The Annals of Applied Probability

A general continuous-state nonlinear branching process

Pei-Sen Li, Xu Yang, and Xiaowen Zhou

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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: \begin{eqnarray*}X_{t}&=&x+\int_{0}^{t}\gamma_{0}(X_{s})\,\mathrm{d}s+\int_{0}^{t}\int_{0}^{\gamma_{1}(X_{s-})}W(\mathrm{d}s,\mathrm{d}u)\\&&{}+\int_{0}^{t}\int_{0}^{\infty}\int_{0}^{\gamma_{2}(X_{s-})}z\tilde{N}(\mathrm{d}s,\mathrm{d}z,\mathrm{d}u),\end{eqnarray*} where $W(\mathrm{d}t,\mathrm{d}u)$ and $\tilde{N}(\mathrm{d}s,\mathrm{d}z,\mathrm{d}u)$ 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 $\mathbb{R}_{+}$ 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.

Article information

Ann. Appl. Probab., Volume 29, Number 4 (2019), 2523-2555.

Received: March 2018
Revised: October 2018
First available in Project Euclid: 23 July 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 60G57: Random measures
Secondary: 60G17: Sample path properties 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Continuous-state branching process nonlinear branching competition extinction explosion coming down from infinity weighted total population Foster–Lyapunov criterion stochastic differential equation


Li, Pei-Sen; Yang, Xu; Zhou, Xiaowen. A general continuous-state nonlinear branching process. Ann. Appl. Probab. 29 (2019), no. 4, 2523--2555. doi:10.1214/18-AAP1459.

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