A Note on the Conditional Probabilities of the Telegraph Process

TL;DR

This paper analyzes the conditional probabilities of the telegraph process with two velocities and reversal rates, providing new proofs and detailed distributional insights, especially in symmetric cases, for understanding its stochastic behavior over time.

Contribution

It introduces a new inductive proof of the conditional probability law and offers detailed distributional analysis of the process at different times, including symmetric cases.

Findings

Derived explicit conditional probability laws.

Provided detailed distributional results for the process at time t.

Analyzed joint distributions of position, maximum, and minimum in symmetric cases.

Abstract

We consider the telegraph process with two velocities, $a_1>a_2\in\mathbb{R}$, and two rates of reversal, $\lambda_1,\lambda_2>0$. We study some of its features with respect to the conditional probability measure where both the initial speed and the number of changes of direction are known. We exhibit a new proof by induction of the (conditional) probability law and a detailed study of the distribution of the motion at time $t>0$ conditioned on its position at a previous time $0<s<t$. In the case of a symmetric process, we present some results on the joint distribution of the position of the motion at time $t>0$, its maximum and its minimum up to that moment.