Weak limit theorems for stochastic integrals and stochastic differential equations. (English) Zbl 0742.60053

Kurtz, Thomas G.; Protter, Philip

Ann. Probab. 19, No. 3, 1035-1070 (1991).

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)].

Reviewer: T.Inglot (Wrocław)

MSC:

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

Keywords:

weak convergence; stochastic differential equations; cadlag processes; Skorokhod topology; stochastic integrals

Citations:

Zbl 0655.60026; Zbl 0638.60049; Zbl 0673.60065