Convergence solution for some Harmonic Stochastic Differential Equations with Application

Abstract

The purpose of this paper is to provide an introduction to the theory, computation, and application of stochastic differential equations and also we study the exact and approximate solution for some harmonic stochastic differential equations , by using Ito integral formula and numerical approximation(the Euler-Maruyama method and the Milstein method) in order discuss the convergence accuracy of their solution. Also we proposed Intermediate points for the generalization to Ito integral formula and stratonovich formula. Milstein method is more accurate than Euler Maruyama method . By looking at the convergence rates of both methods , we find that Euler-Maruyama method is strongly convergent with and weakly convergent with , whereas Milstein method is strongly and weakly convergent with .