IDEAS home Printed from https://ideas.repec.org/p/bie/wpaper/714.html
   My bibliography  Save this paper

A singular stochastic control problem with interconnected dynamics

Author

Listed:
  • Federico, Salvatore

    (Center for Mathematical Economics, Bielefeld University)

  • Ferrari, Giorgio

    (Center for Mathematical Economics, Bielefeld University)

  • Schuhmann, Patrick

    (Center for Mathematical Economics, Bielefeld University)

Abstract

In this paper we study a Markovian two-dimensional bounded-variation stochastic control problem whose state process consists of a diffusive mean-reverting component and of a purely controlled one. The main problem's characteristic lies in the interaction of the two components of the state process: the mean-reversion level of the diffusive component is an affine function of the current value of the purely controlled one. By relying on a combination of techniques from viscosity theory and free-boundary analysis, we provide the structure of the value function and we show that it satisfies a second-order smooth-fit principle. Such a regularity is then exploited in order to determine a system of functional equations solved by the two monotone continuous curves (free boundaries) that split the control problem's state space in three connected regions. Further properties of the free boundaries are also obtained.

Suggested Citation

  • Federico, Salvatore & Ferrari, Giorgio & Schuhmann, Patrick, 2025. "A singular stochastic control problem with interconnected dynamics," Center for Mathematical Economics Working Papers 714, Center for Mathematical Economics, Bielefeld University.
    • Handle: RePEc:bie:wpaper:714
    as

    Download full text from publisher

    File URL: https://pub.uni-bielefeld.de/download/3004915/3004916
    File Function: First Version, 2020
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. M. I. Taksar, 1985. "Average Optimal Singular Control and a Related Stopping Problem," Mathematics of Operations Research, INFORMS, vol. 10(1), pages 63-81, February.
    2. Pui Chan Lon & Mihail Zervos, 2011. "A Model for Optimally Advertising and Launching a Product," Mathematics of Operations Research, INFORMS, vol. 36(2), pages 363-376, May.
    3. Pierre, Erwan & Villeneuve, Stéphane & Warin, Xavier, 2014. "Liquidity Management with Decreasing-returns-to-scale and Secured Credit Line," TSE Working Papers 14-542, Toulouse School of Economics (TSE), revised Jun 2016.
    4. Goran Peskir, 2005. "A Change-of-Variable Formula with Local Time on Curves," Journal of Theoretical Probability, Springer, vol. 18(3), pages 499-535, July.
    5. Erwan Pierre & Stéphane Villeneuve & Xavier Warin, 2016. "Liquidity management with decreasing returns to scale and secured credit line," Post-Print halshs-01522513, HAL.
    6. Erwan Pierre & Stéphane Villeneuve & Xavier Warin, 2016. "Liquidity management with decreasing returns to scale and secured credit line," Finance and Stochastics, Springer, vol. 20(4), pages 809-854, October.
    7. Olivier Jean Blanchard & Stanley Fischer, 1989. "Lectures on Macroeconomics," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262022834, December.
    8. 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.
    Full references (including those not matched with items on IDEAS)

    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. 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.
    2. Etienne Chevalier & Vathana Ly Vath & Alexandre Roch, 2020. "Optimal Dividend and Capital Structure with Debt Covenants," Journal of Optimization Theory and Applications, Springer, vol. 187(2), pages 535-565, November.
    3. Gaïgi, M’hamed & Ly Vath, Vathana & Scotti, Simone, 2022. "Optimal harvesting under marine reserves and uncertain environment," European Journal of Operational Research, Elsevier, vol. 301(3), pages 1181-1194.
    4. Giorgio Ferrari & Tiziano Vargiolu, 2020. "On the singular control of exchange rates," Annals of Operations Research, Springer, vol. 292(2), pages 795-832, September.
    5. Federico, Salvatore & Ferrari, Giorgio & Schuhmann, Patrick, 2020. "Singular Control of the Drift of a Brownian System," Center for Mathematical Economics Working Papers 637, Center for Mathematical Economics, Bielefeld University.
    6. 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.
    7. 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.
    8. Fernando de Holanda Barbosa, 2017. "Hyperinflation: Inflation Tax and Economic Policy Regime," SpringerBriefs in Economics, in: Exploring the Mechanics of Chronic Inflation and Hyperinflation, chapter 0, pages 61-75, Springer.
    9. Stanley Fischer, 1991. "Growth, Macroeconomics, and Development," NBER Chapters, in: NBER Macroeconomics Annual 1991, Volume 6, pages 329-379, National Bureau of Economic Research, Inc.
    10. John Haltiwanger & Scott Schuh, 1999. "Gross job flows between plants and industries," New England Economic Review, Federal Reserve Bank of Boston, issue Mar, pages 41-64.
    11. repec:hum:wpaper:sfb649dp2009-020 is not listed on IDEAS
    12. Fredrik Carlsson & Dinky Daruvala & Olof Johansson‐Stenman, 2005. "Are People Inequality‐Averse, or Just Risk‐Averse?," Economica, London School of Economics and Political Science, vol. 72(287), pages 375-396, August.
    13. Christian Groth & Karl-Josef Koch & Thomas Steger, 2006. "Rethinking the Concept of Long-Run Economic Growth," CESifo Working Paper Series 1701, CESifo.
    14. Buonaguidi, B., 2023. "An optimal sequential procedure for determining the drift of a Brownian motion among three values," Stochastic Processes and their Applications, Elsevier, vol. 159(C), pages 320-349.
    15. J.P.G. Reijnders, 2007. "Impulse or propagation? How the tides turned in Business Cycle Theory," Working Papers 07-07, Utrecht School of Economics.
    16. Thomas Seegmuller, 2005. "Steady state analysis and endogenous fluctuations in a finance constrained model," Cahiers de la Maison des Sciences Economiques v05029, Université Panthéon-Sorbonne (Paris 1).
    17. Smulders, Sjak & Gradus, Raymond, 1996. "Pollution abatement and long-term growth," European Journal of Political Economy, Elsevier, vol. 12(3), pages 505-532, November.
    18. Andersen, Torben M., 2005. "Product market integration, wage dispersion and unemployment," Labour Economics, Elsevier, vol. 12(3), pages 379-406, June.
    19. Xavier Pautrel, 2015. "Abatement Technology and the Environment–Growth Nexus with Education," Environmental & Resource Economics, Springer;European Association of Environmental and Resource Economists, vol. 61(3), pages 297-318, July.
    20. McCallum, Bennett T, 2000. "Theoretical Analysis Regarding a Zero Lower Bound on Nominal Interest Rates," Journal of Money, Credit and Banking, Blackwell Publishing, vol. 32(4), pages 870-904, November.
    21. Radwanski, Juliusz, 2020. "On the Purchasing Power of Money in an Exchange Economy," MPRA Paper 104244, University Library of Munich, Germany.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    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:bie:wpaper:714. 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: Bettina Weingarten (email available below). General contact details of provider: https://edirc.repec.org/data/imbiede.html .

    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.