Learning from the past in an irreversible investment problem

We consider an irreversible investment problem under incomplete information, where the investor decides whether and when to make investments in a project. Upon investment, the investor acquires previously hidden information from the project's past (''learning from the past''), and so the learning rate of the problem is controlled by investing. We set up this original problem as an recursively defined stopping problem, where the learning rate is accelerated after each recursion step. To solve the problem, we show that at each step, there indeed exists a one-sided stopping boundary under general conditions. We proceed to present the optimal investment strategy as a sequence of semi-explicit stopping boundaries derived from smooth fit conditions. Feasibility of our approach is then demonstrated by solving boundaries numerically and by illustrating comparative statistics.