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

Abstract

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.

1. 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 mean-square 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 1⁄2. It is worth noting that the results of mean-square convergence of the stochastic 𝜃-methods and the balanced implicit methods are also new.

2. 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 ℱ₀ contains all the ℙ-null sets). Let 𝑊(𝑡)∶=(𝑊₁(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([𝑡₁,𝑡₂];ℝ^𝑑) is used to represent the family of continuous mappings 𝜓 from [𝑡₁,𝑡₂] to ℝ^𝑑. Finally, 𝐿^𝑝_ℱ𝑡([𝑡₁,𝑡₂];ℝ^𝑑) is used to denote a family of ℱ_𝑡-measurable, 𝐶([𝑡₁,𝑡₂];ℝ^𝑑)-valued random variables 𝜓={𝜓(𝑢)∶𝑡₁≤𝑢≤𝑡₂} such that ‖𝜓‖^𝑝_𝔼∶=sup_{𝑡1≤𝑢≤𝑡2}𝐸|𝜓(𝑢)|^𝑝<∞. 𝔼 denote mathematical expectation with respect to ℙ.

Consider the neutral stochastic delay differential equations (NSDDEs) with Poisson-driven jumps d[𝑥(𝑡)−𝐺(𝑥(𝑡−𝜏))]=𝑓(𝑥(𝑡⁻),𝑥(𝑡⁻−𝜏))d𝑡+𝑔(𝑥(𝑡⁻),𝑥(𝑡⁻(𝑡)(𝑥(𝑡−𝜏))d𝑊+𝑢⁻),𝑥(𝑡⁻−𝜏))d𝑁(𝑡),𝑡∈[0,𝑇], (2.1) with initial data 𝑥(𝑡)=𝜓(𝑡),𝑡∈[−𝜏,0], (2.2) where 𝜓(𝑡)∈𝐿²_ℱ0([−𝜏,0];ℝ^𝑑). Here, 𝜏>0 is a constant, 𝑥(𝑡⁻) denotes lim_{𝑠→𝑡−}𝑥(𝑠), 𝑓∶ℝ^𝑑×ℝ^𝑑→ℝ^𝑑, 𝑔∶ℝ^𝑑×ℝ^𝑑→ℝ^𝑑×𝑏, 𝑢∶ℝ^𝑑×ℝ^𝑑→ℝ^𝑑, and 𝐺∶ℝ^𝑑→ℝ^𝑑.

By the definition of Itô-interpreted stochastic differential equations, the integral version of (2.1) is expressed as follows:𝑥(𝑡)=𝐺(𝑥(𝑡−𝜏))+𝑥(0)−𝐺(𝑥(−𝜏))+^𝑡₀𝑓(𝑥(𝑠⁻),𝑥(𝑠⁻+−𝜏))𝑑𝑠_𝑡0𝑔(𝑥(𝑠⁻),𝑥(𝑠⁻−𝜏))𝑑𝑊(𝑠)+^𝑡₀𝑢(𝑥(𝑠⁻),𝑥(𝑠⁻−𝜏))𝑑𝑁(𝑠),𝑡>0. (2.3)

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;(ii){𝑓(𝑥(𝑡),𝑥(𝑡−𝜏))}∈ℒ¹([0,𝑇];ℝ^𝑑), {𝑔(𝑥(𝑡),𝑥(𝑡−𝜏))}∈ℒ²([0,𝑇];ℝ^𝑑×𝑏),{𝑢(𝑥(𝑡),𝑥(𝑡−𝜏))}∈ℒ¹([0,𝑇];ℝ^𝑑);(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 𝜛∶[0,𝑇]→ℝ^𝑑 such that ∫^𝑇₀|𝜛(𝑡)|^𝑝𝑑𝑡<∞ a.s.
A solution {𝑥(𝑡)} is said to be unique if any other solution {𝑥(𝑡)} is indistinguishable from {𝑥(𝑡)}, that is, ℙ{𝑥(𝑡)=𝑥(𝑡)∀𝑡∈[0𝑇]}=1. (2.4)

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 𝑥₁,𝑥₂,𝑦₁,𝑦₂∈ℝ^𝑑, 𝑣𝑥₁,𝑦₁−𝑣𝑥₂,𝑦₂²⩽𝐾𝑥₁−𝑥₂²+𝑦₁−𝑦₂²,𝑣=𝑓,𝑔or𝑢. (2.5)

Assumption 2 (linear growth condition). There exists a positive constant 𝐿 such that for all 𝑥,𝑦∈ℝ^𝑑, 𝑣𝑥,𝑦²⩽𝐿1+|𝑥|²+𝑦²,𝑣=𝑓,𝑔or𝑢. (2.6)

Assumption 3. There is a constant 𝜂∈(0,1) such that for all 𝑥₁,𝑥₂∈ℝ^𝑑𝐺𝑥₁−𝐺𝑥₂≤𝜂𝑥₁−𝑥₂. (2.7)

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 𝐽₁,𝐽₂,… to denote the constants which are independent of the stepsize ℎ.

Lemma 2.3. If Assumptions 1–3 hold, then (2.1) has a unique strong solution 𝑥(𝑡) on 𝑡≥−𝜏, and the solution of (2.1) satisfies 𝔼sup_{−𝜏≤𝑡≤𝑇}|𝑥(𝑡)|²≤𝐽₁, (2.8) where 𝐽₁ is a positive constant which depends on constants 𝑇, 𝜂, 𝐿, and initial function 𝜓(𝑡).

It is not hard to prove Lemma 2.3 in a similar way as the proof of Theorems 6.2.2 and 6.4.5 in [9].

Lemma 2.4. Let Assumptions 2 and 3 hold. Assume that the initial function 𝜓(𝑡) is uniformly Lipschitz 𝐿²-continuous, that is, there is a positive constant 𝐻 such that 𝔼𝜓𝑢₁−𝜓𝑢₂²≤𝐻𝑢₁−𝑢₂,if−𝜏≤𝑢₂<𝑢₁≤0, (2.9) then 𝔼𝑥𝑡₁−𝑥𝑡₂²≤𝐻𝑡₁−𝑡₂, (2.10) for all 0≤𝑡₁<𝑡₂≤𝑇 with 𝑡₁−𝑠𝜏∈[−𝜏,0], 𝑡₂−𝑠𝜏∈[−𝜏,0], 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 (𝑎+𝑐)²≤𝑎²𝛼+𝑐².1−𝛼 (2.11)

3. 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: 𝑌_𝑛+1=𝐺𝑌_{𝑛+1−𝑚}−𝐺𝑌_𝑛−𝑚+𝑌_𝑛+Φ_𝑓ℎ,𝑌_𝑛,𝑌_𝑛+1,𝑌_𝑛−𝑚,𝑌_{𝑛+1−𝑚},Δ𝑊_𝑛,Δ𝑁_𝑛+Φ_𝑔ℎ,𝑌_𝑛,𝑌_𝑛−𝑚,Δ𝑊_𝑛,Δ𝑁_𝑛+Φ_𝑢ℎ,𝑌_𝑛,𝑌_𝑛−𝑚,Δ𝑊_𝑛,Δ𝑁_𝑛,𝑛=0,1,…,𝑞−1, (3.1) 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, 𝑌𝑡_𝑛+1=𝐺𝑥𝑡_𝑛+1−𝜏−𝐺𝑥𝑡_𝑛−𝜏+𝑥𝑡_𝑛+Φ_𝑓ℎ,𝑥𝑡_𝑛,𝑥𝑡_𝑛+1,𝑥𝑡_𝑛−𝜏,𝑥𝑡_𝑛+1−𝜏,Δ𝑊_𝑛,Δ𝑁_𝑛+Φ_𝑔ℎ,𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏,Δ𝑊_𝑛,Δ𝑁_𝑛+Φ_𝑢ℎ,𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏,Δ𝑊_𝑛,Δ𝑁_𝑛,𝑛=0,1,…,𝑞−1. (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 𝛿_𝑛+1∶=𝑥𝑡_𝑛+1−𝑌𝑡_𝑛+1,𝑛=0,1,…,𝑞−1. (3.3) The global error of method (3.1) is the sequence of random variables 𝜀_𝑛∶=𝑥𝑡_𝑛−𝑌_𝑛,𝑛=0,1,…,𝑞. (3.4)

