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

This paper devises a way to solve by the Bank and El Karoui Representation Theorem a quite complex stochastic, continuous time capacity expansion problem with irreversible investment on the finite time interval $[0, T]$ in the presence of a state dependent scrap value at the terminal time $T$. Standard variational methods are not feasible but the functional to be maximized admits a supergradient, hence the optimal control satisfies some first order conditions which are solved by means of the Representation Theorem. The devise introduced is new in singular stochastic control and of interest in its own right. As far as we know the Representation Theorem has never been applied to this extent. Contrary to the no scrap value case, a non integral term depending on the initial capacity $y$ arises in the supergradient making non trivial the application of the Representation Theorem and the existence of the so-called base capacity $l^{\star}_y(t)$, a positive level depending on $y$ which the optimal investment process is shown to become active at. In the special case of deterministic coefficients, the base capacity equals the boundary ${\hat y}(t)$ obtained by variational methods but only under a further assumption that allows to obtain monotonicity and positiveness of ${\hat y}(t)$ by means of probabilistic methods. Therefore getting a unifying view on the curve at which is optimal to invest is possible even in the presence of scrap value but it requires adding some extra conditions. The advantage is that the integral equation of the base capacity may then be used to characterize ${\hat y}(t)$.