Convergence theorems for set valued and fuzzy valued martingales and smartingales

CONVERGENCE THEOREMS FOR SET VALUED AND FUZZY VALUED MARTINGALES AND SMARTINGALES

SHOUMEI LI AND YUKIO OGURA

ABSTRACT. The purpose of this paperisto give convergence the- orems both for closed convex set valued and relative fuzzy valued- martingales, and sub- and super- martingales. These kinds of mar- tingales, sub- and super-martingales are the extension of classical real valued martingales, sub- and super-martingales. Here we com- pare two kinds of convergences, in the Hausdorff metric and in the Kuratowski-Mosco sense. We also introduce a new convergence for the fuzzy valued case in the graph sense and obtain convergence theorems.

1.

Introduction

In practice, we are often faced with random experiments whose out- comes are not numbers or vectors but are sets or are expressed in inexact linguistic terms. The former case is set valued random variables (also called random set). In the past 50 years, the study of set valued random variables has been developed extensively, with many applications to eccr nomics and optimal control problems, statistical problems and so on, see for examples, Arrow and Debreu [1], Arstein and Vitale [2], Aumman [3, 4], Debreu [8], Hiai and Umegaki [10, 11], Kendall [12], Klein and Thompson [14], Kudo [16], Papageorgiou [27-28], etc.. The concept of set valued random variable has been formalized as an extension of random variables and random vectors (cf. Matheron [23], Molchanov [24]). The latter case is fuzzy set random variables. As a simple example, consider a group of individuals chosen·at random who are questioned about the

Received February24, 1998.

1991 Mathematics Subject Classification: 60G42, 28B20, 60G48, 60D05.

Key words and phrases: set valued martingale, fuzzy valued martingale, sub-, super-martingale, the Kuratowski-Mosco convergence.

Research supported by Grant-in-Aid for Scientific Research No. 09440085.

weather of tomorrow in a particular city on a particular day. Some pos- sible answers would be "a little raining day" , "a heavy raining day", "an extremely heavy raining day" and so on. Whether it will rain tomorrow or not

a random phenomenon but each outcome is an inexact linguis- tic terms. A possible way of handling these inexact linguistic terms is by using the concepts of fuzzy sets (cf. Zadeh [34]). This kind of fuzzy valued random variable was introduced by Puri and Ralescu in [29], and was followed by many works such as strong law of large numbers and central limit theorems (cf. [5, 6, 13, 30] etc.).

Concerning set valued martingales, Hiai and Umegaki [10] have given the definitions and obtained convergence theorems. This theory is the basic foundation of the study of set valued martingale theory and its applications. Their results were used by many authors such as Ban [5, 6J, Luu [22], Papageorgiou [27, 28] etc. The well cultivated results in above theory are, however, mainly on compact set valued random variables.

This comes from the fact that the most typical method in the theory of set valued random variables is to embed the family of all of compact sets into a Banach space by using H. Radstrom's embedding theorem [31], which

only available for the family of all of compact sets. For example, Hiai and Umegaki only got the convergence theorem for the compact set valued martingales by using embedding theorem. Although they obtained a regularity theorem, they did not get the convergence theorem for the closed set valued martingales. Since regular property does not imply convergence in Hausdorff metric for the set valued martingales, we can see that Hausdorff topology

a too strong topology for the study of set valued random variables.

In 1991, Puri and Ralescu [30] built a concept of fuzzy valued martin-

gale and got a convergence theorem for fuzzy valued martingales. The

basic route of this kind of study

to exploit the theory of set valued ran-

dom variables, because fuzzy valued random variables are considered as

a family of set valued random variables satisfying some additional con-

ditions, and there are very rich mathematical properties in the theory of

set valued random variables as that

mentioned above. Most of impor-

tant results in this theory are, however, mainly on fuzzy valued random

variables on the case where the level sets are compact. They still used

the embedding method. We have to notice that many important results

in that theory, such as central limit theorems and convergence theorem

for fuzzy valued martingales, were. obtained under Lipschitz condition,

which is a very strong condition. For example, Lipschitz condition is not satisfied even for simple fuzzy valued random variables. This extra condition is only for keeping the embedding theorem to be available.

Here, we mainly develop the theory of closed convex set valued ran- dom variables, get convergence theorems for the closed convex set valued martingales and sub-, super-martingales, and then use them to study fuzzy valued random variables whose level sets are not compact in the underlying Banach space X.

