In this paper we consider the factor analysis for Lévy-driven multivariate price models with stochastic volatility. Our main aim is to provide conditions on the volatility process under which we can possibly reduce the dimension of the driving Lévy motion. We find that these conditions depend on a particular form of the multivariate Lévy process. In some settings we concentrate on nondegenerate symmetric α-stable Lévy motions.

Many prominent continuous-time stochastic volatility models exhibit certain functional relationships between price jumps and volatility jumps. We show that stochastic volatility models like the Ornstein–Uhlenbeck and other continuous-time CARMA models as well as continuous-time GARCH and EGARCH models all exhibit such functional relations. We investigate the asymptotic behaviour of certain functionals of price and volatility processes for discrete observations of the price process on a grid, which are relevant for estimation and testing problems.

We determine the asymptotic behavior of the realized power variations, and more generally of sums of a given function f evaluated at the increments of a Lévy process between the successive times iΔn for i = 0,1,...,n. One can elucidate completely the first order behavior, that is the convergence in probability of such sums, possibly after normalization and/or centering: it turns out that there is a rather wide variety of possible behaviors, depending on the structure of jumps and on the chosen test function f. As for the associated central limit theorem, one can show some versions of it, but unfortunately in a limited number of cases only: in some other cases a CLT just does not exist.

In this paper we provide an asymptotic analysis of generalized bipower measures of the variation of price processes in financial economics. These measures encompass the usual quadratic variation, power variation, and bipower variations that have been highlighted in recent years in financial econometrics. The analysis is carried out under some rather general Brownian semimartingale assumptions, which allow for standard leverage effects.Ole E. Barndorff-Nielsen's work is supported by the Centre for Analytical Finance (CAF), which is funded by the Danish Social Science Research Council. Neil Shephard's research is supported by the UK's ESRC through the grant “High frequency financial econometrics based upon power variation.” We thank the editor, Peter Phillips, and the referees for their stimulating comments on an earlier version.

We consider a diffusion process X which is observed at times i/n for i = 0,1,...,n, each observation being subject to a measurement error. All errors are independent and centered Gaussian with known variance pn. There is an unknown parameter to estimate within the diffusion coefficient. In this second paper we construct estimators which are asymptotically optimal when the process X is a Gaussian martingale, and we conjecture that they are also optimal in the general case.

We consider a diffusion process X which is observed at times i/n for i = 0,1,...,n, each observation being subject to a measurement error. All errors are independent and centered Gaussian with known variance pn. There is an unknown parameter within the diffusion coefficient, to be estimated. In this first paper the case when X is indeed a Gaussian martingale is examined: we can prove that the LAN property holds under quite weak smoothness assumptions, with an explicit limiting Fisher information. What is perhaps the most interesting is the rate at which this convergence takes place: it is $1/\sqrt{n}$ (as when there is no measurement error) when pn goes fast enough to 0, namely npn is bounded. Otherwise, and provided the sequence pn itself is bounded, the rate is (pn / n)1/4. In particular if pn = p does not depend on n, we get a rate n-1/4.

We show that there exists a pair of Poisson processes which are not stochastically independent, and whose sum is still a Poisson process.