Convergence and stability of balanced methods for stochastic delay integro-differential equations
Introduction
Stochastic integro-differential equations have come to play an important role in a variety of application areas including economy [1], [2], biology [3], population dynamics [4] and so on. The class of stochastic delay integro-differential equations (SDIDEs) that admits explicit solutions is rather limited. Thus, appropriate numerical methods are needed to apply in practice and to study their properties.
The numerical analysis of stochastic delay differential equations (SDDEs) is well studied, for instance, Baker and Buckwar [5], [6], U. Kchler and Platen [7], Mao et al. [8], Liu et al. [9], Wang et.al [10] and so on. As for SDIDEs, there has been much less research of numerical schemes. Mao [12] considered the stability of SDIDEs. Tan et al. [13] investigated the convergence and mean-square stability of the split-step backward Euler method for SDIDEs. Li et al. [14] discussed the mean-square exponential stability of stochastic theta methods for nonlinear SDIDEs.
However, it is already known that the majority of the numerical methods for SDIDEs are explicit or semi-implicit methods which are only implicit in the drift coefficient. These drift-implicit methods are well adapted for stiff systems with small stochastic noise intensity or additive noise. In those case the system with large multiplicative noise, the application of fully implicit methods is unavoidable. One of the most important fully implicit methods are the balanced implicit methods, which were firstly introduced by Milstein, Platen and Schurz [16] and used to solve stiff systems. Some recent papers which consider the balanced implicit methods for the stochastic differential equations include [15], [17], [18], [19].
Consider the following scalar linear stochastic delay integro-differential equationwhere is a positive fixed delay, , is a scalar Brownian motion which defined on an appropriate complete probability space , with a filtration satisfying the usual conditions (i.e. it is increasing and right-continuous while contains all -null sets). Lemma 1.1 [11] For any given , there exist positive numbers and such that the solution of (1) satisfiesfor all ,for any ,for any .
Regarding numerical analysis of (1), Ding et al. [11] discussed the convergence and stability of the semi-implicit Euler method for SDIDEs. However, to the best of our knowledge, there are no stability results of balanced methods for the system (1). The aim of this paper is to investigate the strong mean-square convergence and mean-square stability of the balanced implicit methods of the system (1). The rest of the paper is organized as follows. In the subsequent section, we establish Theorem 2.1, showing the strong balanced implicit methods are convergent with strong order 1/2. Section 3 and Section 4 deal with linear mean-square stabilities of the strong balanced implicit methods and the weak balanced implicit methods. In Section 5, some numerical results are reported to demonstrate the conclusions. Finally, conclusion is made in Section 6.
Section snippets
Convergence of the balanced implicit methods
Given a stepsize , a version of strong balanced methods for (1) is given bywhere is an approximation to with . Here is given byHere is given bywhere the are called control functions. In order to obtain our main results in this paper, we assume that in the Eq. (7) are
Mean-square stability of the strong balanced implicit methods
We investigate the mean-square stability of the strong balanced methods in this section.
The following lemma give the sufficient condition on the stability for the analytic solution of the system (1). Lemma 3.1 Assume that satisfythen the solution of (1) is asymptotically stable in the mean-square, that is, .[11]
Given parameters and stepsize h, we say the balanced implicit methods are mean-square stable if for any . The following theorem will show the
Mean-square stability of the weak balanced implicit methods
In this section, we will investigated the mean-square stability of the weak balanced implicit methods equipped with two-point random variables for the driving process.
Given a stepsize , the weak balanced implicit methods are defined bywhere .
It is not difficult to find that
The following theorem will show that the weak balanced implicit methods (55) can preserve the
Numerical experiments
In this section several numerical examples are given to illustrate our theoretical results in the previous sections. Consider the scalar linear equation
Denoting as the numerical approximation to at step point in ith simulation of all simulations. We use to approximate . For simply, we choose . All the figures are drawn with the vertical axis scaled logarithmically.
We
Conclusion
In this work, we have examined the convergence and the mean-square stability of the balanced methods for the stochastic delay integro-differential equations. It is shown that the strong balanced implicit methods give strong convergence rate of at least 1/2. The forgoing results show that both the strong balanced methods and weak balanced methods can reproduce the mean-square stability of the system with sufficiently small stepsize h. The theory result and the numerical experiment show that
Acknowledgment
This work was supported by: 1. NSF of China (No. 11326238, 11326138 and 11101101); 2. the Project sponsored by the Science Foundation of Jiangxi Provincial Department of Education (No. gjj13374 and gjj13658); 3. the Project sponsored by the Youth Natural Science Foundation of Jiangxi Province (20132BAB211007).
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