Stochastic orderings with respect to a capacity and an application to a financial optimization problem

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Stochastic orderings with respect to a capacity and an

application to a financial optimization problem

Miryana Grigorova

To cite this version:

Miryana Grigorova. Stochastic orderings with respect to a capacity and an application to a financial

optimization problem. 2011. �hal-00614716�

application to a nancial optimization problem

Miryana Grigorova LPMA, Université Paris 7

First version: July 4, 2010 Present version: April30,2011

Abstract

Inananalogouswaytotheclassicalcaseofaprobabilitymeasure,weextendthenotion ofanincreasingconvex(concave)stochasticdominancerelationtothecaseofanormalised monotone (but not necessarily additive) set functionalso called a capacity. We give dif-ferent characterizations of this relation establishing a link to the notions of distribution function and quantile functionwith respect to the givencapacity. The Choquet integral is extensively usedas a tool. Westate a newversion of theclassical upper(resp. lower) Hardy-Littlewood'sinequalitygeneralized tothe case ofa continuousfrombelowconcave (resp. convex)capacity. We apply ourresults to a nancialoptimization problemwhose constraintsare expressedbymeansof theincreasingconvexstochastic dominancerelation withrespecttoacapacity.

Keywords:stochasticorderings,increasingconvexstochasticdominance,Choquetintegral, quantilefunctionwith respectto acapacity, stop-loss ordering,Choquet expectedutility, distortedcapacity,generalizedHardy-Littlewood'sinequalities,distortionriskmeasure, am-biguity

TheauthorisdeeplygratefultoMarie-ClaireQuenezaswellastotwoanonymousrefereesandanassociate editorfortheirhelpfulsuggestionsandremarks.

1 Introduction

Capacitiesandintegrationwith respectto capacities wereintroduced byChoquetandwere af-terwardsapplied in dierent areassuch as economicsand nance among many others(cf. for instance Wang and Yan 2007 for an overview of applications). In economics and nance, ca-pacitiesandChoquetintegralshavebeenused,inparticular,tobuildalternativetheoriestothe "classical"setting ofexpectedutilitymaximizationof VonNeumannandMorgenstern. Indeed, theclassicalexpectedutilityparadigmhasbeenchallengedbyvariousempiricalexperimentsand "paradoxes"(such as Allais's andEllsberg's) thus leadingto thedevelopmentof newtheories. Oneof theproposed new paradigmsisthe Choquetexpected utility(abridged asCEU) where agent'spreferencesarerepresentedbyacapacity

µ

andanon-decreasingreal-valuedfunction

u

. Theagent's"satisfaction"withaclaim

X

isthereforeassessedbytheChoquetintegralof

u(X)

withrespectto thecapacity

µ

. Choquetexpectedutility intervenesin situations where an ob-jectiveprobabilitymeasureisnotgivenandwheretheagentsarenotabletoderiveasubjective probabilityoverthesetofdierentscenarios.

On the other hand, stochastic orders have also been extensively used in the decision theory. They represent partial order relations on the space of random variables on some probability space

(Ω, F , P)

(more precisely, stochastic orders are partial order relations on the set of the corresponding distributionfunctions). Dierent kindsof orders havebeen studied and applied (seeforinstanceMüllerandStoyan2002andShakedandShanthikumar2006forageneral pre-sentation) and linksto the expected utility theory havebeen exploited. Hereafter,wewill call "classical"theresultsonriskordersinthecaseofrandomvariablesona probability space. In theclassicalsettingofrandomvariablesonaprobabilityspace,therearetwoapproachestorisk orderings-economicorderingbasedonclassesofutilityfunctionsandstatisticalorderingwhich isbasedontaildistributions(cf. theexplanationsin WangandYoung1998). Inthe"classical" caseofaprobabilityspacetheyleadtothesameorderingofrisks. Forthepurposeofthispaper wewillfocusontheincreasingconvexordering(orincreasingconvexstochasticdominance rela-tion). Theeconomicapproachtotheclassicalincreasingconvexstochastic dominanceamounts tothefollowingdenition -

X

issaidto bedominatedby

Y

intheincreasingconvexstochastic dominancerelation(denoted

X

≤

icx

Y

)if

E

(u(X)) ≤ E(u(Y ))

forall

u

: R → R

non-decreasing and convexprovidedthe expectations (taken in theLebesgue sense)exist in

R

. Theeconomic

interpretation is then the following:

