Distributional properties of jumps of multi-type CBI processes

TL;DR

This paper analyzes the distributional characteristics of jumps in multi-type CBI processes, providing formulas for jump times, supremum distributions, and conditions under which jumps are negligible.

Contribution

It introduces new formulas for jump time distributions and supremum probabilities in multi-type CBI processes, advancing understanding of their jump behavior.

Findings

Derived the distribution function of the first jump time.

Established the distribution of the supremum of jump norms.

Proved conditions under which jump probabilities are zero.

Abstract

We study the distributional properties of jumps of multi-type continuous state and continuous time branching processes with immigration (multi-type CBI processes). We derive an expression for the distribution function of the first jump time of a multi-type CBI process with jump size in a given Borel set having finite total L\'evy measure, which is defined as the sum of the measures appearing in the branching and immigration mechanisms of the multi-type CBI process in question. Using this we derive an expression for the distribution function of the local supremum of the norm of the jumps of a multi-type CBI process. Further, we show that if $A$ is a nondegenerate rectangle anchored at zero and with total L\'evy measure zero, then the probability that the local coordinate-wise supremum of jumps of the multi-type CBI process belongs to $A$ is zero. We also prove that a converse statement holds.