Stability of Analytical and Numerical Solutions for Nonlinear Stochastic Delay Differential Equations with Jumps

Abstract

This paper is concerned with the stability of analytical and numerical solutions for nonlinear stochastic delay differential equations (SDDEs) with jumps. A sufficient condition for mean-square exponential stability of the exact solution is derived. Then, mean-square stability of the numerical solution is investigated. It is shown that the compensated stochastic θ methods inherit stability property of the exact solution. More precisely, the methods are mean-square stable for any stepsize Δ𝑡=𝜏/𝑚 when 1/2≤𝜃≤1, and they are exponentially mean-square stable if the stepsize Δ𝑡∈(0,Δ𝑡₀) when 0≤𝜃<1. Finally, some numerical experiments are given to illustrate the theoretical results.

1. Introduction

Models that incorporate jumps have become increasingly popular in finance and several areas of science and engineering. In particular, they are used in mathematical finance in order to simulate asset prices, interest rates, and volatilities [1, 2]. Jump models also arise in many other application areas and have proved successful at describing unexpected, abrupt changes of state [3]. So, it is valuable to investigate the properties of the solutions of these problems.

As is well known, explicit solutions of stochastic differential equations can rarely be obtained. It is necessary to construct efficient numerical methods to solve these equations. In recent years, many researchers worked on the construction of numerical schemes for stochastic ordinary differential equations (SODEs) (see [4, 5], and their references) and stochastic delay differential equations (SDDEs), see, for example, [6–11] and references therein. For SODEs with jumps, the strong convergence and mean-square stability of some semi-implicit numerical methods are investigated in [12–15]. A compensated split-step backward Euler method for SODEs with jumps is introduced in [12] and proved to satisfy a better stability property than the split-step backward Euler method.

For SDDEs with jumps, most of the existing work is concerned about convergence property of numerical methods, see, for example, [16–19]. There are few results on stability property, which motivates our work. In [20], Tan and Wang investigated the mean-square stability of the explicit Euler method for linear SDDEs with jumps. The aim of our paper is to investigate the mean-square stability of the compensated stochastic 𝜃 methods for nonlinear SDDEs with jumps.

This paper is organized as follows. In Section 2, we obtain a stability result for the analytical solution of (2.1). In Section 3, the compensated stochastic 𝜃 methods are constructed to solve problem (2.1). In Section 4, our main results will be stated and proved. It is shown that the compensated stochastic 𝜃 methods inherit mean-square stability of the exact solution. More precisely, the methods are mean-square stable for any stepsize Δ𝑡=𝜏/𝑚 when 1/2≤𝜃≤1, and they are exponentially mean-square stable if the stepsize Δ𝑡∈(0,Δ𝑡₀) when 0≤𝜃<1. Moreover, when 𝜃=1, the method is exponentially mean-square stable for every stepsize Δ𝑡=𝜏/𝑚. Finally, some numerical experiments are reported to illustrate the theoretical results.

2. Stability of the Analytical Solution

Throughout this paper, we let 𝒞([−𝜏,0]; 𝑅^𝑑) denote the family of continuous functions from [−𝜏,0] to 𝑅^𝑑 equipped with the norm ‖𝜑‖∶=sup_{−𝜏≤𝑠≤0}|𝜑(𝑠)|, where |⋅| is the Euclidean norm in ℝ^𝑑. Denoted by 𝒞^𝑏_ℱ0([−𝜏,0];ℝ^𝑑) the family of all bounded, ℱ₀ measurable, 𝒞([−𝜏,0]; 𝑅^𝑑) valued stochastic variables. The inner product of 𝑥,𝑦 in ℝ^𝑑 is denoted by ⟨𝑥,𝑦⟩. If 𝐴 is a vector or matrix, its transpose is denoted by 𝐴^𝑇. If 𝐴 is a matrix, its trace norm is denoted by |𝐴|=trace(𝐴^𝑇𝐴).

We consider the nonlinear SDDEs with jumps in Itô’s sense of the form:(𝑡)(𝑥(𝑡)((𝑥(𝑡)((𝑡)(𝑥(𝑡𝑑𝑥=𝑓,𝑥𝑡−𝜏))𝑑𝑡+𝑔,𝑥𝑡−𝜏))𝑑𝑊+ℎ⁻),𝑥(𝑡⁻−𝜏))𝑑𝑁(𝑡),𝑡>0, (2.1) with initial data 𝑥(𝑡)=𝜑(𝑡),𝑡∈[−𝜏,0] and 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×𝑚, ℎ∶ℝ^𝑑×ℝ^𝑑→ℝ^𝑑 are continuous functions, and 𝜑(𝑡)∈𝒞^𝑏_ℱ0([−𝜏,0];ℝ^𝑑). 𝑊(𝑡) is an 𝑚-dimensional Wiener process defined on the complete probability space (Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝑃) with a filtration satisfying the usual conditions (i.e., it is increasing and right continuous while ℱ₀ contains all 𝑃-null sets). 𝑁(𝑡) is a scalar Poisson process with parameter 𝜆 defined on the same probability space. Assume that 𝑊(𝑡) and 𝑁(𝑡) are independent of ℱ₀. Moreover, we assume that 𝜑(𝑡) is ℱ₀-measurable and right continuous with 𝔼‖𝜑‖²<∞. We also assume that 𝑓(0,0)=0,𝑔(0,0)=0 and ℎ(0,0)=0, so problem (2.1) admits a zero solution 𝑥(𝑡)≡0.