X

is dominated by

Y

in the increasing convexstochastic dominanceif

Y

ispreferredto

X

byalldecisionmakerswhoprefermorewealthtolessandwho arerisk-seeking. Thestatistical approach providesan equivalentcharacterization:

X

≤

icx

Y

if and only if

R

+∞

x

P

(X > u)du ≤

R

+∞

x

P

(Y > u)du

,

∀x ∈ R

, provided the integralsexist in

R

. Moreover,we have another characterization which establishes a link between the icx ordering relation andstop-loss premia in reinsurance (cf. Dhaeneet al.2006):

X

≤

icx

Y

if and onlyif

E

_{((X − b)}

+

_{) ≤ E((Y − b)}

+

_{), ∀b ∈ R}

,providedtheexpectationsexist in

R

.

Therstaimofthispaperisto generalizethenotionofincreasingconvexstochasticdominance tothecasewherethemeasurable space

(Ω, F )

isendowedwithagivencapacity

µ

whichisnot necessarilyaprobabilitymeasureandtoinvestigategeneralizationsofthepreviouslymentioned resultsto this setting. Of course, in our case"ordinary"expectations (in theLebesgue sense) have to be replaced by Choquet expectations. We obtainthat analogous kinds of characteri-zations as the previously mentioned in the case of a probability measure remain valid in our moregeneralsettingifweassumethatthecapacity

µ

hascertaincontinuityproperties(namely continuityfrombelowandcontinuityfromabove). Nevertheless,letusremarkthatin allproofs butonetheassumptionofcontinuityfrombelowandfromaboveisnotneeded.

The second aim of this paper is to givean application of the results we obtain to a nancial probleminspiredbyDana(2005)(seealsoDanaandMeilijson2003andthereferencestherein). Theauthorsstudythefollowingproblem:

(

D

˜

)

Minimize

E

(ZC)

undertheconstraints:

C

∈ L

∞

_{(Ω, F , P)}

suchthat

X

≤

icv

C

where theabbreviation

icv

standsfor theincreasingconcaveorderingrelation (withrespect to the probability measure

P

), the symbol

E

denotes Lebesgue expectation (with respect to the probability measure

P

) and where

Z

and

X

are given (see below for more details). Let us just recallthat the increasingconcavestochastic dominancerelationis dened similarlyto the increasingconvexstochasticdominance,theclassofnon-decreasingconvexreal-valuedfunctions inthedenitionbeingreplacedbytheclassofnon-decreasingconcavereal-valuedfunctions. The authorsinterpretthevaluefunction oftheaboveproblem(

D

˜

)asbeingtheminimalexpenditure togetacontingentclaimamongthosewhichdominatethecontingentclaim

X

in theincreasing concave ordering. The value function of the problem is linked to the notion of risk measure

wherewearegivena(continuousfrombelowconcave)capacity

µ

andanon-negativenuméraire

Z

: (D) Maximize

C∈A(X)

E

_µ

_(ZC),

thesymbol

E

µ

denotingtheChoquetintegralwithrespectto

µ

and

A(X)

standingforthe set of allnon-negative, bounded contingentclaims

C

whichprecede agiven non-negativebounded contingentclaim

X

intheincreasingconvexorderingwithrespecttothecapacity

µ

(cf. hereafter for aprecise formulation of theproblem). We caninterpret thefunctional

C

7→ E

µ

(ZC)

asa pricing functional, the measurable function

Z

being interpreted as a discount factor; in fact, theideaofusingChoquetintegralsasnon-linearpricingfunctionals ininsuranceandnanceis notnew(cf. for instancetheoverviewin Wangand Yan 2007andthereferencestherein). The functional

E

µ

(.)

can be seenasa riskmeasure as well (see below for moredetails); non-linear pricing functionals and risk measures have already been connected in the work of Bion-Nadal (2009)and KlöppelandSchweizer(2007).

We also givean interpretation of the valuefunction ofour problem in terms of aclass of risk measures which wecall "generalized"distortion risk measures(as wellasin termsof premium principles in insurance). Inorder to solveproblem (D) , we stateanewversionofthe classical Hardy-Littlewood's upper(resp. lower) inequalitygeneralizedto thecaseofacontinuous from belowconcave(resp. convex)capacity. Then, using thisresult, wecomputethevaluefunction ofproblem(D) .

