A GENERAL CONTINUOUS-STATE NONLINEAR BRANCHING PROCESS

• 成果类型: Article
• 署名作者: Li, Pei-Sen; Yang, Xu; Zhou, Xiaowen
• 署名单位: Renmin University of China; North Minzu University; Concordia University - Canada
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/18-AAP1459
• 发表日期: 2019
• 页码: 2523-2555
• 关键词: STOCHASTIC DIFFERENTIAL-EQUATIONS; ray-knight representation; continuous-time; population; FLOWS; extinction; speed; trees

摘要

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: X-t = x + integral(t)(0) gamma 0(X-s)ds + integral(t)(0)integral(gamma 1(Xs-))(0) W(ds, du) +integral(t)(0)integral(infinity)(0)integral(gamma 2(Xs-))(0) z (N) over tilde (ds, dz, du) where W(dt, du) and (N) over tilde (ds, dz, du) 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 R+ with both gamma 1 and gamma 2 taking nonnegative values. Intuitively, this process can be identified as a continuousstate 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.