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

2023/12/01 by Potiron, Yoann
#60H05 #FOS: Mathematics #Primary 60J65 #Probability (math.PR) #secondary 60G40

paper · doi:10.48550/arxiv.2312.00287

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.

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