For these purposes, we must first make up a theory on non-compact set valued random variables F, and then apply them to that on fuzzy valued ones. In more words, we firstly discuss the properties of Au- mann integrals, conditional expectations for closed convex set valued random variables and give some sufficient conditions for Aumann inte- grals and relative conditional expectations to be closed, which are very necessary results to discuss sub- and super-martingales. Secondly we prove a representation theorem and then obtain convergence theorems for closed convex set valued martingales, sub- and super-martingales in the Kuratowski-Mosco sense under weaker conditions than that in for- mer works. Then we find the relations between the theory of set valued random variables and fuzzy valued random variables so that we can ex- tend the results to fuzzy cases.

Since the embedding theorem can not be used for the closed set val- ued random variables and relative fuzzy valued random variables, we must find a new method to solve the problems above mentioned. The main method here is the systematic use of the associated the set of measurable selections SF of closed set valued random variable F, which lies in £l rn, X], the space of X-valued integrable random variable. This makes the objects more tractable, since £l[n, Xl is already a Banach space. To get convergence theorems for closed set valued martingales, sub- and super-martingales, we exploit the martingale selections. For the results of fuzzy valued random variables, we first use the cut sets of the fuzzy valued random variables and get the system of set valued random variables. Using the selection technique, we got relative results on the system of cut sets and then bring them back to the fuzzy valued cases. This is the first advantage.

Our second advantage is to make use of the Kuratowski-Mosco topol-

ogy in place of Hausdorff topology, which was a main tool in most of

former works. There is a rich history of Kuratowski-Mosco topology

ter the celebrated paper [25] (see also [26], [32, 33J e.g.). Actually, Both notions of convergence for set valued in a metric space, the Hausdorff metric convergence and the Kuratowski-Mooco convergence, are emi- nently useful in several areas of mathematics and applications such as optization and control, stochastic and integral geometry, mathematical economics. However, in an normed space, especially for infinite dimen- sional cases, the Kuratowski-Mooco convergence is more tractable than the Hausdorff one.

We should notice that Papageorgiou got the convergence theorems for closed convex set valued martingales in Kuratowski-Mosco sense in his papers. However the assumption there that the conjugate functions were uniformly lower equi-continuous is not easy to check, especially when we apply them to fuzzy valued martingales. In this paper, we succeed in getting rid of this assumption by exploiting martingale selections. This is our third advantage.

We obtain the convergence theorems for both closed convex set val- .ued sub- and super-martingales in Kuratowski-Mosco sense, since we successfully give the sufficient conditions for the Aumann integral and relative conditional expectation to be closed. Then we extend them to the fuzzy case. There were not any results on this topic before.

Finally, we introduce a new convergence called convergence in graph sense for fuzzy valued random variables and obtain convergence theo- rems, since it makes the topology clearly for the space of fuzzy valued random variables.

We organize this paper as follows: in section 2, we firstly give the basic definitions and results for set valued random variables, and then we state our main theorems for set valued martingales, sub- and su- permartingales. In section 3, we discuss convergences for fuzzy valued cases. In section 4, we introduce convergence in graph for fuzzy random variables, give the equivalent definitions and then obtain some conver- gence theorems for a sequence of fuzzy valued random variables and martingales..

The proofs of theorems in sections 2 and 3 were given in our papers

[17J - [21J. The proofs of theorems in section 4 were given in our preprint

paper "Convergence in graph for fuzzy valued random variables and

martingales" .

2. Convergence theorems for set valued martingales

Throughout this paper, (0, A, J-L) is a complete, probability space.

Denote by (X,

IIx) a real separable Banach space, and by K{X) the family of all nonempty, closed subsets of X, by Kc{X) the family of

closed, convex subsets of X, and Kcc(X) the family of all compact, convex subsets of X.

Define two operations in K{X) as follows:

(2.1) (2.2)

A + ^B = cl{a + ^{b; a EA, bE B},}

AA

{Aa; a EA}, where A is a real number.

REMARK

2.1. We do not define the Minkowski addition at (2.1) in the ordinary way, because {a + b; a

A, b

B} is not a closed set in general, even if A and B are bounded closed sets (cr. Li and Ogura [17]).

