Hans Foellmer "Spatial risk measures: Local specification, aggregation, and phase transition"

The quantification of financial risk in terms of convex risk measures is closely related to the microeconomic theory of preferences in the face of risk and Knightian uncertainty. We discuss some of these connections and then turn to the implications of dynamic or spatial consistency for convex risk measures on product spaces. In the spatial setting of a large network, the local specification of convex risk measures can be seen as a non-linear extension of the local specification of equilibrium states in Statistical Mechanics. We discuss the corresponding aggregation problem of passing from local to global risk measures andthe appearance of phase transitions, using a combination of arguments from preference theory and from Dynkin's boundary theory for Markov processes.