Local times of deterministic paths with finite variation
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866929352166342656 |
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| author | Hove, Darlington Mhlanga, Farai J. Łochowski, Rafał M. Zondi, Phumlani L. |
| author_facet | Hove, Darlington Mhlanga, Farai J. Łochowski, Rafał M. Zondi, Phumlani L. |
| contents | In this note, we define the numbers of level crossings by a c{à}dl{à}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 Itô or Tanaka-Meyer formulas. Some of these formulas are present in [BY14] but we also present some generalizations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_13174 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local times of deterministic paths with finite variation Hove, Darlington Mhlanga, Farai J. Łochowski, Rafał M. Zondi, Phumlani L. Classical Analysis and ODEs In this note, we define the numbers of level crossings by a c{à}dl{à}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 Itô or Tanaka-Meyer formulas. Some of these formulas are present in [BY14] but we also present some generalizations. |
| title | Local times of deterministic paths with finite variation |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2405.13174 |