Definition 2.1 (see [21]). The zero solution of (2.1) is said to be pth moment exponentially stable if there is a pair of positive constants 𝜆 and 𝐶 such that 𝔼|𝑥(𝑡)|^𝑝≤𝐶𝜑exp^{−𝜆(𝑡−𝑡0)},𝑡≥𝑡₀, (2.2) for all 𝜑(𝑡)∈𝒞^𝑏_ℱ0([−𝜏,0];ℝ^𝑑). When 𝑝=2, it is usually said to be exponentially mean-square stable.

Now, we establish a mean-square stability condition for problem (2.1).

Theorem 2.2. Suppose that there are some constants 𝛼_𝑖,𝛽_𝑖,𝛾_𝑖,𝑖=1,2, such that 𝑥₁−𝑥₂,𝑓𝑥₁,𝑦−𝑓𝑥₂,𝑦≤𝛼₁𝑥₁−𝑥₂²,𝑓𝑥,𝑦₁−𝑓𝑥,𝑦₂≤𝛼₂𝑦₁−𝑦₂𝑔𝑥₁,,𝑦₁−𝑔𝑥₂,𝑦₂²≤𝛽₁𝑥₁−𝑥₂²+𝛽₂𝑦₁−𝑦₂²ℎ𝑥₁,,𝑦₁−ℎ𝑥₂,𝑦₂²≤𝛾₁𝑥₁−𝑥₂²+𝛾₂𝑦₁−𝑦₂², (2.3)(2.4)(2.5)(2.6) for all 𝑥₁,𝑦₁,𝑥₂,𝑦₂∈ℝ^𝑑. If 𝛼=2𝛼₁+2𝛼₂+𝛽₁+𝛽₂+𝜆1+2𝛾₁+2𝛾₂<0, (2.7) then the zero solution of (2.1) is exponentially mean-square stable.

Proof. Let 𝑡≥0, 𝛿>0, it follows from Itô’s formula that |𝑥(𝑡+𝛿)|²=|𝑥(𝑡)|²+_𝑡^𝑡+𝛿2𝑥(𝑠),𝑓(𝑥(𝑠),𝑥(𝑠−𝜏))+𝑔(𝑥(𝑠),𝑥(𝑠−𝜏))²+𝑑𝑠_{𝑡𝑡+𝛿}2𝑥(𝑠)(𝑥(𝑠)((𝑠)+,𝑔,𝑥𝑠−𝜏))𝑑𝑊_{𝑡𝑡+𝛿}2⟨𝑥(𝑠⁻),ℎ(𝑥(𝑠⁻),𝑥(𝑠⁻−𝜏))⟩+|ℎ(𝑥(𝑠⁻),𝑥(𝑠⁻−𝜏))|²𝑑𝑁(𝑠)+𝜆_𝑡^𝑡+𝛿2⟨𝑥(𝑠),ℎ(𝑥(𝑠),𝑥(𝑠−𝜏))⟩+|ℎ(𝑥(𝑠),𝑥(𝑠−𝜏))|²𝑑𝑠, (2.8) where 𝑁(𝑡)=𝑁(𝑡)−𝜆𝑡. Taking expectation and using the properties of Itô integral give 𝔼|𝑥(𝑡+𝛿)|²=𝔼|𝑥(𝑡)|²+𝔼_𝑡^𝑡+𝛿2𝑥(𝑠),𝑓(𝑥(𝑠),𝑥(𝑠−𝜏))+𝑔(𝑥(𝑠),𝑥(𝑠−𝜏))²𝑑𝑠+𝜆𝔼_𝑡^𝑡+𝛿2⟨𝑥(𝑠),ℎ(𝑥(𝑠),𝑥(𝑠−𝜏))⟩+|ℎ(𝑥(𝑠),𝑥(𝑠−𝜏))|²𝑑𝑠. (2.9) From (2.3) and (2.4), we have 𝑥(𝑠)(𝑥(𝑠)(𝑥(𝑠)(𝑥(𝑠)(((𝑥(𝑠)((2𝔼,𝑓,𝑥𝑠−𝜏))=2𝔼,𝑓,𝑥𝑠−𝜏))−𝑓0,𝑥𝑠−𝜏))+2𝔼,𝑓0,𝑥𝑠−𝜏))≤2𝛼₁𝔼|𝑥(𝑠)|²+2𝛼₂𝔼|𝑥(𝑠)‖𝑥(𝑠−𝜏)|≤2𝛼₁𝔼|𝑥(𝑠)|²+2𝛼₂sup_{𝑠−𝜏≤𝑢≤𝑠}𝔼|𝑥(𝑢)|². (2.10) Similarly, (2.5) and (2.6) yield 𝔼𝑔(𝑥(𝑠),𝑥(𝑠−𝜏))²≤𝛽₁+𝛽₂sup_{𝑠−𝜏≤𝑢≤𝑠}𝔼|𝑥(𝑢)|²,𝔼|ℎ(𝑥(𝑠),𝑥(𝑠−𝜏))|²≤𝛾₁+𝛾₂sup_{𝑠−𝜏≤𝑢≤𝑠}𝔼|𝑥(𝑢)|²,2𝔼⟨𝑥(𝑠),ℎ(𝑥(𝑠),𝑥(𝑠−𝜏))⟩≤1+𝛾₁+𝛾₂sup_{𝑠−𝜏≤𝑢≤𝑠}𝔼|𝑥(𝑢)|². (2.11) Substituting (2.10)-(2.11) into (2.9) yields 𝔼|𝑥(𝑡+𝛿)|²≤𝔼|𝑥(𝑡)|²+_𝑡^𝑡+𝛿2𝛼₁𝔼|𝑥(𝑠)|²+2𝛼₂+𝛽₁+𝛽₂+𝜆1+2𝛾₁+2𝛾₂sup_{𝑠−𝜏≤𝑢≤𝑠}𝔼|𝑥(𝑢)|²𝑑𝑠. (2.12) Let 𝜈(𝑡)=𝔼|𝑥(𝑡)|², 𝛽=2𝛼₂+𝛽₁+𝛽₂+𝜆(1+2𝛾₁+2𝛾₂), we have 𝐷⁺𝜈(𝑡)≤2𝛼₁𝜈(𝑡)+𝛽sup_{𝑡−𝜏≤𝑢≤𝑡}𝜈(𝑢), (2.13) where 𝐷⁺𝜈(𝑡)=limsup_𝛿↘0𝜈(𝑡+𝛿)−𝜈(𝑡)𝛿. (2.14) Moreover, 𝛼<0 implies −2𝛼₁>𝛽≥0. By Lemma  1.1 in [22], there exist positive constants 𝑣 and 𝑘 such that 𝜈(𝑡)≤𝑘𝑒^−𝑣𝑡,𝑡≥0. (2.15) Hence, the theorem is proven.

