Local times of deterministic paths with finite variation

Abstract

In this note, we define the numbers of level crossings by a c{à}dl{à}g (RCLL) real function $x:[0,+\infty )\to R$ 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 $f:R\to R$, we derive change of variable formulas, which may be seen as analogous of the Itô or Tanaka-Meyer formulas. Some of these formulas are present in [BY14] but we also present some generalizations.