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

Dalalyan, Arnak; Yoshida, Nakahiro

Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 3, pp. 748-789.

Dans cet article, nous considérons le problème d'estimation de la covariation de deux processus de diffusion observés de façon asynchrone. Nous nous plaçons dans le cadre présenté dans [Bernoulli 11 (2005) 359-379, Ann. Inst. Statist. Math. 60 (2008) 367-406] et établissons un développement asymptotique au second ordre de la loi de l'estimateur de Hayashi-Yoshida. Ce développement est valable pour les drifts aléatoires non-anticipatifs et pour des pas d'échantillonnage irréguliers, éventuellement aléatoires, mais indépendant des processus observés. L'approche utilisée pour obtenir les principaux résultats peut être décomposée en trois étapes. La première consiste à établir un développement au second-ordre de la loi de l'estimateur dans le cadre gaussien. La deuxième est l'obtention d'une décomposition stochastique de l'estimateur lui-même et la dernière est l'évaluation de la covariance de Malliavin. A titre d'exemple, nous calculons les constantes du développement au second ordre dans le cas où l'échantillonnage est obtenu par deux processus de Poisson indépendants.

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 [Bernoulli 11 (2005) 359-379, Ann. Inst. Statist. Math. 60 (2008) 367-406], 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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