On SDE associated with continuous-state branching processes conditioned to never be extinct

TL;DR

This paper provides a stochastic differential equation framework for continuous-state branching processes conditioned on survival, revealing explicit mechanisms for immigration through Girsanov transformations.

Contribution

It introduces a novel SDE-based approach to describe conditioned CSBPs, highlighting how to explicitly construct the immigration component via jump selection.

Findings

Explicit SDE representation for conditioned CSBPs

Mechanism to construct immigration term via jump selection

Potential applicability to general diffusion-jump process transforms

Abstract

We study the pathwise description of a (sub-)critical continuous-state branching process (CSBP) conditioned to be never extinct, as the solution to a stochastic differential equation driven by Brownian motion and Poisson point measures. The interest of our approach, which relies on applying Girsanov theorem on the SDE that describes the unconditioned CSBP, is that it points out an explicit mechanism to build the immigration term appearing in the conditioned process, by randomly selecting jumps of the original one. These techniques should also be useful to represent more general h-transforms of diffusion-jump processes.