Second-order asymptotic expansion for a non-synchronous covariation estimator

Arnak S. Dalalyan 1, 2 Nakahiro Yoshida 3
1 IMAGINE [Marne-la-Vallée]
LIGM - Laboratoire d'Informatique Gaspard-Monge, CSTB - Centre Scientifique et Technique du Bâtiment, ENPC - École des Ponts ParisTech
Abstract : In this paper, we consider the problem of estimating the covariation of two diffusion processes when observations are subject to non-synchronicity. Building on recent papers \cite{Hay-Yos03, Hay-Yos04}, we derive second-order asymptotic expansions for the distribution of the Hayashi-Yoshida estimator in a fairly general setup including random sampling schemes and non-anticipative random drifts. The key steps leading to our results are a second-order decomposition of the estimator's distribution in the Gaussian set-up, a stochastic decomposition of the estimator itself and an accurate evaluation of the Malliavin covariance. To give a concrete example, we compute the constants involved in the resulting expansions for the particular case of sampling scheme generated by two independent Poisson processes.
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Arnak S. Dalalyan, Nakahiro Yoshida. Second-order asymptotic expansion for a non-synchronous covariation estimator. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, Institute Henri Poincaré, 2011, 47 (3), pp.748-789. ⟨10.1214/10-AIHP383⟩. ⟨hal-00270287v2⟩

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