University of Munich

 One of the most defining features of modern financial networks is their inherent complex and intertwined structure. In particular the often observed core-periphery structure plays a prominent role. Here we study and quantify the impact that the complexity of networks has on contagion effects and system stability, and our focus is on the channel of default contagion that describes the spread of initial distress via direct balance sheet exposures. We present a general approach describing the financial network by a random graph, where we distinguish vertices (institutions) of different types - for example core/periphery - and let edge probabilities and weights (exposures) depend on the types of both the receiving and the sending vertex. Our main result allows to compute explicitly the systemic damage caused by some initial local shock event, and we derive a complete characterization of resilient respectively non-resilient financial systems in terms of their global statistical characteristics. Due to the random graphs approach these results bear a considerable robustness to local uncertainties and small changes of the network structure over time. Applications of our theory demonstrate that indeed the features captured by our model can have significant impact on system stability; we derive resilience conditions for the global network based on subnetwork conditions only. 

We consider the stationary flow of Prandtl-Eyring fluids in two

dimensions. This model is a good approximation of perfect plasticity.

The corresponding potential is only slightly super linear. Thus, many

severe problems arise in the existence theory of weak solutions. These

problems are overcome by use of a divergence free Lipschitz

truncation. As a second application of this technique, we generalize

the concept of almost harmonic functions to the Stokes system.