By Vol 7

ISBN-10: 4431243321

ISBN-13: 9784431243328

ISBN-10: 443127233X

ISBN-13: 9784431272335

Loads of monetary difficulties may be formulated as limited optimizations and equilibration in their strategies. numerous mathematical theories were providing economists with fundamental machineries for those difficulties coming up in financial thought. Conversely, mathematicians were encouraged through quite a few mathematical problems raised via financial theories. The sequence is designed to collect these mathematicians who're heavily attracted to getting new difficult stimuli from monetary theories with these economists who're seeking effective mathematical instruments for his or her examine. The editorial board of this sequence includes 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 kingdom Univ.), A. Ioffe (Israel Institute of Technology), S. Iwamoto (Kyushu Univ.), ok. Kamiya (Univ. Tokyo), ok. Kawamata (Keio Univ.), N. Kikuchi (Keio Univ.), H. Matano (Univ. Tokyo), ok. 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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Theorem 7. Let (1^, ^ , P) be a probability space with P atomless and p : The following are equivalent: Hi) p is a law invariant convex risk measure satisfying axioms (mon) and (ti); iv) there exist a non empty convex set M ofprobability measures and a convex functional G : L^ -^ RU {+00} with infg^^i G {(fq) = 0 such that p (X) = sup i Pa (X) ruQ {da) - G ( ^ Q ) V , QeM [7(0,1] VX 6 L^, (9) J where, for any Q e My TUQ is a probability measure on (0,1] defined by rriQ (da) = a dZ^^ (1 - <^)Note that the definition of law invariance depends on the probability measure P given a priori, hence it is not surprising that in the representation of law invariant convex risk measures the set M of generalized scenarios will be dependent on P.

Proposition 5. (Proposition 22 Kusuoka (2001)) For any a G [0,1], pa is a law invariant coherent risk measure satisfying (Isc). Theorem 6. (Proposition 15 Kusuoka (2(X)1)). Let (fi, ^ , P) be a probability space with P atomless and p : L^ -^ E. The following are equivalent: i) p is a law invariant coherent risk measure satisfying (Isc); ii) there exists a set SPTQ ofprobability measures on (0,1] such that p {X)= ia)l, sup { / Pa {X)m (da) ^ , VX G L^. meOHo I ^(0,1] (8) Law invariant convex risk measures 37 In Theorem 7 we will provide the characterization of the larger class of law invariant convexriskmeasures (with some further axioms).

Also we study demand functions that can be rationalized by continuous (Theorem 5) and by smooth (Theorem 6) utility functions. In that Section, all the results are based on nonemptiness conditions for the set QO(CA) and its subset QI(CA) := {u e QO(CA) : fip) ^ f{p') ^ u(jp) - u{p') < ^x{p,p')}. Section 4 is devoted to demand functions that are rationalized by positive homogeneous utility functions. Here the cost function ^ is used. In Theorem 7, that generalizes the corresponding variant of the AfriatVarian theory, necessary and sufficient conditions are given for an insatiate demand function / to be rationalized by a positive homogeneous (continuous) concave utility function, which is strictly positive on / ( P ) .

### Advances in Mathematical Economics by Vol 7

by Ronald

4.4