Advances in mathematical economics kusuoka maruyama volume 13

Ezzaki Some various convergence results for multivalued A certain limit of iterated conditional tail expectation 99 Convexity of the lower partition range of a concave vector Pythagorean

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www.SolutionManual.info

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Jean-Michel Grandmont

CREST-CNRSMalakoff, FRANCE

Norimichi Hirano

Yokohama NationalUniversity

Seiichi Iwamoto

Kyushu UniversityFukuoka, JAPAN

Marcel K Richter

University of MinnesotaMinneapolis, U.S.A

Yoichiro Takahashi

Kyoto UniversityKyoto, JAPAN

Makoto Yano

Kyoto UniversityKyoto, JAPAN

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S Kusuoka, T Maruyama (Eds.)

Advances in

Mathematical Economics Volume 13

123

www.elsolucionario.org

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Shigeo Kusuoka

Professor

Graduate School of Mathematical Sciences

The University of Tokyo

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DOI 10.1007/978-4-431-99490-9

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Table of Contents

Research Articles

F Akhiat, C Castaing, and F Ezzaki

Some various convergence results for multivalued

A certain limit of iterated conditional tail expectation 99

Convexity of the lower partition range of a concave vector

Pythagorean mathematical idealism and the framing of

Instructions for Authors

Subject Index

Historical Perspective

J Honda and S.-I Takekuma

S Kusuoka

T.Q Bao and B.S Mordukhovich

N Sagara and M Vlach

A.J Zaslavski

201207M.A Khan and A.J Zaslavski

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Adv Math Econ 13,1 33(2010)

Some various convergence results

for multivalued martingales

Fettah Akhiat1, Charles Castaing2, and Fatima Ezzaki1

1 Laboratoire modélisation et calcul scientifique, Département de Mathématiques,Faculté des Sciences et Techniques, BP 2202, Université Sidi Mohamed BenAbdellah, Fes, Morocco

(e-mail: akhiatfettah@yahoo.fr, fatimaezzaki@yahoo.fr)

2 Département de Mathématiques, Université Montpellier II, 34095 MontpellierCedex 5, France

topol-Key words: martingale, submartingale, supermartingale, mil, conditional

expecta-tion, Mosco convergence, linear topology, Pettis

1 Introduction

The purpose of this paper is to present various convergence results for gales, submartingales, supermartingales and mils with respect to the Moscotopology and the linear topology both in Bochner integration and Pettis in-tegration The paper is organized as follows In 2, we set our notation anddefinitions, and summarize needed results In 3, we state some convergence

martin-We heartily thank the referee for helpful comments and suggestions.

S Kusuoka, T Maruyama (eds.), Advances in Mathematical Economics Volume 13, 1 DOI: 10.1007/978-4-431-99490-9 1,

c

Springer 2010

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2 F Akhiat et al.

theorems for convex weakly compact valued submartingales, mils and bounded closed convex supermartingales in Bochner integration In 4, wepresent various convergence theorems for convex weakly compact valuedPettis-integrable multifunctions Several existence theorems of conditionalexpectation for convex weakly compact valued Pettis-integrable multifunc-tions and an integral representation theorem for a convex weakly compactvalued multifunction defined on L1 or more generally on a K¨othe spaceare also provided In 5, we provide some versions of Levy’s theorem forPettis-integrable multifunctions Specific applications to the convergence ofmultivalued Pettis-integrable martingales are given at the end of 5

un-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 andmils The results presented here are motivated by some applications in Mathe-matical Economics and the Law of Large Numbers (see, e.g [8,9,12,15,26])

2 Preliminaries and background

Let ;F; P / be a complete probability space, Fn/n2N an increasing quence of sub -algebras of F such that F is the -algebra generated by

se-[n1Fn Let E be a separable Banach space, Ethe topological dual of E,

D D x

j/j 2Na dense sequence in Ewith respect to the Mackey ogy E; E/, BE (resp BE) the closed unit ball of E (resp E) Let

topol-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 2 cc.E/, the distance function and the support function associated with

C are defined respectively by

d.x; C /D inffkx yk; y 2 C g x 2 E/

ı.x; C /D supf< x; y >; y2 C g x2 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 2

For each n 2 N[f1g, we denote byL1

cwk.E/.Fn/ (withF1 D F) the space

of all Fn-measurable cwk.E/-valued multifunctions X W ! cwk.E/

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Some various convergence results for multivalued martingales 3

such that ! ! jX.!/j is integrable A sequence Xn/n2N of cc.E/-valuedmultifunctions (mappings for short) is adapted if each XnisFn-measurable

A sequence Xn/n2NinL1

cwk.E/.F/ is bounded (resp uniformly integrable)

if the sequence jXnj/n2Nis 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 intersec-

tion with any closed ball in E is weakly compact A cc.E/-valued sequence

.Xn/n2NMosco-converges [36] to a closed convex set X1if

Beer [3] showed that the topology Lis stronger than the Mosco topology

If the strong dual Eb of E is separable and X is an element of

L1

cwk.E/.F/, for each n 2 N, it is known that the conditional

expecta-tion of X with respect toFn, EFnX , is the unique element (for = a.s.) of

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where H stands for the Hausdorff distance on cwk.E/.

