Approximation of the Ornstein-Uhlenbeck local time by harmonic oscillators

Abstract

We consider a particle of mass 1/β submitted to the action of an harmonic oscillator. If we add a white-noise external force, it is well known that the trajectories of the particle, for β tending to infinity, converge to an Ornstein-Uhlenbeck process. Using the number of crossings of the particle with a fixed level u, we construct a consistent estimator of the Ornstein-Uhlenbeck local time, giving an estimate of the speed of this convergence.