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August 2019 A general continuous-state nonlinear branching process
Pei-Sen Li, Xu Yang, Xiaowen Zhou
Ann. Appl. Probab. 29(4): 2523-2555 (August 2019). DOI: 10.1214/18-AAP1459

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: \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.

Citation

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Pei-Sen Li. Xu Yang. Xiaowen Zhou. "A general continuous-state nonlinear branching process." Ann. Appl. Probab. 29 (4) 2523 - 2555, August 2019. https://doi.org/10.1214/18-AAP1459

Information

Received: 1 March 2018; Revised: 1 October 2018; Published: August 2019
First available in Project Euclid: 23 July 2019

zbMATH: 07120715
MathSciNet: MR3983343
Digital Object Identifier: 10.1214/18-AAP1459

Subjects:
Primary: 60G57
Secondary: 60G17, 60J80

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.29 • No. 4 • August 2019
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