Semi-analytical solution of a McKean-Vlasov equation with feedback through hitting a boundary
This work provides a mathematical framework for modeling default contagion in large interconnected banking systems, but the results are incremental and primarily methodological.
The paper studies a McKean-Vlasov equation with feedback through hitting a boundary, deriving a coupled system of Volterra integral equations and providing an approximation for small interaction parameters. Numerical tests are conducted.
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.