From the definition, it is easy to check that, for each x2 BE, the sequence

Proposition 3.1 Let E be a Banach space Let An/n2N be a sequence in

cwk.E/ Assume that there exist A12 cwk.E/ and a sequence Bn/n2Nin

cwk.E/ which satisfy:

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Proposition 3.2 Let E be a separable Banach space Let D1D fe

j; j 2 Ng

be a dense sequence inBEwith respect to Mackey topology E; E/ Let

.An/n2Nbe a sequence in cwk.E/ and A1WD s-li An2 cwk.E/ Assume

that

limn!1ı

.ej; An/D ı.ej; A1/ 8j 2 N:

Then the following holds

limn!1d.x; An/D d.x; A1/ 8x 2 E:

Proof For each n 2 N and for each x 2 E we have

(3.2.1) d.x; An/D sup

j 2NŒ< e

j; x >ı.ej; An/:

For each j 2 N, the sequence < ej; x > ı.ej; An//n2N converges to

< ej; x >ı.ej; A1/ By (3.2.1), for each j 2 N and for each n 2 N, we

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6 F Akhiat et al.

Remark If An and A1are convex compact, we may replace in tion3.2the set D1by H1 D h

Proposi-j/j 1where hj/j 2Nis a dense sequence

in the closed unit ball BE, with respect to the topology E; E/ that

coin-cides on BEwith the topology of compact convergence

Now we are ready to state the almost surely convergence with respect tothe Mosco topology and the Ltopology for cwk.E/-valued martingales andmils

Theorem 3.1 Assume thatEbis separable Let.Xn/n2N be a bounded mil

in L1

cwk.E/.F/ and X1 2 L1

cwk.E/.F/ such that for each x 2 BE,

limn!1ı.x; Xn/D ı.x; X1/ a:s: Then the following hold:

(i) M - limn!1EFnX1D X1a.s.

(ii) limn!1H.Xn; EFnX1/D 0 a:s:

Proof We will proceed in several steps

Step 1 ClaimM - limn!1EFnX1D X1a.s

We have to prove that

w-ls EFnX1.!/ X1.!/ s-li EFnX1.!/

for a.s ! 2 Let us check the inclusion

X1.!/ s-li EFnX1.!/ a:s:

Let f 2 S1

X 1 By Levy’s theorem for regular martingale EFnf , we have

that limn!1EFnf !/ D f !/ a.s with respect to the norm topology

Since EFnf !/ 2 EFnX1.!/, it follows that f !/ 2 s-li EFnX1.!/

a.s Taking a Castaing representation of X1 (see [16, Theorem III-37]) wededuce that X1.!/ s-li EFnX1.!/ a.s Now we prove the inclusionw-ls EFnX1.!/ X1.!/ a.s Let DD x

j/j 2Nbe a dense sequence in

Ewith respect to the Mackey topology E; E/ Applying Levy’s

theo-rem to the L1-bounded real-valued martingales EFnı.xj; X1/ and taking

the well-known property for multivalued conditional expectation [43] givelim

n!1ı

.xj; EFnX1/D lim

Fnı.xj; X1/D ı.xj; X1/

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a.s Now let ! 2 be fixed but arbitrary for which the preceding ity holds and let x 2 w-ls EFnX1.!/ There is a sequence xk k2N in

equal-EFnkX1.!/ such that xk k2N weakly converges to x For each j 2 N

According to Proposition III-35 in [16], we deduce that x 2 X1.!/

Step 2 Claim limn!1H.Xn; EFnX1/D 0 a.s If Xn/n2Nis an E-valuedbounded mil in L1E.F/, then jjXn EFnX1jj goes to 0 a.s by virtue of a

result of Talagrand [41, Theorem 6, p 1193], because for each x 2 BE,the real-valued L1-bounded mil hx; Xn EFnX1i/n2N converges to 0a.s In the multivalued case, the claim (ii) is true by using the Definition2.1and a careful adaptation of the techniques of Talagrand developed in [41,Theorem 6, p 1193], namely limn!1H.Xn; EFnX1/ D 0 a.s (see [6,7]for details)

Step 3 limn!1ı.x; Xn/D ı.x; X1/ a:s: 8x2 BE:

Let fj/j 2Nbe a dense sequence in the closed unit ball BE with respect tothe topology of the dual norm For each x2 BElet us write

jı.x; Xn/ ı.x; X1/j jı.x; Xn/ ı.x; EFnX1/j

Cjı

.x; EFnX1/ ı

.x; X1/j:

From i i /, it is obvious that the first term jı.x; Xn/ ı.x; EFnX1/j

goes to 0 a.s for all x2 BE when n goes to 1 and so is the second term

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The following result is concerned with submartingales.

