On Mean Square Stability and Dissipativity of Split-Step Theta Method for Nonlinear Neutral Stochastic Delay Differential Equations

A split-step theta (SST) method is introduced and used to solve the nonlinear neutral stochastic delay differential equations (NSDDEs). The mean square asymptotic stability of the split-step theta (SST) method for nonlinear neutral stochastic delay differential equations is studied. It is proved that under the one-sided Lipschitz condition and the linear growth condition, the split-step theta method with θ ∈ (1/2, 1] is asymptotically mean square stable for all positive step sizes, and the split-step theta method with θ ∈ [0, 1/2] is asymptotically mean square stable for some step sizes. It is also proved in this paper that the split-step theta (SST) method possesses a bounded absorbing set which is independent of initial data, and the mean square dissipativity of this method is also proved.


Introduction
Stochastic functional differential equations (SFDEs) play important roles in science and engineering applications, especially for systems whose evolutions in time are influenced by random forces as well as their history information.When the time delays in SFDEs are constants, they turn into stochastic delay differential equations (SDDEs).Both the theory and numerical methods for SDDEs have been well developed in the recent decades; see [1][2][3][4][5][6][7][8].Recently, many dynamical systems not only depend on the present and the past states but also involve derivatives with delays; they are described as the neutral stochastic delay differential equations (NSDDEs).Compared to the stochastic differential equations and the stochastic delay differential equations, the study of the neutral stochastic delay differential equations has just started.In 1981, Kolmanovskii and Myshkis [9] took the environmental disturbances into account, introduced the neutral stochastic delay differential equations (NSDDEs), and gave their applications in chemical engineering and aeroelasticity.The analytical solutions of NSDDEs are hard to obtain; many authors have to study the numerical methods for NSDDEs.Wu and Mao [10] studied the convergence of the Euler-Maruyama method for neutral stochastic functional differential equations under the one-side Lipschitz conditions and the linear growth conditions.In 2009, Zhou and Wu [11] studied the convergence of the Euler-Maruyama method for NSDDEs with Markov switching under the one-side Lipschitz conditions and the linear growth conditions.The convergence of -method and the mean square asymptotic stability of the semi-implicit Euler method for NSDDEs were studied by Gan et al. [12], Zhou and Fang [13], and Yin and Ma [14], respectively.Later, the almost sure exponential stability of Euler-Maruyama method for NSDDEs was studied in [15] with the discrete semimartingale convergence theorem.
To the best of our knowledge, most of these studies have focused on the convergence of numerical solutions for NSD-DEs; the stability and dissipativity of numerical solutions for them are rarely concerned.
The aim of this paper is to study the mean square stability and dissipativity of the split-step theta method with some conditions and the step constrained for NSDDEs.
The paper is organized as follows.In Section 2, some stability definitions about the analytic solutions for NSDDEs are introduced; some notations and preliminaries are also presented in this section.In Section 3, the split-step theta method is introduced and used to solve the NSDDEs; the asymptotic stability of the split-step theta method is proved.In Section 4, the long time behavior of numerical solution is studied and the mean square dissipativity result of the method is illustrated.In Section 5, some numerical experiments are given to confirm the theoretical results.

Exponential Mean Square Stability of Analytic Solution
Let with the notation E denoting the mathematical expectation with respect to P.
The following conditions (a1) and (a2) are standard for the existence and uniqueness of the solution for (1).
As an especial case of Theorem 3.1 in Mao's monograph (see [6]), we can easily know that under hypothesis (a1) and (a2), system (1) has a global unique continuous solution on  ≥ −, which is denoted by ().
Now we recall some stability concepts for the solution of (1).
Definition 1 (see [6]).The trivial solution of ( 1) is said to be exponentially mean square stable, if there exists a pair of constants  > 0 and  > 0, such that, whenever sup Lemma 2. Assume that there exist a symmetric, positive definite  ×  matrix  and positive constants  1 ,  2 , and  ∈ (0, 1) such that for all (, , ) hold, then the trivial solution of ( 1) is exponentially mean square stable.
By Lemma 2, the following result can easily be obtained.Theorem 4. Suppose ( 6) holds.Assume that there are positive constants  1 ,  2 , and , such that, for all ,  ∈   , hold, then the trivial solution of ( 1) is exponentially mean square stable.

The Stability of the Split-Step Theta Method
The split-step theta method is proved to be able to keep the mean square asymptotic stability of the exact solution under the sufficient conditions of the asymptotic stability of the exact solution, so in this paper we use the split-step theta method to solve the NSDDE.Appling the split-step theta (SST) method into problem (1) gives the following form: where the step size Δ = /,  is an integer,   is an approximation to (  ),   = Δ,  = 1, 2, . .., and   =   = (Δ) for  = −, − + 1, . . ., 0.  ∈ [0, 1] is a fixed parameter, and Δ  fl (( + 1)Δ) − (Δ) is the Brownian increment.
When  = 0 the split-step theta method is simplified into the split-step forward Euler method and when  = 1 the split-step theta method is simplified into the split-step backward Euler method.They were discussed for stochastic differential equations in [17][18][19][20].In order to consider the stability property of scheme ( 13)-( 15) we should give some stability concepts for numerical methods firstly.
Definition 5 (see [16]).For a given step size Δ, a numerical method is said to be exponentially mean square stable if there is a pair of positive constants  and  such that for any initial data () the numerical solution   produced by the method satisfies Definition 6 (see [16]).For a given step size Δ, a numerical method is said to be asymptotically mean square stable if for any initial data () the numerical solution   produced by the method satisfies Theorem 7. Assume that system (1) satisfies (7) with − 1 +  2 < 0; then the SST method ( 13)- (15) then, for any  ∈ [0, 1/2), there exists a constant Δ 0 depending on  such that the method is asymptotically mean square stable for Δ ∈ (0, Δ 0 ).
By induction, the following results are obtained from (24): Using condition ( 14), we can get the following inequality: Therefore, It can be deduced from ( 28) and (25) that −( 2 − ((1 −  2 ) 2 +  2  1 ) − ) ≤ 0, so, we can have the following inequality: then we get define the following inequality could be deduced from (31): which implies that the method is asymptotically mean square stable.
Similar to the proof of (31), we can prove that when Δ ∈ (0, Δ 0 ) the method is asymptotically mean square stable; the proof of theorem is completed.
Remark 8.For system (1) with  = 0, it becomes a stochastic delay differential equation; the mean square stability of the theta method has been studied in [16]; Theorem 7 can be regarded as an extension of Theorem 3.4 presented in [16].
Remark 9.For the NSDDEs, the mean square asymptotic stability of the BEM method has been studied by Wang and Chen in [15]; it has shown that BEM method can reproduce the mean square stability of the exact solutions; Theorem 7 improves the result in [15].

Mean Square Dissipativity
The numerical solutions' long time dynamic behavior will be studied in this section.Before it, we make the following hypothesis: assume that there exist a symmetric, positive definite  ×  matrix  and positive constants  1 ,  2 , and  such that, for all (, , ) ∈  + ×   ×   , the following inequality exists:
Theorem 11 means that the discrete system possesses a bounded absorbing set in the sense of mean square.The numerical solution trajectory from any initial date will enter the set in a finite time and thereafter remain inside.It is called mean square dissipativity.
Remark 12.For the study of the dissipativity of numerical methods for deterministic delay differential equations with constant delays, Huang and Chang studied the dissipativity of Runge-Kutta methods and multistep Runge-Kutta methods in [21,22].