Weak Limit Theorems for Stochastic Integrals and Stochastic Differential Equations

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 \ Xₙ dYₙ\ converges in distribution to X dY. 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 appropriate 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.