On the Smoothness of Value Functions and the Existence of Optimal Strategies∗
نویسندگان
چکیده
In economic models where uncertainty is driven by a diffusion process, the smoothness of agents’ value functions and the existence of optimal strategies play a crucial role in the analysis. This paper shows, firstly, that the value function for the optimal control of any timehomogeneous, one-dimensional diffusion is twice continuously differentiable under Lipschitz, growth, and non-vanishing volatility conditions. Secondly, it provides sufficient conditions, based on comparative statics and differential methods, for the strong existence of an optimal control. Thirdly, it shows, under similar conditions, that the value function of any optimal stopping problem is continuously differentiable. The results are applied to growth, experimentation, and dynamic contracting models and to perform sensitivity analysis in parameterized optimal stopping problems.
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