A stochastic free boundary problem

Motivated by stochastic models for order books in stock exchanges we consider stochastic partial differential equations with a free boundary condition. Such equations can be considered generalizations of the classic (deterministic) Stefan problem of heat condition in a two-phase medium. 

Extending results by Kim, Zheng & Sowers we allow for non-linear boundary interaction, general Robin-type boundary conditions and fairly general drift and diffusion coefficients. Existence of maximal local and global solutions is established by transforming the equation to a fixed-boundary problem and solving a stochastic evolution equation in suitable interpolation spaces. Based on joint work with Marvin Mueller.