Published online by Cambridge University Press: 14 July 2016
In this paper, we derive uniform convergence rates of nonparametric estimators for continuous time diffusion processes. In particular, we consider kernel-based estimators of the Nadaraya–Watson type, introducing a new technical device called a damping function. This device allows us to derive sharp uniform rates over an infinite interval with minimal requirements on the processes: The existence of the moment of any order is not required and the boundedness of relevant functions can be significantly relaxed. Restrictions on kernel functions are also minimal: We allow for kernels with discontinuity, unbounded support, and slowly decaying tails. Our proofs proceed by using the covering-number technique from empirical process theory and exploiting the mixing and martingale properties of the processes. We also present new results on the path-continuity property of Brownian motions and diffusion processes over an infinite time horizon. These path-continuity results, which should also be of some independent interest, are used to control discretization biases of the nonparametric estimators. The obtained convergence results are useful for non/semiparametric estimation and testing problems of diffusion processes.
This is a substantially revised version of a chapter in my Ph.D. dissertation at the University of Wisconsin-Madison. I am indebted to my supervisor, Bruce E. Hansen, for his guidance and support. I am very grateful to Dennis Kristensen and Jack R. Porter for their valuable advice and encouragements. I would also like to thank the Editor, Peter C. B. Phillips, the Co-Editor, Oliver B. Linton, and two anonymous referees for their constructive and valuable comments, which have greatly improved the original version of this paper. In particular, I would like to express my sincere gratitude to Professor Phillips for generous support and outstanding editorial input into this paper, which were considerable and far in excess of what I could expect. I also thank Martin Browning, Xiaohong Chen, Valentina Corradi, Bonsoo Koo, Bent Nielsen, Andrew Patton, Olivier Scaillet, Neil Shephard, and seminar participants at University of Wisconsin-Madison and University of Oxford for helpful comments and suggestions. I gratefully acknowledge support from CREATES, Center for Research in Econometric Analysis of Time Series, funded by the Danish National Research Foundation (DNRF78), and from the Danish Council for Independent Research, Social Sciences (grant no. DFF - 4182-00279). Part of this research was conducted while I was visiting the Institute of Economic Research at Kyoto University (under the Joint Research Program of the KIER), the support and hospitality of which are gratefully acknowledged.