The Fundamental Theorem of Asset Pricing for Unbounded Stochastic

Home , Asset pricing, Local martingale, Predictable process, Semimartingale

THE FUNDAMENTAL THEOREM OF ASSET PRICING

FOR UNBOUNDED STOCHASTIC PROCESSES

F. Delbaen and W. Schachermayer

Departementfur Mathematik, Eidgenossische Technische Ho chschule Zuric h,

ETH-Zentrum, CH-8092 Zuric h, Schweiz,

Institut fur Statistik der Universitat Wien,

Brunnerstr. 72, A-1210 Wien, Austria.

Abstract. The Fundamental Theorem of Asset Pricing states - roughly sp eaking -

that the absence of arbitrage p ossibilities for a sto chastic pro cess S is equivalentto

the existence of an equivalent martingale measure for S . It turns out that it is quite

hard to give precise and sharp versions of this theorem in prop er generality,ifone

insists on mo difying the concept of \no arbitrage" as little as p ossible. It was shown

in [DS94] that for a local ly bounded R -valued semi-martingale S the condition of No

Free Lunch with Vanishing Risk is equivalent to the existence of an equivalent local

martingale measure for the pro cess S . It was asked whether the lo cal b oundedness

assumption on S may b e dropp ed.

In the present pap er we showthatifwe drop in this theorem the lo cal b oundedness

assumption on S the theorem remains true if we replace the term equivalent local

martingale measure bythetermequivalent sigma-martingale measure. The concept

of sigma-martingales was intro duced by Chou and Emery | under the name of

\semimartingales de la classe ( )".

Weprovide an example whichshows that for the validity of the theorem in the

non lo cally b ounded case it is indeed necessary to pass to the concept of sigma-

martingales. On the other hand, we also observe that for the applications in Mathe-

matical Finance the notion of sigma-martingales provides a natural framework when

working with non lo cally b ounded pro cesses S .

The duality results whichwe obtained earlier are also extended to the non lo cally

b ounded case. As an application wecharacterize the hedgeable elements.

1. Introduction

The topic of the present pap er is the statemant and pro of of the subsequent Fun-

damental Theorem of Asset Pricing in a general version for not necessarily local ly

bounded semi-martingales:

1.1 Main Theorem. Let S =(S ) bean R -valued semi-martingale de ned

t t2R

on the stochastic base ( ; F ; (F ) ; P).

t t2R

1991 Mathematics Subject Classi cation. Primary 60G44; Secondary 46N30,46E30,90A09, 60H05.

Key words and phrases. Sigma martingale, Arbitrage, fundamental theorem of asset pricing,

Free Lunch.

Typ eset by A S-T X

M E 1

Then S satis es the condition of No Free Lunch with Vanishing Risk if and only

if there exists a probability measure Q P such that S is a sigma-martingale with

respect to Q .

This theorem has b een proved under the additional assumption that the pro cess S

is lo cally b ounded in [DS94]. Under this additional assumption one may replace

the term \sigma-martingale" ab oveby the term \lo cal martingale".

We refer to [DS94] for the history of this theorem, whichgoesback to the seminal

work of Harrison, Kreps and Pliska ([HK79], [HP81], [K81]) and whichisofcentral

imp ortance in the applications of sto chastic calculus to Mathematical Finance. We

also refer to [DS94] for the de nition of the concept of No Free Lunch with Vanishing

Risk which is a mild strengthening of the concept of No Arbitrage.

On the other hand, to the best of our knowledge, the second central concept in

the ab ove theorem, the notion of a sigma-martingale (see def. 2.1 b elow) has not

b een considered previously in the context of Mathematical Finance. In a way, this

is surprising, as we shall see in 2.4 b elow that this concept is very well-suited for

the applications in Mathematical Finance, where one is interested not so muchin

the pro cess S itself but rather in the family (H S ) of stochastic integrals on the

pro cess S ,whereH runs through the S -integrable predictable pro cesses satisfying

a suitable admissibility condition (see [HP81], [DS94] and section 4 and 5 b elow).

The concept of sigma-martingales, which relates to martingales similarly as sigma-

nite measures relate to nite measures, has b een intro duced by C.S. Chou and

M. Emery ([C77], [E78]) under the name \semi-martingales de la classe ( )".

We shall show in section 2 b elow (in particular in example 2.3) that this concept

is indeed natural and unavoidable in our context if we consider pro cesses S with

unb ounded jumps.

The pap er is organized as follows: In section 2we recall the de nition and basic

prop erties of sigma-martingales. In section 3 we present the idea of the pro of

of the main theorem by considering the (very) sp ecial case of a two-step pro cess

S =(S ;S )=(S ) . This presentation is mainly for exp ository reasons in order

0 1 t

t=0

to present the basic idea without burying it under the technicalities needed for the

pro of in the general case. But, of course, the consideration of the two-step case

only yields the (n + 1)'th pro of of the Dalang-Morton-Willinger theorem [DMW90],

i.e., the fundamental theorem of asset pricing in nite discrete time (for alternative

pro ofs see [S92], [KK94], [R94]). We end section 3by isolating in lemma 3.3 the

basic idea of our approach in an abstract setting.

Section 4 is devoted to the pro of of the main theorem in full generality. Weshall

use the notion of the jump measure asso ciated to a sto chastic pro cess and its com-

pensator as presented, e.g., in [JS87].

Section 5 is devoted to a generalization of the duality results obtained in [DS95].

These results are then used to identify the hedgeable elements as maximal elements

in the cone of w -admissible outcomes. The concept of w -admissible integrand is

a natural generalization to the non lo cally b ounded case of the previously used

concept of admissible integrand.

For unexplained notation and for further background on the main theorem we refer

to [DS94]. 2

2. Sigma-Martingales

In this section we recall a concept which has b een intro duced byC.S.Chou[C77]

and M. Emery [E78] under the name \semi-martingales de la classe ( )". This

notion will play a central role in the presentcontext. Wetake the lib erty to baptize

this notion as \sigma-martingales". Wecho ose this name as the relation b etween

martingales and sigma-martingales is somewhat analogous to the relation b etween

nite and sigma- nite measures (compare [E78], prop. 2).

2.1 De nition. An R -valued semimartingale X = (X ) is called a sigma-

t t2R

martingale if there exists an R -valued martingale M and an M -integrable pre-

dictable R -valued pro cess ' such that X = ' M .

We refer to ([E78], prop. 2) for several equivalent reformulations of this de nition

and we now essentially repro duce the basic example given by M. Emery ([E 78],

p.152) which highlights the di erence b etween the notion of a martingale (or, more

generally, a lo cal martingale) and a sigma-martingale.

2.2 Example. [E78]:A sigma-martingale which is not a local martingale.

Let the sto chastic base ( ; F ; P)besuch that there are two indep endent stopping

times T and U de ned on it, b oth having an exp onential distribution with param-

eter 1.

De ne M by

0 for t

M = 1 for t T ^ U and T = T ^ U

1 for t T ^ U and U = T ^ U

It is easy to verify that M is almost surely well-de ned and is indeed a martingale

with resp ect to the ltration (F ) generated by M . The deterministic (and

t t2R

therefore predictable) pro cess ' = is M -integrable (in the sense of Stieltjes) and

X = ' M is well-de ned:

0 for t

X = 1=(T ^ U ) for t T ^ U and T = T ^ U

1=(T ^ U ) for t T ^ U and U = T ^ U

But X fails to be a martingale as E [jX j] = 1, for all t > 0, and it is not hard

to see that X also fails to be a lo cal martingale (see [E78]), as E [jX j]= 1 for

each stopping time T that is not identically zero. But, of course, X is a sigma-

martingale.

Weshallbeinterested in the class of semimartingales S which admit an equivalent

measure under which they are a sigma-martingale. We shall present an example

of an R -valued pro cess S which admits an equivalent sigma-martingale measure

(which in fact is unique) but which do es not admit an equivalent lo cal martingale

measure. This example will b e a slight extension of Emery's example.

The reader should note that in Emery's example 2.2 ab ove one may replace the

measure P by an equivalent measure Q such that X is a true martingale under 3

Q . For example, cho ose Q suchthat under this new measure T and U are inde-

p endent and distributed according to a law on R suchthat is equivalentto

the exp onential law (i.e., equivalent to Leb esgue-measure on R ) and such that

E [ ] < 1.

2.3 Example. A sigma-martingale S which does not admit an equivalent local

martingale measure.

2 1 2

With the notation of the ab ove example de ne the R -valued pro cess S =(S ;S )

1 2

by letting S = X and S the comp ensated jump at time T ^ U i.e.,

2t for t

S =

1 2(T ^ U ) for t T ^ U

(Observe that T ^ U is exp onentially distributed with parameter 2).

Clearly S is a martingale with resp ect to the ltration (F ) generated by S .

t t2R

Denoting by(G ) the ltration generated by S it is a well-known prop ertyof

t t2R

the Poisson-pro cess (c.f. [J 79], p. 347) that on G the restriction of P to G = _ G

t2R

is the unique probability measure equivalenttoP under which S is a martingale.

F equivalent to P It follows that P is the only probability measure on F = _

t2R

1 2

under which S =(S ;S ) is a sigma-martingale.

As S fails to be a lo cal martingale under P (it's rst co ordinate fails to be so)

wehave exhibited a sigma-martingale for which there do es not exist an equivalent

martingale measure.

2.4 Remark. In the applications to Mathematical Finance and in particular in the

context of pricing and hedging derivative securities by no-arbitrage arguments the

ob ject of central interest is the set of stochastic integrals H S on a given sto ck

price pro cess S ,whereH runs through the S -integrable predictable pro cesses such

that the pro cess H S satis es appropriate regularity condition. In the present

context this regularity condition is the admissibility condition H S M for

some M 2 R (see [HP81], [DS94] and section 4 b elow). In di erentcontexts one

might imp ose an L (P)-b oundedness condition on the sto chastic integral H S (see,

e.g., [K81], [DH86], [St90], [DMSSS96]). In section 5, we shall deal with a di erent

notion of admissibility, which is adjusted to the case of big jumps.

