High-dimensional stochastic differential equations under sparsity constraints

研究的对象是高维随机微分方程（SDEs）在稀疏约束下泛函的推理和自适应估计。在高维统计问题中（参数维数与样本量相比非常大），参数的可估计性很差。因此，越来越多的兴趣是在泛函的统计推断是可能的。作为一个概念上的新的方面，我们考虑的数学分析的三角形阵列的SDES的扩散系数应该是一个矩阵的秩大大低于的过程中，这是应该增长的时间间隔的观察的长度。这是一个降维或稀疏性假设。在非参数统计学中，人们通常假设，在某些平滑类，例如保持器类型，但这并不一定保证存在一个强的解决方案的考虑中的平滑的漂移。我们的目标是确定的最佳收敛速度的漂移依赖于秩和几何的扩散coeffcient以及建设完全数据驱动的估计的某些泛函。结果应该是离散时间obsen/ations扩展。在高维问题中，算法方面变得越来越重要。特别是，应该制定出使我们的理论发现成为可能的估计程序。

The object are inference and adaptive estimation of functionals of high-dimensional stochasticdifferential equations (SDEs) under so-called sparsity constraints. In high-dimensional statisticalproblems (parameter dimension is very large as compared to the sample size), the parameteris poorly estimable. Hence, increasing interest is in functionals about which statistical inference is possible. As a conceptually new aspect, we consider for the mathematical analysis a triangular array of SDEs where the diffusion coefficient is supposed to be a matrix of substantially lower rank than the dimension of the process, which is supposed to grow with the length of the time interval of observations. This is a dimension reduction or sparsity assumption. One typically assumes in nonparametric statistics that the drift of the SDE belongs to some smoothness class, for instance of the Holder type, which however does not necessarily guarantee that there exists a strong solution of the SDE under consideration. Our goal is to determine the optimal rates of convergence for certain functionals of the drift in dependence of rank and geometry of the diffusion coefftcient as well as the construction of fully data-driven estimators. The results are supposed to be extended for discrete-time obsen/ations. In high-dimensional problems, algorithmic aspects are of increased importance. In particular, estimation procedures shall be worked out which make our theoretical findings possible.