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 $\left(Y_t\right)$ is observed at $n$ discrete times with regular sampling interval $\Delta$. The unobserved coordinate $\left(V_t\right)$ is an ergodic diffusion which rules the diffusion coefficient (or volatility) of $\left(Y_t\right)$. 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 $\left(Y_t\right)$. 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.