Three Problems in Stochastic Control and Applications

This thesis mainly concludes three different projects that I am devoted to: Recombining Tree Approximations for Optimal Stopping for Diffusions(Chapter 2), Continuity of Utility Maximization under Weak Convergence(Chapter 3) and Disorder Detection with Costly Observations(Chapter 4). The first two projects are related work. The third one is based on [16]. In Chapter 2,we develop two numerical methods for optimal stopping in the framework of one dimensional diffusion. Both of the methods use the Skorohod embedding in order to construct recombining tree approximations for diffusions with general coefficients. This technique allows us to determine convergence rates and construct nearly optimal stopping times which are optimal at the same rate. Finally, we demonstrate the efficiency of our schemes on several models. In Chapter 3, we find sufficient conditions for the continuity of the utility maximization problem from terminal wealth under convergence in distribution of the underlying processes. We provide several examples which illustrate that without these conditions, we cannot generally expect continuity to hold. Finally, we apply our continuity results to numerical computations of the shortfall risk in the Heston model. In Chapter 4, we study the Wiener disorder detection problem where each observation is associated with a positive cost. In this setting, a strategy is a pair consisting of a sequence of observation times and a stopping time corresponding to the declaration of disorder. We characterize the minimal cost of the disorder problem with costly observations as the unique fixed point of a certain jump operator, and we determine the optimal strategy.PhDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/166133/1/guojia_1.pd