Advances in Mathematical Economics

131 Pages·2001·4.524 MB·English

by  Houcine Benabdellah, Charles Castaing (auth.), Shigeo Kusuoka, Toru Maruyama (eds.)

Advances in
MATHEMATICAL
ECONOMICS
Managing Editors
Shigeo Kusuoka Toru Maruyama
University of Tokyo Keio Univeristy
Tokyo, JAPAN Tokyo, JAPAN
Editors
Robert Anderson Jean-Michel Grandmont Norio Kikuchi
University of California, CREST-CNRS Keio Univeristy
Berkeley Malakoff, FRANCE Yokohama, JAPAN
Berkeley, U.S.A. Norimichi Hirano Hiroshi Matano
Charles Castaing
Yokohama National University of Tokyo
Universite Montpellier II University
Tokyo, JAPAN
Montpellier, FRANCE Yokohama, JAPAN
Frank H. Clarke Leonid Hurwicz Kazuo Nishimura
Universite de Lyon I University of Minnesota Kyoto University
Villeurbanne, FRANCE Minneapolis, U.S.A. Kyoto, JAPAN
Gerard Debreu Tatsuro Ichiishi Marcel K. Richter
University of California, Ohio State University University of Minnesota
Berkeley
Ohio, U.S.A. Minneapolis, U.S.A.
Berkeley, U.S.A.
Alexander Ioffe Yoichiro Takahashi
Egbert Dierker
Israel Institute of Kyoto University
University of Vienna
Technology
Kyoto, JAPAN
Vienna, AUSTRIA
Haifa, ISRAEL
Darrell Duffie Michel Valadier
Seiichi Iwamoto
Stanford University Universite Montpellier II
Kyushu University
Stanford, U.S.A. Montpellier, FRANCE
Fukuoka, JAPAN
Lawrence C. Evans
Kazuya Kamiya Akira Yamazaki
University of California,
Hitotsubashi University
Berkeley University of Tokyo
Tokyo, JAPAN
Berkeley, U.S.A. Tokyo, JAPAN
Takao Fujimoto Kunio Kawamata Makoto Yano
Okayama University Keio Univeristy Keio Univeristy
Okayama, JAPAN Tokyo, JAPAN Tokyo, JAPAN
Aims and Scope. The project is to publish Advances in Mathematical
Economics once a year under the auspices of the Research Center of Math
ematical Economics. It is designed to bring together those mathematicians
who are seriously interested in obtaining new challenging stimuli from eco
nomic 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 Oil rigorous mathematical reason
ing.
- Mathematical methods (e.g., analysis, algebra, geometry, probability) mo
tivated 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.
Springer Japan KK
s.
Kusuoka, T. Maruyama
(Eds.)
Advances in
Mathematical Economics
Volume 3
, Springer
Shigeo Kusuoka
Professor
Graduate School of Mathematical Sciences
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
ISBN 978-4-431-65937-2 ISBN 978-4-431-67891-5 (eBook)
DOI 10.1007/978-4-431-67891-5
© Springer Japan 2001
Originally publisbed by Springer-Verlag Tokyo Berlin Heidelberg New York in 2001
Softcover reprint of the hardcover 1st edition 2001
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SPIN: 10789575
Table of Contents
Research Articles
H. Benabdellah, C. Castaing
Weak compactness and convergences in L}" [E] 1
A.Ioffe
Abstract convexity and non-smooth analysis 45
s. Iwamoto
Recursive method in stochastic optimization under
compound criteria 63
s. Kusuoka
On law invariant coherent risk measures 83
v. L. Levin
The Monge-Kantorovich problems and stochastic
preference relations 97
Subject Index 125
Adv. Math. Econ. 3, 1-44 (2001) Advances In
MATHEMATICAL
ECONOMICS
eSpringer-Verlag 2001
Weak compactness and convergences in L1,[E]
Houcine Benabdellah 1 and Charles Castaing2
1 Departement de Mathematiques, Universite Cadi Ayyad, Faculte des Sciences
Semlalia, B.P. SIS, Marrakech, Maroc
2 Departement de Mathematiques, case 051, Universite Montpellier II, F-340!}5
Montpellier cedex 5, France
Received: May 9, 2000
Revised: July 25, 2000
Mathematics Subject Classification (2000): 46E40, 28B05, 28B20
Abstract. Suppose that (0, F, {t) is a complete probability space, E is a Banach
space, E' is the topological dual of E and p is a lifting in .cOO ({t). We state several
convergences and weak compactness results in the Banach space (L1, [EJ, N d of
weak· -scalarly integrable E'-valued functions via the Banach space (L ~;[EJ, N1,p)
associated to the lifting p. Applications to Young measures, Mathematical Eco
nomics, Minimization problems and Set-valued integration are also presented.
