On the Optimal Boundary of a Three-Dimensional Singular Stochastic Control Problem Arising in Irreversible Investment

41 Pages Posted: 26 Jun 2014 Last revised: 5 Dec 2014

See all articles by Tiziano De Angelis

Tiziano De Angelis

University of Manchester

Salvatore Federico

University of Milan

Giorgio Ferrari

Bielefeld University - Center for Mathematical Economics

Date Written: June 19, 2014

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 free-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.

Keywords: irreversible investment, singular stochastic control, optimal stopping, free-boundary problems, nonlinear integral equations

JEL Classification: C02, C73, E22, D92

Suggested Citation

De Angelis, Tiziano and Federico, Salvatore and Ferrari, Giorgio, On the Optimal Boundary of a Three-Dimensional Singular Stochastic Control Problem Arising in Irreversible Investment (June 19, 2014). Institute of Mathematical Economics Working Paper No. 509, Available at SSRN: https://ssrn.com/abstract=2458910 or http://dx.doi.org/10.2139/ssrn.2458910

Tiziano De Angelis

University of Manchester ( email )

Oxford Rd. M13 9PL
Manchester
United Kingdom

Salvatore Federico

University of Milan ( email )

Via Festa del Perdono, 7
Milan, 20122
Italy

Giorgio Ferrari (Contact Author)

Bielefeld University - Center for Mathematical Economics ( email )

Postfach 10 01 31
Bielefeld, D-33501
Germany

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