L\'evy Processes, Generalized Moments and Uniform Integrability

TL;DR

This paper provides new proofs relating the existence of generalized moments of Lévy processes to uniform integrability, with implications for martingale properties and extensions to additive processes.

Contribution

It introduces novel proofs connecting generalized moments and uniform integrability for Lévy processes, extending results to additive processes and offering new characterizations.

Findings

Existence of generalized g-moments is equivalent to uniform integrability.

Certain integrable local martingales are true martingales.

New proofs for lattice distribution characterization and process transience.

Abstract

We give new proofs of certain equivalent conditions for the existence of generalized moments of a L\'evy process $(X_t)_{t\geq 0}$; in particular, the existence of a generalized $g$-moment is equivalent to the uniform integrability of $(g(X_t))_{t\in [0,1]}$. As a consequence, certain functions of a L\'evy process which are integrable and local martingales are already true martingales. Our methods extend to moments of stochastically continuous additive processes, and we give new, short proofs for the characterization of lattice distributions and the transience of L\'evy processes.