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

Akihiko Takahashi, Toshihiro Yamada

TL;DR

This work introduces a novel fractional-order asymptotic expansion for expectations of multidimensional Wiener functionals using Malliavin calculus, under a weakened nondegeneracy condition on the Malliavin covariance. It extends prior expansion schemes by representing expansion coefficients via iterative Malliavin derivatives and inner products, enabling a uniform error bound and applicability to irregular functionals of Gaussian processes. The authors apply the framework to rough differential equations driven by fractional Brownian motion with $H<1/2$, delivering explicit distributional expansions and demonstrating improved accuracy over normal approximations in a numerical example. The results have potential practical impact for high-dimensional Gaussian models and may inform improved Monte Carlo and discretization methods in rough-path contexts.

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.

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

TL;DR

This work introduces a novel fractional-order asymptotic expansion for expectations of multidimensional Wiener functionals using Malliavin calculus, under a weakened nondegeneracy condition on the Malliavin covariance. It extends prior expansion schemes by representing expansion coefficients via iterative Malliavin derivatives and inner products, enabling a uniform error bound and applicability to irregular functionals of Gaussian processes. The authors apply the framework to rough differential equations driven by fractional Brownian motion with , delivering explicit distributional expansions and demonstrating improved accuracy over normal approximations in a numerical example. The results have potential practical impact for high-dimensional Gaussian models and may inform improved Monte Carlo and discretization methods in rough-path contexts.

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 , 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.
Paper Structure (4 sections, 3 theorems, 98 equations, 1 figure)

This paper contains 4 sections, 3 theorems, 98 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Asymptotic expansion of expectation of Wiener functionals
  3. Asymptotic expansion formula of expectation of solution of rough differential equation driven by fractional Brownian motion with $H<1/2$
  4. Concluding remarks

Key Result

Theorem 1

Let $\{ F^{\varepsilon} \}_{\varepsilon \in (0,1]} \subset (\mathbb{D}^\infty)^e$ be a family of Wiener functionals such that $F^{\varepsilon}$ has an asymptotic expansion in $(\mathbb{D}^\infty)^e$: where $F^0,F_1,F_2,\ldots \in (\mathbb{D}^\infty)^e$ and $\{ \kappa_i ; i\in \mathbb{N}\}$ satisfies $0<\kappa_1<\kappa_2<\cdots$, in the sense that for any $m\geq 1$, and assume that the Malliavin

Figures (1)

  • Figure 1: Accuracy of asymptotic expansion for probability distribution function of solution to rough differential equation driven by fractional differential equation with Hurst index $H=0.4$

Theorems & Definitions (10)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Theorem 2
  • Remark 3
  • Remark 4
  • Theorem 3
  • Remark 5
  • Remark 6
  • Remark 7