Spatial Risk Measures: Local Specification and Boundary Risk

We study a mathematical consistency problem motivated by the interplay between local and global risk assessment in a large financial network. In analogy to the theory of Gibbs measures in Statistical Mechanics, we focus on the structure of global convex risk measures which are consistent with a given family of local conditional risk measures. Going beyond the locally law-invariant (and hence entropic) case studied in [11], we show that a global risk measure can be characterized by its behavior on a suitable boundary field. In particular, a global risk measure may not be uniquely determined by its local specification, and this can be seen as a source of “systemic risk”, in analogy to the appearance of phase transitions in the theory of Gibbs measures. The proof combines the spatial version [10] of Dynkin’s method for constructing the entrance boundary of a Markov process with the non-linear extension [14] of backwards martingale convergence.