Complete monotonicity of time-changed L\'evy processes at first passage

TL;DR

This paper characterizes spectrally positive Le9vy processes, time-changed by inverse integral functionals, whose first-passage time Laplace transforms are completely monotone, and identifies a dense subfamily with explicit formulas.

Contribution

It provides a characterization of when time-changed spectrally positive Le9vy processes have completely monotone first-passage time Laplace transforms, including explicit formulas for a dense subfamily.

Findings

Identified conditions for complete monotonicity of first-passage Laplace transforms.

Derived closed-form expressions for a dense subfamily.

Enhanced understanding of time-changed Le9vy processes at first passage.

Abstract

We consider the class of (possibly killed) spectrally positive L\'evy process that have been time-changed by the inverse of an integral functional. Within this class we characterize the family of those processes which satisfy the following property: as functions of point of issue, the Laplace transforms of their first-passage times downwards are completely monotone. A wide (dense, in a sense) subfamily of this family admits closed form expressions for said Laplace transforms.