Title :Interacting Particles, Systemic Failure, and Blow-Up Times

Abstract: Interacting particle systems are increasingly used to model phenomena in a diverse range of fields, such as mathematical finance, neuroscience, and economic game theory. These high-dimensional models can capture the interrelated dynamics of financial firms engaged in borrowing and lending, cells firing within a brain, competing economic decision makers, and more. Herein, we model the loss-absorbing capital of financial firms engaged in a bail-in using a particle system interacting through local time reflection terms. Depending on the level of friction or subsidy, we show that the interval of existence for the bail-in arrangement is either infinite, or almost surely finite. In the large system limit, empirical averages are replaced with expectations, and one obtains a mean-field equation which captures large-scale behaviour. We relate systemic failure of the particle system to finite-time blowup in the mean-field equation, and show connections to free boundary PDEs and systemic risk models involving hitting times. Based on joint work with Philipp Jettkant (Imperial) and Ben Hambly (Oxford).