"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,