On local linear approximations to diffusion processes
Summary: 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 of Kutoyants (2004; Zbl 1038.62073), Liptser and Shiryayev (1977; Zbl 0364.60004), Kushner and Dupuis (2001; Zbl 0968.93005), and Prakasa Rao (1999; Zbl 0952.62077), and also the recent works by Aït-Sahalia (1999; Zbl 1013.91505), and P. A. Mykland (2004; Zbl 1062.62155), (2002; Zbl 1104.62323), and so forth. Shoji and Ozaki (1998; Zbl 0912.60078) 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.
scientific article; zbMATH DE number 5964016
scientific article