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

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• dc.contributor.author Amorino, Chiara
• dc.contributor.author Gloter, Arnaud
• dc.date.accessioned 2025-11-04T17:27:35Z
• dc.date.available 2025-11-04T17:27:35Z
• dc.date.issued 2023
• dc.date.updated 2025-11-04T17:27:34Z
• dc.description.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.
• dc.description.sponsorship The author gratefully acknowledges financial support of ERC Consolidator [grant number 815703] "STAMFORD: Statistical Methods for High Dimensional Diffusions".
• dc.format.mimetype application/pdf
• dc.identifier.citation Amorino C, Gloter A. Estimation of the invariant density for discretely observed diffusion processes: impact of the sampling and of the asynchronicity. Statistics. 2023;57(1):213-59. DOI: 10.1080/02331888.2023.2166047
• dc.identifier.doi http://dx.doi.org/10.1080/02331888.2023.2166047
• dc.identifier.issn 0233-1888
• dc.identifier.uri http://hdl.handle.net/10230/71768
• dc.language.iso eng
• dc.publisher Taylor & Francis
• dc.relation.ispartof Statistics. 2023;57(1):213-259
• dc.relation.projectID info:eu-repo/grantAgreement/EC/H2020/815703
• dc.rights 2023 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
• dc.rights.accessRights info:eu-repo/semantics/openAccess
• dc.rights.uri http://creativecommons.org/licenses/by/4.0/
• dc.subject.keyword Non-parametric estimation
• dc.subject.keyword Stationary measure
• dc.subject.keyword Discrete observation
• dc.subject.keyword Convergence rate
• dc.subject.keyword Ergodic diffusion
• dc.subject.keyword Anisotropic density estimation
• dc.subject.keyword Asynchronous framework
• dc.title Estimation of the invariant density for discretely observed diffusion processes: impact of the sampling and of the asynchronicity
• dc.type info:eu-repo/semantics/article
• dc.type.version info:eu-repo/semantics/publishedVersion