An Irreversible Investment Problem with Incomplete Information about Profitability
Fabian Gierens, Berenice Anne Neumann
TL;DR
The paper studies an irreversible investment problem where profits $X_t$ follow a geometric Brownian motion with a drift drawn from a two-state unknown parameter. By introducing the belief process, the authors recast the problem as a two-dimensional Markovian optimal stopping problem with payoff $g(x,\pi)$ and determine the optimal policy as the first entrance into a boundary $D$, which is described by a boundary function $b(\pi)$. They prove that $b$ is decreasing and continuous, bounded between known-drift thresholds $x_0^*$ and $x_1^*$, and they show that $b$ is the unique solution of a nonlinear integral equation, providing a fixed-point scheme for numerical computation. Numerical experiments illustrate how learning speeds up decisions and quantify the value of information, which tends to be modest and declines with higher signal-to-noise ratios.
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.