Fabian Gierens

• We consider irreversible investment problems where an agent decides if and when to invest in a project which generates a perpetual flow of profits after paying some investment cost. The agent observes the profits which follow a geometric Brownian motion with possibly time-dependent drift. Moreover, she discounts future profits as well as the investment cost. In the classical model introduced by Dixit and Pindyck (1994, Chapter 6, Section 1), the drift is constant and known to the agent. Even though this assumption is convenient for their analysis, it is generally unrealistic. In this thesis, we consider three variants of the classical model which turn out to be two-dimensional optimal stopping problems: Unknown drift, a jump of the drift at a fixed time, and a jump of the drift at an exponentially distributed time. For all variants, the optimal investment decision only depends on the current profits and the current state of the respective additional feature, which is either the current updated belief on the true state of the drift, the remaining time until the drift jumps, or the current state of the drift. Precisely, we characterize the optimal stopping time by a continuous and monotonic boundary function mapping the state of the additional feature on the minimal stopping trigger for the profit process. To achieve these results and to derive the boundary function, we employ the following two approaches: For unknown drift and a jump of the drift at a fixed time, we compute the boundary function numerically by a nonlinear integral equation. This approach requires establishing several properties of the boundary function and the value function such as monotonicity and continuity. Since the agent’s payoff not only depends on the current state of the profit process but also on the state of the additional feature, many standard arguments to derive these properties are not applicable. Instead, our proofs build on a method to connect the value functions for different initial profit levels and initial states of the additional feature. Based on this, we show that the value function solves the associated free boundary problem. Next, we derive a nonlinear integral equation whose unique solution is the boundary function. Utilizing this, we compute the boundary function numerically via a fixed-point iteration in combination with a Monte-Carlo method. For a jump of the drift at an exponentially distributed time and in a special case for unknown drift, we determine the boundary function by a nonlinear equation which, for the latter variant, even yields an analytical expression. We reduce both problems to a family of parameterized one-dimensional optimal stopping problems whose optimal thresholds characterize the boundary function. For each sub-problem, we solve the associated free boundary problem and verify that its solution corresponds to the value function. This way, we establish a nonlinear equation for the threshold. Additionally, we numerically analyze the influence of parameter variation on the optimal investment behavior for every variant. Particularly, we analyze the value of information in the case of unknown drift and, for the same expected jump time, compare the models whose drift jumps at a fixed or an exponentially distributed time.