Weak convergence of stochastic integrals

Abstract

In this paper we provide sufficient conditions for sequences of stochastic processes of the form ${\int }_{[0,t]}{f}_n(u)\hat{I}{¸}_n(u)du$, to weakly converge, in the space of continuous functions over a closed interval, to integrals with respect to the Brownian motion, ${\int }_{[0,t]}f(u)W(du)$, where $\{f_n{\}}_n$ is a sequence satisfying some integrability conditions converging to $f$ and $\{\hat{I}{¸}_n{\}}_n$ is a sequence of stochastic processes whose integrals ${\int }_{[0,t]}\hat{I}{¸}_n(u)du$ converge in law to the Brownian motion (in the sense of the finite dimensional distribution convergence), in the multidimensional parameter set case.

19 pages