On the Smoothness of Value Functions

Strulovici, Bruno and Szydlowski, Martin (2012): On the Smoothness of Value Functions.

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.

• Item Type: MPRA Paper
• Language: English
• Keywords: Super Contact; Smooth Pasting; HJB Equation; Optimal Control; Markov Control; Comparative Statics; Supermodularity; Single-Crossing; Interval Dominance Order
• Subjects: C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C61 - Optimization Techniques ; Programming Models ; Dynamic Analysis

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