D AVID G. H OBSON
The maximum maximum of a martingale
Séminaire de probabilités (Strasbourg), tome 32 (1998), p. 250-263
<http://www.numdam.org/item?id=SPS_1998__32__250_0>
© Springer-Verlag, Berlin Heidelberg New York, 1998, tous droits réservés.
L’accès aux archives du séminaire de probabilités (Strasbourg) (http://portail.
mathdoc.fr/SemProba/) implique l’accord avec les conditions générales d’utili- sation (http://www.numdam.org/conditions). Toute utilisation commerciale ou im- pression systématique est constitutive d’une infraction pénale. Toute copie ou im- pression de ce fichier doit contenir la présente mention de copyright.
Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques
http://www.numdam.org/
David G. Hobson
Department
of MathematicalSciences, University
ofBath,
Claverton
Down, Bath,
BA2 7AY. UK.Abstract
Let be any
martingale
with initial lawMo N
and ter-minal law pi and let S = Mt. Then there is an upper
bound,
with respect to stochasticordering
ofprobability
measures, on the law of S.An
explicit description
of the upper bound isgiven, along
with amartingale
whose maximum attains the upper bound.1 Introduction
Let po and /~i be
probability
measures on R with associated distribution func- tions - - oo,x~ )
. Now let M = be the space of all martin-gales
with initial law ~o and terminal law For such amartingale
M E M
let S
= supot 1Mt
and denote the law of Sby
v. In this short arti- cle we are interestedin the
set P ~ P( 0, 1) _{v;
M E ofpossible
lawsv, and in
particular
we find a least upper bound for P. The fact that M is amartingale imposes quite
restrictive conditions on v.Clearly
M isempty
unless the random variablescorresponding
to thelaws i
have the same finite mean, and henceforth we will assume without loss ofgenerality
that this mean iszero.
Moreover asimple application
ofJensen’s
inequality
shows that a further necessary condition for the space to benon-empty
is that(i) 100 (y -
~~x (y
- ’dx.These conditions are also
sufficient,
see forexample
Strassen[16,
Theorem2]
or
Meyer [10, Chapter XI].
The
question
described in theopening paragraph
is aspecial
case of aproblem
first considered in Blackwell and Dubins[4]
and Dubins and Gilat[7].
There the authors derive conditions on the
possible
laws v of the supremumS
of a
martingale
whose terminal distribution~cl
isgiven,
but whoseinitial law
~co
is notspecified. Let ~
denote stochasticordering
onprobability
measures,
(so
thatp ~
~r if andonly
if >b’x,
with the obvious notationalconvention)
and letp*
denote theHardy
transform of aprobability
measure p. Then it follows from
[4]
and[7]
that(2)
Indeed,
Kertz and Rösler[9]
have shown that the converse to(2)
also holds:for any
probability
measure psatisfying ~ui -~ p -~
there is amartingale
with terminal distribution
~ci
whose maximum has law p.If,
moreover, p is concentrated on[0, oo)
then themartingale M
can be taken to have initial lawconsisting
of the unit mass at 0. See alsoRogers [14]
for aproof
of theseresults based on excursion
theory,
and Vallois[17]
for a discussion of the casewhere
M
is a continuousmartingale.
Thus if ~uo -~o (the
unit mass at0)
then our
problem
is solved andOtherwise,
as Kertz and Rösler[9,
Remark3.3] observe,
(3) ~
V ~1~
v~
In a sense Kertz and Rösler
[9,
Theorem3.4]
answer ourquestion
of inter-est also.
They
describe necessary and sufficient conditions for a candidateprobability
measure v to be a member of These conditions involvedisplaying
apair
of bivariate densities withmarginals
andv)
andmay be
thought
of as a restatement of theproblem.
