Parameter estimation in optional semimartingale regression models

The paper is devoted to the problem of parameter estimation in a multivariate optional semimartingale regression model. The family of optional semimartingales is a rich class of stochastic processes that contains cadlag semimartingales. In general, such processes do not admit cadlag modifications, i.e. right-continuous with finite left-limits. The weighted least squares estimator is derived, and its strong consistency is proved under general conditions on regressors. Furthermore, sequential least squares estimates are systematically studied. It is shown that such estimates have a nice statistical property called fixed accuracy. Sequential estimation procedure developed in the paper works without restrictions on dimensions of unknown parameter and of observation process. The paper contains several examples of multivariate regressions to demonstrate our results and proposed techniques.