Title
Particle Systems and McKean-Vlasov Dynamics with Singular Interaction through Local Times
Abstract
In this talk, I present a particle system in the positive orthant with reflection at the faces. The particles interact through the average of their accumulated local time, driving them towards the boundary. This system models the liquidity distribution in a financial system, with particles representing the banks’ liquidity positions. If a bank’s liquidity is exhausted, it will engage in liquidity-raising activities (modelled by the local time), which draw liquidity from the remaining banks (interaction through the average local time).
I will study the finite particle system and derive a corresponding mean-field limit. Interestingly, we can distinguish three regimes, depending on the interaction strength. While the system is globally well-posed in the subcritical and critical regime, a blow-up must occur if the interaction is too powerful. The latter corresponds to a systemic event, where the liquidity in the financial system dries up. I conclude the talk by reformulating the mean-field limit as the derivative of the supercooled Stefan problem.
This is joint work with Graeme Baker (Columbia University) and Ben Hambly (University of Oxford).