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May 2024 Malliavin calculus techniques for local asymptotic mixed normality and their application to hypoelliptic diffusions
Masaaki Fukasawa, Teppei Ogihara
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Bernoulli 30(2): 983-1006 (May 2024). DOI: 10.3150/23-BEJ1621

Abstract

We study sufficient conditions for a local asymptotic mixed normality property of statistical models. We accommodate the framework of Jeganathan [Sankhyā Ser. A 44 (1982) 173–212] to a triangular array of variable dimension to, in particular, treat high-frequency observations of stochastic processes. When observations are smooth in the Malliavin sense, with the aid of Malliavin calculus techniques by Gobet [Bernoulli 7 (2001) 899–912], we further give tractable sufficient conditions which do not require Aronson-type estimates of the transition density function. The transition density function is even allowed to have zeros. For an application, we prove the local asymptotic mixed normality property of 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.

Funding Statement

The second author was supported partially by Japan Science and Technology Agency, PRESTO Grant Number JPMJPR15E2, and partially by Japan Society for he Promotion of Science KAKENHI Grant Number 19K14604.

Acknowledgements

The authors would like to thank the anonymous referees, an Associate Editor and the Editor for their constructive comments that improved the quality of this paper.

Citation

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Masaaki Fukasawa. Teppei Ogihara. "Malliavin calculus techniques for local asymptotic mixed normality and their application to hypoelliptic diffusions." Bernoulli 30 (2) 983 - 1006, May 2024. https://doi.org/10.3150/23-BEJ1621

Information

Received: 1 November 2020; Published: May 2024
First available in Project Euclid: 31 January 2024

Digital Object Identifier: 10.3150/23-BEJ1621

Keywords: Hypoelliptic diffusion processes , integrated diffusion processes , L2 regularity condition , local asymptotic mixed normality , Malliavin calculus , partial observations

Vol.30 • No. 2 • May 2024
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