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 $\delta>0$ between the time when a decision for intervention is taken and the time when the intervention is actually carried out. We show that this problem can be transformed into a no-delay impulse control problem and there is an explicit relation between the solutions of these two problems. A similar connection is obtained in the case when only a given finite number of interventions is allowed. In this case the problem can be transformed into a sequence of iterated no-delay optimal stopping 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 $\delta$ time units after the (irreversible) order has been placed