On the sojourn time of a Generalized Brownian meander

TL;DR

This paper investigates the distribution of the time a drifted Brownian motion spends positive up to time t under specific boundary conditions, providing explicit results especially at zero drift and barrier.

Contribution

It introduces explicit distributional formulas for the sojourn time of a generalized Brownian meander with boundary constraints, including the zero drift case.

Findings

Explicit distributional results for the sojourn time at zero barrier.

Analysis of the weak limit as initial point approaches the barrier.

Distributional formulas for the process with and without drift.

Abstract

In this paper we study the sojourn time on the positive half-line up to time $ t $ of a drifted Brownian motion with starting point $ u $ and subject to the condition that $ \min_{ 0\leq z \leq l} B(z)> v $, with $ u > v $. This process is a drifted Brownian meander up to time $ l $ and then evolves as a free Brownian motion. We also consider the sojourn time of a bridge-type process, where we add the additional condition to return to the initial level at the end of the time interval. We analyze the weak limit of the occupation functional as $ u \downarrow v $. We obtain explicit distributional results when the barrier is placed at the zero level, and also in the special case when the drift is null.