Valentine Genon-Catalot

Stochastic volatility models as hidden Markov models and statistical applications

This paper deals with the fixed sampling interval case for stochastic volatility models. We consider a two-dimensional diffusion process (Y-t, V-t), where only (Y-t) is observed at n discrete times with regular sampling interval Delta. The unobserved coordinate (V-t) is ergodic and rules the diffusion coefficient (volatility) of (Y-t). We study the ergodicity and mixing properties of the observations (Y-i Delta). For this purpose, we first present a thorough review of these properties for stationary diffusions. We then prove that our observations can be viewed as a hidden Markov model and inherit the mixing properties of (V-t). When the stochastic differential equation of (V-t) depends on unknown parameters, we derive moment-type estimators of all the parameters, and show almost sure convergence and a central limit theorem at rate n(1/2). Examples of models coming from finance are fully treated. We focus on the asymptotic variances of the estimators and establish some links with the small sampling interval case studied in previous papers.

Maximnm contrast estimation for diffusion processes from discrete observations

We consider a one-dimensional diffusion process (Xt) with drift b{θ&, u) depending on an unknown parameter # and small known diffusion coefficient , s. The sample path is observed at times k△A, =0, 1,…,.N up to T=NA, for fixed T. We study maximum contrast estimators (m.c.e.) of θ based on this observation with asymptotic results as e and ^ go to 0 simultaneously. We specify conditions on △A under which the ni.e.e. are asymptotically normal and asymptotically equivalent to the maximum Hkelihood estimator of

Penalized nonparametric mean square estimation of the coefficients of diffusion processes

In this paper, we study nonparametric estimation of the Levy density for Levy processes, first without then with Brownian component. For this, we consider 2n (resp. 3n) discrete time observations with step

Adaptive Laguerre density estimation for mixed Poisson models

\Delta

Nonparametric estimation for a stochastic volatility model

. The asymptotic framework is: n tends to infinity,

A non-linear explicit filter

\Delta=\Delta_n

Asymptotic equivalence of nonparametric diffusion and Euler scheme experiments

tends to zero while

Nonparametric Laguerre estimation in the multiplicative censoring model

n\Delta_n

Estimation for stochastic differential equations with mixed effects

tends to infinity. We use a Fourier approach to construct an adaptive nonparametric estimator and to provide a bound for the global L2-risk. Estimators of the drift and of the variance of the Gaussian component are also studied. We discuss rates of convergence and give examples and simulation results for processes fitting in our framework.