Nowmake the trivial (but nevertheless crucial) observation: passing from S to ' S ,

where ' is a strictly p ositive S -integrable predictable pro cess, does not change the

set of stochastic integrals. Indeed, wemay write

H S =(H' ) (' S )

d 1

where the predictable R -valued pro cess H is S -integrable i H' is 'S -integrable.

The moral of this observation: when we are interested only in the set of stochastic

integrals H S the requirementthat S is a sigma-martingale is just as go o d as the

requirementthat S is a true martingale.

We end this section with two observations which are similar to the results in [E78].

The rst one stresses the distinction b etween the notions of a lo cal martingale and

a sigma-martingale. 4

2.5 Prop osition. For a sigma-martingale X the fol lowing assertions areequiva-

lent.

(i) X is a local martingale.

(ii) X = ' M where the M -integrable, predictable R -valued process ' is in-

creasing and M is a local martingale.

(iii) X = ' M where the M -integrable, predictable R -valued process ' is in-

creasing and M is a martingale.

(iv) X = ' M where the M -integrable, predictable R -valued process ' is in-

creasing and M is a martingale in H .

Proof. (i) ) (iv ): Supp ose that X is a lo cal martingale and assume w.l.g. that

X =0. Then there is a sequence of stopping times T =0; (T ) increasing to

0 0 n

n=1

T 1

in nitysuch that the stopp ed pro cesses X are martingales in H (see, e.g., [J79],

IV.1.d). Find a sequence of scalars > 0suchthat

T T n

n1 n

k X k 2 : n =1; 2;:::

T T

n1 n

Here X =1I X denotes the pro cess \started at T and stopp ed

]]T ;T ]]

n1 n

at T ". Then

T T

n1 n

M = X

n=1

is a martingale in H and wemaywrite X = ' M with

' = 1I :

]]T ;T ]]

n1 n

n=1

(iv ) ) (iii) ) (ii)isobvious.

(ii) ) (i) Under the assumption (ii) wemay nd a sequence (T ) of stopping

n=1

1 T

is in H and ' is b ounded times such that, for each n 2 N , the stopp ed pro cess M

1 T T

n n

is in H which readily implies that X is a lo cal = ' M on [[0;T ]]. Hence X

martingale.

For later use we observe the analogous result to prop osition 2.5 for the case of

sigma-martingales. The pro of is similar and left to the reader

2.6 Prop osition. For a semi-martingale X the fol lowing areequivalent

(i) X is a sigma martingale.

(ii) X = ' M wheretheM -integrable, predictable R -valuedprocess ' is strictly

positive and M isalocal martingale.

(iii) X = ' M wheretheM -integrable, predictable R -valuedprocess ' is strictly

positive and M is a martingale.

(iv) X = ' M wheretheM -integrable, predictable R -valuedprocess ' is strictly

positive and M is a martingale in H . 5

3. One-period processes

In this section we shall present the basic idea of the pro of of the main theorem

in the easy context of a pro cess consisting only of one jump. Let S 0 and

0 d 1

S 2 L ( ; F ; P; R )begiven and consider the sto chastic pro cess S =(S ) ;as

1 t

t=0

ltration wecho ose (F ) where F = F and F is some sub- -algebra of F . At

t 1 0

t=0

a rst stage we shall in addition make the simplifying assumption that F is trivial,

i.e., consists only of null-sets and their complements. In this setting the de nition

of the No-Arbitrage condition (NA) (see [DMW90] or [DS94]) for the pro cess S b oils

down to the requirement that, for x 2 R , the condition (x; S ) 0 a.s. implies

that (x; S ) = 0 a.s., where (; ) denotes the inner pro duct in R .

From the theorem of Dalang-Morton-Willinger [DMW 90] we deduce that the No

Arbitrage condition (NA) implies the existence of an equivalent martingale measure

for S , i.e., a measure Q on ( ; F ); Q P,suchthat E [S ]=0.

Q 1

By now there are several alternative pro ofs of the Dalang-Morton-Willinger theorem

known in the literature ([S92], [KK94], [R94]) and we shall presentyet another pro of

of this theorem in the subsequent lines. While some of the known pro ofs are very

elegant (e.g., [R94]) our subsequent pro of is rather clumsy and heavy. But it is

this metho d which will be extensible to the general setting of an R -valued (not

necessarily lo cally b ounded) semi-martingale and will allowustoprovethe main

theorem in full generality.

Let us x some notation: by Adm we denote the convex cone of admissible elements

d d

of R which consists of those x 2 R such that the random variable (x; S ) is (almost

surely) uniformly b ounded from b elow.

By K we denote the convex cone in L ( ; F ; P) formed by the admissible sto chastic

integrals on the pro cess S , i.e.,

K = f(x; S ):x 2 Admg (3.1)

and we denote by C the convex cone in L ( ; F ; P) formed by the uniformly

b ounded random variables dominated by some elementofK ,i.e.,

0 1

C =(K L ( ; F ; P)) \ L ( ; F ; P)

= ff 2 L ( ; F ; P): there is g 2 K; f g g: (3.2)

Under the assumption that S satis es (NA), i.e. K \ L = f0g,wewantto nd

an equivalent martingale measure Q for the pro cess S . The rst argumentiswell-

known in the presentcontext (compare [S92] and theorem 4.1 b elow for a general

version of this result; we refer to [S94] for an account on the history of this result,

in particular on the work of J.A. Yan [Y80] and D. Kreps [K81]):

3.1 Lemma. If S satis es (NA) the convex cone C is weak-star closed in

1 1

L ( ; F ; P), and C \ L ( ; F ; P) = f0g. Therefore there is a probability mea-

sure Q on F ; Q P such that

1 1

E [f ] 0 for f 2 C:

In the case, when S is uniformly b ounded, the measure Q is already the desired

1 1

equivalent martingale measure. Indeed, in this case the cone Adm of admissible

elements is the entire space R and therefore

E [(x; S )] 0; for x 2 R ;

Q 1

1 6

which implies that

E [(x; S )] = 0; for x 2 R ;

Q 1

whence E [S ]=0.

But if Adm is only a sub-cone of R (p ossibly reduced to f0g), we can only say

much less: rst of all, S need not be Q -integrable. But even assuming that

1 1

E [S ] exists we cannot assert that this value equals zero; we can only assert that

Q 1

(x; E [S ]) = E [(x; S )] 0; for x 2 Adm;

Q 1 Q 1

1 1

which means that E [S ] lies in the cone Adm polar to Adm, i.e.,

Q 1

0 d

E [S ] 2 Adm = fy 2 R :(x; y ) 1forx 2 Admg

Q 1

= fy 2 R :(x; y ) 0forx 2 Admg:

The next lemma will imply that, by passing from Q to an equivalent probability

measure Q with distance kQ Q k in total variation norm less than ">0, wemay

remedy b oth p ossible defects of Q : under Q the exp ectation of S is well-de ned

1 1

and it equals zero.

The idea for the pro of of this lemma go es back in the sp ecial case d = 1 and

Adm = f0g to the work of D. McBeth [M91].

3.2 Lemma. Let Q bea probability measure as in lemma 3.1 and "> 0. Denote

by B the set of barycenters

B = fE [S ]:Q >probability on F ; Q P; kQ Q k <"; and S is Q -integrable g

Q 1 1 1

Then B is a convex subset of R containing 0 in its relative interior. In particular,

thereisQ Q ; kQ Q k <",suchthat E [S ]=0.

1 1 Q 1

Proof. Clearly B is convex. Let us also remark that it is nonempty. To see this let

exp( kS k)

. Clearly S is Q integrable = us take >0 and let us de ne

dQ E [exp( kS k)]

1 1

and from Leb esgue's dominated convergence theorem we deduce that for small

enough kQ Q k <".

If0were not in the relativeinterior of B wecould ndby the Minkowski separation

theorem, an element x 2 R ,suchthat B is contained in the halfspace H = fy 2

R : (x; y ) 0g and such that (x; y ) > 0 for some y 2 B . In order to obtain a

contradiction we distinguish two cases:

Case 1: x fails to be admissible, i.e., (x; S ) fails to be (essentially) uniformly

b ounded from b elow.

First nd, as ab ove, a probability measure Q P; kQ Q k <"=2 and such that

2 1 2

E [S ]iswell-de ned.

Q 1

By assumption the random variable (x; S ) is not (essentially) uniformly b ounded

from b elow, i.e., for M 2 R , the set

= f! :(x; S (! )) < M g

M 1 7

has strictly p ositive Q -measure. For M 2 R de ne the measure Q by

2 +

(

1 "=4 on n

1 "=4+ on :

4Q ( )

2 M

M M

It is straightforward to verify that Q is a probability measure, Q P;

M M 1

kQ Q k <"=2, dQ =dQ 2 L and suchthat

2 2

M M

: [(x; S )] (x; E [S ]) = E [(x; S )] (1 "=4)E

1 1 Q

Q Q

For M>0 big enough the right hand side b ecomes negativewhichgives the desired

contradiction.

Case 2: x is admissible, i.e., (x; S ) is (essentially) uniformly b ounded from b elow.

In this case weknow from the Bepp o-Levi theorem that the random variable (x; S )

is Q -integrable and that E [(x; S )] 0; (note that, for each M 2 R ,wehave

1 Q 1 +

that (x; S ) ^ M is in C and therefore E [(x; S ) ^ M ] 0).

1 Q 1

Also note that (x; S ) cannot b e equal to 0 a.s., b ecause as wesawabove there is

a y 2 B suchthat(x; y ) > 0 and hence (x; S ) cannot equal zero a.s. either.

We next observe that for all > 0 the variable exp ( (x; S ) ) is b ounded. The

exp (x;S )

) (

measure Q , given by = is therefore well de ned, S is Q -

2 1 2

dQ E [exp( (x;S ) )]

1 1

integrable and for small enough we also havethat kQ Q k <". But Q also

2 1 2

satis es:

E [(x; S )] < 0:

Q 1

Indeed:

exp (x; S ) (x; S ) E

1 1 Q

(x; S ) exp (x; S ) (x; S ) + E = E

1 1 1 Q Q

1 1

< E (x; S ) + E (x; S ) 0:

Q 1 Q 1

1 1

The measure Q do es not necessarily satisfy the requirementthat E [kS k] < 1.

