L\'evy Processes, Generalized Moments and Uniform Integrability

David Berger; Franziska Kühn; René L. Schilling

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arXiv · 2102.09004

L\'evy Processes, Generalized Moments and Uniform Integrability

David Berger·Franziska Kühn·René L. Schilling

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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.

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David Berger, Franziska Kühn, René L. Schilling. 2021-02-17. L\'evy Processes, Generalized Moments and Uniform Integrability. https://arxiv.org/abs/2102.09004

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