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

Abstract

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 functions in $W^{1,2,p}$ and the nonlinear operator is required only to be measurable in the state and time variables. In particular, the results hold for all convex obstacles. Applied to stopping problems, they provide general conditions under which a 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 when the obstacle is the maximum of finitely many functions in $W^{1,2,p}$.