Based on the above result, we are going to study the stability of numerical methods for (2.1) in the following sections.

3. Compensated Stochastic 𝜃 Methods for Nonlinear SDDEs with Jumps

Since the compensated Poisson process 𝑁(𝑡)=𝑁(𝑡)−𝜆𝑡 is a martingale satisfying the property𝔼𝑁(𝑡+𝑠)−𝑁(𝑡)=0,𝔼𝑁(𝑡+𝑠)−𝑁(𝑡)²=𝜆𝑠,𝑡,𝑠≥0, (3.1) we rewrite problem (2.1) in an equivalent form:(𝑡)=𝑓(𝑥(𝑡)((𝑥(𝑡)((𝑡)(𝑥(𝑡𝑑𝑥,𝑥𝑡−𝜏))𝑑𝑡+𝑔,𝑥𝑡−𝜏))𝑑𝑊+ℎ⁻),𝑥(𝑡⁻−𝜏))𝑑𝑁(𝑡),𝑡>0, (3.2) where 𝑓(𝑥,𝑦) is defined as𝑓𝑥,𝑦∶=𝑓𝑥,𝑦+𝜆ℎ𝑥,𝑦. (3.3) Applying the stochastic 𝜃 methods to (3.2) leads to the following compensated stochastic 𝜃 methods:𝑋_𝑛+1=𝑋_𝑛+(1−𝜃)𝑓𝑋_𝑛,𝑋_𝑛−𝑚Δ𝑡+𝜃𝑓𝑋_𝑛+1,𝑋_{𝑛−𝑚+1}𝑋Δ𝑡+𝑔_𝑛,𝑋_𝑛−𝑚Δ𝑊_𝑛+ℎ𝑋_𝑛,𝑋_𝑛−𝑚Δ𝑁_𝑛. (3.4) Here, 𝑋_𝑛 denotes that the approximation to 𝑥(𝑡_𝑛), 𝜃 is a parameter with 0≤𝜃≤1, Δ𝑡∶=𝑡_𝑛+1−𝑡_𝑛 is the stepsize which satisfies 𝜏=𝑚Δ𝑡 for a positive integer 𝑚, Δ𝑊_𝑛∶=𝑊(𝑡_𝑛+1)−𝑊(𝑡_𝑛) and Δ𝑁_𝑛∶=𝑁(𝑡_𝑛+1)−𝑁(𝑡_𝑛). In particular,𝑋_𝑙=𝜑(𝑙Δ𝑡),𝑙≤0. (3.5) Note that for 𝜃>0, the numerical solutions in (3.4) are defined by implicit equations. However, due to the one-sided Lipschitz condition (2.3), (3.4) has a unique solution, with probability one, for all 𝜃Δ𝑡(𝛼₁+𝜆√𝛾₁)<1, see, for example, [23, Theorem  14.2] and (19) in [12].

