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

TL;DR

The paper addresses the problem of characterizing the joint distribution of the first jump times of CBI processes when jump sizes lie in multiple prescribed Borel sets with finite total Lévy measure. It builds a multi-type CBI framework by proving that the augmented process (X_t, J_t(A_1), ..., J_t(A_k)) is itself a (k+1)-type CBI process with explicitly defined parameters, and it derives a tractable joint Laplace transform through a system of deterministic differential equations. This Laplace-transform approach yields precise expressions for the joint distribution of jump counts and first-jump times, including a general result for the joint survival function and a special equal-time case with two independent proofs. Overall, the results generalize existing single-set findings and provide a versatile analytic tool for jump-timing analysis in multi-type CBI processes via branching and immigration mechanisms. The findings have potential applications in stochastic population models and related applied probability contexts where jump timing and sizes are of interest.

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 Lévy 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 Lévy measure.