Almost Surely Exponential Stability of Numerical Solutions for Stochastic Pantograph Equations

Abstract

Our effort is to develop a criterion on almost surely exponential stability of numerical solution to stochastic pantograph differential equations, with the help of the discrete semimartingale convergence theorem and the technique used in stable analysis of the exact solution. We will prove that the Euler-Maruyama (EM) method can preserve almost surely exponential stability of stochastic pantograph differential equations under the linear growth conditions. And the backward EM method can reproduce almost surely exponential stability for highly nonlinear stochastic pantograph differential equations. A highly nonlinear example is provided to illustrate the main theory.

1. Introduction

Stochastic unbounded delay systems play an important role in a variety of application areas, including biology, epidemiology, mechanic, economics, and finance. The systems provide powerful models, such as infinite delay Kolmogorov-type systems in mathematic biology [1–4], stochastic neural networks [5–8], and stochastic pantograph equations in science and engineering. The pantograph equation which is a very special unbounded delay equation was used by Ockendon and Tayler [9] in 1971 to study how the electric current is collected by the pantograph of an electric locomotive, from where it gets its name. Such systems have received an increasing attention (see e.g., [3, 10–13]).

Unfortunately, most of stochastic differential equations cannot be solved explicitly. Especially, explicit solutions can rarely be obtained for nonlinear stochastic pantograph equations, so numerical methods have recently received more and more attention (see [6, 8, 13–15]). Most of research efforts have been devoted to the convergence and mean-square stability of various numerical methods for the linear delay systems [14, 16–19]. Recently, several authors were devoted to the convergence in probability of the Euler-Maruyama (EM) method for the nonlinear delay systems. For example, Mao [20] and Milošević [21] developed the convergence in probability of the EM approximate solution for nonlinear SDDE and neutral SDDE under the Khasminskii-type conditions, respectively. Zhou et al. [22, 23] established the convergence in probability of the EM approximation for neutral stochastic functional differential equation under the polynomial growth conditions.

Stability theory of numerical solution is one key problem in numerical analysis. Compared with the convergence, the study on stability of numerical methods for delay systems is relatively scarce due to infinite time-delay that is often the source of instability. Research efforts have been devoted to various stabilities of numerical methods for SDEs. For example, Higham et al. [15, 24–26] investigated stability of numerical methods for SDEs. Pang et al. [27] showed that the EM discretization can capture almost surely and moment exponential stability for all sufficiently small time-step under appropriate conditions for linear hybrid SDE. Mao et al. [28] showed that the backward EM method can reproduce almost surely exponential stability of nonlinear hybrid SDE. Mao and Szpruch [29] developed almost surely asymptotic properties of implicit numerical methods for nonlinear SDEs, via a stochastic version of LaSalle principle. However, the stability of numerical method for highly nonlinear stochastic delay system is less studied, due to time-delay that is often the source of instability (see [2, 30, 31]), which is the main topic of the present paper.

The stability of numerical method is inspired by Wu et al.’s paper [32], in which they first studied that the backward EM method can reproduce almost surely exponential stability of nonlinear stochastic differential delay system d𝑥(𝑡)=𝑓(𝑥(𝑡),𝑥(𝑡−𝜏))d𝑡+𝑔(𝑥(𝑡),𝑥(𝑡−𝜏))d𝑤(𝑡), (1) by using of the discrete semi-martingale convergence theorem, under the following conditions 2𝑥^𝑇𝑓(𝑥,0)≤−𝜆₁|𝑥|²,𝑓(𝑥,𝑦)−𝑓(𝑥,0)²≤𝜆₂𝑦²,𝑔(𝑥,𝑦)²≤𝜆₃|𝑥|²+𝜆₄𝑦². (2)(3)(4) Wu et al.’s work is a very important contribution to numerical SDDE theory. Certainly, we also see that the theory imposes the one-sided linear growth on the drift coefficient (i.e., (3)) and the linear growth on the diffusion coefficient (i.e., (4)), which are rather strong so that many highly nonlinear SDDEs are excluded. Recently, Zhou et al. [33] studied exponential stability of stochastic functional differential equation under polynomial growth condition. To the best of author’s knowledge, there is no work on stability of numerical method for stochastic pantograph differential equations. In the paper, our effort is to develop a new criterion on almost surely exponential stability of numerical solution to stochastic pantograph differential equations, with the help of the discrete semimartingale convergence theorem and the technique used in stability analysis of the exact solution. We will prove that the Euler-Maruyama method can preserve the almost surely exponential stability of stochastic pantograph differential equations under the linear growth conditions. The backward EM method can preserve almost surely exponential stability for highly nonlinear stochastic pantograph differential equations under sufficiently small step size.

2. The Global Solution

Throughout this paper, unless otherwise specified, let |𝑥| be the Euclidean norm in 𝑥∈ℝ^𝑛. 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), while its operator norm is denoted by ‖𝐴‖=sup{|𝐴𝑥|∶|𝑥|=1}. 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 increasing and right continuous and ℱ₀ contains all 𝑃-null sets). Let 𝑤(𝑡)=(𝑤₁(𝑡),𝑤₂(𝑡),…,𝑤_𝑚(𝑡)) be 𝑚-dimensional Brownian motion.

Consider an 𝑛-dimensional stochastic pantograph equation d𝑥(𝑡)=𝑓𝑥(𝑡),𝑥𝑞𝑡d𝑡+𝑔𝑥(𝑡),𝑥𝑞𝑡d𝑤(𝑡),0<𝑞<1 (5) on 𝑡≥0 with the initial data 𝑥₀∈𝑅^𝑛); 𝑓∶ℝ^𝑛×ℝ^𝑛→ℝ^𝑛,  𝑔∶ℝ^𝑛×ℝ^𝑛→ℝ^𝑛 are locally Lipschitz continuous.

(H1) The Local Lipschitz Condition. For each integer 𝑅≥1, there exists a positive constant 𝐾_𝑅 such that 𝑓𝑥₁,𝑦₁−𝑓𝑥₂,𝑦₂∨𝑔𝑥₁,𝑦₁−𝑔𝑥₂,𝑦₂≤𝐾_𝑅𝑥₁−𝑥₂+𝑦₁−𝑦₂ (6) for all 𝑥_𝑘,𝑦_𝑘∈ℝ^𝑛 with |𝑥_𝑘|∨|𝑦_𝑘|≤𝑅 (𝑘=1,2).

