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

TL;DR

This paper derives the joint distribution of the first hitting time and position of the running maximum of Brownian motion, with applications to animal foraging models and extensions to Brownian motion with drift.

Contribution

It provides explicit formulas for the joint distribution of the first hit time and the maximum position, extending previous results to include drifted Brownian motion.

Findings

Derived the joint distribution of $T_m$ and $B_{T_m}$ for standard Brownian motion.

Extended results to Brownian motion with drift.

Applications to animal foraging models.

Abstract

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.