2022/03/02 by Chiara Amorino, Amorino, Chiara, Arnaud Gloter +1 · 1 citation
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Mathematical Biology Tumor Growth #Statistical Methods and Inference #Statistics Theory (math.ST) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2203.01055
openalex publication_date 2022/03/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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. 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 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.