Theorem 3.2 Assume that Eb is separable Let Xn/n2N be a bounded submartingale in L1

cwk.E/.F/ and X1 2 L1

cwk.E/.F/ such that for each

x2 BE, limn!1ı.x; Xn/D ı.x; X1/ a:s:

Then the following hold:

(i)M - limn!1EFnX1D X1a.s.

(ii) limn!1H.Xn; EFnX1/D 0 a:s:

Proof Step 1 Claim M - limn!1EFnX1D X1a.s The proof is the same

as the one of Theorem3.1and is omitted

Step 2 Claim limn!1H.Xn; EFnX1/D 0 a.s

Step 3 limn!1ı.x; Xn/D ı.x; X1/ a:s: 8x2 BE:

Let fj/j 2Nbe a dense sequence in the closed unit ball BE with respect tothe topology of the dual norm For each x2 BElet us write

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From i i /, it is obvious that the first term jı.x; Xn/ ı.x; EFnX1/j

goes to 0 a.s for all x2 BE when n goes to 1 and so is the second term

cwk.E/.F/ and the

relationship with the cwk.E/-valued regular martingale EFnX1, namely

limn!1H.Xn; EFnX1/D 0 a:s:

In Theorem 3.2we provide an alternative proof of this property for martingales via Lemma V.2.9 in [39], while in the case of mils, Talagrand’stechniques are needed to obtain this result

sub-We end this section by providing a new version of Mosco convergence

results for unbounded supermartingales For this purpose, in the remainder

of this section, the conditional expectation is taken in the sense of Hiai–Umegaki [27] IfB is a sub--algebra of F, F W ) E is an integrable F-measurable multifunction, Hiai and Umegaki [27] showed the existence

of aB-measurable and integrable multifunction G such that

SG1.B/ D clfE B W f 2 S1

F.F/g;

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10 F Akhiat et al.

where SF1.F/ and S1

G.B/ is the set of all F-measurable (resp B-measurable)

integrable selections of F and G respectively, the closure being taken in

L1E.;F; P / G is the multivalued conditional expectation of F relative to

B The conditional expectation G WD E BF of Hiai and Umegaki can be

de-fined as the essential supremum of fEB W f 2 S1

F.F/g For more

informa-tion on the Hiai–Umegaki condiinforma-tional expectainforma-tion, see [27] A cc.E/-valuedintegrable sequence Xn/n2N is a supermartingale if Xn isFn-measurable

for each n 2 N and EFnXnC1 Xnfor each n 2 N For the convenience

of the reader we recall and summarize a tightness condition in the space

L1

cwk.E/.;F; P / A sequence Xn/n2N inL1

cwk.E/.F/ is cwk.E/-tight

if, for every " > 0; there is a cwk.E/-valuedF-measurable multifunction

Proof See, e.g [9, Proposition 3.3(i)]

We need a preliminary lemma

Lemma 3.1 Let Xn/n2N be a uniformly integrable supermartingale in

(b) limn!1d.x; Xn/D d.x; X1/ a.s for all x 2 E.

(c) EFmX1 Xma.s for allm2 N.

Proof (a) Let D1 D e

j/j 2Nbe a dense sequence in BE for the Mackeytopology E; E/ As Xn/n2N is a uniformly integrable cwk.E/-valuedsupermartingale in L1

cwk.E/.F/, for each j 2 N, ı.ej; Xn//n2N is abounded real-valued supermartingale in L1R.F/ So it converges a.s for every

j 2 N to a function mj in L1 Applying [14, Theorem 6.1] to the uniformly

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integrable cwk.E/-tight sequence Xn/n2Nprovided a subsequence Xn0/n2N

and X12 L1

cwk.E/.F/ such that for each v 2 L1

E .F/ the following hold

limn!1

thereby proving (c) (see [16, Proposition III-35])

The following result is an extension of a similar one due Choukairi [18,Theorem 2.14] dealing with reflexive separable Banach space and is a vari-ant of a result due to Hess [25, Theorem 5.12] Compare with similar resultsobtained in [18,32,33] for bounded martingales and sub- and supermartin-gales in Banach spaces with RNP property and strongly separable dual usingthe method of selection martingales (see, e.g [18, Proposition 2.7, Theo-rem 2.8]) We stress the fact that here we deal with unbounded supermartin-gales in separable Banach spaces without RNP property

Theorem 3.3 Let Xn/n2N be a closed convex integrable supermartingale

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Proof We will proceed in several steps.

