Local times of deterministic paths with finite variation

TL;DR

This paper introduces a way to measure level crossings of functions with finite variation and derives change of variable formulas similar to Itô and Tanaka-Meyer formulas, extending previous work.

Contribution

It defines level crossing counts for finite variation paths and generalizes change of variable formulas akin to Itô and Tanaka-Meyer formulas.

Findings

Level crossing counts are densities of occupation measures.

Derived generalized change of variable formulas.

Extended previous formulas to broader classes of functions.

Abstract

In this note, we define the numbers of level crossings by a c{\`a}dl{\`a}g (RCLL) real function $x: [0,+\infty) \rightarrow R$ and, in analogy to the work of Bertoin and Yor [BY14] we prove that for $x$ with locally finite total variation these numbers are densities of relevant occupation measures associated with $x$. Next, depending on the regularity of $x$ and $f: R \rightarrow R$, we derive change of variable formulas, which may be seen as analogous of the It\^o or Tanaka-Meyer formulas. Some of these formulas are present in [BY14] but we also present some generalizations.