A Bipolar Theorem for ^0 +

A BIPOLAR THEOREM FOR L ( ; F ; P)

W. Brannath and W. Schachermayer

Institut fur Statistik der Universitat Wien,

Brunnerstr. 72, A-1210 Wien, Austria.

Abstract. A consequence of the Hahn-Banach theorem is the classical bip olar the-

orem which states that the bip olar of a subset of a lo cally convex vector space equals

its closed convex hull.

The space L ( ; F ; P) of real-valued random variables on a probability space

( ; F ; P) equipp ed with the top ology of convergence in measure fails to b e lo cally

convex so that | a priori | the classical bip olar theorem do es not apply. In this

note weshow an analogue of the bip olar theorem for subsets of the p ositive orthant

0 0

L ( ; F ; P), if we place L ( ; F ; P) in duality with itself, the scalar pro duct now

+ +

taking values in [0; 1]. In this setting the order structure of L ( ; F ; P)plays an

imp ortant role and we obtain that the bip olar of a subset of L ( ; F ; P) equals its

closed, convex and solid hull.

In the course of the pro of we show a decomp osition lemma for convex subsets

of L ( ; F ; P)into a \b ounded" and a \hereditarily unb ounded" part, whichseems

interesting in its own right.

1. The Bipolar Theorem

Let ( ; F ; P) b e a probability space and denote by L ( ; F ; P)thevector space of

(equivalence classes of ) real-valued measurable functions de ned on ( ; F ; P) which

we equip with the top ology of convergence in measure (see [KPR 84], chapter I I,

section 2). Recall the wellknown fact (see, e.g.,[KPR 84], theorem 2.2) that, for

a di use measure P, the top ological dual of L ( ; F ; P) is reduced to f0g so that

there is no counterpart to the duality theory, whichworks so nicely in the context

of lo cally convex spaces (compare [Sch 67], chapter IV).

0 0

By L ( ; F ; P)we denote the p ositive orthantof L ( ; F ; P), i.e.,

0 0

L ( ; F ; P)= ff 2 L ( ; F ; P);f 0g:

The research of this pap er was nancially supp orted by the Austrian Science Foundation

(FWF) under grant SFB#10 ('Adaptive Information Systems and Mo delling in Economics and

Management Science')

1991 Mathematics Subject Classi cation. Primary: 62B20; 28A99; 26A20; 52A05; 46A55 Sec-

ondary: 46A40; 46N10; 90A09.

Key words and phrases. Convex sets of measurable functions; Bip olar theorem; b ounded in

probability; hereditarily unb ounded.

Typ eset by A S-T X

M E 1

2 W. BRANNATH AND W. SCHACHERMAYER

0 0

Wemay consider the dual pair of convex cones hL ( ; F ; P);L ( ; F ; P)i where

+ +

we de ne the scalar pro duct hf; gi by

hf; gi = E [fg]; f; g 2 L ( ; F ; P):

Of course, this is not a scalar pro duct in the usual sense of the wordasitmay assume

the value +1. But the expression hf; gi is a wellde ned elementof[0; 1] and the

application (f; g) !hf; gi has | mutatis mutandis | the obvious prop erties of

a bilinear function.

The situation is similar to the one encountered at the very foundation of measure

theory: to overcome the diculty that E [f ] do es not make sense for a general

0 1

element f 2 L ( ; F ; P) one may either restrict to elements f 2 L ( ; F ; P)orto

elements f 2 L ( ; F ; P), admitting in the latter case the p ossibility E [f ]=+1.

In the presentnotewe adopt this second p oint of view.

1.1 De nition. We call a subset C L solid,iff 2 C and 0 g f implies

that g 2 C . The set C is said to b e closedinprobability or simply closed, ifitis

closed with resp ect to the top ology of convergence in probability.

0 0

1.2 De nition. For C L we de ne the polar C of C by

0 0

C = fg 2 L : E [fg] 1; for each f 2 C g

0 0

1.3 Bip olar Theorem. For a set C L ( ; F ; P) the p olar C is a closed,

convex, solid subset of L ( ; F ; P).

The bip olar

00 0 0

C = ff 2 L : E [fg] 1; for each g 2 C g

is the smallest closed, convex, solid set in L ( ; F ; P) containing C .

To prove theorem 1.3 we need a decomp osition result for convex subsets of L we

present in the next section. The pro of of theorem 1.3 will b e given in section 3.

