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

Maria B. Chiarolla

TL;DR

The paper addresses irreversible investment in a stochastic, time-inhomogeneous capacity expansion problem on a finite horizon with a terminal scrap value. It introduces a novel application of the Bank and El Karoui Representation Theorem to obtain a base capacity process $l^{\star}_y(t)$ and an associated investment strategy, despite a non-integral scrap-term in the first-order conditions. In the deterministic-coefficient setting, the base capacity coincides with the investment boundary ${\hat y}(t)$, yielding a unifying view and enabling boundary characterization via a single integral equation. The results provide regularity properties (monotonicity, positivity, continuity) and a practical numerical route for computing the optimal boundary, with potential applicability to complex production settings.

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.

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

TL;DR

The paper addresses irreversible investment in a stochastic, time-inhomogeneous capacity expansion problem on a finite horizon with a terminal scrap value. It introduces a novel application of the Bank and El Karoui Representation Theorem to obtain a base capacity process and an associated investment strategy, despite a non-integral scrap-term in the first-order conditions. In the deterministic-coefficient setting, the base capacity coincides with the investment boundary , yielding a unifying view and enabling boundary characterization via a single integral equation. The results provide regularity properties (monotonicity, positivity, continuity) and a practical numerical route for computing the optimal boundary, with potential applicability to complex production settings.

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 , despite the presence of a state dependent scrap value associated with the production facility at the terminal time . 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 , 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 , 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 obtained by variational methods.
Paper Structure (6 sections, 16 theorems, 97 equations)

This paper contains 6 sections, 16 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. The Model
  3. Solving by the Bank-El Karoui Representation Theorem approach
  4. The case of deterministic coefficients
  5. A unifying view on the optimal investment boundary
  6. Variational approach review

Key Result

Proposition 2.2

There exists a constant $K_{\cal J}$ depending on $T, \kappa_L, \kappa_w, \kappa_K, \kappa_r, \kappa_f, k_f$ only such that

Theorems & Definitions (19)

  • Remark 2.1
  • Proposition 2.2
  • Remark 2.3
  • Proposition 3.1
  • Proposition 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Proposition 3.5
  • Corollary 3.6
  • Theorem 3.7
  • ...and 9 more