Managing Editors Shigeo Kusuoka Toru Maruyama The University of Tokyo Tokyo, JAPAN Keio University Tokyo, JAPAN Editors Jean-Michel Grandmont Robert Anderson CREST-CNRS University of California, Malakoff, FRANCE Berkeley Berkeley, U.S.A Kunio Kawamata Keio University Tokyo, JAPAN Hiroshi Matano Norimichi Hirano Charles Castaing The University of Tokyo Yokohama National Université Montpellier II Tokyo, JAPAN University Montpellier, FRANCE Yokohama, JAPAN Kazuo Nishimura Francis H Clarke Kyoto University Université de Lyon I Kyoto, JAPAN Villeurbanne, FRANCE Tatsuro Ichiishi The Ohio State University Ohio, U.S.A Egbert Dierker Yoichiro Takahashi University of Vienna The University of Tokyo Vienna, AUSTRIA Tokyo, JAPAN Alexander D Ioffe Israel Institute of Darrell Duffie Akira Yamazaki Technology Stanford University Hitotsubashi University Haifa, ISRAEL Stanford, U.S.A Tokyo, JAPAN Lawrence C Evans Makoto Yano University of California, Seiichi Iwamoto Kyushu University Kyoto University Berkeley Fukuoka, JAPAN Kyoto, JAPAN Berkeley, U.S.A Takao Fujimoto Fukuoka University Fukuoka, JAPAN Kazuya Kamiya The University of Tokyo Tokyo, 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 Shigeo Kusuoka • Toru Maruyama Editors Advances in Mathematical Economics Volume 19 123 Editors Shigeo Kusuoka Professor Emeritus The University of Tokyo Tokyo, Japan Toru Maruyama Professor Emeritus Keio University Tokyo, Japan ISSN 1866-2226 ISSN 1866-2234 (electronic) Advances in Mathematical Economics ISBN 978-4-431-55488-2 ISBN 978-4-431-55489-9 (eBook) DOI 10.1007/978-4-431-55489-9 Springer Tokyo Heidelberg New York Dordrecht London © Springer Japan 2015 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, 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 The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made Printed on acid-free paper Contents On the Integration of Fuzzy Level Sets Charles Castaing, Christiane Godet-Thobie, Thi Duyen Hoang, and P Raynaud de Fitte A Theory for Estimating Consumer’s Preference from Demand Yuhki Hosoya 33 Least Square Regression Methods for Bermudan Derivatives and Systems of Functions Shigeo Kusuoka and Yusuke Morimoto Discrete Time Optimal Control Problems on Large Intervals Alexander J Zaslavski 57 91 Index 137 v Adv Math Econ 19, 1–32 (2015) On the Integration of Fuzzy Level Sets Charles Castaing, Christiane Godet-Thobie, Thi Duyen Hoang, and P Raynaud de Fitte Abstract We study the integration of fuzzy level sets associated with a fuzzy random variable when the underlying space is a separable Banach space or a weak star dual of a separable Banach space In particular, the expectation and the conditional expectation of fuzzy level sets in this setting are presented We prove the SLLN for pairwise independent identically distributed fuzzy convex compact valued level sets through the SLLN for pairwise independent identically distributed convex compact valued random set in separable Banach space Some convergence results for a class of integrand martingale are also presented JEL Classification: C01, C02 Mathematics Subject Classification (2010): 28B20, 60G42, 46A17, 54A20 C Castaing Département de Mathématiques, Case courrier 051, Université Montpellier II, 34095 Montpellier Cedex 5, France e-mail: charles.castaing@gmail.com C Godet-Thobie Laboratoire de Mathématiques de Bretagne Atlantique, Université de Brest, UMR CNRS 6295, 6, avenue Victor Le Gorgeu, CS 9387, F-29238 Brest Cedex3, France e-mail: christiane.godet-thobie@univ-brest.fr T.D Hoang Quang Binh University, Quang Binh, Viet Nam e-mail: hoangduyen267@gmail.com P Raynaud de Fitte ( ) Laboratoire Raphaël Salem, UFR Sciences, Université de Rouen, UMR CNRS 6085, Avenue de l’Université, BP 12, 76801 Saint Etienne du Rouvray, France e-mail: prf@univ-rouen.fr C Castaing et al Keywords Conditional expectation • Fuzzy convex • Fuzzy martingale • Integrand martingale • Level set • Upper semicontinuous Article type: Research Article Received: November 5, 2014 Revised: December 1, 