Almost Surely Asymptotic Stability of Exact and Numerical Solutions for Neutral Stochastic Pantograph Equations

Abstract

We study the almost surely asymptotic stability of exact solutions to neutral stochastic pantograph equations (NSPEs), and sufficient conditions are obtained. Based on these sufficient conditions, we show that the backward Euler method (BEM) with variable stepsize can preserve the almost surely asymptotic stability. Numerical examples are demonstrated for illustration.

1. Introduction

The neutral pantograph equation (NPE) plays important roles in mathematical and industrial problems (see [1]). It has been studied by many authors numerically and analytically. We refer to [1–7]. One kind of NPEs reads𝑥(𝑡)−𝑁𝑥𝑞𝑡^′=𝑓𝑡,𝑥(𝑡),𝑥𝑞𝑡.(1.1) Taking the environmental disturbances into account, we are led to the following neutral stochastic pantograph equation (NSPE)𝑑𝑥(𝑡)−𝑁𝑥𝑞𝑡=𝑓𝑡,𝑥(𝑡),𝑥𝑞𝑡𝑑𝑡+𝑔𝑡,𝑥(𝑡),𝑥𝑞𝑡𝑑𝐵(𝑡),(1.2) which is a kind of neutral stochastic delay differential equations (NSDDEs).

Using the continuous semimartingale convergence theorem (cf. [8]), Mao et al. (see [9, 10]) studied the almost surely asymptotic stability of several kinds of NSDDEs. As most NSDDEs cannot be solved explicitly, numerical methods have become essential. Efficient numerical methods for NSDDEs can be found in [11–13]. The stability theory of numerical solutions is one of fundamental research topics in the numerical analysis. The almost surely asymptotic stability of numerical solutions for stochastic differential equations (SDEs) and stochastic functional differential equations (SFDEs) has received much more attention (see [14–19]). Corresponding to the continuous semimartingale convergence theorem (cf. [8]), the discrete semimartingale convergence theorem (cf. [17, 20]) also plays important roles in the almost surely asymptotic stability analysis of numerical solutions for SDEs and SFDEs (see [17–19]). To our best knowledge, no results on the almost surely asymptotic stability of exact and numerical solutions for the NSPE (1.2) can be found. We aim in this paper to study the almost surely asymptotic stability of exact and numerical solutions to NSPEs by using the continuous semimartingale convergence theorem and the discrete semimartingale convergence theorem. We prove that the backward Euler method (BEM) with variable stepsize can preserve the almost surely asymptotic stability under the conditions which guarantee the almost surely asymptotic stability of the exact solution.

In Section 2, we introduce some necessary notations and elementary theories of NSPEs (1.2). Moreover, we state the discrete semimartingale convergence theorem as a lemma. In Section 3, we study the almost surely asymptotic stability of exact solutions to NSPEs (1.2). Section 4 gives the almost surely asymptotic stability of the backward Euler method with variable stepsize. Numerical experiments are presented in the finial section.

2. Neutral Stochastic Pantograph Equation

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 ℒ¹([0,𝑇];𝑅^𝑛) denote the family of all 𝑅^𝑛-value measurable ℱ_𝑡-adapted processes 𝑓={𝑓(𝑡)}_{0≤𝑡≤𝑇} such that ∫^𝑇₀|𝑓(𝑡)|𝑑𝑡<∞ w.p.1. Let ℒ²([0,𝑇];𝑅^𝑛) denote the family of all 𝑅^𝑛-value measurable ℱ_𝑡-adapted processes 𝑓={𝑓(𝑡)}_{0≤𝑡≤𝑇} such that ∫^𝑇₀|𝑓(𝑡)|²𝑑𝑡<∞ w.p.1.

Consider an 𝑛-dimensional neutral stochastic pantograph equation𝑑𝑥(𝑡)−𝑁𝑥𝑞𝑡=𝑓𝑡,𝑥(𝑡),𝑥𝑞𝑡𝑑𝑡+𝑔𝑡,𝑥(𝑡),𝑥𝑞𝑡𝑑𝐵(𝑡),(2.1) on 𝑡≥0 with ℱ₀-measurable bounded initial data 𝑥(0)=𝑥₀. Here 0<𝑞<1, 𝑓∶𝑅⁺×𝑅^𝑛×𝑅^𝑛→𝑅^𝑛, 𝑔∶𝑅⁺×𝑅^𝑛×𝑅^𝑛→𝑅^𝑛, and 𝑁∶𝑅^𝑛→𝑅^𝑛.

