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B1137
**Title: **Estimation of invariant density for a discretely observed diffusion: Impact of the sampling and of the asynchronicity
**Authors: **Arnaud Gloter - Universite d Evry Val d Essonne (France) **[presenting]**

Chiara Amorino - Universite du Luxembourg (Luxembourg)

**Abstract: **The purpose is to estimate in a non-parametric way the density $\pi$ of the stationary distribution of a $d$-dimensional stochastic differential equation $(X_t)_{t \in [0,T]}$, for $d \ge 2$, from the discrete observations of a finite sample $X_{t_0}$,\ldots, $X_{t_n}$ with $0=t_0<t_1<\ldots<t_n=T$. We propose a kernel density estimator, and we study its convergence rates for the pointwise estimation of the invariant density under anisotropic H\"older smoothness constraints. We first find some conditions on the discretization step that ensures it is possible to recover the same rates as if the continuous trajectory of the process was available. As proven recently, such rates are minimax optimal and new in the context of density estimator. If such a condition on the discretization step is not satisfied, we also identify the convergence rate for the estimation of the invariant density. When the data are asynchronous, meaning that different components can be observed at different instants, the computation of the variance of the estimator is more difficult. We find conditions ensuring that this variance is comparable to the one of the continuous case. We also exhibit that the non-synchronicity of the data introduces additional bias terms in the study of the estimator.

Chiara Amorino - Universite du Luxembourg (Luxembourg)