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SCHUR CONVEX FUNCTIONALS: FATOU PROPERTY AND REPRESENTATION
Author(s) -
Grechuk Bogdan,
Zabarankin Michael
Publication year - 2012
Publication title -
mathematical finance
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.98
H-Index - 81
eISSN - 1467-9965
pISSN - 0960-1627
DOI - 10.1111/j.1467-9965.2010.00464.x
Subject(s) - mathematics , invariant (physics) , pure mathematics , probability measure , regular polygon , quantile , property (philosophy) , mathematical analysis , econometrics , geometry , philosophy , epistemology , mathematical physics
The Fatou property for every Schur convex lower semicontinuous (l.s.c.) functional on a general probability space is established. As a result, the existing quantile representations for Schur convex l.s.c. positively homogeneous convex functionals, established on for either p = 1 or p =∞ and with the requirement of the Fatou property, are generalized for , with no requirement of the Fatou property. In particular, the existing quantile representations for law invariant coherent risk measures and law invariant deviation measures on an atomless probability space are extended for a general probability space.