LAMN property for hidden processes: the case of integrated diffusions

TL;DR

This paper establishes the LAMN property for a diffusion process observed through local means, showing that the asymptotic information matches that of standard discrete sampling, despite the data's non-Markovian and non-Gaussian nature.

Contribution

The paper proves the LAMN property for integrated diffusion observations using Malliavin calculus, providing explicit likelihood expressions and asymptotic analysis.

Findings

Asymptotic information matches standard discrete sampling

Explicit likelihood expression derived using Malliavin calculus

LAMN property established for non-Markovian, non-Gaussian data

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 $ \int_0^1 X_{\frac{s+i}{n}} \dd \mu (s)$ for $i=0,...,n-1$ and the unknown parameter appears in the diffusion coefficient of the process $X$ only. Although the data are nor Markovian neither 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 $X$.