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
- 62M05 Markov processes: estimation
- 62M15 Spectral analysis

**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