Approximate Moment Functions for Logistic Stochastic Differential Equations

Abstract

In this paper, we introduce an iterative procedure for approximate moment functions of logistic stochastic differential equations. We first reduce the solutions of the moment functions of such an equation to an infinite system of linear ordinary differential equations. Then, we determine certain upper and lower bounds on the moment functions, and utilize these bounds to solve the corresponding systems approximately via some truncations, iterations and a local improvement step. By focusing on the stochastic Verhulst systems, we compare these moment approximations with numerical solutions via simulation-based methods that include discretizations of the pathwise solutions as well as convergent numerical procedures like partially implicit Euler methods. In particular, we discuss performance of these approximations for certain parameter values.