Therestofthispaperisorganizedasfollows. Insection2wextheterminologyandthenotation bygivingsomewell-knowndenitionsaboutcapacitiesandChoquetintegrals;inparticular,the notionsofcomonotonicmeasurablefunctionsandquantilefunctionwithrespecttoacapacityare recalled. In section3wedenethenotionofincreasingconvex(concave) stochasticdominance with respect to a capacity and explore dierent characterizations analogous to those existing in the classical case of a probability measure. Insection 4westate the "generalized" Hardy-Littlewood's inequalities. In section 5 we formulate and solveour optimization problem (D); in twosubsections weprovideaninterpretation ofthevaluefunction in termsof riskmeasures in nance and premium principles in insurance. Finally, in section 6wepresentdirections for our ongoing and future research concerning some related problems. The Appendix contains three parts: some well-known results about Choquet integration which are used in the paper

arerecalledin AppendixA; AppendixBisdevotedtotheproofsofalemma andaproposition from section 3; the proof of the "generalized" Hardy-Littlewood's inequalities is presented in Appendix C.

2 Notation and denitions

ThedenitionsandresultsofthissectioncanbefoundinthebookbyFöllmerandSchied(2004) (cf. section4.7ofthisreference)and/orin theonebyDenneberg(1994).

Let

(Ω, F )

be a measurable space. Wedenote by

χ

the space of measurable, real-valued and boundedfunctionson

(Ω, F )

.

Denition2.1 Let

(Ω, F )

be a measurable space. A set function

µ

: F −→ [0, 1]

is called a capacity if it satises

µ(∅) = 0

,

µ(Ω) = 1

(normalisation) and the following monotonicity property:

A, B

∈ F, A ⊂ B ⇒ µ(A) ≤ µ(B)

.

We recall the denition of the Choquetintegralwith respect to a capacity

µ

(cf. Denneberg 1994).

Denition2.2 For a measurable real-valued function

X

on

(Ω, F )

, the Choquet integral with respect toacapacity

µ

isdenedasfollows

E

_µ

_{(X) :=}

Z

+∞

0

µ(X > x)dx +

Z

0

−∞

(µ(X > x) − 1)dx.

Note thatthe Choquetintegralin thepreceding denition maynotexist (namely ifoneofthe two(Riemann)integralsontherightsideisequalto

+∞

andtheotherto

−∞

),maybein

R

or maybeequalto

+∞

or

−∞

. TheChoquetintegralalwaysexistsifthefunction

X

isbounded frombeloworfromabove. TheChoquetintegralexistsandisniteif

X

isin

χ

.

Thuswecometothenotionofthe(non-decreasing)distributionfunctionof

X

withrespecttoa capacity

µ

.

Denition2.3 Let

X

be ameasurable function denedon

(Ω, F )

. Wecalladistribution func-tionof

X

with respectto

µ

the non-decreasing function

G

X

denedby

Thenon-decreasingnessof

G

X

indenition 2.3isduetothemonotonicityof

µ

.

In thecase where

µ

is aprobabilitymeasure, thedistribution function

G

X

coincides with the usualdistributionfunction

F

X

of

X

denedby

F

X

(x) := µ(X ≤ x),

∀x ∈ R.

Letusnowdenethegeneralizedinverseofthefunction

G

X

.

Denition2.4 For a measurable real-valued function

X

dened on

(Ω, F )

and for a capacity

µ

,let

G

X

denotethe distributionfunction of

X

withrespectto

µ

. Wecallaquantilefunction of

X

withrespectto

µ

every function

r

X

: (0, 1) −→ ¯

R

verifying

sup{x ∈ R | G

X

(x) < t} ≤ r

X

(t) ≤ sup{x ∈ R | G

X

(x) ≤ t}, ∀t ∈ (0, 1),

wherethe convention

sup{∅} = −∞

isused. The functions

r

l

X

and

r

u

X

denedby

r

l

X

(t) := sup{x ∈ R | G

X

(x) < t}, ∀t ∈ (0, 1)

and

r

u

X

(t) := sup{x ∈ R | G

X

(x) ≤ t}, ∀t ∈ (0, 1)

arecalled thelower andupperquantile functionsof

X

with respectto

µ

.

Fornotationalconvenience,weomitthedependenceon

µ

inthenotation

G

X

and

r

X

whenthere isnoambiguity.

Remark2.1 Let

µ

beacapacityandlet

X

beameasurablereal-valuedfunction suchthat

(2.1)

lim

x→−∞

G

X

(x) = 0

and

lim

x→+∞

G

X

(x) = 1.

