  Moving boundary problems allow to model systems with phase transition at an
inner boundary. Driven by problems in economics and finance, in particular
modeling of limit order books, we consider a stochastic and non-linear
extension of the classical Stefan-problem in one space dimension, where the
paths of the moving interface might have unbounded variation. Working on the
distribution space, Ito-Wentzell formula for SPDEs allows to transform these
moving boundary problems into partial differential equations on fixed domains.
Rewriting the equations into the framework of stochastic evolution equations,
we apply results based on stochastic maximal $L^p$-regularity to obtain
existence, uniqueness and regularity of local solutions. Moreover, we observe
that explosion might take place due to the boundary interaction even when the
coefficients of the original problem have linear growths.
