A stochastic Stefan-type problem under first-order boundary conditions

TL;DR

This paper develops a stochastic extension of the classical Stefan problem, modeling phase transitions with unbounded boundary variation, and establishes existence, uniqueness, and regularity of solutions using advanced stochastic PDE techniques.

Contribution

It introduces a stochastic, non-linear Stefan-type problem in one dimension, transforming it into fixed domain PDEs via Ito-Wentzell formula, and analyzes solution properties with stochastic maximal regularity.

Findings

Existence and uniqueness of local solutions are proven.

Solutions exhibit regularity properties under stochastic maximal $L^p$-regularity.

Boundary explosion phenomena can occur even with linear growth coefficients.

Abstract

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.