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微积分的技巧，提出了一种关于多维Wiener泛函期望的新型通用渐近展开公式。在目标Wiener泛函的Malliavin协方差矩阵满足弱化条件的假设下，证明了渐近展开的一致估计。特别地，该方法避免了奇异核的复杂分数阶积分运算，为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不规则泛函期望提供了易处理的展开形式。数值实验表明，相较于正态逼近，该展开方法能显著提升概率分布函数的近似精度，从而验证了所提方法的有效性。