequence of random variables The global error of method 3.1 is the sequence of random variables Definition 3.3.The numerical method 3.1 is said to be consistent with order p 1 in the mean and with order p 2 in the mean square sense if the following estimates hold with p 2 1/2 and where the constants H 0 and H 1 do not depend on h but may depend on T and the initial data.Here, z L 2 : E|z| 2 1/2 .Definition 3.4.The numerical method 3.1 is said to be convergent with order p in the mean square sense, on the mesh points, if the following estimate holds: where the constant H 2 does not depend on h but may depend on T and the initial data.
In order to obtain the main result, the following properties of the increment functions are required.There exist positive constants L f such that for and there exist the positive constants L g , L u , L u such that for all

3.11
Now, we state our result on the convergence of the one-step method 3.1 .
Theorem 3.5.Suppose that Assumption 3 and the conditions 3.7 -3.11 hold.Assume that the one-step method 3.1 is consistent with order p 1 in the mean and order p 2 in the mean square sense, then the method 3.1 is convergent with order p p 2 − 1/2 in the mean square sense.
Proof.By 3.4 , we have where ε n 1 is defined as follows:

3.13
Squaring and taking expectation on both sides of 3.12 , using Assumption 3 and 2.11 , we have

3.16
Squaring and taking expectation on both sides of 3.15 yields 3.17 We will now estimate the separate terms in 3.17 individually.Without loss of generality, we can assume that 0 < h < 1.We notice that the method 3.1 is consistent with order p 2 in the mean square sense; thus, there exists a constant J 2 such that

3.20
where Since method 3.1 is consistent with order p 1 in the mean square, there exists a constant J 4 such that

3.21
Noticing that ε n is F t n -measurable and by 3.21 , we arrive at

3.23
Here, the fact used has been that Using 3.14 and 3.23 , we arrive at

3.25
The following proof is analogous to that of Theorem 3.1 in 5 ; thus, it is not hard to derive the convergence result.The proof is completed.
Remark 3.6.Notice that if u • 0 in 2.1 , then 2.1 reduces to the NSDDEs without jumps, our Theorem 3.5 coincides with Theorem 3.1 in 5 , that is to say, Theorem 3.5 generalizes Theorem 3.1 in 5 to the case of NSDDEs with jumps.

The Examples
Theorem 3.5 presents the convergence result about the general implicit one-step methods for NSDDEs with jumps.In this section, we discuss the applicability of the theory presented in the previous section.We will give the convergence orders of the stochastic θ-methods and the balanced implicit methods.
where the compensated Poisson process N t N t − λt satisfies Hence, the stochastic θ-methods 4.1 are consistent with order 3/2 in the mean and order 1 in the mean square sense.The proof is completed.Proof.Lemma 4.2 shows that the stochastic θ-methods 4.1 are consistent with order p 1 3/2 in the mean and order p 2 1 in the mean square sense.In order to prove that the stochastic θ-methods 4.1 are convergent with order p 1/2 in the mean square sense, by Theorem 3.5, we only need to verify the conditions 3.7 -3.11 .
From 4.1 , we find that

4.7
For the random variables x i , y i ∈ R d i 1, 2, 3, 4 , by Assumption 1, we have x i − y i .

4.10
From 4.8 -4.10 , we see that the increment functions of the stochastic θ-methods 4.1 satisfy the inequalities 3.7 -3.9 with L f √ K, L g bK, and L u λ 1 λT K. Noting that ΔW n , ΔN n are independent of F t n and x 1 , x 3 , y 1 , and y 3 are F t n -measurable, then using

4.11
Here, the fact used has been that EΔW n 0 and EΔN n λh.From above, we find that the increment functions of the stochastic θ-methods 4.1 satisfy the estimations 3.10 and 3.11 with L u λ √ K. Therefore, the conditions 3.7 -3.11 are satisfied.By Lemma 4.2 and Theorem 3.5, it is not difficult to find that the stochastic θ-methods 4.1 are convergent with order p 1/2 in the mean square sense.The proof is completed.Remark 4.4.For the case of G • 0, 2.1 reduces to the stochastic delay differential equations with jumps where ψ ∈ L 2 F 0 −τ, 0 ; R d .Theorem 4.3 implies that the stochastic θ-methods for 4.12 are convergent with order p 1/2, which coincides with Theorem 3.2 in 11 .
Example 4.5.Consider the balanced implicit methods for system 2.1 ,

4.13
Here, the are, in general matrix, called control functions which are often chosen as constants.
Furthermore, the control functions must satisfy some conditions.
Assumption 4. The C 0 x, y and C 1 x, y represent bounded d × d-matrix-valued functions for x, y ∈ R d .For any real numbers α 0 ∈ 0, α , α 1 ≥ 0, where α ≥ h for all step sizes h considered and x, y ∈ R d , the matrix M x, y I α 0 C 0 x, y α 1 C 1 x, y has an inverse and satisfies the condition

4.15
Discrete Dynamics in Nature and Society Here, I is the unit matrix, and H is a positive constant.Notice that if C 0n , C 1n satisfy Assumption 4, the methods 4.13 are well defined and can be rewritten as
For simplicity, from now on, we suppose that C 0n , C 1n in 4.14 are constants, that is to say, C 0n C 0 , C 1n C 1 .
Lemma 4.7.Let Assumptions 1, 2, and 4 hold, then the balanced implicit methods 4.13 are consistent with order p 1 3/2 in the mean and order p 2 1 in the mean square sense.
Proof.Without loss of generality, we can assume that 0 < h < 1.From Lemma 4.2, we know that the Euler-Maruyama method is consistent with order 3/2 in the mean, thus using 4.4 , we have 4.17 where Y E t n 1 and Y B t n 1 are defined as follows:

