A unifying view on the irreversible investment exercise boundary in a stochastic, time-inhomogeneous capacity expansion problem

This paper studies the investment exercise boundary erasing in a stochastic, continuous time capacity expansion problem with irreversible investment on the finite time interval $[0, T]$ and a state dependent scrap value associated with the production facility at the finite horizon $T$. The capacity process is a time-inhomogeneous diffusion in which a monotone nondecreasing, possibly singular, process representing the cumulative investment enters additively. The levels of capacity, employment and operating capital contribute to the firm's production and are optimally chosen in order to maximize the expected total discounted profits. Two different approaches are employed to study and characterize the boundary. From one side, some first order condition are solved by using the Bank and El Karoui Representation Theorem, and that sheds further light on the connection between the threshold which the optimal policy of the singular stochastic control problem activates at and the optional solution of Representation Theorem. Its application in the presence of the scrap value is new. It is accomplished by a suitable fictitious extension to $+ \infty$ of the firm's horizon and a devise to overcome the difficulties due to the presence of a non integral term in the maximizing functional. The optimal investment process is shown to become active at the so-called ``base capacity'' level, which is given as the unique solution of an integral equation. On the other hand, when the coefficients of the uncontrolled capacity process are deterministic, the optimal stopping problem classically associated to the original capacity problem is resumed and, without invoking variational teckniques but only by means of probabilistic methods, some essential properties of the investment exercise boundary, the ``free boundary'' of its continuation region, are obtained. Despite the lack of knowledge of boundary's continuity, the optimal investment process is proved to be continuous, except for a possible initial jump. Finally, unifying approaches and views, the exercise boundary is shown to coincide with the base capacity, and hence it is characterized by an integral equation not requiring any a priori regularity.