Advances in Mathematical Economics, 1st Edition by Vol 7

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Loads of financial difficulties will be formulated as limited optimizations and equilibration in their ideas. numerous mathematical theories were offering economists with essential machineries for those difficulties coming up in financial idea. Conversely, mathematicians were influenced through a variety of mathematical problems raised via financial theories. The sequence is designed to collect these mathematicians who're heavily attracted to getting new demanding stimuli from financial theories with these economists who're seeking effective mathematical instruments for his or her study. The editorial board of this sequence contains the next well known economists and mathematicians: Managing Editors : S. Kusuoka (Univ. Tokyo), T. Maruyama (Keio Univ.). Editors : R. Anderson (U.C. Berkeley), C. Castaing (Univ. Montpellier), F.H. Clarke (Univ. Lyon I), G. Debreu (U.C. Berkeley), E. Dierker (Univ. Vienna), D. Duffie (Stanford Univ.), L.C. Evans (U.C. Berkeley), T. Fujimoto (Okayama Univ.), J.-M. Grandmont (CREST-CNRS), N. Hirano (Yokohama nationwide Univ.), L. Hurwicz (Univ. of Minnesota), T. Ichiishi (Ohio nation Univ.), A. Ioffe (Israel Institute of Technology), S. Iwamoto (Kyushu Univ.), ok. Kamiya (Univ. Tokyo), okay. Kawamata (Keio Univ.), N. Kikuchi (Keio Univ.), H. Matano (Univ. Tokyo), okay. Nishimura (Kyoto Univ.), M.K. Richter (Univ. Minnesota), Y. Takahashi (Kyoto Univ.), M. Valadier (Univ. Montpellier II), A. Yamaguti (Kyoto Univ./Ryukoku Univ.), M. Yano (Keio Univ.).

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Indeed, if /9 is a law invariant coherent risk measure satisfying (Isc), then G{(fQ) = 0 for any Q e M. So, by setting MQ = {rriQ : Q € M}, the representation in (9) can be rewritten as p{X) = sup M Pa iX)m{da) \ , yX e L^ meMo I 7(0,1] J Proof of Proposition 8. s. constant and such that p {Y) = —p {—¥). From Theorem 6, we know that there exists a set SUIQ of probability measures on (0,1] such that p{X) = sup M mefmo I J(o,i] Pa {X)m{da) V , VX G L° J Hence -p{-¥) = inf \ / -pai-Y) mSOTo I 7(0,1] m (da) \ < inf \ / I "»€OTo I 7(0,1] p^{Y) m (da) \ , J 44 M.

Finally, Section 5 is devoted to a stronger variant of demand rationalizing. In that variant, the budget constraint is rejected and the gain to be maximized by a consumer equals utility minus expences. We say that / is induced (resp. strictly induced) by a utility function U if f{p) e ArgmaxC/P \/p e P (resp. if f{p) = argmaxC/^ Vp G P ) , where the gain U^{q) := U{q) — p - q. J: that are induced by upper semi-continuous (use) concave utility functions U with domU 2 f{P)- Here cost functions C and C"^ are used.

34 M. Frittelli, E. Rosazza Gianin The main result of this paper (see Theorem 7) is the representation of the wider class of law invariant convex risk measures. Moreover, thanks to the characterization of law invariant coherent and convex risk measures (the former by Kusuoka (2001), the latter proved in Section 2), we will study two degenerate classes of law invariant risk measures. Proposition 8 is a minor extension of a result of Castagnoli et al. (2004) and characterizes a degenerate class of coherent risk measures.

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