4.21
It follows from Assumption 2, Lemma 2.3, and 4.21 that

4.23
which implies that the balanced implicit methods 4.13 are consistent with order 3/2 in the mean.In the following, we will show that the balanced implicit methods 4.13 are consistent with order 1 in the mean square sense.Using Assumptions 2 and 4, Lemma 2.3, n λh 1 λh , and 4.19 , we compute that

4.24
Theorem 4.3 implies that the Euler-Maruyama method is convergent with order p 1/2.Thus, by 4.5 and 4.24 , we have

4.25
The proof is completed.

4.33
From Theorem 4.8, it is not difficult to find that the balanced implicit methods for 4.33 are convergent with order p 1/2 in the mean square sense, which coincides with Theorem 2.1 in 13 .To show the convergence of the θ-methods 4.1 and the balanced implicit methods 4.16 , we choose θ 0.5, C 0 1, and C 1 0.5.In all the numerical experiments, we identify the numerical solution using very small stepsize h Δt as the exact solution and compare this with the numerical approximations using h 4Δt, 8Δt, 16Δt, 32Δt, 64Δt for Δt 2 −14 over 2000 different discretized Brownian paths.The mean-square errors ε r ε 2 1 r Δt , r 1, 2, 3, 4, 5, all measured at time T 1, are estimated by trajectory averaging, that is,

Theorem 4 . 3 .
Let Assumption 1 hold, then the stochastic θ-methods 4.1 are convergent with order p 1/2 in the mean square sense.
Definition 2.1 see 9 .An R d -valued stochastic process x t on −τ ≤ t ≤ T is called a solution to 2.1 with initial data 2.2 if it has the following properties: i {x t } 0≤t≤T is continuous and F t -adapted; t n , x t n − τ h g x t n , x t n − τ ΔW n u x t n , x t n − τ ΔN n , 4.18 where C n C 0 h C 1 |ΔW n |.By 4.18 , we have• f x t n , x t n − τ h g x t n , x t n − τ ΔW n u x t n , x t n −τ ΔN n .4.19Noticing that x t n , x t n − τ are F t n -measurable, ΔW n is independent of F t n , and using Lemma 4.6, we find thatE I C n −1 • C n • g x t n , x t n − τ ΔW n | F t n 0.4.20Wenotice that C 0 and C 1 are constants; thus, there exists a positive constant B, such that|C i | ≤ B i 0,1 .Since ΔW n , ΔN n are independent of F t n , by Assumption 4, 4.19 , 4.20 , |E x | F | ≤ E |x| | F , E|ΔW n | ≤ √ bh, and EΔN n λh, we obtain Theorem 4.8.Let Assumptions 1-4 hold, then the balanced implicit methods 4.16 are convergent with order p 1/2 in the mean square sense.Proof.By 4.16 , the increment functions Φ f , Φ g , and Φ u of the balanced implicit methods 4.16 are given as follows: 3, 4 , by Assumptions 1-4, 4.26 , we arrive at λh , the random variables x 1 , x 3 , y 1 , and y 3 are F t n -measurable and using Assumptions 1-4, 4.27 , we obtainE Φ g h, x 1 , x 3 , ΔW n , ΔN n − Φ g h, y 1 , y 3 , ΔW n , ΔN n 2 E I C n −1 g x 1 , x 3 ΔW n − I C n −1 g y 1 , y 3 ΔW n Φ u h, x 1 , x 3 , ΔW n , ΔN n − Φ u h, y 1 , y 3 , ΔW n , ΔN n 2 E I C n −1 u x 1 , x 3 ΔN n − I C n −1 u y 1 , y 3 ΔN n 2 ≤ H 2 Kλ 1 λT h E x 1 − y 1 Since ΔW n ,ΔN n are independent of F t n and the random variables x 1 , x 3 , y 1 , and y 3 are F t nmeasurable; thus, by the inequality |E x | F | ≤ E |x| | F , EΔN n λh, and Lemma 4.6, we compute thatE Φ g h, x 1 , x 3 , ΔW n , ΔN n − Φ g h, y 1 , y 3 , ΔW n , ΔN n | F t n E I C n −1 g x 1 , x 3 ΔW n | F t n − E I C n −1 g y 1 , y 3 ΔW n | F t n Φ u h, x 1 , x 3 , ΔW n , ΔN n − Φ u h, y 1 , y 3 , ΔW n , ΔN n | F t n ≤ E I C n −1 u x 1 , x 3 ΔN n − I C n −1 u y 1 , y 3 ΔN n | F t n ≤ u x 1 , x 3 − u y 1 , y 3 • E I C n −1 • ΔN n | F tn From 4.28 -4.32 , we see that the increment functions of the balanced implicit methods 4.13 satisfy the conditions 3.7 -3.11 with L f H √ K, L g H 2 Kb, L u H 2 Kλ 1 λT , and L u H √ Kλ.A combination of Lemma 4.7 and Theorem 3.5 leads to the conclusion that the balanced implicit methods 4.13 are convergent with order p 1/2 in the mean square sense.The proof is completed.
In this section, several numerical examples are given to illustrate our theoretical results in the previous sections.Consider the nonlinear equation