On Local Linear Approximations to Diffusion Processes
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 par...
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| Format: | Article |
| Language: | English |
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Wiley
2011-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2011/906846 |
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| author | X. L. Duan Z. M. Qian W. A. Zheng |
| author_facet | X. L. Duan Z. M. Qian W. A. Zheng |
| author_sort | X. L. Duan |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-a9d8be68782c439b9a993eb8420f93de |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2011-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-a9d8be68782c439b9a993eb8420f93de2025-02-03T05:58:48ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252011-01-01201110.1155/2011/906846906846On Local Linear Approximations to Diffusion ProcessesX. L. Duan0Z. M. Qian1W. A. Zheng2Department of Mathematics, Mathematical Institute, University of Oxford, 24-29 St Giles', Oxford OX1 3LB, UKDepartment of Mathematics, Mathematical Institute, University of Oxford, 24-29 St Giles', Oxford OX1 3LB, UKDepartment of Statistics, SFS, ITCS, East China Normal University, Shanghai 200062, ChinaDiffusion 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.http://dx.doi.org/10.1155/2011/906846 |
| spellingShingle | X. L. Duan Z. M. Qian W. A. Zheng On Local Linear Approximations to Diffusion Processes International Journal of Mathematics and Mathematical Sciences |
| title | On Local Linear Approximations to Diffusion Processes |
| title_full | On Local Linear Approximations to Diffusion Processes |
| title_fullStr | On Local Linear Approximations to Diffusion Processes |
| title_full_unstemmed | On Local Linear Approximations to Diffusion Processes |
| title_short | On Local Linear Approximations to Diffusion Processes |
| title_sort | on local linear approximations to diffusion processes |
| url | http://dx.doi.org/10.1155/2011/906846 |
| work_keys_str_mv | AT xlduan onlocallinearapproximationstodiffusionprocesses AT zmqian onlocallinearapproximationstodiffusionprocesses AT wazheng onlocallinearapproximationstodiffusionprocesses |