2 Q 1

exp( kS k)

We therefore make a last tansformation and we de ne dQ = dQ .

[exp( kS k)] E

For >0 tending to zero we obtain that kQ Q k tends to 0 and E [(x; S )] tends

2 Q 1

to E [(x; S )]which is strictly negative. So for small enough we nd a probability

Q 1

mesaure Q such that Q P, kQ Q k <", E [kS k] < 1 and E [(x; S )] < 0, a

1 Q 1 Q 1

contradiction to the choice of x.

Lemma 3.2 in conjunction with lemma 3.1 implies in particular that, given the

with S 0andF trivial, wemay nd a probability sto chastic pro cess S =(S )

0 0 t

t=0

measure Q P such that S is a Q -martingale. We obtained the measure Q in

two steps: rst (lemma 3.1) we found Q P which to ok care of the admissible

integrands,which means that

E [(x; S )] 0; for x 2 Adm :

Q 1

1 8

In a second step (lemma 3.2) we found Q P suchthat Q to ok care of al l integrands,

i.e.,

(x; E [S ]) = E [(x; S )] 0 for x 2 R

Q 1 Q 1

and therefore

E [S ]=0;

Q 1

which means that S is a Q -martingale.

In addition, we could assert in lemma 3.2 that kQ Q k <", a prop erty whichwill

b e crucial in the sequel.

The strategy for proving the main theorem will b e similar to the ab ove approach.

Given a general semi-martingale S = (S ) de ned on ( ; F ; (F ) ; P) we

t t2R t t2R

+ +

rst replace P by Q P suchthat Q \takes care of the admissible integrands",

1 1

i.e.,

E [(H S ) ] 0; for H admissible:

Q 1

For this rst step, the necessary technology has b een develop ed in [DS94] and may

b e carried over almost verbatim.

The new ingredientdevelop ed in the present pap er is the second step whichtakes

care of the \big jumps" of S . By rep eated application of an argumentasinlemma

3.2 ab ove we would like to change Q into a measure Q ; Q P, such that S

b ecomes a Q -martingale. A glance at example 2.3 ab ove reveals that this hop e

is, in the general setting, to o optimistic and we can only try to turn S into a Q -

sigma-martingale. This will indeed b e p ossible, i.e., we shall b e able to nd Q and

a strictly p ositive predictable pro cess ', such that, for every | not necessarily

admissible | predictable R -valuedprocess H satisfying kH k d ',wehave that

H S is a Q -martingale. In particular ' S will b e a Q -martingale.

In order to attack this program we shall isolate in Lemma 3.3 b elow, the argument

proving lemma 3.2 in the appropriate abstract setting. In particular weshow that

the construction in the pro of of lemma 3.2 may b e parameterized to dep end in a

measurable way on a parameter varying in a measure space (E; E ;). The pro of of

this lemma is standard but long. One has to check a lot of measurability prop erties

in order to apply the measurable selection theorem. After reading lemma 3.3, the

reader mightwant, at a rst reading, to skip the rest of this section.

3.3 The Crucial Lemma. Let (E; E ;) be a probability measure space and let

(F ) be a family of probability measures on R such that the map ! F is

d d

E -measurable in the sense that, for A 2B(R ) (the Borel sigma algebraonR ),

! F [A]

is E -measurable.

Let us assume that F satis es the property that for each measurable map x: E !

R ; ! x with the property that for every 2 E we have (x ;y) 1, for F

almost every y , we also have that F (dy )(x ;y) 0.

Let ": E ! R nf0g be E measurable and strictly positive.

Then, we may nd an E -measurable map ! G from E to the probability measures

on B (R ) such that, for -almost every 2 E ,

(i) F G and kF G k <" ,

(ii) E [ky k] < 1 and E [y ]=0:

G G

Before proving this lemma, we need some observations and some extra notation

that wewillkeep xed for the rest of this section. We rstassumethat(E; E ;)

is a probability space that is saturated for the null sets, i.e. if A B 2 E and

if (B ) = 0 then A 2 E . The probability can easily be changed into a -

nite p ositive measure, but in order not to overload the statements we skip this

straightforward generalisation. We recall that a Polish space X is a top ological

space that is homeomorphic to a complete separable metrisable space. The Borel

sigma algebra of X is denoted by B (X ). We will mainly be working in a space

E X where X is aPolish space. The canonical pro jection of E X onto E is

denoted by pr . If A 2E B(X )then pr (A) 2E, see [Au65] and [D72]. Furthermore

there is a countable family (f ) of measurable functions f : pr (A) ! X such

n n1 n

that

(1) for each n 1 the graph of f is a selection of A, i.e. f(; f ( )) j 2

n n

pr (A)gA,

(2) for each 2 pr (A) the set ff ( ) j n 1g is dense in A = fx j (; x) 2 Ag.

Wecallsuch a sequence a countable dense selection of A.

The measurability assumption on F can b e reformulated as follows: for each C

function g , de ned on R and with compact supp ort we have that the mapping

! g (y ) dF (y ) is E measurable. This is easily seen using monotone class

arguments. Using suchagiven measurable function F as a kernel, we can de ne a

d d

probability measure on E R as follows. For an element D 2E B(R )of

the form D = A B ,we de ne (D )= F (B ) (d ).

For each 2 E we de ne the set Supp(F ) as the supp ort of the measure F ,i.e.

the smallest closed set of full F -measure. The set S is de ned as f; x) j x 2

Supp(F )g. The set is an element of E B(R ). Indeed, take a countable base

(U ) of the top ology of R and write the complementas:

n n1

[

S = (f j F (U )=0gU ) :

n n

If x: E ! R is a measurable function then ': E ! R [f+1g de ned as '( )=

k(x ;:) k is E measurable. Indeed take a countable dense selection (f )

n n1

L (F )

of S and observe that '( ) = inf f(x ;f ( )) j n 1g.

d d

For each 2 E we denote by Adm( ) the cone in R consisting of elements x 2 R

so that (x; :) 2 L (F ). The set Adm is then de ned as f(; x) j x 2Adm( )g.

This set is certainly in E B(R ). Indeed Adm = f(; x) j inf (x; f ) > 1g

n1 n

where the sequence (f ) is a countable dense selection of S .

n n1

Wesay that a probability measure on R satis es the NA prop erty if for every

x 2 R wehave (fa j (x; a) < 0g) > 0assoonas(fa j (x; a) > 0g) > 0.

In the next lemma weintro duce two equivalentversions of a parametrised form of

the No Arbitrage prop erty.

Lemma. If F is a measurable mapping from (E; E ;) into the probability measures

on R , then the fol lowing areequivalent:

(1) For almost every 2 E , the probability measure F satis es the No Arbi-

trage property

(2) For every measurable selection x of Adm, we have [(; a) j x (a) < 0] >

0 as soon as [(; a) j x (a) > 0] > 0.

F 10

Proof. The implication 1 ) 2 is almost obvious since for each 2 E we have

that F [fa j (x ;a) < 0g] > 0 as so on as F [fa j (x ;a) > 0g] > 0. Therefore if

[(; a) j x (a) > 0] > 0, wehave that (B ) > 0 where B is the set

B = f 2 E j F [fa j (x ;a) > 0g] > 0g :

For the elements 2 B we then also have that F [fa j (x ;a) < 0g] > 0 and

integration with resp ect to then gives the result:

[(; a) j x (a) < 0] = (d )F [fa j (x ;a) < 0g] > 0:

Let us now prove the reverse implication 2 ) 1.

We consider the set

A = f(; x) j F [a j (x; a) 0] = 1 and F [a j (x; a) > 0] > 0g :

The reader can check that this set is in E B(R ) and therefore the set B = P(A) 2

E . Supp ose that (B ) > 0 and take a measurable selection x of A. Outside B

we de ne x =0. Clearly ( f(; a) j (x ;a) > 0g) > 0and hence wehave that

( f(; a) j (x ;a) < 0g) > 0, a contradiction since (x ;a) 0, a.s.. So we see

F F

that B = ; a.s. or what is the same for almost every 2 E the measure F satis es

the No Arbitrage prop erty.

Wenow nally can give a pro of of lemma 3.3. The set P (R ) of probability measures

on R ,endowed with the weak top ology is a Polish space. Weshow that the set

Z Z

(; ) j kxk d < 1; xd =0;F ; k F k <"( )

d d

R R

is in E B(P (R )). Since, by lemma 3.2, for each 2 E the vertical section is

nonempty,we can nd a measurable selection G and this will end the pro of. The

pro of of this measurabilityproperty is easy but requires some arguments.

First we observe that the set M = f j kxk d < 1g is in B (P (R )). This

follows from the fact that ! kxk d is Borel measurable as it is an increasing

limitoftheweak continuous functionals ! min(kxk;n) d.

xd is Borel measurable. Next we observethat M : ! R ; !

The third observation is that f(; ) jk F k <"( )g is in E B(P (R )).

Finally weshow that f(; ) j F g is also in E B(P (R )). This will then end

the pro of of the measurabilityproperty.

We take an increasing sequence of nite sigma-algebras D such that B (R ) is

generated by [ D . For each n we see that the mapping ( ; ; x) ! (x)=

n n

d d

q ( ; ; x)isE B(P (R )) B(R ) measurable. The mapping

q ( ; ; x)=liminfq ( ; ; x)

d d

is clearly E B(P (R )) B(R ) measurable. By the martingale convergence theorem

wehave that for each , the mapping q de nes the Radon-Niko dym densityofthe

part of that is absolutely continuous with resp ect to F . Nowwehavethat

Z Z

f(; ) j F g = (; ) j q ( ; ; x) dF (x)=1;lim nq ^ 1 dF (x)=1

d d

R R 11

and this shows that f(; ) j F g is in E B(P (R )).

Next supp ose that (E; E ;) is not necessarily complete. In that case we rst com-

plete the space (E; E ;)by replacing the sigma-algebra E by E generated by E and

~ ~

all the null sets. We then obtain an E measurable mapping F which can easily b e

replaced byaE measurable mapping F suchthat almost surely F = F .

Remark. Wehave not striven for maximal generality in the formulation of lemma

3.3: for example, we could replace the probability measures F by nite non-

negative measures on R . In this case wemay obtain the G in sucha way that

d d

the total mass G (R ) equals F (R );-almost surely.

