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

### Stochastic volatility and fractional Brownian motion

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 study a discretization problem associated to fractional Brownian motion (FBM). If $(W_t^H)_{t \in [0,1]}$ is a standard FBM with Hurst parameter $H\in [\frac{1}{2},1)$, we first consider the convergence problem for a suitably normalized quadratic variation related to some approximation of the sampled values $W^H_{i/N}$, $i=1,\ldots, N$. We obtain a convergence result with speed $N^{-1/2}$ as the sampling frequency $N \rightarrow \infty$. Extensions are proposed when replacing $W^H_t$ by $X_t=f(W_t^H)$ and, more generally, for $X$ solution of the stochastic differential equation $$\d X_t=b(X_t)\,\d W_t^H + \mu(X_t)\, \d t,\;t\in [0,1].$$ Our method relies on suitable approximation of the FBM quadratic variation together with expansions of the FBM paths in the Schauder basis. \\ In a second part, we can apply this result to the following hidden-Markov statistical problem: We consider a diffusion model of the form $$\nonumber \d Y_t=\sigma_t \,\d W_t, \;t\in [0,1],$$ where the driving Brownian motion $(W_t)_{t \in [0,1]}$ is independent of the stochastic volatility $$\sigma_t=\sigma_0+\Phi(\theta, W_t^H).$$ By taking the Hurst parameter $H$ lie in $(\frac{1}{2},1)$, we can model long range dependence in the volatility. Based on discrete data $Y_{i/n},i=1,\ldots,n$, we prove that, for a regular parametrization $\theta \mapsto \Phi(\theta,\cdot)$, the rate $n^{-1/(4H+2)}$ is optimal for estimating the one-dimensional parameter $\theta$, as the number of observations $n \rightarrow \infty$. We propose a family of rate optimal contrast estimators. Some consequences of this result are: the qualitative difference of this stochastic volatility Model compared to other related hidden Markov chain problems; the influence of the Hurst parameter $H$ in the estimation problem when the sampling frequency $n$ is high.

Mots Clés: Stochastic volatility models ; discrete samplings; high frequency data ; fractional Brownian motion ; contrast estimators

Date: 2002-07-04

Prépublication numéro: PMA-746

Pdf file : PMA-746.pdf