Identifying the Free Boundary of a Stochastic, Irreversible Investment Problem via the Bank-El Karoui Representation Theorem

TL;DR

This paper characterizes the free boundary in a finite-horizon stochastic investment problem using the Bank-El Karoui representation, linking it to an optimal stopping boundary and providing explicit solutions in special cases.

Contribution

It introduces a novel approach to identify the free boundary in finite-horizon singular control problems via the Bank-El Karoui representation theorem, with explicit solutions for specific models.

Findings

The base capacity process is deterministic and equals the free boundary.

The free boundary can be characterized by an integral equation.

Explicit solutions are obtained for the infinite horizon case with Cobb-Douglas production.

Abstract

We study a stochastic, continuous time model on a finite horizon for a firm that produces a single good. We model the production capacity as an Ito diffusion controlled by a nondecreasing process representing the cumulative investment. The firm aims to maximize its expected total net profit by choosing the optimal investment process. That is a singular stochastic control problem. We derive some first order conditions for optimality and we characterize the optimal solution in terms of the base capacity process, i.e. the unique solution of a representation problem in the spirit of Bank and El Karoui (2004). We show that the base capacity is deterministic and it is identified with the free boundary of the associated optimal stopping problem, when the coefficients of the controlled diffusion are deterministic functions of time. This is a novelty in the literature on finite horizon singular stochastic control problems. As a subproduct this result allows us to obtain an integral equation for the free boundary, which we explicitly solve in the infinite horizon case for a Cobb-Douglas production function and constant coefficients in the controlled capacity process.