The Stochastic -Method for Nonlinear Stochastic Volterra Integro-Differential Equations

The stochastic -method is extended to solve nonlinear stochastic Volterra integro-differential equations. The mean-square convergence and asymptotic stability of the method are studied. First, we prove that the stochastic -method is convergent of order in mean-square sense for such equations. Then, a sufficient condition for mean-square exponential stability of the true solution is given. Under this condition, it is shown that the stochastic -method is mean-square asymptotically stable for every stepsize if and when , the stochastic -method is mean-square asymptotically stable for some small stepsizes. Finally, we validate our conclusions by numerical experiments.


Introduction
In this paper we study the numerical solution of thedimensional nonlinear stochastic Volterra integro-differential equation (SVIDE) with convolution kernels with initial data (0) = 0 . Here ( ) is a scalar Brownian motion, and , , , and are given functions which map In particular, (1) can be regarded as a stochastically perturbed problem of the deterministic nonlinear Volterra integro-differential equation (VIDE) In the last decades, VIDEs have received a great deal of attention. A well-known example of this type is the Volterra population equation For a general theory of VIDEs, we refer the reader to the classical book by Burton [1]. Also, many efficient numerical methods such as linear multistep methods, Runge-Kutta methods, and collocation methods have been constructed for VIDEs (see [2][3][4] and the extensive bibliography therein). However, many real-world phenomena are subject to some random environmental effects. In recent years, the study of stability for SVIDEs has attracted the attention of many authors. For example, in 2000, Mao [5] studied the stability of the stochastic integro-differential equation as follows: ( ) = ( ( ) , ) + (∫ 0 ( − ) ( ) , ) ( ) , and several criteria on mean-square exponential stability and 2 -stability have been obtained. Later, Mao and Riedle [6]

Mean-Square Convergence of the Stochastic Θ-Method
Throughout this paper, we use the following notations. Let (Ω, F, {F } ≥0 , ) be a complete probability space with a filtration {F } ≥0 satisfying normal conditions, namely, {F } ≥0 is increasing and right continuous with F 0 containing allnull sets. And let ( ) be a scalar Brownian motion defined on the probability space. Let ⟨⋅, ⋅⟩ be the Euclidean inner product in space and | ⋅ | be the corresponding norm. The matrix norm is denoted by ‖ ⋅ ‖, and denotes the transpose of a matrix . (⋅) stands for the mathematical expectation operator. For any vectors , ∈ , we denote ⟨ , ⟩ = ⋅ .
For the mean-square convergence analysis, we consider the autonomous -dimensional nonlinear SVIDE on [0, ] where < ∞ with initial data (0) = 0 . We assume that the functions and are globally Lipschitz continuous in all variables and satisfy linear growth condition, that is, there exist positive constants and , such that for all , , , ∈ , where ∨ denotes the maximum of and . Also, we assume that the functions and are continuous and satisfy for all , ∈ [0, ], which implies where and̂are positive constants. The above conditions (7)-(9) can guarantee that SVIDE (6) has a unique solution ( ).
The following lemma will be used in our convergence analysis.
Proof. The proof of this lemma is similar to that of Lemma 5.5.2 in [19], and that of Theorem 2.3 in [20]. Now we apply the stochastic Θ-method to SVIDE (6) and obtain the following numerical scheme: where ℎ > 0 is a given stepsize, and is an approximation to ( ) with = ℎ, = 1, . . . , = /ℎ. The increment Δ = ( +1 ) − ( ) is an (0, ℎ)-distributed Gaussian random variable. And the arguments and denote approximations to In this paper, we choose the composite left rectangular formula to discretize the integral terms and obtain the following schemes Moreover we denote that In this section, without loss of generality, we assume 0 < ℎ < 1.
For the convergence analysis, the local truncation error is defined by and the global error is defined by It is obvious that is F -measurable since both ( ) and are F -measurable random variables.
Definition 3. The stochastic Θ-method (15)-(17) is said to be consistent with order 1 in the mean and with order 2 in the mean-square sense if the following estimations hold with 2 ≥ 1/2 and 1 ≥ 2 + (1/2): where the constant does not depend on ℎ but may depend on and the initial data.
Proof. Using the definition of the global error, we have Squaring both sides of (40), taking expectation and absolute values, yields Abstract and Applied Analysis We now estimate the separate terms in (42) individually and in sequence. Due to the consistency condition in the meansquare sense of the method, we have By using the inequality ( + + ) 2 ≤ 3 2 + 3 2 + 3 2 and condition (7), we obtain Note that then we have Due to properties of conditional expectation and the consistency condition in the mean sense, Using the Lipschtiz condition (7) and properties of conditional expectation, we obtain Substituting (43) and (46)-(48) into (42), we have (50) Since 0 ≤ ≤ 1, for sufficiently small stepsize ℎ, there exists a constant ∈ (0, 1), such that Denotê= 2 +̃3 + 2̃4 , and note that 0 = 0, thus we obtain From the above, the theorem is proved.

