A Computational Method for Stochastic Impulse Control Problems

We consider the instantaneous control of a diffusion process on the real line. Two types of costs are incurred. The holding cost rate, incurred at all times, is modeled by a convex function. Transactions costs have both fixed and proportional components, making it an impulse control problem. The objective is to minimize the expected infinite horizon discounted cost. The solution to a quasi-variational inequality, which takes the form of a free-boundary problem, can be shown to be the optimal solution. We develop a methodology that converts the free-boundary problem into a sequence of fixed boundary problems. We show that the arising sequence is monotonic and converges. Provided the converged solution is C¹, we show its optimality. We also provide an epsilon-optimality result. Finally, we illustrate a couple of popular applications of this model.