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

Given {W(m) (t), t ∈ [0, T]}m≥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(t1, . . . , tn) = f1(t1) · · · fn(tn)I{t1≤···≤tn},
where for each i ∈ {1, . . . , n}, fi has continuous derivatives in [0, T]. We apply this result to approximations obtained from uniform transport processes.