Wedenoteby

G

X

(x−)

and

G

X

(x+)

theleft-handandright-handlimitsof

G

X

at

x

. Afunction

r

X

isaquantilefunctionof

X

(withrespectto

µ

)ifandonlyif

(G

X

(r

X

(t)−) ≤ t ≤ G

X

(r

X

(t)+),

∀t ∈ (0, 1).

Inthiscase

r

X

isreal-valued. Notethatthecondition(2.1)issatisedif

X

∈ χ

and

µ

isarbitrary. Thecondition(2.1)issatisedforanarbitrary

X

if

µ

iscontinuousfrombelowandfrom above (seedenition2.5).

Werecallsomewell-knowndenitionsaboutcapacitieswhichwillbeneededlateron.

Denition2.5 Acapacity

µ

iscalledconvex(orequivalently, supermodular) if

Acapacity

µ

iscalledconcave(or submodular,or2-alternating) if

A, B

∈ F ⇒ µ(A ∪ B) + µ(A ∩ B) ≤ µ(A) + µ(B).

Acapacity

µ

iscalledcontinuousfrombelowif

(A

n

) ⊂ F

suchthat

A

n

⊂ A

n+1

,

∀n ∈ N ⇒ lim

n→∞

µ(A

n

) = µ(∪

∞

n=1

A

n

).

Acapacity

µ

iscalledcontinuousfromaboveif

(A

n

) ⊂ F

suchthat

A

n

⊃ A

n+1

,

∀n ∈ N ⇒ lim

n→∞

µ(A

n

) = µ(∩

∞

n=1

A

n

).

Werecallthenotionofcomonotonicfunctions (cf. FöllmerandSchied2004).

Denition2.6 Twomeasurablefunctions

X

and

Y

on

(Ω, F )

arecalledcomonotonic if

(X(ω) − X(ω

0

))(Y (ω) − Y (ω

0

)) ≥ 0, ∀(ω, ω

0

) ∈ Ω × Ω.

For reader's convenience and in order to x the terminology, we summarize someof the main properties ofChoquet integralsin thefollowing propositions (cf. proposition5.1 in Denneberg 1994)andwemaketheconventionthat thepropertiesarevalid provided theexpressionsmake sense(whichisalwaysthecasewhenwerestrainourselvesto elementsin

χ

).

Proposition 2.1 Let

µ

beacapacityon

(Ω, F )

and

X

and

Y

bemeasurablereal-valuedfunctions on

(Ω, F )

,thenwehave theproperties:

•

(positive homogeneity)

E

µ

(λX) = λE

µ

(X), ∀λ ∈ R

+

•

(monotonicity)

X

≤ Y ⇒ E

µ

(X) ≤ E

µ

(Y )

•

(translationinvariance)

E

µ

(X + b) = E

µ

(X) + b, ∀b ∈ R

•

(asymmetry)

E

µ

(−X) = −E

µ

¯

(X)

,where

µ

¯

isthedual capacityof

µ

(

µ(A)

¯

isdenedby

µ(A) = 1 − µ(A

¯

c

_{), ∀A ∈ F)}

•

(comonotonicadditivity) If

X

and

Y

arecomonotonic, then

E

_µ

_{(X + Y ) = E}

_µ

_{(X) + E}

_µ

_{(Y )}

.

Proposition 2.2 Let

µ

beaconcave capacityon

(Ω, F )

and

X

and

Y

bemeasurablereal-valued functions on

(Ω, F )

such that

E

µ

(X) > −∞

and

E

µ

(Y ) > −∞

, then we have the following property

(sub-additivity)

E

µ

(X + Y ) ≤ E

µ

(X) + E

µ

(Y ).

Remark2.2 Werefer the reader to Denneberg (1994) for aslightly weakerassumption than theonegivenin thepreviousproposition.

Remark2.3 Thereadershouldnotbemisledbythevocabularyusedinthepaper. We empha-size thatwhen thecapacity

µ

is concavein thesense ofdenition 2.5,the functional

E

µ

(.)

is a convexfunctionalon

χ

(intheusualsense).

Otherwell-knownresultsaboutChoquetintegrals,quantilefunctionswithrespecttoacapacity andcomonotonicfunctions whichwill beusedinthesequelcanbefoundin theAppendixA.

We end this sectionby twoexamples ofacapacity. Therst exampleis well-knownin the decisiontheory(thinkforinstanceoftherank-dependentexpectedutilitytheory-Quiggin1982 orof theYarii'sdistorted utility theoryin Yaari 1997); thesecond is aslightgeneralizationof therst.

