Compatibility Problems of Probability Measures for Stochastic Processes

Abstract

In this thesis, we address three topics in the area of compatibility for probability measures. By "compatibility", we mean the problems concerning the existence of random variables/stochastic processes which generate certain given probability distributions in some predetermined way. First, we study a compatibility problem for distributions on the real line and probability measures on a measurable space. For a given set of probability measures and a corresponding set of probability distributions, we propose sufficient and necessary conditions for the existence of a random variable, such that under each measure, the distribution of this random variable coincides with the corresponding distribution on the real line. Various applications in optimization and risk management are discussed. Secondly, we investigate a compatibility problem involving periodic stationary processes. We consider a family of random locations, called intrinsic location functionals, of periodic stationary processes. We show that the set of all possible distributions of intrinsic location functionals for periodic stationary processes is the convex hull generated by a specific group of distributions. Two special subclasses of these random locations, invariant intrinsic location functionals and first-time intrinsic location functionals, are studied in more detail. Along this direction, we proceed to propose a unified framework for random locations exhibiting some probabilistic symmetries. A theorem of Noether's type is proved, which gives rise to a conservation law describing the change of the density function of a random location as the interval of interest changes. We also discuss the boundary and near boundary behavior of the distribution of the random locations.