# 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

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