Systemic Robustness: A Mean-Field Particle System Approach

Abstract

This paper is concerned with the problem of capital provision in a large particle system modeled by stochastic differential equations involving hitting times, which arises from considerations of systemic risk in a financial network. Motivated by Tang and Tsai, we focus on the number or proportion of surviving entities that never default to measure the systemic robustness. First we show that the mean-field particle system and its limit McKean–Vlasov equation are both well-posed by virtue of the notion of minimal solutions. We then establish a connection between the proportion of surviving entities in the large particle system and the probability of default in the McKean–Vlasov equation as the size of the interacting particle system $N$ tends to infinity. Finally, we study the asymptotic efficiency of capital provision for different drift $\beta$, which is linked to the economy regime: The expected number of surviving entities has a uniform upper bound if ; it is of order $\sqrt{N}$ if $\beta =0$; and it is of order $N$ if $\beta >0$, where the effect of capital provision is negligible.