IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v130y2020i5p3120-3153.html
   My bibliography  Save this article

Optimal switching problems with an infinite set of modes: An approach by randomization and constrained backward SDEs

Author

Listed:
  • Fuhrman, Marco
  • Morlais, Marie-Amélie

Abstract

We address a general optimal switching problem over finite horizon for a stochastic system described by a differential equation driven by Brownian motion. The main novelty is the fact that we allow for infinitely many modes (or regimes, i.e. the possible values of the piecewise-constant control process). We allow all the given coefficients in the model to be path-dependent, that is, their value at any time depends on the past trajectory of the controlled system. The main aim is to introduce a suitable (scalar) backward stochastic differential equation (BSDE), with a constraint on the martingale part, that allows to give a probabilistic representation of the value function of the given problem. This is achieved by randomization of control, i.e. by introducing an auxiliary optimization problem which has the same value as the starting optimal switching problem and for which the desired BSDE representation is obtained. In comparison with the existing literature we do not rely on a system of reflected BSDE nor can we use the associated Hamilton–Jacobi–Bellman equation in our non-Markovian framework.

Suggested Citation

  • Fuhrman, Marco & Morlais, Marie-Amélie, 2020. "Optimal switching problems with an infinite set of modes: An approach by randomization and constrained backward SDEs," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 3120-3153.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:5:p:3120-3153
    DOI: 10.1016/j.spa.2019.09.008
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414918306847
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2019.09.008?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. repec:dau:papers:123456789/5733 is not listed on IDEAS
    2. Bandini, Elena & Fuhrman, Marco, 2017. "Constrained BSDEs representation of the value function in optimal control of pure jump Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 127(5), pages 1441-1474.
    3. Elie, Romuald & Kharroubi, Idris, 2010. "Probabilistic representation and approximation for coupled systems of variational inequalities," Statistics & Probability Letters, Elsevier, vol. 80(17-18), pages 1388-1396, September.
    4. Gassiat, Paul & Kharroubi, Idris & Pham, Huyên, 2012. "Time discretization and quantization methods for optimal multiple switching problem," Stochastic Processes and their Applications, Elsevier, vol. 122(5), pages 2019-2052.
    5. Carr, Peter, 1998. "Randomization and the American Put," The Review of Financial Studies, Society for Financial Studies, vol. 11(3), pages 597-626.
    6. Dixit, Avinash K, 1989. "Entry and Exit Decisions under Uncertainty," Journal of Political Economy, University of Chicago Press, vol. 97(3), pages 620-638, June.
    7. Cosso, Andrea & Fuhrman, Marco & Pham, Huyên, 2016. "Long time asymptotics for fully nonlinear Bellman equations: A backward SDE approach," Stochastic Processes and their Applications, Elsevier, vol. 126(7), pages 1932-1973.
    8. Bandini, Elena & Cosso, Andrea & Fuhrman, Marco & Pham, Huyên, 2019. "Randomized filtering and Bellman equation in Wasserstein space for partial observation control problem," Stochastic Processes and their Applications, Elsevier, vol. 129(2), pages 674-711.
    9. Avinash K. Dixit & Robert S. Pindyck, 1994. "Investment under Uncertainty," Economics Books, Princeton University Press, edition 1, number 5474.
    10. Said Hamadène & Monique Jeanblanc, 2007. "On the Starting and Stopping Problem: Application in Reversible Investments," Mathematics of Operations Research, INFORMS, vol. 32(1), pages 182-192, February.
    11. Hamadène, Said & Zhang, Jianfeng, 2010. "Switching problem and related system of reflected backward SDEs," Stochastic Processes and their Applications, Elsevier, vol. 120(4), pages 403-426, April.
    12. repec:dau:papers:123456789/12195 is not listed on IDEAS