Remark 3.1. Since 𝛼<0 implies 𝛼₁+𝜆√𝛾₁<0, then the compensated stochastic 𝜃 methods (3.4) produce a well-defined, unique solution if the stability condition 𝛼<0 holds.

Definition 3.2. For a give stepsize Δ𝑡=𝜏/𝑚, a numerical method on the nonlinear SDDEs with jumps (2.1) is said to be exponentially mean-square stable, if there exist positive constants 𝑁 and 𝛾, such that the numerical solution 𝑋_𝑛 produced by this method satisfies 𝔼𝑋_𝑛²≤𝑁𝜑𝑒^{−𝛾(𝑡𝑛−𝑡0)}, (3.6) for all initial data 𝜑∈𝒞^𝑏_ℱ0([−𝜏,0];ℝ^𝑑).

Definition 3.3. For a give stepsize Δ𝑡=𝜏/𝑚, a numerical method on the nonlinear SDDEs with jumps (2.1) is said to be mean-square stable if the numerical solution 𝑋_𝑛 produced by this method satisfies lim_𝑛→∞𝔼𝑋_𝑛²=0. (3.7)

4. Stability Analysis of the Numerical Solutions

In this section, we study mean-square stability and exponentially mean-square stability of the compensated stochastic 𝜃 methods (3.4). Now, we present the main results of the paper.

Theorem 4.1. Suppose that (2.3)–(2.7) hold. If 1/2≤𝜃≤1, then the compensated stochastic 𝜃 methods are mean-square stable for every stepsize Δ𝑡=𝜏/𝑚.

