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On an optimal extraction problem with regime switching

Part of: Mathematical economics Hamilton-Jacobi theories, including dynamic programming Mathematical finance Stochastic processes Markov processes Stochastic systems and control

Published online by Cambridge University Press:  16 November 2018

Giorgio Ferrari  and
Shuzhen Yang
Giorgio Ferrari*
Affiliation:
Bielefeld University
Shuzhen Yang*
Affiliation:
Shandong University
*
* Postal address: Center for Mathematical Economics, Bielefeld University, Universitätsstrasse 25, D-33615 Bielefeld, Germany. Email address: giorgio.ferrari@uni-bielefeld.de
** Postal address: Institution of Financial Studies, Shandong University, Jinan, Shandong, 250100, P. R. China. Email address: yangsz@sdu.edu.cn
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Abstract

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In this paper we study a finite-fuel two-dimensional degenerate singular stochastic control problem under regime switching motivated by the optimal irreversible extraction problem of an exhaustible commodity. A company extracts a natural resource from a reserve with finite capacity and sells it in the market at a spot price that evolves according to a Brownian motion with volatility modulated by a two-state Markov chain. In this setting, the company aims at finding the extraction rule that maximizes its expected discounted cash flow, net of the costs of extraction and maintenance of the reserve. We provide expressions for both the value function and the optimal control. On the one hand, if the running cost for the maintenance of the reserve is a convex function of the reserve level, the optimal extraction rule prescribes a Skorokhod reflection of the (optimally) controlled state process at a certain state and price-dependent threshold. On the other hand, in the presence of a concave running cost function, it is optimal to instantaneously deplete the reserve at the time at which the commodity's price exceeds an endogenously determined critical level. In both cases, the threshold triggering the optimal control is given in terms of the optimal stopping boundary of an auxiliary family of perpetual optimal selling problems with regime switching.

Keywords

Singular stochastic controloptimal stoppingregime switchingHamilton‒Jacobi‒Bellman equationfree boundarycommodity extractionoptimal selling

MSC classification

Primary: 93E20: Optimal stochastic control
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 49L20: Dynamic programming method 60J27: Continuous-time Markov processes on discrete state spaces 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems) 91B76: Environmental economics (natural resource models, harvesting, pollution, etc.)

Information

Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 3 , September 2018 , pp. 671 - 705
Copyright
Copyright © Applied Probability Trust 2018 

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