In contrast the solutionpresented
here is bothexplicit
and constructive.The main results of this article are that the set is bounded above
by
aprobability
measure(in
the sense that if v ~ ~then ~ ~
and that this upper bound is attained. Moreover we
provide
anexplicit
construction
of this upper bound: we do so now for the nicespecial
case wherehas a continuous distribution. For i =
0,1 define ~i
= andlet
a(z)
be the solution witha(z)
> z to theequation
~lo
(a(z) )
= ~1(z)
+(a(z) - (z) ~
Pictorially a(z)
is the x-eo-ordinate of thepoint
where thetangent
to r~l at z intersects thegraph
of SeeFigure
1. Then is definedby
In Section 2 we prove this result and ex-
tend
toarbitrary
measures /~i.Clearly
one non-constructive definition of is via its distributionfunction .
’
(4)
’
Fo,i [x]
_inf
Figure
1: The functions anda(z).
There seems no reason a
priori why
the measurecorresponding
to(4)
shouldbe an element of Indeed if the
greatest
lower bound is defined via its distribution function(5)
= supvE~
then need not be an element of
P;
inparticular
it is not ingeneral
truethat P. See Section 3.4 for a
simple example showing
non-attainment of the lower bound.The Skorokhod
embedding
theorem concerns theembedding
of agiven
law in Brownian motion
by
construction of asuitably
minimalstopping
time.(Skorokhod embeddings
for Brownian motion and other processes remains an active area ofresearch;
see the recent paperby
Bertoin and Le Jan[3]
fora new class of suitable
stopping rules.)
We show that onemartingale
whosemaximum attains the upper bound is a
(time-change of)
Brownianmotion,
and this
explains why
theprescient
reader willrecognise
in thearguments expounded
below elements of the Chacon and Walsh[5]
and Azema and Yor[2]
proofs
of the Skorokhod theorem(and
also theRogers [12]
excursion theoreticversion of the Azéma-Yor
argument).
An incidental remark in Section 3.3 indicates how these alternative derivations of the Skorokhod theorem are in factclosely
related.Finally
some brief words on a motivation forstudying
thisproblem.
LetMt
be theprice
process of a financialasset,
and suppose that interest rates are zero. Then standardarguments
from thetheory
ofcomplete
markets show that whenpricing contingent
claims or derivative securities it is natural to treat Mas if it were a
martingale.
Thesimplest
and mostliquidly
tradedcontingent
claims are
European
calloptions which,
atmaturity T,
havepayoff (MT - k)+.
Suppose
now that instead ofattempting
to modelMt
and thence topredict
theprices
of calloptions,
we assume that theprices
of calls arefairly
determinedby
the market. Fromknowledge
of callprices
for all strikes k it ispossible
toinfer the law
(at
least under the measure used forpricing derivatives)
ofMT.
Bounds on the
prices
of ’exotic’ derivatives can be obtainedby maximising
the
expected payoff
of the exoticoption
over the space ofmartingales
withthe
given (or
rather theinferred)
terminal distribution. These boundsdepend
on the market
prices
of calloptions,
butthey
do notrely
on anymodelling assumptions
whichattempt
to describe theunderlying price
process.As an
example,
the lookbackoption
is asecurity
which atmaturity
T has value
ST,
the maximumprice
attainedby
the asset over the interval[o, TJ.
Given the set ofprices
of calloptions
withmaturity
T we can deducethe
(implied)
law ofMT
under thepricing
measure. SinceMo
is fixed andMt
is a
martingale
under thepricing
measure, theproblem
ofcharacterising
thepossible prices
of a lookbacksecurity
is solved once thepossible
laws of themaximum
ST
have been determined.This
is theproblem
under consideration in Blackwell and Dubins[4],
and moregenerally
here. See Hobson[8]
for amore detailed
analysis
of the lookbackoption,
the derivation ofnon-parametric
bounds on the lookback
price
and thedescription
of an associatedhedging
strategy.
