Rate of estimation for the stationary distribution of jump-processes over anisotropic holder classes
We study the problem of the non-parametric estimation for the density - of the stationary distribution of the multivariate stochastic differential equation with jumps (X), when the dimension d is such that d - 3. From the continuous observation of the sampling path on [0,T]we show that, under anisot...
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| Format: | Article |
| Status: | Published version |
| Publication date: | 2021 |
| Country: | España |
| Institution: | Universitat Pompeu Fabra |
| Repository: | Repositorio Digital de la UPF |
| OAI Identifier: | oai:repositori.upf.edu:10230/71772 |
| Online Access: | http://hdl.handle.net/10230/71772 http://dx.doi.org/10.1214/21-EJS1913 |
| Access Level: | Open access |
| Keyword: | Convergence rate Density estimation Ergodic diffusion with jumps Lévy driven SDE Non-parametric statistics |
| Summary: | We study the problem of the non-parametric estimation for the density - of the stationary distribution of the multivariate stochastic differential equation with jumps (X), when the dimension d is such that d - 3. From the continuous observation of the sampling path on [0,T]we show that, under anisotropic Hölder smoothness constraints, kernel based estimators can achieve fast convergence rates. In particular, they are as fast as the ones found by Dalalyan and Reiss [11] for the estimation of the invariant density in the case without jumps under isotropic Hölder smoothness constraints. Moreover, they are faster than the ones found by Strauch [32] for the invariant density estimation of continuous stochastic differential equations, under anisotropic Hölder smoothness constraints. Furthermore, we obtain a minimax lower bound on the L2-risk for pointwise estimation, with the same rate up to a log(T ) term. It implies that, on a class of diffusions whose invariant density belongs to the anisotropic Holder class we are considering, it is impossible to find an estimator with a rate of estimation faster than the one we propose. |
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