Proof. It follows from (3.4) that 𝑋_𝑛+1−𝜃Δ𝑡𝑓𝑋_𝑛+1,𝑋_{𝑛−𝑚+1}²=𝑋_𝑛−𝜃Δ𝑡𝑓𝑋_𝑛,𝑋_𝑛−𝑚²+2Δ𝑡𝑋_𝑛,𝑓𝑋_𝑛,𝑋_𝑛−𝑚+(Δ𝑡)²(1−2𝜃)𝑓𝑋_𝑛,𝑋_𝑛−𝑚²+𝑔𝑋_𝑛,𝑋_𝑛−𝑚Δ𝑊_𝑛²+ℎ𝑋_𝑛,𝑋_𝑛−𝑚Δ𝑁_𝑛²+𝑀_𝑛, (4.1) where 𝑀_𝑛=2𝑋_𝑛+(1−𝜃)Δ𝑡𝑓𝑋_𝑛,𝑋_𝑛−𝑚,𝑔𝑋_𝑛,𝑋_𝑛−𝑚Δ𝑊_𝑛+2𝑋_𝑛+(1−𝜃)Δ𝑡𝑓𝑋_𝑛,𝑋_𝑛−𝑚,ℎ𝑋_𝑛,𝑋_𝑛−𝑚Δ𝑁_𝑛+2𝑔𝑋_𝑛,𝑋_𝑛−𝑚Δ𝑊_𝑛,ℎ𝑋_𝑛,𝑋_𝑛−𝑚Δ𝑁_𝑛. (4.2) Thus, for 1/2≤𝜃≤1, we have 𝑋_𝑛+1−𝜃Δ𝑡𝑓𝑋_𝑛+1,𝑋_{𝑛−𝑚+1}²≤𝑋_𝑛−𝜃Δ𝑡𝑓𝑋_𝑛,𝑋_𝑛−𝑚²+2Δ𝑡𝑋_𝑛,𝑓𝑋_𝑛,𝑋_𝑛−𝑚+𝑔𝑋_𝑛,𝑋_𝑛−𝑚Δ𝑊_𝑛²+ℎ𝑋_𝑛,𝑋_𝑛−𝑚Δ𝑁_𝑛²+𝑀_𝑛. (4.3) It follows from (2.3), (2.4), and (2.6) that 2𝑋_𝑛,𝑓𝑋_𝑛,𝑋_𝑛−𝑚=2𝑋_𝑛,𝑓𝑋_𝑛,𝑋_𝑛−𝑚+2𝜆𝑋_𝑛,ℎ𝑋_𝑛,𝑋_𝑛−𝑚≤2𝛼₁𝑋_𝑛²+𝛼₂𝑋_𝑛²+𝛼₂𝑋_𝑛−𝑚²+𝜆𝑋_𝑛²+𝛾₁𝑋_𝑛²+𝛾₂𝑋_𝑛−𝑚². (4.4) Note that 𝔼(Δ𝑊_𝑛)=0, 𝔼(Δ𝑁_𝑛)=0 and 𝔼(Δ𝑁_𝑛)²=𝜆Δ𝑡. Furthermore, 𝑋_𝑛 and 𝑋_𝑛−𝑚 are all ℱ_𝑡𝑛-measurable. Therefore, we can easily obtain 𝔼𝑔𝑋_𝑛,𝑋_𝑛−𝑚Δ𝑊_𝑛²=Δ𝑡𝔼𝑔𝑋_𝑛,𝑋_𝑛−𝑚²,𝔼ℎ𝑋_𝑛,𝑋_𝑛−𝑚Δ𝑁_𝑛²=𝜆Δ𝑡𝔼ℎ𝑋_𝑛,𝑋_𝑛−𝑚²,𝔼𝑀_𝑛=0. (4.5)(4.6)(4.7) Taking expectation on both sides of (4.3) and substituting (4.4)–(4.7) into (4.3), we have 𝔼𝑋_𝑛+1−𝜃Δ𝑡𝑓𝑋_𝑛+1,𝑋_{𝑛−𝑚+1}²≤𝔼𝑋_𝑛−𝜃Δ𝑡𝑓𝑋_𝑛,𝑋_𝑛−𝑚²+Δ𝑡2𝛼₁+𝛼₂+𝛽₁+𝜆+2𝜆𝛾₁𝔼𝑋_𝑛²+Δ𝑡𝛼₂+𝛽₂+2𝜆𝛾₂𝔼𝑋_𝑛−𝑚². (4.8) Consequently, by the recursion of inequality (4.8), we have 𝔼𝑋_𝑛+1−𝜃Δ𝑡𝑓𝑋_𝑛+1,𝑋_{𝑛−𝑚+1}²≤𝔼𝑋_𝑛−𝜃Δ𝑡𝑓𝑋_𝑛,𝑋_𝑛−𝑚²+Δ𝑡2𝛼₁+𝛼₂+𝛽₁+𝜆+2𝜆𝛾₁𝔼𝑋_𝑛²+Δ𝑡𝛼₂+𝛽₂+2𝜆𝛾₂𝔼𝑋_𝑛−𝑚²≤𝔼𝑋_𝑛−1−𝜃Δ𝑡𝑓𝑋_𝑛−1,𝑋_{𝑛−𝑚−1}²+Δ𝑡2𝛼₁+𝛼₂+𝛽₁+𝜆+2𝜆𝛾₁^𝑛_{𝑗=𝑛−1}𝔼𝑋_𝑗²+Δ𝑡𝛼₂+𝛽₂+2𝜆𝛾₂^𝑛_{𝑗=𝑛−1}𝔼𝑋_𝑗−𝑚²𝑋≤⋯≤𝔼₀−𝜃Δ𝑡𝑓𝑋₀,𝑋_−𝑚²+Δ𝑡2𝛼₁+𝛼₂+𝛽₁+𝜆+2𝜆𝛾₁^𝑛_𝑗=0𝔼𝑋_𝑗²+Δ𝑡𝛼₂+𝛽₂+2𝜆𝛾₂^𝑛_𝑗=0𝔼𝑋_𝑗−𝑚². (4.9) Noting that ∑^𝑛_𝑗=0𝔼|𝑋_𝑗−𝑚|²=∑⁻¹_𝑗=−𝑚𝔼|𝑋_𝑗|²+∑^𝑛−𝑚_𝑗=0𝔼|𝑋_𝑗|² and 𝜏=𝑚Δ𝑡, we derive from (4.9) that 𝔼𝑋_𝑛+1−𝜃Δ𝑡𝑓𝑋_𝑛+1,𝑋_{𝑛−𝑚+1}²≤𝔼𝑋₀−𝜃Δ𝑡𝑓𝑋₀,𝑋_−𝑚²+Δ𝑡2𝛼₁+2𝛼₂+𝛽₁+𝛽₂+𝜆1+2𝛾₁+2𝛾₂^𝑛_𝑗=0𝔼𝑋_𝑗²+𝜏𝛼₂+𝛽₂+2𝜆𝛾₂×max_{−𝑚≤𝑗≤0}𝔼𝑋_𝑗². (4.10) Rearranging (4.10) and using the notation 𝛼 in (2.7), we obtain 𝔼𝑋_𝑛+1−𝜃Δ𝑡𝑓𝑋_𝑛+1,𝑋_{𝑛−𝑚+1}²−𝛼Δ𝑡^𝑛_𝑗=0𝔼𝑋_𝑗²≤𝐸𝑋₀−𝜃Δ𝑡𝑓𝑋₀,𝑋_−𝑚²+𝛼₂+𝛽₂+2𝜆𝛾₂𝜏max_{−𝑚≤𝑗≤0}𝔼𝑋_𝑗². (4.11)Since 𝔼‖𝜑‖²<∞ and 𝛼<0, we then derive that the series ∑^∞_𝑗=0𝔼|𝑋_𝑗|² is convergent, which implies lim_𝑛→∞𝔼|𝑋_𝑛|²=0. Consequently, for 1/2≤𝜃≤1, the compensated stochastic 𝜃 methods are mean-square stable for any stepsize Δ𝑡=𝜏/𝑚.

