Almost Surely Asymptotic Stability of Numerical Solutions for Neutral Stochastic Delay Differential Equations

Abstract

We investigate the almost surely asymptotic stability of Euler-type methods for neutral stochastic delay differential equations (NSDDEs) using the discrete semimartingale convergence theorem. It is shown that the Euler method and the backward Euler method can reproduce the almost surely asymptotic stability of exact solutions to NSDDEs under additional conditions. Numerical examples are demonstrated to illustrate the effectiveness of our theoretical results.

1. Introduction

The neutral stochastic delay differential equation (NSDDE) has attracted much more attention, and much work (see [1–4]) has been done. For example, Mao [2] studied the existence and uniqueness, moment and pathwise estimates, and the exponential stability of the solution to the NSDDE. Moreover, Mao et al. [4] studied the almost surely asymptotic stability of the NEDDE with Markovian switching:𝑑[𝑥(𝑡)−𝑁(𝑥(𝑡−𝜏),𝑟(𝑡))]=𝑓(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏),𝑟(𝑡))𝑑𝑡+𝑔(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏),𝑟(𝑡))𝑑𝐵(𝑡). (1.1)

Since most NSDDEs cannot be solved explicitly, numerical solutions have become an important issue in the study of NSDDEs. Convergence analysis of numerical methods for NSDDEs can be found in [5–7]. On the other hand, stability theory of numerical solutions is one of the fundamental research topics in the numerical analysis. For stochastic differential equations (SDEs) as well as stochastic delay differential equations (SDDEs), moment stability and asymptotic stability of numerical solutions have received much more attention (e.g., [8–13] for moment stability and [12–14] for asymptotic stability). Recently, Wang and Chen [15] studied the mean-square stability of the semi-implicit Euler method for NSDDEs. We aim in this paper to study the almost surely asymptotic stability of Euler-type methods for NSDDEs using the discrete semimartingale convergence theorem. The discrete semimartingale convergence theorem (cf. [16, 17]) plays an important role in the almost surely asymptotic stability analysis of numerical solutions to SDEs and SDDEs [17–19]. Using the discrete semimartingale convergence theorem, we show that Euler-type methods for NSDDEs can preserve the almost surely asymptotic stability of exact solutions under additional conditions.

In Section 2, we introduce some necessary notations and state the discrete semimartingale convergence theorem as a lemma. In Section 3, we study the almost surely asymptotic stability of exact solutions to NSDDEs. Section 4 gives the almost surely asymptotic stability of the Euler method. In Section 5, we discuss the almost surely asymptotic stability of the backward Euler method. Numerical experiments are presented in Section 6.

2. Preliminaries

Throughout this paper, unless otherwise specified, we use the following notations. Let (Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝑃) be a complete probability space with filtration {ℱ_𝑡}_𝑡≥0 satisfying the usual conditions (i.e., it is right continuous and ℱ₀ contains all 𝑃-null sets). 𝐵(𝑡) is a scalar Brownian motion defined on the probability space. |⋅| denotes the Euclidean norm in 𝑅^𝑛. The inner product of 𝑥,𝑦 in 𝑅^𝑛 is denoted by ⟨𝑥,𝑦⟩ or 𝑥^𝑇𝑦. 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). Let 𝜏>0 and 𝐶([−𝜏,0];𝑅^𝑛) denote the family of all continuous 𝑅^𝑛-valued functions on [−𝜏,0]. Let 𝐶^𝑏_ℱ0([−𝜏,0];𝑅^𝑛) be the family of all ℱ₀-measurable bounded 𝐶([−𝜏,0];𝑅^𝑛)-valued random variables 𝜉={𝜉(𝜃)∶−𝜏≤𝜃≤0}.

