Analysis of stability for stochastic delay integro-differential equations

In this paper, we concern stability of numerical methods applied to stochastic delay integro-differential equations. For linear stochastic delay integro-differential equations, it is shown that the mean-square stability is derived by the split-step backward Euler method without any restriction on step-size, while the Euler–Maruyama method could reproduce the mean-square stability under a step-size constraint. We also confirm the mean-square stability of the split-step backward Euler method for nonlinear stochastic delay integro-differential equations. The numerical experiments further verify the theoretical results.


Introduction
Stochastic delay integro-differential equations, as the mathematical model, widely apply in biology, physics, economics and finance [1,2]. Because of the stochastic delay integrodifferential equations themselves, it is not easy to obtain an explicit solution for these kinds of equations, so it is necessary to research the numerical methods for numerical solution of stochastic delay integro-differential equations [3,4]. Stability is the basic and important property of numerical methods for stochastic systems.
There are few results on the numerical methods to stochastic delay integro-differential equations. Ding et al. [5] dealt with the stability of the semi-implicit Euler method for linear stochastic delay integro-differential equations. Rathinasamy and Balachandran [6] proved mean-square stability of the Milstein method for linear stochastic delay integrodifferential equations with Markovian switching under suitable conditions on the integral term. The condition under which the split-step backward Euler method was meansquare stable has been obtained by Tan and Wang [7,8]. Rathinasamy and Balachandran [9] also analyzed T-stability of the split-step-θ -methods for linear stochastic delay integro-differential equations. Wu [10] investigated the mean-square stability for stochastic delay integro-differential equations by the strong balanced methods and the weak balanced methods with sufficiently small step-size. Numerical researches for stochastic delay integro-differential equations are not perfect enough. Therefore, it is extremely essential to develop the stability of the numerical methods to stochastic equations.
The paper is organized as follows. In Sect. 2 we will introduce related symbols and definitions. Some suitable conditions will be given to guarantee stability of the Euler-Maruyama method for stochastic delay integro-differential equations in Sect. 3. In Sect. 4, the splitstep backward Euler method will be used to prove general mean-square stability of numerical solutions. In Sect. 5, we will discuss stability of nonlinear stochastic delay integrodifferential equations. Furthermore, numerical experiments are provided in Sect. 6.

Preliminaries
Throughout this paper, unless otherwise specified, let ( , F, P) be a complete probability space with a filtration (F t ) t≥0 , which satisfies the usual conditions (i.e., it is increasing and right continuous while F 0 contains all P-null sets). Let | · | be the Euclidean norm, W (t) is Wiener process defined on the probability space, which be F t -adapted and independent of F 0 . Let τ > 0 and C([-τ , 0]; R) denote the family of all continuous R-valued functions on [-τ , 0], C([-τ , 0]; R d ) denote the family of all continuous functions ξ from [-τ , 0] to R d , ξ is defined by ξ = sup -τ ≤t≤0 |ξ (t)|. We assume ξ (t), t ∈ [-τ , 0] is the initial function, which is F 0 -measurable and right continuous, As a matter of convenience, we first consider the following form of linear stochastic delay integro-differential equations: where ξ (t) is initial function, and ξ (t) ∈ C([-τ , 0]; R), α, β, γ , λ, μ, η ∈ R, W (t) is a standard one-dimensional Wiener process and τ is the delay term.
Under the above assumptions, Eq. (1) has a unique solution x(t). In order to analyze mean-square stability of two numerical methods, we introduce the following lemma [11].

