Application Of A Developed Numerical Method for Solving One-Dimensional Stochastic Differential Equations

Abstract

In this work, we present a developed numerical method for solving one-dimensional stochastic differential equations. The method is based on a modification of the Milstein scheme through the introduction of an adaptive correction coefficient θ() that depends on the time step, aiming to achieve a balance between numerical accuracy and stability, particularly when using medium to large time steps.

We studied the numerical properties of the developed  method, including stability, consistency, strong convergence, and weak convergence. The effectiveness of the developed method was tested on both linear and nonlinear benchmark problems, and a numerical comparison was conducted with the Euler–Maruyama, Milstein, and Runge–Kutta methods.The experiments were implemented using Mathematica programming language, and the results demonstrated that the developed method outperformed the Euler–Maruyama, Milstein, and Runge–Kutta methods in terms of accuracy and stability