Definition 3.3. The numerical method (3.1) is said to be consistent with order 𝑝₁ in the mean and with order 𝑝₂ in the mean square sense if the following estimates hold with 𝑝₂⩾1/2 and 𝑝₁⩾𝑝₂+1/2: max_{0≤𝑛≤𝑞−1}𝔼𝛿_𝑛+1∣ℱ_{𝑡𝑛𝐿2}≤𝐻₀ℎ^𝑝1,asℎ⟶0,max_{0≤𝑛≤𝑞−1}𝛿_𝑛+1𝐿2≤𝐻₁ℎ^𝑝2,asℎ⟶0, (3.5) where the constants 𝐻₀ and 𝐻₁ do not depend on ℎ but may depend on 𝑇 and the initial data. Here, ‖𝑧‖_𝐿2∶=(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: max_{1≤𝑛≤𝑞}𝜀_𝑛𝐿2≤𝐻₂ℎ^𝑝,asℎ⟶0, (3.6) where the constant 𝐻₂ 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 𝑥_𝑖,𝑦_𝑖∈ℝ^𝑑 (𝑖=1,2,3,4),Φ_𝑓ℎ,𝑥₁,𝑥₂,𝑥₃,𝑥₄,Δ𝑊_𝑛,Δ𝑁_𝑛−Φ_𝑓ℎ,𝑦₁,𝑦₂,𝑦₃,𝑦₄,Δ𝑊_𝑛,Δ𝑁_𝑛≤ℎ𝐿_𝑓⁴_𝑖=1𝑥_𝑖−𝑦_𝑖, (3.7) and there exist the positive constants 𝐿_𝑔,𝐿_𝑢,𝐿_𝑢 such that for all ℱ_𝑡𝑛-measurable random variables 𝑥₁,𝑥₃,𝑦₁,𝑦₃∈ℝ^𝑑, 𝔼Φ_𝑔ℎ,𝑥₁,𝑥₃,Δ𝑊_𝑛,Δ𝑁_𝑛−Φ_𝑔ℎ,𝑦₁,𝑦₃,Δ𝑊_𝑛,Δ𝑁_𝑛²≤ℎ𝐿_𝑔𝔼𝑥₁−𝑦₁²+𝔼𝑥₃−𝑦₃², (3.8)𝔼Φ_𝑢ℎ,𝑥₁,𝑥₃,Δ𝑊_𝑛,Δ𝑁_𝑛−Φ_𝑢ℎ,𝑦₁,𝑦₃,Δ𝑊_𝑛,Δ𝑁_𝑛²≤ℎ𝐿_𝑢𝔼𝑥₁−𝑦₁²+𝔼𝑥₃−𝑦₃², (3.9)𝔼Φ_𝑔ℎ,𝑥₁,𝑥₃,Δ𝑊_𝑛,Δ𝑁_𝑛−Φ_𝑔ℎ,𝑦₁,𝑦₃,Δ𝑊_𝑛,Δ𝑁_𝑛∣ℱ_𝑡𝑛=0, (3.10)𝔼Φ_𝑢ℎ,𝑥₁,𝑥₃,Δ𝑊_𝑛,Δ𝑁_𝑛−Φ_𝑢ℎ,𝑦₁,𝑦₃,Δ𝑊_𝑛,Δ𝑁_𝑛∣ℱ_𝑡𝑛≤ℎ𝐿_𝑢𝑥₁−𝑦₁+𝑥₃−𝑦₃. (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 𝑝₁ in the mean and order 𝑝₂ in the mean square sense, then the method (3.1) is convergent with order 𝑝=𝑝₂−1/2 in the mean square sense.

