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

TL;DR

This paper introduces a novel application of the Bank and El Karoui Representation Theorem to determine the investment boundary in a complex stochastic capacity expansion problem with irreversible investments and terminal scrap value, providing new insights and unifying existing results.

Contribution

It develops a new method using the Representation Theorem to analyze the investment boundary in a stochastic, time-inhomogeneous setting with terminal scrap value, extending beyond standard variational approaches.

Findings

Existence of a base capacity level $l^{ ext{*}}_y(t)$ that triggers investment.

Unification of the investment boundary with the base capacity in deterministic cases.

Application of the Representation Theorem to a complex singular stochastic control problem.

Abstract

This paper devises a way to apply the Bank and El Karoui Representation Theorem to find the investment boundary of a rich stochastic, continuous time capacity expansion problem with irreversible investment on the finite time interval $[0, T]$, despite the presence of a state dependent scrap value associated with the production facility at the terminal time $T$. Standard variational methods are not feasible for the proposed singular stochastic control problem but it admits some first order conditions, complicated however by an extra, non integral term involving the scrap value function and depending on the initial capacity $y$, which are solved by devising a way to apply the Representation Theorem. Such devise, new and of interest in its own right, provides the existence of the base capacity $l^{\star}_y(t)$, a positive level which the optimal investment process is shown to become active at. As far as we know the Representation Theorem has never been applied to this extent. In the special case of deterministic coefficients, under a further assumption specific to the scrap value case, a unifying view on the curve at which it is optimal to invest emerges: the base capacity equals the investment boundary ${\hat y}(t)$ obtained by variational methods.