Interacting Particle Systems for Systemic Risk

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Systemic Risk: Motivation Stochastic Model Simulating Systemic Shocks Examples of Rare EventsModel FeaturesThe system is self-stabilizing: when N(t) is large there are moredefaults causing X(t) to fall; when N(t) is small, birth rate ishigher.Lifetime of each individual bank is finite.High turnover microscopically; stable macroscopically.If λ − (x) > 0 is bounded away from zero then with positiveprobability the system will eventually completely collapse andregenerate. Large shocks are intrinsic.Numerous add-ons are possible: diffusion terms in X i (t); morecorrelations; default cascades; inhomogeneous dynamics, etc.9LudkovskiSystemic Risk
Systemic Risk: Motivation Stochastic Model Simulating Systemic Shocks Examples of Rare EventsRecurrencePropositionUnder technical conditions on λ − and θ, the system has an invariantdistribution on ∪ n R n .Rule out explosion (intrinsic defaults + interaction); births allowregeneration.Sufficient condition: ∫ ∞0 λ− (x 0 e rt ) dt = +∞ for all x 0 .10LudkovskiSystemic Risk
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Interacting Particle Systems for Systemic Risk

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