Proof. By (3.4), we have 𝜀_𝑛+1=𝜀̂_𝑛+1+𝐺𝑥𝑡_𝑛+1−𝜏−𝐺𝑌_{𝑛+1−𝑚}, (3.12) where 𝜀̂_𝑛+1 is defined as follows: 𝜀̂_𝑛+1=𝑥𝑡_𝑛+1−𝐺𝑥𝑡_𝑛+1−𝜏−𝑌_𝑛+1−𝐺𝑌_{𝑛+1−𝑚}. (3.13) Squaring and taking expectation on both sides of (3.12), using Assumption 3 and (2.11), we have 𝔼𝜀_𝑛+1²≤1𝔼𝜀̂1−𝜂^𝑛+12+1𝜂𝔼𝐺𝑥𝑡_𝑛+1−𝜏−𝐺𝑌_{𝑛+1−𝑚}²≤1𝔼𝜀̂1−𝜂^𝑛+12+𝜂𝔼𝑥𝑡_𝑛+1−𝜏−𝑌_{𝑛+1−𝑚}²=1𝔼𝜀̂1−𝜂^𝑛+12+𝜂𝔼𝜀_{𝑛+1−𝑚}². (3.14) It follows from (3.1), (3.2), and (3.13) that 𝜀̂_𝑛+1=𝑥𝑡_𝑛+1−𝐺𝑥𝑡_𝑛+1−𝜏−𝑌𝑡_𝑛+1+𝑌𝑡_𝑛+1−𝑌_𝑛+1−𝐺𝑌_{𝑛+1−𝑚}=𝑥𝑡_𝑛+1−𝑌𝑡_𝑛+1+𝑥𝑡_𝑛−𝐺𝑥𝑡_𝑛−𝜏+Φ_𝑓ℎ,𝑥𝑡_𝑛,𝑥𝑡_𝑛+1,𝑥𝑡_𝑛−𝜏,𝑥𝑡_𝑛+1−𝜏,Δ𝑊_𝑛,Δ𝑁_𝑛+Φ_𝑔ℎ,𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏,Δ𝑊_𝑛,Δ𝑁_𝑛+Φ_𝑢ℎ,𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏,Δ𝑊_𝑛,Δ𝑁_𝑛−𝑌_𝑛+𝐺𝑌_𝑛−𝑚−Φ_𝑓ℎ,𝑌_𝑛,𝑌_𝑛+1,𝑌_𝑛−𝑚,𝑌_{𝑛+1−𝑚},Δ𝑊_𝑛,Δ𝑁_𝑛−Φ_𝑔ℎ,𝑌_𝑛,𝑌_𝑛−𝑚,Δ𝑊_𝑛,Δ𝑁_𝑛−Φ_𝑢ℎ,𝑌_𝑛,𝑌_𝑛−𝑚,Δ𝑊_𝑛,Δ𝑁_𝑛=𝛿_𝑛+1+𝜀_𝑛̂+𝑅_𝑛, (3.15) where 𝑅_𝑛=Φ_𝑓ℎ,𝑥𝑡_𝑛,𝑥𝑡_𝑛+1,𝑥𝑡_𝑛−𝜏,𝑥𝑡_𝑛+1−𝜏,Δ𝑊_𝑛,Δ𝑁_𝑛−Φ_𝑓ℎ,𝑌_𝑛,𝑌_𝑛+1,𝑌_𝑛−𝑚,𝑌_{𝑛+1−𝑚},Δ𝑊_𝑛,Δ𝑁_𝑛+Φ_𝑔ℎ,𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏,Δ𝑊_𝑛,Δ𝑁_𝑛−Φ_𝑔ℎ,𝑌_𝑛,𝑌_𝑛−𝑚,Δ𝑊_𝑛,Δ𝑁_𝑛+Φ_𝑢ℎ,𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏,Δ𝑊_𝑛,Δ𝑁_𝑛−Φ_𝑢ℎ,𝑌_𝑛,𝑌_𝑛−𝑚,Δ𝑊_𝑛,Δ𝑁_𝑛. (3.16) Squaring and taking expectation on both sides of (3.15) yields 𝔼𝜀̂_𝑛+1²≤𝔼𝜀̂_𝑛²+2𝔼𝛿_𝑛+1²+2𝔼𝑅_𝑛²+2𝔼𝛿_𝑛+1,𝜀_𝑛̂+2𝔼𝜀_𝑛̂,𝑅_𝑛. (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 𝑝₂ in the mean square sense; thus, there exists a constant 𝐽₂ such that 𝔼𝛿_𝑛+1²≤𝐽₂ℎ^2𝑝2. (3.18) By (3.7)–(3.9), we obtain 𝔼𝑅_𝑛²≤3𝔼Φ_𝑓ℎ,𝑥𝑡_𝑛,𝑥𝑡_𝑛+1,𝑥𝑡_𝑛−𝜏,𝑥𝑡_𝑛+1−𝜏,Δ𝑊_𝑛,Δ𝑁_𝑛−Φ_𝑓ℎ,𝑌_𝑛,𝑌_𝑛+1,𝑌_𝑛−𝑚,𝑌_{𝑛+1−𝑚},Δ𝑊_𝑛,Δ𝑁_𝑛²+3𝔼Φ_𝑔ℎ,𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏,Δ𝑊_𝑛,Δ𝑁_𝑛−Φ_𝑔ℎ,𝑌_𝑛,𝑌_𝑛−𝑚,Δ𝑊_𝑛,Δ𝑁_𝑛²+3𝔼Φ_𝑢ℎ,𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏,Δ𝑊_𝑛,Δ𝑁_𝑛−Φ_𝑢ℎ,𝑌_𝑛,𝑌_𝑛−𝑚,Δ𝑊_𝑛,Δ𝑁_𝑛²≤3ℎ²𝐿²_𝑓𝔼𝑥𝑡_𝑛−𝑌_𝑛+𝑥𝑡_𝑛+1−𝑌_𝑛+1+𝑥𝑡_𝑛−𝜏−𝑌_𝑛−𝑚+𝑥𝑡_𝑛+1−𝜏−𝑌_{𝑛+1−𝑚}²+3ℎ𝐿_𝑔𝔼𝑥𝑡_𝑛−𝑌_𝑛²+𝔼𝑥𝑡_𝑛−𝜏−𝑌_𝑛−𝑚²+3ℎ𝐿_𝑢𝔼𝑥𝑡_𝑛−𝑌_𝑛²+𝔼𝑥𝑡_𝑛−𝜏−𝑌_𝑛−𝑚²≤12ℎ²𝐿²_𝑓+3ℎ𝐿_𝑔+3ℎ𝐿_𝑢𝔼𝑥𝑡_𝑛−𝑌_𝑛²+𝔼𝑥𝑡_𝑛−𝜏−𝑌_𝑛−𝑚²+12ℎ²𝐿²_𝑓𝔼𝑥𝑡_𝑛+1−𝑌_𝑛+1²+𝔼𝑥𝑡_𝑛+1−𝜏−𝑌_{𝑛+1−𝑚}²=12ℎ²𝐿²_𝑓+3ℎ𝐿_𝑔+3ℎ𝐿_𝑢𝔼𝜀_𝑛²+𝔼𝜀_𝑛−𝑚²+12ℎ²𝐿²_𝑓𝔼𝜀_𝑛+1²+𝔼𝜀_{𝑛+1−𝑚}², (3.19) which, by the inequality (3.14), yields 𝔼𝑅_𝑛²≤12ℎ²𝐿²_𝑓+3ℎ𝐿_𝑔+3ℎ𝐿_𝑢1𝔼𝜀̂1−𝜂^𝑛2+𝜂𝔼𝜀_𝑛−𝑚²+12ℎ²𝐿²_𝑓+3ℎ𝐿_𝑔+3ℎ𝐿_𝑢𝔼𝜀_𝑛−𝑚²+12ℎ²𝐿²_𝑓1𝔼𝜀̂1−𝜂^𝑛+12+𝜂𝔼𝜀_{𝑛+1−𝑚}²+12ℎ²𝐿²_𝑓𝔼𝜀_{𝑛+1−𝑚}²≤12ℎ²𝐿²_𝑓+3ℎ𝐿_𝑔+3ℎ𝐿_𝑢𝔼𝜀̂_𝑛²+𝔼𝜀_𝑛−𝑚²1max1−𝜂,1+𝜂+12ℎ²𝐿²_𝑓𝔼𝜀̂_𝑛+1²+𝔼𝜀_{𝑛+1−𝑚}²1max≤1−𝜂,1+𝜂12𝑇𝐿²_𝑓+3𝐿_𝑔+3𝐿_𝑢ℎ𝔼𝜀̂_𝑛²+𝔼𝜀_𝑛−𝑚²+𝔼𝜀̂_𝑛+1²+𝔼𝜀_{𝑛+1−𝑚}²1max1−𝜂,1+𝜂=𝐽₃ℎ𝔼𝜀̂_𝑛²+𝔼𝜀_𝑛−𝑚²+𝔼𝜀̂_𝑛+1²+𝔼𝜀_n+1−𝑚², (3.20) where 𝐽₃=(12𝑇𝐿²_𝑓+3𝐿_𝑔+3𝐿_𝑢)max{1/1−𝜂,1+𝜂}.
