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International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 906846, 26 pages
http://dx.doi.org/10.1155/2011/906846
Research Article

On Local Linear Approximations to Diffusion Processes

1Department of Mathematics, Mathematical Institute, University of Oxford, 24-29 St Giles', Oxford OX1 3LB, UK
2Department of Statistics, SFS, ITCS, East China Normal University, Shanghai 200062, China

Received 27 December 2010; Revised 29 April 2011; Accepted 27 June 2011

Academic Editor: Shyam Kalla

Copyright © 2011 X. L. Duan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Diffusion models have been used extensively in many applications. These models, such as those used in the financial engineering, usually contain unknown parameters which we wish to determine. One way is to use the maximum likelihood method with discrete samplings to devise statistics for unknown parameters. In general, the maximum likelihood functions for diffusion models are not available, hence it is difficult to derive the exact maximum likelihood estimator (MLE). There are many different approaches proposed by various authors over the past years, see, for example, the excellent books and Kutoyants (2004), Liptser and Shiryayev (1977), Kushner and Dupuis (2002), and Prakasa Rao (1999), and also the recent works by Aït-Sahalia (1999), (2004), (2002), and so forth. Shoji and Ozaki (1998; see also Shoji and Ozaki (1995) and Shoji and Ozaki (1997)) proposed a simple local linear approximation. In this paper, among other things, we show that Shoji's local linear Gaussian approximation indeed yields a good MLE.