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Irreversible investment under Lévy uncertainty: an equation for the optimal boundary

Part of: Stochastic processes Mathematical economics Stochastic systems and control

Published online by Cambridge University Press:  24 March 2016

Giorgio Ferrari  and
Paavo Salminen
Giorgio Ferrari*
Affiliation:
Bielefeld University
Paavo Salminen*
Affiliation:
Åbo Akademi University
*
* Postal address: Faculty of Science and Engineering, Åbo Akademi University, Fänriksgatan 3 B, FIN-20500 Åbo, Finland. Email address: phsalmin@abo.fi" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink">phsalmin@abo.fi">phsalmin@abo.fi
** Postal address: Center for Mathematical Economics, Bielefeld University, Universitätstrasse 25, 33615 Bielefeld, Germany. Email address: giorgio.ferrari@uni-bielefeld.de
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Abstract

We derive a new equation for the optimal investment boundary of a general irreversible investment problem under exponential Lévy uncertainty. The problem is set as an infinite time-horizon, two-dimensional degenerate singular stochastic control problem. In line with the results recently obtained in a diffusive setting, we show that the optimal boundary is intimately linked to the unique optional solution of an appropriate Bank–El Karoui representation problem. Such a relation and the Wiener–Hopf factorization allow us to derive an integral equation for the optimal investment boundary. In case the underlying Lévy process hits any point in R with positive probability we show that the integral equation for the investment boundary is uniquely satisfied by the unique solution of another equation which is easier to handle. As a remarkable by-product we prove the continuity of the optimal investment boundary. The paper is concluded with explicit results for profit functions of Cobb–Douglas type and CES type. In the former case the function is separable and in the latter case nonseparable.

Keywords

Free-boundaryirreversible investmentsingular stochastic controloptimal stoppingLévy processBank and El Karoui's representation theorembase capacity

MSC classification

Primary: 93E20: Optimal stochastic control
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 60G51: Processes with independent increments; L'evy processes 91B70: Stochastic models
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 1 , March 2016 , pp. 298 - 314
Copyright
Copyright © Applied Probability Trust 2016 

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