On the relation between ordinary and stochastic differential equations

Abstract

The following problem is considered in this paper: Let x t be a solution to the stochastic differential equation: dx t = m[ x t , t] dt+ σ[ x t , t] dy t where y t is the Brownian motion process. Let x t ( n) be the solution to the ordinary differential equation which is obtained from the stochastic differential equation by replacing y t with y t ( n) where y t ( n) is a continuous piecewise linear approximation to the Brownian motion and y t ( n) converges to y t as n → ∞. If x t is the solution to the stochastic differential equation (in the sense of Ito) does the sequence of the solutions x t ( n) converge to x t ? It is shown that the answer is in general negative. It is however, shown that x t ( n) converges in the mean to the solution of another stochastic differential equation which is: dx t = m[x t, t] dt + 1 2 σ[x t, t](∂σ[x t, t]/∂x t)dt+ σ[x t, t]dy t .