Consider an 𝑛-dimensional NSDDE𝑑[𝑥(𝑡)−𝑁(𝑥(𝑡−𝜏))]=𝑓(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏))𝑑𝑡+𝑔(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏))𝑑𝐵(𝑡), (2.1) on 𝑡≥0 with initial data {𝑥(𝜃)∶−𝜏≤𝜃≤0}=𝜉∈𝐶^𝑏_ℱ0([−𝜏,0];𝑅^𝑛). Here 𝑁∶𝑅^𝑛→𝑅^𝑛, 𝑓∶𝑅⁺×𝑅^𝑛×𝑅^𝑛→𝑅^𝑛, and 𝑔∶𝑅⁺×𝑅^𝑛×𝑅^𝑛→𝑅^𝑛.

Let 𝐶^1,2(𝑅⁺×𝑅^𝑛;𝑅⁺) denote the family of all nonnegative functions 𝑉(𝑡,𝑥) on 𝑅⁺×𝑅^𝑛 which are continuously once differentiable in 𝑡 and twice differentiable in 𝑥. For each 𝑉∈𝐶^1,2(𝑅⁺×𝑅^𝑛;𝑅⁺), define an operator LV from 𝑅⁺×𝑅^𝑛×𝑅^𝑛 to 𝑅 by LV𝑡,𝑥,𝑦=𝑉_𝑡𝑡,𝑥−𝑁𝑦+𝑉_𝑥𝑦𝑓+1𝑡,𝑥−𝑁𝑡,𝑥,𝑦2trace𝑔^𝑇𝑡,𝑥,𝑦𝑉_𝑥𝑥𝑡,𝑥−𝑁𝑦𝑔𝑡,𝑥,𝑦, (2.2) where 𝑉_𝑡()=()𝑡,𝑥𝜕𝑉𝑡,𝑥𝜕𝑡,𝑉_𝑥()=()𝑡,𝑥𝜕𝑉𝑡,𝑥𝜕𝑥₁(),…,𝜕𝑉𝑡,𝑥𝜕𝑥_𝑛,𝑉_𝑥𝑥()=𝜕𝑡,𝑥²𝑉(𝑡,𝑥)𝜕𝑥_𝑖𝜕𝑥_{𝑗𝑛×𝑛}. (2.3)

As a standing hypothesis, we impose the following assumption on the coefficients 𝑁,𝑓, and 𝑔.

Assumption 2.1. Assume that both 𝑓 and 𝑔 satisfy the local Lipschitz condition. That is, for each integer 𝑖>0, there exists a positive constant 𝐾_𝑖 such that 𝑓𝑡,𝑥,𝑦−𝑓𝑡,𝑥,𝑦²∨𝑔𝑡,𝑥,𝑦−𝑔𝑡,𝑥,𝑦²≤𝐾_𝑖|𝑥−𝑥|²+𝑦−𝑦² (2.4) for 𝑥,𝑦,𝑥,𝑦∈𝑅^𝑛 with |𝑥|∨|𝑥|∨|𝑦|∨|𝑦|≤𝑖 and 𝑡∈𝑅⁺. Assume also that there is a constant 𝜅∈(0,1) such that 𝑁(𝑥)−𝑁𝑦≤𝜅𝑥−𝑦,∀𝑥,𝑦∈𝑅^𝑛. (2.5) Assume moreover that for all 𝑡∈𝑅⁺, 𝑁(0)=0,𝑓(𝑡,0,0)=0,𝑔(𝑡,0,0)=0. (2.6)

The following discrete semimartingale convergence theorem (cf. [16, 17]) will play an important role in this paper.

Lemma 2.2. Let {𝐴_𝑖} and {𝑈_𝑖} be two sequences of nonnegative random variables such that both 𝐴_𝑖 and 𝑈_𝑖 are ℱ_𝑖-measurable for 𝑖=1,2,…, and 𝐴₀=𝑈₀=0 a.s. Let 𝑀_𝑖 be a real-valued local martingale with 𝑀₀=0 a.s. Let 𝜁 be a nonnegative ℱ₀-measurable random variable. Assume that {𝑋_𝑖} is a nonnegative semimartingale with the Doob-Mayer decomposition 𝑋_𝑖=𝜁+𝐴_𝑖−𝑈_𝑖+𝑀_𝑖. (2.7) If lim_𝑖→∞𝐴_𝑖<∞ a.s., then for almost all 𝜔∈Ωlim_𝑖→∞𝑋_𝑖<∞,lim_𝑖→∞𝑈_𝑖<∞, (2.8) that is, both 𝑋_𝑖 and 𝑈_𝑖 converge to finite random variables.

