Discrete sampling of an integrated diffusion process and parameter estimation of the diffusion coefficient

Université de Marne-la-Vallée, Équipe d'Analyse et de Mathématiques Appliquées, 5 boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée Cedex 2, France; e-mail: gloter@math.univ-mlv.fr

Received: 6 January 1999
Revised: 5 April 2000

Abstract

Let (X_t) be a diffusion on the interval (l,r) and Δ_n a sequence of positive numbers tending to zero. We define J_i as the integral between iΔ_n and (i + 1)Δ_n of X_s. We give an approximation of the law of (J₀,...,J_n-1) by means of a Euler scheme expansion for the process (J_i). In some special cases, an approximation by an explicit Gaussian ARMA(1,1) process is obtained. When Δ_n = n^-1 we deduce from this expansion estimators of the diffusion coefficient of X based on (J_i). These estimators are shown to be asymptotically mixed normal as n tends to infinity.