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" |
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Author Name | Akihiko Takahashi and Toshihiro Yamada |
Date | June 2023 |
Full Paper | PDF file |
Remarks |
Revised in February,April and October 2024; Published in Asymptotic Analysis, vol. pp.1-22, May 2024. |
Abstract |
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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 |