Impulse Control Problems under Non-Constant Volatility

The objective of this dissertation is to study impulse control problems in situations where the volatility of the underlying process is not constant. First, we explore the case where the dynamics of the underlying process are modified for a fixed (or random with known probability distribution) period of time after each intervention of the impulse control. We propose a modified intervention operator to be used in the Quasi-Variational Inequalities approach for solving impulse control problems, and we formulate and prove a verification theorem for finding the Value Function of the problem and the optimal control. Secondly, we use a perturbation approach to tackle impulse control problems when the volatility of the underlying process is stochastic but mean-reverting. The perturbation method permits to approximate the Value Function and the parameters of the optimal control. Finally, we present a numerical scheme to obtain solutions to impulse control problems with constant and stochastic volatility. Throughout the thesis we find explicit solutions to practical applications in financial mathematics; specifically, in optimal central bank intervention of the exchange rate and in optimal policy dividend payments.