Let 𝐶(𝑅^𝑛;𝑅⁺) denote the family of continuous functions from 𝑅^𝑛 to 𝑅⁺. 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)

To be precise, we first give the definition of the solution to (2.1) on 0≤𝑡≤𝑇.

Definition 2.1. A 𝑅^𝑛-value stochastic process 𝑥(𝑡) on 0≤𝑡≤𝑇 is called a solution of (2.1) if it has the following properties:(1){𝑥(𝑡)}_{0≤𝑡≤𝑇} is continuous and ℱ_𝑡-adapted;(2)𝑓(𝑡,𝑥(𝑡),𝑥(𝑞𝑡))∈ℒ¹([0,𝑇];𝑅^𝑛), 𝑔(𝑡,𝑥(𝑡),𝑥(𝑞𝑡))∈ℒ²([0,𝑇];𝑅^𝑛):(3)𝑥(0)=𝑥₀, and (2.1) holds for every 𝑡∈[0,𝑇] with probability 1.A solution 𝑥(𝑡) is said to be unique if any other solution 𝑥(𝑡) is indistinguishable from it, that is, 𝑃{𝑥(𝑡)=𝑥(𝑡),0≤𝑡≤𝑇}=1.(2.4)

To ensure the existence and uniqueness of the solution to (2.1) on 𝑡∈[0,𝑇], we impose the following assumptions on the coefficients 𝑁,𝑓, and 𝑔.

Assumption 2.2. Assume that both 𝑓 and 𝑔 satisfy the global Lipschitz condition and the linear growth condition. That is, there exist two positive constants 𝐿 and 𝐾 such that for all 𝑥,𝑦,𝑥,𝑦∈𝑅^𝑛, and 𝑡∈[0,𝑇], 𝑓𝑡,𝑥,𝑦−𝑓𝑡,𝑥,𝑦²∨𝑔𝑡,𝑥,𝑦−𝑔𝑡,𝑥,𝑦²≤𝐿|𝑥−𝑥|²+𝑦−𝑦²,(2.5) and for all 𝑥,𝑦∈𝑅^𝑛, and 𝑡∈[0,𝑇], 𝑓𝑡,𝑥,𝑦²∨𝑔𝑡,𝑥,𝑦²≤𝐾1+|𝑥|²+𝑦².(2.6)

Assumption 2.3. Assume that there is a constant 𝜅∈(0,1) such that 𝑁(𝑥)−𝑁𝑦≤𝜅𝑥−𝑦,∀𝑥,𝑦∈𝑅^𝑛.(2.7)
Under Assumptions 2.2 and 2.3, the following results can be derived.

Lemma 2.4. Let Assumptions 2.2 and 2.3 hold. Let 𝑥(𝑡) be a solution to (2.1) with ℱ₀-measurable bounded initial data 𝑥(0)=𝑥₀. Then 𝐸sup_{0≤𝑡≤𝑇}|𝑥(𝑡)|²≤()√1+1−𝜅𝜅+31−𝜅1−√𝜅²()𝐸𝑥1−𝜅⁰²()𝑇exp6𝐾𝑇+4(1−𝜅)1−√𝜅.(2.8)

The proof of Lemma 2.4 is similar to Lemma 6.2.4 in [21], so we omit the details.

Theorem 2.5. Let Assumptions 2.2 and 2.3 hold, then for any ℱ₀-measurable bounded initial data 𝑥(0)=𝑥₀, (2.1) has a unique solution 𝑥(𝑡) on 𝑡∈[0,𝑇].

Based on Lemma 6.2.3 in [21] and Lemma 2.4, this theorem can be proved in the same way as Theorem 6.2.2 in [21], so the details are omitted.

The discrete semimartingale convergence theorem (cf. [17, 20]) will play an important role in this paper.

Lemma 2.6. 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.(2.9)Iflim_𝑖→∞𝐴_𝑖<∞ a.s., then for almost all 𝜔∈Ω: lim_𝑖→∞𝑋_𝑖<∞,lim_𝑖→∞𝑈_𝑖<∞,(2.10) that is, both 𝑋_𝑖 and 𝑈_𝑖 converge to finite random variables.