(H2) The Polynomial Growth Conditions. For all 𝑥∈ℝ^𝑛, there exist positive constants 𝛼,𝛽,𝛾,𝑎,𝑎,𝑎,𝑏,𝑏,𝑏̃ such that 𝑥(𝑠)𝑥(𝑠)|𝑥(𝑠,𝑓,𝑥𝑞𝑠≤−𝑎)|^𝛼+2+𝑎𝑥(𝑞𝑠)^𝛽+2+|𝑥(𝑠)|^𝛽+2−𝑎|𝑥(𝑠)|²,𝑔(𝑥(𝑠),𝑥(𝑞𝑠))²≤𝑏|𝑥(𝑠)|^𝛾+2+𝑏𝑥(𝑞𝑠)^𝛾+2+𝑏̃|𝑥(𝑠)|². (7)(8)

Lemma 1 (see [1]). Let 𝛼,𝑏>0, 𝜅(𝑥)∈𝐶(ℝ₊;ℝ); if 𝜅(𝑥)=𝑜(|𝑥|^𝛼)(|𝑥|→∞), then sup_𝑥∈ℝ+𝜅(𝑥)−𝑏|𝑥|^𝛼<∞. (9)

Theorem 2. Let (H1) and (H2) hold with 𝛼≥𝛽∨𝛾, 2𝑎>𝑏̃, 2𝑎>2𝑎(1+1/𝑞)+𝑏+𝑏/𝑞. Then for any initial data 𝑥₀, there almost surely exists a unique global solution 𝑥(𝑡) to (5) on 𝑡≥0.

Proof. Under (H1), applying the standing truncation technique to (5) for any initial data 𝑥₀, there exists a unique maximal local strong solution 0<𝑡<𝜈_𝑒, where 𝜈_𝑒 is the explosion time. To show this solution is global, we only need to show that 𝜈_𝑒=∞ a.s. Let 𝑘₀>0 be sufficiently large such that 𝑘₀>|𝑥₀|. For each integer 𝑘≥𝑘₀, define the stopping time 𝜈_𝑘=inf𝑡∈0,𝜈_𝑒∶|𝑥(𝑡)|≥𝑘,𝑘∈𝑁, (10) where throughout this paper, we set inf0/=∞ (as usual, 0/ = the empty set). By the definition of the stopping time 𝜈_𝑘, it is obvious that 𝜈_𝑘 is an increasing function with 𝑘, so 𝜈_𝑘→𝜈_∞≤𝜈_𝑒 (𝑘→∞) a.s. If we can show that 𝜈_∞=∞ a.s., then 𝜈_𝑒=∞ a.s. which implies that 𝑥(𝑡) is global. In other words, we only prove that 𝑃(𝜈_𝑘≤𝑡)→0 (𝑘→∞,𝑡>0). Define 𝑉(𝑥)=|𝑥|², by 𝑃(𝜈_𝑘≤𝑡)𝑉(𝑥(𝜈_𝑘))≤𝐸𝑉(𝑥(𝑡∧𝜈_𝑘)); we only need to prove that 𝐸𝑉(𝑥(𝑡∧𝜈_𝑘))<+∞ according to 𝑉(𝑥(𝜈_𝑘))=|𝑥(𝜈_𝑘)|²=𝑘²→∞. For 𝜀>0, by the Itô formula, we have 𝑉𝑥𝑡∧𝜈_𝑘=𝑉(𝑥(0))+^{𝑡∧𝜈𝑘}₀2𝑥^𝑇𝑓𝑥(𝑠)+𝑔𝑥(𝑠),𝑥𝑞𝑠,𝑥𝑞𝑠²(𝑡),d𝑠+𝑀 (11) where 𝑀(𝑡)=∫^{𝑡∧𝜈𝑘}₀e^𝜀𝑠𝑉_𝑥(𝑥(𝑠))𝑔(𝑥(𝑠),𝑥(𝑞𝑠))d𝑤(𝑠)d𝑠 is a real-valued continuous local martingale with 𝑀(0)=0. By (7) and (8), we may also estimate 2𝑥^𝑇𝑓𝑥(𝑠),𝑥𝑞𝑠+𝑔(𝑥(𝑠),𝑥(𝑞𝑠))²≤−2𝑎|𝑥(𝑠)|^𝛼+2+2𝑎𝑥(𝑞𝑠)^𝛽+2+|𝑥(𝑠)|^𝛽+2−2𝑎|𝑥(𝑠)|²+𝑏|𝑥(𝑠)|^𝛾+2+𝑏𝑥(𝑞𝑠)^𝛾+2+𝑏̃|𝑥(𝑠)|²≤2𝑎𝑞𝑞𝑥(𝑞𝑠)^𝛽+2−|𝑥(𝑠)|^𝛽+2+𝑏𝑞𝑞𝑥(𝑞𝑠)^𝛾+2−|𝑥(𝑠)|^𝛾+2−2𝑎|𝑥(𝑠)|^𝛼+2−2𝑎−𝑏̃|𝑥(𝑠)|²+2𝑎11+𝑞|𝑥(𝑠)|^𝛽+2+𝑏+𝑏𝑞|𝑥(𝑠)|^𝛾+2. (12) Denoted by 𝐼(𝑥(𝑠))=2𝑎−𝑏̃|𝑥(𝑠)|²+2𝑎|𝑥(𝑠)|^𝛼+2−2𝑎11+𝑞|𝑥(𝑠)|^𝛽+2−𝑏+𝑏𝑞|𝑥(𝑠)|^𝛾+2. (13) Noting that 2𝑎>𝑏̃, 2𝑎>2𝑎(1+1/𝑞)+𝑏+𝑏/𝑞, 𝛼≥𝛽∨𝛾, by Lemma 1, there exists a positive 𝑐₀ such that 𝐼(𝑥(𝑠))≥𝑐₀|𝑥(𝑠)|². Substitution for this and (12) into (11) yields 𝑉𝑥𝑡∧𝜈_𝑘(𝑥(0(𝑡)+2≤𝑉))+𝑀𝑎𝑞^{𝑡∧𝜈𝑘}₀𝑞𝑥(𝑞𝑠)^𝛽+2−|𝑥(𝑠)|^𝛽+2+d𝑠𝑏𝑞^{𝑡∧𝜈𝑘}₀𝑞𝑥(𝑞𝑠)^𝛾+2−|𝑥(𝑠)|^𝛾+2d𝑠−𝑐₀^{𝑡∧𝜈𝑘}₀|𝑥(𝑠)|²d𝑠. (14) Making use of the property of the integral, we may estimate ^𝑡₀e^𝜀𝑠𝑞𝑥(𝑞𝑠)^𝛽+2−|𝑥(𝑠)|^𝛽+2=d𝑠_𝑡0𝑞𝑥(𝑞𝑠)^𝛽+2−|𝑥(𝑠)|^𝛽+2≤d𝑠_0𝑞𝑡|𝑥(𝑠)|^𝛽+2d𝑠−^𝑡₀|𝑥(𝑠)|^𝛽+2d𝑠=−^𝑡_𝑞𝑡|𝑥(𝑠)|^𝛽+2d𝑠. (15) Similarly, ^𝑡₀𝑞𝑥(𝑞𝑠)^𝛾+2−|𝑥(𝑠)|^𝛾+2d𝑠≤−^𝑡_𝑞𝑡|𝑥(𝑠)|^𝛾+2d𝑠. (16) Substituting (15) and (16) into (14) and taking expectation yield 𝔼𝑉𝑥𝑡∧𝜈_𝑘≤𝔼𝜉²−2𝑎𝑞^{𝑡∧𝜈𝑘}_𝑞𝑡|𝑥(𝑠)|^𝛽+2−d𝑠𝑏𝑞^{𝑡∧𝜈𝑘}_{𝑞(𝑡∧𝜈𝑘)}|𝑥(𝑠)|^𝛾+2ds−𝑐₀^{𝑡∧𝜈𝑘}₀|𝑥(𝑠)|²d𝑠≤𝔼𝜉², (17) which implies that there exists almost surely a unique global solution. The proof is complete.