Key words: compact, conditionally weakly compact, Fatou, lifting, tight, Young
measure
Introduction
Throughout E will denote a Banach space, E' the topological dual of
E, (n, T, J.L) a complete probability space. Weak compactness and conver
gences in the Banach space L1(J.L) of Bochner integrable E-valued functions
has been studied extensively (see, for example, ([2], [10], [11], [12], [14], [16],
[17], [18], [19], [20], [21], [22], [23], [26], [40], [43]). However, not much study
has been done for the Banach space (L1, [E], Nl) of weak* -scalarly integrable
E'-valued functions. We denote by £1,[E] the vector space of scalarly mea
surable functions I : n -> E' such that there exists a positive integrable
function h (depending on f) such that Vw E n, 11/(w)11 :S h(w). A semi-norm
£1,
on [E] is defined by
In*
N1(f) = 11/(w)11 J.L(dw) = inf{ln hdJ.L: h integrable; h 2: II/II}·
Two functions I,g E £1,[E] are equivalent (shortly 1== g(w*)) if, (f(.),x) =
(g(.),x) a.e. for every x E E. The equivalence class of I is denoted by f. The
quotient space L1, [E] is equipped with the norm N 1 given by
2 H. Benabdellah, C. Castaing
Let p be the lifting in LE',[E] associated to a lifting p in LR(J.L)( [42, chap.
VI. 4]. We denote by L~~[E] the vector space of all mappings 1 ELk, [E]
such that there exists a sequence (An)n2:1 in F satisfying:
U
An = nand Ifn ~ 1, lAni E C£, [E] and p(IAnI) = IP(Anl f.
n2:1
If 1 E L~~[E], we will show [Prop.3.1(c)] that 111(.)11 is measurable and we
will endow the quotient space L ~nE] with the norm
We will prove [Theorem 3.2] that there is a linear isometric isomorphism
p:
(Lk, [E], N I) -+ (L ~~[E], N1,p), a result which allows us to obtain natural
characterizations of weak compact and conditionally weakly compact subsets
and new convergences results in these spaces. Our approach is quite different
from that employed in the case of Lebesgue-Bochner integrable space Lk(J.L)
because of the lack of characterization of the topological dual of Lk, [E]. Here
several sophisticated techniques are introduced. Our first task is to state a
duality formula [Theorem 4.1] involving the spaces (Lk, [E])' and L~(J.L). Let
7
E Lk,[E]. There is a unique linear weak*-weak continuous mapping Aj
from (Lk,[E])' to L~(J.L) such that
10
Ifl E (Lk,[E])', Ifh E LR(J.L), h Aj(l) dJ.L = l(hl).
This result permits to characterize weakly Cauchy (resp. weakly convergent)
sequences in (Lk, [E], NI) [Theorems 4.4-4.5]. See also Schliichtermann [37]
for the study of weak Cauchy sequences in Loo(J.L, X).
