Probability Theory and Related Fields

, Volume 164, Issue 1–2, pp 361–400 | Cite as

Exact adaptive pointwise drift estimation for multidimensional ergodic diffusions

  • Claudia StrauchEmail author


The problem of pointwise adaptive estimation of the drift coefficient of a multivariate diffusion process is investigated. We propose an estimator which is sharp adaptive on scales of Sobolev smoothness classes. The analysis of the exact risk asymptotics allows to identify the impact of the dimension and other influencing values—such as the geometry of the diffusion coefficient—of the prototypical drift estimation problem for a large class of multidimensional diffusion processes. We further sketch generalizations of our results to arbitrary diffusions satisfying suitable Bernstein-type inequalities.


Ergodic diffusion Minimax drift estimation Exact constants in nonparametric smoothing Sharp adaptivity  Pointwise risk 

Mathematics Subject Classification

62M05 62G07 62G20 



The author is grateful to her Ph.D. advisor, Angelika Rohde, and Enno Mammen for constant encouragement and constructive advise. She also would like to thank two anonymous referees for helpful comments which led to a substantial improvement of the paper. This work was partially supported by the DFG Priority Program SPP 1324 (Project: RO 3766/2-1).


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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Universität HamburgHamburgGermany

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