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 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 Aı̈t-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.