The HausdorfJ metric on K(X)

defined as follows:

(2.3)

max{sup inf Ila - bl1x, sup inf Ila - _{bll x}}

aEA bEB bEB aEA

for A, B

K{X). But if A or B are unbounded,

B) may be infinite. It is well-known (cf. Kuratowski [16, p.214, p.407]) that the family of

bounded elements in K{X) is a complete space with respect to the Hausdorff metric

and the family of

bounded elements in Kc{X), Kcc{X) are closed subsets of this complete space. For A

K{X), IIAIlK denotes the norm of A defined as IIAIIK = sup Ilallx.

aEA

A set valued random variable F : 0

K{X) is a measurable mapping, that is, for every B

K(X), F(-I){B)

{w

0; F{w) n B # 0}

A (cr. Hiai and Umegaki [10], for a few equivalent definitions).

A measurable mapping f : 0

X is called a measurable selection of F if f{w)

F{w) for all w

O. Denote by Ll[O, X] the Banach space of

measurable mappings g: 0---+ X such that the norm Ilgll£

In ^{IIg(w)llx d J-L}

finite. For a measurable set valued random variable F,

define the set .

SF = {f

L

[0, X] : j{w)

F{w) a.e.{J-L)}.

For a sub-a-field Al, denote by SF{Ao) the set of all Al-measurable

mappings in SF'

A set random variable F :

K(X) is called integrably bounded iff the real valued random variable IIF(w)IIK

integrable. Let Ll[Q,A, j.L;

K(X)J denote the space of all integrably bounded set random variables where two set random variables Fl , F

£I[Q, A, j.L; K(X)] are considered to be identical if Fl(w)

F

(w), a.e.(j.L).

The following Theorem is about the relation between

set valued random variable F

Ll[Q,A,j.L;K(X)] and SF (cf. Hiai and Umegaki [10]).

THEOREM

2.1. (1) Let FE Ll[n,A, j.L; K(X)], then

(a) SF is a nonempty, closed and bounded subset in Ll[Q,X],

(b) there exists a sequence {fn} contained in SF such that F(w) =

cl{fn(w)} for all

(c) SF is convex if and only if F( w) is convex for almost every w

(2) Let M be a nonempty closed subset of £I[Q;X]. Then there exists a set random variable F

Ll[Q,A,j.L;K(X)] such that M

SF if and only if

is nonempty, closed, bounded and decomposable, i.e.

h

IAf + In\A9 belongs to M for all A

A and f,9

M.

Define the subset of Ll[Q, A, j.L; K(X)] as follows:

Ll[Q, A,p; B] = {F

Ll[Q, A, j.L; K(X)] : F(w)

B, a.e.(j.L)}, and if Ao is a sub-u-field of A, we denote

Ll[Q, Al, j.L; B] = {F

Ll[Q, A, j.L; B] : F is Ao - measurable}, where B is a subset of K(X).

For each F

Ll[Q, A, j.L; K(X)], Aumann integral (cf. Aumann [4]) of F is

(2.4) 1 ^{Fdj.L =} {1 ^{fdj.L : f}

^{SF } ,}

where In ^fdj.L

the usual Bochner integral. Define lA Fdj.L

{fA fdj.L :

f

SF}, for A

A.

If F

a compact convex valued random variable, the Aumann integral and Debreu integral are equivalent

Klein and Thompson [15]).

Let Ao be a sub-u-field of A. The conditional expectation E[FIAo] of

an F

Ll[Q,A,tt; Kc(X)] is determined as a Al-measurable element of

Ll[O, A, p;; Kc(X)] by

(2.5) SE[PIAoJ(Ao) = cl{E(fIAo) : f

Sp},

where the closure is taken in the Ll[O, X] (cf.Hiai and Umegaki [10]). If

r is separable, this is equivalent to the formula

(2.6) cl i ^{Fdp; = cl} i ^{E[FIAo]dp; for A}

^Al.

REMARK

2.2. The integral In ^{Fdp; of F} and the set {E(fIAo) : f

S p} are not necessary closed in general (cf. counterexample in Li and Ogura [21]).

In the following, we will give some sufficient conditions for the closed- ness of Aumann integral of compact convex set value random vari- able and of closed convex set valued random variable F, and the set {E(fIAo) : f ESp} concerning the conditional expectations E[FIAo] of F (cf. Li and Ogura [21]). They are important for our proof of the set valued submartingale convergence theorem.

A Banach space X is said to have the Radon-Nikodym property (RNP) with respect to a finite measure space.(O, A, p;) if for each p;-continuous X-valued measure

A

X of bounded variation, there exists an integrable mapping f : °

^{X such that m(A) = I f dp; for all AEA.}

A

It

is known that every separable dual space and every reflexive space has the RNP (cf. Chatterji [7]).

