Conditioning the logistic continuous-state branching process on non-extinction via its total progeny

TL;DR

This paper studies how to condition a logistic continuous-state branching process on non-extinction by using its total progeny, leading to a new process with finite lifetime characterized by stochastic equations and duality methods.

Contribution

It introduces a novel conditioning method for logistic CB processes based on total progeny and explicitly characterizes the resulting conditioned process.

Findings

Non-extinction is equivalent to infinite total progeny.

The conditioned process has finite lifetime and is described by a stochastic equation with jumps.

The process can start from zero and is related to a time-changed Ornstein-Uhlenbeck process.

Abstract

The problem of conditioning a continuous-state branching process with quadratic competition (logistic CB process) on non-extinction is investigated. We first establish that non-extinction is equivalent to the total progeny of the population being infinite. The conditioning we propose is then designed by requiring the total progeny to exceed arbitrarily large exponential random variables. This is related to a Doob's $h$-transform with an explicit excessive function $h$. The $h$-transformed process, i.e. the conditioned process, is shown to have a finite lifetime almost surely (it is either killed or it explodes continuously). When starting from positive values, the conditioned process is furthermore characterized, up to its lifetime, as the solution to a certain stochastic equation with jumps. The latter superposes the dynamics of the initial logistic CB process with an additional density-dependent immigration term. Last, it is established that the conditioned process can be starting from zero. Key tools employed are a representation of the logistic CB process through a time-changed generalized Ornstein-Uhlenbeck process, as well as Laplace and Siegmund duality relationships with auxiliary diffusion processes.