ABSTRACT
This chapter aims to develop a self-contained theory of impulse control for Piecewise-Deterministic Processes. It analyses the methods already developed for the optimal stopping problem to obtain more explicit characterizations of the value function of the impulse control problem, with a view to obtaining insight into the problem as well as an avenue to computational techniques. The main results concern regularity of the value function, existence of optimal strategies, convergence of approximating schemes and characterization of optimality in terms of so-called quasi-variational inequalities. The chapter explores the interpretation of the control problems that is based on the idea of randomized stopping. In all stochastic optimization problems one can think interchangeably in terms of maximizing rewards or minimizing costs. It seems most natural to formulate optimal stopping in the former terms and impulse control in the latter.