Lemma 2.1 If
the solution of Eq. (1) is said to be mean-square stable, that is,

Mean-square stability of the Euler-Maruyama method
Now, the Euler-Maruyama method applied to Eq. (1) one gets X n+1 = X n + (αX n + βX n-m + γX n )h + (λX n + μX n-m + ηX n ) W n , where ξ = X 0 , X n is an approximation to the analytical solution x(t n ),n which is F t nmeasurable, h > 0 is the given step-size, which satisfies h = τ m for a positive integer m, t n = nh, n = -m, . . . , 0, and we get X n = ξ (t n ) when t n ≤ 0, W n = W (t n+1 ) -W (t n ) are independent N(0, h) distributed stochastic variables.X n approaches the integral term, this paper will choose a composite trapezoidal rule as the tool of the disperse integral to solve this case. We havē Proof From Eq. (4), we obtain Squaring both sides of Eq. (6), we have According to the inequality (a 1 + a 2 + · · · + a n ) 2 ≤ n(a 2 1 + a 2 2 + · · · + a 2 n ), In a similar way We note that E( W n ) = 0, E[( W n ) 2 ] = h, and X n , X n-1 , . . . , X n-m are F t n -measurable. Substituting (7), (8), (9) into the above equation and taking expectations, Hence let By the condition (2), we know that h 1 > 0, h 2 > 0. If h 0 ∈ (0, h 1 ), we have On the other side, we address the case 1 + αh > 0. If h 0 ∈ (0, h 2 ), we get Let h 0 ∈ max{h 1 , h 2 }; when h ∈ (0, h 0 ], P + Q + R < 1 always holds, then lim n→∞ Y n = lim n→∞ E|X n | 2 = 0 then the Euler-Maruyama method applied to Eq. (1) is mean-square stable. The theorem is completed.
then the numerical method applied to Eq. (1) is said to be general mean-square stable. (2), assume 1αh = 0, the split-step backward Euler method applied to Eq. (1) is generally mean-square stable.

Mean-square stability of the split-step backward Euler method for nonlinear stochastic systems
In this section, we will discuss the mean-square stability of the split-step backward Euler method for nonlinear stochastic delay integro-differential equations. Considering the following nonlinear stochastic equation: is an mdimensional Wiener process and τ is a delay term. If f and g are sufficiently smooth and satisfy the Lipschitz condition and the linear growth condition, Eq. (13) has a unique strong solution x(t), t ∈ [-τ , ∞) and x(t) is a measurable, sample-continuous and F t adapted process [12,13].
The split-step backward Euler method applied to Eq. (13) yields X * n = X n + f (X * n , X n-m ,X n )h, X n+1 = X * n + g(X * n , X n-m ,X n ) W n ; X n , X * n ,X n , h, W n are defined in Sects. 3 and 4. there exist constant a 1 , a 2 , a 3 , b 1 , b 2 , b 3 , for all x, u, v ∈ R d , we have
Proof From the second equation of (14), we obtain Note that E( W n ) = 0, E[( W n ) 2 ] = h, and X n , X n-m ,X are F t n -measurable, hence E X * n , g X * n , X n-m ,X n W n = 0, E g X * n , X n-m ,X n 2 W 2 n = g X * n , X n-m ,X n 2 h.
Combining condition (17) and taking expectations on both sides of the above equation, Next, we should derive the E|X * n | 2 by the first equation of (14), Squaring both sides of Eq. (19), one gets Through the conditions (15), (16), we have 2 X * n , f X * n , X n-m ,X n = 2 X * n , f X * n , 0, 0 + 2 X * n , f X * n , X n-m ,X nf X * n , 0, 0 It is easily to see that for X * nX n from Sect. 3 Substituting these into Eq. (20) and taking expectations In particular Hence We can write So E|X n+1 | 2 ≤ (P + Q + R) E|X n | 2 , E|X n-m | 2 , max n-m≤i≤n E|X i | 2 .
If P + Q + R < 1, it is easily to see that E|X n | 2 → 0 when n → ∞. By the condition (18), we have Namely Therefore, for every step-size h ∈ (0, h 0 ], lim n→∞ E|X n | 2 = 0 holds, the split-step backward Euler method for nonlinear stochastic equations is mean-square stable. The proof is complete.
We should calculate the step-size h 0 ≈ 0.03 from Theorem 5.1, the data used in all figures are plotted by 200 trajectories. It is proved that the split-step backward Euler method has mean-square stability when h = 0.01, while h dissatisfied (0, h 0 ], that is, h = 0.1 > h 0 , the split-step backward Euler method is unstable. This is shown in Fig. 3.

Conclusion
In this paper, we investigate the mean-square stability and general mean-square stability of two numerical methods for a class of linear stochastic delay equations. By comparison, we know that the split-step backward Euler method achieves superiority over the Euler-Maruyama method in terms of mean-square stability. The mean-square stability of numerical method for nonlinear stochastic delay integro-differential equations is eventually confirmed by us.