Asymptotic expansion of the weighted power variation with second order differences of a stochastic differential equation driven by fBm

TL;DR

This paper derives the asymptotic expansion of the distribution of weighted power variations involving second order differences for solutions of SDEs driven by fractional Brownian motion with Hurst index greater than 1/2, including convergence rates.

Contribution

It extends the asymptotic expansion theory of Skorohod integrals to weighted power variations of SDEs driven by fBm, providing explicit formulas and convergence rates.

Findings

Derived the asymptotic expansion formula for the distribution.

Included the rate of convergence in the expansion.

Utilized the theory of exponents for functional estimates.

Abstract

We study a process satisfying a one-dimensional stochastic differential equation driven by fractional Brownian motion with Hurst index $H>1/2$, and consider the weighted power variation based on the second order differences of the process. We derive the asymptotic expansion formula of its distribution based on the theory of expansion of Skorohod integrals by Nualart and Yoshida. The formula includes the rate of convergence as a corollary. To facilitate the application of the general expansion theory, we employ the theory of exponents from arXiv:2407.02254 to obtain estimates of functionals.