1. Let

µ

be aprobability measure on

(Ω, F )

and let

ψ

: [0, 1] → [0, 1]

be a non-decreasing function on

[0, 1]

such that

ψ(0) = 0

and

ψ(1) = 1

. Thenthe set function

ψ

◦ µ

dened by

ψ

◦ µ(A) := ψ(µ(A)), ∀A ∈ F

is acapacityinthesense ofdenition 2.1. Thefunction

ψ

iscalled adistortionfunction andthecapacity

ψ

◦ µ

iscalledadistorted probability. If

thedistortion function

ψ

isconcave,thecapacity

ψ

◦ µ

is aconcavecapacityin thesense ofdenition 2.5.

2. Let

µ

beacapacityon

(Ω, F )

andlet

ψ

beadistortionfunction. Thenthesetfunction

ψ◦µ

isacapacitywhich,byanalogytothepreviousexample,willbecalledadistortedcapacity. Moreover,we havethe following property : if

µ

is a concavecapacityand

ψ

is concave, then

ψ

◦ µ

isconcave. Theproofusesthesameargumentsastheproofofproposition4.69 in FöllmerandSchied(2004)andisleft tothereader(seealsoexercice2.10inDenneberg 1994).

3 Stochastic orderings with respect to a capacity

Theconcept ofstochastic dominanceis awell-known anduseful conceptin decisiontheory. It consistsofintroducingapartialorderinthespaceofrandomvariablesonsomeprobabilityspace

(Ω, F , P)

(moreprecisely,stochasticdominancerelationsarepartialorderrelationsonthesetof

thecorrespondingdistributionfunctions). Theaimofthissection isto "extend"theconceptof stochasticorderings tothecasewhere theprobability

P

isreplaced bythemoregeneralnotion of capacity; for the purposes of this article, the stress is placed on the generalizations of the results on the increasing convex and the increasing concavestochastic dominance relations to thecaseofacapacity. Asusuallydoneintheclassicalcase,weemphasizethelinksbetweenan economicapproachto stochastic orderingsbased onnumericalrepresentationsof theeconomic agents'preferencesandastatisticalapproachbasedonapointwisecomparisonofthedistribution functions or of some other performance functions constructed from the distribution functions. Ourdenitionsareanalogoustothe"classical"caseofaprobabilitymeasure.

3.1 The increasing convex stochastic dominance with respect to a ca-pacity

µ

Analogously to the "classical" denition of an increasing convex stochastic dominance (with respectto aprobabilitymeasure),wedene thenotionofanincreasing convexstochastic dom-inancerelation (or equivalentlyan increasingconvexordering)with respect to acapacity

µ

as follows:

Denition3.1 Let

X

and

Y

betwomeasurablefunctionson

(Ω, F )

andlet

µ

be acapacity on

(Ω, F )

. We saythat

X

is smaller than

Y

in the increasing convexordering (withrespecttothe

capacity

µ

) denotedby

X

≤

icx

Y

if

E

_µ

_{(u(X)) ≤ E}

_µ

_{(u(Y ))}

for allfunctions

u

: R → R

which arenon-decreasing andconvex, providedthe Choquetintegralsexistin

R

.

This denition coincides with theusual denition of theincreasing convex order whenthe ca-pacity

µ

isaprobabilitymeasureon

(Ω, F )

. (cf. Shakedand Shanthikumar2006for detailsin

Remark3.1 Theeconomic interpretation of the icx ordering with respect to a capacity

µ

is the following:

X

≤

icx

Y

if all the agents whose preferences are described by the (common) capacity

µ

andanon-decreasingconvexutilityfunction prefertheclaim

Y

totheclaim

X

. As explained in Kaaset al. (2001), the"classical"stochastic orderings allow to comparerisks (or nancialpositions) accordingtothe expected utility(EU) paradigm. Thestochastic orderings with respect to a capacity studied here allow to compare nancial positions according to the Choquet expected utility (CEU) theory. The

≤

icx

,µ

relation and the

≤

icv

,µ

relation (dened below)derivefrom theCEUtheoryasthecorresponding"classical"stochastic orderingsderive fromtheEUmodel.

LetusmentionthataneconomicsettingwherealltheagentsareCEU-maximizerscharacterized byacommoncapacity

µ

andanon-decreasingconvex(resp. concave)utilityfunctionhasalready beenconsideredin theliteraturein thestudy ofPareto-optima(cf. Chateauneufetal.2000).

Forthesakeofcompleteness,wedene thenotionofanincreasingconcavestochastic domi-nance(orequivalentlyanincreasingconcaveordering)withrespecttoacapacity

µ

.