2014 Introduction The study of fuzzy set-valued variables was initiated by Feron [10], Kruse [18], Kwakernaak [19, 20], Puri and Ralescu [25], Zadeh [30] In particular, Puri and Ralescu [25] introduced the notion of fuzzy set valued random variables whose underlying space is the d -dimensional Euclidean space Rd Concerning the convergence theory of fuzzy set-valued random variables and its applications we refer to [15, 21–23, 25–27] In this paper we present a study of a class of random fuzzy variables whose underlying space is a separable Banach space E or a weak star dual Es of a separable Banach space The paper is organized as follows In Sect we summarize and state the needed measurable results in the weak star dual of a separable Banach space In particular, we present the expectation and the conditional expectation of convex weak star compact valued Gelfand-integrable mappings In Sect we present the properties of random fuzzy convex upper semi continuous integrands (variables) in Es In Sect 4, the fuzzy expectation and the fuzzy conditional expectation for random fuzzy convex upper semi continuous variables are provided in this setting Section is devoted to the SLLN for fuzzy convex compact (compact) valued random level sets through the SLLN for convex compact (compact) valued random sets The above results lead to a new class of integrand martingales that we develop in Sect Some convergence results for integrand martingales are provided Our paper provides several issues in Fuzzy set theory, but captures different tools from Probability and Set-Valued Analysis and shows the relations among them with a comprehensive concept Integration of Convex Weak Star Compact Sets in a Dual Space Throughout this paper, ; F ; P / is a complete probability space, E is a Banach space which we generally assume to be separable, unless otherwise stated, D1 D ek /k2N is a dense sequence in the closed unit ball of E, E is the topological On the Integration of Fuzzy Level Sets dual of E, and B E (resp B E ) is the closed unit ball of E (resp E ) We denote by cc.E/ (resp cwk.E/) (resp ck.E/) the set of nonempty closed convex (resp weakly compact convex) (resp compact convex) subsets of E Given C cc.E/, the support function associated with C is defined by ı x ; C / D supf< x ; y >; y C g x E /: We denote by dH the Hausdorff distance on cwk.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: We denote by Lcwk.E/ F / the space of all F -measurable cwk.E/-valued multifunctions X W ! cwk.E/ such that ! ! jX.!/j is integrable A sequence Xn /n2N in Lcwk.E/ F / is bounded if the sequence jXn j/n2N is bounded in L1R F / A F -measurable closed convex valued multifunction X W ! cc.E/ is integrable if it admits an integrable selection, equivalently if d.0; X / is integrable We denote by Es , (resp Eb ), (resp Ec ) the vector space E endowed with the topology E ; E/ of pointwise convergence, alias w -topology (resp the topology s associated with the dual norm jj:jjEb ), (resp the topology c of compact convergence) and by Em the vector space E endowed with the topology m D E ; H /, where H is the linear space of E generated by D1 , that is the Hausdorff locally convex topology defined by the sequence of semi-norms Pn x / D maxfjhek ; x ij W k Ä ng; x E ; n N: Recall that the topology m is metrizable, for instance, by the metric dE x ; y / WD m X jhek ; x i 2k hek ; y ij ^ 1/; x ; y E : kD1 w We assume from now on that dE is held fixed Further, we have m m s : On the other hand, the restrictions of m , w , c to any bounded subset c of E coincide and the Borel tribes B.Es /, B.Em / and B.Ec / associated with Es , Em , Ec , are equal, but the consideration of the Borel tribe B.Eb / associated with the topology of Eb is irrelevant here Noting that E is the countable union of closed balls, we deduce that the space Es is a Lusin space, as well as the metrizable topological space Em Let K D cwk.Es / be the set of all nonempty convex weak compact subsets in E A K -valued multifunction (alias mapping for short) X W à Es is scalarly F -measurable if, 8x E, the support function ı x; X.:// is F -measurable, hence its graph belongs to F ˝ B.Es / Indeed, let fk /k2N be a sequence in E which separates the points of E , then we have C Castaing et al x X.!