Bayesian Optimal Prediction: A Bayesian approach to Optimal Prediction of the ultimate maximum with asymmetric loss functions

In this thesis, we study prediction of the ultimate maximum of a Brownian motion on a finite time horizon with sequential observation of the process. The problem is twofold in the sense that we study prediction of the time at which the ultimate maximum is achieved and prediction of the value of the maximum. We employ a Bayesian decision-theoretic view to find optimal estimators for both the time and space formulation of the problem. In particular, we focus on asymmetric loss functions such as weighted linear, entropy, and LINEX loss. We find Bayes estimators by using methods from the field of optimal stopping. In many cases, the Bayes estimator can be described by an explicit stopping time, while in some cases we resort to investigating free-boundary equations in the form of nonlinear Volterra equations of the second kind. Finally, we compare the estimators from a mean squared error perspective and by investigating the admissibility of estimators. We find that even though the space and time problems are closely related one has to be careful of using rules from one formulation in another since inadmissibility is possible. To our knowledge, the explicit Bayes estimator of the time problem under entropy loss and the proof of an optimal estimator of the maximal value based on the running maximum is completely new. Furthermore, the use of LINEX loss and comparison of estimators are important new contributions to the literature.