Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 60F17 Functional limit theorems; invariance principles
- 60G44 Martingales with continuous parameter

**Résumé:** 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
$V(Y;r,s)^n_t=n^{{r+s\over2}-1}\sum_{i=1}^{[nt]}
|Y_{i\over n}-Y_{i-1\over n}|^r|Y_{i+1\over n}-Y_{i\over n}|^s$,
where $r$ and $s$ are nonnegative reals with $r+s>0$.
We prove that $V(Y;r,s)^n_t$ converges locally uniformly in time, in
probability, to a limiting process $V(Y;r,s)_t$ (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 $\rn~(V(Y;r,s)^n-V(Y;r,s))$
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 $\si$. We also
provide a multivariate version of these results.

**Mots Clés:** *Central limit theorem ; quadratic variation ; bipower variation*

**Date:** 2004-09-07

**Prépublication numéro:** *PMA-927*