3. Almost Surely Asymptotic Stability of the Exact Solution

In this section, we will study the almost surely asymptotic stability of exact solutions to (2.1). To be precise, let us give the definition on the almost surely asymptotic stability of exact solutions.

Definition 3.1. The solution 𝑥(𝑡) to (2.1) is said to be almost surely asymptotically stable if lim_𝑡→∞𝑥(𝑡)=0a.s. (3.1) for any initial data 𝜉∈𝐶^𝑏_ℱ0([−𝜏,0];𝑅^𝑛).

Theorem 3.2. Let Assumption 2.1 hold. Assume that there are four positive constants 𝜆₁−𝜆₄ such that 2𝑥−𝑁𝑦^𝑇𝑓𝑡,𝑥,𝑦≤−𝜆₁|𝑥|²+𝜆₂𝑦²,𝑔𝑡,𝑥,𝑦²≤𝜆₃|𝑥|²+𝜆₄𝑦² (3.2) for 𝑡≥0 and 𝑥,𝑦∈𝑅^𝑛. If 𝜆₁−𝜆₃>𝜆₂+𝜆₄, (3.3) then, for any initial data 𝜉∈𝐶^𝑏_ℱ0([−𝜏,0];𝑅^𝑛), there exists a unique global solution 𝑥(𝑡) to (2.1) and the solution 𝑥(𝑡) is almost surely asymptotically stable.

Proof. Let 𝑈(𝑡,𝑥)=|𝑥|². Using (3.2) and (3.3), we have LU𝑡,𝑥,𝑦=2𝑥−𝑁𝑦^𝑇𝑓𝑡,𝑥,𝑦+𝑔𝑡,𝑥,𝑦²≤−𝜆₁−𝜆₃|𝑥|²+𝜆₂+𝜆₄𝑦²≤−𝜆₁−𝜆₃|𝑥|²+𝜆₁−𝜆₃𝑦². (3.4) Then, from Theorem 3.1 in [4], we conclude that there exists a unique global solution 𝑥(𝑡) to (2.1) for any initial data 𝜉∈𝐶^𝑏_ℱ0([−𝜏,0];𝑅^𝑛). According to (3.3), there is a constant 𝛼>0 such that 𝜆₁−𝜆₃−2𝛼≥𝜆₂+𝜆₄+2𝛼𝑒^𝛼𝜏∀0≤𝛼≤𝛼. (3.5) Let 𝑉(𝑡,𝑥)=𝑒^𝛼𝑡𝑈(𝑡,𝑥). Here 𝛼∈(0,𝛼]∩(0,(2/𝜏)log(1/𝜅)). Then LV𝑡,𝑥,𝑦=𝑒^𝛼𝑡𝛼𝑈𝑡,𝑥−𝑁𝑦+LU(𝑡,𝑥)≤𝑒^𝛼𝑡𝛼𝑥−𝑁𝑦²−𝜆₁−𝜆₃|𝑥|²+𝜆₂+𝜆₄𝑦²≤𝑒^𝛼𝑡−𝜆₁−𝜆₃−2𝛼|𝑥|²+𝜆₂+𝜆₄+2𝛼𝑦²≤−𝑊(𝑡,𝑥)+𝑊𝑡−𝜏,𝑦, (3.6) where 𝑊(𝑡,𝑥)=(𝜆₁−𝜆₃−2𝛼)𝑒^𝛼𝑡|𝑥|². By Theorem 4.1 in [4], we can obtain that the solution 𝑥(𝑡) is almost surely asymptotically stable. The proof is completed.

Theorem 3.2 gives sufficient conditions of the almost surely asymptotic stability of the NSDDE (2.1). Based on these sufficient conditions, we will investigate the almost surely asymptotic stability of Euler-type methods in the following sections.

