WEAK LIMIT-THEOREMS FOR STOCHASTIC INTEGRALS AND STOCHASTIC DIFFERENTIAL-EQUATIONS

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 {integral X(n) dY(n)} converges in distribution to integral X dY. Examples of applications are given drawn from statistics and filtering theory. In particular, assuming that (U(n), Y(n)) double-line-arrow-pointing-right (U, Y) and that F(n) --> 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.