3. Almost Surely Asymptotic Stability of Neutral Stochastic Pantograph Equations

In this section, we investigate the almost surely asymptotic stability of (2.1). We assume (2.1) has a continuous unique global solution for given ℱ₀-measurable bounded initial data 𝑥₀. Moreover, we always assume that 𝑓(𝑡,0,0)=0,𝑔(𝑡,0,0)=0,𝑁(0)=0 in the following sections. Therefore, (2.1) admits a trivial solution 𝑥(𝑡)=0.

To be precise, let us give the definition on the almost surely asymptotic stability of (2.1).

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 bounded ℱ₀-measurable bounded initial data 𝑥(0).

Lemma 3.2. Let 𝜌∶𝑅⁺→(0,∞) and 𝑧∶[0,∞)→𝑅^𝑛 be a continuous functions. Assume that 𝜎₁∶=limsup_𝑡→∞𝜌(𝑡)𝜌<1𝑞𝑡𝜅,𝜎₂∶=limsup_𝑡→∞𝜌(𝑡)𝑧(𝑡)−𝑁𝑧𝑞𝑡<∞.(3.2) Then, limsup_𝑡→∞𝜌(𝑡𝑧(𝑡≤𝜎)|)|²1−𝜅𝜎₁.(3.3)

Proof. Using the idea of Lemma 3.1 in [9], we can obtain the desired result.

Lemma 3.3. Suppose that (2.1) has a continuous unique global solution 𝑥(𝑡) for given ℱ₀-measurable bounded initial data 𝑥₀. Let Assumption 2.3 hold. Assume that there are functions 𝑈∈𝐶^1,2(𝑅⁺×𝑅^𝑛;𝑅⁺), 𝑤∈𝐶(𝑅^𝑛;𝑅⁺), and four positive constants 𝜆₁>𝜆₂,𝜆₃,𝜆₄ such that 𝐿𝑈𝑡,𝑥,𝑦≤−𝜆₁𝑤(𝑥)+𝑞𝜆₂𝑤𝑦,𝑡,𝑥,𝑦∈𝑅⁺×𝑅^𝑛×𝑅^𝑛,𝑈𝑡,𝑥−𝑁𝑦≤𝜆₃𝑤(𝑥)+𝜆₄𝑤𝑦,(𝑡,𝑥)∈𝑅⁺×𝑅^𝑛.(3.4) Then, for any 𝜀∈(0,𝛾^∗)limsup_𝑡→∞𝑡^{(𝛾∗−𝜀)}𝑈𝑡,𝑥(𝑡)−𝑁𝑥𝑞𝑡<∞a.s.,(3.5) where 𝛾^∗ is positive and satisfies 𝜆₁=𝜆₂𝑞^−𝛾∗.(3.6) That is, lim_𝑡→∞𝑈𝑡,𝑥(𝑡)−𝑁𝑥𝑞𝑡=0a.s.(3.7)

Proof. Choose 𝑉(𝑡,𝑥(𝑡))=𝑡^𝛾𝑈(𝑡,𝑥(𝑡)−𝑁(𝑥(𝑞𝑡))) for (𝑡,𝑥)∈𝑅⁺×𝑅^𝑛 and 𝛾>0. Similar to the proof of Lemma 2.2 in [9], the desired conclusion can be obtained by using the continuous semimartingale convergence theorem (cf. [8]).

Theorem 3.4. Suppose that (2.1) has a continuous unique global solution 𝑥(𝑡) for given ℱ₀-measurable bounded initial data 𝑥₀. Let Assumption 2.3 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(3.8) for 𝑡≥0 and 𝑥,𝑦∈𝑅^𝑛. If 𝜆₁−𝜆₃>𝜆₂+𝜆₄𝑞,(3.9) then, the global solution 𝑥(𝑡) to (2.1) is almost surely asymptotically stable.

Proof. Let 𝑈(𝑡,𝑥)=𝑤(𝑥)=|𝑥|². Applying Lemma 3.3 and Lemma 3.2 with 𝜌=1, we can obtain the desired conclusion.

Theorem 3.4 gives sufficient conditions of the almost surely asymptotic stability of NSPEs (2.1). Based on this result, we will investigate the almost surely asymptotic stability of the BEM with variable stepsize for (2.1) in the following section.

