Year: 2024
Type: article
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. As for the one-phase problems, 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 necessarily monotone, so that the large system limit is obtained by a novel argument.
DOI: 10.1214/24-aap2072
Authors: Graeme Baker, Mykhaylo Shkolnikov
Institutions: Columbia University, Carnegie Mellon University
Language: English
Subfield: Finance
Domain: Social Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W4226402333