/ iff hfk ; x i Ä ı fk ; X.!// for all k N Consequently, for any Borel set G B.Es /, the set X G D f! W X.!/ \ G Ô ;g is F -measurable, that is, X G F , this is a consequence of the Projection Theorem (see e.g [8, Theorem III.23]) and of the equality X G D proj fGr.X / \ G/g: In particular if u W ! Es is a scalarly F -measurable mapping, that is, if for every x E, the scalar function ! 7! hx; u.!/i is F -measurable, then the function f W !; x / 7! jjx u.!/jjEb is F ˝ B.Es /-measurable, and for every fixed ! ; f !; :/ is lower semicontinuous on Es , i.e f is a normal integrand Indeed, we have jjx u.!/jjEb D sup jhek ; x u.!/ij: k2N As each function !; x / 7! hek ; x u.!/i is F ˝ B.Es /-measurable and continuous on Es for each ! , it follows that f is a normal integrand Consequently, the graph of u belongs to F ˝ B.Es / Let B be a sub- -algebra of F It is easy and classical to see that a mapping u W ! Es is B; B.Es // measurable iff it is scalarly B-measurable A mapping u W ! Es is said to be scalarly integrable (alias Gelfand integrable), if, for every x E, the scalar function ! 7! hx; u.!/i is F -measurable and integrable We denote by GE1 ŒE.F / the space of all Gelfand integrable mappings and by L1E ŒE.F / the subspace of all Gelfand integrable mappings u such that the function juj W ! 7! jju.!/jjEb is integrable The measurability of juj follows easily from the above considerations 1 More generally, by Gcwk.E ; F ; P / (or Gcwk.E F / for short) we denote the s / s / space of all scalarly F -measurable and integrable cwk.Es /-valued mappings and 1 by Lcwk.E ; F ; P / (or Lcwk.E F / for short) we denote the subspace of s / s / all cwk.Es /-valued scalarly integrable and integrably bounded mappings X , that is, such that the function jX j W ! ! jX.!/j is integrable, here jX.!/j WD supy 2X.!/ jjy jjEb , by the above consideration, it is easy to see that jX j is F measurable For any X Lcwk.E F /, we denote by SX1 F / the set of all Gelfands / integrable selections of X The Aumann-Gelfand integral of X over a set A F is defined by Z Z EŒ1A X D X dP WD f A A f dP W f SX1 F /g: Discrete Time Optimal Control Problems on Large Intervals 123 Together with (83) and Proposition this implies that vN C g/.y0 / X v.y N t ; yt C1 / sup v N xN v ; xN v // vN vN C g/ y0 / C ı1 ; vN y / (95) ı1 : (96) t D0 It follows from (95), (96) and the property (Pi) that there exists an v; N N /-overtaking optimal program f t gt D0 such that t ; xT t/ vN C g/ / D sup D t ; yt / Ä ; t D 0; : : : ; 0: vN C g/; t u Theorem 15 is proved Proof of Proposition 17 Since the mapping v ! v, N v M / is an isometry Proposition 17 follows from Proposition 15 and the following result Proposition 19 Suppose that g Mu X / and v M / is an upper semicontinuous function, xN v X , rNv > 0, cNv > and that assumptions (A1)–(A3) hold Let be an integer and > Then there exist ı > and an integer T0 0 such that for each u Bd v; ı/ \ M / and each h Mu X / satisfying kh gk Ä ı the following property holds: Q Q for each fxt g1 N / there is fyt g1 N such that xt ; yt / Ä t D0 P.h; u t D0 P.g; v/ for all integers t D 0; : : : ; Proof By (A2), for each u M0 / and each integer T 1, u; T; xN u ; xN u / D T u.xN u ; xN u /; u.xN u ; xN u / (97) u.z; z/ for all z X such that z; z/ : (98) We show that together with (98) and Assertion of Theorem this implies that there exists ı0 0; / such that for each u Bd v; ı0 / \ M0 /, xN u ; xN v / Ä rNv =4: (99) By Assertion of Theorem 3, there exist a positive number ıQ < and a natural number LQ such that the following property holds: (i) For each integer T which satisfy Q each fut gT L, t D0 kut M / and each /-program fxt gTtD0 Q t D 0; : : : ; T vk Ä ı; 1; 124 A.J Zaslavski T X1 fut gtTD01 ; 0; T / ut xt ; xt C1 / 2cNv t D0 we