Setwise convergence of solution measures of stochastic differential equations

Abstract

We study the setwise convergence of solution measures corresponding to stochastic differential equations of the form dx t = f( t, x t ) dt + σ( t, x t ) dW t , x 0 = c, t ϵ [0, 1]. (1) Here W t is a Brownian motion defined on some probability space (Ω, A , μ). We show under quite general conditions, that if the drift term coefficients converge, or the initial conditions x n converge to x 0 then the corresponding solution measures μ x n converge setwise to μ x0 .