Stochastic Stefan-type Moving Boundary Problems

Moving boundary problems allow to model macroscopic systems with phase transition at an inner boundary. Recently, a stochastic and non-linear extension of the classical Stefan-problem in one space dimension came up in economics and finance as a price- time continuous model for the limit order book. More precisely, the dynamics on buy and sell side in an electronic financial markets are modeled by respective second order stochastic partial differential equations and are separated by an inner interface: the mid- price. Existence and uniqueness has been discussed for certain specifications of the problem, but is still an open question in many important setting. In this talk, we discuss an analytic framework and the recent results for this class of stochastic partial differential equations.