Electronic Journal of Probability

On the extinction of continuous state branching processes with catastrophes

Abstract

We consider continuous state branching processes (CSBP's) with additional multiplicative jumps modeling dramatic  events in a random environment. These jumps  are described by a Lévy process with bounded variation paths. We construct the associated class of processes as the unique solution of a stochastic differential equation. The quenched branching property of the process allows us to derive quenched and annealed results and make appear new asymptotic behaviors. We characterize the Laplace exponent of the process as the solution of a backward ordinary differential equation and establish when it becomes extinct. For a class of processes for which extinction and absorption coincide (including the $\alpha$ stable CSBP's plus a drift), we determine the speed of extinction. Four regimes appear, as in the case of branching processes in random environment in discrete time and space.The proofs rely on a fine study of the asymptotic behavior of exponential functionals of Lévy processes. Finally, we apply these results to a cell infection model and determine the mean speed of propagation of the infection.

Article information

Source
Electron. J. Probab., Volume 18 (2013), paper no. 106, 31 pp.

Dates
Accepted: 18 December 2013
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465064331

Digital Object Identifier
doi:10.1214/EJP.v18-2774

Mathematical Reviews number (MathSciNet)
MR3151726

Zentralblatt MATH identifier
1286.60083

Rights

Citation

Bansaye, Vincent; Pardo Millan, Juan Carlos; Smadi, Charline. On the extinction of continuous state branching processes with catastrophes. Electron. J. Probab. 18 (2013), paper no. 106, 31 pp. doi:10.1214/EJP.v18-2774. https://projecteuclid.org/euclid.ejp/1465064331

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