Extinction times of multitype, continuous-state branching processes

TL;DR

This paper analyzes the extinction times of multitype continuous-state branching processes, providing conditions for extinction in finite time based on their branching mechanisms, extending classical results to higher dimensions.

Contribution

It extends Grey's extinction condition from one dimension to multitype processes using advanced fluctuation theory and higher-dimensional Lamperti representations.

Findings

Derived an expression for extinction probability in terms of the branching mechanism.

Established a necessary and sufficient condition for finite-time extinction.

Extended classical extinction criteria to multitype processes.

Abstract

A multitype continuous-state branching process (MCSBP) ${\rm Z}=({\rm Z}_{t})_{t\geq 0}$, is a Markov process with values in $[0,\infty)^{d}$ that satisfies the branching property. Its distribution is characterised by its branching mechanism, that is the data of $d$ Laplace exponents of $\mathbb{R}^d$-valued spectrally positive L\'evy processes, each one having $d-1$ increasing components. We give an expression of the probability for a MCSBP to tend to 0 at infinity in term of its branching mechanism. Then we prove that this extinction holds at a finite time if and only if some condition bearing on the branching mechanism holds. This condition extends Grey's condition that is well known for $d=1$. Our arguments bear on elements of fluctuation theory for spectrally positive additive L\'evy fields recently obtained in \cite{cma1} and an extension of the Lamperti representation in higher dimension proved in \cite{cpgub}.