Theoretical analysis reveals exact distributional formulas and tail asymptotics for Brownian motion with exponential resetting, highlighting extremal properties in stochastic systems.
We study the distributional and asymptotic properties of the supremum of Brownian motion with drift and exponential resetting. We obtain an explicit renewal-type formula for the distribution of the supremum and then derive an approximation for its survival function. Moreover, we find the asymptotics of the tail distribution of the supremum and infimum. We also consider the stationary case and give a new explicit expression for the two-dimensional distributions of such processes.
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Dȩbicki et al. (2026) studied this question.
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