Secondly, let S be a completely regular space, D a dense subset of S, Cb(S)
the space of real bounded continuous functions defined on S, and TD the
topology on Cb(S) of pointwise convergence on D. We denote by T'f) the
product topology on the product space Cb(Sl', each copy of Cb(S) being
equipped with the topology TD, and by B(Cb(St,T'f)) the Borel tribe of the
topological space (Cb(st, T'f)). Using the Stone-Cech compactification of S
and a result in ([40], Theorem 13), we have the following [Theorem 4.8]. Let
(fn)n be a sequence of mappings from n to Cb(S) such that (i) : the mapping
I: W (fn(W))nEN from n to Cb(S)N is (F, B(Cb(S)N), T'f)))-measurable and
f-->
(ii) : sUPn J~ Illn(w)lloo J.L(dw) < 00 (where 11·1100 denotes the sup-norm in
Cb(S)). Then there exist a sequence (gn)n with gn E co{fm : m ~ n} and two
measurable sets A and B with J.L(A U B) = 1 such that (a) Ifw E A, (gn(w))n
is weakly Cauchy in (Cb(S), 11.1100) and (b) Ifw E B, there exist kEN such
Weak compactness and convergences in Lk,[E] 3
that the sequence (gn(W))n>k is equivalent to the vector unit basis of [1. The
preceding result allows to obtain a structure theorem for bounded sequences
in L~nE]. [Theorem 4.9]. Let (fn)n be a bounded sequence in L~~[E]. Then
there exist a sequence (gn)n with gn E co{fm : m ~ n} and two measurable
sets A and B in 0 with J.t(A U B) = 1 such that (a) 'Vw E A, (gn(w))n is
a(E',E") Cauchy in E' and (b) 'Vw E B, there exists kEN such that the
sequence (gn(W))n~k is equivalent to the vector unit basis of [1.
The characterizations of weak compact and conditionally weakly compact
subsets in L1, [E] are obtained as follows.
(1) A subset 1i of L1,[E] is relatively a(L1,[E], (L1, [E])') compact iff the
following two conditions are satisfied: (i) 1i is uniformly integrable in L1, [E].
(ii) For any sequence (fn)n in 1i, there exists a sequence (gn)n with gn E
co{fm : m ~ n} such that for a.e wE 0 the sequence (p(gn)(w))n converges
a(E',E") in E'.
(2) A subset 1t of L1,[E] is conditionally a(L1,[El, (L1,[E])') compact iff
the following two conditions are satisfied: (i) 1t is uniformly integrable in
L1, [E]. (ii) For any sequence (fn)n in 1t, there exists a sequence (gn)n with
gn E co{fm : m ~ n} such that for a.e W E 0 the sequence (p(gn)(w))n is
a(E',E") Cauchy in E'.
Finally we prove that "sequential weakly complete" property lifts from
1,
E' to L [E]. To end this paper we present several new applications to Min
imization problems in L1, [El, Young measures, Fatou-type lemma in Math
ematical Economics and and Set-valued integration.
1. Notations and preliminaries
Let (0, F, J.t) be a complete probability space. We denote by J.t* the exterior
measure associated to J.t. For every function f : 0 ---> [0, +00] we set:
VI := {g : 0 ---> [0, +00] : 9 J.t-integrable and 9 ~ J}
and
It is well known ([42], Chap.l) that N enjoys the following properties:
(1.1) N is an upper integral in Ionescu-Tulcea's sense ([42], Chap.l, Def. 1.1).
(1.2) The mapping A N(IA) from P(O) into [0, +00] coincides with the
f->
exterior measure J.t*.
4 H. Benabdellah, C. Castaing
(1.3) If f : n -+ [0, +00] is F-measurable, then f is J.l-integrable iff N(f) <
In
+00, and in this case: N(f) = fdJ.l.
(1.4) If f : n -+ [0, +00] and N(f) < +00, then there exists a J.l-integrable
function g : n -+ [0, +00] such that f ~ g and N(f) = N(g).
If f : n -+ [-00, +00], we set N1(f) := N(lfl)·
Topologies associated to a lifting of £00 (J.l)
We denote by p a lifting of £OO(J.l) ([42], chap.IV). Let us mention the two
following topologies Jp and Tp on n associated to the lifting p :
(1.5) J is the topology on n which has open basis constituted by the family
p
: {peA) : A E F}.