THEOREM

2.2. (1) If X has the RNP and F

Ll[O, A, p;; Kcc(X)] , then the set

1 ^{Fdp; = {1 fdJ1 : f}

^{Sp }}

is closed in X.

(2)

X has the RNp, F

L

[0, A, p;; Kcc(X)] and Ao is countably generated, that is Ao = o-(2l) for a countable sub-class 2l of A, then the set

{E(fIAo) : f ESp}

is closed in L

[0, X].

THEOREM

2.3. (1) If Xis a reflexive Banach space,

L1[0,A,JL;

Kc(X)]. Then the set

1 ^{FdJL = {1 fdJL : f}

^SF}

is closed.

(2) If X is a reflexive Banach space, F

Ll[O, A,JL; Kc(X)] and Ao =

a(~)

is countably generated. Then the set {E(JIAo) : f

SF}

is closed in L1[0,X].

In the following we discuss set valued martingales and smartingales.

Let {.An : n

N} be a family of complete sub-a-fields of A such that

00

.An c .An+! if

N, and Aoo the a-field generated by U .An.

n=1

A system {F

An : n

N} is called a set valued martingale iff (1) Fn

£1[0, An, JL : Kc(X)], n

N,

·(2) Fn = E[Fn+I!AnJ,

N, a.e.(JL).

A sequence of set random variables {F

n 2: I}

called uniformly integrable iff

By using embedding theorem, Hiai and Umegaki [10] obtained the following results.

(1) Let {Fn,An;n 2: I} be a compact convex valued martingale such that Fn = E[FIAn],n 2: 1, where F

Ll[O,A,JL;K

(X)] (or F

L

[0, A, JL; Kc(X)] and X

reflexive). Then

dH(F

F

0, where F

= E[FIAoo].

(2) Let {F

An; n

-I} be a set valued martingale such that F

=

E[FIAn],n

-1, where F_ 1

L1[0, A, JL; K

(X)](orF_ 1

V[O,A,JL;

Kc(X)] and X

reflexive). Then

dH(F

F-

0

as n

where F- oo = E[FIA-ooJ, and V[O, A, JL; Kc(X)] denotes the closure of the set of all simple functions in Ll[O,A,JL;K

(X)] (cf.

Hiai and Umegaki [10]).

REMARK 2.3. The assertions above are not valid for F

L

[Q,A,JL;

Kc(X)], in general.

A set valued martingale {F _n, An; n

N}

called regular if there exists an F

L

[Q, A, JL; Kc(X)] such that Fn ⁼ E[FIAn] for all n.

Now we give the representation theorem for closed convex set valued martingales by exploiting the method of martingale selections (cf. Li and Ogura [17)).

THEOREM 2.4. Assume that X has the RNP and {FmAn;n

1}

is a set valued martingale in LI[O, A, JL; Kc(X)]. If {Fn} is uniformly integrable, then there exists an Foo

L

[O, Aaa, JL; Kc(X)] such that

F

= E[FooIAn] a.e.(JL),

N.

REMARK 2.4. The X-valued martingales with regular property, as we know, imply convergence in almost everywhere with respect to JL. But in the case of set valued martingales , regular property does not imply convergence in the HausdorfI metric (cf. Li and Ogura[20, Example 4.2]). We can see from this fact that the HausdorfI topology is too strong for the study of set valued random variables. Now we introduce the Kuratowski-Mosco convergence.

Let {Bn}nEN is a sequence of closed sets of X. We will say that B

converges to B in the Kumtowski-Mosco sense [25, 32] (denoted by B

~ ^{B) if!}

w-limsupB

B

s-liminfB

n-oo

where

w-limsupB

= {x = w-limxm :

m

Bm,m

MeN}

n->OO

and

s-liminfB

= {x = s-limx

E Bn,n EN}.

n-oo

In [20], we proved the following theorem by using martingale selection

method. Since the Kuratowski-Mosco convergence and the HausdorfI

convergence are equivalent when X is a finite dimensional space, this result also generalizes the result of Hiai and Umegaki above mentioned.

THEOREM

2.5. Assume that X is a Banach space satisfying the RNP with the separable dual X*. Then, for every uniformly integrably set valued martingale {F n,An : ⁿ

^N}, there exists

unique F

L

[0, A, p.; Kc(X)] such that {Fn,An: n

Nu oo} is a martingale and

K-M ( )

F

F

a.e. p. .

