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A Non Convex Singular Stochastic Control Problem and its Related Optimal Stopping Boundaries

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  • Tiziano De Angelis
  • Giorgio Ferrari
  • John Moriarty

Abstract

Equivalences are known between problems of singular stochastic control (SSC) with convex performance criteria and related questions of optimal stopping, see for example Karatzas and Shreve [SIAM J. Control Optim. 22 (1984)]. The aim of this paper is to investigate how far connections of this type generalise to a non convex problem of purchasing electricity. Where the classical equivalence breaks down we provide alternative connections to optimal stopping problems. We consider a non convex infinite time horizon SSC problem whose state consists of an uncontrolled diffusion representing a real-valued commodity price, and a controlled increasing bounded process representing an inventory. We analyse the geometry of the action and inaction regions by characterising their (optimal) boundaries. Unlike the case of convex SSC problems we find that the optimal boundaries may be both reflecting and repelling and it is natural to interpret the problem as one of SSC with discretionary stopping.

Suggested Citation

  • Tiziano De Angelis & Giorgio Ferrari & John Moriarty, 2014. "A Non Convex Singular Stochastic Control Problem and its Related Optimal Stopping Boundaries," Papers 1405.2442, arXiv.org, revised Nov 2014.
    • Handle: RePEc:arx:papers:1405.2442
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    References listed on IDEAS

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    1. Blanchet-Scalliet, Christophette & El Karoui, Nicole & Jeanblanc, Monique & Martellini, Lionel, 2008. "Optimal investment decisions when time-horizon is uncertain," Journal of Mathematical Economics, Elsevier, vol. 44(11), pages 1100-1113, December.
    2. Helyette Geman & A. Roncoroni, 2006. "Understanding the Fine Structure of Electricity Prices," Post-Print halshs-00144198, HAL.
    3. Giorgio Ferrari, 2012. "On an integral equation for the free-boundary of stochastic, irreversible investment problems," Papers 1211.0412, arXiv.org, revised Jan 2015.
    4. Avinash K. Dixit & Robert S. Pindyck, 1994. "Investment under Uncertainty," Economics Books, Princeton University Press, edition 1, number 5474.
    5. Pindyck, Robert S, 1988. "Irreversible Investment, Capacity Choice, and the Value of the Firm," American Economic Review, American Economic Association, vol. 78(5), pages 969-985, December.
    6. repec:dau:papers:123456789/1433 is not listed on IDEAS
    7. Hélyette Geman & Andrea Roncoroni, 2006. "Understanding the Fine Structure of Electricity Prices," The Journal of Business, University of Chicago Press, vol. 79(3), pages 1225-1262, May.
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    Citations

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    Cited by:

    1. Federico, Salvatore & Ferrari, Giorgio & Schuhmann, Patrick, 2019. "A Model for the Optimal Management of Inflation," Center for Mathematical Economics Working Papers 624, Center for Mathematical Economics, Bielefeld University.
    2. Ferrari, Giorgio & Yang, Shuzhen, 2016. "On an optimal extraction problem with regime switching," Center for Mathematical Economics Working Papers 562, Center for Mathematical Economics, Bielefeld University.
    3. de Angelis, Tiziano & Ferrari, Giorgio & Martyr, Randall & Moriarty, John, 2016. "Optimal entry to an irreversible investment plan with non convex costs," Center for Mathematical Economics Working Papers 566, Center for Mathematical Economics, Bielefeld University.
    4. Giorgio Ferrari & Shuzhen Yang, 2016. "On an Optimal Extraction Problem with Regime Switching," Papers 1602.06765, arXiv.org, revised Dec 2017.
    5. de Angelis, Tiziano & Ferrari, Giorgio & Moriarty, John, 2016. "A solvable two-dimensional degenerate singular stochastic control problem with non convex costs," Center for Mathematical Economics Working Papers 531, Center for Mathematical Economics, Bielefeld University.
    6. de Angelis, Tiziano & Ferrari, Giorgio & Moriarty, John, 2016. "A solvable two-dimensional singular stochastic control problem with non convex costs," Center for Mathematical Economics Working Papers 561, Center for Mathematical Economics, Bielefeld University.
    7. Salvatore Federico & Giorgio Ferrari & Patrick Schuhmann, 2019. "A Model for the Optimal Management of Inflation," Department of Economics University of Siena 812, Department of Economics, University of Siena.
    8. Tiziano De Angelis & Erik Ekstrom, 2016. "The dividend problem with a finite horizon," Papers 1609.01655, arXiv.org, revised Nov 2017.
    9. de Angelis, Tiziano & Ferrari, Giorgio, 2016. "Stochastic nonzero-sum games: a new connection between singular control and optimal stopping," Center for Mathematical Economics Working Papers 565, Center for Mathematical Economics, Bielefeld University.
    10. Tiziano De Angelis & Giorgio Ferrari & John Moriarty, 2019. "A Solvable Two-Dimensional Degenerate Singular Stochastic Control Problem with Nonconvex Costs," Mathematics of Operations Research, INFORMS, vol. 44(2), pages 512-531, May.

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    More about this item

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • E22 - Macroeconomics and Monetary Economics - - Consumption, Saving, Production, Employment, and Investment - - - Investment; Capital; Intangible Capital; Capacity
    • D92 - Microeconomics - - Micro-Based Behavioral Economics - - - Intertemporal Firm Choice, Investment, Capacity, and Financing

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