4. Almost Surely Asymptotic Stability of the Backward Euler Method

To define the BEM for (2.1), we introduce a mesh 𝐻={𝑚;𝑡_−𝑚,𝑡_−𝑚+1,…,𝑡₀,𝑡₁,…,𝑡_𝑛,…} as follows. Let ℎ_𝑛=𝑡_𝑛+1−𝑡_𝑛, ℎ_−𝑚−1=𝑡_−𝑚. Set 𝑡₀=𝛾₀>0 and 𝑡_𝑚=𝑞⁻¹𝛾₀. We define 𝑚−1 grid points 𝑡₁<𝑡₂<⋯<𝑡_𝑚−1 in (𝑡₀,𝑡_𝑚) by 𝑡_𝑖=𝑡₀+𝑖Δ₀,for𝑖=1,2,…,𝑚−1,(4.1) where Δ₀=(𝑡_𝑚−𝑡₀)/𝑚 and define the other grid points by 𝑡_{𝑘𝑚+𝑖}=𝑞^−𝑘𝑡_𝑖,for𝑘=−1,0,1,…,𝑖=0,1,2,…,𝑚−1.(4.2) It is easy to see that the grid point 𝑡_𝑛 satisfies 𝑞𝑡_𝑛=𝑡_𝑛−𝑚 for 𝑛≥0, and the step size ℎ_𝑛 satisfies 𝑞ℎ_𝑛=ℎ_𝑛−𝑚,for𝑛≥0,lim_𝑛→∞ℎ_𝑛=∞.(4.3) For the given mesh 𝐻, we define the BEM for (2.1) as follows:𝑌_𝑛+1−𝑁𝑌_{𝑛+1−𝑚}=𝑌_𝑛−𝑁𝑌_𝑛−𝑚+ℎ_𝑛𝑓𝑡_𝑛+1,𝑌_𝑛+1,𝑌_{𝑛+1−𝑚}+𝑔𝑡_𝑛,𝑌_𝑛,𝑌_𝑛−𝑚Δ𝐵_𝑛𝑌,𝑛≥−𝑚,_−𝑚−𝑁𝑌_{−𝑚−𝑚}=𝑥₀−𝑁𝑥₀+ℎ_−𝑚−1𝑓𝑡_−𝑚,𝑌_−𝑚,𝑌_{−𝑚−𝑚}+𝑔0,𝑥₀,𝑥₀𝐵𝑡_−𝑚.(4.4) Here, 𝑌_𝑛(𝑛≥−𝑚) is an approximation value of 𝑥(𝑡_𝑛) and ℱ_𝑡𝑛-measurable. Δ𝐵_𝑛=𝐵(𝑡_𝑛+1)−𝐵(𝑡_𝑛) is the Brownian increment. The approximations 𝑌_𝑛−𝑚(𝑛=−𝑚,−𝑚+1,…,−1) are calculated by the following formulae: 𝑌_𝑛−𝑚=1−𝜃_𝑛𝑥₀+𝜃_𝑛𝑌_−𝑚,𝑛=−𝑚,−𝑚+1,…,−1,(4.5) where 𝜃_𝑛=𝑞𝑡_𝑛/𝑡_−𝑚. As a standard hypothesis, we assume that the BEM (4.4) is well defined.

To be precise, let us introduce the definition on the almost surely asymptotic stability of the BEM (4.4).

Definition 4.1. The approximate solution 𝑌_𝑛 to the BEM (4.4) is said to be almost surely asymptotically stable if lim_𝑛→∞𝑌_𝑛=0a.s.(4.6) for any bounded ℱ₀-measurable bounded initial data 𝑥₀.

Theorem 4.2. Assume that the BEM (4.4) is well defined. Let Assumption 2.3 hold. Let conditions (3.8) and (3.9) hold. Then the BEM approximate solution (4.4) obeys lim_𝑛→∞𝑌_𝑛=0𝑎.𝑠.(4.7) That is, the approximate solution 𝑌_𝑛 to the BEM (4.4) is almost surely asymptotically stable.

