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

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

Nonparametric estimation of scalar diffusions based on low frequency data is ill-posed

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Résumé: We study the problem of estimating the coefficients of a diffusion $(X_{t}, t \geq 0)$; the estimation is based on discrete data $X_{n\Delta}, n=0,1,\ldots, N$. The sampling frequency $\Delta^{-1}$ is constant, and asymptotics are taken as the number of observations tends to infinity. We prove that the problem of estimating both the diffusion coefficient -- the volatility -- and the drift in a nonparametric setting is ill-posed: The minimax rates of convergence for Sobolev constraints and squared-error loss coincide with that of a respectively first and second order linear inverse problem. To ensure ergodicity and limit technical difficulties we restrict ourselves to scalar diffusions living on a compact interval with reflecting boundary conditions. An important consequence of this result is that we can characterize quantitatively the difference between the estimation of a diffusion in the low frequency sampling case and inference problems in other related frameworks: nonparametric estimation of a diffusion based on continuous or high frequency data, but also parametric estimation for fixed $\Delta$, in which case $\sqrt{N}$-consistent estimators usually exist. Our approach is based on the spectral analysis of the associated Markov semigroup. A rate-optimal estimation of the coefficients is obtained via the nonparametric estimation of an eigenvalue-eigenfunction pair of the transition operator of the discrete time Markov chain $(X_{n\Delta},n=0,1,\ldots,N)$ in a suitable Sobolev norm, together with an estimation of its invariant density.

Mots Clés: Diffusion processes ; nonparametric estimation ; discrete sampling ; low frequency data ; spectral approximation ; ill-posed problems

Date: 2002-07-04

Prépublication numéro: PMA-747

Pdf file : PMA-747.pdf