Numerical analysis of a McKean--Vlasov equation with feedback through hitting the boundary by Prof. Vadim Kaushanskii (UCLA)
Abstract: We study a McKean--Vlasov equation arising from a mean-field model with positive feedback from hitting a boundary and provide an interpretation as a structural credit risk model with default contagion in a large interconnected banking system.
First, we develop an Euler-type particle method for the simulation of the equation. We establish convergence of our method before and after the jump. Before the jump we establish the convergence of order $1/2$. Moreover, we give a modification of the scheme using Brownian bridges and local mesh refinement, which improves the order to $1$. After the jump we show that the convergence might be arbitrary slow. We confirm our theoretical results with numerical tests and empirically investigate cases with blow-up.
Second, we study the corresponding non-linear diffusion equation. 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.