Since method (3.1) is consistent with order 𝑝₁ in the mean square, there exists a constant 𝐽₄ such that 𝔼𝔼𝛿_𝑛+1∣ℱ_𝑡𝑛^21/2≤𝐽₄ℎ^𝑝1. (3.21) Noticing that 𝜀_𝑛̂ is ℱ_𝑡𝑛-measurable and by (3.21), we arrive at 2𝔼𝜀_𝑛̂,𝛿_𝑛+1=2𝔼𝔼𝜀_𝑛̂,𝛿_𝑛+1∣ℱ_𝑡𝑛≤2𝔼𝔼𝜀_𝑛̂,𝛿_𝑛+1∣ℱ_𝑡𝑛=2𝔼𝜀_𝑛̂,𝔼𝛿_𝑛+1∣ℱ_𝑡𝑛≤2ℎ𝔼𝜀̂_𝑛^21/2ℎ⁻¹𝔼𝔼𝛿_𝑛+1∣ℱ_𝑡𝑛^21/2≤ℎ𝔼𝜀̂_𝑛²+ℎ⁻¹𝔼𝔼𝛿_𝑛+1∣ℱ_𝑡𝑛²≤ℎ𝔼𝜀̂_𝑛²+𝐽₄²ℎ^2𝑝1−1=ℎ𝔼𝜀̂_𝑛²+𝐽₅ℎ^2𝑝1−1, (3.22) where 𝐽₅=(𝐽₄)². Applying the inequality |𝔼(𝑥∣ℱ)|²≤𝔼(|𝑥|²∣ℱ), (3.7), (3.10), and (3.11) yields 𝔼𝑅_𝑛∣ℱ_𝑡𝑛²=𝔼Φ_𝑓ℎ,𝑥𝑡_𝑛,𝑥𝑡_𝑛+1,𝑥𝑡_𝑛−𝜏,𝑥𝑡_𝑛+1−𝜏,Δ𝑊_𝑛,Δ𝑁_𝑛−Φ_𝑓ℎ,𝑌_𝑛,𝑌_𝑛+1,𝑌_𝑛−𝑚,𝑌_{𝑛+1−𝑚},Δ𝑊_𝑛,Δ𝑁_𝑛+Φ_𝑔ℎ,𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏,Δ𝑊_𝑛,Δ𝑁_𝑛−Φ_𝑔ℎ,𝑌_𝑛,𝑌_𝑛−𝑚,Δ𝑊_𝑛,Δ𝑁_𝑛+Φ_𝑢ℎ,𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏,Δ𝑊_𝑛,Δ𝑁_𝑛−Φ_𝑢ℎ,𝑌_𝑛,𝑌_𝑛−𝑚,Δ𝑊_𝑛,Δ𝑁_𝑛∣ℱ_𝑡𝑛²=𝔼Φ_𝑓ℎ,𝑥𝑡_𝑛,𝑥𝑡_𝑛+1,𝑥𝑡_𝑛−𝜏,𝑥𝑡_𝑛+1−𝜏,Δ𝑊_𝑛,Δ𝑁_𝑛−Φ_𝑓ℎ,𝑌_𝑛,𝑌_𝑛+1,𝑌_𝑛−𝑚,𝑌_{𝑛+1−𝑚},Δ𝑊_𝑛,Δ𝑁_𝑛∣ℱ_𝑡𝑛+𝔼Φ_𝑢ℎ,𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏,Δ𝑊_𝑛,Δ𝑁_𝑛−Φ_𝑢ℎ,𝑌_𝑛,𝑌_𝑛−𝑚,Δ𝑊_𝑛,Δ𝑁_𝑛∣ℱ_𝑡𝑛²≤2𝔼Φ_𝑓ℎ,𝑥𝑡_𝑛,𝑥𝑡_𝑛+1,𝑥𝑡_𝑛−𝜏,𝑥𝑡_𝑛+1−𝜏,Δ𝑊_𝑛,Δ𝑁_𝑛−Φ_𝑓ℎ,𝑌_𝑛,𝑌_𝑛+1,𝑌_𝑛−𝑚,𝑌_{𝑛+1−𝑚},Δ𝑊_𝑛,Δ𝑁_𝑛²∣ℱ_𝑡𝑛+2𝔼Φ_𝑢ℎ,𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏,Δ𝑊_𝑛,Δ𝑁_𝑛−Φ_𝑢ℎ,𝑌_𝑛,𝑌_𝑛−𝑚,Δ𝑊_𝑛,Δ𝑁_𝑛∣ℱ_𝑡𝑛²≤2ℎ²𝐿_𝑓²𝔼𝑥𝑡_𝑛−𝑌_𝑛+𝑥𝑡_𝑛+1−𝑌_𝑛+1+𝑥𝑡_𝑛−𝜏−𝑌_𝑛−𝑚+𝑥𝑡_𝑛+1−𝜏−𝑌_{𝑛+1−𝑚}²∣ℱ_𝑡𝑛+2ℎ²𝐿_𝑢²𝑥𝑡_𝑛−𝑌_𝑛+𝑥𝑡_𝑛−𝜏−𝑌_𝑛−𝑚²≤8ℎ²𝐿_𝑓²𝜀_𝑛²+𝔼𝜀_𝑛+1²∣ℱ_𝑡𝑛+𝜀_𝑛−𝑚²+𝜀_{𝑛+1−𝑚}²+4ℎ²𝐿_𝑢²𝜀_𝑛²+𝜀_𝑛−𝑚²=8ℎ²𝐿_𝑓²+4ℎ²𝐿_𝑢²𝜀_𝑛²+8ℎ²𝐿_𝑓²𝔼𝜀_𝑛+1²∣ℱ_𝑡𝑛+8ℎ²𝐿_𝑓²+4ℎ²𝐿_𝑢²𝜀_𝑛−𝑚²+8ℎ²𝐿_𝑓²𝜀_{𝑛+1−𝑚}². (3.23) Here, the fact used has been that 𝑥(𝑡_𝑛), 𝑥(𝑡_𝑛−𝜏), 𝑥(𝑡_𝑛+1−𝜏), 𝑌_𝑛, 𝑌_𝑛−𝑚, 𝑌_{𝑛+1−𝑚} are ℱ_𝑡𝑛-measurable. Using (3.14) and (3.23), we arrive at 2𝔼𝜀_𝑛̂,𝑅_𝑛=2𝔼𝔼𝜀_𝑛̂,𝑅_𝑛∣ℱ_𝑡𝑛=2𝔼𝜀_𝑛̂,𝔼𝑅_𝑛∣ℱ_𝑡𝑛≤2𝔼𝜀_𝑛̂⋅𝔼𝑅_𝑛∣ℱ_𝑡𝑛≤2ℎ𝔼𝜀̂_𝑛^21/2ℎ⁻¹𝔼𝔼𝑅_𝑛∣ℱ_𝑡𝑛^21/2≤ℎ𝔼𝜀̂_𝑛²+ℎ⁻¹𝔼𝔼𝑅_𝑛∣ℱ_𝑡𝑛²≤ℎ𝔼𝜀̂_𝑛²+8ℎ𝐿_𝑓²+4ℎ𝐿_𝑢²𝔼𝜀_𝑛²+8ℎ𝐿_𝑓²𝔼𝜀_𝑛+1²+8ℎ𝐿_𝑓²+4ℎ𝐿_𝑢²𝔼𝜀_𝑛−𝑚²+8ℎ𝐿_𝑓²𝔼𝜀_{𝑛+1−𝑚}²≤ℎ𝔼𝜀̂_𝑛²+8ℎ𝐿_𝑓²+4ℎ𝐿_𝑢²1𝔼𝜀̂1−𝜂^𝑛2+𝜂𝔼𝜀_𝑛−𝑚²+8ℎ𝐿_𝑓²1𝔼𝜀̂1−𝜂^𝑛+12+𝜂𝔼𝜀_{𝑛+1−𝑚}²+8ℎ𝐿_𝑓²+4ℎ𝐿_𝑢²𝔼𝜀_𝑛−𝑚²+8ℎ𝐿_𝑓²𝔼𝜀_{𝑛+1−𝑚}²=1+8𝐿_𝑓²+4𝐿_𝑢²1𝜀̂1−𝜂ℎ𝔼_𝑛²+8𝐿_𝑓²1𝜀̂1−𝜂ℎ𝔼_𝑛+1²+8𝐿_𝑓²+4𝐿_𝑢²1+𝜂ℎ𝔼𝜀_𝑛−𝑚²+8𝐿_𝑓²1+𝜂ℎ𝔼𝜀_{𝑛+1−𝑚}²≤1+8𝐿_𝑓²+4𝐿_𝑢²1max𝔼𝜀̂1−𝜂,1+𝜂ℎ,_𝑛2+𝔼𝜀̂_𝑛+1²+𝔼𝜀_{𝑛+1−𝑚}²+𝔼𝜀_𝑛−𝑚²=𝐽₆ℎ𝔼𝜀̂_𝑛²+𝔼𝜀̂_𝑛+1²+𝔼𝜀_{𝑛+1−𝑚}²+𝔼𝜀_𝑛−𝑚², (3.24) where 𝐽₆=1+(8(𝐿_𝑓)²+4(𝐿_𝑢)²)max{1/(1−𝜂),1+𝜂}. Inserting (3.18), (3.20), (3.22), and (3.24) into (3.17) yields 𝔼𝜀̂_𝑛+1²≤𝔼𝜀̂_𝑛²+2𝐽₂ℎ^2𝑝2+2𝐽₃ℎ𝔼𝜀̂_𝑛²+𝔼𝜀_𝑛−𝑚²+𝔼𝜀̂_𝑛+1²+𝔼𝜀_{𝑛+1−𝑚}²+ℎ𝔼𝜀̂_𝑛²+𝐽₅ℎ^2𝑝1−1+𝐽₆ℎ𝔼𝜀̂_𝑛²+𝔼𝜀̂_𝑛+1²+𝔼𝜀_{𝑛+1−𝑚}²+𝔼𝜀_𝑛−𝑚²=2𝐽₃+𝐽₆ℎ𝔼𝜀̂_𝑛+1²+1+1+2𝐽₃+𝐽₆ℎ𝔼𝜀̂_𝑛²+2𝐽₃+𝐽₆ℎ𝔼𝜀_{𝑛+1−𝑚}²+𝔼𝜀_𝑛−𝑚²+2𝐽₂+𝐽₅ℎ^2𝑝2. (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.