have Card.ft f0; : : : ; T g W Q xt ; xN v / > rNv g/ < L: Choose a positive number ı0 such that Q ı0 < ; ı0 < 2L/ Q 1: ı0 < ı; (100) u Bd v; ı0 / \ M0 /: (101) Assume that In view of (101), relation (98) holds Set t D xN u ; t D 0; 1; : : : : (102) We show that (99) holds Assume the contrary Then t ; xN v / > rNv =4; t D 0; 1; : : : : (103) By (100), (101), (103) and property (i), for each integer k 0, Q kC1/L X u t ; t C1 / Q < u; 0; L/ 2cNv 4: (104) Q t Dk L It follows from (100), (101) and (A2) that Q xN v ; xN v / C cNv C ı0 LQ Q Ä v; 0; L/ Q C ı0 LQ Ä Lv .u; 0; L/ Q xN v ; xN v / C cNv C 2ı0 LQ Ä Lu Q xN v ; xN v / C cNv C 1: Ä Lu Relations (104) and (105) imply that for each integer k 0, Q kC1/L X u t ; t C1 / Q xN v ; xN v / Ä Lu cNv Q t Dk L Together with (102) this implies that Q xN v ; xN v / Q xN u ; xN u / Ä Lu Lu cNv 3; 3: (105) Discrete Time Optimal Control Problems on Large Intervals 125 u.xN u ; xN u / < u.xN v ; xN v /: This contradicts (98) The contradiction we have reached proves (99) Thus we have shown that for each function u satisfying (101) relation (99) is valid By Theorem 15, there exist ı 0; ı0 / and a natural number T0 such that the following property holds: (ii) For each integer T T0 , each h M X / satisfying kh fut g0T M0 / satisfying kut vk Ä ı; t D 0; : : : ; T gk Ä ı, each and each /-program fzt gTtD0 which satisfies z0 ; xN v / Ä rNv =2; h.zT / C T X1 ut zt ; zt C1 / h; fut gtTD01 ; 0; T; z0 / ı t D0 Q there exists fxt g1 N such that t D0 P.g; v/ zT t ; xt / Ä ; t D 0; : : : ; : Assume that u M /; ku vk Ä ı; h Mu X /; kh Q N /: fxt g1 t D0 P.h; u gk Ä ı; (106) (107) In view of (106), (107) and (A3), lim xt D xN u : t !1 (108) It follows from (100), (106) and the choice of ı0 that (99) holds By (108), there exists an integer S0 > T0 such that xS0 ; xN u / Ä rNv =4: Together with (99) this implies that xS0 ; xN v / Ä rNv =2: (109) zt D xS0 t ; t D 0; : : : ; S0 : (110) Set 126 A.J Zaslavski By (109) and (110), z0 ; xN v / Ä rNv =2: (111) is an /-program We show that It is clear that fzt gSt D0 h.zS0 / C SX u.zt ; zt C1 / D h; u; S0 ; z0 /: (112) t D0 Let fyt gSt D0 be an /-program satisfying y0 D z0 : (113) In order to prove (112) it is sufficient to show that SX h.zS0 / C u.zt ; zt C1 / h.yS0 / C SX t D0 u.yt ; yt C1 /: t D0 It follows from (110) that h.zS0 / C SX u.zt ; zt C1 / D h.x0 / C t D0 SX uN xS0 t ; xS0 t /: (114) t D0 Set yNt D yS0 t ; t D 0; : : : ; S0 : (115) In view of (115), h.yS0 / C SX u.yt ; yt C1 / D h.yN0 / C t D0 SX uN yNt ; yNt C1 /: (116) t D0 In view of (107), (109), (110), (113)–(116), Proposition 4, Corollary and Nu; N /overtaking optimality of fxt g1 t D0 (see (107)), h.zS0 / C SX u.zt ; zt C1 / Œh.yS0 / C SX t D0 D h.x0 / C SX t D0 u.yt ; yt C1 / t D0 uN xt ; xt C1 / Œh.yN0 / C SX t D0 uN yNt ; yNt C1 / Discrete Time Optimal Control Problems on Large Intervals D h.x0 / C SX ŒNu.xt ; xt C1 / 127 u.xN u ; xN u / uN xt / C uN uN yNt / C uN xt C1 / t D0 C h.yN0 / C uN uN x0 / SX ŒNu.yNt ; yNt C1 / xS0 / u.xN u ; xN u / yNt C1 / t D0 uN C D h.x0 / C SX ŒNu.yNt ; yNt C1 / yN0 / uN uN h.yN0 / x0 / yNS0 // u.xN u ; xN u / uN uN yN0 / yNt / C uN yNt C1 / t D0 h.x0 / C uN x0 / h.yN0 / uN yN0 / 0: Thus (112) holds By (106), (111), (112), the inequality S0 > T0 and the property (ii) Q applied to fzt gSt D0 there exists an fxt g1 N such that for all t D 0; : : : ; , t D0 P.g; v/ zS0 t ; xt / D xt ; xt /: Proposition 19 is proved 10 Proof of Theorem 17 In the sequel we use the following auxiliary results Proposition 20 (Proposition 12.1 of [39]) Let v M /, fxt g1 t D0 be a v; /overtaking optimal and v; /-good program and t2 > t1 be nonnegative integers such that xt1 D xt2 Then xt D xN v for all integers t D t1 ; : : : ; t2 Proposition 21 (Proposition 12.2 of [39]) Let v M /, fxt g1 t D0 be a v; /overtaking optimal program such that x0 D xN v Then xt D xN v for all integers t For any x; y/ X X and any nonempty set D X X put x; y/; D/ D inff x; z1 / C y; z2 / W z1 ; z2 / Dg: Since the mapping v ! v, N v M / is