Denition3.2 Let

X

and

Y

betwomeasurablefunctionson

(Ω, F )

andlet

µ

be acapacity on

(Ω, F )

. Wesaythat

X

issmaller than

Y

inthe increasing concave ordering(withrespecttothe

capacity

µ

) denotedby

X

≤

icv

Y

if

E

_µ

_{(u(X)) ≤ E}

_µ

_{(u(Y ))}

for allfunctions

u

: R → R

which arenon-decreasing andconcave, providedthe Choquetintegralsexistin

R

.

Remark3.2 Asin theprevioussection, thedependence onthecapacity

µ

in thenotationfor thestochasticdominancerelations

≤

icx and

≤

icv

isintentionallyomitted. Nevertheless,weshall note

≤

icx,µ

and

≤

icx,µ

whenanexplicitmentionofthecapacitytowhichwereferisneeded.

Asintheclassicalcasewherethecapacity

µ

isaprobabilitymeasure,theorderingrelations

≤

icx and

≤

icv

arelinkedtoeachotherin thefollowingmanner:

Proposition 3.1 Let

X

and

Y

be twomeasurable functions. Then

where

µ

¯

denotes the dual capacity ofthe capacity

µ

(thedualcapacity

µ

¯

beingdenedby

µ(A) = 1 − µ(A

¯

c

_{), ∀A ∈ F}

).

Proof: Theproofisbasedonthefactthatafunction

x

7→ u(x)

isnon-decreasingandconvexin

x

ifandonlyifthefunction

x

7→ −u(−x)

isnon-decreasingandconcavein

x

andontheproperty

ofasymmetryoftheChoquetintegral;thedetailsarestraightforward.



Noticethatintheclassicalcasewherethecapacity

µ

isaprobabilitymeasure,thedualcapacity istheprobabilityitselfandsoproposition3.1reducestoawell-knownresultfromthestochastic orderliterature.

Theaimofthefollowingpropositionsistoobtaincharacterizationsofthestochasticdominance relations

≤

icx and

≤

icv

. Duetoproposition3.1,weneedtoconsiderthecaseof

≤

icx

only.

Proposition 3.2 Let

µ

beacapacity. Then wehave thefollowing statements:

(i) If

X

≤

icx

,µ

Y

,then

E

µ

((X − b)

+

_{) ≤ E}

µ

((Y − b)

+

), ∀b ∈ R

, providedthe Choquetintegrals existin

R

.

(ii) If the capacity

µ

has the additional propertiesofcontinuityfrombelow andcontinuityfrom above,thenthe converse implicationholds true,namely,

if

E

µ

((X − b)

+

_{) ≤ E}

µ

((Y − b)

+

), ∀b ∈ R

, provided the Choquet integrals existin

R

,then

X

≤

icx

,µ

Y

.

Proof: Theproofadaptstheproofoftheorem1.5.7. inMüllerandStoyan(2002)toourcase. Theproofofassertion(i)istrivial,thefunction

x

7→ (x − b)

+

beingnon-decreasingandconvex forall

b

∈ R

.

Letus nowprovethe assertion(ii). Let

u

bea non-decreasingandconvexfunction (such that

E

_µ

_(u(X))

existsin

R

and

E

µ

(u(Y ))

existsin

R

). Weconsiderthree cases:

1.

lim

x→−∞

u(x) = 0.

Then it is well-known that

u

can be approximated from below bya sequence

(u

n

)

offunctionsofthefollowingform(cf. MüllerandStoyan2002):

u

n

(x) =

n

X

i=1

where

a

in

≥ 0

and

b

in

∈ R

. Let us now remark that all the functions in the family

(a

in

(X − b

in

)

+

)

i∈{1,··· ,n}

arepairwisecomonotonic(thanksto propositionA.2)andsofor

all

i

∈ {2, · · · , n}

,

a

in

(X −b

in

)

+

iscomonotonicwith

P

i−1

j=1

a

jn

(X −b

jn

)

+

. Hence,usingthe additivityoftheChoquetintegralwithrespectto comonotonicfunctions andthepositive homogeneityoftheChoquetintegral,weobtain

E

_µ

_(u

_n

_{(X)) =}

n

X

i=1

a

in

E

µ

[(X − b

in

)

+

] ≤

n

X

i=1

a

in

E

µ

[(Y − b

in

)

+

] = E

µ

(u

n

(Y )).

