Weak convergence to the multiple Stratonovich integral

Abstract

We have considered the problem of the weak convergence, as ε tends to zero, of the multiple integral processes ∫ 0 t ⋯ ∫ 0 t f(t 1,…,t n) dη ε(t 1)⋯ dη ε(t n),t∈[0,T] in the space C 0([0,T]) , where f∈ L 2([0, T] n ) is a given function, and { η ε ( t)} ε>0 is a family of stochastic processes with absolutely continuous paths that converges weakly to the Brownian motion. In view of the known results when n⩾2 and f( t 1,…, t n )=1 { t 1< t 2<⋯< t n } , we cannot expect that these multiple integrals converge to the multiple Itô–Wiener integral of f, because the quadratic variations of the η ε are null. We have obtained the existence of the limit for any { η ε }, when f is given by a multimeasure, and under some conditions on { η ε } when f is a continuous function and when f( t 1,…, t n )= f 1( t 1)⋯ f n ( t n )1 { t 1< t 2<⋯< t n } , with f i ∈ L 2([0, T]) for any i=1,…, n. In all these cases the limit process is the multiple Stratonovich integral of the function f.