Optimal Boundary Surface for Irreversible Investment with Stochastic Costs

TL;DR

This paper develops a mathematical framework for determining the optimal irreversible investment boundary in a stochastic environment with uncertain costs, using advanced stochastic control and integral equations.

Contribution

It introduces a novel boundary surface characterization for the optimal control in a complex stochastic model with independent diffusions for market and investment costs.

Findings

Explicit boundary surface derived from nonlinear integral equations

Optimal control expressed as a Skorohod reflection problem

Framework applicable to general convex cost functions

Abstract

This paper examines a Markovian model for the optimal irreversible investment problem of a firm aiming at minimizing total expected costs of production. We model market uncertainty and the cost of investment per unit of production capacity as two independent one-dimensional regular diffusions, and we consider a general convex running cost function. The optimization problem is set as a three-dimensional degenerate singular stochastic control problem. We provide the optimal control as the solution of a Skorohod reflection problem at a suitable boundary surface. Such boundary arises from the analysis of a family of two-dimensional parameter-dependent optimal stopping problems and it is characterized in terms of the family of unique continuous solutions to parameter-dependent nonlinear integral equations of Fredholm type.