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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. Cripps, Martin William & Keller, Godfrey & Rady, Sven, 2003. "Strategic Experimentation with Exponential Bandits," CEPR Discussion Papers 3814, C.E.P.R. Discussion Papers.
  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. John Quah & Bruno Strulovici, 2007. "Comparative Statics, Informativeness, and the Interval Dominance Order," Economics Series Working Papers 2007-WO4, University of Oxford, Department of Economics.
  4. 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.
  5. Milgrom, Paul & Shannon, Chris, 1994. "Monotone Comparative Statics," Econometrica, Econometric Society, vol. 62(1), pages 157-80, January.
  6. Manuel Santos & Juan Pablo Rincon-Zapatero, 2007. "Differentiability of the Value Function without Interiority Assumptions," Working Papers 0704, University of Miami, Department of Economics.
  7. John K.‐H. Quah & Bruno Strulovici, 2012. "Aggregating the Single Crossing Property," Econometrica, Econometric Society, vol. 80(5), pages 2333-2348, 09.
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Cited by:
  1. Godfrey Keller & Sven Rady, 2013. "Breakdowns," Levine's Working Paper Archive 786969000000000635, David K. Levine.
  2. Martin Szydlowski, 2012. "Ambiguity in Dynamic Contracts," Discussion Papers 1543, Northwestern University, Center for Mathematical Studies in Economics and Management Science.

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