Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**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*