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arXiv 2002.06266math.PR
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关于多重普通积分到多重Stratonovich积分的strong convergence

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

Xavier Bardina, Carles Rovira

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摘要：

给定$\{W^{(m)}(t), t \in [0,T]\}_{m \ge 1}$一个序列的近似一个标准的布朗运动$W$在$[0,T]$，使$W^{(m)}(t)$几乎肯定收敛到$W(t)$，我们表明，在正规条件下的近似，多个普通积分关于$dW^{(m)}$收敛到多个Stratonovich积分。我们积分功能的类型$$f(x_1,\ldots,x_n)=f_1(x_1)\ldots f_n(x_n) I_{\{x_1\le \ldots \le x_n\}},$$，其中对于每个$i \in \{1,\ldots,n\}$，$f_i$连续衍生物在$[0,T].$我们应用这一结果的近似从均匀输运过程。

英文摘要：

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.

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