Maximum Likelihood Estimations of SDE Dynamics Based on Discrete Time Data How well does the Euler Method Perform?

This paper employs maximum likelihood (ML) estimations to obtain parameters for stochastic differential equations (SDE). Three discretization methods for approximating SDE solutions are applied in the maximum likelihood estimations: the Euler method, the Milstein method and the Ozaki method. A ML estimation based on continuous time data serves as benchmark model for the theoretical treatment of the SDE parameter estimation. It can be approximated by the ML estimation using the Euler method as the observation steps become finer and finer. The performances of the ML estimations using the three discretization methods are compared and evaluated by using the example of \ a SDE model for the short-term-interest-rate. As an evaluation criterion we take the errors of the one-step-ahead predictions. We show that the predictions of the Euler method and of the Ozaki method are equivalent in estimating the parameters of the SDE process of the short-time-interest-rate. Numerically the magnitude of the prediction errors of the Euler method and the Milstein method are quite similarly. As it turns out the Euler method is not inferior to the other two methods for our chosen performance criterion.