Maximum likelihood estimates of a class of one-dimensional stochastic differential equation models from discrete data (Q2759335)

This paper deals with some computational aspects in estimation, using maximum likelihood techniques, of the following class of one-dimensional stochastic differential equation models: \(dX_{t}=k(\theta-X_{t})dt+\sigma X_{t}^{\beta}dW_{t}\), where \(\sigma\) is unknown. The case when \(\beta\) is unknown and has to be estimated is also studied. The author considers three approaches to estimation:NEWLINENEWLINENEWLINE1) estimates are obtained by discretizing the explicit expressions for the estimates which maximize the likelihood function in continuous time; 2) estimates are obtained by discretizing the likelihood function through the quadrature approximation before maximizing it; 3) estimates are obtained by maximizing the likelihood function of the Euler scheme approximation to the underlying continuous process.NEWLINENEWLINENEWLINEThe simulation experiments carried out in this paper indicate that the three maximum likelihood estimators considered, which are asymptotically equivalent, have very similar behaviour even for processes observed over a relatively short time interval.