4. Stability of the Euler Method

Applying the Euler method (EM) to (2.1) yields𝑋_𝑘+1−𝑁𝑋_{𝑘+1−𝑚}=𝑋_𝑘−𝑁𝑋_𝑘−𝑚+𝑓𝑘ℎ,𝑋_𝑘,𝑋_𝑘−𝑚ℎ+𝑔𝑘ℎ,𝑋_𝑘,𝑋_𝑘−𝑚Δ𝐵_𝑘𝑋,𝑘=0,1,2,…,_𝑘=𝜉(𝑘ℎ),𝑘=−𝑚,−𝑚+1,…,0. (4.1) Here ℎ=𝜏/𝑚 (𝑚 is an positive integer) is the stepsize, and Δ𝐵_𝑘=𝐵((𝑘+1)ℎ)−𝐵(𝑘ℎ) represents the Browian motion increment. To be precise, let us introduce the definition on the almost surely asymptotic stability of numerical solutions.

Definition 4.1. The numerical solution 𝑋_𝑘 to (2.1) is said to be almost surely asymptotically stable if lim_𝑘→∞𝑋_𝑘=0a.s. (4.2) for any bounded variables 𝜉(𝑘ℎ), 𝑘=−𝑚,−𝑚+1,…,0.

Theorem 4.2. Let conditions (3.2)-(3.3) hold. Assume that 𝑓 satisfies the linear growth condition, namely, there exists a positive constant 𝐿 such that 𝑓𝑡,𝑥,𝑦²≤𝐿|𝑥|²+𝑦². (4.3) Then there exists a ℎ₀>0 such that if ℎ<ℎ₀, then for any given finite-valued ℱ₀-measurable random variables 𝜉(𝑘ℎ), 𝑘=−𝑚,−𝑚+1,…,0, the EM approximate solution (4.1) is almost surely asymptotically stable.

