Tail behavior of stopped L\'evy processes with Markov modulation

TL;DR

This paper investigates the tail probabilities of Markov-modulated Le9vy processes stopped at a state-dependent rate, revealing exponential decay rates linked to spectral roots, with applications to economic wealth distribution models.

Contribution

It provides a novel analysis of tail decay rates for stopped Markov-modulated Le9vy processes using spectral methods, with an application to economic modeling.

Findings

Tail probabilities decay exponentially at rates from spectral roots.

Application to stationary wealth distribution in economic models.

Methodology links tail behavior to spectral properties of matrix functions.

Abstract

This article concerns the tail probabilities of a light-tailed Markov-modulated L\'evy process stopped at a state-dependent Poisson rate. The tails are shown to decay exponentially at rates given by the unique positive and negative roots of the spectral abscissa of a certain matrix-valued function. We illustrate the use of our results with an application to the stationary distribution of wealth in a simple economic model in which agents with constant absolute risk aversion are subject to random mortality and income fluctuation.