Explicit formula of boundary crossing probabilities for continuous local martingales to constant boundary

• Main Author: Potiron, Yoann
• Format: Journal Article
• Language: English
• Published: 30.11.2023
• Subjects: Mathematics - Probability
• DOI: 10.48550/arxiv.2312.00287

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.