an isometry Theorem 17 follows from Propositions 14, 15 and the following result Proposition 22 Let M be either MN0 / or MNc;0 / and g Mu X / Then there exists a set F M \ M / which is a countable intersection of open everywhere dense subsets of M such that for each v F there exists a unique point zv X 128 A.J Zaslavski such that v C g/.zv / D sup C g/ v v and there exist a unique v; /-overtaking optimal program fzvt g1 t D0 satisfying z0 D zv Let M be either MN0 / or MNc;0 / By Theorem 8, there exists a set F0 M \ M / which is a countable intersection of open everywhere dense subsets of M For each g Mu X / denote by Eg the set of all v M \ M / for which the following property holds: (Pi) There exists a unique point zv X such that v C g/.zv / D sup v C g/ and there exists a unique v; /-overtaking optimal program fzvt g1 t D0 satisfying zv0 D zv We proceed the proof of Proposition 22 with the following auxiliary result Lemma For each g Mu X /, Eg is a an everywhere dense subset of M Proof Let v M \ M / It is sufficient to show that for any neighborhood U of v in M, U \ Eg 6D ; There are two cases: v C g/.xN v / D sup v C g/I (117) v C g/.xN v / < sup v C g/: (118) Assume that (117) holds Let v x; y/ D v.x; y/ 0; 1/ Define x; xN v / C y; xN v //; x; y/ : (119) It is easy to see that v M \ M0 / with xN v D xN v : (120) In view of (119), every v ; /-good program fxt g1 t D0 is v; /-good and lim xt D xN v : t !1 Hence v M /: (121) Discrete Time Optimal Control Problems on Large Intervals 129 Proposition and (119)–(121) imply that v xN v ; xN v / D v.xN v ; xN v /; v C g/.y/ Ä v y/ Ä v v y/; y X; C g/.y/; y X; v v xN v / D C g/.xN v / D v v xN v / D 0; C g/.xN v /: (122) Assume that a point z X satisfies C g/.z/ D sup v C g/ v (123) and that fzt g1 t D0 is an v ; /-overtaking optimal program such that z0 D z: (124) In view of (117), (119)–(124) and Proposition 8, sup D T X1 D lim Œ T !1 v v C g/ g.z/ D sup .z/ D lim T !1 T X1 v xN v ; xN v / Œv zt ; zt C1 / t D0 Ä lim sup g.z/ t D0 T v.xN v ; xN v / v.zt ; zt C1 / C g/ v T X1 zt ; xN v / C zt C1 ; xN v // t D0 T X1 T !1 t D0 v.zt ; zt C1 / v.xN v ; xN v // X zt ; xN v / C zt C1 ; xN v // t D0 Ä v X z/ zt ; xN v / C zt C1 ; xN v // t D0 D X zt ; xN v / C zt C1 ; xN v // C v C g/.z/ g.z/: t D0 This implies that zt D xN v for all integers t and v Eg Assume that (118) holds There exist a point z X which satisfies v C g/.z / D sup v C g/ and an v; /-overtaking optimal program fzt g1 t D0 satisfying z0 D z : By Proposition and (125), (125) 130 A.J Zaslavski v T X1 z / D lim T !1 v.xN v ; xN v /: Œv.zt ; zt C1 / (126) t D0 Assumption (A3) imply that lim z t !1 t D xN v : (127) By (118), (125), (127) and Proposition 10, v C g/.zt / < v C g/.z0 / for all large enough integers t In view of (128), there exists an integer 1: (128) such that v C g/.z / D v C g/.z / v C g/.zt / < v C g/.z / and that for all integers t > v Let We may assume without loss of generality C g/.z0 / D sup v C g/; v 0; 1/ For all x; y/ v x; y/ D v.x; y/ C g/.zt / < v D Thus C g/.z0 / for all integers t 1: (129) define x; y/; f.zt ; zt C1 / W t D 0; 1; : : : g [ f.xN v ; xN v /g//: (130) By (130), v M \ M0 /; v xN v ; xN v / D v.xN v ; xN v /; every v ; /-good program is v; /-good and converges to xN v Hence v M /; xN v D xN v : (131) It follows from (125), (126), (130), (131) and the equality z0 D z that v y/ Ä v y/; y X; z / D v z /; C g/: (132) v ; T; xN v ; xN v / D T v.xN v ; xN v /: (133) sup v C g/ D sup v v By (131), for all natural numbers T , Discrete Time Optimal Control Problems on Large Intervals 131 Proposition and (131) imply that v xN v / D 0: (134) Set K D f.zt ; zt C1 / W t D 0; 1; : : : g [ f.xN v ; xN v /g: (135) Assume that y X satisfies C g/.y/ D sup v C g/ v (136) and that fyt g1 t D0 is an v ; /-overtaking optimal program satisfying y0 D y: (137) By (125), (130)–(132), (135)–(137), Proposition and since the program fyt g1 t D0 is v ; /-overtaking optimal we have sup C g/ D v D lim v C g/.z / D T X1 T !1 T !1 C g/.z / D v C