Applying Brownian motion to the study of birth-death chains

TL;DR

This paper leverages Brownian motion properties to analyze birth-death chains, deriving extinction probabilities and conditions for expected value convergence, with local time theory playing a key role.

Contribution

It introduces a novel application of Brownian motion to derive key properties of birth-death chains, including extinction probability and convergence conditions.

Findings

Calculated extinction probabilities for birth-death chains

Established conditions for expected value convergence

Utilized Brownian motion local time in proofs

Abstract

Basic properties of Brownian motion are used to derive two results concerning birth-death chains. First, the probability of extinction is calculated. Second, sufficient conditions on the transition probabilities of a birth-death chain are given to ensure that the expected value of the chain converges to a limit. The theory of Brownian motion local time figures prominently in the proof of the second result.