Solving Impulse-Control Problems with Control Delays

Abstract

AbstractSeveral classes of stochastic control problems, wherein the controller can bring about an instantaneous change in state, lead to free boundary problems. Moving boundary methods are a class of computational methods that have been developed recently to solve such free boundary problems. The goal of this paper is to provide a detailed description of the methodology. We specifically focus on stochastic impulse-control problems which arise when the cost of control includes a fixed cost. The inclusion of a fixed cost, very common in financial applications, and makes the control effect finite changes in state, bringing about discontinuities in the state evolution. These problems are, hence, more complicated than problems wherein controls have only proportional costs (singular control) or wherein controls simply terminate the process (optimal stopping). We show how the impulse-control problem is transformed to a Quasi Variational Inequality and then describe the moving boundary method. We demonstrate problems with no control delay, fixed delay and stochastic delay. We also review all the theoretical guarantees that have been established. This paper summarizes and presents an implementation focused description of the research presented in (Feng and Muthuraman, A computational method for stochastic impulse control problems. Mathematics of Operations Research 35(4), 830–850, 2010 and Muthuraman et al., Inventory management with stochastic lead times. Working Paper, 2011) for solving impulse-control problems.KeywordsLead TimeInventory LevelFree Boundary ProblemInventory PositionStochastic Control ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.