(1.6) Tp is the topology on n which has open sets all sets A E F such that
A c peA).
It is clear that Tp is finer than Jp• For the convenience of the reader, we recall
some useful properties of these topologies ([42], Chap.V, Theorem 1).
Proposition 1.1. Let us denote by T one of the topologies Tp and Jp. Then
T enjoys the following properties "
(a) T is extremally disconnected (the T-closure of A E Tis peA)).
(b) If A is a T-open set and A == 0, then A = 0.
(c) If f : n -+ JR. is such that there exists a J.l-negligible set M such that the
set fen \ M) is bounded in JR., then f is T-continuous on n iff f E £OO(J.l)
and p(f) = f·
(d) F is the completion of the Borel tribe of the topological space (n, T).
Lifting of Haire-measurable functions with values in a completely
regular space
Let X be a completely regular space and C(X) the space of real-valued con
tinuous functions defined on X. We will use the following notations.
£~ (n, F) is the set of all Baire-measurable mappings f : n -+ X, that is f
is (F,Bo(X))-measurable where Bo(X) is the Baire tribe of X.
M~(n,F) is the set of all mappings f E £~(n,F) such that fen) is rela
tively compact in X.
£~(n,F,J.l) is the set of all mappings f E £~(n,F) such that there exists a
J.l-negligible set P such that fen \ P) is relatively compact in X.
It is obvious that
M~(n,F) c £~(n,F,J.l) c £~(n,F).
k,
Weak compactness and convergences in L [E) 5
A mapping f : 0 ---+ X is Baire-measurable iff for every 'I' E C(X), the
function 'I' 0 f is .1"-measurable. Two mappings f and 9 in .c3c (0,.1") are said
to be weakly equivalent (shortly f := g(w», if
'I' 0 f = 'I' 0 9 /-l-a.e.
for every 'I' E C(X). It is clear that ":= (w)" defines an equivalence relation
on .c3c(0, .1").
If A is a Banach algebra, a nonzero linear form ( on A satisfying Va, b E
A, ((ab) = ((a)((b) is called a character. Let us recall the following ([26],
chap.IV. 6.25) :
Lemma 1.2. Let K be a Hausdorff compact space and C(K) the Banach alge
bra of all real valued continuous functions defined on K. Then the characters
of C(K) are the Dirac measures 8x : r.p t-> r.p(x)(x E K).
Proof. Although this result is well known, we provide an alternative proof for
the sake of completeness. Let ( be a character of C(K). Since ( is continuous
([26], Lemma IV.6.24), there exists a unique Radon measure A on K such
that, V'P E C(K), (('I') = JK'PdA. So it is enough to prove that the support
of A is reduced to a point. Suppose not. Then there exist x and y in SUpp(A)
and two disjoint open neigbourhoods V and W of x and y respectively with
A(V) > 0 and A(W) > O. Since C(K) is dense in L2(IAI), there exist two
sequences f nand gn in C (K) such that f n ---+ 1 v and gn ---+ 1 w for the norm
of L2(IAI). It follows that fngn ---+ 1v1w = 0 for the norm of U(IAj) because
the mapping (I, g) t-> f 9 is continuous from L2 x L2 to L1. Therefore we have
and
((In gn) = ((In) ((gn)
= [fn dA [gn dA ---+ [ 1vdA [ 1wdA
= A(V)A(W) > O.
A contradiction. 0
x
Proposition 1.3. ([42]) For every f E C (O,.1", /-l) there is a unique map
ping fp : 0 ---+ X such that
(1.3.1) V'P E C(X), 'I' 0 fp = p( 'I' 0 f).
Moreover, if P is a /-l-negligible set such that f(O \ P) is relatively compact
in X, then fp(O) C f(O \ P).
Proof. Note that 'I' 0 f E .coo for all 'I' E C(X) so that formula (1.3.1) has a
meaning.