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Continuous-state branching processes with collisions: First passage times and duality

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  • Foucart, Clément
  • Vidmar, Matija

Abstract

We introduce a class of one-dimensional positive Markov processes generalizing continuous-state branching processes (CBs), by taking into account a phenomenon of random collisions. Besides branching, characterized by a general mechanism Ψ, at a constant rate in time two particles are sampled uniformly in the population, collide and leave a mass of particles governed by a (sub)critical mechanism Σ. Such CB processes with collisions (CBCs) are shown to be the only Feller processes without negative jumps satisfying a Laplace duality relationship with one-dimensional diffusions on the half-line. This generalizes the duality observed for logistic CBs in Foucart (2019). Via time-change, CBCs are also related to an auxiliary class of Markov processes, called CB processes with spectrally positive migration (CBMs), recently introduced in Vidmar (2022). We find necessary and sufficient conditions for the boundaries 0 or ∞ to be attracting and for a limiting distribution to exist. The Laplace transform of the latter is provided. Under the assumption that the CBC process does not explode, the Laplace transforms of the first passage times below arbitrary levels are represented with the help of the solution of a second-order differential equation, whose coefficients express in terms of the Lévy–Khintchine functions Σ and Ψ. Sufficient conditions for non-explosion are given.

Suggested Citation

  • Foucart, Clément & Vidmar, Matija, 2024. "Continuous-state branching processes with collisions: First passage times and duality," Stochastic Processes and their Applications, Elsevier, vol. 167(C).
    • Handle: RePEc:eee:spapps:v:167:y:2024:i:c:s0304414923002028
    DOI: 10.1016/j.spa.2023.104230
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    References listed on IDEAS

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    1. Anyue Chen & Junping Li & Jing Zhang, 2020. "Branching Collision Processes with Immigration," Methodology and Computing in Applied Probability, Springer, vol. 22(3), pages 1063-1088, September.
    2. Borovkov, Konstantin & Novikov, Alexander, 2008. "On exit times of Lévy-driven Ornstein-Uhlenbeck processes," Statistics & Probability Letters, Elsevier, vol. 78(12), pages 1517-1525, September.
    3. Li, Pei-Sen, 2019. "A continuous-state polynomial branching process," Stochastic Processes and their Applications, Elsevier, vol. 129(8), pages 2941-2967.
    4. Cox, J. Theodore & Rösler, Uwe, 1984. "A duality relation for entrance and exit laws for Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 16(2), pages 141-156, February.
    5. Hui He & Zenghu Li & Wei Xu, 2018. "Continuous-State Branching Processes in Lévy Random Environments," Journal of Theoretical Probability, Springer, vol. 31(4), pages 1952-1974, December.
    6. Duhalde, Xan & Foucart, Clément & Ma, Chunhua, 2014. "On the hitting times of continuous-state branching processes with immigration," Stochastic Processes and their Applications, Elsevier, vol. 124(12), pages 4182-4201.
    7. David Landriault & Bin Li & Hongzhong Zhang, 2017. "A Unified Approach for Drawdown (Drawup) of Time-Homogeneous Markov Processes," Papers 1702.07786, arXiv.org.
    8. Palau, S. & Pardo, J.C., 2017. "Continuous state branching processes in random environment: The Brownian case," Stochastic Processes and their Applications, Elsevier, vol. 127(3), pages 957-994.
    9. Fu, Zongfei & Li, Zenghu, 2010. "Stochastic equations of non-negative processes with jumps," Stochastic Processes and their Applications, Elsevier, vol. 120(3), pages 306-330, March.
    10. Foucart, Clément & Möhle, Martin, 2022. "Asymptotic behaviour of ancestral lineages in subcritical continuous-state branching populations," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 510-531.
    11. Zenghu Li & Chunhua Ma, 2008. "Catalytic Discrete State Branching Models and Related Limit Theorems," Journal of Theoretical Probability, Springer, vol. 21(4), pages 936-965, December.
    12. Vidmar, Matija, 2023. "Complete monotonicity of time-changed Lévy processes at first passage," Statistics & Probability Letters, Elsevier, vol. 193(C).
    13. Patie, Pierre, 2005. "On a martingale associated to generalized Ornstein-Uhlenbeck processes and an application to finance," Stochastic Processes and their Applications, Elsevier, vol. 115(4), pages 593-607, April.
    14. Rogerio A. R. Junqueira & Reinaldo Morabito, 2015. "Production and Logistics Planning in Seed Corn," International Series in Operations Research & Management Science, in: Lluis M. Plà-Aragonés (ed.), Handbook of Operations Research in Agriculture and the Agri-Food Industry, edition 127, chapter 0, pages 55-77, Springer.
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