Discussion Papers 2023

CIRJE-F-1215	"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
Date	June 2023
Full Paper	PDF file
Remarks	Revised in February and April 2024

Abstract
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

Abstract

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