Extinction properties of multi-type continuous-state branching processes

TL;DR

This paper studies the extinction behavior of multi-type continuous-state branching processes with potentially infinitely many types, modeled as super Markov chains with local and non-local branching mechanisms, and discusses open problems.

Contribution

It introduces a novel approach to multi-type continuous-state branching processes allowing infinitely many types as super Markov chains with complex mechanisms.

Findings

Exploration of extinction properties for these processes.

Discussion of open problems in the field.

Extension beyond finite-type models.

Abstract

Recently in Barczy, Li and Pap (2015), the notion of a multi-type continuous-state branching process (with immigration) having d-types was introduced as a solution to an d-dimensional vector- valued SDE. Preceding that, work on affine processes, originally motivated by math- ematical finance, in Duffie, Filipovic and Schachermayer (2003) also showed the existence of such processes. See also more recent contributions in this direction due to Gabrielli and Teichmann (2014) and Caballero, Perez Garmendia and Uribe Bravo (2015). Older work on multi-type continuous-state branching processes is more sparse but includes Watanabe (1969) and Ma (2013), where only two types are considered. In this paper we take a completely different approach and consider multi-type continuous-state branching process, now allowing for up to a countable infinity of types, defined instead as a super Markov chain with both local and non-local branching mechanisms. In the spirit of Englander and Kyprianou (2004) we explore their extinction properties and pose a number of open problems.