Acknowledgement
It is apleasure
to thank DavidMarles,
Uwe Rösler andan anonymous referee for
helpful
andinsightful
comments on aprevious
version of this paper.2 Main results
In this section we derive conditions
relating
the distribution of the maximumv to the initial and terminal distributions po, 1. First we recall some
simple
bounds which do not
depend
on the initial law .Clearly P[Mi
> >x]
so that v. Define thenon-decreasing
barycentre
functionbi by
’’
bi (x) _ 1
for all x such that
JP>[M1 x]
>0,
andb1(x)
= x otherwise.By
Doob’ssubmartingale inequality
Fixing
c, then at least in the case where has noatoms,
there is some d with >c]
=d].
Moreover it is trivial thatE[Mi - d; A]
E[Mi - d; d]
for all setsA,
so thatE[M1
S >c] JE[M1; M1 d]
=d].
Thus c ,.b1(d)
and it follows that v ~~*,
where~c*,
the
Hardy
transform of has associated distribution functionFi given by
= The
proof
in the case where ~.1 contains atomsrequires only
minor modifications.The above
paragraph,
which follows Blackwell and Dubins[4] closely,
contains a
proof
of(2),
andprovides
many of the essentialarguments
we willuse in
Proposition
2.1 to find an upper bound for P. Our purpose is to consider the effect offixing
the initial law of themartingale.
For i =
0,1
define the functionsby
=
~x(1 - Fi[y])dy
=~x(y - x) i(dy)
=E[(Mi = x)+].
The
functions
arepositive, decreasing
and convex with > -x. Recallfrom the introduction that a necessary and sufficient condition for
~1)
to be
non-empty
is that for all x. Henceforth we assume that this condition is satisfied.Furthermore,
if = thenIE((Ml - x)+; Mo x)
-~)+) - Mo
>x)
=
0,
so that
P(Mi
> x,Mo x)
= 0.Similarly P(Mi
x,Mo > x)
= 0 so that ifassociates any mass with a
point
x E{z : 7?i(~) =
then we must havethat the
martingale
M is constant on the setMo
= x, and ~cl must include acorresponding
atom.By considering
such atomsseparately,
andby dividing
the set I =
{x
> into its constituent intervals we can reduce to the case where I takes the form of asingle
interval I =(i-, i+)
C(-oo, oo).
and assume further that ~,o
places
no mass at theendpoints
of I.We now construct functions a, a and
/3
which willplay a
crucial role insubsequent analysis.
Asmotivation,
supposetemporarily
that ~cl has noatoms. Note that the derivative of r~l is
given by r~i (~)
= >x~
_- 1. For z i- and z >
i+
leta(z)
= z and otherwise definea(z)
to bethe
unique
solution witha(z)
> z to theequation
(6)
_ +~d~z) -
a(z)
is the x-coordinate of thepoint
where thetangent
to 7yi at z, taken in the direction ofincreasing
z, intersects with the function RecallFigure
1.The function a is
non-decreasing,
and on I it satisfies.
~li~z)) = z).
Define to be the distribution function
given by
=
(The composition Fi
o is well defined sinceassigns
no mass to intervalswhere a is
constant.)
Let be the associatedprobability
measure; will be the upper bound on’
For the
general
case where has atoms we let the aboveargument guide
our intuition. For u E(0,1)
define =infix : F1[x]
>u}.
Theparameter u
willplay
the role ofdefining
theslope
of the relevanttangent.
Define
hu
viahu(z) =
+z~l -’~~ - ~n~~u~~ - ~~~~~1 - ~~~
Then hu
is a convexfunction;
if 0 thenhu
has aunique
root in(x, oo).
On =
~o~(~~u))
seta(u) == /?(~),
and on >~o~/~~u))
leta(u)
be the root in
(/3(tt), oo)
ofhu ;
then the function a satisfies(7)
=~ly~~u~~ - ~aO) - ~~~))~1 - u) .
