Explicit formula of boundary crossing probabilities for continuous local martingales to constant boundary
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866910349315276800 |
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| author | Potiron, Yoann |
| author_facet | Potiron, Yoann |
| contents | An explicit formula for the probability that a continuous local martingale crosses a one or two-sided random constant boundary in a finite time interval is derived. We obtain that the boundary crossing probability of a continuous local martingale to a constant boundary is equal to the boundary crossing probability of a standard Wiener process to a constant boundary up to a time change of quadratic variation value. This relies on the constancy of the boundary and the Dambis, Dubins-Schwarz theorem for continuous local martingale. The main idea of the proof is the scale invariant property of the time-changed Wiener process and thus the scale invariant property of the first-passage time. As an application, we also consider an inverse first-passage time problem of quadratic variation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_00287 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Explicit formula of boundary crossing probabilities for continuous local martingales to constant boundary Potiron, Yoann Probability Primary 60J65, secondary 60G40, 60H05 An explicit formula for the probability that a continuous local martingale crosses a one or two-sided random constant boundary in a finite time interval is derived. We obtain that the boundary crossing probability of a continuous local martingale to a constant boundary is equal to the boundary crossing probability of a standard Wiener process to a constant boundary up to a time change of quadratic variation value. This relies on the constancy of the boundary and the Dambis, Dubins-Schwarz theorem for continuous local martingale. The main idea of the proof is the scale invariant property of the time-changed Wiener process and thus the scale invariant property of the first-passage time. As an application, we also consider an inverse first-passage time problem of quadratic variation. |
| title | Explicit formula of boundary crossing probabilities for continuous local martingales to constant boundary |
| topic | Probability Primary 60J65, secondary 60G40, 60H05 |
| url | https://arxiv.org/abs/2312.00287 |