Proof. Set 𝑌_𝑛=𝑌_𝑛−𝑁(𝑌_𝑛−𝑚). For 𝑛≥0, from (4.4), we have 𝑌_𝑛+1−ℎ_𝑛𝑓𝑡_𝑛+1,𝑌_𝑛+1,𝑌_{𝑛+1−𝑚}²=𝑌_𝑛+𝑔𝑡_𝑛,𝑌_𝑛,𝑌_𝑛−𝑚Δ𝐵_𝑛².(4.8) Then, we can obtain that 𝑌_𝑛+1²≤𝑌_𝑛²+2ℎ_𝑛𝑌_𝑛+1,𝑓𝑡_𝑛+1,𝑌_𝑛+1,𝑌_{𝑛+1−𝑚}+𝑔𝑡_𝑛,𝑌_𝑛,𝑌_𝑛−𝑚Δ𝐵_𝑛²+2𝑌_𝑛,𝑔𝑡_𝑛,𝑌_𝑛,𝑌_𝑛−𝑚Δ𝐵_𝑛,(4.9) which subsequently leads to 𝑌_𝑛+1²≤𝑌_𝑛²+2ℎ_𝑛𝑌_𝑛+1,𝑓𝑡_𝑛+1,𝑌_𝑛+1,𝑌_{𝑛+1−𝑚}+𝑔𝑡_𝑛,𝑌_𝑛,𝑌_𝑛−𝑚²ℎ_𝑛+𝑚_𝑛,(4.10) where 𝑚_𝑛=2𝑌_𝑛,𝑔𝑡_𝑛,𝑌_𝑛,𝑌_𝑛−𝑚Δ𝐵_𝑛+𝑔𝑡_𝑛,𝑌_𝑛,𝑌_𝑛−𝑚²Δ𝐵²_𝑛−ℎ_𝑛.(4.11) By conditions (3.8) and (3.9), we have 𝑌_𝑛+1²≤𝑌_𝑛²−𝜆₁ℎ_𝑛𝑌_𝑛+1²+𝜆₂ℎ_𝑛𝑌_{𝑛+1−𝑚}²+𝜆₃𝑌_𝑛²+𝜆₄𝑌_𝑛−𝑚²ℎ_𝑛+𝑚_𝑛.(4.12) Using the equality |𝑎+𝑏|²≤2|𝑎|²+2|𝑏|², we obtain that 𝑌_𝑛+1²≥12𝑌_𝑛+1²−𝑁𝑌_{𝑛+1−𝑚}²,𝑌_𝑛²≤2𝑌_𝑛²+2𝑁𝑌_𝑛−𝑚².(4.13) Inserting these inequalities to (4.12) and using Assumption 2.3 yield 12+𝜆₁ℎ_𝑛𝑌_𝑛+1²≤2+𝜆₃ℎ_𝑛𝑌_𝑛²+𝜅²+𝜆₂ℎ_𝑛𝑌_{𝑛+1−𝑚}²+2𝜅²+𝜆₄ℎ_𝑛𝑌_𝑛−𝑚²+𝑚_𝑛.(4.14) Let 𝐴_𝑛=1+2𝜆₁ℎ_𝑛, 𝐵_𝑛=3−2𝜆₁ℎ_𝑛+2𝜆₃ℎ_𝑛, 𝐶_𝑛=2𝜅²+2𝜆₂ℎ_𝑛, and 𝐷_𝑛=4𝜅²+2𝜆₄ℎ_𝑛. Using these notations, (4.14) implies that 𝑌_𝑛+1²−𝑌_𝑛²≤𝐵_𝑛𝐴_𝑛𝑌_𝑛²+𝐶_𝑛𝐴_𝑛𝑌_{𝑛+1−𝑚}²+𝐷_𝑛𝐴_𝑛𝑌_𝑛−𝑚²+2𝐴_𝑛𝑚_𝑛.(4.15) Then, we can conclude that 𝑌_𝑛²≤𝑌₀²+^𝑛−1_𝑖=0𝐵_𝑖𝐴_𝑖𝑌_𝑖²+^𝑛−1_𝑖=0𝐶_𝑖𝐴_𝑖𝑌_{𝑖+1−𝑚}²+^𝑛−1_𝑖=0𝐷_𝑖𝐴_𝑖𝑌_𝑖−𝑚²+^𝑛−1_𝑖=02𝐴_𝑖𝑚_𝑖.(4.16) Note that ^𝑛−1_𝑖=0𝐶_𝑖𝐴_𝑖𝑌_{𝑖+1−𝑚}²=^𝑛−𝑚_{𝑖=−𝑚+1}𝐶_{𝑖+𝑚−1}𝐴_{𝑖+𝑚−1}𝑌_𝑖²=⁻¹_{𝑖=−𝑚+1}𝐶_{𝑖+𝑚−1}𝐴_{𝑖+𝑚−1}𝑌_𝑖²+^𝑛−1_𝑖=0𝐶_{𝑖+𝑚−1}𝐴_{𝑖+𝑚−1}𝑌_𝑖²−^𝑛−1_{𝑖=𝑛−𝑚+1}𝐶_{𝑖+𝑚−1}𝐴_{𝑖+𝑚−1}𝑌_𝑖²,_{𝑛−1𝑖=0}𝐷_𝑖𝐴_𝑖𝑌_𝑖−𝑚²=^{𝑛−𝑚−1}_𝑖=−𝑚𝐷_𝑖+𝑚𝐴_𝑖+𝑚𝑌_𝑖²=⁻¹_𝑖=−𝑚𝐷_𝑖+𝑚𝐴_𝑖+𝑚𝑌_𝑖²+^𝑛−1_𝑖=0𝐷_𝑖+𝑚𝐴_𝑖+𝑚𝑌_𝑖²−^𝑛−1_{𝑖=𝑛−𝑚}𝐷_𝑖+𝑚𝐴_𝑖+𝑚𝑌_𝑖².