The Tanaka formula for symmetric stable processes with index $\alpha $, $0<\alpha <2$

For a symmetric stable process $Z=(Z_t)_{t\ge0}$ of index $0<\alpha<2$, any $a\in\mathbf{R}$, and $\gamma\in(0,2)$ satisfying $\alpha-1<\gamma<\alpha$, we give the explicit form of the Doob–Meyer decomposition of the submartingale $|Z-a|^\gamma=(|Z_t-a|^{\gamma})_{t\ge0}$, which consists of $|a|^{\gamma}$, a stochastic integral with respect to the compensated Poisson random measure associated with $Z$, and a predictable increasing process. If $1<\alpha<2$, then the case $\gamma=\alpha-1$, 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 $0<\alpha<2$ using 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 $\alpha$, $1<\alpha<2$, manuscript, 1997], and [K. Yamada, Fractional derivatives of local times of $\alpha$-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].