Weak Limit Theorems for Stochastic Integrals and Stochastic Differential Equations

Abstract

Assuming that {(Xn, Yn)) is a sequence of cadlag processes converging in distribution to (X, Y) in the Skorohod topology, conditions are given under which the sequence {JXn dYn} converges in distribution to JXdY. Examples of applications are given drawn from statistics and filtering theory. In particular, assuming that (Un, Yn) =* (U, Y) and that Fn -* F in an appro- priate sense, conditions are given under which solutions of a sequence of stochastic differential equations dXn = dUn + Fn(Xn) dYn converge to a solution of dX = dU + F(X) dY, where Fn and F may depend on the past of the solution. As is well known from work of Wong and Zakai, this last conclusion fails if Y is Brownian motion and the Yn are obtained by linear interpolation; however, the present theorem may be used to derive a generalization of the results of Wong and Zakai and their successors