Estimation of the invariant density for discretely observed diffusion processes: impact of the sampling and of the asynchronicity

Abstract

We aim at estimating in a non-parametric way the density π of the stationary distribution of a d-dimensional stochastic differential equation (𝑋𝑡)𝑡∈[0,𝑇], for 𝑑≥2, from the discrete observations of a finite sample 𝑋𝑡0,…, 𝑋𝑡𝑛 with 0=𝑡0<𝑡1<⋯<𝑡𝑛=:𝑇𝑛. We propose a kernel density estimator and we study its convergence rates for the pointwise estimation of the invariant density under anisotropic Hölder smoothness constraints. First of all, we 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 in the recent work [Amorino C, Gloter A. Minimax rate of estimation for invariant densities associated to continuous stochastic differential equations over anisotropic Holder classes; 2021. arXiv preprint arXiv:2110.02774], such rates are optimal and new in the context of density estimator. Then we deal with the case where such a condition on the discretization step is not satisfied, which we refer to as the intermediate regime. In this new regime we identify the convergence rate for the estimation of the invariant density over anisotropic Hölder classes, which is the same convergence rate as for the estimation of a probability density belonging to an anisotropic Hölder class, associated to n iid random variables 𝑋1,…,𝑋𝑛. After that we focus on the asynchronous case, in which each component can be observed at different time points. Even if the asynchronicity of the observations complexifies the computation of the variance of the estimator, we are able to 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.