(4.17) We, therefore, have 𝑌_𝑛²+^𝑛−1_{𝑖=𝑛−𝑚+1}𝐶_{𝑖+𝑚−1}𝐴_{𝑖+𝑚−1}𝑌_𝑖²+^𝑛−1_{𝑖=𝑛−𝑚}𝐷_𝑖+𝑚𝐴_𝑖+𝑚𝑌_𝑖²≤𝑌₀²+⁻¹_{𝑖=−𝑚+1}𝐶_{𝑖+𝑚−1}𝐴_{𝑖+𝑚−1}𝑌_𝑖²+⁻¹_𝑖=−𝑚𝐷_𝑖+𝑚𝐴_𝑖+𝑚𝑌_𝑖²+^𝑛−1_𝑖=0𝐵_𝑖𝐴_𝑖+𝐶_{𝑖+𝑚−1}𝐴_{𝑖+𝑚−1}+𝐷_𝑖+𝑚𝐴_𝑖+𝑚𝑌_𝑖²+^𝑛−1_𝑖=02𝐴_𝑖𝑚_𝑖.(4.18) Similar to (4.15), from (4.4), we can obtain that 𝑌₀²−𝑌₋₁²≤𝐵₋₁𝐴₋₁𝑌₋₁²+𝐶₋₁𝐴₋₁𝑌_−𝑚²+𝐷₋₁𝐴₋₁21−𝜃₋₁²𝑥₀²+2𝜃²₋₁𝑌_−𝑚²+2𝐴₋₁𝑚₋₁,𝑌_𝑛²−𝑌_𝑛−1²≤𝐵_𝑛−1𝐴_𝑛−1𝑌_𝑛−1²+𝐶_𝑛−1𝐴_𝑛−121−𝜃_𝑛²𝑥₀²+2𝜃²_𝑛𝑌_−𝑚²+𝐷_𝑛−1𝐴_𝑛−121−𝜃_𝑛−1²𝑥₀²+2𝜃²_𝑛−1𝑌_−𝑚²+2𝐴_𝑛−1𝑚_𝑛−1𝑌,−𝑚+1≤𝑛≤−1,_−𝑚2−𝑥₀²≤𝐵_−𝑚−1+𝐷_−𝑚−1𝐴_−𝑚−1𝑥₀²+𝐶_−𝑚−1𝐴_−𝑚−121−𝜃_−𝑚²𝑥₀²+2𝜃²_−𝑚𝑌_−𝑚²+2𝐴_−𝑚−1𝑚_−𝑚−1,(4.19) where 𝐴_𝑛,𝐵_𝑛,𝐶_𝑛,𝐷_𝑛(𝑛=−𝑚−1,…,−1) are defined as before, 𝑚_𝑛=2𝑌_𝑛−𝑁𝑌_𝑛−𝑚,𝑔𝑡_𝑛,𝑌_𝑛,𝑌_𝑛−𝑚Δ𝐵_𝑛+𝑔𝑡_𝑛,𝑌_𝑛,𝑌_𝑛−𝑚²Δ𝐵²_𝑛−ℎ_𝑛,−𝑚≤𝑛≤−1,𝑚_−𝑚−1=2𝑥₀−𝑁𝑥₀,𝑔0,𝑥₀,𝑥₀𝐵𝑡_−𝑚+𝑔0,𝑥₀,𝑥₀²𝐵²𝑡_−𝑚−𝑡_−𝑚.(4.20) From (4.19), we have 𝑌₀²≤𝐴𝑥₀²+𝐵𝑌_−𝑚²+⁻¹_{𝑖=−𝑚+1}𝐵_𝑖𝐴_𝑖𝑌_𝑖²+⁻¹_{𝑖=−𝑚−1}2𝐴_𝑖𝑚_𝑖,(4.21) where 𝐵𝐴=1+^−𝑚−1+𝐷_−𝑚−1𝐴_−𝑚−1+⁻¹_𝑖=−𝑚2𝐷_𝑖𝐴_𝑖1−𝜃_𝑖²+⁻²_{𝑖=−𝑚−1}2𝐶_𝑖𝐴_𝑖1−𝜃_𝑖+1²,𝐵𝐵=^−𝑚𝐴_−𝑚+𝐵₋₁𝐴₋₁+⁻¹_𝑖=−𝑚2𝐷_𝑖𝐴_𝑖𝜃²_𝑖+⁻²_{𝑖=−𝑚−1}2𝐶_𝑖𝐴_𝑖𝜃²_𝑖+1.(4.22) Obviously 𝐴>0. By (4.18) and (4.21), we can obtain that 𝑌_𝑛²+^𝑛−1_{𝑖=𝑛−𝑚+1}𝐶_{𝑖+𝑚−1}𝐴_{𝑖+𝑚−1}𝑌_𝑖²+^𝑛−1_{𝑖=𝑛−𝑚}𝐷_𝑖+𝑚𝐴_𝑖+𝑚𝑌_𝑖²≤𝐴𝑥₀²+𝐷𝐵+⁰𝐴₀𝑌_−𝑚²+^𝑛−1_{𝑖=−𝑚+1}𝐵_𝑖𝐴_𝑖+𝐶_{𝑖+𝑚−1}𝐴_{𝑖+𝑚−1}+𝐷_𝑖+𝑚𝐴_𝑖+𝑚𝑌_𝑖²+𝑀_𝑛,(4.23) where 𝑀_𝑛=∑^𝑛−1_{𝑖=−𝑚−1}(2/𝐴_𝑖)𝑚_𝑖. Similar