Brownian Motion Conditioned to Spend Limited Time Below a Barrier

TL;DR

This paper explicitly characterizes a Brownian motion conditioned to spend at most one unit of time below zero, providing formulas for key distributional properties and analyzing their behavior as the starting point varies.

Contribution

It extends previous work by deriving explicit formulas for the distribution of last zero and occupation time for any starting point, generalizing earlier results.

Findings

Explicit formulas for distributions of last zero and occupation time

Analysis of distributional behavior as starting point tends to infinity

Generalization of previous special case results

Abstract

We condition a Brownian motion with arbitrary starting point $y \in \mathbb{R}$ on spending at most $1$ time unit below $0$ and provide an explicit description of the resulting process. In particular, we provide explicit formulas for the distributions of its last zero $g=g^y$ and of its occupation time $\Gamma=\Gamma^y$ below $0$ as functions of $y$. This generalizes a result of Benjamini and Berestycki from 2011, which covers the special case $y=0$. Additionally, we study the behavior of the distributions of $g^y$ and $\Gamma^y$, respectively, for $y \to \pm\infty$.