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 $$ f(t_1,\dotsc,t_n)=f_1(t_1)\dotsm f_n(t_n) I_{\{t_1\le \dotsb \le t_n\}}, $$ where for each $i \in \{1,\dotsc,n\}$, $f_i$ has continuous derivatives in $[0,T]$. We apply this result to approximations obtained from uniform transport processes.