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:**

- 62M05 Markov processes: estimation
- 60J60 Diffusion processes, See also {58G32}

**Résumé:** Let $(X_t)_{t\geq 0}$ be a weak solution of the one-dimensional
stochastic differential equation:
$$
dX_t=f(X_t)dt+\sigma(X_t)dW_t.
$$
The drift and diffusion coefficient $(f,\sigma)$ may vary in a wide
class $\Sigma$ of function described in
terms of the dynamics of the solution that include
positive recurrence, null recurrence and even transience.
From the continuous observation of the trajectory up to time $T$, we
construct an estimator of $f$ at any given point $x_0$
and prove that if $f$ has H\"older smoothness locally around $x_0$, the error normalized by a random factor related to the local time of $X$ at level $x_0$ is
tight, uniformly over $\Sigma$. The behaviour of the random factor
simultaneously depends on the dynamics of $X$ and on the local
smoothness properties of $f$ at $x_0$. Under fairly general conditions, we show that the
local time of $X$ at level $x_0$ has asymptotically deterministic behaviour, which implies
the optimality of our estimator in a local minimax sense.

**Mots Clés:** *diffusion processes ; local time ; nonparametric estimation ;
Nadaraya-Watson estimator ; random bandwidth ; random normalizing factor*

**Date:** 2002-10-18

**Prépublication numéro:** *PMA-762*

**Pdffile : **PMA-762.pdf