Limit theorems for discretely observed stochastic volatility models

A general set-up is proposed to study stochastic volatility models. We consider here a two-dimensional diffusion process (Y-t, V-t) and assume that only (Y-t) is observed at n discrete times with regular sampling interval Delta. The unobserved coordinate (V-t) is an ergodic diffusion which rules the diffusion coefficient (or volatility) of (Y-t). The following asymptotic framework is used: the sampling interval tends to 0, while the number of observations and the length of the observation time tend to infinity. We study the empirical distribution associated with the observed increments of (Y-t). We prove that it converges in probability to a variance mixture of Gaussian laws and obtain a central limit theorem. Examples of models widely used in finance, and included in this framework, are given.