Proof. Let 𝑌_𝑘=𝑋_𝑘−𝑁(𝑋_𝑘−𝑚). Then, it follows from (4.1) that 𝑌_𝑘+1=𝑌_𝑘+𝑓𝑘ℎ,𝑋_𝑘,𝑋_𝑘−𝑚ℎ+𝑔𝑘ℎ,𝑋_𝑘,𝑋_𝑘−𝑚Δ𝐵_𝑘,𝑘=0,1,2,…. (4.4) Squaring both sides of (4.4), we have 𝑌_𝑘+1²=𝑌_𝑘²+𝑓𝑘ℎ,𝑋_𝑘,𝑋_𝑘−𝑚²ℎ²+𝑔𝑘ℎ,𝑋_𝑘,𝑋_𝑘−𝑚²ℎ+2𝑌_𝑘,𝑓𝑘ℎ,𝑋_𝑘,𝑋_𝑘−𝑚ℎ+2𝑌_𝑘,𝑔𝑘ℎ,𝑋_𝑘,𝑋_𝑘−𝑚Δ𝐵_𝑘+2ℎ𝑓𝑘ℎ,𝑋_𝑘,𝑋_𝑘−𝑚,𝑔𝑘ℎ,𝑋_𝑘,𝑋_𝑘−𝑚Δ𝐵_𝑘+𝑔𝑘ℎ,𝑋_𝑘,𝑋_𝑘−𝑚²Δ𝐵²_𝑘−ℎ. (4.5) Using (3.2) and (4.3), we can obtain that 𝑌_𝑘+1²≤𝑌_𝑘²+𝐿𝑋_𝑘²+𝑋_𝑘−𝑚²ℎ²+𝜆₃𝑋_𝑘²+𝜆₄𝑋_𝑘−𝑚²ℎ+−𝜆₁𝑋_𝑘²+𝜆₂𝑋_𝑘−𝑚²ℎ+𝑚_𝑘, (4.6) where 𝑚_𝑘=2𝑌_𝑘,𝑔𝑘ℎ,𝑋_𝑘,𝑋_𝑘−𝑚Δ𝐵_𝑘+2ℎ𝑓𝑘ℎ,𝑋_𝑘,𝑋_𝑘−𝑚,𝑔𝑘ℎ,𝑋_𝑘,𝑋_𝑘−𝑚Δ𝐵_𝑘+𝑔𝑘ℎ,𝑋_𝑘,𝑋_𝑘−𝑚²Δ𝐵²_𝑘−ℎ. (4.7) It therefore follows that 𝑌_𝑘+1²−𝑌_𝑘²≤−𝜆₁−𝜆₃−𝐿ℎℎ𝑋_𝑘²+𝜆₂+𝜆₄+𝐿ℎℎ𝑋_𝑘−𝑚²+𝑚_𝑘, (4.8) which implies that 𝑌_𝑘²−𝑌₀²≤−𝜆₁−𝜆₃−𝐿ℎℎ^𝑘−1_𝑖=0𝑋_𝑖²+𝜆₂+𝜆₄+𝐿ℎℎ^𝑘−1_𝑖=0𝑋_𝑖−𝑚²+^𝑘−1_𝑖=0𝑚_𝑖. (4.9) Note that ^𝑘−1_𝑖=0𝑋_𝑖−𝑚²=⁻¹_𝑖=−𝑚𝑋_𝑖²+^{𝑘−𝑚−1}_𝑖=0𝑋_𝑖². (4.10) Then, we have 𝑌_𝑘²+𝜆₂+𝜆₄+𝐿ℎℎ^𝑘−1_{𝑖=𝑘−𝑚}𝑋_𝑖²≤𝑌₀²+𝜆₂+𝜆₄+𝐿ℎℎ⁻¹_𝑖=−𝑚𝑋_𝑖²−𝜆₁−𝜆₃−𝜆₂−𝜆₄−2𝐿ℎℎ^𝑘−1_𝑖=0𝑋_𝑖²+𝑀_𝑘, (4.11) where 𝑀_𝑘=∑^𝑘−1_𝑖=0𝑚_𝑖. By [18], 𝑀_𝑘 is a martingale with 𝑀₀=0. From (3.3), we obtain that 𝜆₁−𝜆₃−𝜆₂−𝜆₄−2𝐿ℎ>0,as0<ℎ<ℎ₀, (4.12) where ℎ₀=(𝜆₁−𝜆₃−𝜆₂−𝜆₄)/2𝐿. Hence, from Lemma 2.2, we therefore have lim_𝑘→∞^𝑘−1_𝑖=0𝑋_𝑖²<∞a.s.,0<ℎ<ℎ₀. (4.13) Then, we conclude that lim_𝑘→∞𝑋_𝑘²=0a.s.,0<ℎ<ℎ₀. (4.14) The proof is completed.

Theorem 4.2 shows that if the coefficient 𝑓 obeys the linear growth condition, in addition to the conditions imposed in Theorem 3.2, then the EM approximate solution (4.1) reproduces the almost surely asymptotic stability of exact solutions to (2.1) for sufficiently small stepsize.

5. Stability of the Backward Euler Method

Applying the backward Euler method (BEM) to (2.1) yields𝑋_𝑘+1−𝑁𝑋_{𝑘+1−𝑚}=𝑋_𝑘−𝑁𝑋_𝑘−𝑚+𝑓(𝑘+1)ℎ,𝑋_𝑘+1,𝑋_{𝑘+1−𝑚}ℎ+𝑔𝑘ℎ,𝑋_𝑘,𝑋_𝑘−𝑚Δ𝐵_𝑘𝑋,𝑘=0,1,2,…,_𝑘=𝜉(𝑘ℎ),𝑘=−𝑚,−𝑚+1,…,0. (5.1) As a standing hypothesis, we assume that the BEM (5.1) is well defined. The following theorem shows that if the above assumption and the conditions imposed in Theorem 3.2 hold, then the BEM approximate solution (5.1) inherits the almost surely asymptotic stability of exact solutions to (2.1) without any stepsize restriction.

Theorem 5.1. Let conditions (3.2)-(3.3) hold. Then for any given finite-valued ℱ₀-measurable random variables 𝜉(𝑘ℎ), 𝑘=−𝑚,−𝑚+1,…,0, the BEM approximate solution (5.1) is almost surely asymptotically stable.

