Mathematical Computation June 2014, Volume 3, Issue 2, PP.19-25
Almost Sure Stability of the Weak Backward Euler-Maruyama Method for the Stochastic Lotka-Volterra Model in One Dimension Wei Liu Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, U.K. Email: w.liu@lboro.ac.uk
Abstract The almost sure exponential stability of the weak backward Euler-Maruyama (EM) method is discussed for the stochastic version of the Lotka-Volterra model in one dimension. As the nonlinear term exists in the drift coefficient, the explicit Euler-Maruyama method is not a good candidate [6]. The backward EM method is normally a naturally replacement in the nonlinear case, but we show that in this model the backward EM method may not be well defined. Then we turn to the weak backward EM method, in which the normal distribution is replaced by a two-point distribution, and we prove that this method can reproduce the almost sure exponential stability of the underlying model. Keywords: Almost Sure Exponential Stability; Weak Backward Euler-Maruyama Method; Nonlinear SDEs
1 INTRODUCTION Stochastic differential equations (SDEs) have been employed to model uncertain scenarios in many areas for decades [5, 7] . Due to the difficulties to find explicit solutions to most nonlinear SDEs [5], the research on numerical approximations to SDEs has been blooming in recent years [4, 6]. Among different aspects of numerical analysis for SDEs, the almost sure exponential stability is one of the most popular topics. This paper is devoted to the study on reproducing the almost sure exponential stability of the underlying SDE by using numerical solutions. There already exist many literatures on this direction, and we just mention a few of them here [1, 2, 8] and the references therein. However, few works have been contributed to the weak method, in which the two-point distribution replaces the normally distributed Brownian increment. We refer the readers to [1] for the weak methods in the linear case. In this paper, we are investigating one type of nonlinear SDE, the one dimension stochastic Lotka-Volterra model, which is a stochastic population model [5]. We first show that there is positive probability that the explicit EulerMaruyama (EM) method may blow up while the underlying SDE is stable. In many papers, the backward EM method (also called the semi-implicit EM method) is a naturally good substitute to the explicit method in dealing with stability problems [2]. But this is not the case in this paper, as we will see the backward EM method may even not be well defined for the stochastic Lotka-Volterra model. Then we employ the weak backward EM method, which has been proved converge weakly to the underlying solution [4]. We first show that by properly choosing the stepsize, the weak backward EM can preserve the positivity. This is a desirable property, as the underling equation is a population model and its solution can only be nonnegative. More importantly, the positivity property is essential to the proof of the almost surely exponential stability. This paper is organised in the following way. In Section 2, the mathematical preliminary is briefed as well as the properties of the underlying equation. Section 3, firstly, sees the discussions of the failures of the explicit EM method and the backward EM method then the main result is presented. Some numerical simulations are used to illustrate the theoretical result in Section 4. We conclude this paper by some further discussion in Section 5. - 19 www.ivypub.org/mc