Asymptotic parameter estimation theory for stochastic differential equations.

摘要

We study the asymptotic behaviour of Maximum Likelihood (ML) and Least Squares (LS)-type estimators of parameters from Stochastic Differential Equations(SDE's). First we give a brief review of contributions to the ML theory based on continuous sampling. Then we consider repeated sampling on the time interval (0,T). The observation period T therefore, does not go to infinity as usually is the case, but instead, we allow the number of replications of the process to grow without bound. We prove that this new procedure yields consistent and asymptotically normal estimates. An example is given where the asymptotic covariance matrix is calculated explicitly. In addition, these results are valid not only for independent replications but also for exchangeable interacting systems. An application to testing the hypothesis of non-interaction for this particular example is also considered. Next, we discuss briefly the contribution of Le Breton on discrete sampling.;Finally, we report on simulation studies which yield the empirical Mean Square Errors of the various procedures presented in this study.;On the LS methods, our main contribution is the proof of the strong consistency of the LS estimator for stationary processes. We also consider estimators derived from non-stationary processes. The special case of non-stationary Ornstein-Uhlenbeck process is dealt with in detail and we obtain the strong consistency of the LS estimator.