Advances in Mathematical Economics

188 Pages·2004·3.591 MB·English

by  Charles Castaing, Paul Raynaud de Fitte (auth.), Shigeo Kusuoka, Toru Maruyama (eds.)

Advances in
MATHEMATICAL
ECONOMICS
Managing Editors
Shigeo Kusuoka Toru Maruyama
University of Tokyo Keio University
Tokyo, JAPAN Tokyo, JAPAN
Editors
Robert Anderson Jean-Michel Grandmont Norio Kikuchi
University of California, CREST-CNRS Keio University
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 H urwicz 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,
University of Tokyo Hitotsubashi University
Berkeley
Tokyo, JAPAN Tokyo, JAPAN
Berkeley, U.S.A.
Takao Fujimoto Kunio Kawamata Makoto Yano
Kagawa University Keio University Keio University
Kagawa, JAPAN Tokyo, JAPAN Tokyo, JAPAN
Aims and Scope. The project is to publish Advances in Mathemat
ical Economics once a year under the auspices of the Research Cen
ter of Mathematical Economics. It is designed to bring together those
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The scope of Advances in Mathematical Economics includes, but is
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- Economic theories in various fields based on rigorous mathematical
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motivated by economic theories.
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Authors are asked to develop their original results as fully as pos
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Springer
Tokyo
Berlin
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London
Milan
Palis
s.
Kusuoka, T. Maruyama
(Eds.)
Advances in
Mathematical Economics
Volume 6
, 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-13: 978-4-431-68452-7 e-ISBN-13: 978-4-431-68450-3
DOl: 10.1007/978-4-431-68450-3
Printed on acid-free paper
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SPIN: 10966679
Table of Contents
Research Articles
C. Castaing, P. Raynaud de Fitte
On the fiber product of Young measures with
application to a control problem with measures 1
D. Glycopantis, A. Muir
The compactness of Pr(K) 39
S. Iwamoto
Recursive methods in probability control 55
S. Kusuoka
Approximation of expectation of diffusion processes
based on Lie algebra and Malliavin calculus 69
V. L. Levin
Optimal solutions of the Monge problem 85
H. Nakagawa, T. Shouda
Valuation of mortgage-backed securities based on
unobservable prepayment costs 123
K. Urai, A. Yoshimachi
Fixed point theorems in Hausdorff topological
vector spaces and economic equilibrium theory 149
A. Yamazaki
Monetary equilibrium with buying and selling price
spread without transactions costs 167
Subject Index 185
Instructions for Authoers 187
Adv. Math. Econ. 6, 1-38 (2004) Advances In
MATHEMATICAL
ECONOMICS
©Springer-Verlag 2004
On the fiber product of Young measures with
application to a control problem with measures
Charles Castaing1 and Paul Raynaud de Fitte2
1 Departement de Mathematiques, Case 051, Universite de Montpellier II, 34095
Montpellier Cedex 5, France
2 Laboratoire Raphael Salem, UMR CNRS 6085, UFR Sciences, Universite de Rouen,
76821 Mont Saint Aignan Cedex, France
Received: April 11, 2003
Revised: June 19,2003
JEL classification: C61, C73
Mathematics Subject Classification (2000): 28 A 33, 46 N 10, 49 L 25
Abstract. This paper studies, in the context of separable metric spaces, the stable con
vergence ofthe fiber product for Young measures with applications to a control problem
governed by an ordinary differential equations where the controls are Young measures.
Essentially we study some variational properties of the value functions and the exis
tence of quasi-saddle points of these functions which occurs in this dynamic control
problem, and also their link with the viscosity solution of the associated Hamilton
Jacobi-Bellman equation.
Key words: Young measure, relaxed control, fiber product, dynamic programming,
viscosity solution
1. Introduction
This paper is divided in two parts. In section 2 we state some new convergence
results for Young measures and also the proofs of the fiber product of Young
measures. The third section is devoted to the study of the value functions of
a control problem where the dynamic is governed by an ordinary differential
equation (ODE) where the controls are Young measures. Here the stable con
vergence for the fiber product of Young measures is crucial in the statement
of the variational properties of the value functions in the control problems un
der consideration and the developements of Mathematical Economics (see e.g
[Tat02]). Similar differential games with Young measures governed by some
2 C. Castaing, P. Raynaud de Fitte
classes of evolution inclusions are given in a forthcoming paper. References
for control problems are e.g [EK72 E1l87 ES84 KS88 BJ91].
