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


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.


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Thomas G. Kurtz. Philip Protter. "Weak Limit Theorems for Stochastic Integrals and Stochastic Differential Equations." Ann. Probab. 19 (3) 1035 - 1070, July, 1991.


Published: July, 1991
First available in Project Euclid: 19 April 2007

zbMATH: 0742.60053
MathSciNet: MR1112406
Digital Object Identifier: 10.1214/aop/1176990334

Primary: 60H05
Secondary: 60F17 , 60G44

Keywords: Filtering , Skorohod topology , Stochastic differential equations , stochastic integrals , symmetric statistics , weak convergence , Wong-Zakai correction

Rights: Copyright © 1991 Institute of Mathematical Statistics

Vol.19 • No. 3 • July, 1991
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