Step 1 Here we will use a careful adaptation of a truncation technique

devel-oped in [18, Theorem 2.16] By our assumption there is f 2 L1E.F/ such

that fn D EFnf for all n 2 N For each k 2 N, let us consider the

multi-function

Xnk D Xn\ ŒfnC EFn.jf j C k/BE:

We are going to check that Xnk/n2N is a uniformly integrable valued supermartingale inL1

cwk.E/-cwk.E/.F/ By i/ and Proposition3.3, Xnk/n2N

is cwk.E/-tight Let m < n As Xnk Xn, by supermartingale property and

by monotonicity of conditional expectation one has

nfor all n 2 N, where

hknWD jfnj C EFn.jf j C k/ Further the uniformly integrable submartingale.hkn/n2Nconverges a.s to a positive integrable function hk Hence there exists

a positive constant rk depending on ! 2 such that hkn rk a.s for all

n 2 N So jXk

nj rk a.s for all n 2 N Applying Lemma3.1(a), (b) (c)

to the cwk.E/-tight uniformly integrable supermartingale Xnk/n2Nprovides

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Let f 2 SX11 By (3.3.5) we have EFnf !/2 Xn.!/ By Levy’s theorem

we have limn!1EFnf D f a.s Hence f !/ 2 s-liXn.!/ a.s Taking a

Castaing representation of X1we get X1.!/2 s-liXn.!/ a.s

4 Convergences and conditional expectation

in Pettis integration

We present in this section some new convergence results for Pettis-integrable

cwk.E/-valued multifunctions and also we state the existence of conditional

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14 F Akhiat et al.

expectations for these multifunctions A multifunction X W ) cwk.E/

is Pettis-integrable if X is F-measurable and scalarly integrable (that is

ı.x; X.:// is integrable for every x2 E) and if the scalarly integrable lections of X are Pettis-integrable Let us denote byS1

se-P e.X /.F/ the set of all F-measurable and Pettis-integrable selections of X By [16, Theorem V-13],

S1

P e.X /.F/ is convex and compact with respect to the topology of pointwise

convergence on L1˝ E, namely the topology PE1.F/; L1.F/ ˝ E/,

where PE1.F/ is the space of all F-measurable and Pettis-integrable

E-valued functions defined on ;F; P / The usual Pettis norm jjf jjP e of

cwk.E/.Fn/ Given a sub--algebra B and a

cwk.E/-valued Pettis integrable multifunction X 2 P1

cwk.E/.F/, the Pettis tional expectation of X is, by definition, a B-measurable cwk.E/-valued

condi-Pettis-integrable multifunction denoted by P e-EBXnwhich satisfies

Definition 4.1 An adapted sequence.Xn/n2Nin P1

cwk.E/.F/ is a mil if for every " > 0, there exists p such that for n p, we have

P sup

pqnH.Xq; P e-EFqXn/ > "/ < ";

where H stands for the Hausdorff distance on cwk.E/ and P e-E FqXn is the Pettis conditional expectation ofXnassociated with the -algebra Fq

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Accordingly, our main task is to prove the existence of Pettis conditional pectation forF-measurable cwk.E/-valued Pettis-integrable multifunctions.

ex-We will provide first some preliminary results in multivalued Pettis tion which lead us to the existence of conditional expectation in this class ofmultifunctions It is worth to address the following related question Given a

integra-F-measurable and scalarly integrable cwk.E/-valued (resp ck.E/-valued) mapping X W ) E, find sufficient conditions for which X is cwk.E/- Pet- tis (resp ck.E/- Pettis) in the sense: for each A 2 F there is KA2 cwk.E/

(resp ck.E/) such thatR

Aı.x; X /dP D ı.x; KA/ for all x2 E Inother words, KA is the cwk.E/ (resp ck.E/) Pettis integral of X over A.

