We consider nonparametric invariant density and drift estimation for a class of multidimensional degenerate resp. hypoelliptic diffusion processes, so-called stochastic damping Hamiltonian systems or kinetic diffusions, under anisotropic smoothness assumptions on the unknown functions. The analysis is based on continuous observations of the process, and the estimators' performance is measured in terms of the sup-norm loss. Regarding invariant density estimation, we obtain highly nonclassical results for the rate of convergence, which reflect the inhomogeneous variance structure of the process. Concerning estimation of the drift vector, we suggest both non-adaptive and fully data-driven procedures. All of the aforementioned results strongly rely on tight uniform moment bounds for empirical processes associated to deterministic and stochastic integrals of the investigated process, which are also proven in this paper.
翻译:我们认为,在对未知功能的异常平稳假设下,对某类多维退化性累赘、低弹性扩散过程、所谓的肢解性阻断汉密尔顿系统或动能扩散过程的不参数性密度和漂移估计值,非参数性密度和漂移估计值是非参数性的,根据对未知功能的异常平稳假设进行的分析,根据对过程的连续观察,对估计值的性能进行测量。关于不变化性密度估计,我们从趋同速度中获得了高度非经典的结果,这反映了过程的不均匀差异结构。关于对漂移矢量的估计,我们建议采用不适应性和完全以数据为驱动的程序。上述所有结果都非常依赖与调查过程的确定性和随机性组成部分相关的经验性过程的严格统一时间界限,本文也证明了这一点。