A Meyer-Itô formula for stable processes via fractional calculus

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