The Annals of Probability

Weak Limit Theorems for Stochastic Integrals and Stochastic Differential Equations

Thomas G. Kurtz and Philip Protter

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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.

Article information

Ann. Probab. Volume 19, Number 3 (1991), 1035-1070.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60H05: Stochastic integrals
Secondary: 60F17: Functional limit theorems; invariance principles 60G44: Martingales with continuous parameter

Stochastic integrals stochastic differential equations weak convergence Skorohod topology filtering symmetric statistics Wong-Zakai correction


Kurtz, Thomas G.; Protter, Philip. Weak Limit Theorems for Stochastic Integrals and Stochastic Differential Equations. Ann. Probab. 19 (1991), no. 3, 1035--1070. doi:10.1214/aop/1176990334.

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