When X W ! E isF-measurable and scalarly integrable E-valued

map-ping, the preceding notions coincide with the classical definition of integrability In the following we will provide the answer to this question

Pettis-Proposition 4.1 Let.Xn/n2Nbe a sequence in P1

cwk.E/.F/ and respectively

(ii) For each A 2 F, RAXndP /n2N is Cauchy in.cwk.E/;H/ resp:

.ck.E/;H//, here H is the Hausdorrf distance on cwk.E/ resp:ck.E// Then for eachA2 F, there is KA2 cwk.E/ resp:ck.E// such that

Proof By i i / For each A 2F, RAXndP /n2N is Cauchy in cwk.E/;H/

(resp .ck.E/;H/), hence RAXndP /n1 converges in cwk.E/;H/ (resp.

.ck.E/;H// to a convex weakly compact (resp convex compact) set KA

scalarly integrable mapping satisfying:

(i) For eachA2 F, limn!1R

Aı.x; Xn/dP DR

Aı.x; X /dP for all

x2 E.

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(iii) Xn/n2N is Pettis uniformly integrable, that is, for every " > 0, there

exist > 0 such that every A 2 F with P.A/ < , one has

supn1xsup2BE

A\B k

XndP / ı

.x;Z

A\B k

XmdP /jC

Z

An.A\Bk/jı.x; Xn/ ı.x; Xm/jdP

for each x 2 BE , m; n 2 N, each A 2 F and for each k 2 N Let

" > 0 For k large enough, using (iii) we have P A n A \ Bk// < so that

supn1supx 2BE

R

An.A\B k /jı.x; Xn/jdP < " Hence the last integral is

< 2" Now k being chosen, the integral

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is " for m; n large enough using (ii) By taking the supremum on BE inthe left member of the inequality

jı

.x;Z

AXndP /

ı

.x;Z

cwk.E/.F/ satisfying the Pettis uniformly integrable condition (iii)

Com-pare with Theorem 1 in [37] dealing Vitali theorem for single-valued integrable functions

Pettis-In the following we will provide a simple criteria of Pettis integrabilityvia compactness results in Set-Valued Bochner integration [16]

Theorem 4.2 Let E be a separable Banach space Let X be a cwk.E/ resp:ck.E//-valued Pettis-integrable mapping Then there exist a sequence

further, for each n 2 N, the multifunction Xn WD 1AnX is cwk.E/-valued

(resp ck.E/-valued) F-measurable and integrably bounded, consequently

the set ofF- measurable selections of 1AnX is convex weakly compact in

L1E.F/ (see, e.g [13, Theorem 6.2.3]), henceR

A1AnXdP /n2N is Cauchy in cwk.E/;H/ (resp .ck.E/; H/).

Let " > 0 Let us choose N large enough so that P A n A \ AN/ implies

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se-converges to a convex weakly compact (convex compact) set KA with spect to the Hausdorff distance In particular, we have

re-limn!1ı

.x;Z

In the single valued case, namely X 2 PE1, the above results are reduced

to the classical Pettis integral The following characterisation of Pettis grability is well-known (see, e.g [13,24,38]) An inspection of the proof ofTheorem4.2provides an alternative proof

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inte-Some various convergence results for multivalued martingales 19

Corollary 4.1 LetX W ! E be a scalarly F-measurable and scalarly

integrable mapping Then the following is equivalent

Proof The implication a/ ) b/ is well-known (see, e.g [1]) To prove that

.b/ ) a/ one can repeat the arguments of the proof of Theorem4.2 Let

AnD ŒjXj n for each n 2 N Then An2 F with " limnP An/D 1 Now

we assert that for each A 2F, the sequence RA1AnXdP /n2Nis Cauchy in

E Let " > 0 Let us choose N large enough so that P A n A \ AN/

A1AnXdP

i hx

;Z

A1AmXdP

ij

D jZ

Ahx; 1AnXidP

Z

Ahx; 1AmXidP j

Z

A1AnXdP

Z

A1AmXdPjj < ":

Hence we conclude that R

AXndP /n2N is a Cauchy sequence in E quently R

Conse-AXndP /n2Nconverges to an element kA 2 E with respect to the

norm topology In particular, we have

limn!1hx;

Z

A1AnXdPi D hx; kAi

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of the preceding results can be found in [2,13,24,31].

The above considerations lead to the existence of conditional expectation

of a special class of cwk.E/-valued Pettis-integrable multifunctions pare with [21,42] for details and comments on this subject We begin withsingle-valued Pettis-integrable functions

Com-Theorem 4.3 Assume that E is a separable Banach space Let B be a sub- -algebra of F and X be a Pettis-integrable E-valued function such that EB jXj 2 Œ0; C1Œ Then there exists a unique B-measurable, Pettis-

integrable E-valued function, denoted by P e-E B X , which enjoys the

fol-lowing property: For everyh2 L1.B/, one has

P Z

e-hP e-E

BXdP D P

e-Z

hXdP:

Now we proceed to the existence of conditional expectation in a class of

cwk.E/-valued Pettis-integrable multifunctions Namely

Theorem 4.4 Assume that Eb is separable Let B be a sub--algebra of

F and let X be a cwk.E/-valued Pettis-integrable multifunction such that

EB jXj 2 Œ0; C1Œ Then there exists a unique B-measurable,

cwk.E/-valued Pettis-integrable multifunction, denoted by P e-EB X , which enjoys

the following property: For everyh2 L1.B/, one has

P Z

hXdP denote the cwk.E/-valued

Aumann–Pettis integral ofhP e-EB X and hX respectively.

