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On the smoothness of value functions and the existence of optimal strategies in diffusion models

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  • Strulovici, Bruno
  • Szydlowski, Martin

Abstract

Studies of dynamic economic models often rely on each agent having a smooth value function and a well-defined optimal strategy. For time-homogeneous optimal control problems with a one-dimensional diffusion, we prove that the corresponding value function must be twice continuously differentiable under Lipschitz, growth, and non-vanishing-volatility conditions. Under similar conditions, the value function of any optimal stopping problem is shown to be (once) continuously differentiable. We also provide sufficient conditions, based on comparative statics and differential methods, for the existence of an optimal control in the sense of strong solutions. The results are applied to growth, experimentation, and dynamic contracting settings.

Suggested Citation

  • Strulovici, Bruno & Szydlowski, Martin, 2015. "On the smoothness of value functions and the existence of optimal strategies in diffusion models," Journal of Economic Theory, Elsevier, vol. 159(PB), pages 1016-1055.
  • Handle: RePEc:eee:jetheo:v:159:y:2015:i:pb:p:1016-1055
    DOI: 10.1016/j.jet.2015.03.015
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    Cited by:

    1. Weidong Tian & Zimu Zhu, 2020. "A Portfolio Choice Problem Under Risk Capacity Constraint," Papers 2005.13741, arXiv.org, revised Dec 2021.
    2. Bergemann, Dirk & Pavan, Alessandro, 2015. "Introduction to Symposium on Dynamic Contracts and Mechanism Design," Journal of Economic Theory, Elsevier, vol. 159(PB), pages 679-701.
    3. Rutger-Jan Lange & Coen N. Teulings, 2021. "The option value of vacant land: Don't build when demand for housing is booming," Tinbergen Institute Discussion Papers 21-022/IV, Tinbergen Institute.
    4. Ke, T. Tony & Villas-Boas, J. Miguel, 2019. "Optimal learning before choice," Journal of Economic Theory, Elsevier, vol. 180(C), pages 383-437.
    5. Aislinn Bohren, 2016. "Using Persistence to Generate Incentives in a Dynamic Moral Hazard Problem," PIER Working Paper Archive 16-024, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania, revised 15 Oct 2016.
    6. Jean-Paul Décamps & Stéphane Villeneuve, 2019. "A two-dimensional control problem arising from dynamic contracting theory," Finance and Stochastics, Springer, vol. 23(1), pages 1-28, January.
    7. Kolb, Aaron M., 2019. "Strategic real options," Journal of Economic Theory, Elsevier, vol. 183(C), pages 344-383.
    8. Jonas Al-Hadad & Zbigniew Palmowski, 2020. "Perpetual American options with asset-dependent discounting," Papers 2007.09419, arXiv.org, revised Jan 2021.
    9. Dirk Bergemann & Alessandro Pavan, 2015. "Introduction to JET Symposium Issue on "Dynamic Contracts and Mechanism Design"," Cowles Foundation Discussion Papers 2016, Cowles Foundation for Research in Economics, Yale University.
    10. Gorno, Leandro & Iachan, Felipe S., 2020. "Competitive real options under private information," Journal of Economic Theory, Elsevier, vol. 185(C).
    11. Dylan Possamai & Nizar Touzi, 2020. "Is there a Golden Parachute in Sannikov's principal-agent problem?," Papers 2007.05529, arXiv.org.

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    More about this item

    Keywords

    Optimal control; Optimal stopping; Smooth pasting; Super contact; Markov control; HJB equation;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games
    • D86 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Economics of Contract Law
    • O40 - Economic Development, Innovation, Technological Change, and Growth - - Economic Growth and Aggregate Productivity - - - General

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