# Approximation of occupation time functionals

## TL;DR

This paper investigates the strong $L^2$-approximation of occupation time functionals for $d$-dimensional cdlg processes, providing general bounds that extend to non-Markovian processes like fractional Brownian motion.

## Contribution

It offers new upper bounds on approximation errors under weak assumptions, applicable to a broad class of processes including non-Markovian ones.

## Key findings

- Bounds are sharp up to a log-factor for Brownian motion.
- Results generalize previous literature significantly.
- Applicable to non-Markovian processes like fractional Brownian motion.

## Abstract

The strong $L^2$-approximation of occupation time functionals is studied with respect to discrete observations of a $d$-dimensional c\`adl\`ag process. Upper bounds on the error are obtained under weak assumptions, generalizing previous results in the literature considerably. The approach relies on regularity for the marginals of the process and applies also to non-Markovian processes, such as fractional Brownian motion. The results are used to approximate occupation times and local times. For Brownian motion, the upper bounds are shown to be sharp up to a log-factor.