4. 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.

Example 4.1. Consider the stochastic 𝜃-methods for system (2.1), 𝑌_𝑛+1=𝐺𝑌_{𝑛+1−𝑚}+𝑌_𝑛−𝐺𝑌_𝑛−𝑚+(1−𝜃)𝑓𝑌_𝑛,𝑌_𝑛−𝑚+𝜃𝑓𝑌_𝑛+1,𝑌_{𝑛+1−𝑚}ℎ+𝑔𝑌_𝑛,𝑌_𝑛−𝑚Δ𝑊_𝑛+𝑢𝑌_𝑛,𝑌_𝑛−𝑚Δ𝑁_𝑛,𝑛=0,1,…,𝑞−1, (4.1) where 𝑚=𝜏/ℎ, 0≤𝜃≤1.

Lemma 4.2. Let Assumption 1 hold, then the stochastic 𝜃-methods (4.1) are consistent with order 𝑝₁=3/2 in the mean and order 𝑝₂=1 in the mean square sense.

Proof. Combining (2.1), (3.2) with (4.1) yields 𝛿_𝑛+1=𝑥𝑡_𝑛+1−𝑌𝑡_𝑛+1=^𝑡𝑛+1_𝑡𝑛𝑓(𝑥(𝑠),𝑥(𝑠−𝜏))−𝑓𝑥𝑡_𝑛,𝑥𝑡_𝑛+−𝜏𝑑𝑠_{𝑡𝑛+1𝑡𝑛}𝑔(𝑥(𝑠),𝑥(𝑠−𝜏))−𝑔𝑥𝑡_𝑛,𝑥𝑡_𝑛(𝑠)+−𝜏𝑑𝑊_{𝑡𝑛+1𝑡𝑛}𝑢(𝑥(𝑠),𝑥(𝑠−𝜏))−𝑢𝑥𝑡_𝑛,𝑥𝑡_𝑛(𝑠)𝑓𝑥𝑡−𝜏𝑑𝑁−𝜃ℎ_𝑛+1,𝑥𝑡_𝑛+1−𝜏−𝑓𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏. (4.2) Noticing the compensated Poisson process, 𝑁(𝑡)=𝑁(𝑡)−𝜆𝑡, which satisfies 𝔼^𝑏_𝑎𝑢(𝑠)𝑑𝑁(𝑠)∣ℱ_𝑎=0. (4.3) Using Assumption 1, (4.3), Hölder inequality, Lemma 2.4, the properties of conditional expectation, and Jenson’s inequality: |𝔼(𝑥∣ℱ)|²≤𝔼(|𝑥|²∣ℱ), we compute that 𝔼𝔼𝛿_𝑛+1∣ℱ_𝑡𝑛²≤5𝔼𝔼^𝑡𝑛+1_𝑡𝑛𝑓(𝑥(𝑠),𝑥(𝑠−𝜏))−𝑓𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏𝑑𝑠∣ℱ_𝑡𝑛²+5𝔼𝔼^𝑡𝑛+1_𝑡𝑛𝑔(𝑥(𝑠),𝑥(𝑠−𝜏))−𝑔𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏𝑑𝑊(𝑠)∣ℱ_𝑡𝑛²+5𝔼𝔼^𝑡𝑛+1_𝑡𝑛𝑢(𝑥(𝑠),𝑥(𝑠−𝜏))−𝑢𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏𝑑𝑁(𝑠)∣ℱ_𝑡𝑛²+5𝜆²𝔼𝔼^𝑡𝑛+1_𝑡𝑛𝑢(𝑥(𝑠),𝑥(𝑠−𝜏))−𝑢𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏𝑑𝑠∣ℱ_𝑡𝑛²+5ℎ²𝔼𝔼𝑓𝑥𝑡_𝑛+1,𝑥𝑡_𝑛+1−𝜏−𝑓𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏∣ℱ_𝑡𝑛²≤5𝔼𝔼^𝑡𝑛+1_𝑡𝑛𝑓(𝑥(𝑠),𝑥(𝑠−𝜏))−𝑓𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏𝑑𝑠²∣ℱ_𝑡𝑛+5𝜆²𝔼𝔼^𝑡𝑛+1_𝑡𝑛𝑢(𝑥(𝑠),𝑥(𝑠−𝜏))−𝑢𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏𝑑𝑠²∣ℱ_𝑡𝑛+5ℎ²𝔼𝔼𝑓𝑥𝑡_𝑛+1,𝑥𝑡_𝑛+1−𝜏−𝑓𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏²∣ℱ_𝑡𝑛≤51+𝜆²𝐾ℎ^𝑡𝑛+1_𝑡𝑛𝔼𝑥(𝑠)−𝑥𝑡_𝑛²+𝑥(𝑠−𝜏)−𝑥𝑡_𝑛−𝜏²𝑑𝑠+5ℎ²𝐾𝔼𝑥𝑡_𝑛+1−𝑥𝑡_𝑛²+𝑥𝑡_𝑛+1−𝜏−𝑥𝑡_𝑛−𝜏²≤101+𝜆²𝐾𝐻+10𝐾𝐻ℎ³,𝔼𝛿_𝑛+1²≤4ℎ𝔼^𝑡𝑛+1_𝑡𝑛𝑓(𝑥(𝑠),𝑥(𝑠−𝜏))−𝑓𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏²𝑑𝑠+4𝔼^𝑡𝑛+1_𝑡𝑛𝑔(𝑥(𝑠),𝑥(𝑠−𝜏))−𝑔𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏²𝑑𝑠+8𝜆^𝑡𝑛+1_𝑡𝑛𝔼𝑢(𝑥(𝑠),𝑥(𝑠−𝜏))−𝑢𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏²𝑑𝑠+8𝜆²ℎ^𝑡𝑛+1_𝑡𝑛𝔼𝑢(𝑥(𝑠),𝑥(𝑠−𝜏))−𝑢𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏²𝑑𝑠+4ℎ²𝔼𝑓𝑥𝑡_𝑛+1,𝑥𝑡_𝑛+1−𝜏−𝑓𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏²≤4ℎ+4+8𝜆+8𝜆²ℎ𝐾^𝑡𝑛+1_𝑡𝑛𝔼𝑥(𝑠)−𝑥𝑡_𝑛²+𝑥(𝑠−𝜏)−𝑥𝑡_𝑛−𝜏²𝑑𝑠+4𝐾ℎ²𝔼𝑥𝑡_𝑛+1−𝑥𝑡_𝑛²+𝑥𝑡_𝑛+1−𝜏−𝑥𝑡_𝑛−𝜏²≤8𝐾𝐻1+2𝑇+2𝜆+2𝜆²𝑇ℎ², (4.4)(4.5) where the compensated Poisson process 𝑁(𝑡)=𝑁(𝑡)−𝜆𝑡 satisfies 𝔼^𝑡2_𝑡1𝑢(𝑥(𝑠))𝑑𝑁(𝑠)²=𝜆^𝑡2_𝑡1𝔼|𝑢(𝑥(𝑠))|²𝑑𝑠. (4.6) 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.

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.

