Local time-space calculus for symmetric Lévy processes

We construct a stochastic calculus with respect to the local time process of a symmetric Lévy process X without Brownian component. The required assumptions on the Lévy process are satisfied by the symmetric stable processes with index in ( 1 , 2 ) . Based on this construction, the explicit decomposition of F ( X t , t ) is obtained for F continuous function admitting a Radon–Nikodym derivative ∂ F ∂ t and satisfying some integrability condition. This Itô formula provides, in particular, the precise expression of the martingale and the continuous additive functional present in Fukushima’s decomposition.