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

Abstract

In dynamic models driven by diusion 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 diusion is twice continuously dierentiable, under Lipschitz, growth, and non-vanishing volatility conditions. Under similar conditions, the value function of any optimal stopping problem is continuously dierentiable. 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