Contributions to the theory of stochastic convexity and stochastic majorization.

Abstract

This dissertation presents some contributions to the theory of stochastic convexity and stochastic majorization. In the first part of this dissertation, we develop an operator-analytic approach to study temporal stochastic convexity and concavity of Markov processes. We obtain sufficient and necessary conditions for the process {X(t),t ∊ S} which imply that the expectation Ef(X(t)) is a monotone convex (concave) function of t whenever f is a monotone convex (concave) function. Our operator-analytic approach is quite powerful, but not as intuitive as sample path approaches used in other works. However, using it, we can obtain results that we could not obtain otherwise. In particular, we show that a result of Shaked and Shanthikumar is incorrect and we prove two alternative versions of it. In the second part of this dissertation, we discuss some applications of stochastic convexity. Using the F-monotonicity and F-convexity developed in the first part of this dissertation, we obtain relations among the several notions of stochastic convexity and stochastic majorization, and characterize the relationship between stochastic submodularity and stochastic rearrangement. These results generalize and extend several known results in the literature. Applications of our results to stochastic allocation problems are also discussed.