On telegraph processes, their first passage times and running extrema

TL;DR

This paper explores the distributions of extrema, first passage times, and related processes in telegraph models, deriving explicit formulas for joint distributions using integral equations.

Contribution

It introduces new explicit formulas for joint distributions of extrema, switches, and terminal positions in telegraph processes using coupled integral equations.

Findings

Derived explicit formulas for joint distributions.

Presented techniques for analyzing extrema and passage times.

Enhanced understanding of telegraph process behaviors.

Abstract

In this note, we present some ideas for describing the distributions of the running maximum/minimum, first passage times and telegraphic meanders. Explicit formulae for joint distribution of the extrema, the number of velocity switches and the terminal position are derived using coupled integral equations technique.