TY - GEN
T1 - Spatial risk measures
T2 - Conference on Stochastic Analysis and Applications, 2013
AU - Föllmer, Hans
AU - Klüppelberg, Claudia
N1 - Publisher Copyright:
© Springer International Publishing Switzerland 2014.
PY - 2014
Y1 - 2014
N2 - 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 ofGibbs 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.
AB - 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 ofGibbs 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.
KW - Convex risk measure
KW - Phase transition
KW - Spatial riskmeasure
KW - Systemic risk
UR - http://www.scopus.com/inward/record.url?scp=84919415672&partnerID=8YFLogxK
U2 - 10.1007/978-3-319-11292-3_11
DO - 10.1007/978-3-319-11292-3_11
M3 - Conference contribution
AN - SCOPUS:84919415672
T3 - Springer Proceedings in Mathematics and Statistics
SP - 307
EP - 326
BT - Stochastic Analysis and Applications 2014
A2 - Crisan, Dan
A2 - Hambly, Ben
A2 - Zariphopoulou, Thaleia
PB - Springer New York LLC
Y2 - 23 September 2013 through 27 September 2013
ER -