On the parabolic generator of a general one-dimensional Lévy process. (English) Zbl 1191.60063

Eisenbaum, Nathalie; Kyprianou, Andreas

Electron. Commun. Probab. 13, 198-209 (2008).

Summary: The purpose of this note is twofold. Firstly to complete a recent accumulation of results concerning extended version of Ito’s formula for any one dimensional Lévy processes, \(X\). Secondly, we use the latter to characterise the parabolic generator of \(X\), \({\mathbf A}:=\{(f,g): f(X_\cdot,\cdot)- \int_0^\cdot g(X_s,s)\,ds\}\) is a local martingale. We also establish a necessary condition for a pair of functions to be in the domain of the parabolic generator when \(X\) has a Gaussian component.

MSC:

60G51	Processes with independent increments; Lévy processes
60G44	Martingales with continuous parameter
60H05	Stochastic integrals
60J55	Local time and additive functionals

Keywords:

stochastic calculus; local time-space; Itô formula; parabolic generator