Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### Dynamics adaptive estimation of a scalar diffusion

Auteur(s):

Code(s) de Classification MSC:

• 62M05 Markov processes: estimation
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.