Explicit formula of boundary crossing probabilities for continuous local martingales to constant boundary
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 cross...
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| Main Author | |
|---|---|
| Format | Journal Article |
| Language | English |
| Published |
30.11.2023
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| Subjects | |
| Online Access | Get full text |
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| Abstract | 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. |
|---|---|
| AbstractList | 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. |
| Author | Potiron, Yoann |
| Author_xml | – sequence: 1 givenname: Yoann surname: Potiron fullname: Potiron, Yoann |
| BackLink | https://doi.org/10.48550/arXiv.2312.00287$$DView paper in arXiv |
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| Snippet | 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... |
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| SubjectTerms | Mathematics - Probability |
| Title | Explicit formula of boundary crossing probabilities for continuous local martingales to constant boundary |
| URI | https://arxiv.org/abs/2312.00287 |
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