On the strong convergence of multiple ordinary integrals to multiple Stratonovich integrals

Abstract

Given $\{W^{(m)}(t), t \in [0,T]\}_{m \ge 1}$ a sequence of approximations to a standard Brownian motion $W$ in $[0,T]$ such that $W^{(m)}(t)$ converges almost surely to $W(t)$ we show that, under regular conditions on the approximations, the multiple ordinary integrals with respect to $dW^{(m)}$ converge to the multiple Stratonovich integral. We are integrating functions of the type $$f(x_1,\ldots,x_n)=f_1(x_1)\ldots f_n(x_n) I_{\{x_1\le \ldots \le x_n\}},$$ where for each $i \in \{1,\ldots,n\}$, $f_i$ has continuous derivatives in $[0,T].$ We apply this result to approximations obtained from uniform transport processes.