Limit Theorems for Discretely Observed Stochastic Volatility Models

Abstract

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 Δ . 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.