Reflected diffusions and applications to finance and operations management

Abstract

This dissertation provides explicit solutions to four special stochastic optimal control problems for reflected diffusions and Markov modulated reflected diffusions. The main mathematical tool that we use is the ergodic theory for stochastic differential equations (SDE’s), in particular, the ergodic theory for reflected diffusions. First, we present some basic definitions of reflected diffusions, the Itô’s formulas. After this we study the ergodic theorems, namely, the law of large numbers, for reflected diffusions. The reflected diffusions and Markov modulated reflected diffusions are used to simulate bounded price dynamics regulated by policy makers. Motivated by this, we analyze several stochastic optimal control problems for reflected diffusions (or for Markov modulated reflected diffusions). These stochastic optimal control problems are solved in the sense that we can reduce them to some explicit optimization problems. Numerical computations are also given for the reduced optimization problems. Our work resulted in one accepted paper, one completed paper and two on-going projects.