Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-26T13:55:40.512Z Has data issue: false hasContentIssue false
  • English
  • Français

On an optimal extraction problem with regime switching

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

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: [email protected]
** Postal address: Institution of Financial Studies, Shandong University, Jinan, Shandong, 250100, P. R. China. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Almansour, A. and Insley, M. (2016). The impact of stochastic extraction cost on the value of an exhaustible resource: an application to the Alberta Oil Sands. Energy J. 37, 10.5547/01956574.37.2.aalm.Google Scholar
[2]Bachelier, L. (1900). Theorie de la spéculation. Ann. Sci. Ecole Norm. Sup. (3) 17, 21–86.Google Scholar
[3]Baldursson, F. M. and Karatzas, I. (1996). Irreversible investment and industry equilibrium. Finance Stoch. 1, 69–89.Google Scholar
[4]Bensoussan, A., Yan, Z. and Yin, G. (2012). Threshold-type policies for real options using regime-switching models. SIAM J. Financial Math. 3, 667–689.Google Scholar
[5]Brekke, K. A. and Øksendal, B. (1994). Optimal switching in an economic activity under uncertainty. SIAM J. Control Optimization 32, 1021–1036.Google Scholar
[6]Brennan, M. J. and Schwartz, E. S. (1985). Evaluating natural resource investments. J. Business 58, 135–157.Google Scholar
[7]Buffington, J. and Elliott, R. J. (2002). American options with regime switching. Internat. J. Theoret. Appl. Finance 5, 497–514.Google Scholar
[8]De Angelis, T., Ferrari, G. and Moriarty, J. (2015). A nonconvex singular stochastic control problem and its related optimal stopping boundaries. SIAM J. Control Optimization 53, 1199–1223.Google Scholar
[9]De Angelis, T., Ferrari, G. and Moriarty, J. (2018). A solvable two-dimensional degenerate singular stochastic control problem with nonconvex costs. Math. Operat. Res. Available at https://pubsonline.informs.org/doi/10.1287/moor.2018.0934.Google Scholar
[10]Dellacherie, C. and Meyer, P. A. (1978). Probabilities and Potential. North-Holland, Amsterdam.Google Scholar
[11]Dixit, A. K. and Pindyck, R. S. (1994). Investment Under Uncertainty. Princeton University Press.Google Scholar
[12]Feliz, R. A. (1993). The optimal extraction rate of a natural resource under uncertainty. Econom. Lett. 43, 231–234.Google Scholar
[13]Fenton, C. P. and Nance, P. K. (2011). Subzero commodity prices: why commodity prices fall through the zero bound and where and how it could happen in 2011. J.P. Morgan Global Commodities Research. Available at http://ssrn.com/abstract=1922787.Google Scholar
[14]Fleming, W. H. and Soner, H. M. (2006). Controlled Markov Processes and Viscosity Solutions. Springer, New York.Google Scholar
[15]Geman, H. (2007). Mean reversion versus random walk in oil and natural gas prices. In Advances in Mathematical Finance, Birkhäuser, Boston, MA, pp. 219–228.Google Scholar
[16]Guo, X. (2001). An explicit solution to an optimal stopping problem with regime switching. J. Appl. Prob. 38, 464–481.Google Scholar
[17]Guo, X. and Zervos, M. (2015). Optimal execution with multiplicative price impact. SIAM J. Financial Math. 6, 281–306.Google Scholar
[18]Guo, X. and Zhang, Q. (2004). Closed-form solutions for perpetual American put options with regime switching. SIAM J. Appl. Math. 64, 2034–2049.Google Scholar
[19]Guo, X., Miao, J. and Morellec, E. (2005). Irreversible investment with regime shifts. J. Econom. Theory 122, 37–59.Google Scholar
[20]Hamilton, J. D. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57, 357–384.Google Scholar
[21]Insley, M. (2017). Resource extraction with a carbon tax and regime switching prices: exercising your options. Energy Econom. 67, 1–16.Google Scholar
[22]Karatzas, I. and Shreve, S. E. (1998). Methods of Mathematical Finance. Springer, New York.Google Scholar
[23]Lumley, R. R. and Zervos, M. (2001). A model for investments in the natural resource industry with switching costs. Math. Operat. Res. 26, 637–653.Google Scholar
[24]McDonald, R. and Siegel, D. (1986). The value of waiting to invest. Quart. J. Econom. 101, 707–727.Google Scholar
[25]Meyer, P. A. (ed.) (1976). Séminaire de Probailities X (Lecture Notes Math. 511). Springer, Berlin.Google Scholar
[26]Jiang, Z. (2015). Optimal dividend policy when cash reserves follow a jump-diffusion process under Markov-regime switching. J. Appl. Prob. 52, 209–223.Google Scholar
[27]Jiang, Z. and Pistorius, M. (2012). Optimal dividend distribution under Markov regime switching. Finance Stoch. 16, 449–476.Google Scholar
[28]Peskir, G. and Shiryaev, A. (2006). Optimal Stopping and Free-Boundary Problems. Birkhäuser, Basel.Google Scholar
[29]Pindyck, R. S. (1981). The optimal production of an exhaustible resource when price is exogenous and stochastic. Scand. J. Econom. 83, 277–288.Google Scholar
[30]Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion. Springer, Berlin.Google Scholar
[31]Shreve, S. E. (1988). An introduction to singular stochastic control. In Stochastic Differential Systems, Stochastic Control Theory and Applications, Springer, New York, pp. 513–528.Google Scholar
[32]Skorokhod, A. V. (1989). Asymptotic Methods in the Theory of Stochastic Differential Equations. American Mathematical Society, Providence, RI.Google Scholar
[33]Sotomayor, L. R. and Cadenillas, A. (2011). Classical and singular stochastic control for the optimal dividend policy when there is regime switching. Insurance Math. Econom. 48, 344–354.Google Scholar
[34]Trigeorgis, L. (1996). Real Options: Managerial Flexibility and Strategy in Resource Allocations. MIT Press, Cambridge, MA.Google Scholar
[35]Yin, G. and Xi, F. (2010). Stability of regime-switching jump diffusions. SIAM J. Control Optimization 48, 4525–4549.Google Scholar
[36]Zhang, Q. and Zhu, X. Y. (2009). Valuation of stock loans with regime switching. SIAM J. Control Optimization 48, 1229–1250.Google Scholar
[37]Zhu, C. and Yin, G. (2009). On strong Feller, recurrence, and weak stabilization of regime-switching diffusions. SIAM J. Control Optimization 48, 2003–2031.Google Scholar
[38]Zhu, J. and Yang, H. (2016). Optimal financing and dividend distribution in a general diffusion model with regime switching. Adv. Appl. Prob. 48, 406–422.Google Scholar