Treffer: Sharp adaptive drift estimation and Donsker-type theorems for multidimensional ergodic diffusions ; Scharf adaptive Driftschätzung und Theoreme vom Donsker-Typus für mehrdimensionale ergodische Diffusionen

We consider two problems concerning nonparametric estimation and weak convergence properties of ergodic multidimensional diffusion processes. The problem of sharp adaptive estimation of the drift of a diffusion process, given as a strong solution of an Itô stochastic differential equation, is investigated first. Assuming that a continuous record of observations is available, exact data-driven procedures both for global and pointwise estimation are proposed, attaining the optimal constant under natural smoothness conditions on the drift. Both a connection to the classical Pinsker result on estimation with respect to L2 risk over Sobolev classes and to the problem of optimal recovery are established. The sharp results in particular allow to evaluate the influence of the diffusion matrix. In the second part of the thesis, we study infinite-dimensional extensions of the CLT for additive functionals of ergodic diffusions. Having introduced the notions of Donsker classes and pregaussianness in the context of weak convergence of empirical processes of diffusions, classical results from empirical process theory are revisited. Various parallels to the classical empirical process based on i.i.d. observations are shown to hold for diffusions satisfying a Poincaré inequality. We further establish increased regularity for multivariate ergodic diffusions with finite invariant measure. The effect is disclosed by investigating smoothed versions of the empirical diffusion process. ; In dieser Dissertation werden zwei Problemkreise für multivariate ergodische Diffusionsprozesse betrachtet. Zunächst untersuchen wir die Frage der scharf adaptiven Schätzung der Driftfunktion einer Diffusion, die als starke Lösung einer stochastischen Differentialgleichung gegeben ist. Auf der Grundlage einer stetigen Aufzeichnung von Beobachtungen werden exakte, datenbasierte Verfahren für die globale und für die punktweise Schätzung vorgeschlagen, die die optimale Grenzkonstante unter natürlichen Glattheitsannahmen an den Driftkoeffizienten ...