In order to investigate the exponential stability of the numerical methods, we need the following lemma which is Theorem  1 in [24].

Lemma 4.2 (see [24]). Suppose, for some fixed integer 𝑁≥0, that 𝑡_𝑛=𝑡₀+𝑛Δ𝑡 for some Δ𝑡>0 and {𝑣_𝑛}^∞_−𝑁 is a sequence of positive numbers that satisfies 𝑣_𝑛+1−𝑣_𝑛Δ𝑡≤−𝛼_Δ𝑡𝑣_𝑛+𝛽_Δ𝑡max_𝑗∈𝒥𝑣_𝑛+𝑗for𝑛∈𝒩 (4.12) with 𝑁=0 if 𝛽_Δ𝑡=0, where 𝒥∶={−𝑁,…,−1,0}. If 0≤𝛽_Δ𝑡<𝛼_Δ𝑡,0<𝛼_Δ𝑡Δ𝑡<1, (4.13) then 𝑣_𝑛≤{max_𝑗∈𝒥𝑣_𝑗}exp{−𝑣⁺(𝑡_𝑛−𝑡₀)}, where 𝑣⁺>0 is a constant.

Now, we present the result as follows.

Theorem 4.3. Suppose that (2.3)–(2.7) hold and the drift coefficient 𝑓 satisfies the linear growth condition, that is, there is a constant 𝐷 such that 𝑓𝑥,𝑦²≤𝐷|𝑥|²+𝑦². (4.14) Define Δ𝑡₁=−𝛼/2(1−𝜃)²(2𝐷+𝜆²(𝛾₁+𝛾₂)),Δ𝑡₂=−(2𝛼₁+𝛼₂+𝛽₁+𝜆+2𝜆𝛾₁)/2(1−𝜃)²(𝐷+𝜆²𝛾₁) and Δ𝑡₃=inf{Δ𝑡>0∶𝑃(𝜃,Δ𝑡)<0}, where 𝑃(𝜃,Δ𝑡)=2(1−𝜃)²(𝐷+𝜆²𝛾₁)(Δ𝑡)²+((1−𝜃)(2𝛼₁+𝛼₂+𝜆+𝜆𝛾₁)+𝛽₁+𝜆𝛾₁)Δ𝑡+1. If 0≤𝜃<1, and the stepsize Δ𝑡∈(0,Δ𝑡₀) with Δ𝑡₀=min{Δ𝑡₁,Δ𝑡₂,Δ𝑡₃}, then the compensated stochastic 𝜃 methods are exponentially mean-square stable.