to the proof in [18], we can obtain that 𝑀_𝑛 is a martingale with 𝑀_−𝑚−1=0. Note that ℎ_{𝑖+𝑚−1}≤ℎ_𝑖+𝑚 and ℎ_𝑖+𝑚=ℎ_𝑖/𝑞 for 𝑖≥−𝑚. Then, we have 𝐵_𝑖𝐴_𝑖+𝐶_{𝑖+𝑚−1}𝐴_{𝑖+𝑚−1}+𝐷_𝑖+𝑚𝐴_𝑖+𝑚≤3+6𝜅²−2𝜆₁ℎ_𝑖−𝜆₃ℎ_𝑖−𝜆₂ℎ_{𝑖+𝑚−1}−𝜆₄ℎ_𝑖+𝑚1+2𝜆₁ℎ_𝑖≤11+2𝜆₁ℎ_𝑖3+6𝜅²−2𝜆₁−𝜆₃−𝜆₂𝑞−𝜆₄𝑞ℎ_𝑖.(4.24) Using the condition (3.9) and lim_𝑖→∞ℎ_𝑖=∞, we obtain that there exists an integer 𝑖^∗ such that 𝐵_𝑖𝐴_𝑖+𝐶_{𝑖+𝑚−1}𝐴_{𝑖+𝑚−1}+𝐷_𝑖+𝑚𝐴_𝑖+𝑚≥0,𝑖≤𝑖^∗,𝐵_𝑖𝐴_𝑖+𝐶_{𝑖+𝑚−1}𝐴_{𝑖+𝑚−1}+𝐷_𝑖+𝑚𝐴_𝑖+𝑚<0,𝑖>𝑖^∗.(4.25) Set 𝑈_−𝑚−1=0, 𝑈_−𝑚=𝐷𝐵+⁰𝐴₀𝑌_−𝑚²𝐷if𝐵+⁰𝐴₀𝐷>0,0if𝐵+⁰𝐴₀𝑈≤0,_𝑛=𝑈_−𝑚+^𝑛−1_{𝑖=−𝑚+1}𝐵_𝑖𝐴_𝑖+𝐶_{𝑖+𝑚−1}𝐴_{𝑖+𝑚−1}+𝐷_𝑖+𝑚𝐴_𝑖+𝑚𝑌_𝑖²if−𝑚+1≤𝑛≤𝑖^∗𝑈+1,_−𝑚+^𝑖∗_{𝑖=−𝑚+1}𝐵_𝑖𝐴_𝑖+𝐶_{𝑖+𝑚−1}𝐴_{𝑖+𝑚−1}+𝐷_𝑖+𝑚𝐴_𝑖+𝑚𝑌_𝑖²if𝑛>𝑖^∗𝑉+1,_𝑛=0if−𝑚−1≤𝑛≤𝑖^∗−+1,_{𝑛−1𝑖=𝑖∗+1}𝐵_𝑖𝐴_𝑖+𝐶_{𝑖+𝑚−1}𝐴_{𝑖+𝑚−1}+𝐷_𝑖+𝑚𝐴_𝑖+𝑚𝑌_𝑖²if𝑛>𝑖^∗+1.(4) Obviously, lim_𝑛→∞𝑈_𝑛=𝑈_−𝑚+^𝑖∗_{𝑖=−𝑚+1}𝐵_𝑖𝐴_𝑖+𝐶_{𝑖+𝑚−1}𝐴_{𝑖+𝑚−1}+𝐷_𝑖+𝑚𝐴_𝑖+𝑚𝑌_𝑖²<∞,a.s.(4.27) Moreover, (4.23) implies that 𝑌_𝑛²≤𝐶𝑥₀²+𝑈_𝑛−𝑉_𝑛+𝑀_𝑛.(4.28) Here 𝐶=max{𝐴,1}. According to (4.27), using Lemma 2.6 yields limsup_𝑛→∞𝑌_𝑛²<∞a.s.,lim_𝑛→∞𝑉_𝑛<∞a.s.(4.29) Then, we have lim_𝑖→∞−𝐵_𝑖𝐴_𝑖+𝐶_{𝑖+𝑚−1}𝐴_{𝑖+𝑚−1}+𝐷_𝑖+𝑚𝐴_𝑖+𝑚𝑌_𝑖²=0a.s.(4.30) Note that lim_𝑖→∞−𝐵_𝑖𝐴_𝑖+𝐶_{𝑖+𝑚−1}𝐴_{𝑖+𝑚−1}+𝐷_𝑖+𝑚𝐴_𝑖+𝑚=𝜆₁−𝜆₂−𝜆₃−𝜆₄𝜆₁>0.(4.31) We therefore obtain that lim_𝑛→∞𝑌_𝑛²=0a.s.(4.32) Then, the desired conclusion is obtained. This completes the proof.

