High-dimensional stochastic differential equations under sparsity constraints

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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.