Proof. We derive from (3.4) that 𝑋_𝑛+1−𝜃Δ𝑡𝑓𝑋_𝑛+1,𝑋_{𝑛−𝑚+1}²=𝑋_𝑛+(1−𝜃)Δ𝑡𝑓𝑋_𝑛,𝑋_𝑛−𝑚+𝑔𝑋_𝑛,𝑋_𝑛−𝑚Δ𝑊_𝑛+ℎ𝑋_𝑛,𝑋_𝑛−𝑚Δ𝑁_𝑛². (4.15) Hence, we have 𝑋_𝑛+1²≤𝑋_𝑛²+(1−𝜃)²(Δ𝑡)²𝑓𝑋_𝑛,𝑋_𝑛−𝑚²+𝑔𝑋_𝑛,𝑋_𝑛−𝑚Δ𝑊_𝑛²+ℎ𝑋_𝑛,𝑋_𝑛−𝑚Δ𝑁_𝑛²+2𝜃Δ𝑡𝑋_𝑛+1,𝑓𝑋_𝑛+1,𝑋_{𝑛−𝑚+1}+2(1−𝜃)Δ𝑡𝑋_𝑛,𝑓𝑋_𝑛,𝑋_𝑛−𝑚+𝑀_𝑛, (4.16) where 𝑀_𝑛 is defined as (4.2). From (2.6), (3.3), and (4.14), we obtain 𝑓𝑋_𝑛,𝑋_𝑛−𝑚²=𝑓𝑋_𝑛,𝑋_𝑛−𝑚+𝜆ℎ𝑋_𝑛,𝑋_𝑛−𝑚²≤2𝐷𝑋_𝑛²+𝑋_𝑛−𝑚²+𝜆²𝛾₁𝑋_𝑛²+𝛾₂𝑋_𝑛−𝑚². (4.17) Substituting (4.4)–(4.7) and (4.17) into (4.16), and taking expectation, we have 𝔼𝑋_𝑛+1²≤𝔼𝑋_𝑛²+2(1−𝜃)²(Δ𝑡)²𝐷+𝜆²𝛾₁𝔼𝑋_𝑛²+𝐷+𝜆²𝛾₂𝔼𝑋_𝑛−𝑚²+Δ𝑡𝛽₁𝔼𝑋_𝑛²+𝛽₂𝔼𝑋_𝑛−𝑚²+𝜆Δ𝑡𝛾₁𝔼𝑋_𝑛²+𝛾₂𝔼𝑋_𝑛−𝑚²+𝜃Δ𝑡2𝛼₁+𝛼₂+𝜆1+𝛾₁𝔼𝑋_𝑛+1²+𝛼₂+𝜆𝛾₂𝔼𝑋_{𝑛−𝑚+1}²+(1−𝜃)Δ𝑡2𝛼₁+𝛼₂+𝜆1+𝛾₁𝔼𝑋_𝑛²+𝛼₂+𝜆𝛾₂𝔼𝑋_𝑛−𝑚², (4.18) which yields 1−𝜃Δ𝑡2𝛼₁+𝛼₂+𝜆1+𝛾₁𝔼𝑋_𝑛+1²≤1−𝜃Δ𝑡2𝛼₁+𝛼₂+𝜆1+𝛾₁𝔼𝑋_𝑛²+2𝛼₁+𝛼₂+𝛽₁+𝜆+2𝜆𝛾₁()Δ𝑡+21−𝜃²𝐷+𝜆²𝛾₁(Δ𝑡)²𝔼𝑋_𝑛²+𝛼₂+𝛽₂+2𝜆𝛾₂Δ𝑡+2(1−𝜃)²𝐷+𝜆²𝛾₂(Δ𝑡)²max_{𝑛−𝑚≤𝑖≤𝑛−𝑚+1}𝔼𝑋_𝑖². (4.19) Hence, 𝔼𝑋_𝑛+1²−𝔼𝑋_𝑛²𝑋Δ𝑡≤−𝐴𝔼_𝑛²+𝐵max_{𝑛−𝑚≤𝑖≤𝑛−𝑚+1}𝔼𝑋_𝑖², (4.20) where 𝐴=−2𝛼₁+𝛼₂+𝛽₁+𝜆+2𝜆𝛾₁+2(1−𝜃)²𝐷+𝜆²𝛾₁Δ𝑡1−𝜃Δ𝑡2𝛼₁+𝛼₂+𝜆1+𝛾₁,𝛼𝐵=²+𝛽₂+2𝜆𝛾₂+2(1−𝜃)²𝐷+𝜆²𝛾₂Δ𝑡1−𝜃Δ𝑡2𝛼₁+𝛼₂+𝜆1+𝛾₁. (4.21) By Lemma 4.2, we derive that the methods are exponentially mean-square stable if 0≤𝐵<𝐴,0<𝐴Δ𝑡<1. (4.22) That is, Δ𝑡<−𝛼2(1−𝜃)²2𝐷+𝜆²𝛾₁+𝛾₂,Δ𝑡<−2𝛼₁+𝛼₂+𝛽₁+𝜆+2𝜆𝛾₁2(1−𝜃)²𝐷+𝜆²𝛾₁,𝑃(𝜃,Δ𝑡)>0, (4.23) where 𝑃(𝜃,Δ𝑡)=2(1−𝜃)²(𝐷+𝜆²𝛾₁)(Δ𝑡)²+((1−𝜃)(2𝛼₁+𝛼₂+𝜆+𝜆𝛾₁)+𝛽₁+𝜆𝛾₁)Δ𝑡+1. Since 𝑃(𝜃,0)=1, there must exist Δ𝑡₃>0 such that 𝑃(𝜃,Δ𝑡)>0 when Δ𝑡<Δ𝑡₃. On the other hand, if 𝑃(𝜃,Δ𝑡)≥0 is always true, we then define Δ𝑡₃ as ∞. Therefore, let Δ𝑡₁=−𝛼/2(1−𝜃)²(2𝐷+𝜆²(𝛾₁+𝛾₂)), Δ𝑡₂=−(2𝛼₁+𝛼₂+𝛽₁+𝜆+2𝜆𝛾₁)/2(1−𝜃)²(𝐷+𝜆²𝛾₁), Δ𝑡₃=inf{Δ𝑡>0∶𝑃(𝜃,Δ𝑡)<0} and Δ𝑡₀=min{Δ𝑡₁,Δ𝑡₂,Δ𝑡₃}, then (4.22) holds when Δ𝑡∈(0,Δ𝑡₀), which completes the proof of Theorem 4.3.

By the proof of Theorem 4.3, we can easily obtain the following result.

Theorem 4.4. Suppose that (2.3)–(2.7) hold. If 𝜃=1, then the compensated stochastic 𝜃-method is exponentially mean-square stable for every stepsize Δ𝑡=𝜏/𝑚.

