  In this paper we study the effect of stochastic perturbations on a common
type of moving boundary value PDE's which endorse Stefan boundary conditions,
or Stefan problems, and show the existence and uniqueness of the solutions to a
number of stochastic equations of this kind. Moreover we also derive the space
and time regularities of the solutions and the associated boundaries via
Kolmogorov's Continuity Theorem in an appropriately defined normed space.
  The paper first conveys our previous results where randomness is smoothly
correlated in space and Brownian in time, then introduces a new methodology
that enables us to prove the existence and uniqueness of the solution to the
standard heat equation driven by a space-time Brownian sheet as well as its
boundary regularity, and finally extends it to the stochastic moving boundary
partial PDE's driven by the same type of randomness.
