LAMN property for hidden processes: the case of integrated diffusions

A Gloter, E Gobet - Annales de l'IHP Probabilités et statistiques, 2008 - numdam.org
Annales de l'IHP Probabilités et statistiques, 2008numdam.org
In this paper we prove the Local Asymptotic Mixed Normality (LAMN) property for the
statistical model given by the observation of local means of a diffusion process X. Our data
are given by∫ 1 0 X (s+i)/n dμ (s) for i= 0,..., n− 1 and the unknown parameter appears in the
diffusion coefficient of the process X only. Although the data are neither Markovian nor
Gaussian we can write down, with help of Malliavin calculus, an explicit expression for the
log-likelihood of the model, and then study the asymptotic expansion. We actually find that …
Abstract
In this paper we prove the Local Asymptotic Mixed Normality (LAMN) property for the statistical model given by the observation of local means of a diffusion process X. Our data are given by∫ 1
0 X (s+i)/n dμ (s) for i= 0,..., n− 1 and the unknown parameter appears in the diffusion coefficient of the process X only. Although the data are neither Markovian nor Gaussian we can write down, with help of Malliavin calculus, an explicit expression for the log-likelihood of the model, and then study the asymptotic expansion. We actually find that the asymptotic information of this model is the same one as for a usual discrete sampling of X.
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