Suprema of L\'evy processes with completely monotone jumps: spectral-theoretic approach

TL;DR

This paper develops spectral-theoretic methods to analyze the transition densities and extremal functionals of one-dimensional Lévy processes with completely monotone jumps, including stable and Brownian-influenced processes.

Contribution

It provides integral formulas and eigenfunction expansions for transition densities and extremal distributions of these Lévy processes, extending known results to a broader class.

Findings

Derived integral expressions for transition densities of killed Lévy processes.

Established eigenfunction expansions under regularity and growth conditions.

Obtained formulas for distributions of process extrema, including supremum and infimum.

Abstract

We study spectral-theoretic properties of non-self-adjoint operators arising in the study of one-dimensional L\'evy processes with completely monotone jumps with a one-sided barrier. With no further assumptions, we provide an integral expression for the bivariate Laplace transform of the transition density $p_t^+(x, y)$ of the killed process in $(0, \infty)$, and under a minor regularity condition, a generalised eigenfunction expansion is given for the corresponding transition operator $P_t^+$. Assuming additionally appropriate growth of the characteristic exponent, we prove a generalised eigenfunction expansion of the transition density $p_t^+(x, y)$. Under similar conditions, we additionally show integral formulae for the cumulative distribution functions of the infimum and supremum functionals $\underline{X}_t$ and $\overline{X}_t$. The class of processes covered by our results include many stable and stable-like L\'evy processes, as well as many processes with Brownian components. Our results recover known expressions for the classical risk process, and provide similar integral formulae for some other simple examples of L\'evy processes.