Advances in mathematical economics kusuoka maruyama volume 13

www.elsolucionario.org www.SolutionManual.info Managing Editors Shigeo Kusuoka The University of Tokyo Tokyo, JAPAN Toru Maruyama Keio University Tokyo, JAPAN Editors Robert Anderson University of California, Berkeley Berkeley, U.S.A Charles Castaing Universit´e Montpellier II Montpellier, FRANCE Takao Fujimoto Fukuoka University Fukuoka, JAPAN Jean-Michel Grandmont CREST-CNRS Malakoff, FRANCE Francis H Clarke Universit´e de Lyon I Villeurbanne, FRANCE Norimichi Hirano Yokohama National University Yokohama, JAPAN Egbert Dierker University of Vienna Vienna, AUSTRIA Tatsuro Ichiishi The Ohio State University Ohio, U.S.A Darrell Duffie Stanford University Stanford, U.S.A Alexander Ioffe Israel Institute of Technology Haifa, ISRAEL Lawrence C Evans University of California, Berkeley Berkeley, U.S.A Seiichi Iwamoto Kyushu University Fukuoka, JAPAN Kazuya Kamiya The University of Tokyo Tokyo, JAPAN Kunio Kawamata Keio University Tokyo, JAPAN Hiroshi Matano The University of Tokyo Tokyo, JAPAN Kazuo Nishimura Kyoto University Kyoto, JAPAN Marcel K Richter University of Minnesota Minneapolis, U.S.A Yoichiro Takahashi Kyoto University Kyoto, JAPAN Makoto Yano Kyoto University Kyoto, JAPAN Aims and Scope The project is to publish Advances in Mathematical Economics once a year under the auspices of the Research Center for Mathematical Economics It is designed to bring together those mathematicians who are seriously interested in obtaining new challenging stimuli from economic theories and those economists who are seeking effective mathematical tools for their research The scope of Advances in Mathematical Economics includes, but is not limited to, the following fields: – Economic theories in various fields based on rigorous mathematical reasoning – Mathematical methods (e.g., analysis, algebra, geometry, probability) motivated by economic theories – Mathematical results of potential relevance to economic theory – Historical study of mathematical economics Authors are asked to develop their original results as fully as possible and also to give a clear-cut expository overview of the problem under discussion Consequently, we will also invite articles which might be considered too long for publication in journals www.SolutionManual.info www.elsolucionario.org S Kusuoka, T Maruyama (Eds.) Advances in Mathematical Economics Volume 13 123 Shigeo Kusuoka Professor Graduate School of Mathematical Sciences The University of Tokyo 3-8-1 Komaba, Meguro-ku Tokyo 153-0041, Japan Toru Maruyama Professor Department of Economics Keio University 2-15-45 Mita, Minato-ku Tokyo 108-8345, Japan www.SolutionManual.info ISSN 1866-2226 e-ISSN 1866-2234 ISBN 978-4-431-99489-3 e-ISBN 978-4-431-99490-9 DOI 10.1007/978-4-431-99490-9 Springer Tokyo Dordrecht Heidelberg London New York c Springer 2010 This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Table of Contents Research Articles F Akhiat, C Castaing, and F Ezzaki Some various convergence results for multivalued martingales J Honda and S.-I Takekuma A note on Aumann’s core equivalence theorem without monotonicity 35 M.A Khan and A.J Zaslavski On two classical turnpike results for the Robinson–Solow– Srinivasan model 47 S Kusuoka A certain limit of iterated conditional tail expectation 99 T.Q Bao and B.S Mordukhovich Set-valued optimization in welfare economics 113 N Sagara and M Vlach Convexity of the lower partition range of a concave vector measure 155 A.J Zaslavski Good locally maximal programs for the Robinson–Solow– Srinivasan model 161 Historical Perspective S T Lowry Pythagorean mathematical idealism and the framing of economic and political theory 177 Subject Index 201 Instructions for Authors 207 V www.elsolucionario.org Adv Math Econ 13, 1–33 (2010) Some various convergence results for multivalued martingales Fettah Akhiat1 , Charles Castaing2 , and Fatima Ezzaki1 Laboratoire mod´elisation et calcul scientifique, D´epartement de Math´ematiques, Facult´e des Sciences et Techniques, BP 2202, Universit´e Sidi Mohamed Ben Abdellah, Fes, Morocco (e-mail: akhiatfettah@yahoo.fr, fatimaezzaki@yahoo.fr) D´epartement de Math´ematiques, Universit´e Montpellier II, 34095 Montpellier Cedex 5, France (e-mail: castaing.charles@numericable.fr) Received: April 26, 2009 Revised: October 15, 2009 JEL classification: C01, C02, G12 www.SolutionManual.info Mathematics Subject Classification (2000): 28B20, 60G42, 46A17, 54A20 Abstract We prove various convergence results for multivalued martingales, subsupermartingales and mils with respect to the Mosco topology and the linear topology both in Bochner integration and Pettis integration We also state some existence theorems of Pettis conditional expectation for multivalued Pettis-integrable multifunctions Key words: martingale, submartingale, supermartingale, mil, conditional expectation, Mosco convergence, linear topology, Pettis Introduction The purpose of this paper is to present various convergence results for martingales, submartingales, supermartingales