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

TL;DR

This paper investigates the efficiency and optimality of L2-approximation methods for occupation and local times of symmetric alpha-stable Lévy processes using high-frequency data, providing explicit convergence results.

Contribution

It establishes the asymptotic efficiency of Riemann sum estimators for 0 < alpha ≤ 1 and their rate optimality for 1 < alpha ≤ 2, with explicit convergence constants.

Findings

Riemann sum estimators are asymptotically efficient for 0 < alpha ≤ 1.

Estimators are rate optimal for 1 < alpha ≤ 2.

Explicit constants for convergence are derived.

Abstract

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.