Simulation of particle systems interacting through hitting times

Understand

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.