Discussion Papers 2023


"New Asymptotic Expansion Formula via Malliavin Calculus and Its Application to Rough Differential Equation Driven by Fractional Brownian Motion"

Author Name

Akihiko Takahashi and Toshihiro Yamada


June 2023

Full Paper PDF file

Revised in February and April 2024; forthcoming in Asymptotic Analysis.

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.

Keywords: Asymptotic expansion, Wiener functional, Malliavin calculus, Rough differential equation, Fractional Brownian motion