Distribution of the time at which the deviation of a Brownian motion is maximum before its first-passage time

TL;DR

This paper derives analytical expressions for the probability density of the time at which a Brownian motion reaches its maximum before first hitting zero, revealing different tail behaviors with and without drift.

Contribution

It provides the first analytical calculation of the distribution of the maximum time before first passage for Brownian motion with and without drift.

Findings

Power-law tails in the driftless case: $t_m^{-3/2}$ and $t_m^{-1/2}$.

Exponential decay of $P(t_m)$ with drift towards the origin.

Numerical simulations confirm analytical results.

Abstract

We calculate analytically the probability density $P(t_m)$ of the time $t_m$ at which a continuous-time Brownian motion (with and without drift) attains its maximum before passing through the origin for the first time. We also compute the joint probability density $P(M,t_m)$ of the maximum $M$ and $t_m$. In the driftless case, we find that $P(t_m)$ has power-law tails: $P(t_m)\sim t_m^{-3/2}$ for large $t_m$ and $P(t_m)\sim t_m^{-1/2}$ for small $t_m$. In presence of a drift towards the origin, $P(t_m)$ decays exponentially for large $t_m$. The results from numerical simulations are in excellent agreement with our analytical predictions.