Proof. Lemma 4.2 shows that the stochastic 𝜃-methods (4.1) are consistent with order 𝑝₁=3/2 in the mean and order 𝑝₂=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 Φ_𝑓ℎ,𝑌_𝑛,𝑌_𝑛+1,𝑌_𝑛−𝑚,𝑌_{𝑛+1−𝑚},Δ𝑊_𝑛,Δ𝑁_𝑛=(1−𝜃)𝑓𝑌_𝑛,𝑌_𝑛−𝑚+𝜃𝑓𝑌_𝑛+1,𝑌_{𝑛+1−𝑚}Φℎ,_𝑔ℎ,𝑌_𝑛,𝑌_𝑛−𝑚,Δ𝑊_𝑛,Δ𝑁_𝑛=𝑔𝑌_𝑛,𝑌_𝑛−𝑚Δ𝑊_𝑛,Φ_𝑢ℎ,𝑌_𝑛,𝑌_𝑛−𝑚,Δ𝑊_𝑛,Δ𝑁_𝑛=𝑢𝑌_𝑛,𝑌_𝑛−𝑚Δ𝑁_𝑛. (4.7) For the random variables 𝑥_𝑖,𝑦_𝑖∈ℝ^𝑑 (𝑖=1,2,3,4), by Assumption 1, we have Φ_𝑓ℎ,𝑥₁,𝑥₂,𝑥₃,𝑥₄,Δ𝑊_𝑛,Δ𝑁_𝑛−Φ_𝑓ℎ,𝑦₁,𝑦₂,𝑦₃,𝑦₄,Δ𝑊_𝑛,Δ𝑁_𝑛=(1−𝜃)𝑓𝑥₁,𝑥₃+𝜃𝑓𝑥₂,𝑥₄ℎ−(1−𝜃)𝑓𝑦₁,𝑦₃+𝜃𝑓𝑦₂,𝑦₄ℎ≤√𝐾ℎ⁴_𝑖=1𝑥_𝑖−𝑦_𝑖. (4.8) Noticing that 𝔼|Δ𝑊_𝑛|²=𝑏ℎ, 𝔼|Δ𝑁_𝑛|²=𝜆ℎ(1+𝜆ℎ), and the random variables 𝑥₁,𝑥₃,𝑦₁,𝑦₃ are ℱ_𝑡𝑛-measurable, we derive that 𝔼Φ_𝑔ℎ,𝑥₁,𝑥₃,Δ𝑊_𝑛,Δ𝑁_𝑛−Φ_𝑔ℎ,𝑦₁,𝑦₃,Δ𝑊_𝑛,Δ𝑁_𝑛²≤𝔼𝑔𝑥₁,𝑥₃Δ𝑊_𝑛−𝑔𝑦₁,𝑦₃Δ𝑊_𝑛²≤𝑏Kℎ𝔼𝑥₁−𝑦₁²+𝔼𝑥₃−𝑦₃²,𝔼Φ_𝑢ℎ,𝑥₁,𝑥₃,Δ𝑊_𝑛,Δ𝑁_𝑛−Φ_𝑢ℎ,𝑦₁,𝑦₃,Δ𝑊_𝑛,Δ𝑁_𝑛²≤𝔼𝑢𝑥₁,𝑥₃Δ𝑁_𝑛−𝑢𝑦₁,𝑦₃Δ𝑁_𝑛²≤𝜆(1+𝜆𝑇)𝐾ℎ𝔼𝑥₁−𝑦₁²+𝔼𝑥₃−𝑦₃². (4.9)(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_𝑡𝑛 and 𝑥₁,𝑥₃,𝑦₁, and 𝑦₃ are ℱ_𝑡𝑛-measurable, then using |𝔼(𝑥∣ℱ)|≤𝔼(|𝑥|∣ℱ), we find that 𝔼Φ_𝑔ℎ,𝑥₁,𝑥₃,Δ𝑊_𝑛,Δ𝑁_𝑛−Φ_𝑔ℎ,𝑦₁,𝑦₃,Δ𝑊_𝑛,Δ𝑁_𝑛∣ℱ_𝑡𝑛=𝔼𝑔𝑥₁,𝑥₃Δ𝑊_𝑛−𝑔𝑦₁,𝑦₃Δ𝑊_𝑛∣ℱ_𝑡𝑛=𝑔𝑥₁,𝑥₃−𝑔𝑦₁,𝑦₃𝔼Δ𝑊_𝑛∣ℱ_𝑡𝑛𝔼Φ=0,_𝑢ℎ,𝑥₁,𝑥₃,Δ𝑊_𝑛,Δ𝑁_𝑛−Φ_𝑢ℎ,𝑦₁,𝑦₃,Δ𝑊_𝑛,Δ𝑁_𝑛∣ℱ_𝑡𝑛=𝔼𝑢𝑥₁,𝑥₃Δ𝑁_𝑛−𝑢𝑦₁,𝑦₃Δ𝑁_𝑛∣ℱ_𝑡𝑛=𝑢𝑥₁,𝑥₃−𝑢𝑦₁,𝑦₃⋅𝔼Δ𝑁_𝑛∣ℱ_𝑡𝑛≤𝜆√𝐾ℎ𝑥₁−𝑦₁+𝑥₃−𝑦₃. (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 d𝑥(𝑡)=𝑓(𝑥(𝑡⁻),𝑥(𝑡⁻−𝜏))d𝑡+𝑔(𝑥(𝑡⁻),𝑥(𝑡⁻(𝑡)(𝑥(𝑡−𝜏))d𝑊+𝑢⁻),𝑥(𝑡⁻(𝑡)𝑥(𝑡)(𝑡)−𝜏))d𝑁,0<𝑡<𝑇,=𝜓,−𝜏≤𝑡≤0, (4.12) where 𝜓∈𝐿²_ℱ0([−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), 𝑌_𝑛+1=𝐺𝑌_{𝑛+1−𝑚}+𝑌_𝑛−𝐺𝑌_𝑛−𝑚+𝑓𝑌_𝑛,𝑌_𝑛−𝑚ℎ+𝑔𝑌_𝑛,𝑌_𝑛−𝑚Δ𝑊_𝑛+𝑢𝑌_𝑛,𝑌_𝑛−𝑚Δ𝑁_𝑛+𝐶𝑌_𝑛,𝑌_𝑛−𝑚𝑌_𝑛−𝐺𝑌_𝑛−𝑚−𝑌_𝑛+1+𝐺𝑌_{𝑛+1−𝑚}. (4.13) Here, the 𝑑×𝑑 matrix 𝐶(𝑌_𝑛,𝑌_𝑛−𝑚) is given by 𝐶𝑌_𝑛,𝑌_𝑛−𝑚=𝐶₀𝑌_𝑛,𝑌_𝑛−𝑚ℎ+𝐶₁𝑌_𝑛,𝑌_𝑛−𝑚Δ𝑊_𝑛=𝐶_0𝑛ℎ+𝐶_1𝑛Δ𝑊_𝑛, (4.14) where the 𝐶_0𝑛=𝐶₀(𝑌_𝑛,𝑌_𝑛−𝑚), 𝐶_1𝑛=𝐶₁(𝑌_𝑛,𝑌_𝑛−𝑚) are, in general matrix, called control functions which are often chosen as constants.

Furthermore, the control functions must satisfy some conditions.

Assumption 4. The 𝐶₀(𝑥,𝑦) and 𝐶₁(𝑥,𝑦) represent bounded 𝑑×𝑑-matrix-valued functions for 𝑥,𝑦∈ℝ^𝑑. For any real numbers 𝛼₀∈[0,𝛼],𝛼₁≥0, where 𝛼≥ℎ for all step sizes ℎ considered and 𝑥,𝑦∈ℝ^𝑑, the matrix 𝑀(𝑥,𝑦)=𝐼+𝛼₀𝐶₀(𝑥,𝑦)+𝛼₁𝐶₁(𝑥,𝑦) has an inverse and satisfies the condition 𝑀𝑥,𝑦⁻¹≤𝐻<∞. (4.15) 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 𝑌_𝑛+1=𝐺𝑌_{𝑛+1−𝑚}+𝑌_𝑛−𝐺𝑌_𝑛−𝑚+𝐼+𝐶𝑌_𝑛,𝑌_𝑛−𝑚⁻¹×𝑓𝑌_𝑛,𝑌_𝑛−𝑚ℎ+𝑔𝑌_𝑛,𝑌_𝑛−𝑚Δ𝑊_𝑛+𝑢𝑌_𝑛,𝑌_𝑛−𝑚Δ𝑁_𝑛. (4.16)

Lemma 4.6 (see [12]). If 𝐹(𝑦,𝜔) is independent of ℱ,𝑦∈ℝ^𝑑, 𝜔∈Ω, 𝔼(𝐹(𝑦,𝜔))=𝜑(𝑦), and 𝜍 is ℱ-measurable, then 𝔼(𝐹(𝜍,𝜔)∣ℱ)=𝜑(𝜍).

For simplicity, from now on, we suppose that 𝐶_0𝑛,𝐶_1𝑛 in (4.14) are constants, that is to say, 𝐶_0𝑛=𝐶₀, 𝐶_1𝑛=𝐶₁.

