Distributional properties of first jump times of CBI processes with jump sizes in given Borel sets

TL;DR

This paper derives a general formula for the joint distribution of the first jump times in CBI processes with specified jump size sets, extending previous work to multiple Borel sets with finite Lévy measures.

Contribution

It generalizes earlier results by providing a joint distribution expression for multiple Borel sets with finite Lévy measures in CBI processes.

Findings

Derived a joint distribution function for first jump times in CBI processes.

Extended previous single-set results to multiple Borel sets.

Generalized the understanding of jump behaviors in CBI processes.

Abstract

We derive an expression for the joint distribution function of the first jump times of a continuous state and continuous time branching process with immigration (CBI process) with jump sizes in given Borel sets having finite total L\'evy measures, which is defined as the sum of the measures appearing in the branching and immigration mechanisms of the CBI process in question. Our result generalizes a corresponding result of He and Li (2016), who considered this problem in case of a single Borel set having finite total L\'evy measure.