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.
Citation
Thomas G. Kurtz. Philip Protter. "Weak Limit Theorems for Stochastic Integrals and Stochastic Differential Equations." Ann. Probab. 19 (3) 1035 - 1070, July, 1991. https://doi.org/10.1214/aop/1176990334
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