For stability, we need to impose a stronger condition on the coefficients as follows.

(H3) The Polynomial Growth Conditions. For all 𝑥∈ℝ^𝑛, there exist positive constants 𝛼, 𝛽, 𝛾, 𝑎, 𝑎, 𝑎, 𝑏, 𝑏, 𝑏̃, and 𝜀 with 𝜀≤2𝑎−𝑏̃ such that 𝑥(𝑠)𝑥(𝑠)|𝑥(𝑠,𝑓,𝑥𝑞𝑠≤−𝑎)|^𝛼+2+𝑎e^{−(1−𝑞)𝜀𝑠}𝑥𝑞𝑠^𝛽+2+|𝑥(𝑠)|^𝛽+2−𝑎|𝑥(𝑠)|²,𝑔𝑥(𝑠),𝑥𝑞𝑠²≤𝑏|𝑥(𝑠)|^𝛾+2+𝑏e^{−(1−𝑞)𝜀𝑠}𝑥(𝑞𝑠)^𝛾+2+𝑏̃|𝑥(𝑠)|². (18)(19)

Theorem 3. Let (H1) and (H3) hold with 𝛼≥𝛽∨𝛾, 2𝑎>𝑏̃,   2𝑎>2𝑎(1+1/𝑞)+𝑏+𝑏/𝑞. Then for any initial data 𝑥₀, the solution 𝑥(𝑡) is pth moment and almost surely exponentially stable; that is, limsup_𝑡→∞1𝑡|𝑥(𝑡𝜀log)|≤−2, (20) where 𝜀≤2𝑎−𝑏̃.

Proof. Clearly, (H3) implies that (H2) by letting 𝜀=0, so there exists a unique global solution. Define 𝑉(𝑥)=|𝑥|², for any 𝜀>0; by the Itô formula, we have e^𝜀𝑡𝑉(𝑥(𝑡))=𝑉(𝑥(0))+^𝑡₀e^𝜀𝑠𝑥(𝑠)(𝑥(𝑠𝐿𝑉,𝑥𝑞𝑠+𝜀𝑉)))d𝑠+𝑀(𝑡), (21) where 𝑀(𝑡)=∫^𝑡₀e^𝜀𝑠𝑉_𝑥(𝑥(𝑠))𝑔(𝑥(𝑠),𝑥(𝑞𝑠))d𝑤(𝑠)d𝑠 is a real-valued continuous local martingale with 𝑀(0)=0. By (18) and (19), we may also estimate 𝑥(𝑠)(𝑥(𝑠||𝐿𝑉,𝑥𝑞𝑠+𝜀𝑉))≤−2𝑎𝑥(𝑠)^𝛼+2+2𝑎e^{−(1−𝑞)𝜀𝑠}𝑥(𝑞𝑠)^𝛽+2+|𝑥(𝑠)|^𝛽+2−2𝑎|𝑥(𝑠)|²+𝑏|𝑥(𝑠)|^𝛾+2+𝑏e^{−(1−𝑞)𝜀𝑠}𝑥(𝑞𝑠)^𝛾+2+𝑏̃|𝑥(𝑠)|²+𝜀|𝑥(𝑠)|²≤2𝑎𝑞𝑞e^{−(1−𝑞)𝜀𝑠}𝑥(𝑞𝑠)^𝛽+2−|𝑥(𝑠)|^𝛽+2+𝑏𝑞𝑞e^{−(1−𝑞)𝜀𝑠}𝑥(𝑞𝑠)^𝛾+2−|𝑥(𝑠)|^𝛾+2−2𝑎|𝑥(𝑠)|^𝛼+2−2𝑎−𝑏̃−𝜀|𝑥(𝑠)|²+2𝑎11+𝑞|𝑥(𝑠)|^𝛽+2+𝑏+𝑏𝑞|𝑥(𝑠)|^𝛾+2. (22) Denoted by 𝐼(𝑥(𝑠))=2𝑎−𝑏̃−𝜀|𝑥(𝑠)|²+2𝑎|𝑥(𝑠)|^𝛼+2−2𝑎11+𝑞|𝑥(𝑠)|^𝛽+2−𝑏+𝑏𝑞|𝑥(𝑠)|^𝛾+2. (23) Noting that 2𝑎>𝑏̃, 2𝑎>2𝑎(1+1/𝑞)+𝑏+𝑏/𝑞, 𝜀<2𝑎−𝑏̃, 𝛼≥𝛽∨𝛾, by Lemma 1, there exists a positive 𝑐₀ such that 𝐼(𝑥(𝑠))≥𝑐₀|𝑥(𝑠)|². Substitution for this and (22) into (21) yields e^𝜀𝑡𝑉(𝑥(𝑡))≤𝑉(𝑥(0))+𝑀(𝑡)+2𝑎𝑞^𝑡₀e^𝜀𝑠𝑞e^{−(1−𝑞)𝜀𝑠}𝑥(𝑞𝑠)^𝛽+2−|𝑥(𝑠)|^𝛽+2+d𝑠𝑏𝑞^𝑡₀e^𝜀𝑠𝑞e^{−(1−𝑞)𝜀𝑠}𝑥(𝑞𝑠)^𝛾+2−|𝑥(𝑠)|^𝛾+2d𝑠−𝑐₀^𝑡₀e^𝜀𝑠|𝑥(𝑠)|²d𝑠. (24) Making use of the property of the integral, we may estimate ^𝑡₀e^𝜀𝑠𝑞e^{−(1−𝑞)𝜀𝑠}𝑥(𝑞𝑠)^𝛽+2−|𝑥(𝑠)|^𝛽+2=d𝑠_𝑡0𝑞e^𝜀𝑞𝑠𝑥(𝑞𝑠)^𝛽+2−e^𝜀𝑠|𝑥(𝑠)|^𝛽+2≤d𝑠_0𝑞𝑡e^𝜀𝑠|𝑥(𝑠)|^𝛽+2d𝑠−^𝑡₀e^𝜀𝑠|𝑥(𝑠)|^𝛽+2d𝑠=−^𝑡_𝑞𝑡e^𝜀𝑠|𝑥(𝑠)|^𝛽+2d𝑠. (25) Similarly, ^𝑡₀e^𝜀𝑠𝑞e^{−(1−𝑞)𝜀𝑠}𝑥𝑞𝑠^𝛾+2−|𝑥(𝑠)|^𝛾+2d𝑠≤−^𝑡_𝑞𝑡e^𝜀𝑠|𝑥(𝑠)|^𝛾+2d𝑠. (26) Substituting for (25) and (26) into (24) yields e^𝜀𝑡𝑉(𝑥(𝑡(𝑥(0(𝑡)−2))≤𝑉))+𝑀𝑎𝑞^𝑡_𝑞𝑡e^𝜀𝑠|𝑥(𝑠)|^𝛽+2−d𝑠𝑏𝑞^𝑡_𝑞𝑡e^𝜀𝑠|𝑥(𝑠)|^𝛾+2d𝑠−𝑐₀^𝑡₀e^𝜀𝑠|𝑥(𝑠)|²d𝑠. (27) Applying the nonnegative semimartingale convergence theorem (see [33]) in (27), we have limsup_𝑡→∞e^𝜀𝑡𝑉(𝑥(𝑡))<∞a.s. (28) That is, there is a finite positive random variable 𝐶₀ such that sup_0≤𝑡<∞e^𝜀𝑡𝑉(𝑥(𝑡))≤𝐶₀ a.s. This implies limsup_𝑡→∞(1/𝑡)log|𝑥(𝑡)|≤−𝜀/2 a.s. The proof is complete.