    13. Brennan, Michael J & Schwartz, Eduardo S, 1985. "Evaluating Natural Resource Investments," The Journal of Business, University of Chicago Press, vol. 58(2), pages 135-157, April.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Perninge, Magnus, 2024. "Optimal stopping of BSDEs with constrained jumps and related zero-sum games," Stochastic Processes and their Applications, Elsevier, vol. 173(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Mihail Zervos & Carlos Oliveira & Kate Duckworth, 2018. "An investment model with switching costs and the option to abandon," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 88(3), pages 417-443, December.
    2. El Asri, Brahim, 2013. "Stochastic optimal multi-modes switching with a viscosity solution approach," Stochastic Processes and their Applications, Elsevier, vol. 123(2), pages 579-602.
    3. Hamadène, Said & Zhang, Jianfeng, 2010. "Switching problem and related system of reflected backward SDEs," Stochastic Processes and their Applications, Elsevier, vol. 120(4), pages 403-426, April.
    4. Adkins, Roger & Paxson, Dean, 2019. "Rescaling-contraction with a lower cost technology when revenue declines," European Journal of Operational Research, Elsevier, vol. 277(2), pages 574-586.
    5. Tsekrekos, Andrianos E., 2010. "The effect of mean reversion on entry and exit decisions under uncertainty," Journal of Economic Dynamics and Control, Elsevier, vol. 34(4), pages 725-742, April.
    6. Di Corato, Luca & Moretto, Michele, 2011. "Investing in biogas: Timing, technological choice and the value of flexibility from input mix," Energy Economics, Elsevier, vol. 33(6), pages 1186-1193.
    7. Insley, Margaret, 2017. "Resource extraction with a carbon tax and regime switching prices: Exercising your options," Energy Economics, Elsevier, vol. 67(C), pages 1-16.
    8. Bolton, Patrick & Wang, Neng & Yang, Jinqiang, 2019. "Investment under uncertainty with financial constraints," Journal of Economic Theory, Elsevier, vol. 184(C).
    9. Iraj J. Fooladi & Nargess K. Kayhani, 2003. "Is Entrepreneurship Only About Entering A New Business," Journal of Entrepreneurial Finance, Pepperdine University, Graziadio School of Business and Management, vol. 8(2), pages 1-11, Summer.
    10. Lim, Terence & Lo, Andrew W. & Merton, Robert C. & Scholes, Myron S., 2006. "The Derivatives Sourcebook," Foundations and Trends(R) in Finance, now publishers, vol. 1(5–6), pages 365-572, April.
    11. Moon, Yongma & Baran, Mesut, 2018. "Economic analysis of a residential PV system from the timing perspective: A real option model," Renewable Energy, Elsevier, vol. 125(C), pages 783-795.
    12. Jyh-Bang Jou, 2000. "Irreversible Investment Decisions Under Uncertainty with Tax Holidays," Public Finance Review, , vol. 28(1), pages 66-81, January.
    13. Bulan, Laarni & Mayer, Christopher & Somerville, C. Tsuriel, 2009. "Irreversible investment, real options, and competition: Evidence from real estate development," Journal of Urban Economics, Elsevier, vol. 65(3), pages 237-251, May.
    14. Johnson, Timothy C. & Zervos, Mihail, 2010. "The explicit solution to a sequential switching problem with non-smooth data," LSE Research Online Documents on Economics 29003, London School of Economics and Political Science, LSE Library.
    15. Kou, Ying & Luo, Meifeng, 2018. "Market driven ship investment decision using the real option approach," Transportation Research Part A: Policy and Practice, Elsevier, vol. 118(C), pages 714-729.
    16. Shackleton, Mark B. & Tsekrekos, Andrianos E. & Wojakowski, Rafal, 2004. "Strategic entry and market leadership in a two-player real options game," Journal of Banking & Finance, Elsevier, vol. 28(1), pages 179-201, January.
    17. Davis, Graham A. & Cairns, Robert D., 2017. "The odd notion of “reversible investment”," Journal of Banking & Finance, Elsevier, vol. 81(C), pages 172-180.
    18. Steven R. Grenadier, 2003. "An Equilibrium Analysis of Real Estate," NBER Working Papers 9475, National Bureau of Economic Research, Inc.
    19. Andrianos Tsekrekos, 2013. "Irreversible exit decisions under mean-reverting uncertainty," Journal of Economics, Springer, vol. 110(1), pages 5-23, September.
    20. Feil, Jan-Henning & Musshoff, Oliver, 2013. "Modelling investment and disinvestment decisions under competition, uncertainty and different market interventions," Economic Modelling, Elsevier, vol. 35(C), pages 443-452.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:130:y:2020:i:5:p:3120-3153. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.