Thecapacity

µ

beingcontinuousfrombelow,weapplythemonotoneconvergencetheorem as stated in theorem A.1 in order to pass to the limit in the previous inequality and to obtain

E

_µ

_{(u(X)) ≤ E}

_µ

_{(u(Y )).}

2. Thecasewhen

lim

x→−∞

u(x) = a ∈ R

canbereducedto thepreviousonebyconsidering the function

x

7→ u(x) − a

. Thus, we obtain that

E

µ

(u(X) − a) ≤ E

µ

(u(Y ) − a)

and concludethanksto thetranslationinvarianceoftheChoquetintegral.

3. If

lim

x→−∞

u(x) = −∞

, then thefunction

u

n

(x) := max(u(x), −n)

fulls theconditions ofthesecond caseforany

n

∈ N

(indeed,

u

n

isnon-decreasing,convexandbounded from below),so

E

_µ

_(u

_n

_{(X)) ≤ E}

_µ

_(u

_n

_{(Y )), ∀n ∈ N.}

Moreover,

u

n

decreasesto

u

,or equivalently,

−u

n

increasesto

−u

. A naturalideaisthen to apply themonotoneconvergence theoremin order to passto thelimitin the previous inequalitybyusingtheasymmetryoftheChoquetintegralandbyobservingthatthe con-tinuityfromaboveof

µ

isequivalenttothecontinuityfrombelowof

µ

¯

.

Note that the previousreasoning is rigourous in the case where werestrain ourselves to functions in

χ

. Indeed, if

X

is afunction in

χ

(hence

X

is bounded), then

(u

n

(X))

is a bounded sequence (in fact, it is easily seen that

max u(sup X), 0
≥ u

n

(X) ≥ u(inf X)

for all

n

where

inf X

and

sup X

denote the lower and upper bound of

X

respectively). Therefore,themonotoneconvergencetheoremasstatedintheoremA.1combinedwiththe translationinvarianceoftheChoquetintegralallowsustoconclude.

be-by thesameargumentsasin the proofof theorem 8.1 in Denneberg (1994)(see alsothe proof oflemma C.1in theappendix),weseethat ifasequenceofreal-valuedmeasurable functions

(Z

n

)

convergesfrombelowtoareal-valuedfunction

Z

(denotedby

Z

n

↑ Z

)andif acapacity

ν

iscontinuousfrombelow,thenthesequenceofdistributionfunctions

(G

Z

n

,ν

)

(with respectto

ν

)convergesmonotonelyto thedistribution function

G

Z,ν

of

Z

. So,the correspondingsequenceofquantilefunctions

(r

Z

n

,ν

)

convergesfrombelowto(the)quantile function

r

Z,ν

of

Z

almosteverywhere(seetheproofoflemmaC.2in theappendixforthe same argument). Therefore, we canuse proposition A.1 and the dominatedconvergence theoremforLebesgueintegrals(withrespecttotheLebesguemeasureon[0,1])inorderto conclude. Indeed,itsucestoset

Z

n

:= −u

n

(X)

,

Z

:= −u(X)

and

ν

:= ¯

µ

andtoobserve that

r

Z

0

,ν

≤ r

Z

n

,ν

≤ r

Z,ν

almost everywhere and that the functions

r

Z,ν

and

r

Z

0

,ν

are integrablewithrespecttotheLebesguemeasurebyassumption.

Wegivesomedetailsconcerningtheintegrabilityof

r

Z,ν

and

r

Z

₀

,ν

forreader'sconvenience. Being anon-decreasingfunction on

(0, 1)

, thefunction

r

Z,ν

isintegrablein theLebesgue sense ifandonlyifitsgeneralizedRiemannintegralexistsandisnite. Thus,the integra-bilityof

r

Z,ν

isdueto theequation

Z

1

0

r

Z,ν

dt

=

Z

1

0

r

Z,¯

µ

dt

= E

µ

¯

(Z) = E

µ

¯

(−u(X)) = −E

µ

(u(X)),

where the term

E

µ

(u(X))

belongs to

R

by assumption. The integrability of

r

Z

0

,ν

is a consequenceofthatof

r

Z,ν

andcanbeprovedbymeansofsimilarcalculations.



Observe that in the classical case where

µ

is a probability measure the previous proposition reduces to a well-known characterization of the increasing convex order; it allows to link the increasingconvexordertothenotionofastop-losspremiuminreinsurance. Accordingly,inthe classicalcase,theincreasingconvexorderissometimescalled stop-lossorder.

