New asymptotic expansion formula via Malliavin calculus and its application to rough differential equation driven by fractional Brownian motion (Forthcoming in Asymptotic Analysis)

Abstract

This paper presents a novel generic asymptotic expansion formula of expectations of multidi-mensional 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 differen-tial 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.