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

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 Itô 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 $l^*(t)$, i.e., the unique solution of a representation problem in the spirit of Bank and El Karoui [P. Bank and N. El Karoui, Ann. Probab., 32 (2004), pp. 1030--1067]. We show that the base capacity is deterministic and it is identified with the free boundary $\hat{y}(t)$ 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.