5. Numerical Examples

The purpose of this section is to illustrate our theoretical results presented in the previous section by numerical experiments. We first consider the following nonlinear scalar SDDEs with jumps:𝑑𝑥(𝑡)=−4𝑥(𝑡)−𝑥³(𝑡)+𝑥(𝑡−1)𝑑𝑡+sin(𝑥(𝑡−1))𝑑𝑊(𝑡)−𝑥(𝑡⁻)(𝑡)𝑥(𝑡)[],𝑑𝑁,𝑡>0,=1,𝑡∈−1,0 (5.1) where 𝑁(𝑡) is a scalar Poisson process with parameter 𝜆=1. In this case, (2.3)–(2.6) are satisfied with 𝛼₁=−4,𝛼₂=1,𝛽₁=0,𝛽₂=1,𝛾₁=1,𝛾₂=0, and 𝜏=1. So we have 𝛼=−2 in (2.7), which guarantees mean-square stability of the zero solution of (5.1) by Theorem 2.2.

The following numerical experiments will show how the parameter 𝜃 and the stepsize Δ𝑡 influence the mean-square stability of the compensated stochastic 𝜃 methods. We simulate the expectation of |𝑋_𝑛|² by using 1000 trajectories, that is,𝔼𝑋_𝑛²≈11000¹⁰⁰⁰_𝑖=1𝑋_𝑛𝜔_𝑖². (5.2)

Theorem 4.1 shows that the compensated stochastic 𝜃 methods are mean-square stable for every stepsize Δ𝑡=𝜏/𝑚 when 1/2≤𝜃≤1. In Figure 1, we use (3.4) to solve (5.1) and choose the parameter 𝜃 with different values 0.5 and 0.8, and we take the stepsize Δ𝑡=1/8,1/4,1/2, and 1, respectively. We can find that the compensated stochastic 𝜃 methods are mean-square stable with these stepsizes.

(a)

(b)

Figure 1 

Fixed 𝜃=0.5 (a) and 𝜃=0.8 (b) with different stepsize Δ𝑡=1/8,1/4,1/2,1.

Theorem 4.3 shows that the compensated stochastic 𝜃 methods are exponentially mean-square stable if the stepsize Δ𝑡∈(0,Δ𝑡₀) when 0≤𝜃<1. Now, we consider the following nonlinear scalar SDDEs with jumps:𝑑𝑥(𝑡)=−3𝑥(𝑡)𝑑𝑡+sin(𝑥(𝑡−1))𝑑𝑊(𝑡)−𝑥(𝑡⁻)(𝑡)𝑥(𝑡)[],−1𝑑𝑁,𝑡>0,=1,𝑡∈−1,0 (5.3) where 𝑁(𝑡) is a scalar Poisson process with parameter 𝜆=1. (2.3)–(2.7) and (4.14) are satisfied with 𝛼₁=−3,𝛼₂=0,𝛽₁=0,𝛽₂=1,𝛾₁=0,𝛾₂=1,𝜏=1,𝐷=9, and 𝛼=−2. Therefore, the zero solution of (5.3) is exponentially mean-square stable. By Theorem 4.3, we calculate Δ𝑡₁=1/19(1−𝜃)², Δ𝑡₂=5/18(1−𝜃)², and 𝑃(𝜃,Δ𝑡)=18(1−𝜃)²(Δ𝑡)²−5(1−𝜃)Δ𝑡+1. It is easy to see that 𝑃(𝜃,Δ𝑡)>0 for every Δ𝑡>0, then we get Δ𝑡₃=∞. Therefore, we obtain Δ𝑡₀=min{Δ𝑡₁,Δ𝑡₂,Δ𝑡₃}=1/19(1−𝜃)², which implies that the methods applied to (5.3) have less restrictions on the stepsize as the value of 𝜃 increases. Now, we use (3.4) to solve (5.3) and choose the parameter 𝜃=0 and 𝜃=0.2, and we take the stepsize Δ𝑡=1/20,1/8,1/2, and 1, respectively. By Theorem 4.3, we compute that Δ𝑡₀=0.0526 when 𝜃=0 and Δ𝑡₀=0.0822 when 𝜃=0.2. Figure 2 indicates that both methods are exponentially mean-square stable if the stepsize Δ𝑡=0.05, which is well selected in Δ𝑡∈(0,Δ𝑡₀). It also shows that the methods maybe stable when the stepsize is bigger than Δ𝑡₀, since both methods are stable when stepsize Δ𝑡=0.125, but they are not stable for stepsize Δ𝑡=1. This indicates that the restriction of the stepsize Δ𝑡₀ in Theorem 4.3 is not theoretical optimal. From Figure 2, we also can find that the methods behave better stability when the value of 𝜃 increases and the stepsize Δ𝑡 decreases.

(a)

(b)

Figure 2 

Fixed 𝜃=0 (a) and 𝜃=0.2 (b) with different stepsize Δ𝑡=1/20,1/8,1/2,1.

Acknowledgments

The authors thank the anonymous referee and the editors for their detailed comments and helpful suggestions. This work was supported by NSF of China (no.11171352) and Hunan Provincial Innovation Foundation for Postgraduates (no. CX2010B118).