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Skeletal stochastic differential equations for continuous-state branching processes

Part of: Stochastic processes Stochastic analysis Markov processes

Published online by Cambridge University Press:  11 December 2019

D. Fekete ,
J. Fontbona  and
A. E. Kyprianou
D. Fekete*
Affiliation:
University of Exeter
J. Fontbona*
Affiliation:
Universidad de Chile
A. E. Kyprianou*
Affiliation:
University of Bath
*
*Postal address: College of Engineering, Mathematics and Physical Sciences, University of Exeter, Prince of Wales Road, Exeter, EX4 4SB, UK.
***Postal address: Centre for Mathematical Modelling, DIM CMM, UMI 2807 UChile-CNRS, Universidad de Chile, Santiago, Chile.
****Postal address: Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK.
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Abstract

It is well understood that a supercritical continuous-state branching process (CSBP) is equal in law to a discrete continuous-time Galton–Watson process (the skeleton of prolific individuals) whose edges are dressed in a Poissonian way with immigration which initiates subcritical CSBPs (non-prolific mass). Equally well understood in the setting of CSBPs and superprocesses is the notion of a spine or immortal particle dressed in a Poissonian way with immigration which initiates copies of the original CSBP, which emerges when conditioning the process to survive eternally. In this article we revisit these notions for CSBPs and put them in a common framework using the well-established language of (coupled) stochastic differential equations (SDEs). In this way we are able to deal simultaneously with all types of CSBPs (supercritical, critical, and subcritical) as well as understanding how the skeletal representation becomes, in the sense of weak convergence, a spinal decomposition when conditioning on survival. We have two principal motivations. The first is to prepare the way to expand the SDE approach to the spatial setting of superprocesses, where recent results have increasingly sought the use of skeletal decompositions to transfer results from the branching particle setting to the setting of measure valued processes. The second is to provide a pathwise decomposition of CSBPs in the spirit of genealogical coding of CSBPs via Lévy excursions, albeit precisely where the aforesaid coding fails to work because the underlying CSBP is supercritical.

Keywords

Continuous-state branching processstochastic differential equationskeletal decompositionspine decomposition

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 60G99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 4 , December 2019 , pp. 1122 - 1150
Copyright
© Applied Probability Trust 2019 

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