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

Nuffield College Economics Working Paper No. 2004-W29

Posted: 6 Dec 2004

See all articles by Ole E. Barndorff-Nielsen

Ole E. Barndorff-Nielsen

University of Aarhus - Thiele Centre, Department of Mathematical Sciences

Svend Erik Graversen

University of Aarhus - Department of Mathematical Sciences

Jean Jacod

Université Paris VI Pierre et Marie Curie

Mark Podolskij

University of Heidelberg - Institute of Applied Mathematics

Neil Shephard

Harvard University

Date Written: November 2004

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

Suggested Citation

Barndorff-Nielsen, Ole E. and Graversen, Svend Erik and Jacod, Jean and Podolskij, Mark and Shephard, Neil, A Central Limit Theorem for Realised Power and Bipower Variations of Continuous Semimartingales (November 2004). Nuffield College Economics Working Paper No. 2004-W29. Available at SSRN: https://ssrn.com/abstract=627068

Ole E. Barndorff-Nielsen

University of Aarhus - Thiele Centre, Department of Mathematical Sciences ( email )

Ny Munkegade
Aarhus, DK 8000
Denmark

Svend Erik Graversen

University of Aarhus - Department of Mathematical Sciences ( email )

DK-8000 Aarhus
Denmark

Jean Jacod

Université Paris VI Pierre et Marie Curie ( email )

4, Place Jussieu, B.P. 169
Laboratoire de Probabilites
F-75252-Paris Cedex 05
France
01 44 27 53 21 (Phone)

Mark Podolskij

University of Heidelberg - Institute of Applied Mathematics ( email )

Grabengasse 1
Heidelberg, 69117
Germany
00496221546276 (Phone)

Neil Shephard (Contact Author)

Harvard University ( email )

1875 Cambridge Street
Cambridge, MA 02138
United States

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