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On a first hit distribution of the running maximum of Brownian motion
Julien Randon-Furling, Paavo Salminen, Pierre Vallois · 2021-03-15 · via math.PR updates on arXiv.org

Let $(S_t)_{t\geq 0}$ be the running maximum of a standard Brownian motion $(B_t)_{t\geq 0}$ and $T_m:=\inf\{t; \, mS_t<t\},\, m>0$. In this note we calculate the joint distribution of $T_m$ and $B_{T_m}$. The motivation for our work comes from a mathematical model for animal foraging. We also present results for Brownian motion with drift.

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