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 XX without Brownian component. The required assumptions on the Lévy process are satisfied by the symmetric stable processes with index in (1,2)(1,2). Based on this construction, the explicit decomposition of F(Xt,t)F(Xt,t) is obtained for FF 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.