3. Stability of the Numerical Solution for Linear SPDE

In the section, we will establish almost surely exponential stability of EM method numerical solution under the following linear growth conditions.

(H4) The Linear Growth Conditions. For any 𝑥∈ℝ^𝑛, there exist positive constants 𝑎,𝑎,𝑐,𝑐,𝑏,𝑏 such that 𝑥(𝑠),𝑓𝑥(𝑠),𝑥𝑞𝑠≤−𝑎|𝑥(𝑠)|²+𝑎e^{−(1−𝑞)𝜀𝑠}𝑥(𝑞𝑠)²,𝑓(𝑥(𝑠),𝑥(𝑞𝑠))²≤𝑐|𝑥(𝑠)|²+𝑐e^{−(1−𝑞)𝜀𝑠}𝑥𝑞𝑠²,𝑔(𝑥(𝑠),𝑥(𝑞𝑠))²≤𝑏|𝑥(𝑠)|²+𝑏e^{−(1−𝑞)𝜀𝑠}𝑥(𝑞𝑠)². (29)(30)(31) Clearly, let 𝛼=𝛽=𝛾=𝑎=𝑏̃=0 in (H2) and (H3), which implies (H4). Similar to Theorems 2 and 3, we may obtain the following result.

Theorem 4. Let (H1) and (H4) hold with 2𝑎>(2𝑎+𝑏+𝑐)/𝑞+𝑏+𝑐. Then for any initial data 𝑥₀, there almost surely exists a unique global solution 𝑥(𝑡) to (5) on 𝑡≥0. Moreover, for 𝜀≤2𝑎−𝑏−𝑐−(2𝑎+𝑏+𝑐)/𝑞, the solution is almost surely exponentially stable; that is, limsup_𝑡→∞1𝑡|𝑥(𝑡𝜀log)|≤−2𝑎.𝑠. (32) Now we define the Euler-Maruyama approximate solution for (5). Given a step size Δ∈(0,1), compute the discrete approximations 𝑋_𝑘≈𝑥₀, 𝑡_𝑘=𝑘Δ by setting 𝑋₀=𝑥₀ and performing 𝑋_𝑘+1=𝑋_𝑘+𝑓𝑋_𝑘,𝑋_[𝑞𝑘]Δ+𝑔𝑋_𝑘,𝑋_[𝑞𝑘]Δ𝑤_𝑘, (33) where the increments Δ𝑤_𝑘=𝑤(𝑡_𝑘+1)−𝑤(𝑡_𝑘), 𝑘=1,…, are independent 𝑁(0,Δ)-distributed Gaussian random variables ℱ_𝑡𝑘-measurable at the mesh-point 𝑡_𝑘.

The following discrete semimartingale theorem plays an important role in the section.

Lemma 5 (see [32]). Let {𝐴_𝑖} and {𝑈_𝑖} be two nonnegative random variables such that both 𝐴_𝑖 and 𝑈_𝑖 are ℱ_𝑖−1-measurable for 𝑖=1,2,… with 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. If lim_𝑡→∞𝐴_𝑖<∞ a.s., then for almost all 𝜔∈Ω lim_𝑡→∞𝑋_𝑖<∞, lim_𝑡→∞𝑈_𝑖<∞. That is, all of the three processes 𝑋_𝑖 and 𝑈_𝑖 converge to finite random variables.

