We propose a dynamic mean-field model for ‘systemic risk’ in large financial systems, derived from a system of interacting diffusions on the positive half-line with an absorbing boundary at the origin. These diffusions represent the distances-to-default of financial institutions, and absorption at zero corresponds to default. As a way of modelling correlated exposures and herd behaviour, we consider a common source of noise and a form of mean-reversion in the drift. Moreover, we introduce an endogenous contagion mechanism whereby the default of one institution causes a drop in the distances-to-default of the other institutions. In this way, we aim to capture key ‘system-wide’ effects on risk. The resulting mean-field limit is characterised uniquely by a nonlinear SPDE on the half-line with a Dirichlet boundary condition. The density of this SPDE gives the conditional law of a non-standard ‘conditional’ McKean–Vlasov diffusion, for which we provide a novel upper Dirichlet heat kernel type estimate. Depending on the realisations of the common noise and the rate of mean-reversion, the SPDE can exhibit rapid accelerations in the loss of mass at the boundary. In other words, the contagion mechanism can give rise to periods of significant systemic default clustering.

中文翻译：

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具有内源性传染的系统风险的SPDE模型

我们提出了一个大型金融系统中“系统风险”的动态平均场模型，该模型是从正半线上的扩散与原点具有吸收边界的相互作用的系统得出的。这些扩散代表金融机构的违约距离，零吸收对应于违约。作为对相关暴露和畜群行为进行建模的一种方式，我们考虑了常见的噪声源和漂移中均值回归的形式。此外，我们引入了一种内源性传染机制，其中一个机构的违约导致其他机构的违约距离下降。通过这种方式，我们旨在捕获关键的“全系统”风险影响。由此产生的平均场极限的唯一特征是，在半直线上具有Dirichlet边界条件的非线性SPDE。该SPDE的密度给出了非标准“条件” McKean-Vlasov扩散的条件定律，为此我们提供了一种新颖的上狄利克雷热核类型估计。取决于常见噪声和均值回复率的实现，SPDE可以在边界处的质量损失中显示出快速加速的趋势。换句话说，传染机制可能会导致出现严重的系统默认聚类。