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

TL;DR

This paper develops a new asymptotic expansion formula for expectations of Wiener functionals using Malliavin calculus, improving approximation accuracy for rough differential equations driven by fractional Brownian motion with H<1/2.

Contribution

It introduces a novel asymptotic expansion formula applicable to irregular functionals of rough differential equations, avoiding complex fractional calculus.

Findings

Better approximation of probability distributions than normal approximation

Validates the expansion through numerical experiments

Provides a uniform estimate under weaker conditions

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.