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Current version: August 2024.
(Supersedes and expands on Theorem 3 in Smoothness of Value functions in General Control-Stopping Diffusion Problems)
We establish the existence, uniqueness, and W1,2,p-regularity of solutions to fully nonlinear parabolic obstacle problems when the obstacle is the pointwise supremum of arbitrary functions in W1,2,p, and the operator is only assumed to be measurable in the state and time variables. The results hold for a large class of non-smooth obstacles, including all convex obstacles. Applied to stopping problems, they imply that the decision maker never stops at a convex kink of the stopping payoff. The proof relies on new W1,2,p-estimates for obstacle problems where the obstacle is the maximum of finitely many functions in W1,2,p.
Current version: August 2024.
How do decisions change with the economic environment and with time? This paper studies general nonstationary stopping problems and provides the methodological tools to answer these questions. First, we identify conditions that ensure a monotone relation between decisions’ timing and outcomes. These conditions apply to a prevalent class of economic environments. Second, we develop a theory of monotone comparative statics for stopping problems, offering general and unifying qualitative insights into the decision-maker’s value and stopping behavior. We apply our results to models of information acquisition, bankruptcy, irreversible investment, and option pricing to explain documented patterns at odds with current theories.
Current version: November 2022 (New version coming soon!).
We derive properties of value functions in mixed optimal control and stopping problems in which (i) the state variable may be multi-dimensional, (ii) the domain may be unbounded and irregular, and (iii) primitives may be time-dependent. We show that the value function is the unique Lp-solution of the Hamilton-Jacobi-Bellman equation and that it is twice parabolically differentiable a.e. and continuously differentiable in the state variable under general conditions that accommodate most economic applications. We show that the smooth-pasting property must hold everywhere with respect to all non-time variables and provide sufficient conditions under which smooth pasting must also hold with respect to time. Our results imply that numerical solutions obtained by standard methods converge uniformly to the value function.
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