A Central Limit Theorem for Realised Power and Bipower Variations of Continuous Semimartingales

Ole BARNDORFF--NIELSEN,
Department of Mathematical Sciences,
University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark

Svend Erik GRAVERSEN,
Department of Mathematical Sciences,
University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark

Jean JACOD,
 Laboratoire de Probabilites et
 Modeles Aleatoires (CNRS UMR 7599)
 Universite P. et M. Curie, 4 Place Jussieu,
 75 252 - Paris Cedex,
 France.

Mark PODOLSKIJ,
Dept. of Probability and Statistics,
Ruhr University of Bochum,
Universitatstrasse 150, 44801 Bochum, Germany,

Neil SHEPHARD,
Nuffield College, Oxford OX1 1NF, UK.

Abstract

Consider a semimartingale of the form Y_{t}=Y_0+\int _0^{t}a_{s}ds+\int _0^{t}_{s-} dW_{s}, where a is a locally bounded predictable process and  (the "volatility") is an adapted right--continuous process with left limits and W is a Brownian motion. We define the realised bipower variation process V(Y;r,s)_{t}^n=n^{((r+s)/2)-1} \sum_{i=1}^{[nt]}|Y_{(i/n)}-Y_{((i-1)/n)}|^{r}|Y_{((i+1)/n)}-Y_{(i/n)}|^{s}, where r and s are nonnegative reals with r+s>0. We prove that V(Y;r,s)_{t}n converges locally uniformly in time, in probability, to a limiting process V(Y;r,s)_{t} (the "bipower variation process"). If further  is a possibly discontinuous semimartingale driven by a Brownian motion which may be correlated with W and by a Poisson random measure, we prove a central limit theorem, in the sense that \sqrt(n) (V(Y;r,s)^n-V(Y;r,s)) converges in law to a process which is the stochastic integral with respect to some other Brownian motion W', which is independent of the driving terms of Y and \sigma. We also provide a multivariate version of these results.

Keywords: Central limit theorem, quadratic variation, bipower variation,

 

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