A system {Fm An : n E N} is called a set valued submarlingale (cf. Li and Ogura [21]) ill it satisfies the following two conditions

(1) for each n E N, Fn ^E

An, p.; Kc(X)],

(2) for each n EN, SPn(An)

{E[fIAn] : f E Spn+1(An+r)}.

REMARK

2.5. 1) Condition (2) is equivalent to

(3)

Foreachn,m E Nwithn

m, SPn(An)

{E[fIAn]: f E SPm(Am)}.

2) Condition (2) is stronger than the notion of submartingale in [10], where they use the condition

that is,

(2') SPn(An)

cl{E(JIAn) : f

SPn+l(An+l)}.

3) Condition (2) is equivalent to formula (2') if

S = {E(JIAn) : f

SPn~l (An+l)}

is closed. Concerning its closedness, we have given the sufficient condi- tions in Theorems

and

Now we give a convergence theorem for a set valued submartingale

follows.

THEOREM

2.6. Assume that X is a Banacb space satisfying tbe RNP witb tbe separable dual X*. Then, for every uniformly integrably set valued submartingale {F

An : n

N}, tbere exists a unique F

L

[0, Aoo, ^p.; Kc(X)] sucb tbat F n ~ ^F oo , ^a.e.(p.).

To prove this theorem, we first get a selection representation lemma for the closed convex set valued submartingale (cf. Li and Ogura [21], Lemma 5.1).

A system {F

An : n E N} is called a set valued supermarlingale iff

it satisfies the following two conditions

(1) for each

N, F n

^L

^[0, An, J-L; Kc(.l:)], (2) for each

N, E[Fn

IAn](w)

Fn(w), a.e.(J-L).

REMARK

2.6. Condition (2) is equivalent to E[FmIAn](w) c Fn(w), a.e.(J-L) for each n,

with n

The following

the convergence theorem for set valued super-martingales in Kuratowski-Mosco sense.

THEOREM

2.7. Assume that X is a Banach space satisfying the RNP with the separable dual r, {F

An : n

N} is uniformly integrably set valued supermartingale and

00

(2.5) M = n{f

^L

[0,Aro,J-L; X] : E(JIAn)

SFn(An)}

n=1

is a nonempty set. Then there exists a unique F oo

Aro, ^J-L; Kc(X)]

such that F n ~ ^F oo , ^a.e.(J-L).

3. Convergence theorems for fuzzy valued martingales Let F(X) denote the family of all fuzzy sets v : X

[0,1] such that its each a-cut set

= {x EX: v(x) .

a} is nonempty, closed subset of X, for every ° ^{< a}

^{1 with}

⁼

^EX:

^{> o} is bounded}

subset of X. For two fuzzy sets vI,

E F(X),

:j v2 iff

c

for every a E (0,1]. Obviously, (F(X),:j) is a partial ordered set.

The Hausdorff metric in K(X) can be extended to F(X) by (3.1)

= sup

O<a:::;1

Then we can prove similarly as Puri and Ralescu in [29] that (F(X), d)

a complete metric space.

A fuzzy valued random variable(cf. [17]) or fuzzy random variable is a function X : n

F(X), such that Xa(w) = {x EX: X(w)(x)

a} is a set random variable for every.a

(0,1].

A fuzzy random variable X is called integrably bounded if for every

a

(0,1], the real valued random variable IIXa(w)IIK is integrable. A

fuzzy random variable X

called strongly integrably bounded if there

exists a J-L-integrable function f :

lR such that I\X

(w)IIK

f(w)

for almost every

° and for all a

(0,1].

Let Ll[Q,A,JL;F(X)] be the set of all integrably bounded fuzzy ran- dom variables and L

[Q, A, JL; Fc(X)J denote the set of all fuzzy random variables X

L1[Q,A,J.l;F(X)J such that X a

L1[Q,A,J.l;Kc(X)J ^for all a

(O,IJ. Similarly we have the notation L

[Q,A, JL; Fcc(X)J

For each X, YE L1[Q, A,

F(X)J, we can define the metric function Doo : ^L

[Q,A, JL;F(X)]

L

Q,A,JL;F(X)] - JR by

(3.2) Doo(X, Y)

sup

Ya),

O<a~;l

where

Y

In dH(Xa, Ya)dJL.

