Studies of dynamic economic models often rely on each agent having a smooth value function and a well-defined optimal strategy. For time-homogeneous optimal control problems with a one-dimensional diffusion, we prove that the corresponding value function must be twice continuously differentiable under Lipschitz, growth, and non-vanishing-volatility conditions. Under similar conditions, the value function of any optimal stopping problem is shown to be (once) continuously differentiable. We also provide sufficient conditions, based on comparative statics and differential methods, for the existence of an optimal control in the sense of strong solutions. The results are applied to growth, experimentation, and dynamic contracting settings.
Bibliographical noteFunding Information:
We are grateful for comments from John Quah and Yuliy Sannikov. Part of this research was accomplished while the first author was visiting the Economic Theory Center at Princeton University, whose hospitality is gratefully acknowledged. This author is also thankful for financial support from the NSF under Grant No. 1151410.
- HJB equation
- Markov control
- Optimal control
- Optimal stopping
- Smooth pasting
- Super contact