To illustrate the meaning of the Crucial lemma we note a little observation in the

spirit of [M91] which shows in particular the limitations of the no-arbitrage-theory

when applied e.g. to Gaussian mo dels for the sto ck returns in nite discrete time.

T d

3.4 Prop osition. Let (S ) be an adapted R -valued process based on

t=0

T T

( ; F ; (F ) ; P) such that for every predictable process (h ) we have that (h

t t

t=0 t=1

S ) = h S is unboundedfrom above and from below as soon as (h S ) 6 0.

T t t T

t=1

For example, this assumption is satis ediftheF -conditional distributions of the

jumps S are normal ly distributedonR .

Then, for " > 0, there is a measure Q P; kQ Pk < ", such that S is a Q -

martingale.

As a consequence, the set of equivalent martingale measures is dense with respect

to the variation norm in the set of P-absolutely continuous measures.

Proof. Supp ose rst that T =1. Contrary to the setting of the motivating example

at the b eginning of this section we do not assume that F is trivial.

Let (E; E ;)be( ; F ; P) and denote by(F ) the F -conditional distribution of

0 ! ! 2 0

S = S S . The assumption of lemma 3.3 is (trivially) satis ed as byhyp othesis

1 1 0

the F -measurable functions x(! ) such that P-a.s. wehave(x ;y) 1;F -a.s.,

0 ! !

satisfy (x ;y)=0;F -a.s., for P-a.e. ! 2 .

! !

Cho ose "(! ) " and nd G as in the lemma. To translate the change of the

conditional distributions of S intoachange of the measure P, nd Y : R !

R ,

Y (!; x)= (x) x 2 R ;! 2

such that, for P-a.e. ! 2 ;Y (!; ) isaversion of the Radon-Niko dym derivative

of G with resp ect to F , and suchthat Y (; )isF Borel(R )-measurable.

! ! 0

Letting

(! )=Y (!; S (! ))

we obtain an F -measurable density of a probability measure. Assertion (i) of

^ ^

lemma 3.3 implies that Q P and kQ Pk <". Assertion (ii) implies that

E [kS k jF ] < 1 a.s.

^ 1 0

and

E [S jF ]=0: a.s.

^ 1 0

Q 12

We are not quite nished yet as this only shows that (S ) is a Q -sigma-martingale

t=0

but not necessarily a Q -martingale as it may happ en that E [kS k d ]=1. But it

is easy to overcome this diculty: nd a strictly p ositive F -measurable function

w (! ), normalised so that E [w ] = 1 and suchthat E [w (! )E [kS k jF ]] < 1.

^ ^ 1 0

Q Q

We can construct w is sucha way that the probability measure Q de ned by

dQ ( ! )

= w (! );

dQ (! )

still satis es kQ Pk <". Then

E [kS k ] < 1

Q 1

and

E [S jF ]=0; a.s.,

Q 1 0

i.e., S is a Q -martingale.

To extend the ab ove argument from T = 1 to arbitrary T 2 N we need yet another

small re nement: an insp ection of the pro of of lemma 3.3 ab ove reveals that in

addition to assertions (i) and (ii) of lemma 3.3, and given M > 1, wemaycho ose

G such that

(iii) k k M; - a.s.

1 d

L (R ;F )

We have not mentioned this additional assertion in order not to overload lemma

3.3 and as we shall only need (iii) in the present pro of.

Using (iii), with M =2 say, and, cho osing w ab ove also uniformly b ounded by2,

the argument in the rst part of the pro of yields a probability Q P; kQ Pk <",

suchthat k k 4.

L (P)

T T

Now let T 2 N and (S ) , based on ( ; (F ) ; F ; P), be given. By backward

t t

t=0 t=0

induction on t = T;:::;1 apply the rst part of the pro of to nd F -measurable

(t)

densities Z such that, de ning the probability measure Q by

(t)

= Z ;

(t) t

is a Q -martingale with wehave that the two-step pro cess (S Z )

u v

u=t1

v =t+1

t (t) (t) T 1

resp ect to the ltration (F ) ; Q P; kQ Pk <"4 T , and such that

u 1

u=t1

kZ k 4.

L (P)

De ning

= Z ;

t=1

we obtain a probability measure Q ; Q P suchthat(S ) is a martingale under

t=0

Q . Indeed, 13

E [S jF ]=E [S Z jF ]

Q t t1 P t u t1

u=1

t1 T

Y Y

=( Z ) E [Z S Z jF ]

u P t t u t1

u=1 u=t+1

t1 T

Y Y

=( Z ) E [S Z jF ]=0

(t)

u t u t1

u=1 u=t+1

and

t1 T

Y Y

E [kS k d ] k Z k E [kS Z k d ] < 1:

(t)

Q t u 1 t u

R R

u=1 u=t+1

Finally wemay estimate kQ Pk by

kQ Pk = E [j Z 1j]

1 P t

t=1

T t t1

X Y Y

E [ j Z Z j]

P u u

t=1 u=1 u=1

T t1

X Y

k Z k E [Z 1]

u P t

L (P)

t=1 u=1

T T 1

T 4 "4 T = ":

The pro of of the rst part of prop osition 3.4 is thus nished and wehaveshown in

the course of the pro of that wemay nd Q such that, in addition to the assertions

of the prop osition, is uniformly b ounded.

As regards the nal assertion, let P be any P-absolutely continuous measure. For

00 00 0

given ">0, rst take P P suchthatkP P k <". Now apply the rst assertion

with P replacing P. As a result we get an equivalent martingale measure Q such

00 0

that kQ P k <", hence also kQ P k < 2".

This nishes the pro of of prop osition 3.4

4. The General R -valued Case

In this section S =(S ) denotes a general R -valued cadlag semi-martingale

t t2R

based on ( ; F ; (F ) ; P) where we assume that the ltration (F ) satis es

t t2R t t2R

+ +

the usual conditions of completeness and right continuity.

Similarly as in [DS94] we de ne an S -integrable R -valued predictable pro cess H =

to b e an admissible integrand if the sto chastic pro cess (H )

t t2R

(H S ) = (H ;dS ) t 2 R

t u u +

0 14

is (almost surely) uniformly b ounded from b elow.

It is imp ortant to note that, similarly as in prop osition 3.4 ab ove, it mayhappen

that the cone of admissible integrands is rather small and p ossibly even reduced to

zero: consider, for example, the case when S is a comp ound Poisson pro cess with

(two-sided) unb ounded jumps, i.e., S = X , where (N ) is a Poisson

t i t t2R

i=1

pro cess and (X ) an i.i.d. sequence of real random variables such that kX k =

i 1

i=1

kX k = 1. Clearly, a predictable pro cess H ,suchthatH S remains uniformly

b ounded from b elow, must vanish almost surely.

Continuing with the general setup wedenoteby K the convex cone in L ( ; F ; P)

given by

K = ff =(H S ) : H admissibleg

where this de nition requires in particular that the random variable (H S ) :=

lim (H S ) is (almost surely) well-de ned (compare [DS94], de nition 2.7).

t!1 t

Again we denote by C the convex cone in L ( ; F ; P) formed by the uniformly

b ounded random variables dominated by some elementofK ,i.e.,

0 1

C =(K L ( ; F ; P)) \ L ( ; F ; P)

= ff 2 L ( ; F ; P): there is g 2 K; f g g: (4.1)

We say [(DS94), p.467], that the semi-martingale S satis es the condition of No

Free Lunch with Vanishing Risk (NFLVR) if the closure C of C , taken with resp ect

1 1

to the norm-top ology kk of L ( ; F ; P)intersects L ( ; F ; P) only in 0, i.e.,

C \ L = f0g: S satis es (NFLVR) ()

For the economic interpretation of this concept, whichisavery mild strengthening

of the \no arbitrage" concept, we refer to [DS94].

The subsequent crucial theorem 4.1 was proved in ([DS94], th.4.2) under the addi-

tional assumption that S is b ounded. An insp ection of the pro of given in [DS94]

reveals that | for the validity of the subsequent theorem 4.1 | the b oundedness

assumption on S may b e dropp ed.

4.1 Theorem. Under the assumption (NFLVR) the cone C is weak-star closedin

L ( ; F ; P). Hencethereisaprobability measure Q P such that

E [f ] 0; for f 2 C:

4.2 Remark. In the case, when S is b ounded, Q is already a martingale measure for

S , and when S is lo cally b ounded, Q is a lo cal martingale measure for S (compare

[DS94], theorem 1.1 and corollary 1.2).

Totake care of the non lo cally b ounded case wehavetotake care of the \big jumps"

of S . We shall distinguish b etween the jumps of S o ccurring at accessible stopping

times and those o ccurring at totally inaccessible stopping times.

We start with an easy lemma which will allowustochange the measure Q count-

ably many times without lo osing the equivalence to P. 15

4.3 Lemma. Let (Q ) beasequenceofprobability measures on ( ; F ; P) such

n=1

that each Q is equivalent to P. Suppose further that the sequence of strictly positive

numbers (" ) is such that

n n2

(1) kQ Q k <" ,

n n+1 n+1

n n

(2) if Q [A] " 2 then P[A] 2 .

n n+1

Then the sequence (Q ) converges with respect to the total variation norm to a

n n1

probability measure Q , which is equivalent to P.

Proof of lemma 4.3. Clearly the second assumption implies that " 2 and

n+1

hence the sequence (Q ) converges in variation norm to a probability measure

n n1

Q . We have to show that Q P. For each n we let q be de ned as the

n+1

Radon-Niko dym derivativeof Q with resp ect to Q . Clearly for each n 2we

n+1 n

then have j1 q j dQ " and hence the Markov inequality implies that

n+1 n n+1

n n

Q [j1 q j 2 ] 2 " . The hyp othesis on the sequence (" ) then

n n+1 n+1 n n2

n n

implies that P[j1 q j 2 ] 2 . From the Borel Cantelli lemma it also

n+1

P Q

follows that a.s. the series j1 q j converges and hence the pro duct q

n n

n2 n2

converges to a function q a.s. di erentfrom0. Clearly q = whichshows that

Q Q P.

