A Note on the sharp $L^p$-Covergence rate of Upcrossings to the Brownian local time

TL;DR

This paper establishes a precise $L^p$-convergence rate for the number of upcrossings approximating Brownian local time, offering new $p$-variation bounds for $p$ between 2 and infinity.

Contribution

It provides the first sharp $L^p$-rate of convergence for upcrossings to Brownian local time, extending understanding of their $p$-variation properties.

Findings

Sharp $L^p$-convergence rate established

Novel $p$-variation estimates for upcrossings

Complements existing almost sure convergence results

Abstract

In this note, we prove a sharp $L^p$-rate of convergence of the number of upcrossings to the local time of the Brownian motion. In particular, it provides novel $p$-variation estimates ($2 < p < \infty$) for the number of upcrossings of the Brownian motion. Our result complements the fundamental work of Koshnevisan \cite{kho} who obtains an almost sure exact rate of convergence in the sup norm.