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.