Theorem 6. Let (H1) and (H4) hold with 2𝑎>(2𝑎+𝑚𝑏+𝑐)/𝑞+𝑚𝑏+𝑐. Then for 𝜀≤2𝑎−𝑚𝑏−𝑐−(2𝑎+𝑚𝑏+𝑐)/𝑞, there exists a sufficiently small Δ^∗∈(0,1) such that the approximate solution {𝑋_𝑘} defined by (33) satisfies limsup_𝑘→∞1𝑋𝑘Δlog_𝑘𝜀≤−2𝑎.𝑠. (34)

Proof. By (33) and (H4), we may compute 𝑋_𝑘+1²=𝑋_𝑘+𝑓𝑋_𝑘,𝑋_[𝑞𝑘]Δ+𝑔𝑋_𝑘,𝑋_[𝑞𝑘]Δ𝑤_𝑘,𝑋_𝑘+𝑓𝑋_𝑘,𝑋_[𝑞𝑘]Δ+𝑔𝑋_𝑘,𝑋_[𝑞𝑘]Δ𝑤_𝑘=𝑋_𝑘²+2𝑋_𝑘,𝑓𝑋_𝑘,𝑋_[𝑞𝑘]Δ+𝑓(𝑋_𝑘,𝑋_[𝑞𝑘])²Δ²+𝑔(𝑋_𝑘,𝑋_[𝑞𝑘])Δ𝑤_𝑘²+2𝑋_𝑘+𝑓𝑋_𝑘,𝑋_[𝑞𝑘]Δ,𝑔𝑋_𝑘,𝑋_[𝑞𝑘]Δ𝑤_𝑘=𝑋_𝑘²+2⟨𝑋_𝑘,𝑓𝑋_𝑘,𝑋_[𝑞𝑘]+⟩Δ𝑓(𝑋_𝑘,𝑋_[𝑞𝑘])²Δ+𝑔(𝑋_𝑘,𝑋_[𝑞𝑘])²Δ+𝑔(𝑋_𝑘,𝑋_[𝑞𝑘])²Δ𝑤_𝑘²−𝑚Δ+2⟨𝑋_𝑘+𝑓𝑋_𝑘,𝑋_[𝑞𝑘]Δ,𝑔𝑋_𝑘,𝑋_[𝑞𝑘]Δ𝑤_𝑘⟩=1−2𝑎Δ+𝑐Δ²+𝑚𝑏Δ𝑋_𝑘²+2𝑎+𝑐Δ+𝑚𝑏e^{−(1−𝑞)𝜀𝑘Δ}𝑋_[𝑞𝑘]²Δ+𝑔(𝑋_𝑘,𝑋_[𝑞𝑘])²Δ𝑤_𝑘²𝑋−𝑚Δ+2_𝑘+𝑓𝑋_𝑘,𝑋_[𝑞𝑘]Δ,𝑔𝑋_𝑘,𝑋_[𝑞𝑘]Δ𝑤_𝑘=1−2𝑎Δ+𝑐Δ²+𝑚𝑏Δ𝑋_𝑘²+2𝑎+𝑐Δ+𝑚𝑏e^{−(1−𝑞)𝜀𝑘Δ}𝑋_[𝑞𝑘]²Δ+𝑠_𝑘, (35) where 𝑠_𝑘=|𝑔(𝑋_𝑘,𝑋_[𝑞𝑘])|²(|Δ𝑤_𝑘|²−𝑚Δ)+2⟨𝑋_𝑘+𝑓(𝑋_𝑘,𝑋_[𝑞𝑘])Δ,𝑔(𝑋_𝑘,𝑋_[𝑞𝑘])Δ𝑤_𝑘⟩. By (35), it is easy to obtain e^{𝜀(𝑘+1)Δ}𝑋_𝑘+1²−e^𝜀𝑘Δ𝑋_𝑘²=1−2𝑎Δ+𝑐Δ²+𝑚𝑏Δ−e^−𝜀Δe^{𝜀(𝑘+1)Δ}𝑋_𝑘²+2𝑎+𝑐Δ+𝑚𝑏e^{𝜀(𝑘𝑞+1)Δ}𝑋_[𝑞𝑘]²Δ+e^{𝜀(𝑘+1)Δ}𝑠_𝑘. (36) With the help of recursive method, it is not difficult to get e^𝜀𝑘Δ𝑋_𝑘²=𝑋₀²+1−2𝑎Δ+𝑐Δ²+𝑚𝑏Δ−e^−𝜀Δ×^𝑘−1_𝑖=0e^{𝜀(𝑖+1)Δ}𝑋_𝑖²+2𝑎+𝑐Δ+𝑚𝑏×^𝑘−1_𝑖=0e^{𝜀(𝑖+1)Δ}e^{−(1−𝑞)𝜀𝑖Δ}𝑋_[𝑞𝑖]²−e^𝜀Δ𝑋_𝑖²Δ+2𝑎+𝑐Δ+𝑚𝑏e^{𝜀Δ𝑘−1}_𝑖=0e^{𝜀(𝑖+1)Δ}𝑋_𝑖²Δ+^𝑘−1_𝑖=0e^{𝜀(𝑖+1)Δ}𝑠_𝑖=𝑋₀²−2𝑎−𝑐Δ−𝑚𝑏−1−e^−𝜀ΔΔ−2𝑎+𝑐Δ+𝑚𝑏e^𝜀Δ×^𝑘−1_𝑖=0e^{𝜀(𝑖+1)Δ}𝑋_𝑖²Δ+^𝑘−1_𝑖=0e^{𝜀(𝑖+1)Δ}e^{−(1−𝑞)𝜀𝑖Δ}𝑋_[𝑞𝑖]²−e^𝜀Δ𝑋_𝑖²Δ+^𝑘−1_𝑖=0e^{𝜀(𝑖+1)Δ}𝑠_𝑖. (37) Obviously, ∑^𝑘−1_𝑖=0e^{𝜀(𝑖+1)Δ}𝑠_𝑖 is a martingale. Let [𝑞𝑖]=𝑗; then 𝑗≤𝑞𝑖<𝑗+1, so 𝑞𝑖−1<𝑗≤𝑞𝑖. If 0≤𝑖≤𝑘−1, then −1<𝑗≤𝑞(𝑘−1)≤[𝑞𝑘]+1−𝑞≤[𝑞𝑘]+1. This implies that ^𝑘−1_𝑖=0e^{𝜀(𝑖+1)Δ}e^{−(1−𝑞)𝜀𝑖Δ}𝑋_[𝑞𝑖]²−e^𝜀Δ𝑋_𝑖²Δ=^𝑘−1_𝑖=0e^{𝜀(𝑞𝑖+1)Δ}𝑋_[𝑞𝑖]²−^𝑘−1_𝑖=0e^{𝜀(𝑖+2)Δ}𝑋_𝑖²≤^[𝑞𝑘]+1_𝑖=0e^{𝜀(𝑖+2)Δ}𝑋_𝑖²−^𝑘−1_𝑖=0e^{𝜀(𝑖+2)Δ}|𝑋(𝑖Δ)|^𝛼+2≤−^𝑘−1_{𝑖=[𝑞𝑘]+2}e^{𝜀(𝑖+2)Δ}𝑋_𝑖². (38) Denote by 𝑓(Δ)e=2𝑎−𝑐Δ−𝑚𝑏+^−𝜀Δ−1Δ−e^𝜀Δ2𝑎+𝑐Δ+𝑚𝑏. (39) By the Taylor series, we have e^−𝜀Δ()=1−𝜀Δ+𝜀Δ²2−(𝜀Δ)³()e3!+𝑜𝜀Δ>1−𝜀Δ,_−𝜀Δ−1Δ>−𝜀. (40) Therefore 𝑓(Δ)>2𝑎−𝑐Δ−𝑚𝑏−𝜀−e^𝜀Δ(2𝑎+𝑐Δ+𝑚𝑏). Noting that 2𝑎>(2𝑎+𝑚𝑏+𝑐)/𝑞+𝑚𝑏+𝑐,𝑞∈(0,1), for any given 𝜀, choose a sufficiently small Δ^∗ (Δ^∗<Δ) such that for all Δ<Δ^∗, 2𝑎−𝑐Δ−𝑚𝑏−𝜀−e^𝜀Δ2𝑎+𝑐Δ+𝑚𝑏>0. (41) Substituting for (38) and (41) into (37), Lemma 5 implies that there exists a positive constant 𝐶₀ such that e^𝜀𝑘Δ|𝑋_𝑘|²≤𝐶₀. That is, limsup_𝑘→∞(1/𝑘Δ)log|𝑋_𝑘|≤−𝜀/2 a.s. The proof is complete.

