On the Smoothness of Value Functions and the Existence of Optimal Strategies

45 Pages Posted: 2 Feb 2012 Last revised: 5 Oct 2012

See all articles by Bruno H. Strulovici

Bruno H. Strulovici

Northwestern University

Martin Szydlowski

Hong Kong University of Science & Technology (HKUST) - Department of Economics

Date Written: August 23, 2012

Abstract

In dynamic models driven by di usion 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-speci c arguments or explicit solutions. In this paper, we prove that the value function for the optimal control of any time-homogeneous, one-dimensional di usion is twice continuously di erentiable, under Lipschitz, growth, and non-vanishing volatility conditions. Under similar conditions, the value function of any optimal stopping problem is continuously di erentiable. For the rst problem, we provide sucient 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.

Keywords: Optimal Control, Optimal Stopping, Smooth Pasting, Super Contact, Comparative Statics, Envelope Theorem, Strong Solution, Markov Control, HJB Equation

JEL Classification: C61, D9, D83, D86, E20, G11

Suggested Citation

Strulovici, Bruno H. and Szydlowski, Martin, On the Smoothness of Value Functions and the Existence of Optimal Strategies (August 23, 2012). Available at SSRN: https://ssrn.com/abstract=1996808 or http://dx.doi.org/10.2139/ssrn.1996808

Bruno H. Strulovici (Contact Author)

Northwestern University ( email )

2001 Sheridan Road
Evanston, IL 60208
United States

Martin Szydlowski

Hong Kong University of Science & Technology (HKUST) - Department of Economics ( email )

Clear Water Bay
Kowloon, Hong Kong
China

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