Abstract

Almost sure exponential stability of the split-step backward Euler (SSBE) method applied to an Itô-type stochastic differential equation with time-varying delay is discussed by the techniques based on Doob-Mayer decomposition and semimartingale convergence theorem. Numerical experiments confirm the theoretical analysis.

1. Introduction

In this paper we study the following nonlinear SDDE: for every . Here is a time-varying delay satisfying and . The initial function when . We further assume that the initial data is independent of Wiener measure driving the equation and is a scalar Brownian motion on the complete probability space with a filtration satisfying the usual conditions. Moreover, are Borel-measurable functions.

Stability theory for numerical methods applied to stochastic differential equation (SDE) typically deals with mean-square behavior [1]. The mean-square stability analysis of numerical methods for SDDE has received a great deal of attention (see, e.g., [2, 3] and the references therein). Recently, the almost sure (a.s.) stability (or the trajectory stability) is becoming prevalent in the science literature [4–11]. However, the prior works concerned with SDDE are [7, 8, 10]. Rodkina et al. [7] studied almost sure stability of a drift-implicit -method applied to an SDE with memory. Using the martingale techniques, Wu and his coauthors [8, 10] discussed almost sure exponential stability of the Euler-Maruyama (EM) method for the SDE with a constant delay and stochastic functional differential equation. We note that the two above schemes are all single-stage method; this paper studies the almost sure stability of a two-stage scheme named split-step backward Euler (SSBE) method [12, 13] applied to the nonlinear SDDE (1) with time-varying delay.

Applying the SSBE method (see [12, 13]) to (1) yieldswhere and for , , Here is the step size and denotes the approximation of at time . We remark that in (3) depends on how memory values are handled on nongrid points. The almost sure convergence of SSBE method has been investigated by Guo and Tao [14]; the main aim of this paper is to study the almost sure stability of the SSBE method applied to (1).

2. Preliminary Results

Before stating the main results, we present the essential notation and definitions which are necessary for further consideration. Let be the Euclidean norm in and the family of continuous functions from to , equipped with the supremum norm . Also, denote by the family of bounded, -measurable, -valued random variables. If is a vector or matrix, its transpose is denoted by . The inner product of is denoted by or .

Now we give some definitions on the almost sure exponential stability of SDDEs and its numerical approximation.

Definition 1. The solution to (1) is said to be almost surely exponentially stable if there exists a constant such that for any initial data .

Definition 2. The solution to (2a) and (2b) is said to be almost surely exponentially stable if there exists a constant such that for any bounded variables when .

For the purpose of stability, we assume that , which implies that (1) admits the equilibrium solution corresponding to the initial condition for . As a standing hypothesis, we will impose the following local Lipschitz condition (cf. [11, 12, 14]) on the coefficients and .(A1)For each integer , there exists a positive constant such that, for all with , and all , , where is the maximal operator. To guarantee the almost sure stability of the unique solution to (1), we need the following assumption for the time-varying delay .(A2)Let the delay function be Borel measurable and bounded. In what follows we introduce the result of almost sure stability of SDDEs (1). The proof of the following lemma can be found in [15].

Lemma 3. Let Assumptions (A1) and (A2) hold. Assume that there are four nonnegative constants such that for all and . If then the trivial solution of (1) is almost surely exponentially stable.

To explain our idea, we cite the discrete semimartingale convergence theorem as follows.

Theorem 4 (see [8, 9]). Let be an almost sure nonnegative stochastic sequence of -measurable random variables on probability space . Assume that permits the decomposition where and are two nondecreasing, predictable processes with ; is local -martingale with on . Then, the requirement of (a.s.) implies that for almost all .

3. Almost Sure Asymptotic Exponential Stability of Numerical Solution

In this section, our aim is to examine if the SSBE method can reproduce the almost sure exponential stability of the exact solution of (1). Comparing to the existing results of single-stage methods [8, 10], we need to appropriately estimate the intermediate solution , which also leads to more complex structure of the inner product of , so that the discrete semimartingale convergence theorem is still valid for this case.

Now we give the main result of almost sure stability of the SSBE approximate solution .

Theorem 5. Suppose that conditions of Lemma 3 are satisfied and the drift coefficient satisfies the linear growth condition; namely, there exists a constant such that for all and . Then there exists an such that if , the SSBE approximate solution is almost surely exponentially stable.

Proof. Note that from the SSBE method (2a) and (2b).
By using (6) and (7), we have Equation (13), together with (14), shows that Therefore, by conditions (8) and (12), we have Similarly, under conditions (6), (7), and (12), which implies that By Vieta theorem, because the discriminant of the quadratic equation is positive and , there must exist an such that for any ; then For simplicity, in what follows, the formula is denoted by . Combining (16) and (19) leads us to For any positive constant , we have which yields by using (20). Summing up both sides of inequality (22) from to (), we get Let . Since and is -measurable, we obtain which implies that is a martingale.
Similarly, are also martingales. Therefore, is a martingale with . Then we have Noting that there are two approximating cases of the time dependent delay term in (3), the following analysis will be divided into two situations. First, we have under condition . There exists an such that, for any , , where is the minimal operator. Further, we set for any nonnegative integer , , for , and Therefore, a direct application of Theorem 4 to the sequence yields that Choose the , such that and hence We therefore obtain that, for any , as required.
Now, let us discuss the second situation: . Inequality (27) gives Since we have where There exists an such that, for any , Similarly, the solution is almost surely exponentially stable by using Theorem 4.
Consequently, we conclude that, for any , the SSBE approximate solution is almost surely exponentially stable.