Optimal L2 -approximation of occupation and local times for symmetric stable processes

The L2-approximation of occupation and local times of a symmetric $\alpha$-stable L{\'e}vy process from high frequency discrete time observations is studied. The standard Riemann sum estimators are shown to be asymptotically efficient when 0 < $\alpha$ $\le$ 1, but only rate optimal for 1 < $\alpha$ $\le$ 2. For this, the exact convergence of the L2-approximation error is proven with explicit constants.