A singular two-phase Stefan problem and particles interacting through their hitting times

TL;DR

This paper introduces a probabilistic model for a two-phase Stefan problem involving interacting particles, demonstrating existence of solutions with physically consistent discontinuities, extending previous one-phase results.

Contribution

It extends the existence theory of singular Stefan problems to a two-phase setting with non-monotone free boundaries using a novel large particle system limit approach.

Findings

Existence of solutions with physical discontinuities established.

The model applies to systemic risk in finance with interconnected banks.

A new method for non-monotone free boundary limits is developed.

Abstract

We consider a probabilistic formulation of a singular two-phase Stefan problem in one space dimension, which amounts to a coupled system of two McKean-Vlasov stochastic differential equations. In the financial context of systemic risk, this system models two competing regions with a large number of interconnected banks or firms at risk of default. Our main result shows the existence of a solution whose discontinuities obey the natural physicality condition for the problem at hand. Thus, this work extends the recent series of existence results for singular one-phase Stefan problems in one space dimension that can be found in [DIRT15a], [NS19a], [HLS18], [CRS20]. As therein, our existence result is obtained via a large system limit of a finite particle system approximation in the Skorokhod M1 topology. But, unlike for the previously studied one-phase case, the free boundary herein is not monotone, so that the large system limit is obtained by a novel argument.