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 assu me that only (Y-t) is observed at n discrete times with regular sampli ng interval Delta. The unobserved coordinate (V-t) is an ergodic diffu sion which rules the diffusion coefficient (or volatility) of (Y-t). T he following asymptotic framework is used: the sampling interval tends to 0, while the number of observations and the length of the observat ion time tend to infinity. We study the empirical distribution associa ted with the observed increments of (Y-t). We prove that it converges in probability to a variance mixture of Gaussian laws and obtain a cen tral limit theorem. Examples of models widely used in finance, and inc luded in this framework, are given.