Stability analysis of explicit and implicit stochastic Runge-Kutta methods for stochastic differential equations

This paper concerns to the stability analysis of explicit and implicit stochastic Runge-Kutta methods in approximating the solution of stochastic models. The stability analysis of the schemes in mean-square norm is investigated. Linear stochastic differential equations are used as test equations to demonstrate the efficiency of the proposed schemes.


Introduction
Modelling of the physical and biological systems via stochastic differential equations (SDEs) had been intensively researched over the last few decades. General form of SDEs is 0 0 0 ( ) ( ( )) ( ( )) ( ), ( ) , [ , ], m dy t f y t dt g y t dW t y t y t t T y = + = ∈ ∈ (1) where the deterministic term ( ( )) f y t is the drift coefficient, the stochastic terms ( ( )) g y t is diffusion coefficients and the ( ) W t is independent Wiener processes, which increment ( ) ( ) ( ) W t W t t W t ∆ = + ∆ − is a Gaussian random variable (0, ). N t ∆ In SDEs, the state variable is governed by the uncontrolled fluctuations or environmental noise, hence contribute to the complexity of finding the analytical solution of the systems. Recently, there has been much interest in designing a reliable and efficient numerical integrator for SDEs. Effort has been made to develop the explicit and implicit numerical methods for solving SDEs. The method is called explicit if it is explicit in both deterministic and stochastic components. Meanwhile, if both deterministic and stochastic components are implicit, a numerical integrator is then called as implicit method. Not much work has been done to introduce an implicit numerical integrator of SDEs. Implicit methods have better convergence and stability compare than explicit and semi-implicit methods [1]. Among of the recent works is Milstein et al. [2] who proposed the balanced implicit method for the numerical solutions of SDEs. For balanced implicit methods, the type and degree of implicitness and its stability properties can be chosen by appropriate weights. Then the implicit 2-stage stochastic Runge-Kutta (SRK2) method of SDEs was introduced by Burrage and Tian [3]. Later is an extension work of explicit SRK2 method that was developed in [4]. Rosli et al. [5] investigated the stability analysis of explicit stochastic Runge-Kutta methods with 1.0 and 1.5 order of convergence. In this paper, the stability analysis of 2stage explicit and implicit stochastic Runge-Kutta methods to simulate the solution of SDEs is presented. The arrangement of this paper is as follows; Section 2 considers the numerical schemes of

Stochastic Runge-Kutta for SDEs
A general class of s-stage SRK methods was derived by Rümelin [6] for solving SDEs in equation (1). The general formulation is given by  (2). The method is 1.0 order of convergence and the matrix coefficients obtained are It can be written as Burrage and Tian [3] proposed a stiffly accurate diagonal implicit SRK (SADISRK2) method with matrix coefficients It can be written as SADISRK2 in equation (4) for

MS-Stability of Explicit SRK2 Approximations Scheme
The MS-stability functions and MS-stability region for the discrete time approximation derivative-free method of SRK2 of explicit and implicit schemes are presented in this section. By applying the SRK2 of Burrage's scheme to the linear test in equation (5) Square both sides of equation (7) and then take the expectation of the Stratonovich integrals where The stability region of a stability of equation (9) is plotted in Maple 16 and the region is illustrated in Figure 1.  Based on Figures 1 and 2, it is clear that the implicit SRK2 method shows better stability result compare than the explicit Burrage and Platen's scheme. It can be confirmed by performing numerical experiments that is presented in the next section.

Numerical Experiment
We carried out the numerical experiment to examine the stability properties of the explicit and implicit SRK2 methods. The following numerical experiments show that the step size, h influences the meansquare stability of the corresponding methods. We used linear SDE in equation (5) Figure 3b. This indicates that the implicit method is numerically stable compare than the explicit method.

Conclusion
We have presented the stability function and stability region for explicit and implicit derivative-free SRK2 methods for a linear test in equation (1). It can be seen that, the implicit SRK2 method shows better stability region compare than explicit schemes. The theoretical finding is confirmed by the numerical experiment. For various values of step size, implicit method that was performed to a linear test equation indicates numerical stability. Whereas, the explicit SRK2 methods of Burrage scheme show numerical instability for certain values of step size.