On some transformations of bilateral birth-and-death processes with applications

TL;DR

This paper introduces a method to relate bilateral birth-and-death processes, enabling the derivation of their transition probabilities, first-passage-time densities, and crossing probabilities through simple product-form relationships.

Contribution

It presents a novel approach to connect birth-and-death processes, simplifying the analysis of their probabilistic properties and crossing behaviors.

Findings

Derived explicit relationships among process parameters

Established formulas for transition probabilities and passage times

Demonstrated applications to process analysis

Abstract

A method yielding simple relationships among bilateral birth-and-death processes is outlined. This allows one to relate birth and death rates of two processes in such a way that their transition probabilities, first-passage-time densities and ultimate crossing probabilities are mutually related by some product-form expressions.