g/.y/ v xN v ; xN v / C g.y/ t D0 T X1 D lim Œ Œv yt ; yt C1 / v v.yt ; yt C1 / v.xN v ; xN v / yt ; yt C1 /; K// C g.y/ t D0 Ä v X y/ yt ; yt C1 /; K/ C g.y/: t D0 Together with (118) and (133) this implies that v C g/.y/ D v C g/.z /; yt ; yt C1 / K for all integers t (138) 0: (139) In view of (118), (125), (129), (135), (137)–(139) and the equality z0 D z , y D y0 D z : We show by induction that yt D zt for all integers t zt 6D xN v for all integers t xN v fzt W t D 0; 1; : : : g: (140) There are two cases: 0I (141) (142) 132 A.J Zaslavski Assume that (141) holds By Proposition 20, (125)–(126) and (141), zt1 6D zt2 for all integers t2 > t1 Assume that T 0: (143) is an integer and that yt D zt ; t D 0; : : : ; T: (144) (Note that in view of (140) and the equality z0 D z our assumption holds for T D 0.) By (135), (139), (141), (143) and (144), zT ; yT C1 / D yT ; yT C1 / K D f.zt ; zt C1 / W t D 0; 1; : : : g [ f.xN v ; xN v /g and yT C1 D zT C1 Thus yt D zt for all integers t Assume that (142) holds By (118), (125) and the equality z0 D z , there is a natural number S such that zS D xN v ; zt 6D xN v for all integers t Œ0; S /: (145) Propositions 20 and 21 imply that, zt D xN v for all integers t S; zt2 6D zt1 for all integers t1 ; t2 Œ0; S such that t1 < t2 : Assume that T (146) (147) is an integer and that yt D zt ; t D 0; : : : ; T: (148) (Note that in view of (140), our assumption holds for T D 0.) If T < S , then by (135), (139), (145) and (148), zT ; yT C1 / D yT ; yT C1 / f.zt ; zt C1 / W t D 0; 1; : : : g [ f.xN v ; xN v /g and yT C1 D zT C1 If T S , then by (135), (139), (145), (146) and (148), xN v ; yT C1 / D yT ; yT C1 / f.zt ; zt C1 / W t D 0; 1; : : : g [ f.xN v ; xN v /g and yT C1 D xN v D zT C1 Thus yt D zt for all integers t in the both cases (see (141) and (142)) This implies that v Eg Therefore the inclusion above holds in the both cases (see (117) and (118)) Since v ! v as ! 0C in M we conclude that for any neighborhood U of v in M, U \ Eg 6D ;: Thus Eg is an everywhere dense subset of M Lemma is proved Completion of the proof of Proposition 22 By definition, for every v Eg , there exist a unique v; /-overtaking optimal program fzvt g1 t D0 satisfying Discrete Time Optimal Control Problems on Large Intervals v C g/.zv0 / D sup v 133 C g/: be an integer By Proposition 17, there exist an open Let v Eg and k neighborhood U v; k/ of v in M and an integer T v; k/ k such that the following property holds: Q (Pii) For each u U v; k/ \ M / and each fxt g1 t D0 P.g; u/ we have xt ; zvt / Ä k , t D 0; : : : ; k Set F1 D \1 pD1 [ fU v; k/ W v Eg ; k pg; F D F1 \ F0 : (149) Clearly, F is a countable intersection of open everywhere dense subsets of M and F F0 M / .i / Let u F , p be an integer and fxt g1 t D0 , i D 1; be u; /-overtaking optimal programs such that v i / C g/.x0 / D sup v C g/; i D 1; 2: By (149), there exist vp Eg and an integer kp (150) p such that u U vp ; kp /: i / (151) v In view of (149), (151) and property (Pii), xt ; zt p / Ä kp Ä p i D 1; This implies that natural number we conclude is proved , t D 0; : : : ; p, 1/ 2/ xt ; xt / Ä 2p , t D 0; : : : ; p Since p is any 1/ 2/ 0: Proposition 22 that xt D xt for all integers t t u 11 Proof of Proposition 18 By Theorem 8, there exists a set G0 M\M / which is a countable intersection of open everywhere dense subsets of M Denote by E the set of all v; g/ M \ M // A for which there exists a v;g/ Q unique program fzt g1 t D0 P.g; v/ By Lemma 4, E is a an everywhere dense subset of M A Let v; g/ E and k be an integer By Proposition 17, there exist an open neighborhood U v; g; k/ of v; g/ in M A and an integer T v; g; k/ k such that the following property holds: (i) For each u; h/ U v; g; k/ \ M / v;g/ have xt ; zt / Ä k , t D 0; : : : ; k Q A/ and each fxt g1 t D0 P.h; u/ we 134 A.J Zaslavski Set F D \1 pD1 [ fU v; g; k/ W v; g/ E; k pg \ G0 A/: (152) Clearly, F is a countable intersection of open everywhere dense subsets of M Let u; h/ F , p be an integer and i / Q fxt g1 t D0 P.h; u/; i