Malliavin calculus techniques for local asymptotic mixed normality and their application to degenerate diffusions

TL;DR

This paper develops new Malliavin calculus-based techniques to establish local asymptotic mixed normality for high-frequency observations of degenerate diffusions, even when transition densities have zeros, extending previous results.

Contribution

It introduces tractable sufficient conditions for LAMN in degenerate diffusions using Malliavin calculus without requiring Aronson estimates, applicable to models with zeros in transition densities.

Findings

LAMN property established for hypoelliptic diffusions

Conditions applicable to models with transition density zeros

Extension of results from elliptic to degenerate diffusions

Abstract

We study sufficient conditions for a local asymptotic mixed normality property of statistical models. We develop a scheme with the $L^2$ regularity condition proposed by Jeganathan [\textit{Sankhya Ser. A} \textbf{44} (1982) 173--212] so that it is applicable to high-frequency observations of stochastic processes. Moreover, by combining with Malliavin calculus techniques by Gobet [\textit{Bernoulli} \textbf{7} (2001) 899--912, 2001], we introduce tractable sufficient conditions for smooth observations in the Malliavin sense, which do not require Aronson-type estimates of the transition density function. Our results, unlike those in the literature, can be applied even when the transition density function has zeros. For an application, we show the local asymptotic mixed normality property of degenerate (hypoelliptic) diffusion models under high-frequency observations, in both complete and partial observation frameworks. The former and the latter extend previous results for elliptic diffusions and for integrated diffusions, respectively.