On the Smoothness of Value Functions

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 differentiable. 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.