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Proof Both Theorems4.3and4.4follow from a more general result ing the integral representation theorem (Theorem4.5) for a class of mapping

Let us mention a useful corollary

Corollary 4.2 Under the hypotheses and notations of Theorem 4.3 and 4.4 , the following hold

(1) For every h 2 L1.B/ and for every x 2 E and for every f 2

and henceP e-EBf !/2 P e-EB X.!/ a.s.

(2) For everyh2 L1.B/ and for every x2 E, one has

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22 F Akhiat et al.

Proof

(1) EqualityR

hh ˝ x; P e-EB idP D R

hh ˝ x; fidP follows from

Theorem4.3and equality

j/j 2Nis a dense sequence in Efor the Mackeytopology, we get

a.s for all j 2 N By Proposition III-35 in [16], we get

P e-EBf !/2 P e-EBX.!/ a.s.

(2) Follows from the Strassen formula [16] applied to the Aumann–Pettismulltivalued integrals of the cwk.E/-valued Pettis-integrable X and

Now establish an integral representation theorem for class of mapping

M W L1.B/ ! cwk.E/ (alias multivalued Dunford–Pettis theorem) in the

vein of [16, Theorem V-17], recovering both Theorems4.3and4.4

Theorem 4.5 Assume thatEbis separable Let B be a sub--algebra of F Let us consider a cwk.E/-valued mapping M W L1.B/ ! cwk.E/ satisfying the following conditions:

(i) For eachx 2 E, the scalar functionh 7! ı.x; M.h// is

contin-uous on bounded subset ofL1.B/ for the topology of convergence in probability.

(ii) M.f C g/ D M.f / C M.g/ if fg 0 for f; g 2 L1.B/.

(iii) There is a sequence.Xn/n2Nin L1

cwk.E/.;F; P / and a B-measurable partition.Bn/n2Nof satisfying

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Proof By virtue of (iii) and Theorem 3 (Remark 4) in [43], for each n 2

N, there is a unique cwk.E/-valuedB-measurable and integrably bounded

B n

hndP; 8h 2 L1

.B/:

Let us define .!/ D n.!/ if ! 2 Bn Then isB-measurable Using i/

it is not difficult to check that

for every h 2 L1.B/ and for every n; l 2 N Let us consider an arbitrary

B-measurable selection g of Then we have

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24 F Akhiat et al.

for every x2 E By (ii) the multifunction h 7! M.h/ is scalarly continuous

on bounded subsets of L1.B/ with respect to the convergence in probability,

so that from the above estimate and (4.5.1) the sequence

is a L1 B/; L1.B// Cauchy sequence But the pointwise limit of this

se-quence is hx; gi, therefore by classical property of L1space we have

hgdP 2 M.h/ by passing to the limit when

m goes to 1 in (4.5.2) Now we prove that is Pettis-integrable As any

B-measurable selection g of is Pettis integrable, according to our

defini-tion it is enough to check that is scalarly integrable Let x 2 E Bythe measurable implicit Theorem III-38 in [16], there isB-measurable selec-

tion of such that hx; i D ı.x; / We conclude that the

cwk.E/-valuedB-measurable multifunction is Pettis-integrable Let us denote by

S1

P e./.B/ the set of all B-measurable and Pettis-integrable selections of .

ThenS1

P e./.B/ is nonempty convex .P1

E ŒE.B/; L1.B/˝E/ compact,

by applying Theorem V.14 in [16] We finish the proof by showing that

Assume that there is 2 M.h/ n P e-R

hYdP By Hahn–Banach theorem,there is x2 Esuch that

By the measurable implicit theorem [16], there is aB-measurable and

Pettis-integrable selectioneg of such that

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Using the assumption EB jXj 2 Œ0; C1Œ provides a B-measurable

partition Bn/n2N of and a sequence Xn/n2N WD 1BnX /n2N in

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26 F Akhiat et al.

