英文标题：
《An SPDE Model for Systemic Risk with Endogenous Contagion》
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作者：
Ben Hambly, Andreas Sojmark
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最新提交年份：
2018
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英文摘要：
  We propose a dynamic mean field model for `systemic risk\' in large financial systems, which we derive 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 can cause 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 characterized 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 that is essential to the proofs. Depending on the realizations 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.
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中文摘要：
我们提出了一个大型金融系统中“系统性风险”的动态平均场模型，该模型源自于一个在原点具有吸收边界的正半线上的相互作用扩散系统。这些扩散代表金融机构违约的距离，零吸收对应违约。作为对相关暴露和羊群行为建模的一种方法，我们考虑了一种常见的噪声源和漂移中的一种均值回归形式。此外，我们引入了一种内生传染机制，其中一个机构的违约可以导致其他机构违约距离的下降。通过这种方式，我们旨在捕获对风险的关键“全系统”影响。得到的平均场极限的唯一特征是半直线上具有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，对于该定律，我们提供了一个新的上Dirichlet热核类型估计，这对证明至关重要。根据常见噪声的实现情况和平均回复率，SPDE可以在边界处质量损失中表现出快速加速。换言之，这种传染机制可能会导致一段时期的重大系统性违约集群。
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分类信息：

一级分类：Mathematics        数学
二级分类：Probability        概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Quantitative Finance        数量金融学
二级分类：Mathematical Finance        数学金融学
分类描述：Mathematical and analytical methods of finance, including stochastic, probabilistic and functional analysis, algebraic, geometric and other methods
金融的数学和分析方法，包括随机、概率和泛函分析、代数、几何和其他方法
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