A Meyer-Itô formula for stable processes via fractional calculus

Abstract

The infinitesimal generator of a one-dimensional strictly alpha -stable process can be represented as a weighted sum of (right and left) Riemann-Liouville fractional derivatives of order alpha and one obtains the fractional Laplacian in the case of symmetric stable processes. Using this relationship, we compute the inverse of the infinitesimal generator on Lizorkin space, from which we can recover the potential if alpha in (0,1) and the recurrent potential if alpha in (1,2). The inverse of the infinitesimal generator is expressed in terms of a linear combination of (right and left) Riemann-Liouville fractional integrals of order alpha . One can then state a class of functions that give semimartingales when applied to strictly stable processes and state a Meyer-Itô theorem with a non-zero (occupational) local time term, providing a generalization of the Tanaka formula given by Tsukada [1]. This result is used to find a Doob-Meyer (or semimartingale) decomposition for |X_t - x|^{gamma } with X a recurrent strictly stable process of index alpha and gamma in (alpha -1,alpha ), generalizing the work of Engelbert and Kurenok [2] to the asymmetric case.