In this talk, we discuss how optimal transport, which is a theory for matching different distributions in a cost effective way, is applied to stochastic processes, then to free boundary problems. In particular, we focus on the Stefan problem which is a free boundary problem describing the interface between water and ice.  We consider the case where mass is carried by the stochastic process, and the transportation is determined by a stopping time, a random time for stopping the process. Our approach is related to the Skorokhod problem, a classical problem in probability regarding the Brownian motion.