Two fuzzy random variables X, YE L

[Q, A, JL; F(X)J are considered to be identical if Doo(X, Y)

O. We also can define the metric function D

Ll[Q,A,JL;F(X)J x L

[Q, A, JL; F(X)] - JR. by

Dl(X, Y)

1

~(Xa, ^Ya)dA(a),

where A

the Lebegue measure. Then both of (L

[Q,A,JL;F(X)],D

and (L

[Q, A,

F(X)], D

are complete metric spaces, and £I[Q, A, JL;

Fc(X)] ::) L

[n, A, JL; F cc(X)] are the closed subsets of Ll[Q, A, JL; F(X)J (cf. Li and Ogura [17]).

In the study of fuzzy random variables, the typical method

to use set valued theory, since all of the cut sets of a fuzzy random variable is a family of set valued random variables satisfying additional properties.

The following

the basic Lemma.

LEMMA

3.1. Let {Sa: a

[0, In be a family of subsets of L

[Q, X], Sa be nonempty, closed, bounded and decomposable for every

> 0 and satisfy the following conditions:

(1) a "5: (3 ===* Sp

Sa,

00

(2) al "5: a2 "5: .•• "5: an "5: ... , lim an = a ===* Sa = n ^San'

n->oo

Then, there exists a unique YE L

[Q, A, JL; F(X)] such that for every a,

(3.3) Sa

{f

L

[Q, XJ; f(w)

Ya(w), a.e.(JL)}.

If

{Sa: a

[0, I]} have the above conditions (1) - (2) and (3) for any a

(0,1], Sa is convex.

Then, there exists

unique Y

L1[Q, A, JL; Fc(X)J which satisfies

(3.3)

The expected value of any fuzzy random variable X, denoted by E[X], is a fuzzy set such that, for every a

(O,IJ,

(3.4) (E[X])a

cl 1 ^Xadj.t

^{cl{E(f); f}

^SxJ,

where the closure is taken in X. From the existence theorem, we can get an equivalent definition

follows, .

E(X)(x) = sup{a

[0,1] : x

clE(Xa)}.

From now on, we limit fuzzy random variable in L

[n,Aa,j.t;F

(X)]

only for simplicity of notations.

The conditional expectation of fuzzy valued random variable X, de- noted by E[XIAlJ, is a fuzzy random variable satisfying the following two conditions.

(3.5)

(3.6)

i ^{E[XIAl]dJL =} i ^{XdJL for every}

^Aa·

From [17], the conditional expectation E[XIAl] uniquely exists.

{xn, An; n

N}

called a fuzzy valued martingale or fuzzy martingale ill

(1) xn

L

[n, An,j.t; Fe(X)] , for all n

(2) xn = E[xn+lIAn], for all n

N.

A sequence of fuzzy random variables {X

n

N}

called uniformly integmble if

lim sup J IXn(w)ldj.t

,).->00nEN

{lXn(wlj>,).}

where Ixn(w)1 = d(xn(w), I{o}), and I{o}

the indicator function.

THEOREM

3.2. Let {Xn,An;n

I} be a fuzzy valued martingale such that xn = E[XIAn],n

1, where X

L

[O,A,j.t;F

(X)] (or X E LI[n, A, j.t; Fe (X)J and X is reBexive) and X is strongly integrably bounded. Then

(3.7) D

(X

,X

0,

where X

= E[XIAoo] and L

[O, A, j.t; Fe(X)] is the closure of all simply

fuzzy random variables

[17]).

THEOREM

3.3. Let {xn, An; n ::; -I} be a fuzzy valued martingale such thatXn = E[X-IIAn],n::;

where X-I

Ll[f2,A,Jl;F

(X)]

(or X-I E V[f2,

Jl; Fc(X)] and

is reflexive) and X-I is strongly integrably bounded. Then

(3.8)

as n

where x-oo = ^E[XIA-

THEOREM

3.4. Let

be separable, {xn, An; n 2.:: I} be a fuzzy valued martingale in V[f2,A,Jl;F

(X)]. IfTI ::; T2 ::; ... ::; Tn ::; ... is a sequence

stopping times, and lim

IXnldj.l

0 for every n E N. Then {XTn,

n 2.:: I} forms

martingale. That is, for every n 2.:: 1,

In the above IXTn(w) I = d(XTn(W), I{o}), where I{o} is the indicator func- tion.

