Template-type: ReDIF-Paper 1.0
Author-Name: Ole Barndorff-Nielsen 
Author-Workplace-Name: University of Aarhus
Author-Name: Svend Erik Graversen
Author-Workplace-Name: University of Aarhus
Author-Name: Jean Jacod
Author-Workplace-Name: Universtie P. et M. Curie
Author-Name: Mark Podolskij
Author-Workplace-Name: Ruhr University of Bochum
Author-Name: Neil Shephard
Author-Email: neil.shephard@nuffield.ox.ac.uk
Author-Workplace-Name: Nuffield College, University of Oxford, UK
Title: A Central Limit Theorem for Realised Power and Bipower Variations of Continuous Semimartingales
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.
Length: 35 pages
Creation-Date:2004-11-01
Number:2004-W29
File-URL: http://www.nuff.ox.ac.uk/economics/papers/2004/W29/BN-G-J-P-S_fest.pdf
File-Format: application/pdf
Handle: RePEc:nuf:econwp:0429