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On the Smoothness of Value Functions


  • Bruno Strulovici
  • Martin Szydlowski


In dynamic models driven by diffusion processes, the smoothness of the value function plays a crucial role for characterizing properties of the solution. However, available methods to ensure such smoothness have limited applicability in economics, and economists have often relied on either model-specific arguments or explicit solutions. In this paper, we prove that the value function for the optimal control of any time-homogeneous, one-dimensional diffusion is twice continuously differentiable, under Lipschitz, growth, and non-vanishing volatility conditions. Under similar conditions, the value function of any optimal stopping problem is continuously diferentiable. For the first problem, we provide sufficient conditions for the existence of an optimal control. The optimal control is Markovian and constructed from the Bellman equation. We also establish an envelope theorem for parameterized optimal stopping problems. Several applications are discussed, including growth, dynamic contracting, and experimentation models. JEL Classification Numbers: C61, D9, D83, D86, E20, G11

Suggested Citation

  • Bruno Strulovici & Martin Szydlowski, 2012. "On the Smoothness of Value Functions," Discussion Papers 1542, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  • Handle: RePEc:nwu:cmsems:1542r

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    References listed on IDEAS

    1. Juan Pablo Rinc ́on–Zapatero & Manuel S. Santos, 2010. "Differentiability of the Value Function in Continuous–Time Economic Models," Working Papers 2010-28, University of Miami, Department of Economics.
    2. John K.-H. Quah & Bruno Strulovici, 2009. "Comparative Statics, Informativeness, and the Interval Dominance Order," Econometrica, Econometric Society, vol. 77(6), pages 1949-1992, November.
    3. Godfrey Keller & Sven Rady & Martin Cripps, 2005. "Strategic Experimentation with Exponential Bandits," Econometrica, Econometric Society, vol. 73(1), pages 39-68, January.
    4. John K.‐H. Quah & Bruno Strulovici, 2012. "Aggregating the Single Crossing Property," Econometrica, Econometric Society, vol. 80(5), pages 2333-2348, September.
    5. Rincón-Zapatero, Juan Pablo & Santos, Manuel S., 2009. "Differentiability of the value function without interiority assumptions," Journal of Economic Theory, Elsevier, vol. 144(5), pages 1948-1964, September.
    6. Milgrom, Paul & Shannon, Chris, 1994. "Monotone Comparative Statics," Econometrica, Econometric Society, vol. 62(1), pages 157-180, January.
    7. John K.-H. Quah & Bruno Strulovici, 2013. "Discounting, Values, and Decisions," Journal of Political Economy, University of Chicago Press, vol. 121(5), pages 896-939.
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    Cited by:

    1. Martin Szydlowski, 2014. "Incentives, Project Choice, and Dynamic Multitasking," 2014 Meeting Papers 1240, Society for Economic Dynamics.
    2. repec:spr:reaccs:v:22:y:2017:i:3:d:10.1007_s11142-017-9402-6 is not listed on IDEAS
    3. John K.-H. Quah & Bruno Strulovici, 2013. "Discounting, Values, and Decisions," Journal of Political Economy, University of Chicago Press, vol. 121(5), pages 896-939.
    4. Keller, Godfrey & Rady, Sven, 2015. "Breakdowns," Theoretical Economics, Econometric Society, vol. 10(1), January.
    5. Martin Szydlowski, 2012. "Ambiguity in Dynamic Contracts," Discussion Papers 1543, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    6. Fudenberg, Drew & Romanyuk, Gleb & Strack, Philipp, 2017. "Active learning with a misspecified prior," Theoretical Economics, Econometric Society, vol. 12(3), September.
    7. 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.
    8. J. Aislinn Bohren, 2011. "Stochastic Games in Continuous Time: Persistent Actions in Long-Run Relationships, Second Version," PIER Working Paper Archive 14-033, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania, revised 01 Aug 2014.
    9. Huang, Yao Tung & Kwok, Yue Kuen, 2014. "Analysis of optimal dynamic withdrawal policies in withdrawal guarantee products," Journal of Economic Dynamics and Control, Elsevier, vol. 45(C), pages 19-43.

    More about this item


    Stochastic Control; Super Contact; Smooth Pasting; Value Function;

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis

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