Lemma 4.7. Let Assumptions 1, 2, and 4 hold, then the balanced implicit methods (4.13) are consistent with order 𝑝₁=3/2 in the mean and order 𝑝₂=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 𝔼𝔼𝑥𝑡_𝑛+1−𝑌^𝐵𝑡_𝑛+1∣ℱ_𝑡𝑛²=𝔼𝔼𝑥𝑡_𝑛+1−𝑌^𝐸𝑡_𝑛+1∣ℱ_𝑡𝑛+𝔼𝑌^𝐸𝑡_𝑛+1−𝑌^𝐵𝑡_𝑛+1∣ℱ_𝑡𝑛²≤2𝔼𝔼𝑥𝑡_𝑛+1−𝑌^𝐸𝑡_𝑛+1∣ℱ_𝑡𝑛²+2𝔼𝔼𝑌^𝐸𝑡_𝑛+1−𝑌^𝐵𝑡_𝑛+1∣ℱ_𝑡𝑛²≤201+𝜆²𝐾𝐻+20𝐾𝐻ℎ³+2𝔼𝔼𝑌^𝐸𝑡_𝑛+1−𝑌^𝐵𝑡_𝑛+1∣ℱ_𝑡𝑛², (4.17) where 𝑌^𝐸(𝑡_𝑛+1) and 𝑌^𝐵(𝑡_𝑛+1) are defined as follows: 𝑌^𝐸𝑡_𝑛+1=𝐺𝑥𝑡_𝑛+1−𝜏+𝑥𝑡_𝑛−𝐺𝑥𝑡_𝑛−𝜏+𝑓𝑥𝑡_𝑛,𝑥𝑡_𝑛ℎ𝑥𝑡−𝜏+𝑔_𝑛,𝑥𝑡_𝑛−𝜏Δ𝑊_𝑛+𝑢𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏Δ𝑁_𝑛,𝑌^𝐵𝑡_𝑛+1=𝐺𝑥𝑡_𝑛+1−𝜏+𝑥𝑡_𝑛−𝐺𝑥𝑡_𝑛−𝜏+𝐼+𝐶_𝑛⁻¹×𝑓𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏ℎ+𝑔𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏Δ𝑊_𝑛+𝑢𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏Δ𝑁_𝑛, (4.18) where 𝐶_𝑛=𝐶₀ℎ+𝐶₁|Δ𝑊_𝑛|. By (4.18), we have 𝑌^𝐸𝑡_𝑛+1−𝑌^𝐵𝑡_𝑛+1=𝐼−𝐼+𝐶_𝑛⁻¹⋅𝑓𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏ℎ+𝑔𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏Δ𝑊_𝑛+𝑢𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏Δ𝑁_𝑛=𝐼+𝐶_𝑛⁻¹⋅𝐼+𝐶_𝑛⋅𝑓𝑥𝑡−𝐼_𝑛,𝑥𝑡_𝑛−𝜏ℎ+𝑔𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏Δ𝑊_𝑛+𝑢𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏Δ𝑁_𝑛=𝐼+𝐶_𝑛⁻¹⋅𝐶_𝑛⋅𝑓𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏ℎ+𝑔𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏Δ𝑊_𝑛+𝑢𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏Δ𝑁_𝑛. (4.19) Noticing that 𝑥(𝑡_𝑛),𝑥(𝑡_𝑛−𝜏) are ℱ_𝑡𝑛-measurable, Δ𝑊_𝑛 is independent of ℱ_𝑡𝑛, and using Lemma 4.6, we find that 𝔼𝐼+C_𝑛⁻¹⋅𝐶_𝑛⋅𝑔𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏Δ𝑊_𝑛∣ℱ_𝑡𝑛=0. (4.20) We notice that 𝐶₀ and 𝐶₁ 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_𝑡𝑛, by Assumption 4, (4.19), (4.20), |𝔼(𝑥∣ℱ)|≤𝔼(|𝑥|∣ℱ), 𝔼|Δ𝑊_𝑛|≤√𝑏ℎ, and 𝔼Δ𝑁_𝑛=𝜆ℎ, we obtain 𝔼𝑌^𝐸𝑡_𝑛+1−𝑌^𝐵𝑡_𝑛+1∣ℱ_𝑡𝑛≤𝑓𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏ℎ⋅𝔼𝐼+𝐶_𝑛⁻¹⋅𝐶₀ℎ+𝐶₁Δ𝑊_𝑛∣ℱ_𝑡𝑛+𝑢𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏⋅𝔼𝐼+𝐶_𝑛⁻¹⋅𝐶₀ℎ+𝐶₁Δ𝑊_𝑛⋅Δ𝑁_𝑛∣ℱ_𝑡𝑛≤𝑓𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏𝐻ℎ⋅𝔼𝐶₀ℎ+𝐶₁Δ𝑊_𝑛∣ℱ_𝑡𝑛+𝑢𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏𝐻⋅𝔼𝐶₀ℎ+𝐶₁Δ𝑊_𝑛⋅Δ𝑁_𝑛∣ℱ_𝑡𝑛≤𝑓𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏+𝑢𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏𝜆𝐻ℎ^3/2𝐵√ℎ+√𝑏. (4.21) It follows from Assumption 2, Lemma 2.3, and (4.21) that 𝔼𝔼𝑌^𝐸𝑡_𝑛+1−𝑌^𝐵𝑡_𝑛+1∣ℱ_𝑡𝑛²≤2𝐻²𝐵²√𝑇+√𝑏²ℎ³𝔼𝑓𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏²+𝔼𝑢𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏²𝜆²≤2𝐿𝐻²𝐵²1+𝜆²√𝑇+√𝑏²1+2𝐽₁ℎ³. (4.22) Inserting (4.22) into (4.17) yields 𝔼𝔼𝑥𝑡_𝑛+1−𝑌^𝐵𝑡_𝑛+1∣ℱ_𝑡𝑛²≤201+𝜆²𝐾𝐻+20𝐾𝐻+4𝐿𝐻²𝐵²1+𝜆²√𝑇+√𝑏²1+2𝐽₁ℎ³, (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, |𝐶_𝑖|≤𝐵 (𝑖=0,1), 𝔼|Δ𝑊_𝑛|²=𝑏ℎ, 𝔼|Δ𝑊_𝑛|⁴≤3𝑏²ℎ², 𝔼Δ𝑁²_𝑛=𝜆ℎ(1+𝜆ℎ), and (4.19), we compute that 𝔼𝑌^𝐸𝑡_𝑛+1−𝑌^𝐵𝑡_𝑛+1²≤3ℎ²𝐻²𝔼𝐶₀ℎ+𝐶₁Δ𝑊_𝑛²⋅𝑓𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏²+3𝐻²𝔼𝐶₀ℎ+𝐶₁Δ𝑊_𝑛²⋅𝑔𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏²Δ𝑊_𝑛²+3𝐻²𝔼𝐶₀ℎ+𝐶₁Δ𝑊_𝑛²⋅𝑢𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏²Δ𝑁²_𝑛≤3ℎ²𝐻²𝔼2𝐶₀²ℎ²+2𝐶₁²Δ𝑊_𝑛²⋅𝑓𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏²+3𝐻²𝔼2𝐶₀²ℎ²Δ𝑊_𝑛²+2𝐶₁²Δ𝑊_𝑛⁴⋅𝑔𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏²+3𝐻²𝔼2𝐶₀²ℎ²+2𝐶₁²Δ𝑊_𝑛²⋅𝑢𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏²Δ𝑁²_𝑛≤3ℎ²𝐻²2𝐵²ℎ²+2𝐵²𝑏ℎ𝔼𝑓𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏²+3𝐻²2𝐵²ℎ²𝑏ℎ+2𝐵²⋅3𝑏²ℎ²𝔼𝑔𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏²+3𝐻²𝜆ℎ(1+𝜆ℎ)2𝐵²ℎ²+2𝐵²𝑏ℎ𝔼𝑢𝑥𝑡_𝑛,𝑥𝑡_𝑛−𝜏²≤6𝐻²𝐵²𝐿𝑇²+2𝑏𝑇+3𝑏²+𝜆(1+𝜆𝑇)(𝑇+𝑏)1+2𝐽₁ℎ². (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 𝔼𝑥𝑡_𝑛+1−𝑌^𝐵𝑡_𝑛+1²≤2𝔼𝑥𝑡_𝑛+1−𝑌^𝐸𝑡_𝑛+1²+2𝔼𝑌^𝐸𝑡_𝑛+1−𝑌^𝐵𝑡_𝑛+1²≤16𝐾𝐻1+2𝑇+2𝜆+2𝜆²𝑇+12𝐻²𝐵²𝐿×𝑇²+2𝑏𝑇+3𝑏²+𝜆(1+𝜆𝑇)(𝑇+𝑏)1+2𝐽₁ℎ². (4.25) The proof is completed.

