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**Laplacian Growth Models**## Singular and smoothed models for systemic risk with endogenous contagion

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Christoph Reisinger, University of Oxford
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**Tuesday, June 27, 2023**

**Abstract**: We consider a mean-field model for systemic risk, in which the values of financial firms are given by a system of interacting, absorbed diffusions on the positive half-line, with both idiosyncratic and common noise. Absorption of one component transmits losses to the surviving entities smoothly over a short time $epsilon$. Hambly & Sojmark, Fin. Stoch., 23(3), 2019, show that, as the number of firms goes to infinity, the empirical measure of the underlying particle system converges to the unique solution of a nonlinear SPDE, which can be characterised as the conditional law of the solution to a McKean—Vlasov equation. In this work, we demonstrate that as $epsilon$, which is interpretable as a regularisation parameter, goes to zero, the solution of the McKean—Vlasov equation converges in law to a solution of a singular equation with instantaneous interaction, and where a physical condition gives an upper bound on possible discontinuities in the loss process. This allows the construction of solutions to the limiting equation in greater generality than previously, including variable coefficients with certain state and measure dependence and under common noise. In a special case with constant coefficients and sufficiently regular initial data, we derive a rate of convergence. Numerical results illustrate the theoretical findings. Joint work with Ben Hambly and Aldair Petronilia.