We are now ready to takethe crucial step in the pro of of the main theorem. To

make life easier we still make the simplifying assumption that S do es not jump

at predictable times. In 4.6 b elow we nally shall also deal with the case of the

predictable jumps.

4.5 Prop osition. Let S = (S ) be an R -valued semi-martingale which is

t t2R

quasi-left-continuous, i.e., such that, for every predictable stopping time T we have

S = S almost surely.

T T

Suppose, as in 4.1 above, that Q P is a probability measure verifying

E [f ] 0 for f 2 C:

Then thereis, for ">0,aprobability measure Q P, kQ Q k <", such that S

is a sigma-martingale with respect to Q .

In addition, for every predictable stopping time T , the probabilities Q and Q on

F ,coincide , conditional ly on F , i.e.,

T T

= a:s:

dQ dQ

1 1

j j

F F

T T

De ne the stopping time T by Proof. Step 1:

T =infft : kS k d 1g

and rst supp ose that S remains constant after time T , hence S has at most one

jump bigger than 1.

Similarly as in [JS87], I I.2.4 we decomp ose S into

S = X + X 16

where X equals \S stopp ed at time T ", i.e.,

S for t

X =

S for t T

and X the jump of S at time T , i.e.,

X =S 1I :

t T

[[T;1]]

As X is b ounded, it is a sp ecial semi-martingale, and we can nd its Do ob-Meyer

decomp osition with resp ect to Q

X = M + B

where M is a lo cal Q -martingale and B a predictable pro cess of lo cally nite

variation.

We shall now nd a probability measure Q on F ; Q P, s.t.

2 2

(i) kQ Q k <"=2,

2 1

is F -measurable, (ii) Q j = Q j and

T 2 F 1 F

T T

(iii) S is a sigma-martingale under Q .

Weintro duce the jump measure asso ciated to X ,

(! ; dt; dx)= ;

(T (! );S (! ))

where denotes Dirac-measure at (t; x) 2 R R and we denote by the Q -

t;x + 1

comp ensator of (see [JS87], prop. I I.1.6). Similarly as in ([JS87], prop. I I.2.9)

wemay ndacontinuous, lo cally Q -integrable, predictable and increasing pro cess

A such that

B = b A

(! ; dt; dx)=F (dx)dA (! )

!;t t

i d

where b = (b ) is a predictable pro cess and F (dx) a transition kernel from

!;t

i=1

d d

( R ; P ) into (R ; Borel (R )), i.e., a P -measurable map (!; t) ! F (dx)

+ !;t

from R into the nonnegative Borel measures on R .

Wemay assume that A is constant after T ,continuous, Q -integrable and its inte-

gral is b ounded by one, i.e.,

E [A ]=dA( R ) 1;

Q 1 +

where dA denotes the measure on P de ned by dA(]]T ;T ]]) = E [A A ], for

1 2 Q T T

1 2 1

stopping times T T .

1 2

Wenow shall nd a P -measurable map (!; t) ! G such that for dA-almost each

!:t

(!; t),

d d

(a) F (dx) G (dx);F (R )=G (R ) and kF G k < ,

!;t !;t !;t !;t !;t !;t

(b) E [ky k ] < 1 and E [y ]=b(!; t).

G G

!;t !;t 17

This is a task of the typ e of \martingale problem" or rather \semi-martingale

problem" as dealt with, e.g., in [JS87], def. III.2.4.

We apply lemma 3.3 and remark 3.4 ab ove: as measure space (E; E ;!) we take

( R ; P ;dA) and we shall consider the map

=(!; t) ! F := F

!;t !:t

b(!;t)

where denotes the Dirac measure at b(!; t) 2 R so that F is the measure

!;t

b(!;t)

F on R translated bythevector b(!; t).

!;t

We claim that the family (F ) satis es the assumptions of lemma 3.3

!;t

(!;t)2 R

ab ove. Indeed, let H be any P -measurable function such that dA-almost surely

!;t

H 2 Adm (F ) = Adm (F ).

!;t !;t !;t

By multiplying H with a predictable strictly p ositive pro cess v , if necessary, we

0 0

may assume that kH k d 1 and that H 2 Supp (F ). Here Supp (F )

!;t !;t !;t !;t

is de ned as the p olar of the set S upp(F ), which can also b e de ned as the set

!;t

fx 2 R j (x; y ) 1;F a.s.g.

!;t

The latter assumption translates to the fact that H =(H (! )) is an admissible

t t2R

integrand for the pro cess X ,as

(H X ) 1 a.s., for all t 2 R :

t +

Noting that M is a (lo cally b ounded) lo cal martingale and B is of lo cally b ounded

variation, wemay nd a sequence of stopping times (U ) increasing to in nity,

j =1

such that, for each j 2 N ,

U 1

( ) M is a martingale, b ounded in the Hardy space H (Q ), and

( ) B is of b ounded variation.

Hence, for each predictable set P contained in [[0;U ]], for some j 2 N , we have

that H 1I is an admissible integrand for S and H 1I M is a martingale b ounded

P P

in H (Q ) and therefore

E [(H 1I M ) ]=0:

Q P 1

As byhyp othesis

E [(H 1I S ) ] 0

Q P 1

we obtain

E [(H 1I (X + B )) ] 0:

Q P 1

Using the identities

(H F +(H ;b ))1I dA(!; t) E [(H 1I (X + B )) ]=

!;t !;t !;t !;t P Q P 1

= (H F )1I dA(!; t) 0

!;t !;t P

which hold true for each P 2P contained in [[0;U ]], for some j 2 N , we conclude

that for dA-almost each(!; t)wehave

H F = E [(H ; )] 0;

!;t !;t !;t

!;t 18

i.e., the hyp othesis of lemma 3.3 is satis ed.

Hence wemay nd a transition kernel G as describ ed bylemma3.3|with "

!;t

replaced by "=2 | and letting G = G we obtain a transition kernel

!;t !;t

b(!;t)

satisfying ( ) and ( )above.

We now haveto translate the change of transition kernels from F to G into

t;! t;!

achange of measures from Q to Q on the sigma-algebra F which will b e done

1 2 T

by de ning the Radon-Niko dym derivative . We refer to [JS87, III.3] for a

treatment of the relevantversion of Girsanov's theorem for random measures.

For (!; t) xed denote by Y (!; t ; ) the Radon-Nikodym derivative of G with

!;t

resp ect to F ,i.e.

!;t

(x); x 2 R ; Y (! ; t; x)=

!;t

whichis F -almost surely well-de ned and strictly p ositive. Wemay and do cho ose

!;t

for dA-almost each(!; t),aversion Y (! ; t; x)suchthatY (; ; )isP Borel (R )-

measurable.

Wenow de ne

(! )=Z (! )=Y (!; T (! ); S (! ))1I +1I

T (! ) fT<1g fT =1g

and Z (! )=Y (!; T (! ); S (! ))1I +1I

T (! ) fT tg fT>tg

The intuitiveinterpretation of these formulas go es as follows: for xed ! 2 we

lo ok at time T (! ) which is the unique \big" jump of (S (! )) . The density

t t2R

Y (!; T (! );x) gives the density of the distribution of the comp ensated jump mea-

sure G with resp ect to F , if the jump equals x and therefore we evaluate

!;t !;t

Y (!; T (! );x) at the point x = S (! ) to determine the density of Q with

T (! )

resp ect to Q . If T (! )= 1 the density (! ) is simply equal to 1.

dQ j

2 F

we In order to show that Q is indeed a probability measure and that Z =

2 t

dQ j

1 F

shall show that (Z ) is a uniformly integrable martingale closed by Z .

t t2R 1

WemaywriteZ =(Z ) as

t t2R

Z =1+(Y (!; t; x) 1) :

From the de nition of the comp ensator ([JS87], I I.1.8) we deduce that wemay

write the comp ensator Z of Z

Z =1+(Y (! ; t; x) 1)

=1+((Y (! ; t; x) 1) F ) A

!;t

Noting that, for dA-almost each (!; t) we have that (Y (! ; t; x) 1) F =

!;t

E [Y (!; t; x) 1] = 0 we deduce that the comp ensator Z is constant and there-

!;t

fore Z is a lo cal martingale. As Z is the di erence of the two increasing pro cesses

1+Y (!; t; x) and 1 19

and as for the latter wemay estimate

E [(1 ) ]=Q [fT < 1g] 1

Q 1 1

we infer that Z is a lo cal martingale of Q -integrable variation and therefore a

uniformly integrable martingale under Q . As (Z ) converges a.s. to Z we

1 t t2R 1

conclude that Z closes Z and, in particular, E [Z ]=1.

1 1

To estimate the distance kQ Q k note that

2 1

] kQ Q k = E [ 1

2 1 Q

E [(jY (!; t; x) 1j ) ]

Q 1

E (kF G k A) "=2E [A ] "=2:

Q !;t !;t 1 Q 1

1 1

Next observe that Q j = Q j : indeed, we have to show that Q and Q

2 F 1 F 1 2

T T

coincide on the sets of the form A \fT >tg, where A 2F , as these sets generate

F . Noting that Z is equal to 1 on fT >tg, this b ecomes obvious.

T t

Finally weshowthat S is a sigma-martingale under Q . First note that M remains

a lo cal martingale under Q as M is continuous at time T , i.e., M = M , and

2 T T

Q and Q coincide on F .

1 2 T

As regards the remaining part X + B of the semi-martingale S we have by (b)

ab ove that, for dA-almost each(!; t); E [ky k ] < 1 and E [y ]=b(!; t).

G G

!;t !;t

This does not necessarily imply that X + B is already a martingale (or a lo cal

martingale) under Q as a glance at example 2.2 reveals. Wemay only conclude

that X + B is a Q -sigma-martingale, as we presently shall see.

De ne

' (! )=(E [ky k d ]) ^1;

t G

!;t

which is a predictable dA-almost surely strictly p ositive pro cess. The pro cess '

(X + B ) is a pro cess of Q -integrable variation as

E [var (' (X + B ))] =

E [' (ky k d G A + kb k d A)] 2E [A ]=2E [A ] 2

Q !;t !;t Q 1 Q 1

R R

2 2 1

Hence ' (X + B ) is a pro cess of integrable variation whose comp ensator is constant

and therefore ' (X + B ) is a Q -martingale of integrable variation, whence in

particular a Q -martingale. Therefore X + B as well as S are Q -sigma-martingales.