4. Stability of Numerical Solution for Highly Nonlinear SPDE

In the section, we will prove that the backward EM method can preserve the almost surely exponential stability of the true solution of highly nonlinear stochastic pantograph differential equation.

(H5) The Polynomial Growth Conditions. For any 𝑥∈ℝ^𝑑, there exist positive constants 𝛼,𝑎,𝑎,𝑎,𝑏,𝑏,𝑏̃ such that 𝑥(𝑠)𝑥(𝑠)||,𝑓,𝑥𝑞𝑠≤−𝑎𝑥(𝑠)^𝛼+2+𝑎e^{−(1−𝑞)𝜀𝑠}𝑥𝑞𝑠^𝛼+2−𝑎|𝑥(𝑠)|²,𝑔(𝑥(𝑠),𝑥(𝑞𝑠))²≤𝑏|𝑥(𝑠)|^𝛼+2+𝑏e^{−(1−𝑞)𝜀𝑠}𝑥(𝑞𝑠)^𝛼+2+𝑏̃|𝑥(𝑠)|². (42)(43) By Theorems 2 and 3, (5) has a unique global solution and the solution is almost surely exponentially stable.

Given a step size Δ∈(0,1), and for 𝑡∈[0,𝑇],  𝑀Δ=𝑇 for some positive integer 𝑀. Let 𝑡_𝑘=𝑘Δ (𝑘≥0),  [𝑡/Δ] be the integer part of 𝑡/Δ. Define the backward Euler-Maruyama method as follows: 𝑋_𝑘+1=𝑋_𝑘+𝑓𝑋_𝑘+1,𝑋_{[𝑞(𝑘+1)]}Δ+𝑔𝑋_𝑘,𝑋_[𝑞𝑘]Δ𝑤𝑡_𝑘. (44) To guarantee that this method is well defined, we impose the following one-sided Lipschitz condition on the drift coefficient 𝑓(𝑥,𝑦).

(H6) One-Sided Lipschitz Condition. There exists a positive constant 𝜆>0 such that for any 𝑥_𝑖∈ℝ^𝑛 (𝑖=1,2) 𝑥₁−𝑥₂,𝑓𝑥₁,𝑦−𝑓𝑥₂,𝑦≤𝜆𝑥₁−𝑥₂². (45) Applying a fixed point theorem one can prove that (44) has a unique solution 𝑋_𝑘+1; given 𝑋_𝑘 if 𝜆Δ<1, then the backward EM scheme (44) is well defined (see, e.g., [28]). From now on we always assume that Δ<𝜆⁻¹.

Theorem 7. Let (H1), (H5) and (H6) hold with 2𝑎>𝑚𝑏̃,  2𝑎>𝑚𝑏+(2𝑎+𝑚𝑏)/𝑞; there exists a sufficiently small Δ^∗∈(0,1) such that the approximate solution {𝑋_𝑘} defined by (44) satisfies limsup_𝑘→∞1𝑋𝑘Δlog_𝑘𝜀≤−2𝑎.𝑠., (46) where 𝜀<2𝑎−𝑚𝑏̃.

