A continuous-state polynomial branching process

TL;DR

This paper introduces a new class of continuous-state polynomial branching processes, providing explicit formulas for extinction and explosion probabilities, and establishing their connection as limits of discrete processes.

Contribution

It constructs the process as a unique solution to a stochastic integral equation and derives explicit extinction and explosion probabilities, including conditions for finite-time extinction or explosion.

Findings

Explicit extinction and explosion probabilities derived

Conditions for finite-time extinction or explosion established

Process shown as a limit of discrete-state processes

Abstract

A continuous-state polynomial branching process is constructed as the pathwise unique solution of a stochastic integral equation with absorbing boundary condition. The extinction and explosion probabilities and the mean extinction and explosion times are computed explicitly, which are also new in the classical branching case. We present necessary and sufficient conditions for the process to extinguish or explode in finite times. In the critical or subcritical case, we give a construction of the process coming down from infinity. Finally, it is shown that the continuous-state polynomial branching process arises naturally as the rescaled limit of a sequence of discrete-state processes.