Informally, a(u)
is the x-coordinate of thepoint
where thetangent
to 7yi withgradient -(1 - u)
intersects thegraph
of IfFi
has an atom of mass v ati+
then for all u > 1 - v we havea(u)
= meanwhilea(u)
> on(0,1 - v).
The function a is continuous and has(left)-derivative
.. da a(u) - (3(u)
’ ’ du + 1 -
u’
where
7yo
isagain
a left-derivative.By
theconvexity
of and the definitionof a as the x-coordinate of the
point
where a line with intersects it is clear that for u E(0,1)
the denominator must bepositive. Finally
define the measure via its distribution function
(9)
=inf{u: a(u)
>x}.
Where defined we have that
a(u)
=a(/~(u)),
and the two definitions of agree.Example
2.1Examples always help
to makethings
clearer...Let be the uniform measures on
{-1,1}, {-2, 0, 2} respectively.
Inparticular
is discrete so that a is not welldefined,
and thisexample
illus-trates the
general
method. Then =max{ -x, 0, (1 - ~r)/2}
and 771(~) _ 0, 2(1 - x)/3, (2 - x)/3}.
Furthera(u) =
+2/(3(1 - ~))7(i/3~2/3)
+(4~ - !)/(! - 2~)7~~i/3).
See
Figure
2 for apictorial representation
of the functions a,/3
andNote that a is continuous and
non-decreasing,
and that for u E(0,1), Qf(~) ~ /j(u-f-)
V’
Figure
2: a,,Q
andF-10
for the distributions inExample
2.1.Proposition
2.1 Let M be amartingale
with the desired initial and terminal distributions. Denoteby
v the lawof
the maximum process S. Thendv E ,
so that is an upper bound
for
Proof
We prove that _
a(u)
for all u E(o,1),
whereFv
is the distribution function associated with the law v. SinceF-103BD and
a areincreasing
functionsit is sufhcient to prove that if >
c)
=1- =1- u, then ca(u).
Fix c, then
by
Doob’ssubmartingale inequality
> c,
Mo c]
5’ > c,Mo c].
Similarly, by
themartingale property,
E[Mo ;Mo
>c]
>c, Mo
>c].
Adding
these twoexpressions yields
after someelementary manipulations
>
c] + ~o (c) 1E[Ml ; S
>c].
Now suppose >
c]
= 1 - u,then,
since for any set A it is true thatE[Y; AJ Y
>o]
,IE[Ml ; s > cJ
=1E[Ml - ,Q(u);
S >4
+(1 - E[Mi - A(u)~ Mi >-
+(1- Using (7)
we can summarise theseinequalities:
c(1- u)
+ +(1-
= +
a(~c)(1- u).
It is easy to see
by
theconvexity
of r~4 that~(1 2014 u)
+ isincreasing
onSince
a(u)
> it follows that ca(u).
D
Inspection
of theproof
ofProposition
2.1 reveals that the upper bound is attained if both themartingale
iscontinuous, (so
that there isequality
inDoob’s
inequality),
and the sets(S
>c)
can be identified with sets of the form(Ml
>d).
Guidedby
these observations ourgoal
now is to construct amartingale
M with initial law and terminal law whose maximum has law We ensurecontinuity
of themartingale
Mby basing
theconstruction
on a
stopping
time for a Brownian motion. Moreover thestopping
rule is afunction of the current maximum of the Brownian motion and its current value.
Rösler
[15] provides
an alternative construction of amartingale
whosemaximum attains the upper bound. This construction is in the
spirit
of argu- mentsgiven by
Blackwell and Dubins[4]
and Kertz and Rösler[9].
In somerespects
the Rösler construction issimpler
than the methodspresented below;
the
advantage
of the methods we use is thatthey provide
a solution of the Skorokhodproblem.
