This paper presents a novel generic asymptotic expansion formula of expectations of multidimensional Wiener functionals through a Malliavin calculus technique. The uniform estimate of the asymptotic expansion is shown under a weaker condition on the Malliavin covariance matrix of the target Wiener functional. In particular, the method provides a tractable expansion for the expectation of an irregular functional of the solution to a multidimensional rough differential equation driven by fractional Brownian motion with Hurst index $H<1/2$, without using complicated fractional integral calculus for the singular kernel. In a numerical experiment, our expansion shows a much better approximation for a probability distribution function than its normal approximation, which demonstrates the validity of the proposed method.

翻译：本文通过Malliavin演算技术，提出了一种新颖的多维Wiener泛函期望的通用渐近展开公式。在目标Wiener泛函的Malliavin协方差矩阵满足较弱条件的情况下，证明了该渐近展开的一致估计。特别地，该方法为具有Hurst指数$H<1/2$的分数布朗运动驱动的多维粗糙微分方程解的非常规泛函的期望，提供了一种易于处理的展开式，而无需对奇异核使用复杂的分数阶积分演算。在数值实验中，我们的展开式对概率分布函数的逼近效果远优于其正态逼近，这证明了所提方法的有效性。