Some results on the Brownian meander with drift

TL;DR

This paper investigates the properties of the drifted Brownian meander, including its limiting behavior, distribution of maxima, and first-passage times, extending known results from the driftless case.

Contribution

It provides new analysis and representations for the drifted Brownian meander, including conditions for its construction and extensions of classical results.

Findings

Derived the limiting process as initial point approaches the boundary

Obtained distribution formulas for the maximum and first-passage times

Extended driftless case results to the drifted Brownian meander

Abstract

In this paper we study the drifted Brownian meander, that is a Brownian motion starting from $ u $ and subject to the condition that $ \min_{ 0\leq z \leq t} B(z)> v $ with $ u > v $. The limiting process for $ u \downarrow v $ is analyzed and the sufficient conditions for its construction are given. We also study the distribution of the maximum of the meander with drift and the related first-passage times. The representation of the meander endowed with a drift is provided and extends the well-known result of the driftless case. The last part concerns the drifted excursion process the distribution of which coincides with the driftless case.