Almost Sure Stability of the Weak Backward Euler-Maruyama Method for the Stochastic Lotka-Volterra Model in One Dimension

摘要

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.