Date/heure
25 novembre 2021
10:45 - 11:45
Lieu
Salle de conférences Nancy
Oratrice ou orateur
Chiara Amorino (Université du Luxembourg)
Catégorie d'évènement Séminaire Probabilités et Statistique
Résumé
In this talk, we will discuss some results on the estimation of the invariant density associated to a multivariate diffusion X = (Xt)t≥0, assuming that a continuous record of observations (Xt)0≤t≤T is available. We will see that, when X = (Xt)t≥0 is the solution of a stochastic differential equation with Levy-type jumps, it is possible to find the parametric convergence rate 1/T in the monodimensional case and log(T)/T when the dimension d is equal to 2. For d ≥ 3 we find the convergence rate (log(T)/T)γ, where γ is an explicit exponent depending on the dimension d and on β3, the harmonic mean of the smoothness of the invariant density over the d directions after having removed β1 and β2, which are the smallest. Moreover, we obtain a lower bound on the L2-risk for pointwise estimation, with the rate (1/T)γ. In order to fill the logarithmic gap we consider then X = (Xt)t≥0 as a solution to a continuous stochastic differential equation. One (surprising) finding is that the convergence rate depends on the fact that β2 < β3 or β2 = β3. In particular, we show that kernel density estimators achieve the rate (log(T)/T)γ in the first case and (1/T)γ in the second. Finally, we prove a minimax lower bound on the L2-risk for the pointwise estimation with the same rates (log(T)/T)γ or (1/T)γ, depending on the value of β2 and β3.