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.
Citation
Xavier Bardina. Carles Rovira. "On the strong convergence of multiple ordinary integrals to multiple Stratonovich integrals." Publ. Mat. 65 (2) 859 - 876, 2021. https://doi.org/10.5565/PUBLMAT6522114
Information