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

Randolf Altmeyer (1) , Ronan Le Guével (2, 3)

1 CMS - Faculty of mathematics Centre for Mathematical Sciences [Cambridge]
2 UR - Université de Rennes
3 IRMAR - Institut de Recherche Mathématique de Rennes

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

Mots clés: occupation time; local time; stable process; Lévy process; lower bound

Domaines: Probabilités [math.PR]

Randolf Altmeyer, Ronan Le Guével. Optimal L2 -approximation of occupation and local times for symmetric stable processes. Electronic Journal of Statistics , 2022, 16, pp.2859 - 2883. ⟨10.1214/22-EJS2013⟩. ⟨hal-03324325⟩