Continuous-state Branching Processes in Random Environment
Motivated by the study of negative jumps in finance/biology, we introduce a general continuous-state branching process in random environment (CBRE-process) defined as the strong solution of a stochastic integral equation. The environment is determined by a Levy process with jumps no less than $-1$. We give characterizations of the quenched and annealed transition semigroups of the process in terms of a backward stochastic integral equation driven by another Levy process determined by the environment. The process hits zero with strictly positive probability if and only if its branching mechanism satisfies Grey's condition. In that case, a characterization of the extinction probability is given using a random differential equation with singular terminal condition. The strong Feller property of the CBRE-process is established by a coupling method. We also prove a necessary and sufficient condition for the ergodicity of the subcritical CBRE-process with immigration. Finally, we study the decay rate of the survival probability of CBRE-processes, which is determined by the expectation of some exponential functionals of Levy process. We shall see that five regimes arise from the convergence rate. Both the exact convergence rate and the explicit limiting coefficients are given. This talk is based on a part of Ph.D. thesis.