Local asymptotic quadraticity of statistical experiments connected with a Heston model

TL;DR

This paper investigates the local asymptotic properties of likelihood ratios in Heston models, revealing different behaviors in subcritical, critical, and supercritical regimes, and establishing optimal testing and estimation results.

Contribution

It provides a comprehensive analysis of local asymptotic properties of Heston models, including proofs of normality, quadraticity, and mixed normality, along with optimal testing and estimation strategies.

Findings

Local asymptotic normality in subcritical case

Local asymptotic quadraticity in critical case

Absence of local asymptotic quadraticity in supercritical case

Abstract

We study local asymptotic properties of likelihood ratios of certain Heston models. We distinguish three cases: subcritical, critical and supercritical models. For the drift parameters, local asymptotic normality is proved in the subcritical case, only local asymptotic quadraticity is shown in the critical case, while in the supercritical case not even local asymptotic quadraticity holds. For certain submodels, local asymptotic normality is proved in the critical case, and local asymptotic mixed normality is shown in the supercritical case. As a consequence, asymptotically optimal (randomized) tests are constructed in cases of local asymptotic normality. Moreover, local asymptotic minimax bound, and hence, asymptotic efficiency in the convolution theorem sense are concluded for the maximum likelihood estimators in cases of local asymptotic mixed normality.