2 2

Summing up: Wehaveproved prop osition 4.5 under the additional hyp othesis that

S remains constant after the rst time T when S jumps by at least 1 with resp ect

to k k d .

Step 2: Now we drop this assumption and assume w.l.g. that S =0. Let T =

0 0

0;T = T and de ne inductively the stopping times

T = inf ft> T : kS k 1g; k =2; 3;:::

k k 1 t

R 20

so that (T ) increases to in nity. Let

k =1

(k )

S =1I S k =1; 2;::::

]]T ;T ]]

k 1 k

(1)

Note that S satis es the assumptions of the rst part of the pro of, where wehave

shown that there is a measure Q P, satisfying (i), (ii), (iii) ab ove for T = T .

2 1

Now rep eat the ab ove argumenttocho ose inductively, for k =2; 3;:::, measures

Q P suchthat

k +1

2 Q [A]

k 1 k

(i) kQ Q k <"=2 ^ inf f : A 2F; P[A] 2 g.

k +1 k

P[A]

k +1

(ii) Q j = Q j and is F -measurable.

k +1 F k F T

(T ) (T )

k k k

(k )

(iii) S is a sigma-martingale under Q .

k +1

The condition in (i) ab oveischosen suchthatwemay apply lemma 4.3 to conclude

that

Q = lim Q

k !1

(k )

exists and is equivalenttoP. From (ii) and (iii) it follows that each S is a sigma-

martingale under Q ,whenceS is a sigma-martingale to o. This proves the rst part

of prop osition 4.5.

As regards the nal assertion of prop osition 4.5 note that, for any predictable

stopping time U , the random times

U if T

k 1 k

U =

1 otherwise

are predictable stopping times, for k =1; 2;::::

By our construction and prop erty(ii)abovewe infer that, for k =1; 2;:::,

( U )

a.s. =

dQ dQ

1 1

j j

F F

( U )

which implies that

= a.s.

dQ dQ

1 1

j j

F F

U U

The pro of of prop osition 4.5 is complete now.

Prop osition 4.5 contains the ma jor part of the pro of of the main theorem. The

missing ingredient is still the argument for the predictable jumps of S. The argument

for the predictable jumps given b elow will be similar to (but technically slightly

easier than) the pro of of prop osition 4.5.

4.6 Proof of the Main Theorem. Let S be an R -valued semi-martingale satisfying

the assumption (NFLVR). By theorem 4.1 wemay nd a probability measure Q

P such that,

E [f ] 0 for f 2 C:

1 21

Wealsomay nd a sequence (T ) of predictable stopping times exhausting the

k =1

accessible jumps of S , i.e., such that for each predictable stopping time T with

P[T = T < 1] = 0, for each k 2 N , we have that S = S almost surely.

k T T

We may and do assume that the stopping times (T ) are disjoint, i.e., that

k =1

P[T = T < 1] = 0 for k 6= j .

k j

Denote by D the predictable set

[

D = [[T ]] R

K +

k =1

a i

and split S into S = S + S , where

a i

S =1I S S =1I S

( R )nD

a i

where the letters \a" and \i" refer to \accessible" and \inaccessible". S and S

are well-de ned semimartingales and in view of the ab ove construction S is quasi-

left-continuous.

a i 1 a i

Denote by C and C the cones in L ( ; F ; P) asso ciated by (4.1) to S and S , and

a i

observe that C and C are subsets of C (obtained by considering only integrands

supp orted by D or ( R ) n D resp ectively) hence

a i

E [f ] 0; for f 2 C and for f 2 C :

Hence S satis es the assumptions of prop osition 4.5 with resp ect to the proba-

bility measure Q and we therefore may nd a probability measure, now denoted

^ ^

by Q ; Q P, whichturnsS into a sigma-martingale and such that, for eachpre-

dictable stopping time T ,wehave

= (4.3)

dQ dQ

1 1

jF jF

T T

By assumption wehave, for each k =1; 2;:::, and for each admissible integrand H

supp orted by[[T ]], that

E [(H S ) ]=E [H (S S )] 0:

Q 1 Q T T

(T )

1 1

k k

Noting that the inequality remains true if we replace H by H 1I , for any F -

(T )

measurable set A, and using (4.3) weobtain

E [(H S ) ]=E [H (S S )] 0 (4.4)

^ 1 ^ T T T

k k k

Q Q

for each admissible integrand supp orted by[[T ]].

Wenow shall pro ceed inductively on k : supp ose wehavechosen, for k 0, proba-

~ ^ ~ ~

bility measures Q = Q ; Q ;:::;Q such that

0 1 k

[S jF ]=S j =1;:::;k E

T T T

j j j

k 22

and such that, for

2 Q [A]

j j +1

: A 2F; P[A] 2 g; " = "=2 ^ inf f

P[A]

wehave

~ ~

kQ Q k <" j =0;:::;k 1:

j j +1 j

~ ~

In addition we assume that Q and Q agree \b efore T and after T "; this

j j 1 j j

~ ~

means that Q and Q coincide on the -algebra F and that the Radon-

j j 1 T

~ ~

Niko dym derivative dQ =dQ is F -measurable. Now consider the stopping time

j j 1 T

T : denote on the set fT < 1g by F the jump measure of the jump S

k +1 k +1 !

k +1

S conditional on F . By (4.4) this ( ; F ; P)-measurable family of

T T

k +1 k +1

k +1

probability measures on R satis es the assumptions of lemma 3.3 and we therefore

may ndanF -measurable family of probability measures G , a.s. de ned on

T !

k +1

fT < 1g, such that

k +1

(i) F G and kF G k <"

! ! ! ! k

] < 1 and bary(G )=E [y ]= 0. (ii) E [ky k

! G G

! !

Letting, similarly as in the pro of of prop osition 4.5 ab ove,

Y (!; x)= (x)

be a F B(R )-measurable version of the Radon-Niko dym derivatives

k +1

and de ning

k +1

(! )) + 1I (! )=1I Y (!; S

fT =1g fT <1g

k +1

k +1 k +1

~ ~ ~ ~ ~

we obtain a measure Q P; kQ Q k <" , Q = Q

k +1 k +1 k k k +1 k

j j

F F

(T ) ( T )

k +1 k +1

k +1

b eing F -measurable. For each M 2 R and

T +

k +1

1I S =1I S

[[T ]]\fT <1 and E [ky k]M g [[T ]]\fT <1 and E [ky k]M g

G G

k +1 k +1 k +1 k +1

! !

is a martingale under Q and therefore

k +1

(k +1)

S := 1I S

[[T ]]\fT <1g

k +1 k +1

is a sigma-martingale under Q .

k +1

(k )

Letting Q = lim Q , each of the semi-martingales S =1I S is a Q -sigma-

k !1 k

[[T ]]

martingale. It follows that

a (k )

S = S

k =1

is a Q -sigma-martingale and therefore

a i

S = S + S

is a Q -sigma-martingale to o.

The pro of of the main theorem is complete now.

For later use, let us resume in the subsequent prop osition what wehaveshown in

the ab ove pro of. 23

4.7 Prop osition. Denote by MS the set of probability measures Q equivalent to

P such that for admissible integrands, the process H S becomes a supermartingale.

Moreprecisely

MS = fQ j for each f 2 C : E [f ] 0g :

e e

If S satis es NF LV R,then M is dense in MS .

5. Duality Results and Maximal Elements

In this section we supp ose without further notice that S is an R -valued semi-

martingale that satis es the N F LV R prop erty, so that the set

M = fQ j Q P and S is a Q sigma martingaleg

is nonempty. We remark that when the price pro cess S is lo cally b ounded then the

set M coincides with the set as intro duced in [DS94], i.e. the set of all equivalent

lo cal martingale measures for the pro cess S .

In the case of lo cally b ounded pro cesses weshowed the following duality equality,

(see [DS94], theorem 5.7 for the case of b ounded functions and [DS95], theorem 9

for the case of p ositive functions): for a nonnegative random variable g wehave:

sup E [g ] = inf f j there is H admissible and g +(H S ) g :

Q 1

Q2M

Using this equalitywewere able to deriveacharacterisation of maximal elements,

see [DS95] corollary 14.

In the general case, i.e. when the pro cess S is not necessarily lo cally b ounded,

the set of admissible integrands might b e restricted to the zero integrand, compare

prop osition 3.4 ab ove. Belowwe will show that also in this case the ab ove equality

remains valid, at least for p ositive random variables g . This result do es not imme-

diately follow from the results in [DS95]. Another approachtotheproblemisto

enlarge the concept of admissible integrand in a wayaswas also done in [S94] and

[DS96]. For a random variable w 1wecallanintegrand H to b e w -admissible if

the pro cess H S can b e controlled from b elowby the function w (see 5.3 and 5.4

below for a formal de nition of this concept and of the de nition of feasible weight

function). The aim of this section is to prove a duality equality appropriate for the

setting of w -admissible integrands. More precisely wewillprove the following:

5.1 Theorem. If w isafeasible weight function and g is a random variable such

that g w then:

E [g ]=inff j thereisH w admissible and g +(H S ) g : sup

Q 1

;E [w ]<1 Q2M

If the quantities are nite then the in mum is a minimum.

Remark. The reader can see that even in the case of lo cally b ounded pro cesses S

the result yields more precise information. Indeed we restrict the supremum to

those measures Q 2 M suchthat E [w ] < 1.

For a feasible weight function w ,we denote by K the set

K = f(H S ) j H is w -admissibleg :

w 1

The maximal elements in this set are then characterised as follows: 24

5.2 Theorem. If w 1 is afeasible weight function, if H is w -admissible and if

h =(H S ) , then the fol lowing areequivalent:

(1) h is maximal

(2) thereisQ 2 M such that E [w ] < 1 and E [h]=0

Q Q

(3) thereisQ 2 M such that E [w ] < 1 and H S is Q -unifo rmly integrable

martingale.

As observed ab ove, we have to admit unb ounded weight functions w in order to

guarantee the existence of suciently many admissible integrands. If however we

take w to o big then, e.g., doubling strategies might be admissible and arbitrage

opp ortunities will o ccur. We are therefore lead to the following de nition.

