A Problem of Sequential Entry and Exit Decisions Combined with Discretionary Stopping

We consider a stochastic control problem that has emerged in the economics literature as an investment model under uncertainty. This problem combines features of both stochastic impulse control and optimal stopping. The aim is to discover the form of the optimal strategy. It turns out that this has a priori rather unexpected features. The results that we establish are of an explicit nature. We also construct an example whose value function does not possess C1 regularity.