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.
Highlights
Diffusion processes are used as theoretical models in analyzing random phenomena evolved in continuous time
These models may be described in terms of Ito ’s type stochastic differential equations dXt A Xt, θ dt σ Xt, θ dWt, 1.1 where Wt t≥0 is a Brownian motion, with some unknown parameters θ to be determined in rational ways
We will assume throughout the paper that W is a one-dimensional Brownian motion, and X is real valued
Summary
Diffusion processes are used as theoretical models in analyzing random phenomena evolved in continuous time. In this paper we consider the linear diffusion approximation proposed by Shoji and Ozaki 6 to the diffusion model 1.3 , which leads to the following approximation of the likelihood function L Xt0 , . 1.11 where tj jT/n so that Xtj is a sample with fixed duration δ tj − tj−1 over 0, T , and hj t, x, y is the probability transition density of the following linear diffusion model dXt A Xtj−1 , θ A Xtj−1 , θ Xt − Xtj−1 dt dWt, 1.12 when tj−1 ≤ t < tj and Xtj−1 Xtj−1. The main goal of the paper is to prove Theorem 3.1 which implies that the local linear approximations 1.12 is efficient for the propose of deriving MLE with discrete samples.
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