In [20], we defined the Kuratowski-Mosco convergence for the fuzzy valued random martingales by using every cut set of them to be con- vergent in the Kuratowski-Mosco sense as we defined

section

We proved the following theorem.

THEOREM

3.5. Assume that X is a Banach space with the sep- arable dual

satisfying

Then, for every uniformly integrable fuzzy valued martingale {xn,An;n EN}, there exists a unique X OO E L

[f2, Aoo,Jl; Fc(X)] such that {xn,An; n

(N U oo)} is a fuzzy valued martingale, a.e. for each n

N, xn

E[xooIAn), and xn ~ XOO.

A system {xn, An; n

N} is called a fuzzy valued submartingale (resp.

fuzzy valued supermartingale )

(1) xn

V[f2, An,Jl; Fc(X)], for all

N, (2) xn

E[xn+IIAn], for n

N (resp.

By using the same method as in Theorem 3.5, we can get convergence

theorems for submartingales and supermartingales. However, it is a little

difficult to understand the topology of the space of fuzzy valued random

variables, induced by the family of set valued random variables in the

Kuratowski-Mosco convergence.

4. Graph convergence for sequences of fuzzy valued random variables and martingales

In this section, we introduce convergence in graph for fuzzy random variables.

Let v E F(x), write

Gr v

{(x, y)

x

[0,1], v(x) ;:::: y}.

It

clear that Gr v denotes the domain between the curve of v and x-axis if x is lR. We call it the graph of v. Since all the cut set of v are nonempty closed set, the graph of v is a closed set of space X x [O,IJ.

For v

v E F(x),

is called to converge to v in graph (denoted by

Vn

~ ^{v) iff Gr}

converges to Gr v in x x [O,lJ in the Kuratowski- Mosco sense.

REMARK 4.1.

for any

E [0,1], (vn)a ~ ^(v)a, we can prove easily that

~ v. But the opposite is not correct. We can see it from the following example.

EXAMPLE 4.1. Let x = lRI, a < b < c and

and

v(x) = { 0, 1/2, 1,

x < a,x >

<

b ::; x ::;

{

0,

x < a, x >

vn(x) = 1/2 - 1/2n, a::; x < b,

1, b::; x::;

Then

~ ^{v, but}

does not converge to

in the Kuratowski- Mosco sense.

THEOREM 4.2. Let v

v E F(x), then

~ v iffthe following two conditions are satisfied,

(1) for any x

x, there exists

sequence {x

n

N} of x ^converging

to x in strong topology of X such that

liminfvn(xn) ;:::: lI(x),

n--->oo

(2) for any given subsequence {link} of {lI n} ^{and any} sequence {x nk } which converges to x in the weakly topology of X, we have

limsup link (X nk ) ::; ^lI(X).

k-->oo

In fact, we can prove that (1) is equivalent to

Gr II

s-liminfGr lI

in X x [O,IJ,

n-->oo

and (2) is equivalent to

w-limsup Gr lI

Gr lI,

n-->oo

in X

[O,IJ.

By using Theorem 4.2, we can prove the following theorem.

A sequence of fuzzy random variables {X

n

N} is called uniformly integrably bounded iff there exists an JL-integrable function f :

JR such that IIX;:(w)IIK ::; f(w) for almost every wEn, for

a E (O,IJ and all

N.

THEOREM

4.3. Assume that

sequence of fuzzy random variables {xn : n

N} is uniformly integrably bounded. Then, if {Xn : n

N}

converges to an integrably bounded fuzzy random variable X in graph, we have E[xnJ _~ E[X]. Furthermore, if Ao is a sub-u-filed, then E[xnIAoJ ~ ^E[XIAoJ·

Now we give the following convergence theorems for fuzzy valued mar- tingale, sub- and super-martingales.

THEOREM

4.4. Assume that X is a reflexive Banach space. Then, for every uniformly integrably fuzzy valued martingales or submartingale {xn,An;n EN}, there exists

unique X

E £I[Q,Aoo,JL;Fc(X)J such that xn _~ XOO a.e.(JL).

THEOREM

4.5. Assume that X is a reflexive Banach space, {xn, An; n

E

N} is uniformly integrably fuzzy valued supermartingale and for each aE(O,IJ,

00

n=l

is a nonempty set. Then there exists a unique xoo

£I[Q, Aoo, JL; Fc(X)J

.L Gr

SUC11

that xn

XOO a.e ..

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Department of Mathematics Saga University

1 Honjo-Machi

Saga 840-8502, Japan