Suppose
that an initialpoint
xo is chosenaccording
to the distribution For motivation consider first the case where ~cl has a continuous distribu- tion function and the function a and its inverse are well defined. Let B be aBrownian motion started at xo and define
sf
=sup Bu
0ut
T =
: Bu
Note that T is almost
surely
finite forif y
> xo andHy
denotes the firsthitting
time of the Brownian motion B at
level y
thenT inf{t > Hy : Bt
We show
that,
whenaveraged
over the law of thestarting point, Br
hasthe law Then also
>
~]
= >x1,
and the distribution function of
SB
isgiven by
(10)
=It will follow
(in Corollary
2.1below)
that defined via(9)
is an elementof
~1).
Ourgoal
now is to prove theabove
claim thatBT
has law inthe
setting
of ageneral probability
measure .Proposition
2.2 For a Brownian motion B with initial lawdefine SB
=sUPO$u$t
Bu
and T =inf{u
:F1[Bu]
ThenBr
~ 1 andSB
~Proof
This follows
directly
from Chacon and Walsh[5] although
here we pro- vide a directproof,
similar inspirit
to Azema and Yor[2].
For a connectionbetween these two
approaches,
see Section 3.3.Suppose
that has an atom of size v ati+,
and thence that hasan atom of at least this size there also.
’
With and T all as above define the random variable Z via Z = Then > Z >
Fl(BT-)
so that(3(Z)
-Br and,
since a iscontinuous,
We find the law of Z: it is sufficient to show that Z has the uniform distribution on[0,1],
or moreparticularly,
and to allow foratoms in
~,o,l,
it is sufficient to show thatP(Z u)
= u, for 0 u 1 - v.Then
x)
=x)
= =F’yx>>
and
similarly x)
=. Return to the consideration of the law of Z: for a test function ~
(continuous
andcompactly supported) set g
== ~ oa-1
and defineG(x) =
fo g(u)du.
Note thatG(a(x))
=fo
Ito calculus shows thatNt -
G(SB)-(SB-Bt)g(SB)
is a continuous localmartingale,
moreover thestopped martingale
NT is bounded(see [2]
fordetails).
Therefore one haso =
No1
-(SB
-BT)s(sB)
-G(So )1
(11)
=E[G(a(Z)) - G(Bo)] - E[(a(Z) - a(Z))9(~(Z))l
Let ~r denote the law of
Z,
and let = 1 - =~r((x, oo)).
Note that03C0 has
support
contained in the interval[0,1].
Then=
f o
=.
f o
For the second term a
simple
transformation of variableyields
E[G(B0)] = 10F0[03B1(u)]03A6(u)d03B1(u) and
(11)
becomes10(F03C0[u] - F0[03B1(u)])03A6(u)d03B1(u) =
10(03B1(u) - 03B2(u))03A6(u)03C0(du).Since 03A6 is
arbitrary,
03C0 mustsatisfy
theidentity
(12) (a(u) -
_ -Substituting
from(8) gives
that at least for x1 - v, (whence a(u)
>for all u E
(O,x))
F03C0[x] - x
= / 2014 1 ! dtZ = 2014 / du.
and it follows that = x for x 1 - v. O
Corollary
2.1 The measure is an elementof
Proof
It suffices to show that
Mt
=B(T
A(t/(1 - t))
is a truemartingale
and notjust
a localmartingale,
orequivalently
that isuniformly integrable.
This follows
by
astraightforward
extension to Lemma 2.3 inRogers [14]
andan
appeal
to Theorem 1 inAzéma, Gundy
and Yor[1].
. D3 Remarks
3.1 The
case
=8o
and reduction toprevious
results.If -
do
then = x- =(-x)
V 0.Suppose
that 1 has a continuous distribution function then sincea(x)
> 0 the formula(6)
becomes_ ~lu~) -~~li~~)
~"
(x) I
.
The function
a(x)
iseasily
shown toequal
thebarycentre
functionbl(X) and,
for
general also,
the results of theprevious
section become the twin state- ments that is both an elementof,
and an upper boundfor, ~(bo,
.3.2
Excursion-theoretic arguments
The
defining equation
for ~rgiven
in(12)
can be derivedusing
excursion ar-guments.
