Fluctuation theory for one-sided L\'evy processes with a matrix-exponential time horizon

TL;DR

This paper extends key fluctuation identities for one-sided Lévy processes from exponential to matrix-exponential time horizons, preserving structure and enabling broader applications including deterministic times.

Contribution

It generalizes fluctuation formulas to matrix-exponential time horizons, replacing scalar parameters with matrices while maintaining core properties.

Findings

Formulas generalize to matrix-exponential horizons

Eigenvalues of matrices lie in the right half-plane

Numerical illustrations demonstrate applicability

Abstract

There is an abundance of useful fluctuation identities for one-sided L\'evy processes observed up to an independent exponentially distributed time horizon. We show that all the fundamental formulas generalize to time horizons having matrix exponential distributions, and the structure is preserved. Essentially, the positive killing rate is replaced by a matrix with eigenvalues in the right half of the complex plane which, in particular, applies to the positive root of the Laplace exponent and the scale function. Various fundamental properties of thus obtained matrices and functions are established, resulting in an easy to use toolkit. An important application concerns deterministic time horizons which can be well approximated by concentrated matrix exponential distributions. Numerical illustrations are also provided.