Föllmer, Hans Spatial risk measures and their local specification: the locally law-invariant case. (English) Zbl 1345.91011 Stat. Risk. Model. 31, No. 1, 79-101 (2014). Summary: We consider convex risk measures in a spatial setting, where the outcome of a financial position depends on the states at different nodes of a network. In analogy to the theory of Gibbs measures in Statistical Mechanics, we discuss the local specification of a global risk measure in terms of conditional local risk measures for the single nodes of the network, given their environment. Under a condition of local law invariance, we show that a consistent local specification must be of entropic form. Even in that case, a global risk measure may not be uniquely determined by the 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. Cited in 2 ReviewsCited in 7 Documents MSC: 91B30 Risk theory, insurance (MSC2010) 91G70 Statistical methods; risk measures 91G40 Credit risk 91B72 Spatial models in economics 60G30 Continuity and singularity of induced measures Keywords:convex risk measure; spatial risk measure; entropic risk measure; phase transition; systemic risk PDFBibTeX XMLCite \textit{H. Föllmer}, Stat. Risk. Model. 31, No. 1, 79--101 (2014; Zbl 1345.91011) Full Text: DOI Link