Proof. By (44) and (H5), we may compute 𝑋_𝑘+1²=𝑋_𝑘+1,𝑋_𝑘+𝑓𝑋_𝑘+1,𝑋_{[𝑞(𝑘+1)]}Δ+𝑔𝑋_𝑘,𝑋_[𝑞𝑘]Δ𝑤_𝑘=𝑋_𝑘+1,𝑓𝑋_𝑘+1,𝑋_{[𝑞(𝑘+1)]}Δ+𝑋_𝑘+1,𝑋_𝑘+𝑔𝑋_𝑘,𝑋_[𝑞𝑘]Δ𝑤_𝑘≤−𝑎𝑋_𝑘+1^𝛼+2Δ+𝑎e^{−𝜀(1−𝑞)(𝑘+1)Δ}𝑋_{[𝑞(𝑘+1)]}^𝛼+2Δ−𝑎𝑋_𝑘+1²1Δ+2𝑋_𝑘+1²+12𝑋_𝑘²+12𝑔𝑋_𝑘,𝑋_[𝑞𝑘]²Δ𝑤_𝑘²+𝑋_𝑘,𝑔𝑋_𝑘,𝑋_[𝑞𝑘]Δ𝑤_𝑘=12−𝑎Δ𝑋_𝑘+1²+𝑎e^{−𝜀(1−𝑞)(𝑘+1)Δ}𝑋_{[𝑞(𝑘+1)]}^𝛼+2Δ−𝑎𝑋_𝑘+1^𝛼+21Δ+2𝑋_𝑘²+12𝑏𝑋_𝑘^𝛼+2+𝑏e^{−𝜀(1−𝑞)𝑘Δ}𝑋_[𝑞𝑘]^𝛼+2+𝑏̃𝑋_𝑘²×Δ𝑤_𝑘²+⟨𝑋_𝑘,𝑔𝑋_𝑘,𝑋_[𝑞𝑘]⟩Δ𝑤_𝑘. (47) That is, (1+2𝑎Δ)𝑋_𝑘+1²=1+𝑚𝑏̃Δ𝑋_𝑘²+2𝑎e^{−𝜀(1−𝑞)(𝑘+1)Δ}𝑋_{[𝑞(𝑘+1)]}^𝛼+2Δ−2𝑎𝑋_𝑘+1^𝛼+2Δ+𝑚𝑏𝑋_𝑘^𝛼+2+𝑚𝑏e^{−𝜀(1−𝑞)𝑘Δ}𝑋_[𝑞𝑘]^𝛼+2×Δ+𝑠^Δ_𝑘, (48) where 𝑠^Δ_𝑘=𝑏𝑋_𝑘^𝛼+2+𝑏e^{−𝜀(1−𝑞)𝑘Δ}𝑋_[𝑞𝑘]^𝛼+2+𝑏̃𝑋_𝑘²×Δ𝑤_𝑘²−𝑚Δ+2𝑋_𝑘,𝑔𝑋_𝑘,𝑋_[𝑞𝑘]Δ𝑤_𝑘. (49) According to (48), we may obtain (1+2𝑎Δ)e^{𝜀(𝑘+1)Δ}𝑋_𝑘+1²−e^𝜀𝑘Δ𝑋_𝑘²=1+𝑚𝑏̃Δ−(1+2𝑎Δ)e^−𝜀Δe^{𝜀(𝑘+1)Δ}𝑋_𝑘²+2𝑎e^{𝜀𝑞(𝑘+1)Δ}𝑋_{[𝑞(𝑘+1)]}^𝛼+2Δ−2𝑎e^{𝜀(𝑘+1)Δ}𝑋_𝑘+1^𝛼+2Δ+𝑚𝑏e^{𝜀(𝑘+1)Δ}𝑋_𝑘^𝛼+2Δ+𝑚𝑏e^{−𝜀(𝑞𝑘+1)Δ}𝑋_[𝑞𝑘]^𝛼+2Δ+e^{𝜀(𝑘+1)Δ}𝑠^Δ_𝑘. (50) With the help of recursive method, compute (1+2𝑎Δ)e^𝜀𝑘Δ𝑋_𝑘²=(1+2𝑎Δ)𝑋₀²+1+𝑚𝑏̃Δ−(1+2𝑎Δ)e^−𝜀Δ×^𝑘−1_𝑖=0e^{𝜀(𝑖+1)Δ}𝑋_𝑖²−2𝑎^𝑘−1_𝑖=0e^{𝜀(𝑖+1)Δ}𝑋_𝑖+1^𝛼+2Δ+2𝑎^𝑘−1_𝑖=0e^{𝜀𝑞(𝑖+1)Δ}𝑋_{[𝑞(𝑖+1)]}^𝛼+2Δ+𝑚𝑏^𝑘−1_𝑖=0e^{𝜀(𝑖+1)Δ}𝑋_𝑖^𝛼+2Δ+𝑚𝑏^𝑘−1_𝑖=0e^{𝜀(𝑞𝑖+1)Δ}𝑋_[𝑞𝑖]^𝛼+2Δ+^𝑘−1_𝑖=0e^{𝜀(𝑖+1)Δ}𝑠^Δ_𝑖=(1+2𝑎Δ)𝑋₀²−1+2𝑎Δ−1+𝑚𝑏̃Δe^𝜀Δ×^𝑘−1_𝑖=0e^𝜀𝑖Δ𝑋_𝑖²−2𝑎^𝑘_𝑖=1e^𝜀𝑖Δ𝑋_𝑖^𝛼+2Δ+𝑚𝑏^𝑘−1_𝑖=0e^{𝜀(𝑖+1)Δ}𝑋_𝑖^𝛼+2Δ+2𝑎^𝑘−1_𝑖=0e^{𝜀(𝑖+1)Δ}𝑋_𝑖^𝛼+2Δ+𝑚𝑏^𝑘−1_𝑖=0e^{𝜀(𝑖+2)Δ}𝑋_𝑖^𝛼+2Δ+2𝑎^𝑘−1_𝑖=0e^{𝜀(𝑖+1)Δ}e^{−𝜀(1−𝑞)(𝑖+1)Δ}𝑋_{[𝑞(𝑖+1)]}^𝛼+2−𝑋_𝑖^𝛼+2Δ+𝑚𝑏^𝑘−1_𝑖=0e^{𝜀(𝑖+1)Δ}e^{−𝜀(1−𝑞)𝑖Δ}𝑋_[𝑞𝑖]^𝛼+2−e^𝜀Δ𝑋_𝑖^𝛼+2Δ+𝑆_𝑘, (51) where 𝑆_𝑘=∑^𝑘−1_𝑖=0e^{𝜀(𝑖+1)Δ}𝑠^Δ_𝑘. Obviously, 𝑆_𝑘 is a martingale. Let [𝑞𝑖]=𝑗; then 𝑗≤𝑞𝑖<𝑗+1, so 𝑞𝑖−1<𝑗≤𝑞𝑖. If 0≤𝑖≤𝑘−1; then −1<𝑗≤𝑞(𝑘−1)≤[𝑞𝑘]+1−𝑞≤[𝑞𝑘]+1. This implies ^𝑘−1_𝑖=0e^{𝜀(𝑖+1)Δ}e^{−𝜀(1−𝑞)𝑖Δ}𝑋_[𝑞𝑖]^𝛼+2−e^𝜀Δ𝑋_𝑖^𝛼+2=^𝑘−1_𝑖=0e^{𝜀(1+𝑞𝑖)Δ}𝑋_[𝑞𝑖]^𝛼+2−^𝑘−1_𝑖=0e^{𝜀(𝑖+2)Δ}𝑋_𝑖^𝛼+2=^[𝑞𝑘]+1_𝑖=0e^{𝜀(𝑖+2)Δ}𝑋_𝑖^𝛼+2−^𝑘−1_𝑖=0e^{𝜀(𝑖+2)Δ}𝑋_𝑖^𝛼+2≤−^𝑘−1_{𝑖=[𝑞𝑘]+2}e^{𝜀(𝑖+2)Δ}𝑋_𝑖^𝛼+2. (52) Similarly, ^𝑘−1_𝑖=0e^{−𝜀(𝑖+1)Δ}e^{𝜀(1−𝑞)(𝑖+1)Δ}𝑋_{[𝑞(𝑖+1)]})^𝛼+2−𝑋_𝑖^𝛼+2≤−^𝑘−1_{𝑖=[𝑞𝑘]+2}e^{𝜀(𝑖+1)Δ}𝑋_𝑖^𝛼+2. (53) Substituting for (52) and (53) into (51) yields (1+2𝑎Δ)e^𝜀𝑘Δ𝑋_𝑘²=(1+2𝑎Δ)𝑋₀²−1+2𝑎Δ−1+𝑚𝑏̃Δe^𝜀Δ×^𝑘−1_𝑖=0e^𝜀𝑖Δ𝑋_𝑖²+−2𝑎+𝑚𝑏e^𝜀Δ+2𝑎e^𝜀Δ+𝑚𝑏e^2𝜀Δ×^𝑘−1_𝑖=1e^𝜀𝑖Δ𝑋_𝑖^𝛼+2Δ−2𝑎e^𝜀𝑘Δ𝑋_𝑘^𝛼+2Δ+2𝑎+𝑚𝑏+𝑚𝑏e^𝜀Δe^𝜀Δ𝑋₀^𝛼+2Δ−2𝑎^𝑘−1_{𝑖=𝑞𝑘+2}e^{𝜀(𝑖+1)Δ}𝑋_𝑖^𝛼+2Δ−𝑚𝑏^𝑘−1_{𝑖=𝑞𝑘+2}e^{𝜀(𝑖+2)Δ}𝑋_𝑖^𝛼+2Δ+𝑆_𝑘. (54) Denote by 𝑓(Δ)=1+2𝑎Δ−(1+𝑚𝑏̃Δ)e^𝜀Δ. Differentiating with respect to Δ yields 𝑓^′(Δ)=2𝑎−𝑚𝑏̃e^𝜀Δ−1+𝑚𝑏̃Δ𝜀e^𝜀Δ,𝑓^′′(Δ)=−2𝑚𝑏̃𝜀e^𝜀Δ−1+𝑚𝑏̃Δ𝜀²e^𝜀Δ. (55) Clearly, 𝑓^′(0)=2𝑎−𝑚𝑏̃−𝜀>0, 𝑓^′′(0)<0; then there exists a Δ>0 such that 𝑓^′(Δ)=0. 𝑓(Δ) is an increase function for Δ<Δ and noting that 𝑓(0)=0 therefore there exists a sufficiently small Δ^∗ (Δ^∗<Δ) such that for all Δ<Δ^∗, 1+2𝑎Δ−1+𝑚𝑏̃Δe^𝜀Δ>0. (56) On the other hand, since 2𝑎>(2𝑎+𝑚𝑏)/𝑞+𝑏,   𝑞∈(0,1), then 2𝑎>2𝑎+𝑚𝑏+𝑚𝑏̃, and there exists sufficiently small Δ<Δ^∗ such that 2𝑎−2𝑎e^𝜀Δ−𝑚𝑏+𝑏e^𝜀Δe^𝜀Δ>0. (57) Substituting for (56) and (57) into (54), Lemma 5 implies that there exists a positive constant 𝐶₀ such that (1+2𝑎Δ)e^𝜀𝑘Δ𝑋_𝑘²≤𝐶₀. (58) That is, limsup_𝑘→∞1𝑋𝑘Δlog_𝑘𝜀≤−2a.s. (59) The proof is complete.

