Learning from the past in an irreversible investment problem

TL;DR

The paper addresses irreversible investment under incomplete information by introducing learning from the past, where investment actions accelerate the rate of information revelation. It reformulates the problem as a recursively defined stopping problem with a belief process $\Pi_t$ and a sequence of one-sided boundaries $\{b_n\}$ that determine optimal investment times, derived via smooth-fit conditions. The authors prove convexity properties, construct semi-explicit candidate value functions, and establish verification results showing that the optimal policy indeed consists of hitting boundaries at each step, with boundaries computed numerically. Numerical experiments highlight how boundaries respond to learning rate, signal-to-noise ratio, discount rate, and the total number of investment opportunities, revealing a nuanced trade-off between accruing information and immediate earnings. This work advances irreversible investment theory under Bayesian learning by providing a tractable, recursively solvable framework and insights into how learning dynamics shape optimal timing strategies.

Abstract

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.

Figures (8)

• Figure 1: Functions $V_1(\pi)$ and $F_1(\pi)$ , together with their shared upper bound $(1-k)\pi$ and lower bound $(\pi-k)^+$. Parameters used are: $\mu_0=-1$, $\mu_1=1$, $\sigma=4$, $r=0.1$, $N=10$, and $N\varepsilon =1$.
• Figure 2: Functions $V_n(\pi)$, $F_n(\pi)$, and $g_n(\pi)$ for $n=1,2,3$, where we notate $g_1(\pi):=\pi-k$, and boundaries $b_3,b_2,$ and $b_1$. Parameters used are: $\mu_0=-1$, $\mu_1=1$, $\sigma=4$, $r=0.1$, $N=10$, and $N\varepsilon =1$.
• Figure 3: The boundaries $b_{10},\ldots,b_1$ corresponding to the control rates $u_{1},\ldots,u_{10}$. The boundaries $\{b_n\}_{n=2}^{10}$ are solved numerically whereas the boundary $b_1$ is given in \ref{['c']}. Parameters used are: $\mu_0=-1$, $\mu_1=1$, $\sigma=4$, $r=0.1$, $N=10$, and $N\varepsilon =1$. From the choice of $\mu_0$, $\mu_1$, and $\sigma$, the values $k=0.5$ and $\rho=0.5$ follow.
• Figure 4: Comparison of boundaries $b_{10},\ldots,b_1$ with different total learning rates. Parameters used are: $\mu_0=-1$, $\mu_1=1$, $\sigma=4$, $r=0.1$, $N=10$, and $N\varepsilon =1$ and $N\varepsilon=10$, respectively.
• Figure 5: Comparison of boundaries $b_{10},\ldots,b_1$ with different values of $\sigma$. Parameters used are: $\mu_0=-1$, $\mu_1=1$, $r=0.1$, $N=10$, $N\varepsilon =1$, and $\sigma=1$, $\sigma=4$, and $\sigma=10$, respectively corresponding to signal-to-noise ratios of $\rho=2$, $\rho=0.5$, and $\rho=0.2$.

Theorems & Definitions (22)

• Remark 2.1
• Lemma 3.1
• proof
• Remark 3.2
• Remark 3.3
• Lemma 4.1
• proof
• Remark 4.2
• Proposition 4.4
• proof