5.3 De nition. Arandom variable w : ! R such that w 1 is cal ledafeasible

weight function for the process S ,if

(1) there is a strictly positve boundedpredictable process ' such that the maximal

function of the R -valuedstochastic integral ' S satis es (' S ) w .

(2) thereisanelementQ 2 M such that E [w ] < 1.

If no confusion can arise to which pro cess the feasibility condition refers, then we

will simply say that the weight function is feasible. The rst item in the de nition

requires that w is big enough in order to allowintegrands H suchthatboth H and

H are w -admissible. The second item requires w to be not to o big and as we

will see this will avoid arbitrage opp ortunities. It follows from prop osition 2.6, that

b ecause M is nonempty, the existence of feasible weight functions is guaranteed.

5.4 De nition. If w 1 is a random variable, if there is Q 2 M such that

E [w ] < 1 and if a is a nonnegative real number, then we say that the integrand

H is a w -admissible if for each element Q 2 M and each t 0, we have

(H S ) a E [w jF ]. If H is a w -admissible for some a, then we simply say

t Q t

that H is w -admissible.

Remark. If weput w =1 we again nd the usual concept of admissible integrands.

Remark. The present notion of admissible integrand is more suitable for our pur-

p oses than the one intro cuded in [DS96].

For feasible weight functions w ,itmight happ en that for some elements Q 2 M

we have that E [w ] = 1, see the example 5.11 below. The next lemma, based

on a well known stability prop erty of the set M , shows that in the inequality

(H S ) E [w jF ], it do es not harm to restrict to elements Q 2 M such that

t Q t

E [w ] < 1.

5.5 Lemma. Let w 0 be such that E [w ] < 1 for some Q 2 M . Suppose

Q 0

that for some Q 2 M and some real constant k the set A = fE [w jF ] k g is

Q t

nonempty, then thereisQ 2 M such that on A we have E [w jF ]=E [w jF ]

1 Q t Q t

and E [w ] < 1.

h i

Proof. Let Z be a cadlag version of the density pro cess Z = E . Now jF

t t Q t

0 25

weput

Z =1 for s

Z =1 for s t and ! 2= A

for s t and ! 2 A: Z =

1 e

Clearly the probability measure Q de ned by dQ = Z dQ is in the set M and

1 1 0

satis es the required prop erties. Indeed on the set A wehave E [w jF ]=E [w j

Q t Q

F ]andE [w ] E [w ]+k<1.

t Q Q

1 0

Remark. In the sequel wewillmake frequent use of Theorem D and corollary 4.12

of [DS96]. These two results were proved for the slightly more restrictive notion of

admissibility, but the reader can go through the pro ofs and check that the results

remain valid for the present notion of w -admissible integrands. Indeed the lower

b ound H S w is only used to control the negative parts of the p ossible jumps

in the sto chastic integral. This can also be achieved by the inequality H S

E [w j F ] where E [w ] < 1. Compare the formulation of Theorems B and C in

Q t Q

[DS96]. For the convenience of the reader let us rephrase the results of [DS96] in

the present setting.

Theorem D. Let Q be a probability measure, equivalent to P. Let M be an R -

valued Q -local martingale and w 1 a Q -integrable function.

n d

Given a sequence (H ) of M -integrable R -valuedpredictable processes such that

(H M ) E [w jF ]; for al l n; t;

t Q t

then thereareconvex combinations

n n n+1

K 2 convfH ;H ;:::g;

and there is a super-martingale (V ) ;V 0, such that

t t2R 0

lim lim (K M ) = V for t 2 R ; a.s., (i)

s t +

n!1

s!t;s2Q

and an M -integrable predictable process H such that

((H M ) V ) is increasing. (ii)

t t t2R

In addition, H M is a local martingale and a super-martingale.

d e

Corollary 4.12. Let S be an R - valued semi-martingale such that M (S ) 6= ;

(S ) with E [w ] < 1. and w 1 a weight function such that thereissomeQ 2 M

Then the convex cone

fg j thereisa1 w admissible integrand H such that g (H S ) g

1 26

is closedinL ( ) with respect to the topology of convergenceinmeasure.

Let w 1besuch that there is Q 2 M , with E [w ] < 1. The set

K = f(H S ) j H is w -admissibleg

w 1

is a cone in the space of measurable functions L . As in [DS94] we need the cone

of all elements that are dominated by outcomes of w -admissible integrands:

C = fg j g (H S ) where, H is w -admissibleg :

If H is w -admissible and E [w ] < 1 for some Q 2 M , then it follows from the

results in [AS92] that the pro cess H S is a Q -sup ermartingale. Therefore the limit

(H S ) exists and E [(H S ) ] 0. It also follows that for elements g 2C ,

1 Q 1

wehavethat 1 E [g ] 0. We will use this result frequently. We also remark

that if H is w -admissible and if (H S ) w then H is already 1 w admissible.

Indeed b ecause of the sup ermartingale prop ertyof H S wehave that (at least for

those Q 2 M such that E [w ] < 1):

(H S ) E [(H S ) jF ] E [w jF ]:

t Q 1 t Q t

By lemma 5.5 this means that H is 1 w -admissible.

5.6 Theorem. If w 1 and if there is some Q 2 M is such that E [w ] < 1,

then

1 1 0

C = h j h 2 L and hw 2C

w w

is weak closedinL (Q ).

Proof. This is corollary 4.12 cited ab ove.

Wenowprove the duality result stated in theorem xx. The pro of is broken up into

several lemmata. As we will work with functions w 1 that are not necessarily

feasible weight functions we will make use of a larger class of equivalent measures

namely:

= fQ P j E [w ] < 1 and for each h 2C : E [h] 0g : MS

Q w Q

5.7 Lemma. If w 1 has a nite expectation for at least one element Q 2 M ,

if g is a random variable such that g w then:

sup E [g ] inf f j thereisH w admissible and g +(H S ) g :

Q 1

Q2MS ;E [w ]<1

Proof. The pro of follows the same lines as the pro of of theorem 9 in [DS95]. If w 1

and Q 2 MS then as observed ab ove, the pro cess H S is a Q -sup er martingale

for eachthatH is w -admissible. Therefore the inequality g +(H S ) implies

that E [g ] .

Remark. If, under the hyp othesis of the theorem, sup E [g ] = 1,

Q2M ;E [w ]<1

then also inf f j there is H w admissible and g +(H S ) g = 1. This

simply means that no matter how big the constant a is taken, there is no w -

admissible integrand H such that g a +(H S ) .

1 27

5.8 Lemma. If w 1,ifforsomeQ 2 M we have E [w ] < 1,if is bounded,

then thereisa probability measure Q 2 MS such that E [g ] > .

Proof. The hyp othesis on means that:

1 1

\C = f0g: + L

w +

Because the set C is weak closed, we can apply Yan's separation theorem [Y80]

and we obtain a strictly p ositive measure , equivalentto P suchthat

(1) E [ ] > 0

(2) for all h 2C wehave E [h] 0.

If we normalize so that the measure Q de ned as dQ = d b ecomes a probability

measure, then we nd that

(1) Q P and E [w ] < 1,

(2) E [g ] > ,

(3) for all h 2C wehave that E [hw ] 0.

The latter inequality together with the Bepp o-Levi theorem then implies that for

each w -admissible integrand H wehavethat E [(H S ) ] 0.

Q 1

5.9 Lemma. If w 1,ifsomeQ 2 M we have E [w ] < 1,if g w then

0 Q

sup E [g ] inf f j thereisH w admissible and g +(H S ) g :

Q 1

Q2MS

Moreover if the quantity on the right hand side is nite then the in mum is a

minimum.

g ^n

Proof. For each n 1, wehave that is b ounded and hence we can apply the

previous lemma. This tells us that, for each n 2 N ,

=supfE [g ^ n] j Q 2 MS g

n Q

inf f j there is H w admissible and g ^ n +(H S ) g :

Because there is nothing to prove when lim = 1 wemay supp ose that sup =

n n n

lim = <1. So, for each n,we takeaw -admissible integrand H such that

n n

n e

such that E [w ] < 1. +(H S ) . Let us now x Q 2 M g ^ n +

Q 1 0 n

n n n+1

From Theorem D in [DS94] we deduce the existence of K 2 conv fH ;H ;:::g

as well as H ,such that

(1) V = lim lim (K M ) exists a.s., for all t 0,

t s!t;s2Q n!1 s

(2) (H S ) V is increasing,

t t

(3) V 0.

0 n

From this it follows that (H S ) V + V V . Since H is w -admissible (and

t 0 t t

hence 1 w -admissible) we have that K is 1 w -admissible and hence we nd

that V E [w jF ] for all Q 2 M such that E [w ] < 1. It is now clear that

t Q t Q

H is w -admissible. Since the sequence is increasing we also obtain that for all

t and all Q 2 M with E [w ] < 1:

1 1

n n

(K S ) + + E [(K S ) jF ]+ + E [g ^ n jF ] :

t n Q 1 t n Q t

n n

This yields that for all t and all n

1 1

V + + E [g ^ n jF ] : (H S ) + +

t n Q t t n

n n

If t tends to in nity this gives (H S ) + + g ^ n for all n. By taking the

1 n

limit over n we nally nd that

(H S ) + g:

This shows the desired inequality and at the same time also shows that the in mum

is a minimum.

We are now ready to prove the duality results. We start with the case of admissible

integrands thus extending theorem 9 of [DS95] to the case of non lo cally b ounded

pro cesses S . Recall that we assume throughout this section that S is an R valued

semi martingale satisfyling N F LV R .

5.10 Theorem. For a nonnegative random variable g we have:

E [g ] = inf f j thereisH admissible and g +(H S ) g : sup

Q 1

Q2M

Proof. From the previous lemmata it follows that weonlyhavetoshowthat

sup E [g ]= sup E [g ]:

Q Q

Q2M Q2S

This follows from prop osition 4.7 and the fact that g is b ounded from b elow.