2. Stable convergence versus convergence in probability
2.1 Young measures
For simplicity, the topological spaces we consider in this work are only metric
spaces. Most of the convergence results on Young measures in this work can be
extended to the case of completely regular Suslin spaces (which includes e.g.
weak topologies of separable Banach spaces, or spaces of distributions). This
will be detailed in a forthcoming work. All metric spaces we shall consider are
assumed to be separable, or, more generally, they do not contain any discrete
subset with measurable cardinal: this ensures that every Borel measure f.1 on
such a metric space § has a separable support and, consequently, f.1 is inner
regular w.r.t. the totally bounded subsets of § (the reader interested in measur
able cardinals can read [Bi168, Appendix III]; recall that it is consistent with
the usual axioms of logic to assume that measurable cardinals do not exist).
Actually, in the convergence results presented here, we only need that the limit
has separable support.
If (§, d) is a metric space, we denote by Cb (§) the set of all real-valued
bounded continuous functions defined on § and we denote by Bs the Borel
Mt
a-algebra of §. The set of probability measures on Bs is denoted by (§).
Mt
We endow (§) with the narrow (or weak) topology, that is, the coarsest
topology such that, for each f E Cb (§), the mapping f.1 f-+ f.1(1), Mt (§) -+
JR., is continuous.
Let BL(§, d) = BL( d) be the set of all mappings f : § -+ JR. which satisfy
If(x) - f(y)1
IlfIIBL(d) := sup If(x)1 + sup d() < +00.
xES x,yES, xopy X, Y
The space BL(d), endowed with the norm 1I.IIBL(d)' is a Banach space. Dudley
Mt
[Dud66, Theorem 6 and Theorem 8] has shown that (§) embeds homeo
morphically in the strong dual BL( d) * of BL( d). Let us denote by BLI (d) the
Mt
unit ball of BL(d). The topology of (§) is induced by the metric DBL(d)
defined by
DBL( d) (f.1, v) = sup (f.1(1) - v(l) .
JEBL,(d)
Let (§, d) be a metric space and let (n, S, P) be a probability space. We
denote by y(n, S, Pi §) the set of measurable mappings
Fiber product of Young measures, application to a control problem 3
. {n
M~ (§)
I->
f.1. w f.1w
I->
Each element f.1 of y(n, S, P; §) can be identified with the measure /1 on
IA
(n x §, S ® Bs) defined by /1(A x B) = f.1w(B) dP(w) (and the mapping
f.1 I-> /1 is onto if § has the Radon property, see e.g. [Va173]). In the sequel, we
shall use freely this identification. For instance, if f: n x § -+ ~ is a bounded
In
measurable mapping, the notation f.1(f) denotes f.1w (f (w, .)) d P( w). The el
ements of y(n, S, P; §) are called Young measures on n x §.
The set y(n, S, P; §) is endowed with the coarsest topology such that,
for each A E S and each f E Cb (§), the mapping f.1 I-> f.1( lA ® I),
y(n, S, P; §) -+ ~ is continuous. Convergence in this topology is sometimes
called stable convergence, we will follow this tradition.
°
Let L (n; §) be the set of random elements of § defined on n (we identify
random elements which are equal P-almost everywhere). To each element X
of LO(n; §), we associate the Young measure fix : w I-> 6X(w), where, for
any x E §, 6x denotes the probability concentrated on x. Young measures of
the form fix are called degenerate Young measures. If P has no atoms and if §
is Suslin, the set of degenerate Young measures is dense in y(n, S, P; §), see
[BaI84b].
We call integrand on n x § any measurable mapping f : n x § -+ R
If furthermore f(w,.) is continuous for every wEn, we say that f is a
Caratheodory integrand. In the case when § is separable, a sufficient condi
tion for a mapping f : n x § -+ ~ to be a Caratheodory integrand is that
f (w, .) be continuous for every wEn and f (., x) be measurable for every
x E § [CV77, Lemma III. 14]. If f is an integrand such that there exists a mea
surable P-integrable function <J>: n -+ ~+ such that If(w, .) I :::; <J>(w) for each
wEn, we say that f is Ll-bounded.
It is well known that, if § is a Suslin metrizable space and if (f.10:)0: is
a net in y(n,S,p;§) which stably converges to some f.100 E y(n,S,p;§),
then we have lim info: f.10:(f) ?: f.100(f) for every integrand f ?: 0 such that
f(w,.) is l.s.c. for each wEn (this is called the Portmanteau Theorem, or the
Semicontinuity Theorem, see e.g. [BalDO] for a proof for sequences - the result
also holds true for nets, see [RdF03] for a reasoning with nets). In particular,
we have the following characterization of stable convergence.
Stable convergence and Caratheodory integrands Assume that § is Suslin
metrizable, let (f.10:)0: be a net in y(n,S,p;§) and let f.100 E y(n,S,p;§).
Then (f.10:)0: stably converges to f.100 if and only if limo: f.10: (f) = f.100 (f) for
every Ll-bounded Caratheodory integrand f on n x §.
