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

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.