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.
Highlights
Introduction and preliminariesStochastic differential equations (SDE) representing continuous-state branching processes (CSBP) or CSBP with immigration (CBI) have attracted increasing attention in the last years, as powerful tools for studying pathwise and distributional properties of these processes as well as some scaling limits, see e.g. Dawson and Li [5], [6], Lambert [19], Fu and Li [11] and Caballero et al [4].In this note, we are interested in SDE representations for-critical CSBP conditioned to never be extinct
We study the pathwise description of acritical 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 the use of 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
Summary
Stochastic differential equations (SDE) representing continuous-state branching processes (CSBP) or CSBP with immigration (CBI) have attracted increasing attention in the last years, as powerful tools for studying pathwise and distributional properties of these processes as well as some scaling limits, see e.g. Dawson and Li [5], [6] , Lambert [19], Fu and Li [11] and Caballero et al [4]. The enlargement of the probability space and the marking procedure are both inspired in a construction of Lambert [19] on stable Lévy processes. They are reminiscent of the sized biased tree representation of measure changes for Galton-Watson trees (Lyons et al [26]) or for branching Brownian motions (see e.g. Kyprianou [17] and Englänger and Kyprianou [9]), but we do not aim at fully developing those ideas in the present framework. We start by recalling some definitions and classic results about CSBPs and Lévy processes along the lines of [18, Chap. 1,2 and 10], in particular the relationship between them through the Lamperti transform. (We refer the reader to Le Gall [22] and Li [23] for further background on CSBP)
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