This paper deals with the ®xed 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 Ä. The unobserved coordinate (V t ) is ergodic and rules the diffusion coef®cient (volatility) of (Y t ). We study the ergodicity and mixing properties of the observations (Y iÄ ). For this purpose, we ®rst 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 1a2 . Examples of models coming from ®nance 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.

Abstract.  We consider N independent stochastic processes (Xi (t), t ∈  [0,Ti]), i=1,…, N, defined by a stochastic differential equation with drift term depending on a random variable φi. The distribution of the random effect φi depends on unknown parameters which are to be estimated from the continuous observation of the processes Xi. We give the expression of the exact likelihood. When the drift term depends linearly on the random effect φi and φi has Gaussian distribution, an explicit formula for the likelihood is obtained. We prove that the maximum likelihood estimator is consistent and asymptotically Gaussian, when Ti=T for all i and N tends to infinity. We discuss the case of discrete observations. Estimators are computed on simulated data for several models and show good performances even when the length time interval of observations is not very large.

We consider a one-dimensional diffusion process (Xt) which is observed at n + 1 discrete times with regular sampling interval ∆. Assuming that (Xt) is strictly stationary, we propose nonparametric estimators of the drift and diffusion coefficients obtained by a penalized least squares approach. Our estimators belong to a finite-dimensional function space whose dimension is selected by a data-driven method. We provide non-asymptotic risk bounds for the estimators. When the sampling interval tends to zero while the number of observations and the length of the observation time interval tend to infinity, we show that our estimators reach the minimax optimal rates of convergence. Numerical results based on exact simulations of diffusion processes are given for several examples of models and illustrate the qualities of our estimation algorithms. This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2007, Vol. 13, No. 2, 514-543. This reprint differs from the original in pagination and typographic detail.

In this paper, we study nonparametric estimation of the Lévy density for pure jump Lévy processes. We consider n discrete time observations with step ∆. The asymptotic framework is: n tends to infinity, ∆ = ∆n tends to zero while n∆n tends to infinity. First, we use a Fourier approach ("frequency domain"): this allows to construct an adaptive nonparametric estimator and to provide a bound for the global L 2-risk. Second, we use a direct approach ("time domain") which allows to construct an estimator on a given compact interval. We provide a bound for L 2-risk restricted to the compact interval. We discuss rates of convergence and give examples and simulation results for processes fitting in our framework.

This paper is concerned with nonparametric estimation of the Lévy density of a pure jump Lévy process. The sample path is observed at n discrete instants with fixed sampling interval. We construct a collection of estimators obtained by deconvolution methods and deduced from appropriate estimators of the characteristic function and its first derivative. We obtain a bound for the L 2 -risk, under general assumptions on the model. Then we propose a penalty function that allows to build an adaptive estimator. The risk bound for the adaptive estimator is obtained under additional assumptions on the Lévy density. Examples of models fitting in our framework are described and rates of convergence of the estimator are discussed. November 2, 2018

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In this paper, we study nonparametric estimation of the Lévy density for Lévy processes, with and without Brownian component. For this, we consider n discrete time observations with step ∆. The asymptotic framework is: n tends to infinity, ∆ = ∆n tends to zero while n∆n tends to infinity. We use a Fourier approach to construct an adaptive nonparametric estimator of the Lévy density and to provide a bound for the global L 2 -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.where b = EL 1 has a statistical meaning (contrary tob).In Section 2, we present our main assumptions and some preliminary properties. In Section 3, we assume that σ = 0 and study the estimation of the function h(x) = x 2 n(x). Using a sample of size 2n, we build two collections of estimators (ĥ m ,h m ) m>0 indexed by a cut-off parameter m. The ESTIMATION FOR LÉVY PROCESSES 3 collections are obtained by Fourier inversion of two different estimators of the Fourier transform h * of the function h. The estimators of h * are built using empirical estimators of the characteristic function ψ ∆ and its first two derivatives. First, we give a bound for the L 2 -risk of (ĥ m ,h m ) for fixed m. Then, introducing an adequate penalty, we propose a data-driven choice of the cut-off parameter which yields an estimator (ĥm,hm) for each collection. The L 2 -risk of these estimators is studied. We discuss the rates of convergence reached on Sobolev classes of regularity for the function h. In Section 4, we consider the general case. To reach the Lévy density and get rid of the unknown σ 2 , we must now use derivatives of ψ ∆ up to the order 3 and we estimate the function p(x) = x 3 n(x) developing the Fourier inversion approach and adaptive choice of the cut-off parameter as for h. It is worth stressing that the point of view of small sampling interval is crucial to our study. Indeed, it helps obtaining simple estimators of ψ ∆ and its successive derivatives which are used to estimate the Fourier transform p * of p. Section 5 is devoted to the estimation of (b, σ). We study classical empirical means of the observations. This gives an estimator of b but cannot give estimators of σ. To estimate σ, we consider power variation estimators, introduced in Woerner (2006), Barndorff-Nielsen, Shephard and Winkel (2006), Jacod (2007), Aït-Sahalia and Jacod (2007), under the asymptotic framework of high frequency data within a long time interval. In Section 6, we give examples of Lévy models satisfying our set of assumptions. We provide numerical simulation results in Section 7. Section 8 contains the main proofs. In the Appendix, two classical results, used in proofs, are recalled.

This paper deals with parameter estimation for stochastic volatility models. We consider a twodimensional diffusion process (Y t , V t). Only (Y t) is observed at n discrete times with a regular sampling interval. The unobserved coordinate (V t) rules the diffusion coef®cient (volatility) of (Y t) and is an ergodic diffusion depending on unknown parameters. We build estimators of the parameters present in the stationary distribution of (V t), based on appropriate functions of the observations. Consistency is proved under the asymptotic framework that the sampling interval tends to 0, while the number of observations and the length of the observation time tend to in®nity. Asymptotic normality is obtained under an additional condition on the rate of convergence of the sampling interval. Examples of models from ®nance are treated, and numerical simulation results are given.