and mils with respect to the Mosco topology and the linear topology both in Bochner integration and Pettis integration The paper is organized as follows In  2, we set our notation and definitions, and summarize needed results In  3, we state some convergence We heartily thank the referee for helpful comments and suggestions S Kusuoka, T Maruyama (eds.), Advances in Mathematical Economics Volume 13, DOI: 10.1007/978-4-431-99490-9 1, c Springer 2010 F Akhiat et al theorems for convex weakly compact valued submartingales, mils and unbounded closed convex supermartingales in Bochner integration In  4, we present various convergence theorems for convex weakly compact valued Pettis-integrable multifunctions Several existence theorems of conditional expectation for convex weakly compact valued Pettis-integrable multifunctions and an integral representation theorem for a convex weakly compact valued multifunction defined on L1 or more generally on a Kăothe space are also provided In  5, we provide some versions of Levy’s theorem for Pettis-integrable multifunctions Specific applications to the convergence of multivalued Pettis-integrable martingales are given at the end of  To our knowledge, not many results in this area are known We refer to [17–21, 25, 27–30, 32–35, 37, 40–42] for related results on martingales and mils The results presented here are motivated by some applications in Mathematical Economics and the Law of Large Numbers (see, e.g [8,9,12,15,26]) Preliminaries and background Let ; F ; P / be a complete probability space, Fn /n2N an increasing sequence of sub -algebras of F such that F is the -algebra generated by [n Fn Let E be a separable Banach space, E the topological dual of E, D D xj /j 2N a dense sequence in E with respect to the Mackey topology E ; E/, B E (resp B E ) the closed unit ball of E (resp E ) Let cc.E/ (resp cwk.E/) (resp ck.E/) be the set of nonempty closed convex (resp weakly compact convex) (resp compact convex) subsets of E Given C cc.E/, the distance function and the support function associated with C are defined respectively by d.x; C / D inffkx yk; y C g x E/ ı x ; C / D supf< x ; y >; y C g x E /: A cc.E/-valued mapping C W ! cc.E/ is F -measurable, if its graph belongs to F ˝ B.E/, where B.E/ is the Borel tribe of E For any C cc.E/, we set jC j D supfkxk W x C g: For any A; B cwk.E/, the Hausdorff distance between A and B is denoted by H.A; B/ D sup jı x ; A/ ı x ; B/j: x 2B E For each n N[f1g, we denote by L1cwk.E / Fn / (with F1 D F ) the space ! cwk.E/ of all Fn -measurable cwk.E/-valued multifunctions X W Some various convergence results for multivalued martingales such that ! ! jX.!/j is integrable A sequence Xn /n2N of cc.E/-valued multifunctions (mappings for short) is adapted if each Xn is Fn -measurable A sequence Xn /n2N in L1cwk.E / F / is bounded (resp uniformly integrable) if the sequence jXn j/n2N is bounded (resp uniformly integrable) in L1R F / A F -measurable closed convex valued multifunction X W ) E is integrable if it admits an integrable selection, equivalently if d.0; X / is integrable A closed convex set C in E is ball-weakly compact, if its intersection with any closed ball in E is weakly compact A cc.E/-valued sequence Xn /n2N Mosco-converges [36] to a closed convex set X1 if X1 D s-liXn D w-lsXn ; where s-li Xn D fx E W jjxn xjj ! 0I xn Xn g and w-ls Xn D fx E W x D w- lim xj ; xj Xnj g j !1 and s (resp w) is the strong (resp weak) topology in E If Xn /n2N Moscoconverges to X1 in cc.E/, we write M - lim Xn D X1 : n!1 The linear topology ing topologies: L [3] on cc.E/ is the upper bounded of the two follow- www.SolutionManual.info (a) The topology of pointwise convergence of support functions on E (b) The topology of pointwise convergence of distance functions on E Proposition 2.1 Let Cn /n2N[f1g be a sequence in cc.E/, then C1 D limn!1 Cn () L i / ı x ; C1 / D limn!1 ı x ; Cn / 8x E i i / d.x; C1 / D limn!1 d.x; Cn / 8x E Beer [3] showed that the topology L is stronger than the Mosco topology If the strong dual Eb of E is separable and X is an element of L1cwk.E / F /, for each n N, it is known that the conditional expectation of X with respect to Fn , E Fn X , is the unique element (for = a.s.) of L1cwk.E / Fn / such that Z Z Fn ı u.!/; E X.!//dP !/ D ı u.!/; X.!//dP !/ for all u L1 E Fn / See [43, Remark of Theorem 3] For more information for the conditional expectation of multifunctions, we refer to [27, 43] ... invite articles which might be considered too long for publication in journals www.SolutionManual.info www.elsolucionario.org S Kusuoka, T Maruyama (Eds.) Advances in Mathematical Economics Volume. .. are seeking effective mathematical tools for their research The scope of Advances in Mathematical Economics includes, but is not limited to, the following fields: – Economic theories in various... Grant -in- Aid for Scientific Research (No 18610003) from the Ministry of Education, Culture, Sports, Science and Technology, Japan S Kusuoka, T Maruyama (eds.), Advances in Mathematical Economics Volume