Non-Local Solvable Birth-Death Processes

TL;DR

This paper investigates non-local difference-differential equations related to birth-death processes, providing spectral solutions, stochastic representations, and analysis of their invariant, limit distributions, and correlation structures.

Contribution

It introduces a novel spectral decomposition approach for non-local equations linked to birth-death processes and offers stochastic representations and distributional analyses.

Findings

Spectral decomposition in orthogonal polynomials for non-local equations

Stochastic representation via time-changed birth-death processes

Characterization of invariant and limit distributions

Abstract

In this paper we study strong solutions of some non-local difference-differential equations linked to a class of birth-death processes arising as discrete approximations of Pearson diffusions by means of a spectral decomposition in terms of orthogonal polynomials and eigenfunctions of some non-local derivatives. Moreover, we give a stochastic representation of such solutions in terms of time-changed birth-death processes and study their invariant and their limit distribution. Finally, we describe the correlation structure of the aforementioned time-changed birth-death processes.