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: Φ_𝑓ℎ,𝑌_𝑛,𝑌_𝑛+1,𝑌_𝑛−𝑚,𝑌_{𝑛+1−𝑚},Δ𝑊_𝑛,Δ𝑁_𝑛=𝐼+𝐶𝑌_𝑛,𝑌_𝑛−𝑚⁻¹𝑓𝑌_𝑛,𝑌_𝑛−𝑚Φℎ,_𝑔ℎ,𝑌_𝑛,𝑌_𝑛−𝑚,Δ𝑊_𝑛,Δ𝑁_𝑛=𝐼+𝐶𝑌_𝑛,𝑌_𝑛−𝑚⁻¹𝑔𝑌_𝑛,𝑌_𝑛−𝑚Δ𝑊_𝑛,Φ_𝑢ℎ,𝑌_𝑛,𝑌_𝑛−𝑚,Δ𝑊_𝑛,Δ𝑁_𝑛=𝐼+𝐶𝑌_𝑛,𝑌_𝑛−𝑚⁻¹𝑢𝑌_𝑛,𝑌_𝑛−𝑚Δ𝑁_𝑛. (4.26)(4.27) For 𝑥_𝑖,𝑦_𝑖∈ℝ^𝑑 (𝑖=1,2,3,4), by Assumptions 1–4, (4.26), we arrive at Φ_𝑓ℎ,𝑥₁,𝑥₂,𝑥₃,𝑥₄,Δ𝑊_𝑛,Δ𝑁_𝑛−Φ_𝑓ℎ,𝑦₁,𝑦₂,𝑦₃,𝑦₄,Δ𝑊_𝑛,Δ𝑁_𝑛=𝐼+𝐶₀ℎ+𝐶₁Δ𝑊_𝑛⁻¹𝑓𝑥₁,𝑥₃ℎ−𝐼+𝐶₀ℎ+𝐶₁Δ𝑊_𝑛⁻¹𝑓𝑦₁,𝑦₃ℎ≤𝐻√𝐾ℎ𝑥₁−𝑦₁+𝑥₃−𝑦₃. (4.28) Noticing that 𝔼|Δ𝑊_𝑛|²=𝑏ℎ, 𝔼Δ𝑁²_𝑛=𝜆ℎ(1+𝜆ℎ), the random variables 𝑥₁, 𝑥₃, 𝑦₁, and 𝑦₃ are ℱ_𝑡𝑛-measurable and using Assumptions 1–4, (4.27), we obtain 𝔼Φ_𝑔ℎ,𝑥₁,𝑥₃,Δ𝑊_𝑛,Δ𝑁_𝑛−Φ_𝑔ℎ,𝑦₁,𝑦₃,Δ𝑊_𝑛,Δ𝑁_𝑛²=𝔼[𝐼+𝐶_𝑛)]⁻¹𝑔𝑥₁,𝑥₃Δ𝑊_𝑛−𝐼+𝐶_𝑛⁻¹𝑔𝑦₁,𝑦₃Δ𝑊_𝑛²≤𝐻²𝑏𝐾ℎ𝔼𝑥₁−𝑦₁²+𝔼𝑥₃−𝑦₃²,𝔼Φ_𝑢ℎ,𝑥₁,𝑥₃,Δ𝑊_𝑛,Δ𝑁_𝑛−Φ_𝑢ℎ,𝑦₁,𝑦₃,Δ𝑊_𝑛,Δ𝑁_𝑛²=𝔼𝐼+𝐶_𝑛⁻¹𝑢𝑥₁,𝑥₃Δ𝑁_𝑛−𝐼+𝐶_𝑛⁻¹𝑢𝑦₁,𝑦₃Δ𝑁_𝑛²≤𝐻²𝐾𝜆(1+𝜆𝑇)ℎ𝔼𝑥₁−𝑦₁²+𝔼𝑥₃−𝑦₃². (4.29)(4.30) Since Δ𝑊_𝑛,Δ𝑁_𝑛 are independent of ℱ_𝑡𝑛 and the random variables 𝑥₁, 𝑥₃, 𝑦₁, and 𝑦₃ are ℱ_𝑡𝑛-measurable; thus, by the inequality |𝔼[𝑥∣ℱ]|≤𝔼[|𝑥|∣ℱ], 𝔼Δ𝑁_𝑛=𝜆ℎ, and Lemma 4.6, we compute that 𝔼Φ_𝑔ℎ,𝑥₁,𝑥₃,Δ𝑊_𝑛,Δ𝑁_𝑛−Φ_𝑔ℎ,𝑦₁,𝑦₃,Δ𝑊_𝑛,Δ𝑁_𝑛∣ℱ_𝑡𝑛=𝔼𝐼+𝐶_𝑛⁻¹𝑔𝑥₁,𝑥₃Δ𝑊_𝑛∣ℱ_𝑡𝑛−𝔼𝐼+𝐶_𝑛⁻¹𝑔𝑦₁,𝑦₃Δ𝑊_𝑛∣ℱ_𝑡𝑛𝔼Φ=0,_𝑢ℎ,𝑥₁,𝑥₃,Δ𝑊_𝑛,Δ𝑁_𝑛−Φ_𝑢ℎ,𝑦₁,𝑦₃,Δ𝑊_𝑛,Δ𝑁_𝑛∣ℱ_𝑡𝑛≤𝔼𝐼+𝐶_𝑛⁻¹𝑢𝑥₁,𝑥₃Δ𝑁_𝑛−𝐼+𝐶_𝑛⁻¹𝑢𝑦₁,𝑦₃Δ𝑁_𝑛∣ℱ_𝑡𝑛≤𝑢𝑥₁,𝑥₃−𝑢𝑦₁,𝑦₃⋅𝔼𝐼+𝐶_𝑛⁻¹⋅Δ𝑁_𝑛∣ℱ_𝑡𝑛≤𝐻√𝐾𝜆ℎ𝑥₁−𝑦₁+𝑥₃−𝑦₃. (4.31)(4.32) 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=𝐻²𝐾𝑏, 𝐿_𝑢=𝐻²𝐾𝜆(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.

Remark 4.9. For the case of 𝐺(⋅)=0 and 𝜏=0, (2.1) reduces to the stochastic differential equations with jumps d𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡⁻))d𝑡+𝑔(𝑡,𝑥(𝑡⁻))d𝑊(𝑡)+𝑢(𝑡,𝑥(𝑡⁻(𝑡)𝑥(0))d𝑁,0<𝑡<𝑇,₋)=𝑥₀. (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].

5. Numerical Experiments

In this section, several numerical examples are given to illustrate our theoretical results in the previous sections. Consider the nonlinear equationd𝑥(𝑡)(=(𝑡)+𝑥()[(−𝑥𝑡−1))]−𝑥𝑡−11+𝑥²()[(𝑥(𝑡(𝑥((𝑡)+1𝑡−1d𝑡+sin))+cos𝑡−1))]d𝑊2𝑥(𝑡)d𝑁(𝑡),𝑡∈[0,𝑇], (5.1)with initial data𝑥(𝑡)≡1,𝑡∈[−1,0]. (5.2)

To show the convergence of the 𝜃-methods (4.1) and the balanced implicit methods (4.16), we choose 𝜃=0.5, 𝐶₀=1, and 𝐶₁=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=𝜀₂^1+𝑟_Δ𝑡,𝑟=1,2,3,4,5, all measured at time 𝑇=1, are estimated by trajectory averaging, that is, 𝜀^𝑟≅1/2000∑²⁰⁰⁰_𝑗=1|𝑥_𝑟,𝑇(𝜔_𝑗)−𝑌_𝑟,𝑞(𝜔_𝑗)|². We plot our approximation to √𝜀^𝑟 against ℎ on a log-log scale in Figure 1. For reference, a dashed line of slope one-half is added in two graphs. In Figure 1., we show the convergence of the 𝜃-method (4.1) in the left picture and the balanced implicit methods (4.16) in the right picture, respectively.

(a)

(b)

Figure 1 

The convergence of the numerical methods. (a) Stochastic 𝜃-method (4.1); (b) the balanced implicit methods (4.16).

We see that the slopes of the two curves appear to match well in two pictures in Figure 1, which is consistent with the strong order of one-half implied in Theorem 4.3 and Theorem 4.8.

6. Conclusion

In this paper, we consider a family of implicit one-step methods for the NSDDEs with jumps. A relationship between the consistent order and the convergence order is established. A general framework for mean-square convergence of the methods is provided. The applicability of the mean-square convergence theory is illustrated by the stochastic 𝜃-methods and the balanced implicit methods. We have generalized the existing results. The examples presented in Section 4 show that the main result in this paper can be applied not only to semi-implicit methods but also to full implicit methods.

Acknowledgments

This research is supported with funds provided by the National Natural Science Foundation of China (no. 10871207 and 11171352) and the project sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.