On a first hit distribution of the running maximum of Brownian motion

Abstract

Let $(S_t{)}_{t\ge 0}$ be the running maximum of a standard Brownian motion $(B_t{)}_{t\ge 0}$ and $T_m:=\inf \{t;m{S}_t<t\},m>0$. 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.

24 pages, 2 figures, submitted