Stochastic volatility models as hidden Markov models and statistical applications

This paper deals with the fixed sampling interval case for stochastic volat ility models. We consider a two-dimensional diffusion process (Y-t, V-t), w here only (Y-t) is observed at n discrete times with regular sampling inter val Delta. The unobserved coordinate (V-t) is ergodic and rules the diffusi on coefficient (volatility) of (Y-t). We study the ergodicity and mixing pr operties of the observations (Y-i Delta). For this purpose, we first presen t a thorough review of these properties for stationary diffusions. We then prove that our observations can be viewed as a hidden Markov model and inhe rit the mixing properties of (V-t). When the stochastic differential equati on of (V-t) depends on unknown parameters, we derive moment-type estimators of all the parameters, and show almost sure convergence and a central limi t theorem at rate n(1/2). Examples of models coming from finance are fully treated. We focus on the asymptotic variances of the estimators and establi sh some links with the small sampling interval case studied in previous pap ers.