Stochastic integral representation of the $L^{2}$ modulus of Brownian local time and a central limit theorem

TL;DR

This paper proves a central limit theorem for the $L^2$-modulus of continuity of Brownian local time using stochastic analysis techniques, including Knight's theorem and Clark-Ocone formula.

Contribution

It introduces a stochastic integral representation of the $L^2$-modulus of Brownian local time and establishes a new CLT using advanced stochastic analysis methods.

Findings

Established a CLT for the $L^2$-modulus of Brownian local time

Developed a stochastic integral representation of the local time modulus

Applied Knight's theorem and Clark-Ocone formula in the proof

Abstract

The purpose of this note is to prove a central limit theorem for the $L^2$-modulus of continuity of the Brownian local time obtained in \cite{CLMR}, using techniques of stochastic analysis. The main ingredients of the proof are an asymptotic version of Knight's theorem and the Clark-Ocone formula for the $L^2$-modulus of the Brownian local time.