Edgeworth expansion for functionals of continuous diffusion processes

TL;DR

This paper develops advanced Edgeworth expansions for high-frequency functionals of continuous diffusion processes, enhancing the accuracy of approximations for power variations and studentized statistics using sophisticated stochastic calculus techniques.

Contribution

It introduces second order Edgeworth expansions for power variations of diffusion processes, employing martingale embedding, Malliavin calculus, and stable CLTs.

Findings

Derived asymptotic expansions for weighted functionals of Brownian motion

Provided second order Edgeworth expansion for power variation

Demonstrated density expansion for studentized statistics

Abstract

This paper presents new results on the Edgeworth expansion for high frequency functionals of continuous diffusion processes. We derive asymptotic expansions for weighted functionals of the Brownian motion and apply them to provide the second order Edgeworth expansion for power variation of diffusion processes. Our methodology relies on martingale embedding, Malliavin calculus and stable central limit theorems for semimartingales. Finally, we demonstrate the density expansion for studentized statistics of power variations.