A general continuous-state nonlinear branching process

TL;DR

This paper introduces a generalized continuous-state nonlinear branching process with population-dependent rates, analyzing its extinction, explosion, and stability behaviors using martingale and Foster-Lyapunov techniques.

Contribution

It develops a comprehensive framework for nonlinear branching processes with state-dependent rates, providing sharp conditions for key behaviors and explicit results for power function cases.

Findings

Sharp conditions for extinction and explosion

Criteria for coming down from infinity

Explicit results for power function rate cases

Abstract

In this paper we consider the unique nonnegative solution to the following generalized version of the stochastic differential equation for a continuous-state branching process. \beqnn X_t \ar=\ar x+\int_0^t\gamma_0(X_s)\dd s+\int_0^t\int_0^{\gamma_1(X_{s-})} W(\dd s,\dd u)\cr \ar\ar\qquad+\int_0^t\int_{0}^\infty\int_0^{\gamma_2(X_{s-})} z\tilde{N}(\dd s, \dd z, \dd u), \eeqnn where $W(\dd t,\dd u) $ and $\tilde{N}(\dd s, \dd z, \dd u)$ denote a Gaussian white noise and an independent compensated spectrally positive Poisson random measure, respectively, and $\gamma_0,\gamma_1$ and $\gamma_2$ are functions on $\mbb{R}_+$ with both $\gamma_1$ and $\gamma_2$ taking nonnegative values. Intuitively, this process can be identified as a continuous-state branching process with population-size-dependent branching rates and with competition. Using martingale techniques we find rather sharp conditions on extinction, explosion and coming down from infinity behaviors of the process. Some Foster-Lyapunov type criteria are also developed for such a process. More explicit results are obtained when $\gamma_i, i=0, 1, 2$ are power functions.