Numerical solution of stiff random ordinary differential equations via averaged schemes

Abstract

Random ordinary differential equations (RODEs) are important differential equations that contain a stochastic process in their vector field functions. Examples of such stochastic processes are Brownian motion (i.e., Wiener process) and fractional Brownian motion. Taking a path from the stochastic process, these equations can be considered as ordinary differential equations, but numerical schemes for solving ordinary differential equations don't necessarily preserve their convergence order for these equations. In this paper, an implicit ϑ‐averaged scheme for numerical solution of RODEs is introduced. The proposed method involves averaging the noise inside the vector field and leads to integrals of the noise over discretization subintervals. Pathwise convergence and B‐stability of the above scheme are proved. Eventually, the desired averaged scheme is implemented on a medical model.