Optimal Stochastic Impulse Control with Delayed Reaction

We study impulse control problems of jump diffusions with delayed reaction. This means that there is a delay δ>0 between the time when a decision for intervention is taken and the time when the intervention is actually carried out. We show that under certain conditions this problem can be transformed into a sequence of iterated no-delay optimal stopping problems and there is an explicit relation between the solutions of these two problems. The results are illustrated by an example where the problem is to find the optimal times to increase the production capacity of a firm, assuming that there are transaction costs with each new order and the increase takes place δ time units after the (irreversible) order has been placed.