Gaussian Bridges : Modeling and Inference

Abstract: This thesis consists of a summary and five papers, dealing with the modeling of Gaussian bridges and membranes and inference for the α-Brownian bridge.In Paper I we study continuous Gaussian processes conditioned that certain functionals of their sample paths vanish. We deduce anticipative and non-anticipative representations for them. Generalizations to Gaussian random variables with values in separable Banach spaces are discussed. In Paper II we present a unified approach to the construction of generalized Gaussian random fields. Then we show how to extract different Gaussian processes, such as fractional Brownian motion, Gaussian bridges and their generalizations, and Gaussian membranes from them.In Paper III we study a simple decision problem on the scaling parameter in α-Brownian bridges. We generalize the Karhunen-Loève theorem and obtain the distribution of the involved likelihood ratio based on Karhunen-Loève expansions and Smirnov's formula. The presented approach is applied to a simple decision problem for Ornstein-Uhlenbeck processes as well. In Paper IV we calculate the bias of the maximum likelihood estimator for the scaling parameter and propose a bias-corrected estimator. We compare it with the maximum likelihood estimator and two alternative Bayesian estimators in a simulation study. In Paper V we solve an optimal stopping problem for the α-Brownian bridge. In particular, the limiting behavior as α tends to zero is discussed.