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

TL;DR

This paper develops a semi-analytical approach to solve a McKean-Vlasov equation modeling interconnected banking systems with default contagion, using heat potentials and integral equations.

Contribution

It introduces a novel semi-analytical method combining heat potentials and Volterra equations for this class of non-linear diffusion equations.

Findings

Derived coupled Volterra integral equations for transition density and loss

Provided an approximation method for small interaction parameters

Developed a numerical algorithm and validated it computationally

Abstract

In this paper, we study the non-linear 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 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.