Semi-analytical solution of a McKean–Vlasov equation with feedback through hitting a boundary

• Published in: European journal of applied mathematics Vol. 32; no. 6; pp. 1035 - 1068
• Main Authors: LIPTON, ALEXANDER, KAUSHANSKY, VADIM, REISINGER, CHRISTOPH
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.12.2021
• Subjects: Absorption; Algorithms; Applied mathematics; Approximation; Banking industry; Banks; Default; Exact solutions; Interaction parameters; Methods; Partial differential equations; Phase transitions; Vlasov equations; Volterra integral equations; 60G99; 65R20; Volterra equations; semi-analytic solution; 35C20; structural credit contagion model; 35K58; supercooled Stefan problem; McKean–Vlasov equations
• ISSN: 0956-7925; 1469-4425
• DOI: 10.1017/S0956792519000342

In this paper, we study the nonlinear diffusion equation associated with a particle system where the common drift depends on the rate of absorption of particles at a boundary. We provide an interpretation of this equation, which is also related to the supercooled Stefan problem, as a structural credit risk model with default contagion in a large interconnected banking system. Using the method of heat potentials, we derive a coupled system of Volterra integral equations for the transition density and for the loss through absorption. An approximation by expansion is given for a small interaction parameter. We also present a numerical solution algorithm and conduct computational tests.