Proof. Let 𝑌_𝑘=𝑋_𝑘−𝑁(𝑋_𝑘−𝑚). Then, it follows from (5.1) that 𝑌_𝑘+1−𝑓(𝑘+1)ℎ,𝑋_𝑘+1,𝑋_{𝑘+1−𝑚}ℎ=𝑌_𝑘+𝑔𝑘ℎ,𝑋_𝑘,𝑋_𝑘−𝑚Δ𝐵_𝑘,𝑘=0,1,2,…. (5.2) Squaring both sides of (5.2), we have 𝑌_𝑘+1²+𝑓(𝑘+1)ℎ,𝑋_𝑘+1,𝑋_{𝑘+1−𝑚}²ℎ²=𝑌_𝑘²+2𝑌_𝑘+1,𝑓(𝑘+1)ℎ,𝑋_𝑘+1,𝑋_{𝑘+1−𝑚}ℎ+|𝑔𝑘ℎ,𝑋_𝑘,𝑋_𝑘−𝑚|²ℎ+2𝑌_𝑘,𝑔𝑘ℎ,𝑋_𝑘,𝑋_𝑘−𝑚Δ𝐵_𝑘+𝑔𝑘ℎ,𝑋_𝑘,𝑋_𝑘−𝑚²Δ𝐵²_𝑘−ℎ. (5.3) Using (3.2), we can obtain that 𝑌_𝑘+1²≤𝑌_𝑘²+−𝜆₁𝑋_𝑘+1²+𝜆₂𝑋_{𝑘+1−𝑚}²ℎ+𝜆₃𝑋_𝑘²+𝜆₄𝑋_𝑘−𝑚²ℎ+𝑚_𝑘, (5.4) where 𝑚_𝑘=2𝑌_𝑘,𝑔𝑘ℎ,𝑋_𝑘,𝑋_𝑘−𝑚Δ𝐵_𝑘+𝑔𝑘ℎ,𝑋_𝑘,𝑋_𝑘−𝑚²Δ𝐵²_𝑘−ℎ. (5.5) It therefore follows that 𝑌_𝑘+1²−𝑌_𝑘²≤−𝜆₁ℎ𝑋_𝑘+1²+𝜆₃ℎ𝑋_𝑘²+𝜆₂ℎ𝑋_{𝑘+1−𝑚}²+𝜆₄ℎ𝑋_𝑘−𝑚²+𝑚_𝑘, (5.6) which implies that 𝑌_𝑘²−𝑌₀²≤−𝜆₁ℎ^𝑘−1_𝑖=0𝑋_𝑖+1²+𝜆₃ℎ^𝑘−1_𝑖=0𝑋_𝑖²+𝜆₂ℎ^𝑘−1_𝑖=0𝑋_{𝑖+1−𝑚}²+𝜆₄ℎ^𝑘−1_𝑖=0𝑋_𝑖−𝑚²+^𝑘−1_𝑖=0𝑚_𝑖. (5.7) Note that ^𝑘−1_𝑖=0𝑋_𝑖−𝑚²=⁻¹_𝑖=−𝑚𝑋_𝑖²+^{𝑘−𝑚−1}_𝑖=0𝑋_𝑖²,_{𝑘−1𝑖=0}𝑋_{𝑖+1−𝑚}²=⁻¹_{𝑖=−𝑚+1}𝑋_𝑖²+^𝑘−𝑚_𝑖=0𝑋_𝑖². (5.8) Then, we have 𝑌_𝑘²≤𝑌₀²+𝜆₂ℎ⁻¹_{𝑖=−𝑚+1}𝑋_𝑖²+𝜆₄ℎ⁻¹_𝑖=−𝑚𝑋_𝑖²−𝜆₁ℎ^𝑘_𝑖=1𝑋_𝑖²+𝜆₃ℎ^𝑘−1_𝑖=0𝑋_𝑖²+𝜆₂ℎ^𝑘−𝑚_𝑖=0𝑋_𝑖²+𝜆₄ℎ^{𝑘−𝑚−1}_𝑖=0𝑋_𝑖²+^𝑘−1_𝑖=0𝑚_𝑖. (5.9) Namely, 𝑌_𝑘²+𝜆₁ℎ𝑋_𝑘²+𝜆₂ℎ^𝑘−1_{𝑖=𝑘−𝑚+1}𝑋_𝑖²+𝜆₄ℎ^𝑘−1_{𝑖=𝑘−𝑚}𝑋_𝑖²≤𝑌₀²+𝜆₁ℎ𝑋₀²+𝜆₂ℎ⁻¹_{𝑖=−𝑚+1}𝑋_𝑖²+𝜆₄ℎ⁻¹_𝑖=−𝑚𝑋_𝑖²−𝜆₁−𝜆₃−𝜆₂−𝜆₄ℎ^𝑘−1_𝑖=0𝑋_𝑖²+𝑀_𝑘, (5.10) where 𝑀_𝑘=∑^𝑘−1_𝑖=0𝑚_𝑖. By [18], 𝑀_𝑘 is a martingale with 𝑀₀=0. From (3.3), we obtain that 𝜆₁−𝜆₃−𝜆₂−𝜆₄>0. (5.11) Using Lemma 2.2 yields lim_𝑘→∞^𝑘−1_𝑖=0𝑋_𝑖²<∞a.s.,ℎ>0. (5.12) Then, we conclude that lim_𝑘→∞𝑋_𝑘²=0a.s.,ℎ>0. (5.13) The proof is completed.

