The Annals of Applied Probability

A general continuous-state nonlinear branching process

Pei-Sen Li, Xu Yang, and Xiaowen Zhou

Full-text: Open access

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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1563869049

Digital Object Identifier
doi:10.1214/18-AAP1459

Mathematical Reviews number (MathSciNet)
MR3983343

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

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

Citation

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. https://projecteuclid.org/euclid.aoap/1563869049


Export citation

References

  • Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.
  • Bansaye, V., Méléard, S. and Richard, M. (2016). Speed of coming down from infinity for birth-and-death processes. Adv. in Appl. Probab. 48 1183–1210.
  • Berestycki, J., Berestycki, N. and Limic, V. (2010). The $\Lambda$-coalescent speed of coming down from infinity. Ann. Probab. 38 207–233.
  • Berestycki, J., Fittipaldi, M. C. and Fontbona, J. (2018). Ray–Knight representation of flows of branching processes with competition by pruning of Lévy trees. Probab. Theory Related Fields 172 725–788.
  • Berestycki, J., Döring, L., Mytnik, L. and Zambotti, L. (2015). Hitting properties and non-uniqueness for SDEs driven by stable processes. Stochastic Process. Appl. 125 918–940.
  • Bertoin, J. and Le Gall, J.-F. (2003). Stochastic flows associated to coalescent processes. Probab. Theory Related Fields 126 261–288.
  • Bertoin, J. and Le Gall, J.-F. (2005). Stochastic flows associated to coalescent processes. II. Stochastic differential equations. Ann. Inst. Henri Poincaré Probab. Stat. 41 307–333.
  • Bertoin, J. and Le Gall, J.-F. (2006). Stochastic flows associated to coalescent processes. III. Limit theorems. Illinois J. Math. 50 147–181.
  • Chen, R.-R. (1997). An extended class of time-continuous branching processes. J. Appl. Probab. 34 14–23.
  • Chen, A. (2002). Uniqueness and extinction properties of generalised Markov branching processes. J. Math. Anal. Appl. 274 482–494.
  • Chen, M.-F. (2004). From Markov Chains to Non-equilibrium Particle Systems, 2nd ed. World Scientific, River Edge, NJ.
  • Cherny, A. S. and Engelbert, H.-J. (2005). Singular Stochastic Differential Equations. Lecture Notes in Math. 1858. Springer, Berlin.
  • Dawson, D. A. and Li, Z. (2006). Skew convolution semigroups and affine Markov processes. Ann. Probab. 34 1103–1142.
  • Dawson, D. A. and Li, Z. (2012). Stochastic equations, flows and measure-valued processes. Ann. Probab. 40 813–857.
  • Dong, Y. (2018). Jump stochastic differential equations with non-Lipschitz and superlinearly growing coefficients. Stochastics 90 782–806.
  • Duhalde, X., Foucart, C. and Ma, C. (2014). On the hitting times of continuous-state branching processes with immigration. Stochastic Process. Appl. 124 4182–4201.
  • Foucart, C., Li, P. S. and Zhou, X. (2019). Time-changed spectrally positive Lévy processes starting from infinity. Preprint.
  • Fu, Z. and Li, Z. (2010). Stochastic equations of non-negative processes with jumps. Stochastic Process. Appl. 120 306–330.
  • Grey, D. R. (1974). Asymptotic behaviour of continuous time, continuous state-space branching processes. J. Appl. Probab. 11 669–677.
  • Höpfner, R. (1985). On some classes of population-size-dependent Galton–Watson processes. J. Appl. Probab. 22 25–36.
  • Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, 2nd ed. North-Holland, Amsterdam.
  • Karlin, S. and Taylor, H. M. (1981). A Second Course in Stochastic Processes. Academic Press, New York.
  • Kawazu, K. and Watanabe, S. (1971). Branching processes with immigration and related limit theorems. Teor. Veroyatn. Primen. 16 34–51.
  • Klebaner, F. C. (1984). Geometric rate of growth in population-size-dependent branching processes. J. Appl. Probab. 21 40–49.
  • Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Universitext. Springer, Berlin.
  • Lambert, A. (2005). The branching process with logistic growth. Ann. Appl. Probab. 15 1506–1535.
  • Le, V. (2014). Branching process with interaction. Ph.D. thesis, Reading Aix-Marseille Univ.
  • Le, V. and Pardoux, E. (2015). Height and the total mass of the forest of genealogical trees of a large population with general competition. ESAIM Probab. Stat. 19 172–193.
  • Le, V., Pardoux, E. and Wakolbinger, A. (2013). “Trees under attack”: A Ray–Knight representation of Feller’s branching diffusion with logistic growth. Probab. Theory Related Fields 155 583–619.
  • Li, Y. (2006). On a continuous-state population-size-dependent branching process and its extinction. J. Appl. Probab. 43 195–207.
  • Li, Y. (2009). A weak limit theorem for generalized Jiřina processes. J. Appl. Probab. 46 453–462.
  • Li, Z. (2011). Measure-Valued Branching Markov Processes. Probability and Its Applications (New York). Springer, Heidelberg.
  • Li, Z. (2012). Continuous-state branching processes. Available at arXiv:1202.3223v1.
  • Li, P. S. (2018). A continuous-state polynomial branching process. Stochastic Process. Appl. To appear. Available at arXiv:1609.09593v3.
  • Li, Z. and Mytnik, L. (2011). Strong solutions for stochastic differential equations with jumps. Ann. Inst. Henri Poincaré Probab. Stat. 47 1055–1067.
  • Li, P. S., Yang, X. and Zhou, X. (2018). The discrete approximation of a class of continuous-state nonlinear branching processes. Sci. Sin., Math. (In Chinese). To appear.
  • Meyn, S. P. and Tweedie, R. L. (1993). Stability of Markovian processes. III. Foster–Lyapunov criteria for continuous-time processes. Adv. in Appl. Probab. 25 518–548.
  • Pakes, A. G. (2007). Extinction and explosion of nonlinear Markov branching processes. J. Aust. Math. Soc. 82 403–428.
  • Pardoux, É. (2016). Probabilistic Models of Population Evolution. Scaling Limits, Genealogies and Interactions. Mathematical Biosciences Institute Lecture Series. Stochastics in Biological Systems 1. Springer, Cham.
  • Pardoux, E. and Wakolbinger, A. (2015). A path-valued Markov process indexed by the ancestral mass. ALEA Lat. Am. J. Probab. Math. Stat. 12 193–212.
  • Protter, P. E. (2005). Stochastic Integration and Differential Equations, 2nd edition. Stochastic Modelling and Applied Probability 21. Springer, Berlin.
  • Sevast’janov, B. A. and Zubkov, A. M. (1974). Controlled branching processes. Teor. Veroyatn. Primen. 19 15–25.
  • Wang, L., Yang, X. and Zhou, X. (2017). A distribution-function-valued SPDE and its applications. J. Differential Equations 262 1085–1118.
  • Zorich, V. A. (2004). Mathematical Analysis. I. Universitext. Springer, Berlin.