Continuous-state branching processes with spectrally positive migration

TL;DR

This paper introduces a generalization of continuous-state branching processes with immigration by allowing the immigration process to be any spectrally positive Lévy process, and derives explicit formulas for key probabilistic quantities.

Contribution

It extends the class of CSBPs with immigration to include spectrally positive Lévy processes and provides explicit Laplace transform formulas for important first passage and explosion times.

Findings

Derived explicit Laplace transform formulas for first passage times.

Established properties of the generalized processes.

Highlighted differences from classical CBIs.

Abstract

Continuous-state branching processes (CSBPs) with immigration (CBIs), stopped on hitting zero, are generalized by allowing the process governing immigration to be any L\'evy process without negative jumps. Unlike the CBIs, these newly introduced processes do not appear to satisfy any natural affine property on the level of the Laplace transforms of the semigroups. Basic properties are noted. Explicit formulae (on neighborhoods of infinity) for the Laplace transforms of the first passage times downwards and of the explosion time are derived.