Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### Rate of convergence for parametric estimation in a stochastic volatility model

Auteur(s):

Code(s) de Classification MSC:

• 62G99 None of the above but in this section
• 62F99 None of the above but in this section
• 62M99 None of the above but in this section

Résumé: We consider the following hidden Markov chain problem: estimate the finite-dimensional parameter $\theta$ in the equation $v_t =v_0 + \int_0^t\sigma(\theta,v_s)dW_s + \hbox{drift}$, when we observe discrete data $X_{i/n}$ at times $i=0,\ldots,n$ from the diffusion $X_t =x_0 + \int_0^t v_s dB_s + \hbox{drift}$. The processes $(W_t)_{t \in [0,1]}$ and $(B_t)_{t \in [0,1]}$ are two independent Brownian motions; asymptotics are taken as $n \rightarrow \infty$. This stochastic volatility model has been paid some attention lately, especially in financial mathematics. We prove in this note that the rate $n^{-1/4}$ is a lower bound for estimating $\theta$. This rate is indeed optimal, since Gloter, [5], exhibited $n^{-1/4}$ consistent estimators. This result shows in particular the significant difference between the high frequency data'' framework and stochastic volatility in an ergodic framework (compare Genon-Catalot, Jeantheau and Laredo, [2], [3], [4], and also S\orensen [12]).

Mots Clés: Stochastic volatility models ; Discrete sampling ; High frequency data ; Nonparametric Bayesian estimation

Date: 2001-04-24

Prépublication numéro: PMA-650