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A Numerical Method for Solving Singular Stochastic Control Problems

Author

Listed:
  • Sunil Kumar

    (Graduate School of Business, Stanford University, Stanford, California 94305)

  • Kumar Muthuraman

    (School of Industrial Engineering, Purdue University, West Lafayette, Indiana 47907)

Abstract

Singular stochastic control has found diverse applications in operations management, economics, and finance. However, in all but the simplest of cases, singular stochastic control problems cannot be solved analytically. In this paper, we propose a method for numerically solving a class of singular stochastic control problems. We combine finite element methods that numerically solve partial differential equations with a policy update procedure based on the principle of smooth pasting to iteratively solve Hamilton-Jacobi-Bellman equations associated with the stochastic control problem. A key feature of our method is that the presence of singular controls simplifies the procedure. We illustrate the method on two examples of singular stochastic control problems, one drawn from economics and the other from queueing systems.

Suggested Citation

  • Sunil Kumar & Kumar Muthuraman, 2004. "A Numerical Method for Solving Singular Stochastic Control Problems," Operations Research, INFORMS, vol. 52(4), pages 563-582, August.
    • Handle: RePEc:inm:oropre:v:52:y:2004:i:4:p:563-582
    DOI: 10.1287/opre.1030.0107
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    References listed on IDEAS

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    Cited by:

    1. Pierre, Erwan & Villeneuve, Stéphane & Warin, Xavier, 2016. "Numerical approximation of a cash-constrained firm value with investment opportunities," TSE Working Papers 16-637, Toulouse School of Economics (TSE).
    2. Erwan Pierre & St'ephane Villeneuve & Xavier Warin, 2016. "Numerical approximation of a cash-constrained firm value with investment opportunities," Papers 1603.09049, arXiv.org, revised Oct 2016.
    3. Arun Chockalingam & Kumar Muthuraman, 2011. "American Options Under Stochastic Volatility," Operations Research, INFORMS, vol. 59(4), pages 793-809, August.
    4. Dabadghao, Shaunak S. & Chockalingam, Arun & Soltani, Taimaz & Fransoo, Jan, 2021. "Valuing Switching options with the moving-boundary method," Journal of Economic Dynamics and Control, Elsevier, vol. 127(C).
    5. René Caldentey & Lawrence M. Wein, 2006. "Revenue Management of a Make-to-Stock Queue," Operations Research, INFORMS, vol. 54(5), pages 859-875, October.
    6. Muthuraman, Kumar, 2008. "A moving boundary approach to American option pricing," Journal of Economic Dynamics and Control, Elsevier, vol. 32(11), pages 3520-3537, November.
    7. Amy R. Ward & Sunil Kumar, 2008. "Asymptotically Optimal Admission Control of a Queue with Impatient Customers," Mathematics of Operations Research, INFORMS, vol. 33(1), pages 167-202, February.
    8. Balikcioglu, Metin & Fackler, Paul L., 2018. "A Numerical Method for Multidimensional Impulse and Barrier Control Problems," CEnREP Working Papers 277666, North Carolina State University, Department of Agricultural and Resource Economics.
    9. Min Dai & Yue Kuen Kwok & Jianping Zong, 2008. "Guaranteed Minimum Withdrawal Benefit In Variable Annuities," Mathematical Finance, Wiley Blackwell, vol. 18(4), pages 595-611, October.
    10. Ghosh, Arka P. & Weerasinghe, Ananda P., 2010. "Optimal buffer size and dynamic rate control for a queueing system with impatient customers in heavy traffic," Stochastic Processes and their Applications, Elsevier, vol. 120(11), pages 2103-2141, November.
    11. Haolin Feng & Kumar Muthuraman, 2010. "A Computational Method for Stochastic Impulse Control Problems," Mathematics of Operations Research, INFORMS, vol. 35(4), pages 830-850, November.
    12. Pierre, Erwan & Villeneuve, Stéphane & Warin, Xavier, 2016. "Numerical approximation of a cash-constrained firm value with investment opportunities," IDEI Working Papers 860, Institut d'Économie Industrielle (IDEI), Toulouse.
    13. Vijay V. Desai & Vivek F. Farias & Ciamac C. Moallemi, 2012. "Approximate Dynamic Programming via a Smoothed Linear Program," Operations Research, INFORMS, vol. 60(3), pages 655-674, June.
    14. H. Dharma Kwon, 2010. "Invest or Exit? Optimal Decisions in the Face of a Declining Profit Stream," Operations Research, INFORMS, vol. 58(3), pages 638-649, June.
    15. Melda Ormeci Matoglu & John Vande Vate, 2011. "Drift Control with Changeover Costs," Operations Research, INFORMS, vol. 59(2), pages 427-439, April.
    16. Jing-Sheng Song & Paul Zipkin, 2013. "Supply Streams," Manufacturing & Service Operations Management, INFORMS, vol. 15(3), pages 444-457, July.
    17. Jodi Dianetti & Giorgio Ferrari & Renyuan Xu, 2024. "Exploratory Optimal Stopping: A Singular Control Formulation," Papers 2408.09335, arXiv.org, revised Oct 2024.
    18. Dabadghao, Shaunak S. & Chockalingam, Arun & Soltani, Taimaz & Fransoo, Jan C., 2021. "Valuing switching options with the moving-boundary method," Other publications TiSEM 45fe7e78-129f-4d41-ac2f-5, Tilburg University, School of Economics and Management.
    19. Takeshi Nagae & Takashi Akamatsu, 2006. "Dynamic Revenue Management of a Toll Road Project under Transportation Demand Uncertainty," Networks and Spatial Economics, Springer, vol. 6(3), pages 345-357, September.

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