5. Numerical Experiments

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

Consider the following scalar problem:𝑑𝑥(𝑡)−12𝑥(=((𝑡)((𝑥((𝑡)𝑥(0)0.5𝑡)−8𝑥+𝑥0.5𝑡))𝑑𝑡+sin0.5𝑡))𝑑𝐵,𝑡≥0,=𝑥₀.(5.1) For the test (5.1), we have 𝜆₁=11, 𝜆₂=4, 𝜆₃=0, and 𝜆₄=1 corresponding to Theorem 3.4. By Theorem 3.4, the solution to (5.1) is almost surely asymptotically stable.

Theorem 4.2 shows that the BEM approximation to (5.1) is almost surely asymptotically stable. In Figure 1, We compute three different paths (𝑌_𝑛(𝜔₁),𝑌_𝑛(𝜔₂),𝑌_𝑛(𝜔₃)) using the BEM (4.4) with 𝑥₀=2,𝑡₀=0.01,𝑚=2. In Figure 2, three different paths (𝑌_𝑛(𝜔₁),𝑌_𝑛(𝜔₂),𝑌_𝑛(𝜔₃)) of BEM approximations are computed with 𝑥₀=10,𝑡₀=1,𝑚=1. The results demonstrate that these paths are asymptotically stable.

Figure 1 

Almost surely asymptotic stability with 𝑥0=2,𝑡0=0.01,𝑚=2.

Figure 2 

Almost surely asymptotic stability with 𝑥0=10,𝑡0=1,𝑚=1.

Acknowledgments

The author would like to thank the referees for their helpful comments and suggestions.