Matrix Representations for Scale Functions of Spectrally Negative L\'evy Processes with Rational Jumps

TL;DR

This paper extends the matrix representation of scale functions for spectrally negative Lévy processes from phase-type to matrix-exponential jumps using a novel embedding into rational arrival process-modulated fluid processes.

Contribution

It introduces a new embedding approach into rational arrival process-modulated fluid processes, enabling explicit scale function formulas for general matrix-exponential jumps.

Findings

Provides explicit formulas for scale functions in the general case.

Demonstrates the utility of orbit process representations beyond phase-type distributions.

Recovers known results in the phase-type special case.

Abstract

For a spectrally negative L\'evy process with Laplace transform $\psi$, the $q$-scale function is characterized as the function whose Laplace transform is $(\psi(\cdot)-q)^{-1}$. It has applications in fluctuation theory, for example, exit problems and first hitting probabilities. It is also used in areas like ruin theory, risk theory, continuous state branching processes and optimal control. In this paper, we extend the scale function representation of Ivanovs (2021) from spectrally negative L\'evy processes with phase-type jumps to the general case of matrix-exponential jumps. The extension is non-trivial because the probabilistic arguments employed by Ivanovs rely on an embedding to a Markov-modulated Brownian motion, a framework that does not accommodate the algebraic generality of matrix-exponential distributions. We overcome this limitation by embedding the L\'evy process into a stochastic fluid process modulated by a rational arrival process (RAP), a class of continuous-valued Markov processes driven by orbit processes. This approach yields iterative schemes related to those of Ivanovs (2021) to provide a simple and explicit formula for the scale function. Our method gives the same fixed point when restricted to the phase-type case, and demonstrates the utility of orbit representations in analytical problems beyond the phase-type setting.