We nish this intro ductory section by giving an easy extension of the bip olar

0 0

). Recall that, with the theorem 1.3 to subsets of L (as opp osed to subsets of L

usual de nition of solid sets in vector lattices (see [Sch 67], chapter V, section 1),

a set D L is de ned to b e solid in the following way.

0 0

1.4 De nition. A set D L is solid,if f 2 D and h 2 L with jhj jf j implies

h 2 D .

Note that a set D L is solid if and only if the set of its absolut values jD j =

0 0 0

fjhj : h 2 D gL form a solid subset of L as de ned in 1.1 and D = fh 2 L :

+ +

jhj2 jD jg. Hence the second part of theorem 1.3 implies:

1.5 Corollary. Let C L and jC j = fjf j : f 2 C g. Then the smallest closed,

0 0 00

convex, solid set in L containing C equals ff 2 L : jf j2jC j g.

0 0

Proof. Let D b e the smallest closed, convex, solid set in L containing jC j and

D = ff : jf j2D g. One easily veri es that D is the smallest closed, convex

and solid subset of L containing C . Applying theorem 1.3 to jC j,we obtain that

0 00 0 00

D = jC j , which implies that D = ff 2 L : jf j2jC j g.

For more detailed results in the line of corollary 1.5 concerning more general

subsets of L we refer to [B 97].

A BIPOLAR THEOREM FOR L ( ; F ; P) 3

2. A Decomposition Lemma for Convex Subsets of L ( ; F ; P)

Recall that a subset of a top ological vector space X is bounded if it is absorb ed

byevery zero-neighborhood of X ([Sch 67], Chapter I, Section 5). In the case of

L ( ; F ; P) this amounts to the following well-known concept.

2.1 De nition. A subset C L ( ; F ; P)is boundedinprobability if, for ">0,

there is M>0 suchthat

P[jf j >M] <"; for f 2 C:

Wenowintro duce a concept which describ es a strong form of unb oundedness in

L ( ; F ; P).

2.2 De nition. A subset C L ( ; F ; P) is called hereditarily unboundedin

probability on a set A 2F, if, for every B 2F;B A; P[B ] > 0wehave that

C j = ff : f 2 C g fails to b e a b ounded subset of L ( ; F ; P).

B B

Wenow are ready to formulate the decomp osition result:

2.3 Lemma. Let C b e a convex subset of L ( ; F ; P). There exists a partition of

into disjointsets ; 2F such that

u b

(1) The restriction C j of C to is b ounded in probability.

(2) C is hereditarily unb ounded in probabilityon .

The partition f ; g is the unique partition of satisfying (1) and (2) (up to

u b

null sets). Moreover

(3) If P[ ] > 0 wemay nd a probability measure Q equivalent to the restric-

b b

tion Pj of P to such that C is b ounded in L ( ; F ;Q ). In fact, we

b b

maycho ose Q such that is uniformly b ounded.

(4) For ">0 there is f 2 C s.t.

P[ \ff<" g] <":

(5) Denote by D the smallest closed, convex, solid set containing C .ThenD

has the form

D = D j L j ;

b + u

0 0

( ; F ; P) g. j = f v : v 2 L where D j = f u : u 2 D g and L

+ + u u b b

Proof. Noting that the lemma holds true for C i it holds true for the solid hull of

C wemay assume w.l.g. that C is solid and convex.

Wenow use a standard exhausting argument to obtain . Denote by B the

family of sets B 2F; P[B ] > 0, verifying

for ">0 there is f 2 C; s.t. P[B \ff<" g] <":

Note that B is closed under countable unions: indeed, for (B ) is B and ">0,

n=1

nd elements (f ) in C such that

n=1

n 1 n

P[B \ff < 2 " g] < 2 ":

n n

4 W. BRANNATH AND W. SCHACHERMAYER

Then, by the convexity and solidityof C

F = 2 f

N n

n=1

is in C and, for N large enough,

P[B \fF <" g] <":

Hence there is a set of maximal measure in B , whichwe denote by and which

is unique up to null-sets. Let = n .

b u

(1) and (3): If P[ ] = 0 assertions (1) and (3) are trivially satis ed; hence we

may assume that P[ ] > 0. Wewanttoverify (3). Note, since C is a solid subset

0 1 0

, the convex set C = C \ L ( ; F ; Pj of L ) is dense in C with resp ect to the

+ b

convergence in probability Pj ; hence, byFatou's Lemma, it is enough to nd a

0 1

probability measure Q Pj such that C is b ounded in L (Q ). To this end

b b

0 0

we apply Yan's theorem ([Y 80], theorem 2) to C .For convex, solid subsets C of

L (Pj ), this theorem states, that the following two assertions are equivalent:

+ b

(i) for each A 2F with Pj [A]=P[ \ A] > 0, there is M>0 such that

1 0

M is not in the L ( ; F ; Pj )-closure of C ;

(ii) there exists a probability measure Q equivalentto Pj suchthat C is a

b ounded subset of L ( ; F ;Q ). In addition, wemaycho ose Q such that

b b

is uniformly b ounded.

