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

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.


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.

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 F 0 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 T y.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 T A .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 F 0 −τ, 0 ; R n be the family of all F 0 -measurable bounded C −τ, 0 ; R n -valued random variables ξ {ξ θ : −τ ≤ θ ≤ 0}.
Consider an n-dimensional NSDDE 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.

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 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 A 0 U 0 0 a.s.Let M i be a real-valued local martingale with M 0 0 a.s.Let ζ be a nonnegative F 0 -measurable random variable.Assume that {X i } is a nonnegative semimartingale with the Doob-Mayer decomposition that is, both X i and U i converge to finite random variables.

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.

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 F 0 −τ, 0 ; R n .According to 3.3 , there is a constant α > 0 such that where W t, x λ 1 − λ 3 − 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.

Stability of the Euler Method
Applying the Euler method EM to 2.1 yields

4.1
Here h τ/m m is an positive integer is the stepsize, and Squaring both sides of 4.4 , we have

4.5
Using 3.2 and 4.3 , we can obtain that where

4.7
It therefore follows that

4.10
Then, we have where where Then, we conclude that lim 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.

Stability of the Backward Euler Method
Applying the backward Euler method BEM to 2.1 yields

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.Proof.Let Y k X k − N X k−m .Then, it follows from 5.1 that Squaring both sides of 5.2 , we have

5.3
Using 3.2 , we can obtain that where It therefore follows that which implies that

5.8
Then, we have

5.9
Namely, where M k k−1 i 0 m i .By 18 , M k is a martingale with M 0 0. From 3.3 , we obtain that Then, we conclude that The proof is completed.

Numerical Experiments
In this section, we present numerical experiments to illustrate the theoretical results presented in the previous sections.Consider the following scalar linear problem

6.1
For test 6.1 , we have that L 72, λ 1 11, λ 2 4, λ 3 0, and λ 4 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 ω 1 , X n ω 2 , X n ω 3 using EM 4.1 to approximate 6.1 with the stepsize h 1/25.Figure 1 shows that X n ω 1 , X n ω 2 , X n ω 3 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 ω 1 , X n ω 2 , X n ω 3 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.

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.

Figure 1 :
Figure 1: Almost surely asymptotic stability of the EM approximate solution X n with the stepsize h 1/25.

Figure 2 :
Figure 2: Almost surely asymptotic stability of the BEM approximate solution X n with the stepsize h 0.1.
Let Assumption 2.1 hold.Assume that there are four positive constants λ 1 − λ 4 such that for any initial data ξ ∈ C b F 0 −τ, 0 ; R n .Theorem 3.2.
kh represents the Browian motion increment.To be precise, let us introduce the definition on the almost surely asymptotic stability of numerical solutions.
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 thatf t, x, y 2 ≤ L |x| 2 y 2 .4.3Then there exists a h 0 > 0 such that if h < h 0 , then for any given finite-valued F 0 -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