The tanaka formula for symmetric stable processes with index α, 0<α<2

For a symmetric stable process Z =(Z_t)_t≥0 of index 0 <α<2, any a ∈ R, and γ ∈ (0, 2) satisfying α − 1 <γ<α, we give the explicit form of the Doob–Meyer decomposition of the submartingale |Z − a|^γ =(|Z_t − a|^γ)_t≥0, which consists of |a|^γ, a stochastic integral with respect to the compensated Poisson random measure associated with Z, and a predictable increasing process. If 1 <α<2, then the case γ = α − 1, corresponding to the famous Tanaka formula, is also considered. This extends results of Salminen and Yor [Tanaka formula for symmetric Lévy processes, in Séminaire de Probabilités XL, Springer, 2007, pp. 265–285] to general indexes 0 <α<2using a different approach. Related works are [H. Tanaka, Z. Wahrsch. Verw. Geb., 1 (1963), pp. 251–257], [P. Fitzsimmons and R. K. Getoor, Ann. Inst. H. Poincaré Probab.Statist., 28 (1992), pp. 311–333], [T. Yamada, Tanaka Formula for Symmetric Stable Processes of Index α, 1 <α<2, manuscript, 1997], and [K. Yamada, Fractional derivatives of local times of α-stable Lévy processes as the limits of occupation time problems, in Limit Theorems in Probability and Statistics, Vol. II, János Bolyai Math. Soc., 2002, pp. 553–573].