Stability Analysis
In this section, we will discuss the analytical stability and numerical stability of SVIDE (1). First, the analytical stability is derived by the technique in [5]. We extend the exponential stability result to a more general type. And then the meansquare asymptotic stability of the stochastic Θ-method is investigated. Let us give the definition of the mean-square exponential stability (see Mao [19]).
for all initial data 0 ∈ .

Theorem 9.
Assume that there exist six positive constants 1 , 2 , 3 , 4 , , and such that for all , ∈ . And the convolution kernel functions and decay exponentially fast, that is If then for any given initial data 0 , there exists a pair of positive constants and such that the solution ( ) of SVIDE (1) has property that which implies (57).
Proof. The proof of this theorem is similar to that of Theorem 2.1 in [5] by choosing the Lyapunov function ( , ) = 1 | | 2 .
Remark 10. If we apply Theorem 9 to problem (5), then the corresponding mean-square stability condition is This condition is in accordance with the assumption of Theorem 2.1 in Mao's papaer [5].
In the following, we study the mean-square asymptotic stability of the stochastic Θ-method for SVIDE (1).

Abstract and Applied Analysis
Definition 11. The numerical solution { } ≥0 to SVIDE (1) is said to be mean-square asymptotically stable if the numerical solution has the property that for any initial data 0 .
Proof. By (15), we have Consequently Taking expectation on both sides of (68) and by 1/2 ≤ ≤ 1, it follows that By conditions (58)-(60), we have And by (61) and the Jensen inequality, we derive that Substituting (71) into (70), we have which implies Therefore by condition (62), we have for any > 0, where which shows that the stochastic Θ-method is mean-square asymptotically stable. The proof of the theorem is completed.

Remark 14.
In recent years, there are so many valuable results on the stability analysis of numerical methods for stochastic delay integro-differential equations. However, these results can not be applied to stochastic Volterra integro-differential equations directly since they have essential differences. Moreover, for the deterministic cases, these two issues are studied separately in the past few decades; please refer to the book [2] by Brunner.

Numerical Experiments
In this section, some numerical examples are given to validate our conclusions. First we illustrate the theoretical convergence order of stochastic Θ-method. Here we consider the following linear SVIDE: with the initial data (0) = 1. And we choose the parameters = −1, = 0.5, = 0.5, = 0.5, and = 1. Since the true solution of (82) can not be obtained, thus we compute a reference solution with the trapezoidal rule ( = 0.5) under a very small stepsize ℎ = 2 −15 . The mean-square error is denoted as follows: where denotes the number of steps and is the number of sample paths. In our experiments, we choose = 500. Here we choose four different numerical methods with = 0 (classical Euler-Maruyama method), = 0.3, = 0.7, and = 1 (Backward Euler method), and the numerical results are presented in Table 1. From the table, one can easily receive that the mean-square convergence order of the stochastic Θmethod for SVIDEs is 1/2. Also, we illustrate the convergence result by Figure 1, where the reference dotted curve has a slope 1/2.
In the rest of this section, some numerical experiments are given to validate our stability results. Consider the following nonlinear SVIDE: with the initial data (0) = 1. We choose the parameters = −10, = 3, = 2, and = 1.    Therefore, which implies that the true solution of (84) is mean-square asymptotically stable. In the following numerical experiments, we will show the numerical mean-square asymptotic stability of the stochastic (ii) if = 0.3, ℎ 0 = 93/520. When the stepsize ℎ ∈ (0, 93/520), the method is mean-square asymptotically stable.
In Figures 2 and 3, one can find that when the stepsize ℎ ∈ (0, ℎ 0 ), | | 2 tends to zero, but it tends to infinity when the stepsize is even large. Since the stability conditions are sufficient, one may find that the restriction of the stepsize in Theorem 13 is not optimal.
Theorem 12 says that when ∈ [1/2, 1], the stochastic Θmethod is mean-square asymptotically stable for any stepsize, and the following numerical experiments demonstrate this for a selection stepsize (see Figure 4).

Conclusions
In this paper, we first studied the mean-square convergence the stochastic Θ-method for stochastic Volterra integrodifferential equations and proved that the stochastic Θmethod has mean-square convergence order 1/2 when the functions and in (1) are globally Lipschitz continuous in all variables and satisfy linear growth condition. And then the mean-square stability of such method has been considered.
It should be mentioned that the assumptions of the meansquare convergence in the paper are slightly strong. In the future, we will look for more relaxed conditions to establish new convergence results. In addition, as we mentioned in the introduction section, there are many efficient numerical methods to solve stochastic ordinary differential equations in the existing literature, such as stochastic linear multistep method and stochastic Runge-Kutta methods. Therefore, the adaptation of such methods for stochastic Volterra integrodifferential equations is worth studying in the future.