Readers who would like an introduction to excursiontheory
arereferred to
Rogers [13].
Forsimplicity
consider the case where has noatoms and as before let B be a Brownian motion with maximum process
SB
such that the initialpoint Bo
is chosenaccording
to the law DefineT =
inf {u :
Consider
splitting
the Brownianpath
into excursions below its max-imum.
Imagine plotting
the excursions from the maximumsf
= s, in xyspace, in such a way that the
x-component
isalways Bt,
and they-component
is chosen tokeep
the two-dimensional process on thetangent
to whichjoins
with
(a-1(s),r~l(a-1(s))).
As the maximum sincreases,
so the linealong
which excursions areplotted changes.
SeeFigure
3. T is then the firsttime that one of these excursions first meets the curve
(x,
~1(x)).
By
constructiony]
=P[Bo a(y)] - a(y), SB
>a(y)~.
Now for the event
(Br
Edy)
to occur it must be true that both(Bo a(y))
and
(Sf
>a(y)),
andthen,
before the maximum rises toa(y+dy),
there mustbe an excursion down from the maximum of relative
depth
at leasta(y) -
y.Using Lévy’s identity
in law of thepairs (SB, SB - B)
and(LB,
and thefact that the local time rate of excursions of
height
in modulus at least x isx-1,
it follows that.
dy] = P[Bo - a(y), ST - a(y)]a(dy) a(y) -
y
°
Figure
3: Some of the excursions below5’~
= splotted along
thetangents
to7/1 at
a-1 (s)
Define
1/J(y)
=y] -
We wish to show that - 0. Now+ = E
= (F0[a(y)]
- F1[y]
- 03C8(y))a(dy) a(y) - y - 1(dy)
= -
03C8(y) F0[a(y)] - F1[y]1(dy),
where this last line follows from the
identity
a(dy) - pi (dy)
.~)-2/
-
Fo[a(y)] -
°
It must follow that
~(~/) =
0.3.3 The
Skorokhod Embedding Theorem
The
proof
of the Skorokhodembedding
theoremgiven
in Chacon and Walsh[5]
can be
expressed pictorially
in a similar manner to Sections 2 and 3.2. Oneversion of their
algorithm, (see
inparticular
Dubins[6]),
involves a sequenceof the
following steps: firstly
choose a value x and draw thetangent
to yyi at x;and
secondly
run a Brownian motion until it leaves some interval defined via thistangent (and
thehistory
of the construction todate).
Our construction isa
special
case in which the values x chosen at eachstep
are as small aspossible.
By
the remarks in Section3.1,
when ~,o -do
the functiona(x)
isequivalent
to the
barycentre
functionb1(x)
and theargument
of Section 2 reduces to theAzéma-Yor
proof.
Thus the Azema and Yor[2]
andRogers [14] proofs
of theSkorokhod
embedding
theorem are seen to bespecial
cases of theproof
dueto Chacon and Walsh
[5].
3.4 The
minimum maximum
of amartingale
.A maximal
(respectively minimal)
element of a set of measures S is a measurefor which there does not exist v E S with v ~- p
(respectively v « p).
The results of Section 2 show that 1 is the
unique
maximal element ofN~1).
However it is not ingeneral
true that P has aunique
minimalelement.
It is clear that if M is any
martingale
with the desired initial and ter- minallaws,
and ifM
is themartingale consisting
of asingle jump
such thatMt
=Mo
for0 t
1 andMl,
then the law of the maximum of Mstochastically
dominates that ofM.
Thus thestudy
of minimal elements reduces to astudy
of discreteparameter martingales
at thetimepoints
0 and1.
Suppose
theprobability
measures and /~i are the Uniform measures,
on
{-1,1}
and{-2, o, 2} respectively.