D 1; 2: By (152), there exist vp ; gp / E and an integer kp u; h/ U vp ; gp ; kp /: i / A (153) p such that (154) gp ;vp / / Ä kp Ä p , t D 0; : : : ; p, i D 1; This 1/ 2/ implies that xt ; xt / Ä 2p , t D 0; : : : ; p Since p is any natural number we 1/ 2/ conclude that xt D xt for all integers t 0: Proposition 18 is proved t u In view of (152)–(154), xt ; zt References Anderson BDO, Moore JB (1971) Linear optimal control Prentice-Hall, Englewood Cliffs Aseev SM, Veliov VM (2012) Maximum principle for infinite-horizon optimal control problems with dominating discount Dyn Contin Discret Impuls Syst Ser B 19:43–63 Aubry S, Le Daeron PY (1983) The discrete Frenkel-Kontorova model and its extensions I Physica D 8:381–422 Baumeister J, Leitao A, Silva GN (2007) On the value function for nonautonomous optimal control problems with infinite horizon Syst Control Lett 56:188–196 Blot J (2009) Infinite-horizon Pontryagin principles without invertibility J Nonlinear Convex Anal 10:177–189 Blot J, Cartigny P (2000) Optimality in infinite-horizon variational problems under sign conditions J Optim Theory Appl 106:411–419 Blot J, Hayek N (2014) Infinite-horizon optimal control in the discrete-time framework Springer briefs in optimization Springer, New York Cartigny P, Michel P (2003) On a sufficient transversality condition for infinite horizon optimal control problems Autom J IFAC 39:1007–1010 Gaitsgory V, Rossomakhine S, Thatcher N (2012) Approximate solution of the HJB inequality related to the infinite horizon optimal control problem with discounting Dyn Contin Discret Impuls Syst Ser B 19:65–92 10 Hayek N (2011) Infinite horizon multiobjective optimal control problems in the discrete time case Optimization 60:509–529 11 Jasso-Fuentes H, Hernandez-Lerma O (2008) Characterizations of overtaking optimality for controlled diffusion processes Appl Math Optim 57:349–369 12 Kolokoltsov V, Yang W (2012) The turnpike theorems for Markov games Dyn Games Appl 2:294–312 13 Leizarowitz A (1985) Infinite horizon autonomous systems with unbounded cost Appl Math Opt 13:19–43 14 Leizarowitz A (1986) Tracking nonperiodic trajectories with the overtaking criterion Appl Math Opt 14:155–171 Discrete Time Optimal Control Problems on Large Intervals 135 15 Leizarowitz A, Mizel VJ (1989) One dimensional infinite horizon variational problems arising in continuum mechanics Arch Ration Mech Anal 106:161–194 16 Lykina V, Pickenhain S, Wagner M (2008) Different interpretations of the improper integral objective in an infinite horizon control problem J Math Anal Appl 340:498–510 17 Makarov VL, Rubinov AM (1977) Mathematical theory of economic dynamics and equilibria Springer, New York 18 Malinowska AB, Martins N, Torres DFM (2011) Transversality conditions for infinite horizon variational problems on time scales Optim Lett 5:41–53 19 Marcus M, Zaslavski AJ (1999) The structure of extremals of a class of second order variational problems Ann Inst Henri Poincaré Anal Non Lineare 16:593–629 20 McKenzie LW (1976) Turnpike theory Econometrica 44:841–866 21 Mordukhovich BS (2011) Optimal control and feedback design of state-constrained parabolic systems in uncertainly conditions Appl Anal 90:1075–1109 22 Mordukhovich BS, Shvartsman I (2004) Optimization and feedback control of constrained parabolic systems under uncertain perturbations In: de Queiroz MS, Malisoff M, Wolensk P (eds) Optimal control, stabilization and nonsmooth analysis Lecture notes in control and information sciences Springer, Berlin/New York, pp 121–132 23 Ocana Anaya E, Cartigny P, Loisel P (2009) Singular infinite horizon calculus of variations Applications to fisheries management J Nonlinear Convex Anal 10:157–176 24 Pickenhain S, Lykina V, Wagner M (2008) On the lower semicontinuity of functionals involving Lebesgue or improper Riemann integrals in infinite horizon optimal control problems Control Cybern 37:451–468 25 Porretta A, Zuazua E (2013) Long time versus steady state optimal control SIAM J Control Optim 51:4242–4273 