Lévy processes, generalized moments and uniform integrability
D. Berger
F. Kühn
R. L. Schilling
Abstract:
We give new proofs of certain equivalent conditions for the existence
of generalized moments of a Lévy process (Xt)t ≥0;
in particular, the existence of a generalized g-moment is equivalent to the
uniform integrability of (g(Xt))t ∈ [0,1].
As a consequence, certain functions of a Lévy 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 Le’vy processes.
2010 AMS Mathematics Subject Classification: 60G51, 60G44, 60G40, 26A12, 26B35.
Keywords and phrases: Lévy process, additive process, Dynkin's formula,
generalized moment, Gronwall's inequality, local martingale, condition D, condition DL.