Assertion (i) is satis ed b ecause otherwise we could nd a subset A 2F;A

; P[A] > 0 b elonging to the family B , in contradiction to the construction of

b u

ab ove.

Hence assertion (ii) holds true which implies assertion (3) of the lemma. Obvi-

ously (3) implies assertion (1).

(2) and (4): As is an elementofB we infer that (4) holds true which in turn

implies (2).

(5): Obviously D D j L j .To show the reverse inclusion let f = v + w

b + u

with v 2 D j and w 2 L j .Wehave to show that f 2 D . Prop erty (2) implies

b + u

that, for every n 2 N,we nd an f 2 C such that P[ff n g\ ] (1=n).

n n u

Since h =(1 (1=n)) v +(1=n)(f ^ (nw)) 2 D and h ! v + w in probability,

n n n

it follows that f 2 D .

According to (2), C is unb ounded in probabilityin L ( ; F ; Pj ) for each B

B u

with P[B ] > 0; the uniqueness of the decomp osition = [ (uptonull sets)

u b

with resp ect to the assertions (1) and (2) immediately follows from this.

3. The Proof of the Bipolar Theorem 1.3

To prove the rst assertion of theorem 1.3 x a set C L ( ; F ; P) and note

0 0

that the convexity and solidityof C are obvious and the closedness of C follows

from Fatou's lemma.

To prove the second assertion of the theorem denote by D the intersection of all

closed, convex and solid sets in L containing C . Clearly D is closed, convex and

00 00

solid, which implies the inclusion D C .Wehave to show that C D .

A BIPOLAR THEOREM FOR L ( ; F ; P) 5

Using assertion (5) of lemma 2.3 wemay decomp ose into = [ such that

b u

D = D j L j and (if P[ ] > 0) we nd a probability measure Q supp orted

b b

b + u

by and equivalent to the restriction Pj of P to such that D is b ounded in

b b

L ( ; F ;Q ) (assertion (2)).

Now supp ose that there is f 2 C nD and let us work towards a contradiction.

Let f = f denote the restriction of f to . It is enough to showthat f is

b 0 0 b b

in D . Let us denote by D = ff : f 2 D g the restriction of D to and by

b b

1 1

D = D L ( ; F ;Q )=fh 2 L ( ; F ;Q )g : 9 f 2 D s.t. h f; Q a.s.g

b b b b b b

the set of elements of L (Q ) dominated by an elementof D . It is straightforward

b b

1 1 1

(Q )andL (Q ) to verify that D and D are L (Q )-closed, convex subsets of L

b b b b b

resp ectively, and that D is b ounded in L (Q ).

b b

To show that f is contained in D (equivalently in D or in D ) it suces to

b b b

show that f ^ M is in D , for each M 2 R . Indeed, bytheL (Q)-b oundedness

b b +

1 1

and L (Q)-closedness of D this will imply that f = L (Q) lim f ^ M is

b b M !1 b

in D .

So we are reduced to assuming that f is an elementofL (Q ) which is not an

b b

elementofD . Nowwemay apply a version of the Hahn-Banach theorem (the

separation theorem [Sch 67], theorem 9.2) to the Banach space L (Q ) to nd an

element g 2 L (Q ) such that

E [f g ] > 1whileE [fg] 1; for f 2 D :

b b

As D contains the negative orthantof L (Q )we conclude that g 0. Con-

b b

sidering g as an elementof L ( ; F ; P)by letting g equal zero on we therefore

0 00

have that g 2 C and the rst inequalityabove implies that f 2= C and so that

f=2 C ,acontradiction nishing the pro of.

4. Notes and Comments

4.1 Note: Our motivation for the formulation of the bip olar theorem 1.3 ab ove

comes from Mathematical Finance: in the language of this theory there often comes

up a duality relation b etween a set of contingent claims and a set of state price densi-

ties, i.e., Radon-Niko dym derivatives of absolutely continuous martingale measures.

