On sequential estimation of parameters in semimartingale regression models with continuous time parameter

Abstract

We consider the problem of parameter estimation for multidimensional continuous-time linear stochastic regression models with an arbitrary finite number of unknown parameters and with martingale noise. The main result of the paper claims that the unknown parameters can be estimated with prescribed mean-square precision in this general model providing a unified description of both discrete and continuous time process. Among the conditions on the regressors there is one bounding the growth of the maximal eigenvalue of the design matrix with respect to its minimal eigenvalue. This condition is slightly stronger as compared with the corresponding conditions usually imposed on the regressors in asymptotic investigations but still it enables one to consider models with different behavior of the eigenvalues. The construction makes use of a two-step procedure based on the modified least-squares estimators and special stopping rules.