Existence, uniqueness, and regularity of solutions to nonlinear and non-smooth parabolic obstacle problems

We establish the existence, uniqueness, and W^1, 2, p-regularity of solutions to fully nonlinear parabolic obstacle problems when the obstacle is the pointwise supremum of arbitrary functions in W^1, 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 W^1, 2, p-estimates for obstacle problems where the obstacle is the maximum of finitely many functions in W^1, 2, p.