Weak Limit Theorems for Stochastic Integrals and Stochastic Differential Equations

Abstract

Assuming that $\{(X_n,Y_n)\}$ is a sequence of cadlag processes converging in distribution to $(X,Y)$ in the Skorohod topology, conditions are given under which the sequence $\{\int X_n dY_n\}$ converges in distribution to $\int X dY$. Examples of applications are given drawn from statistics and filtering theory. In particular, assuming that $(U_n,Y_n) \Rightarrow (U,Y)$ and that $F_n \rightarrow F$ in an appropriate sense, conditions are given under which solutions of a sequence of stochastic differential equations $dX_n = dU_n + F_n(X_n)dY_n$ converge to a solution of $dX = dU + F(X)dY$, where $F_n$ 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 $Y_n$ 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.