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arXiv · 2405.13174

Local times of deterministic paths with finite variation

Darlington Hove·Farai J. Mhlanga·Rafał M. Łochowski·Phumlani L. Zondi
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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.

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Darlington Hove, Farai J. Mhlanga, Rafał M. Łochowski, Phumlani L. Zondi. 2024-05-21. Local times of deterministic paths with finite variation. https://arxiv.org/abs/2405.13174

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