# A central limit theorem for realised power and bipower variations of continuous semimartingales

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1 Dept. of Mathematical Science 2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires 3 Dept. of Probability and Statistics 4 Nuffield College

Abstract : Consider a semimartingale of the form $Y t=Y 0+\int 0^ta sds+\int 0^t\si {s-}~dW s$, where $a$ is a locally bounded predictable process and $\si$ the ``volatility- is an adapted right-continuous process with left limits and $W$ is a Brownian motion. We define the realised bipower variation process $VY;r,s^n t=n^{{r+s\over2}-1}\sum {i=1}^{nt} |Y {i\over n}-Y {i-1\over n}|^r|Y {i+1\over n}-Y {i\over n}|^s$, where $r$ and $s$ are nonnegative reals with $r+s>0$. We prove that $VY;r,s^n t$ converges locally uniformly in time, in probability, to a limiting process $VY;r,s t$ the -bipower variation process-. If further $\si$ 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 $ n~VY;r,s^n-VY;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 $\si$. We also provide a multivariate version of these results.

Keywords : bipower variation Central limit theorem quadratic variation

Author: ** Ole Barndorff-Nielsen - Svend Erik Graversen - Jean Jacod - Mark Podolskij - Neil Shephard - **

Source: https://hal.archives-ouvertes.fr/