Local times for functions with finite variation: two versions of Stieltjes change of variables formula

TL;DR

This paper introduces two types of occupation measures for finite variation functions, showing they are absolutely continuous and can be described by a Meyer-Tanaka like formula, extending change-of-variables formulas.

Contribution

It presents two natural notions of occupation measures for finite variation functions and establishes their absolute continuity and local time representations.

Findings

Both measures are absolutely continuous with respect to Lebesgue measure

Occupation densities are characterized by a Meyer-Tanaka like formula

Two versions of change-of-variables formula are compared

Abstract

We introduce two natural notions for the occupation measure of a function $V$ with finite variation. The first yields a signed measure, and the second a positive measure. By comparing two versions of the change-of-variables formula, we show that both measures are absolutely continuous with respect to Lebesgue measure. Occupation densities can be thought of as local times of $V$, and are described by a Meyer-Tanaka like formula.