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On the Smoothness of Value Functions

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  • Strulovici, Bruno
  • Szydlowski, Martin

Abstract

We prove that under standard Lipschitz and growth conditions, the value function of all optimal control problems for one-dimensional diffusions is twice continuously differentiable, as long as the control space is compact and the volatility is uniformly bounded below, away from zero. Under similar conditions, the value function of any optimal stopping problem is continuously differentiable. For the first problem, we provide sufficient conditions for the existence of an optimal control, which is also shown to be Markov. These conditions are based on the theory of monotone comparative statics.

Suggested Citation

  • Strulovici, Bruno & Szydlowski, Martin, 2012. "On the Smoothness of Value Functions," MPRA Paper 36326, University Library of Munich, Germany, revised 31 Jan 2012.
  • Handle: RePEc:pra:mprapa:36326
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    References listed on IDEAS

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    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.
    8. Peter M. Demarzo & Michael J. Fishman & Zhiguo He & Neng Wang, 2012. "Dynamic Agency and the q Theory of Investment," Journal of Finance, American Finance Association, vol. 67(6), pages 2295-2340, December.
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    Citations

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    Cited by:

    1. Dmitry Livdan & Alexander Nezlobin, 2017. "Accounting rules, equity valuation, and growth options," Review of Accounting Studies, Springer, vol. 22(3), pages 1122-1155, September.
    2. 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.
    3. Szydlowski, Martin, 2019. "Incentives, project choice, and dynamic multitasking," Theoretical Economics, Econometric Society, vol. 14(3), September.
    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. Martin Szydlowski, 2012. "Ambiguity in Dynamic Contracts," Discussion Papers 1543, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    6. 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.
    7. 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.
    8. Fudenberg, Drew & Romanyuk, Gleb & Strack, Philipp, 2017. "Active learning with a misspecified prior," Theoretical Economics, Econometric Society, vol. 12(3), September.
    9. 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.
    10. 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.
    11. Nezlobin, Alexander, 2018. "Dynamic Investment Models in Accounting Research," Foundations and Trends(R) in Accounting, now publishers, vol. 12(3), pages 216-297, December.
    12. Keller, Godfrey & Rady, Sven, 2015. "Breakdowns," Theoretical Economics, Econometric Society, vol. 10(1), January.
    13. 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.

    More about this item

    Keywords

    Super Contact; Smooth Pasting; HJB Equation; Optimal Control; Markov Control; Comparative Statics; Supermodularity; Single-Crossing; Interval Dominance Order;

    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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