Continuous state branching processes in random environment: The Brownian case

Motivated by the works of Böinghoff and Huzenthaler [6] and Bansaye et al. [1], we introduce continuous state branching processes in a Brownian random environment. Roughly speaking, a process in this class behaves as a continuous state branching process but its dynamics are perturbed by an independent Brownian motion with drift. More precisely, we define a continuous state branching process in a Brownian random environment as the unique strong solution of a stochastic differential equation that satisfies a quenched branching property. In this paper, we are interested in the long-term behaviour of such class of processes. In particular, we provide a necessary condition under which the process is conservative, i.e. that it does not explode a.s. at a finite time, and we also study the event of extinction conditionally on the environment. In the particular case where the branching mechanism is stable, we compute explicitly extinction and explosion probabilities. We show that three regimes arise for the speed of explosion in the case when the scaling index is negative. In the case when the scaling index is positive, the process is conservative and we prove that five regimes arise for the speed of extinction, as for discrete (time and space) branching processes in random environment and branching diffusions in random environment (see [6]). The precise asymptotics for the speed of extinction allow us to introduce the so called process conditioned to be never extinct or Q-process. The proofs of the speed of explosion and extinction are based on the precise asymptotic behaviour of exponential functionals of Brownian motion. At the end of this paper, we discuss the immigration case which in particular provides an extension of the Cox-Ingersoll-Ross model in a random environment. ∗Centro de Investigación en Matemáticas A.C. Calle Jalisco s/n. 36240 Guanajuato, México. E-mail: [email protected]. Corresponding author †Centro de Investigación en Matemáticas A.C. Calle Jalisco s/n. 36240 Guanajuato, México. E-mail: [email protected]