Numerical Solution of Nonlinear Stochastic Processes Under Gaussian White Noise

Abstract

In this paper Methodological approach based on the Euler-Maruyama method and the Radial Basis Functions are elaborated for instationnary probability density functions associated to nonlinear stochastic processes under Gaussian white noise. Based on the Euler-Maruyama time solution and a Kernel estimation method the probability density of the considered non-linear stochastic equation is obtained. On the other hand and for more general problems a time-scheme is coupled with Radial Basis Function (RBF) and applied to the associated Fokker-Planck (FP) equation. An implicit-time approach is elaborated and time-dependent probability density function (PDF) is computed for various nonlinear stochastic equations. The obtained results using Radial Basis Functions to get the probability density function are in good agreement when time is increasing comparing with the used Euler-Maryuma associated to nonlinear stocahstic processes The applicability and efficiency of the elaborated mathematical and numerical approaches are demonstrated.