This paper deals with the ®xed sampling interval case for stochastic volatility models. We consider a two-dimensional diffusion process (Y t , V t ), where only (Y t ) is observed at n discrete times with regular sampling interval Ä. The unobserved coordinate (V t ) is ergodic and rules the diffusion coef®cient (volatility) of (Y t ). We study the ergodicity and mixing properties of the observations (Y iÄ ). For this purpose, we ®rst present a thorough review of these properties for stationary diffusions. We then prove that our observations can be viewed as a hidden Markov model and inherit the mixing properties of (V t ). When the stochastic differential equation of (V t ) depends on unknown parameters, we derive moment-type estimators of all the parameters, and show almost sure convergence and a central limit theorem at rate n 1a2 . Examples of models coming from ®nance are fully treated. We focus on the asymptotic variances of the estimators and establish some links with the small sampling interval case studied in previous papers.

Abstract.  We consider N independent stochastic processes (Xi (t), t ∈  [0,Ti]), i=1,…, N, defined by a stochastic differential equation with drift term depending on a random variable φi. The distribution of the random effect φi depends on unknown parameters which are to be estimated from the continuous observation of the processes Xi. We give the expression of the exact likelihood. When the drift term depends linearly on the random effect φi and φi has Gaussian distribution, an explicit formula for the likelihood is obtained. We prove that the maximum likelihood estimator is consistent and asymptotically Gaussian, when Ti=T for all i and N tends to infinity. We discuss the case of discrete observations. Estimators are computed on simulated data for several models and show good performances even when the length time interval of observations is not very large.