New asymptotic expansion formula via Malliavin calculus and its application to rough differential equation driven by fractional Brownian motion

Article type: Research Article

Authors: Takahashi, Akihiko^a | Yamada, Toshihiro^{b; *}

Affiliations: [a] University of Tokyo, Japan | [b] Hitotsubashi University, Japan

Correspondence: [*] Corresponding author. E-mail: toshihiro.yamada@r.hit-u.ac.jp; Tel.: +81-42-580-8788.

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

DOI: 10.3233/ASY-241910

Journal: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-22, 2024