Birth-death processes are time-changed Feller's Brownian motions

TL;DR

This paper establishes a precise, constructive connection between Feller's Brownian motions and birth-death processes through a specific time change, linking their parameters and providing a new framework for understanding these stochastic processes.

Contribution

It introduces a method to transform Feller's Brownian motions into birth-death processes and vice versa, clarifying their parameter correspondence and filling a gap in the literature.

Findings

Any Feller's Brownian motion can be transformed into a birth-death process via a specific time change.

The transformation provides a constructive framework linking the parameters of both processes.

The approach offers a new perspective on the relationship between diffusions and Markov chains.

Abstract

A Feller's Brownian motion is a diffusion process on the half-line with general boundary behavior at the origin, described by four parameters. A birth-death process, on the other hand, is a continuous-time Markov chain on the nonnegative integers, characterized by three parameters reflecting its behavior at infinity. This paper aims to build a connection between the two: we show that any Feller's Brownian motion can be transformed into a birth-death process via a specific time change, and vice versa. The transformation identifies a precise correspondence between their parameters. Our approach is based on a pathwise representation of the Feller process and offers a constructive framework for birth-death processes, filling a gap in the existing literature.