An Irreversible Investment Problem with Incomplete Information about Profitability

TL;DR

This paper studies an irreversible investment problem with unknown profitability, where the decision depends on a belief about the profit's drift, and characterizes the optimal investment threshold through a nonlinear integral equation.

Contribution

It introduces a novel analysis of investment under incomplete information with direct payoff dependence on the unknown parameter, deriving structural properties of the optimal threshold.

Findings

Optimal investment threshold depends on current belief about profitability.

Thresholds are monotonic and continuous functions of the belief.

Numerical methods are used to compute the boundary and assess the value of information.

Abstract

We analyze an irreversible investment decision for a project which yields a flow of future operating profits given by a geometric Brownian motion with unknown drift. In contrast to similar optimal stopping problems with incomplete information, the agent's payoff now depends directly on the unknown drift and not only indirectly through the underlying dynamics. Hence, many standard arguments are not applicable. Nonetheless, we show that it is optimal to invest in the project if the current profit level exceeds a threshold depending on the current belief for the true state of the unknown drift. These thresholds are described by a boundary function, for which we establish structural properties like monotonicity and continuity. To prove these, we identify a central class of stopping times with useful features. Moreover, we characterize the boundary function as the unique solution of a nonlinear integral equation. Building on this characterization we compute the boundary function numerically and investigate the value of information.