26 Samuelson PA (1965) A catenary turnpike theorem involving consumption and the golden rule Am Econ Rev 55:486–496 27 Zaslavski AJ (2006) Turnpike properties in the calculus of variations and optimal control Springer, New York 28 Zaslavski AJ (2007) Turnpike results for a discrete-time optimal control systems arising in economic dynamics Nonlinear Anal 67:2024–2049 29 Zaslavski AJ (2009) Two turnpike results for a discrete-time optimal control systems Nonlinear Anal 71:902–909 30 Zaslavski AJ (2010) Stability of a turnpike phenomenon for a discrete-time optimal control systems J Optim Theory Appl 145:597–612 31 Zaslavski AJ (2011) Turnpike properties of approximate solutions for discrete-time control systems Commun Math Anal 11:36–45 32 Zaslavski AJ (2011) Structure of approximate solutions for a class of optimal control systems J Math Appl 34:1–14 33 Zaslavski AJ (2011) Stability of a turnpike phenomenon for a class of optimal control systems in metric spaces Numer Algebra Control Optim 1:245–260 34 Zaslavski AJ (2011) The existence and structure of approximate solutions of dynamic discrete time zero-sum games J Nonlinear Convex Anal 12:49–68 35 Zaslavski AJ (2013) Nonconvex optimal control and variational problems Springer optimization and its applications Springer, New York 36 Zaslavski AJ (2014) Turnpike properties of approximate solutions of nonconcave discrete-time optimal control problems J Convex Anal 21:681–701 37 Zaslavski AJ (2014) Turnpike phenomenon and infinite horizon optimal control Springer optimization and its applications Springer, New York 38 Zaslavski AJ (2014) Turnpike properties for nonconcave problems Adv Math Econ 18:101– 134 39 Zaslavski AJ (2014) Structure of solutions of discrete time optimal control roblems in the regions close to the endpoints Set-Valued Var Anal 22:809–842 Index A Asymptotic turnpike property, 93 Aumann-Gelfand integral, X over set A, Aumann integral, 9, 10 B Baire category, 93 Banach space, 100 Bermudan derivatives, 58, 59 Bolza problem, 92 C Cardinality, 96 cc (E), ck (E), Closed convergence topology, 34, 35, 39, 41, 45 Compact metric space, 92 Conditional expectation, level sets, 11, 12, 13, 14, 19 cwk (E), D Demand function, 34–37, 41, 43–45 Discrete-time optimal control system, 92 E Estimate error, 34, 35, 41, 45 Expectation of random fuzzy convex integrands, 18 Expectation of the level sets, 11, 12 Expected value (or expectation), 26 F F-measurable, Fuzzy conditional expectation, 18, 19, 26 Fuzzy convex upper semicontinuous variable, Fuzzy expectation, 18, 19 Fuzzy martingale, 26 Fuzzy set valued random variables, Fuzzy submartingale, 26 Fuzzy supermartingale, 26 G Gelfandintegrable selections, ( /-Good program, 93 H Hörmander type diffusion process, 61, 78–81 I Infinite horizon optimal control problem, 93 Infinite interval, 92 Integrability theory, 3, 34 Integrably bounded mappings, Integrand martingales, 27 Inverse demand function, 35, 36, 38, 41, 45, 52 137 138 K K -valued random set, L Lagrange problem, 92 Least square regression methods, 58 Local C topology, 35, 41–43, 45 Local uniform topology, 41, 45 Lower semicontinuous integrand martingale, 27 M Martingales, 27 Monte Carlo methods, 58 N Normal integrand, Index R Random fuzzy convex upper semicontinuous integrands, level sets, 9, 12 Random fuzzy convex upper semicontinuous variable, level sets, 8–9 Random lower semicontinuous integrands, Random systems of piece-wise polynomials, 60 Random upper semicontinuous integrands, Re-simulation, 71–73 Revealed preference theory, 34 S Scalarly F-measurable mapping, 3, Scalarly integrable, Sequential weak lower limit w -liXn , Sequential weak upper limit w -lsXn , Support function, O Optimal control problem, 92 Overtaking optimal program, 98, 99 T Turnpike phenomenon, 93 P Perturbation, 98 Preference relation, 34, 35, 38, 42, 43, 45 ( /-Program, 92 p-transitive, 34, 36, 38, 42, 43 W Weak axiom of revealed preference, 34 Weakly compactly generated (WCG), Weak star (w K for short) converges, w -topology,