In this setting it turns out that L ( ; F ; P) often is the natural space to work in

(as opp osed to L ( ; F ; P) for some p> 0), as it remains unchanged under the

passage from P to an equivalent measure Q (while L ( ; F ; P)doeschange, for

duality relations and to [KS 97] for an applications of the bip olar theorem 1.3.

4.2 Note: Lemma 2.3 may b e viewed as a variation of theorem 1 in [Y 80],

which is a result based on previous work of Mokob o dzki (as an essential step in

Dellacherie's pro of of the semimartingale characterization theorem due to Bichteler

and Dellacherie; see [Me 79] and [Y 80]). The pro of of Yan's theorem is a blend of

a Hahn-Banach and an exhaustion argument (see, e.g., [S 94] for a presentation of

this pro of and [Str 90], [S 94] for applications of Yan's theorem to Mathematical

6 W. BRANNATH AND W. SCHACHERMAYER

Finance) In fact, these arguments have their ro ots in the pro of of the Halmos-Savage

theorem [HS 49] and the theorems of Nikishin and Maurey [N 70], [M 74].

4.3 Note: In the course of the pro of of lemma 2.3 wehave shown that a convex

subset C of L ( ; F ; P) is hereditarily unb ounded in probabilityonasetA 2F

i , for ">0, there is f 2 C with

P[A \ff<" g] <";

which seems a fact worth noting in its own right.

00 0

4.4 Note: Notice that by theorem 1.3 the bip olar C of a given set C L ,

although originally de ned with resp ect to P,doesnotchange if we replace P

by an equivalent measure Q. This may also b e seen directly (without applying

theorem 1.3) in the following way: If Q P are equivalent probability measures

and h = dQ=dP is the Radon-Niko dym derivativeof Q with resp ect to P, then

0 0 0

the p olar C (Q) ofagiven convex set C L with resp ect to Q equals C (Q)=

1 0 0

h C (P), where C (P) is the dual of C with resp ect to P. On the other hand

1 1 0 00

E [fg]=E [fhh g ]=E [fh g ] for all g 2 L and therefore the p olar C (Q)

P P Q

0 00 0

of C (Q) (de ned with resp ect to Q) coincides with the p olar C (P)of C (P)

(de ned with resp ect to P).

References

[B 97].W. Brannath, No Arbitrage and Martingale Measures in Option Pricing, Dissertation, Uni-

versity of Vienna (1997).

[DSF. 94]. Delbaen, W. Schachermayer, AGeneral Version of the Fundamental Theorem of Asset

Pricing, Math. Annalen 300 (1994), 463 | 520.

[HSHalmos, 49]. P.R., Savage, L.J. (1949), Application of the Radon-Nikodym Theorem to the Theory

of Sucient Statistics, Annals of Math. Statistics 20, 225{241..

[KSD. 97]. Kramkov, W. Schachermayer, A Condition on the Asymptotic Elasticity of Utility Func-

tions and Optimal Investment in Incomplete Markets, Preprint (1997).

[KPRN.J. 84]. Kalton, N.T. Peck, J.W. Rob erts, An F-space Sampler, London Math. So c. Lecture Notes

89 (1984).

[M 74].B. Maurey, Theoremes de factorisation pour les operateurs lineaires a valeurs dans un espace

L , Asterisque 11 (1974).

[MeP 79]..A., Meyer, Caracterisation des semimartingales, d'apres Del lacherie,Seminaire de Proba-

bilites XI I I, Lect. Notes Mathematics 721 (1979), 620 | 623.

[N 70].E.M. Nikishin, Resonance theorems and superlinear operators, Usp ekhi Mat. Nauk 25, Nr. 6

(1970), 129 | 191.

[S 94].W. Schachermayer, Martingale measures for discrete time processes with in nite horizon,

Math. Finance 4 (1994), 25 | 55.

[SchSc 67].haefer, H.H. (1966), Topological Vector Spaces, Springer Graduate Texts in Mathematics.

[StrStric 90]. ker, C., Arbitrage et lois de martingale, Ann. Inst. Henri. Pincare Vol. 26, no. 3 (1990),

451{460.

1 1

[Y 80].J. A. Yan, Caracterisation d' une classe d'ensembles convexes de L ou H ,Seminaire de

Probabilites XIV, Lect. Notes Mathematics 784 (1980), 220{222.