Example 8. Consider the following nonlinear scalar SPDE d𝑥(𝑡)=−0.5𝑥(𝑡)−5𝑥⁵(𝑡)+3e^{−0.25𝜀𝑡}𝑥⁵𝑞𝑡d𝑡+𝑥³(𝑡)d𝑤(𝑡), (60) where 𝑤(𝑡) is scalar Brownian motion. Define 𝑓(𝑥,𝑦,𝑡)=−0.5𝑥−5𝑥⁵+3e^{−0.25𝜀𝑡}𝑦⁵, 𝑔(𝑥,𝑦,𝑡)=𝑥³. Compute 𝑓𝑥₁,𝑦,𝑡−𝑓𝑥₂𝑥,𝑦,𝑡≤−0.5₁−𝑥₂−5𝑥⁵₁−𝑥⁵₂≤−0.5𝑥₁−𝑥₂1+10𝑥⁴₁+𝑥³₁𝑥₂+𝑥²₂𝑥²₂+𝑥₁𝑥³₂+𝑥⁴₂. (61) Noting that 𝑎²+𝑏²≥(𝑎+𝑏)²/2, compute 𝑥⁴₁+𝑥³₁𝑥₂+𝑥²₂𝑥²₂+𝑥₁𝑥³₂+𝑥⁴₂≥𝑥²₁+𝑥²₂²2+𝑥₁𝑥₂𝑥²₁+𝑥²₂+𝑥₁𝑥₂²≥𝑥²₁+𝑥²₂2+𝑥₁𝑥₂², (62) which implies 𝑥₁−𝑥₂,𝑓𝑥₁,𝑦,𝑡−𝑓𝑥₂,𝑦,𝑡≤−0.5𝑥₁−𝑥₂². (63) That implies that 𝑓(𝑥,𝑦,𝑡) satisfies the one-sided growth condition. By using the inequality 𝑎^𝑝𝑏^𝑞≤(𝑝/(𝑝+𝑞))𝑎^𝑝+𝑞+(𝑞/(𝑝+𝑞))𝑏^𝑝+𝑞, it is easy to compute 𝑥,𝑓𝑥,𝑦,𝑡≤−0.5𝑥²−5𝑥⁶+3e^{−0.25𝜀𝑡}𝑦⁵𝑥≤−0.5𝑥²−4.5𝑥⁶+2.5𝑦⁶ (64) and |𝑔(𝑥(𝑡),𝑦(𝑡))|²≤𝑥⁶. By Theorems 2 and 7, (60) has unique global solution and the solution is almost surely exponentially stable.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author expresses their sincere gratitude to two anonymous referees for their detailed comments and helpful suggestions. This project is supported by NNSF (11301198) and Fundamental Research Funds for the Central Universities (2011QN167).