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

Aiming at studying the investment exercise boundary, this paper devises a way to apply the Bank and El Karoui Representation Theorem to a quite general stochastic, continuous time capacity expansion problem with irreversible investment on the finite time interval and including a state dependent scrap value associated with the production facility at the terminal time T. The capacity process is a time-inhomogeneous diffusion in which a monotone non-decreasing, possibly singular, control process representing the cumulative investment enters additively. The functional to be maximized admits a supergradient, hence the optimal control satisfies some first order conditions which are solved by means of the Bank and El Karoui Representation Theorem. Its application in the case of non-zero scrap value at time T is not obvious and, as far as we know, it is new in the literature on singular stochastic control. In fact, due to the scrap value, in the supergradient appears also a non integral term. This challenge is overcome by suitably extending the horizon. The optimal investment process is shown to become active at the so-called base capacity level, given in terms of the optional solution of the Representation Theorem. Contrary to what happens in the no scrap value case, here the base capacity depends on the initial capacity y. Hence, a priori, it is not clear if and how it is related to the investment exercise boundary associated to the capacity expansion problem. Under the assumption of deterministic coefficients, discount factor, conversion factor, wage rate and interest rate, the investment boundary is shown to coincide with the base capacity. Therefore, unifying views, the base capacity is deterministic and independent of y, and its integral equation may be used to characterize the investment boundary, without any a priori regularity of it.