Occupation densities in solving exit problems for Markov additive processes and their reflections

TL;DR

This paper advances the understanding of exit problems for spectrally negative Markov additive processes by constructing a scale matrix and exploring its properties through probabilistic methods, extending classical results to a more general setting.

Contribution

It provides a probabilistic construction of the scale matrix for Markov additive processes and generalizes the relation between the scale function and excursion measures.

Findings

Constructed the scale matrix probabilistically.

Identified the transform of the scale matrix.

Extended the relation between scale function and excursion measure.

Abstract

This paper solves exit problems for spectrally negative Markov additive processes and their reflections. A so-called scale matrix, which is a generalization of the scale function of a spectrally negative \levy process, plays a central role in the study of exit problems. Existence of the scale matrix was shown in Thm. 3 of Kyprianou and Palmowski (2008). We provide a probabilistic construction of the scale matrix, and identify the transform. In addition, we generalize to the MAP setting the relation between the scale function and the excursion (height) measure. The main technique is based on the occupation density formula and even in the context of fluctuations of spectrally negative L\'{e}vy processes this idea seems to be new. Our representation of the scale matrix $W(x)=e^{-\Lambda x}\eL(x)$ in terms of nice probabilistic objects opens up possibilities for further investigation of its properties.