Advances in Applied Probability

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

Giorgio Ferrari and Paavo Salminen

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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.

Article information

Source
Adv. in Appl. Probab., Volume 48, Number 1 (2016), 298-314.

Dates
First available in Project Euclid: 8 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.aap/1457466167

Mathematical Reviews number (MathSciNet)
MR3473579

Zentralblatt MATH identifier
1341.93108

Subjects
Primary: 93E20: Optimal stochastic control
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60G51: Processes with independent increments; Lévy processes 91B70: Stochastic models

Keywords
Free-boundary irreversible investment singular stochastic control optimal stopping Lévy process Bank and El Karoui's representation theorem base capacity

Citation

Ferrari, Giorgio; Salminen, Paavo. Irreversible investment under Lévy uncertainty: an equation for the optimal boundary. Adv. in Appl. Probab. 48 (2016), no. 1, 298--314. https://projecteuclid.org/euclid.aap/1457466167


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