On the extinction of Continuous State Branching Processes with catastrophes

By Vincent Bansaye, Juan Carlos Pardo Millan, C. Smadi. 2013

57 citations recorded by the source corpus.

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 Levy 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 Levy processes. Finally, we apply these results to a cell infection model and determine the mean speed of propagation of the infection.

Subfields: Finance, Management Science and Operations Research, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1d322541737b472d5f4b12d11bc0827c0e99764c

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Citations in the corpus

Stochastic Models for Structured Populations: Scaling Limits and Long Time Behavior (2015, background): “The proof in [7] uses a discretization of ∫ t 0 exp(−βKs)ds of the form ∑n i=0 Π i j=0Ai and the different regimes are inherited from local limit theorems for semi-direct products [60].”
Asymptotic results for exponential functionals of Levy processes (2016, methodology): “Bansaye et al. (2013) studied the problem in the case where the environment is given by a Lévy process with bounded variations and showed some interesting applications of the results to a cell infection model.”