  We study the nonlocal Stefan problem, where the phase transition is described
by a nonlocal diffusion as well as the change of enthalpy functions. By using a
stochastic optimization approach introduced for the local case, we construct
global-time weak solutions and give a probabilistic interpretation for the
solutions. An important ingredient in our analysis is a probabilistic
interpretation of the enthalpy and temperature variables in terms of a particle
system. Our approach in particular establishes the connection between the
parabolic obstacle problem and the Stefan Problem for the nonlocal diffusions.
For the melting problem, we show that our solution coincides with those studied
in the literature.
