Approximation of some discrete-time stochastic processes by differential equations

Abstract

This work deals with solutions of ordinary differential equations as approximations of some discrete-time stochastic processes. Similarly, these stochastic processes may be seen as schemes of approximation for this solution. Indeed, these stochastic schemes are defined and their convergence to the solution of a differential equation is proven. Moreover, the asymptotic distribution of the fluctuations about the limit solution is studied. This fact gives the asymptotic distribution of a random global error of approximation. Main results are illustrated by means of the so called SIS epidemic model and numerical simulations are carried out.