Numerical Simulation and Methodology Based on Improved Split Step Method for Studying Stochastic Models

In this paper, we improved the split step $ vartheta $ method to solve the stochastic differential equations. The strong convergence of this approximation for stochastic differential equations, whose drift and diffusion coefficients are globally Lipschitz continuous, are investigated. Furthermore, we analyze the stability in the mean square sense of our scheme by scalar stochastic differential equation with multi dimensional Wiener processes. The study of stability shows the mean square stability of the method for $ vartheta in [1/2, 1] $. Finally, we present some numerical examples to describe the methodology and implementation of the split step $ vartheta $ method to solve linear and nonlinear one dimensional stochastic differential equations and the Lotka-Volterra stochastic system.