Simulation of a simple particle system interacting through hitting times

We develop an Euler-type particle method for the simulation of a McKean–Vlasov equation arising from a mean-field model with positive feedback from hitting a boundary. Under assumptions on the parameters which ensure differentiable solutions, we establish convergence of order 1/2 in the time step. Moreover, we give a modification of the scheme using Brownian bridges and local mesh refinement, which improves the order to 1. We confirm our theoretical results with numerical tests and empirically investigate cases with blow-up.