A probabilistic approach to continuous differentiability of optimal stopping boundaries

We obtain the first probabilistic proof of continuous differentiability of time-dependent optimal boundaries in optimal stopping problems. The underlying stochastic dynamics is a one-dimensional, time-inhomogeneous diffusion. The gain function is also time-inhomogeneous and not necessarily smooth. Moreover, we include time-and-state-dependent discount rate and the time-horizon can be either finite or infinite. Our arguments of proof are of a local nature that allows us to obtain the result under more general conditions than those used in the PDE literature. As a byproduct of our main result we also obtain the first probabilistic proof of the link between the value function of an optimal stopping problem and the solution of the Stefan problem.