Pokern, Yvo, Papaspiliopoulos, Omiros, Roberts, Gareth O. and Stuart, A. M. (2009) Non parametric Bayesian drift estimation for one-dimensional diffusion processes. Working Paper. University of Warwick. Centre for Research in Statistical Methodology, Coventry.

Preview

PDF WRAP_Pokern_09-29w.pdf - Published Version - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader Download (1175Kb)

Abstract

We consider diffusions on the circle and establish a Bayesian estimator for the drift function based on observing the local time and using Gaussian priors. Given a standard Girsanov likelihood, we prove that the procedure is well-defined and that the posterior enjoys robustness against small deviations of the local time. A simple method for estimating the local time from high-frequency discrete time observations yielding control of the L2 error is proposed. Complemented by a finite element implementation this enables error-control for a fixed random sample all the way from high-frequency discrete observation to the numerical computation of the posterior mean and covariance. An empirical Bayes procedure is suggested which allows automatic selection of the smoothness of the prior in a given family. Some numerical experiments extend our observations to subsets of the real line other than circles and exhibit more probabilistic convergence properties such as rates of posterior contraction.

Item Type:	Working or Discussion Paper (Working Paper)
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics Faculty of Science > Statistics
Library of Congress Subject Headings (LCSH):	Diffusion processes, Bayesian statistical decision theory
Series Name:	Working papers
Publisher:	University of Warwick. Centre for Research in Statistical Methodology
Place of Publication:	Coventry
Date:	2009
Volume:	Vol.2009
Number:	No.29
Number of Pages:	48
Status:	Not Peer Reviewed
Access rights to Published version:	Open Access
References:	[1] G.O. Roberts A. Beskos, O. Papaspiliopoulos. Retrospective exact simulation of diffusion sample paths with applications. Bernoulli, 12(6):1077–1098, 2006. [2] R. A. Adams. Sobolev Spaces. Academic Press, 1975. [3] Federico M. Bandi and Peter C. B. Phillips. Fully nonparametric estimation of scalar diffusion models. Econometrica, 71(1):241–283, 2003. [4] Alexandros Beskos, Omiros Papaspiliopoulos, Gareth O. Roberts, and Paul Fearnhead. Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes. J. R. Stat. Soc. Ser. B Stat. Methodol., 68(3):333–382, 2006. With discussions and a reply by the authors. [5] J. P. N. Bishwal. Parameter estimation in stochastic differential equations, volume 1923 of Lecture Notes in Mathematics. Springer, Berlin, 2008. [6] V. I. Bogachev. Gaussian Measures. AMS, 1998. [7] D. Braess. Finite Elemente, Schnelle L¨oser und Anwendungen in der Elastizit ¨atstheorie. Springer, 1997. [8] H. Br´ezis. Analyse fonctionelle – Th´eorie et applications. Dunod, 1999. [9] Fabienne Comte, Valentine Genon-Catalot, and Yves Rozenholc. Penalized nonparametric mean square estimation of the coefficients of diffusion processes. Bernoulli, 13(2):514–543, 2007. [10] B.R.Brooks et al. Charmm: A program for macromolecular energy, minmization and dynamics calculations. J. Comp. Chem., 4:187–217, 1983. [11] L. C. Evans. Partial Differential Equations. AMS, 1998. [12] A. W. van der Vaart F. H. van der Meulen and J. H. van Zanten. Convergence rates of posterior distributions for Brownian semimartingale models. Bernoulli, 12(5):863– 888, 2006. [13] J. Zabczyk G. Da Prato. Stochastic Equations in Infinite Dimensions. CUP, 1992. [14] A. van der Vaart H. v. Zanten. Rates of contraction of posterior distributions based on gaussian process priors. Annals of Statistics, 36:1435–1463, 2008. [15] W. Hackbusch. Elliptic differential equations : theory and numerical treatment. Springer, 1992. [16] J. Jacod. Rates of convergence to the local time of a diffusion. Annales de L’institut Henri Poincar´e, Probabilit´es et statistique, 34:505–544, 1998. [17] S. Chib O. Elerian and N. Shephard. Likelihood inference for discretely observed nonlinear diffusions. Econometrica, 69(4):959–993, 2001. [18] N. G. Polson and G. O. Roberts. Bayes factors for discrete observations from diffusion processes. Biometrika, 81(1):11–26, 1994. [19] J.N. Reddy. An Introduction to the Finite Element Method. McGraw-Hill, 1984. [20] G. O. Roberts and O. Stramer. On inference for partially observed nonlinear diffusion models using the metropolis-hastings algorithm. Biometrika, 88(3):603–621, 2001. [21] J. C. Robinson. Infinite-dimensional Dynamical Systems. CUP, 2001. [22] T. Schlick. Molecular Modeling and Simulation, an Interdisciplinary Guide. Springer, New York, 2002. [23] Y.A.Kutoyants. Statistical Inference for Ergodic Diffusion Processes. Springer, 2004.
URI:	http://wrap.warwick.ac.uk/id/eprint/35217

Request changes to a record

Actions (login required)

View Item