A Computational Method for Stochastic Impulse Control Problems
32 Pages Posted: 26 Sep 2010 Last revised: 2 Oct 2010
Date Written: September 26, 2010
We consider the instantaneous control of a diffusion process on the real line. Two types of costs are incurred: the holding and transaction costs. The holding cost is incurred at all times at the rate modeled by a convex function of the state. Transactions costs have both a fixed component and a proportional component, making it an impulse control problem. The objective is to choose a control policy that minimizes the expected infinite horizon discounted costs. The solution to a Quasi-Variational-Inequality (QVI) can be shown to be the optimal solution to this impulse control problem. The QVI takes the form of a free-boundary ordinary differential equation (ODE) wherein the boundaries of the domain in which the ODE needs to be solved are themselves unknown. In this paper we develop a methodology that converts the free-boundary problem into a sequence of fixed boundary problems. We show that the arising sequence has monotonically improving solutions and that the sequence converges. Provided the converged solution is C1, we show that it is the optimal solution and that the optimal policy takes the form of a control band policy which has a simple and intuitive representation. We also provide an ǫ-optimality result that provides an upper bound on the error when the sequence is terminated after convergence to within a tolerance. Finally we illustrate a couple of popular applications of this model in finance and operations management.
Keywords: Impulse Control, Free boundary problems
JEL Classification: C61
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