Various topics in stochastic control and measure theory: singular stochastic control, optimal stopping and decompositions of measures

Abstract

Chapter II describes my doctoral work on the problem of optimally stopping the exponential of a Brownian bridge which was posed by Ernst and Shepp in their paper [42] and was motivated by bond selling with non-negative prices. We show how to obtain an optimal stopping rule for this problem and we prove regularity of the value function and of the optimal boundary. Chapter III describes my doctoral work on problems of singular control with discretionary stopping. We develop a rigorous probabilistic study for this class of problems which leads to the dynamic programming principle. Chapter IV describes my doctoral work on some decompositions of measures and processes. In particular, we study a decomposition of measures introduced by Dellacherie from which we show how to obtain well-known decompositions of measures and processes and which we extend to vector measures.