UC Berkeley

This thesis analyzes a class of impulse control problems for multi-dimensional jump diffusions in a finite time horizon. Following the basic mathematical setup from Stroock and Varadhan, this paper first establishes rigorously an appropriate form of the Dynamic Programming Principle (DPP). It then shows that the value function is a viscosity solution for the associated Hamilton-Jacobi-Belleman (HJB) equation involving integro-differential operators. Finally, it proves the regularity of the viscosity solution for HJB with first-order jump diffusions. Furthermore, it proposes a new regularity framework for second-order jump diffusions in the optimal stopping problem.