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Loads of monetary difficulties should be formulated as restricted optimizations and equilibration in their recommendations. numerous mathematical theories were delivering economists with imperative machineries for those difficulties bobbing up in monetary conception. Conversely, mathematicians were prompted via quite a few mathematical problems raised by way of fiscal theories. The sequence is designed to assemble 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 learn. The editorial board of this sequence includes the next favourite 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 country 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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It remains to show that p satisfies convexity and lower semi-continuity. x {x) Z^^ [x) dx = sup [EQ [-X]; 0 < P : ^^ ~ ^ Q } . (23) Law invariant convexriskmeasures 43 Convexity: from identity (23) it follows that for any X,Y e L°° and for any a € [0,1] piaX+{l-a)Y) = snp\ Sup

A^|,([0,1]),CE([0,1])) which shows that the vector measure —u — W,dt belongs to the subdifferential dlf^ of the of the convex functional integral / / ^ defined on CE{[0, 1]) by If^ {v) := f^ f^{t, v{t)) dt, Wv G CE{[0, 1]) . Using Assumption 4), we have sup^/^ fn{t^Un{t))dt < -hoo. Consequently l i m / fnit, Unit)) dt = / fooit,u{t)) dt <-\-oo. 13) with w = u^ integrating on [0,1], taking Assumption 2) into account, and passing to the limit when n goes to oo, gives the required inequality because (un) converges uniformly to u and {—ün{-) — Wn{^)) is bounded in Li;([0,1]).

Atti Sem. Mat Fis. : Optimal control problems and variational problems. Tech. : Control problems governed by functional evolution inclusions with Young measures. J. Nonlinear Convex Anal. : On the fiber product of Young measures with application to a control problem with measures. Adv. Math. Econ. : Young measures on Topological Spaces. With Applications in Control Theory and Probability Theory. : Dunford-Pettis-types theorem and convergences in set-valued integration. J. Nonlinear Convex Anal.

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