Investment timing under incomplete information

We study the decision of when to invest in a project whose value is perfectly observable but driven by a parameter that is unknown to the decision-maker ex-ante. This problem is equivalent to an optimal stopping problem for a bivariate Markov process. Using filtering and martingale techniques, we show that the optimal investment region is characterized by a continuous and non-decreasing boundary in the value-belief state space. This generates path-dependency in the optimal investment strategy. We further show that the decision-maker always benefit from an uncertain drift relative to an average drift situation, and that the value of the option to invest is not globally increasing with respect to the volatility of the value process