Thejoint
laws of all(single jump)
’
martingales
with these initial and terminal distributions areparameterised by 8, (0
81/12),
in thefollowing
table:Table 1: The
joint
law ofMB
If
SB
denotes the supremum of themartingale MB parameterised by
8then >
x]
= VMf
>x].
Inparticular 0]
=(3/4) -
0 and>
0]
=(1/2)
+ 8. It isimpossible
to minimise both theseexpressions simultaneously.
For thissimple example
there is anon-degenerate family
ofminimal elements of and the
greatest
lower bound of~(~o,
isnot attained. Further /~o V is not an element of
~1).
3.5 The
minimum maximum
of acontinuous martingale
If the
martingales
M are further constrained to becontinuous,
then as Perkins[11]
hasshown,
if the initial law istrivial,
then there is aunique
minimalelement to the set of
possible
laws of the maximum.Specifically,
let have zero mean and letMC
--_ be thespace of all continuous
martingales
which are null at0,
and have terminal law . Let~c(~o,
be the set of laws of the associated maxima. ThenPc
has aunique
minimalelement,
which can berepresented
as a Skorokhod
embedding
of a Brownian motion. See[11]
for further detailsof this construction. -
References
[1] AZÉMA, J., GUNDY,
R.F. ANDYOR, M;
Surl’integrabilité
uniforme desmartingales continues,
Séminaire deProbabilités, XIV, 53-61,
1980.[2] AZÉMA,
J ANDYOR, M;
Une solutionsimple
auproblème
deSkorokhod,
Séminaire de
Probabilités, XIII, 90-115,
1979.[3] BERTOIN,
J. AND LEJAN, Y.; Representation
of measuresby balayage
from a
regular
recurrentpoint,
Annalsof Probability, 20, 538-548,
1992.[4] BLACKWELL,
D. ANDDUBINS, L.D.;
A Converse to the Dominated Con- vergenceTheorem,
Illinois Journalof Mathematics, 7, 508-514,
1963.[5] CHACON,
R. ANDWALSH, J.B.;
One-dimensionalpotential embedding,
Séminaire de
Probabilités, X, 19-23,
1976.[6] DUBINS, L.D.;
On a theorem ofSkorokhod,
Annalsof
MathematicalStatistics, 39, 2094-2097,
1968.[7] DUBINS,
L.D. ANDGILAT, D.;
On the distribution of maxima of mar-tingales, Proceedings of
the American MathematicalSociety, 68, 337-338,
1978.
[8] HOBSON, D.G.;
Robusthedging
of the lookbackoption,
To appear in Finance andStochastics,
1998.[9] KERTZ,
R.P. ANDRÖSLER, U.; Martingales
withgiven
maxima andterminal
distributions,
Israel J.Math., 69, 173-192,
1990.[10] MEYER, P.A.; Probability
andPotentials, Blaidsell, Waltham, Mass.,
1966.
[11] PERKINS, E.;
The Cereteli-Davis solution to theH1-embedding problem
and an
optimal embedding
in Brownianmotion,
Seminar on StochasticProcesses, 1985, Birkhauser, Boston, 173-223,
1986.[12] ROGERS, L.C.G.;
Williams’ characterisation of the Brownian excursion law:proof
andapplications,
Séminaire deProbabilités, XV, 227-250,
1981.
[13] ROGERS, L.C.G.;
Aguided
tourthrough excursions,
Bull. London Math.Soc., 21, 305-341,
1989.[14] ROGERS, L.C.G.;
Thejoint
law of the maximum and the terminal value of amartingale,
Prob. Th. Rel.Fields, 95, 451-466,
1993.[15] RÖSLER, U.;
Personal communication.[16] STRASSEN, V.;
The existence ofprobability
measures withgiven marginals,
Ann. Math.Statist., 36, 423-439,
1965.[17] VALLOIS, P.;
Sur la loi du maximum et dutemps
local d’unemartingale
continue uniformément
intégrable,
Proc. London Math.Soc., 3, 399-427,
1994.