Moloche, Guillermo (2001): Local Nonparametric Estimation of Scalar Diffusions.

PDF
MPRA_paper_46154.pdf Download (357kB)  Preview 
Abstract
This paper studies the functional estimation of the drift and diffusion functions for recurrent scalar diffusion processes from equally spaced observations using the local polynomial kernel approach. Almost sure convergence and a CLT for the estimators are established as the sampling frequency and the time span go to infinity. The asymptotic distributions follow a mixture of normal laws. This theory covers both positive and null recurrent diffusions.
Almost sure convergence rates are sometimes path dependent but expected rates can always be characterized in terms of regularly varying functions.
The general theory is specialized for positive recurrent diffusion processes, and it is shown in this case that the asymptotic distributions are normal.
We also obtain the limit theory for kernel density estimators when the process is positive recurrent, namely, requiring only that the invariant probability measure exists. Nonetheless, it is also shown that such an estimator paradoxically vanishes almost surely when the invariant measure is fat tailed and nonintegrable, that is, in the null recurrent case.
Item Type:  MPRA Paper 

Original Title:  Local Nonparametric Estimation of Scalar Diffusions 
English Title:  Local Nonparametric Estimation of Scalar Diffusions 
Language:  English 
Keywords:  Nonparametric estimation, Diffusion processes 
Subjects:  C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C14  Semiparametric and Nonparametric Methods: General C  Mathematical and Quantitative Methods > C2  Single Equation Models ; Single Variables > C22  TimeSeries Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes C  Mathematical and Quantitative Methods > C5  Econometric Modeling > C58  Financial Econometrics 
Item ID:  46154 
Depositing User:  Guillermo Moloche 
Date Deposited:  13 Apr 2013 09:59 
Last Modified:  10 Oct 2019 06:29 
References:  F.M. Bandi and G. Moloche (2001) On the functional estimation of multivariate diffusion processes, Working Paper. F.M. Bandi and P.C.B. Phillips (2001) Fully Nonparametric Estimation of Scalar Diffusion Models, Working Paper, Cowles Foundation for Research in Economics. D.A. Darling and M. Kac (1957) On Occupation Times for Markoff Processes, Transactions of the American Mathematical Society, 84, 1957, 444458. K. Ito and H.P. McKean (1974) Diffusion processes and their sample paths, Second printing, corrected, SpringerVerlag, Berlin. J. Fan and I. Gijbels (1996) Local Polynomial Modelling and Its Applications, Chapman & Hall. W. Feller (1971) An introduction to probability theory and its applications. Vol. II. , Second, John Wiley & Sons Inc., New York. D. FlorensZmirou (1993) On Estimating the Diffusion Coefficient from Discrete Observations, Journal of Applied Probability 30, 790804. S.A. Geman (1979) On a comon sense estimator for the drift of a diffusion, Working Paper. I. Karatzas and S. E. Shreve (1991) Brownian Motion and Stochastic Calculus, SpringerVerlag. Y. Kasahara and S. Kotani and H. Watanabe (1980) On the Green Functions of 1Dimensional Diffusion Processes, Publications of the Research Institute for Mathematical Sciences, Kyoto Univ., 175188. Y. Kasahara (1975) Spectral Theory of Generalized Second Order Differential Operators and its Applications to Markov Processes, Japanese Journal of Mathematics 1, 6784. Kutoyants, Y. A. (1997) On density estimation by the observations of ergodic diffusion processes, Statistics and control of stochastic processes (Moscow, 1995/1996), 253274, World Scientific Publishing, River Edge, NJ. G. Moloche (2001) Kernel Regression for NonStationary HarrisRecurrent Processes, Working Paper. P.C.B. Phillips and J.Y. Park, (1999) Nonstationary Kernel Regression and Density Estimation, Working Paper, Cowles Foundation for Research in Economics. H. Tanaka (1958) Certain limit theorems concerning onedimensional diffusion processes, Memoirs of the Faculty of Science Series A Mathematics 12, 111. D. Revuz and M. Yor (1994) Continuous Martingales and Brownian Motion, SpringerVerlag. Yakowitz, S. (1989) Nonparametric density and regression estimation for Markov sequences without mixing assumptions, Journal of Multivariate Analysis 30, 124136. Zirbel, C. L. (1997) Mean occupation times of continuous onedimensional Markov processes, Stochastic Processes and their Applications 69, 161178. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/46154 