(3) Variants of Theorems4.3–4.5are available when dealing with the space

L1EŒE.;F; P / and the space L1

E ŒE.;F; P / (see [15,43]) ther it is possible to formulate the integral representation 4.5 on a Kothespace instead of L1 in the vein of Theorem V-17 in [16] Howeverthe study of the convergence of Pettis conditional expectation for un-bounded cc.E/-valued Pettis-integrable multifunctions in the same style

Fur-as in [9,16,27,43] (dealing with Bochner integration) is an open problem.(4) We will show in next section that convergence of cwk.E/-valued Pettis-integrable martingales with respect to the Mosco topology or lineartopology is now available, using the above techniques and the conver-gence of regular martingales of the form XnD P e-EFnX , where X is

a cwk.E/-valued Pettis-integrable multifunction and P e-EFnX is the

Pettis conditional expectation of X For this purpose, we will provide

in 5 some convergence results for the Pettis conditional expectation ofthe form P e-EFnX with X 2 P1

cwk.E/.F/ and its applications to the

convergence of Pettis-integrable martingales and mils

5 Convergence of cwk.E/-valued Pettis-integrable

martingales

We present first some versions of Levy’s theorem for a cwk.E/-valuedPettis-integrable multifunction This study is a starting point for furtherapplications in the Mosco convergence of Pettis-integrable multivalued mar-tingales, compare with [21,37,42] dealing vector-valued Pettis-integrablemartingales

Proposition 5.1 Assume that E is a separable Banach space and X is a

Pettis-integrable E-valued mapping such that C/ W E FnjXj 2 Œ0; C1Œ for

eachn2 N Then we have

limn!1P e-E

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for each k 2 N We claim that

limn!1P e-E

FnX D lim

n!1P e-E

FnX

a.s., thus proving the claim and completes the proof

The following is a nontrivial extension of Proposition5.1

Proposition 5.2 Assume that E is a separable Banach space Let Y be a

real-valued positive F-measurable function such that E FnY 2 Œ0; C1Œ for

eachn 2 N Let Xn/n2Nbe a sequence of Pettis-integrable E-valued

func-tions Assume that the following conditions are satisfied:

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28 F Akhiat et al.

We have the estimate

jP e-EFnXn X1j jP e-EFnXn P e-EFnX1j C jP e-EFnX1 X1j

EFnjXn X1j C jP e-EFnX1 X1j:

From the above estimate and /, /, the result follows

Now we provide a multivalued version of the Levy’s theorem for

cwk.E/-valued Pettis-integrable multifunctions extending Proposition5.1

Theorem 5.1 Assume that Eb is separable Let X be a cwk.E/-valued

Pettis-integrable multifunction such thatEFnjXj 2 Œ0; C1Œ for each n 2 N.

Claim 1X s-li P e-EFnX a:s:

As EFnjXj 2 Œ0; C1Œ for each n 2 N, by Theorem4.4the conditionalexpectations P e-EFnX is Fn-measurable and Pettis-integrable Now let

f 2 S1

P e.X /.F/ By Theorem 4.3 and Corollary 4.2(1), P e-EFnf

is Fn-measurable and Pettis-integrable and satisfies P e-EFnf !/ 2

P e-EFnX.!/ a.s Furthermore, by Proposition5.1, limn!P e-EFnf D f

a.s So we conclude that f 2 s-li P e-EFnX a.s Since this is true for any

f 2 S1

P e.X /.F/, by invoking /, we see that Claim 1 is true

Claim 2w-ls P e-EFnX X a:s:

Let xj/j 2Nbe a dense sequence in Efor the Mackey topology E; E/

Then the calculus of support functions in the integral representation formula

of Theorem4.4[cf Corollary4.2(3)] imply

a.s for all j 2 N Let ! 2 be such that the preceding relations are

satisfied Let x 2 w-ls P e-EFnX.!/ Then xk ! x weakly for some

xk 2 P e-EFnk.X /.!/ and hence

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So x 2 X.!/ because X is convex weakly compact valued [16, Prop III-35]

We end this paper with some new applications to the L convergence

of cwk.E/-valued Pettis-integrable martingales illustrating the above niques Compare with Theorem 3.1 in [21] dealing with Pettis-integrablevector-valued martingales

tech-Theorem 5.2 Assume thatEbis separable and E is such that c0is not morphic to a subspace of E c0 6,! E/ Let Xn;Fn/n2N be an adapted sequence of cwk.E/-valued Pettis-integrable multifunctions satisfying:

iso-(i) EFqjXnj < 1 for each n 2 N and each 1 q < n.

(ii) Xn;Fn/n2N is a cwk.E/-valued Pettis-integrable martingale, that is,

P e-EFnXnC1D Xnfor alln2 N.

