  We consider a general tractable model for default contagion and systemic risk
in a heterogeneous financial network, subject to an exogenous macroeconomic
shock. We show that, under some regularity assumptions, the default cascade
model could be transferred to a death process problem represented by
balls-and-bins model. We also reduce the dimension of the problem by
classifying banks according to different types, in an appropriate type space.
These types may be calibrated to real-world data by using machine learning
techniques. We then state various limit theorems regarding the final size of
default cascade over different types. In particular, under suitable assumptions
on the degree and threshold distributions, we show that the final size of
default cascade has asymptotically Gaussian fluctuations. We next state limit
theorems for different system-wide wealth aggregation functions and show how
the systemic risk measure, in a given stress test scenario, could be related to
the structure and heterogeneity of financial networks. We finally show how
these results could be used by a social planner to optimally target
interventions during a financial crisis, with a budget constraint and under
partial information of the financial network.
