Local times of deterministic paths with finite variation
Volume 180 / 2026
Colloquium Mathematicum 180 (2026), 255-271
MSC: Primary 60G17; Secondary 60G44, 60F17, 60H05
DOI: 10.4064/cm9372-11-2025
Published online: 21 May 2026
Abstract
We define the numbers of level crossings by a càdlàg (RCLL) real function $x\colon [0,+\infty ) \rightarrow \mathbb R$ and, in analogy to the work of Bertoin and Yor (2014), 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 $f:\mathbb R \to \mathbb R$, we derive change of variable formulas, which may be seen as analogs of the Itô or Tanaka–Meyer formulas. Some of these formulas were already given by Bertoin and Yor (2014), but we also present some generalizations.