In this section, we shall also use a different result, which holds for nonneces
sarily Suslin spaces. Let us first define the spaces BL( d) and BL' (d).
4 C. Cast aing, P. Raynaud de Fitte
We denote by BL( d) the set of all integrands I on 0 x § ---+ JR such that
there exists a measurable mapping ¢ : 0 ---+ JR+ with 11/(w, .)IIBL(d) ::; ¢(w)
for every w E 0 (if § is separable, BL( d) is simply the set of integrands I such
that I(w,.) E BL(d) for each w EO). We denote by BL'(d) the set of elements
I of BL( d) which have the form
n
L
I(w,x) = 1Ai(W)!i(X),
i=l
where (AI, ... , An) is a measurable partition of 0 and each Ii is in BL( d). Let
L~L(d) be the space of Bochner integrable functions defined on (0, S, P) with
values in BL( d). We have
BL'(d) C L~L(d) c {J E BL(d); I is L1-bounded}.
Lemma 2.1.1. Let (§, d) be a metric space. Let (1l,c')aEA be a net in Y(O, S, P; §)
and let J-too E Y(O, S, P; §). Thefollowing conditions are equivalent.
(a) (J-ta)a stably converges to J-too.
(b) For each L1-bounded integrand I E BL(d), we have lima J-ta(f)
J-too(f).
(c) For each I E L~L(d)' we have lima J-ta(f) = J-too(f).
(d) For each integrand IE BL'(d), we have lima J-ta(f) = J-too(f).
Proof. Assume (a). Let E > O. Let I be an L1-bounded element ofBL(d). There
exists a measurable function ¢: 0 ---+ JR+ such that 11/(w, .)IIBL(d) ::; ¢(w) for
every w E O. Furthermore, there exists a P-integrable function 'P : 0 ---+ JR+
such that I/(w, x)1 ::; 'P(w) for each (w, x) E 0 x §. We can thus find Of E S
and M > 0 such that
r
P(O \ Of) < E, ¢ 10, ::; M, and 'PdP < E.
}O\O,
1
Define E BL( d) by
f(w, x) = { oitl(w, x) if wE Of
ifw E 0 \ Of
We have f(w,.) E BLl(d) for every w E O. Moreover, for any J-t E
Y(O, S, P; §), we have
(2.1.1)
Now, from our general hypothesis on metric spaces (see the Introduction),
the measure J-tOO(O x .) E M~ (§) has a separable support, thus it is inner
Fiber product of Young measures, application to a control problem 5
regular w.r.t. the totally bounded subsets of §. There exists a totally bounded
subset K of § such that poo (n x K) > 1 - ElM. Recall that every Lipschitz
function h : K -> [0, 1] can be extended to a Lipschitz function defined on
§, with same Lipschitz coefficient, and with values in [0,1] (see e.g. [Dud66]).
For any continuous function h on § and any B c §, let us denote IlhilB :=
SUPxEB Ih(x)l· The set of restrictions to K of elements of BL1(d) is totally
bounded for II.II (it is a subset of the compact space BL1 (R, d), where R is
K
the d-completion of K). e There exist thus gl, ... ,gn E BL1 (d) such that, for
each h E BLI (d), we have infi=l, ... ,n Ilh - gillK :::; ElM.
For each wEn, we can find N(w) E {I, ... , n} such that 111(w,.) -
gN(w) 11K :::; ElM. Furthermore, we can assume that N is measurable, because
Lipschitz functions on K are determined by their values on a countable (dense)
subset of K. For i = 1, ... , n, let Ai = {N(w) = i}, and let 9 = L~=l 1Ai Q9
gi. We have 9 E BL'(d) and 111(w,.) - g(w, .)IIK :::; ElM for every wEn.
Let Kf = {x E §; d(x, K) < ElM}. For each wEn, as l(w,.) and
g(w,.) are I-Lipschitz, we have
(2.1.2)
Let h : °§ -> [0,1] be a Lipschitz mapping such that hex) = 1 if x E K and
hex) = if x rf. Kf (we can take e.g. hex) = (1 - (MIE)d(x, K)) V 0). We
have
limpa( 10 Q9 (1- h)) = pOO( 10 Q9 (1 - h)) :::; pOO(n x (§ \ K)) :::; ElM,
a
thus there exists ao E A such that
(2.1.3)
Furtherm~re, we have l1cw, x) - g(w,x)1 :::; 2 for every (w,x) E n x §,
because f and 9 are bounded by 1. From (2.1.2) and (2.1.3) we thus have, for
a 2: ao,
(1- 1 11-
I(pa - pOO) g) < (pa + pOO) gl
(/1- h))
< (pa + pOO) gl (10 Q9
+ (pa + pOO) (2( 10 Q9 (1- h)))