6. Numerical Experiments

In this section, we present numerical experiments to illustrate the theoretical results presented in the previous sections.

Consider the following scalar linear problem𝑑𝑥(𝑡)−12(𝑥(=((𝑡)(𝑥(()(𝑡)𝑥(𝑡)sin𝑡−2))−8𝑥+sin𝑡−2)))𝑑𝑡+𝑥𝑡−2𝑑𝐵,𝑡≥0,=𝑡+1,−2≤𝑡≤0. (6.1) For test (6.1), we have that 𝐿=72, 𝜆₁=11, 𝜆₂=4, 𝜆₃=0, and 𝜆₄=1. By Theorem 3.2, the exact solution to (6.1) is almost surely asymptotically stable.

Theorem 4.2 shows that the EM approximate solution to (6.1) can preserve the almost surely asymptotic stability of exact solutions for ℎ<1/24. In Figure 1, we compute three different paths (𝑋_𝑛(𝜔₁),𝑋_𝑛(𝜔₂),𝑋_𝑛(𝜔₃)) using EM (4.1) to approximate (6.1) with the stepsize ℎ=1/25. Figure 1 shows that 𝑋_𝑛(𝜔₁),𝑋_𝑛(𝜔₂),𝑋_𝑛(𝜔₃) are asymptotically stable. Theorem 5.1 shows that the BEM approximate solution to (6.1) reproduces the almost surely asymptotic stability of exact solutions for any ℎ>0. In Figure 2, three different paths (𝑋_𝑛(𝜔₁),𝑋_𝑛(𝜔₂),𝑋_𝑛(𝜔₃)) are computed by using the BEM (5.1) to approximate (6.1) with the stepsize ℎ=0.1. Figure 2 demonstrates that these paths are asymptotically stable.

Figure 1 

Almost surely asymptotic stability of the EM approximate solution 𝑋𝑛 with the stepsize ℎ=1/25.

Figure 2 

Almost surely asymptotic stability of the BEM approximate solution 𝑋𝑛 with the stepsize ℎ=0.1.

7. Conclusions

This paper deals with the almost surely asymptotic stability of Euler-type methods for NSDDEs by using the discrete semimartingale convergence theorem. We show that the EM reproduces the almost surely asymptotic stability of exact solutions to NSDDEs under an additional linear growth condition. If we assume the BEM is well defined, the BEM can also preserve the almost surely asymptotic stability without the additional linear growth condition.

Acknowledgments

The authors would like to thank the referees for their helpful comments and suggestions. This work is supported by the NSF of China (no.11071050).