We now complete the pro of for the case of feasible weight functions w and w -

admissible integrands:

e e

Proof of Theorem 5.1. In this case weshow that M = MS . We already observed

e e e

that M MS . Takenow Q 2 MS .

w w

Since w is now supp osed to b e a feasible weight function, wehave the existence of

a strictly p ositive predictable function ' such that (' S ) w . It follows that

i i i i

outcomes of the form 1I ' S ' S or 1I ' S ' S , where s < t,

A A

t s t s

A 2 F and (S ) are the co ordinates of S , are outcomes of w -admissible

s i=1:::d

. integrands. Therefore ' S is a Q martingale and Q 2 M

Remark. An interesting question is whether by taking the supremum in theorem

such that for the feasible weight 5.1, wehave to restrict to those elements Q 2 M

function w wehave E [w ] < 1. More precisely is there a contingent claim g w

suchthat

E [g ] > sup E [g ]: sup

Q Q

Q2M

Q2M ;E [w ]<1

An insp ection of the pro of of the ab ove theorem shows that we used the Q -

intgerability of the feasible weight function w in order to conclude that the w -

admissible integrand H de ned a Q -sup ermartingale H S . 29

5.11 Example. There is a continuous process S , S = 0, satisfying NF LVR

andsuch that

(1) P 2 M ,

(2) S is a P uniformly integrable martingale

(3) S is boundedabove and hence for al l Q 2 M the process S is a submartin-

gale

(4) for some Q 2 M , the process S S is not a Q supermartingale

This example will then , as we will see, also solve negatively the question whether

the two suprema are the same.

The example is based on [DS97]. There wegave an example of a one dimensional,

continuous, strictly p ositive, price pro cess X , X = 1, such that the set M is

nonempty, P 2 M and X is a P-uniformly integrable martingale, whereas for

some other element Q 2 M wehave that E [X ] < 1. Wenow take the following

Q 1

elements w = jX j +1 and S = 1 X . Clearly E [w ] < 1 and since X is

1 P

continuous, we can nd a predictable, strictly p ositive pro cess ' such that ' S

remains b ounded by1. It follows that w is feasible. Clearly the pro cess S is then

the gains pro cess of the w admissible integrand H = 1. If we put g = S we

trivially have that E [g ] > 0. This shows the following two assertions:

(1) the pro cess S is a not a Q sup ermartingale however it is a P uniformly

integrable martingale and a Q lo cal martingale.

E [g ] > sup E [g ]=0: (2) sup

e e

Q Q

Q2M ;E [w ]<1 Q2M

Wenowturntothecharacterisation of maximal and of attainable elements. The

approachis di erent from the one used in [DS95], which was based on a change

of numeraire technique. In order not to overload the statements we henceforth

supp ose that w is a feasible weight function.

5.12 De nition. An element g 2K is cal led maximal if h 2K and h g imply

w w

that h = g .

5.13 Lemma. If g 2K ,thenthere is a maximal element h 2K such that h g .

w w

Proof. It is sucienttoshow that every increasing sequence in K has an upp er

b ound in K . So let h , h = g , b e an increasing sequence in K . For each n take

w n 1 w

n n

H , w -admissible so that h =(H S ) . As in the previous pro of we then nd,

n 1

as an application of Theorem D in [DS96], that there is H , w -admissible such that

(H S ) lim h . This concludes the pro of of the lemma.

1 n n

Proof of Theorem 5.2. If E [w ] < 1 then H S is a Q sup ermartingale and hence

(2) and (3) are equivalent. Also it is clear that (2) implies (1). Indeed if g is the

such that also result of a w -admisible integrand then E [g ] 0 for each Q 2 M

E [w ] < 1. It follows that h is necessarily maximal.

The only remaining part is that (1) implies (2). Since always E [h] 0 for Q 2 M

such that also E [w ] < 1,we obtain already that for measures Q satisfying these

+ +

assumptions, h is Q -integrable. So x such a measure Q . Nowlet w = h + w .

Clearly w is a feasible weight function. We will work with the set K . The

1 w

problem is, however, that we do not (yet) know that h is still maximal in the

bigger cone K . From the construction of w it follows that for elements Q 2

w 1

1 30

M , E [w ] < 1 if and only if E [w ] < 1. Now let g h be the result of a

Q 1 Q

w -admissible integrand. Hence g = (K S ) where K is w -admissible. Since

1 1 1

(K S ) g h w and since K is w -admissible wehavethat K is already

1 1

w -admissible. (Rememb er that E [w ] < 1 if and only if E [w ] < 1)From the

Q 1 Q

maximalityof h in K it then follows that g = h,i.e. h is maximal in K . This

w w

can then b e translated into

1 1

+ L \C = f0g:

+ w

Using Yan's separation theorem in the same way as in the pro of of theorem 5.1

ab ove, we nd a measure Q such that E [w ] < 1, Q 2 M and E [h] 0.

1 Q 1 1 Q

1 1

The following theorem generalises a result due to Ansel-Stricker and Jacka, [Jk92]

and [AS94].

5.14 Theorem. Let w be a feasible weight function and let f w . Then are

equivalent

(1) thereisameasure Q 2 M such that E [w ] < 1 and such that

E [f ]= sup [f ] < 1

R2M ;E [w ]<1

(2) f can be hedged, i.e. there is 2 R, Q 2 M such that E [w ] < 1,

a w -admissible integrand H such that H S is a Q uniformly integrable

martingale, and such that f = +(H S ) .

Proof. Clearly (2) implies (1) by the previous theorem.

For the reverse implication take now Q as in (1), then the duality result gives a

2 R as well as a w -admissible integrand H such that f +(H S ) , where

=sup [f ]. Here we use explicitly that the in mum in the duality

R2M ;E [w ]<1

theorem is a minimum. But then it follows from E [w ] < 1 and from the equality

E [f ] = that f = +(H S ) and that H S is a Q uniformly integrable

Q 1

martingale.

References

[AS94] J. P. Ansel, C. Stricker, Couverture des actifs contingents et prix maximum, Ann.

Inst. Henri Poincare 30 (1994), 303{315.

[Au65] R. Aumann, Integrals of Set-ValuedFunctions, Journal of Mathematical Analysis and

Applications 12 (1965), 1{12.

[C77] C.S. Chou, Caracterisation d'une classe de semimartingales,Seminaire de Probabilite

XI I I, LNM, Springer 721 (1977/78), 250-252.

[D72] C. Dellacherie, Capacites et processus stochastiques, Springer, Ergebnisse der Mathe-

matik 67, Berlin (1972).

[DH86] D. Due and C. F. Huang, Multiperiodsecurity markets with di erential information,

J. Math. Economics 15 (1986), 283{303.

[DMSSS96] F. Delbaen, P. Monat, W. Schachermayer, M. Schweizer, C. Stricker, Weighted Norm

Inequalities and Closedness of a SpaceofStochastic Integrals, to app ear in Finance

and Sto chastics 1 (1997).

[DMW90] R. C. Dalang and A. Morton and W. Willinger, Equivalent Martingale measures and

no-arbitrage in stochastic,Stochastics and Sto chastic Rep orts 29 (1990), 185{201. 31

[DS94] F. Delbaen, W. Schachermayer, AGeneral Version of the Fundamental Theorem of

Asset Pricing, Math. Annalen 300 (1994), 463{520.

[DS95] F. Delbaen, W. Schachermayer, The No-Arbitrage Property under a Change of Nume-

raire,Stochastics and Sto chastic Rep orts 53 (1995), 213{226.

[DS96b] F. Delbaen, W. Schachermayer, ACompactness Principle for BoundedSequences of

Martingales, preprint (1996).

[DS97a] F. Delbaen, W. Schachermayer, A simple counter-example to several problems in the

theory of asset pricing, which arises generical ly in incomplete markets., Mathematical

Finance (1997).

[E78] M. Emery, Compensation de processus a variation nie non localement integrables,

Sem. de Probabilite XIV, LNM 784 (1978/79), 152{160.

[HK79] M. J. Harrison, D. M. Kreps, Martingales and Arbitrage in Multiperiod Securities

Markets, Journal of Economic Theory 20 (1979), 381-408.

[HP81] J. M. Harrison, S. R. Pliska, Martingales and Stochastic Integrals in the Theory of

Continous Trading, Sto chastic Pro cesses and their Applications 11 (1981), 215{260.

[J79] J. Jaco d, Calcul Stochastique et Problemes de Martingales, Lecture Notes in Mathe-

matics 714 (1979).

[Jk92] S. Jacka, A Martingale Representation Result and an Application to Incomplete Fi-

nancial Markets, Mathematical Finance 2 (1992), 239{250.

[JS87] J. Jaco d, A. Shiryaev, Limit Theorems for Stochastic Processes, Springer (1987).

[K81] D. M. Kreps, Arbitrage and Equilibrium in Economics with in nitely many Com-

modities, Journal of Mathematical Economics 8 (1981), 15{35.

[KK94] Y. Kabanov, D. Kramkov, No arbitrage and equivalent martingale measures: An

elementary proof of the Harrison-Pliska theorem, Theory Prob. Appl. 39 (1994).

[M91] D. W. Mcb eth, On the Existenceofequivalent local Martingale Measures, Thesis at

Cornell University (1991), 70 p.

[R94] L.C.G. Rogers, Equivalent martingale measures and no-arbitrage, Sto chastics and

Sto chastic Rep orts 51 (1994), 41{49.

[S92] W. Schachermayer, A Hilbert Space Proof of the Fundamental Theorem of Asset

Pricing in Finite Discrete Time, Insurance, Math. and Econ. 11 (1992), 249{257.

[S94] W. Schachermayer, Martingale Measures for discrete time processes with in nite hori-

zon, Math. Finance 4 (1994), 25{55.

[St90] C. Stricker, Arbitrage et Lois de Martingale, Annales de l'Institut Henri Poincare 26

(1990), 451{460.

1 1

[Y80] J. A. Yan, Caracterisation d' une classe d'ensembles convexes de L ou H , Seminaire

de Probabilites XIV, Lect. Notes Mathematics 784 (1978/79), 220{222.

Departement fur Mathematik, ETH Zurich, CH-8092 Zurich, Switzerland

E-mail address : [email protected]

Institut fur Statistik, Universitat Wien, Brunnerstr. 72, A-1210 Wien, Austria.

E-mail address : [email protected] 32