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Weak limit theorems for stochastic integrals and stochastic differential equations. (English) Zbl 0742.60053
Assuming that \(\{(X_ n,Y_ n)\}\) is a sequence of cadlag processes converging in distribution to \((X,Y)\) in the Skorokhod topology, conditions are given under which the sequence \(\{\int X_ n dY_ n\}\) of stochastic integrals converges in distribution to \(\int X dY\). This result is related to that of A. Jakubowski, J. Mémin and G. Pages [Probab. Theory Relat. Fields 81, No. 1, 111-137 (1989; Zbl 0638.60049)]. Several examples of applications are given drawn from statistics and filtering theory. As a particular application conditions are found under which solutions of a sequence of stochastic differential equations \(dX_ n=dU_ n+F_ n(X_ n)dY_ n\) converge in distribution to a solution of \(dX=dU+F(X)dY\). This generalizes results of L. Słomiński [Stochastic Processes Appl. 31, No. 2, 173-202 (1989; Zbl 0673.60065)].

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