The first passage time on the (reflected) Brownian motion with broken drift hitting a random boundary

TL;DR

This paper derives formulas for the joint Laplace transform of hitting times and positions for a reflected Brownian motion with broken drift hitting a random boundary, advancing understanding of first rendezvous times in stochastic processes.

Contribution

It provides new explicit formulas for the joint Laplace transform in a complex stochastic setting involving broken drift and random boundaries.

Findings

Derived formulas for joint Laplace transforms of hitting times and positions.

Extended previous work on rendezvous times of Brownian motion and compound Poisson processes.

Contributes to the mathematical understanding of boundary crossing problems in stochastic processes.

Abstract

In this paper we consider a (reflected) Brownian motion with broken drift hitting a random boundary. Some dedicated calculations allow us to obtain the formula on the joint Laplace transform of the hitting time and hitting position. These develop the research on first rendezvous times of (reflected) Brownian motion and compound Poisson-type processes by Perry et al. (2004).