A Stefan-type stochastic moving boundary problem

TL;DR

This paper investigates a class of stochastic PDEs with Stefan-type boundaries motivated by financial applications, establishing existence, uniqueness, and conditions for global solutions through advanced mathematical techniques.

Contribution

It introduces a novel approach to handle stochastic Stefan-type problems by transforming them into fixed boundary stochastic evolution equations, extending previous results.

Findings

Proved local existence and uniqueness of strong solutions.

Derived conditions for the existence of global solutions.

Analyzed potential finite-time blow-up behavior.

Abstract

Motivated by applications in economics and finance, in particular to the modeling of limit order books, we study a class of stochastic second-order PDEs with non-linear Stefan-type boundary interaction. To solve the equation we transform the problem from a moving boundary problem into a stochastic evolution equation with fixed boundary conditions. Using results from interpolation theory we obtain existence and uniqueness of local strong solutions, extending results of Kim, Zheng and Sowers. In addition, we formulate conditions for existence of global solutions and provide a refined analysis of possible blow-up behavior in finite time.