On the explosion of a class of continuous-state nonlinear branching processes

TL;DR

This paper studies the explosion behavior of a class of continuous-state nonlinear branching processes, showing how they tend to infinity in finite time and determining the explosion speed under various conditions.

Contribution

It introduces a new analysis of explosion phenomena in generalized continuous-state branching processes via Lamperti transformations and scale function asymptotics.

Findings

Processes explode to infinity in finite time along a deterministic curve

Explosion speed is characterized for different rate function regimes

New asymptotic results for spectrally positive Lévy process scale functions

Abstract

In this paper, we consider a class of generalized continuous-state branching processes obtained by Lamperti type time changes of spectrally positive L\'evy processes using different rate functions. When explosion occurs to such a process, we show that the process converges to infinity in finite time asymptotically along a deterministic curve, and identify the speed of explosion for rate function in different regimes. To prove the main theorems, we also establish a new asymptotic result for scale function of spectrally positive L\'evy process.