Approximation via regularization of the local time of semimartingales and Brownian motion

TL;DR

This paper introduces approximation schemes for the local time of continuous semimartingales and Brownian motion using regularization, providing convergence results and rates in various senses.

Contribution

It presents new regularization-based approximation schemes for local time with proven convergence and rate results, extending to a broad class of processes.

Findings

Convergence in ucp sense for the approximation schemes.

Established L^2 convergence rates for Brownian motion.

Almost sure convergence of some schemes.

Abstract

Through a regularization procedure, few approximation schemes of the local time of a large class of one dimensional processes are given. We mainly consider the local time of continuous semimartingales and reversible diffusions, and the convergence holds in ucp sense. In the case of standard Brownian motion, we have been able to determine a rate of convergence in $L^2$, and a.s. convergence of some of our schemes.