Optimal time to invest under uncertainty with a scale change

In this paper, we investigate the optimal investment problem to maximize expected discounted payoff of a project whose value is given by a product of two processes: a geometric Brownian motion representing continuous fluctuation over time and a Markov process which gives a discontinuous scale change. It turns out that the optimal policy is of threshold type whose thresholds depend on the current state of the Markov process. For 2-state case, the problem can be solved explicitly by using Bellman equation and smooth pasting conditions. On the other hand, the problem becomes much involved when there are multiple states. We exploit the structure of the optimal policy and the form of the value functions which enables us to develop a simple numerical procedure for computing the optimal policy and the value functions.