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On the Smoothness of Value Functions

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  • Bruno Strulovici
  • Martin Szydlowski

Abstract

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

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Paper provided by Northwestern University, Center for Mathematical Studies in Economics and Management Science in its series Discussion Papers with number 1542.

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Date of creation: 23 Aug 2012
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Handle: RePEc:nwu:cmsems:1542r

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

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  1. John K.-H. Quah & Bruno Strulovici, 2007. "Comparative Statics, Informativeness, and the Interval Dominance Order," Economics Papers 2007-W04, Economics Group, Nuffield College, University of Oxford.
  2. Milgrom, P. & Shannon, C., 1991. "Monotone Comparative Statics," Papers 11, Stanford - Institute for Thoretical Economics.
  3. Godfrey Keller & Martin Cripps, 2003. "Strategic Experimentation with Exponential Bandits," Economics Series Working Papers 143, University of Oxford, Department of Economics.
  4. Manuel Santos & Juan Pablo Rincon-Zapatero, 2007. "Differentiability of the Value Function without Interiority Assumptions," Working Papers 0704, University of Miami, Department of Economics.
  5. Juan Pablo Rincón-Zapatero & Manuel S. Santos, 2010. "Differentiability of the value function in continuous-time economic models," Economics Working Papers we1022, Universidad Carlos III, Departamento de Economía.
  6. John K.‐H. Quah & Bruno Strulovici, 2012. "Aggregating the Single Crossing Property," Econometrica, Econometric Society, vol. 80(5), pages 2333-2348, 09.
  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, 2012. "Ambiguity in Dynamic Contracts," Discussion Papers 1543, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  2. Keller, Godfrey & Rady, Sven, 2012. "Breakdowns," Discussion Paper Series of SFB/TR 15 Governance and the Efficiency of Economic Systems 396, Free University of Berlin, Humboldt University of Berlin, University of Bonn, University of Mannheim, University of Munich.

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