(iii) supn2Nsupx 2BE

R

jı.x; Xn/jdP < 1

(iv) There is a partition An/n2N in[1

nD1Fn such that for eachm 2 N,

.XnjAm/n2Nis bounded in L1

cwk.E/.Am/.

(v) XnjAm/n2Nis cwk.E/-tight for each m 2 N.

Then there is a cwk.E/-valued Pettis-integrable multifunction X1such that

Shortly.Xn/n2NL-converges a.s toX1.

Proof By i / and Theorem4.4the Pettis conditional expectations

P e-EFqXn 1 q < n/

exist and belong to P1

cwk.E/.F/ Accordingly the cwk.E/-valued

Pettis-integrable martingale given in i i / exists Now for each m 2 N, let n.m/ 2 N

be such that Am 2 Fn.m/ Then XnjAm;FnjAm/nn.m/is a cwk.E/-valuedmartingale inL1

cwk.E/.Am/ It follows that, for each x2 BE,

.ı.x; XnjAm/;FnjAm/nn.m/

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30 F Akhiat et al.

is a real-valued L1-bounded martingale, so it converges a.s to a function

hmxin L1.Am/ As XnjAm;FnjAm/nn.m/is bounded and cwk.E/-tight in

L1

cwk.E/.Am/ by iv/ and v/, from [14, Theorem 6.1], we can find a quence Xn0jAm/nn.m/and X1m 2 L1

subse-cwk.E/.Am/ such that Xn0jAm/nn.m/

weakly biting converges to X1m 2 L1

cwk.E/.Am/ so that by identifying thelimits we get

mD1X1m1Am Then obviously Xn/n2N L-converges a.s to

X1 It is easy to check that X1isF-measurable By iii/ it follows that, for

every x 2 BE, ı.x; Xn//n2Nis a real-valued L1-bounded martingale,

so it converges a.s to a function in L1 Hence X1is scalarly integrable, andsince c0 6,! E/ any F-measurable and scalarly integrable selection of X1

is Pettis-integrable Accordingly X1is Pettis-integrable

We finish this paper with an application of Theorems4.3–5.2

Corollary 5.1 Assume that E is separable and is such that c0is not phic to a subspace ofE c0 6,! E/ Let Xn;Fn/n2Nbe an adapted sequence

isomor-of E-valued Pettis-integrable functions satisfying:

(i) EFqjXnj < 1 for each n 2 N and each 1 q < n.

(ii) Xn;Fn/n2Nis an E-valued Pettis-integrable martingale, that is, XnD

P e-EFnXnC1for alln2 N.

R

jhx; XnijdP < 1.

.XnjAm/n2Nis bounded inL1E.Am/.

(v) XnjAm/n2Nis cwk.E/-tight for each m 2 N.

Then there is an E-valued Pettis-integrable function X1such that.Xn/n2N

norm converges a.s toX1 If.Xn/ is Pettis-uniformly integrable in PE1.F/, then limn!1jjXn X1jjP1

E D 0.

Proof Under our assumption any L1-bounded martingale in L1E.F/

satis-fying the cwk.E/-tightness condition norm converges a.s to a function in

Trang 38

L1E.F/, so the result follows from the arguments given in the proof of

The-orem 5.2while the convergence in Pettis norm follows from [1,

Remarks (1) If Ebis separable and if E have the RNP, then one can stitute the tightness condition v/ in Theorem5.2by the following: for each

sub-m2 N, for each A 2 Am\ F, [n2NR

AXndP is relatively weakly compact

in E When E have the RNP, one may recover a former result due to Egghe[21, Corollary 3.2], namely

Corollary 5.2 Assume that E is separable and have the RNP Let

.Xn;Fn/n2N be an adapted sequence of E-valued Pettis-integrable

func-tions satisfying:

(i) EFqjXnj < 1 for each n 2 N and each 1 q < n.

(ii) Xn;Fn/n2Nis an E-valued Pettis-integrable martingale, that is, XnD

P e-EFnXnC1for alln2 N.

R

jhx; XnijdP < 1.

.XnjAm/n2Nis bounded inL1E.Am/.

Then there is an E-valued Pettis-integrable function X1such that.Xn/n2N

norm converges a.s toX1 If.Xn/ is Pettis-uniformly integrable in PE1.F/, then limn!1jjXn X1jjP1

E D 0.

(2) The results presented in 5 lead us to address the following question:

is it possible to prove a version of Theorem5.1(resp Theorem5.2) for bounded closed convex valued Pettis-integrable mappings (resp closed con-

un-vex valued Pettis-integrable supermartingales)

(3) The weak star Kuratowski convergence for sub- supermartingales andmils in a weak star dual of a separable Banach space can be found in a forth-coming paper [10]

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