Letusnowestablishalinkbetweentheincreasingconvexstochasticdominancewithrespectto acapacity

µ

andthedistributionfunction withrespecttothecapacity

µ

.

Proposition 3.3 Let

µ

be a capacity and let

X

and

Y

be two measurable functions. The fol-lowing twostatements areequivalent:

(i)

E

µ

((X − b)

+

_{) ≤ E}

µ

((Y − b)

+

), ∀b ∈ R

, providedthe Choquetintegrals existin

R

.

(ii)

R

+∞

x

µ(X > u)du ≤

R

+∞

x

µ(Y > u)du

,

∀x ∈ R

, providedthe integralsexistin

R

.

Proof: Using thedenition of theChoquet integralandachangeof variables, we haveforall

b

∈ R

,

E

_µ

_{((X − b)}

+

_{) =}

Z

+∞

0

µ((X − b)

+

_{> u)du =}

Z

+∞

0

µ(X > u + b)du =

Z

+∞

b

µ(X > u)du

which provesthedesiredresult.



Now,wearereadytorelatethepreviousresultstothenotionofaquantilefunctionwithrespect to

µ

. WereferthereadertoShakedandShanthikumar(2006)foraproofofthefollowingresult in the classical case of a probability measure and to Ogryczak and Ruszczynski (2001) for a dierentproof ofthesameresultbasedonconvexduality; seealsolemma A.22 inFöllmerand Schied(2004). Ourproofisinspiredbythelasttworeferences.

Proposition 3.4 Let

µ

bea capacity andlet

X

and

Y

betworeal-valuedmeasurablefunctions such that

R

1

0

|r

X

(t)|dt < +∞

and

R

1

0

|r

Y

(t)|dt < +∞

where

r

X

and

r

Y

denote (the) quantile functionsof

X

and

Y

withrespectto

µ

. The following twostatementsareequivalent:

(i)

G

(2)

X

(x) :=

R

+∞

x

µ(X > u)du ≤

R

+∞

x

µ(Y > u)du =: G

(2)

Y

(x)

,

∀x ∈ R

. (ii)

R

1

y

r

X

(t)dt ≤

R

1

y

r

Y

(t)dt, ∀y ∈ [0, 1]

.

Inorderto provethepropositionweneedthefollowinglemmawhichcorrespondsto lemma A.22in FöllmerandSchied(2004)in theclassicalcase.

Lemma3.1 Let

µ

be acapacity on

(Ω, F )

andlet

X

bea measurable function on

(Ω, F )

such that the quantile function

r

X

of

X

with respect to

µ

is integrable (with respectto the Lebesgue measureon

[0, 1]

). Denethe function

G

(2)

X

by

G

(2)

_X

(x) :=

Z

+∞

x

µ(X > u)du =

Z

+∞

x

(1 − G

X

(u))du, x ∈ R.

Then the conjugatefunction of

G

(2)

X

isgiven by

r

(2)

_X

(y) := sup

x

∈R

(xy − G

(2)

_X

(x)) =











−

R

1

y+1

r

X

(t)dt,

if

y

∈ [−1, 0]

+∞,

otherwise

.

Proof of the lemma: TheargumentsoftheproofbeingalmostthesameasthoseofFöllmer andSchied (2004),theproofisplacedintheAppendixB.

Wearereadytoproveproposition3.4. Proofof proposition 3.4:

Theproofisbasedonlemma3.1. Suppose that(i)holdstruei.e.

G

(2)

X

(x) ≤ G

(2)

Y

(x), ∀x ∈ R

. Thenforall

y

∈ R

,

r

(2)

_X

(y) := sup

x∈R

(xy − G

(2)

_X

(x)) ≥ sup

x∈R

(xy − G

(2)

_Y

(x)) =: r

(2)

_Y

(y),

which implies,inparticular,that

−

R

1

y+1

r

X

(t)dt ≥ −

R

1

y+1

r

Y

(t)dt,

forall

y

∈ [−1, 0],

or equiva-lently,

Z

1

y

r

X

(t)dt ≤

Z

1

y

r

Y

(t)dt,

forall

y

∈ [0, 1].

Theconverseimplicationcanbeobtainedbymeansof asimilarargumentafter observing that the function

G

(2)

X

is the conjugatefunction of

r

(2)

X

. Indeed, this followsfrom the fact that the function

G

(2)

X

isconvex,properandlower-semicontinuous(cf. theorem24.2inRockafellar1972) andfromthebidualitytheorem(cf. theorem12.2inRockafellar1972).



Weconcludethissectionbyestablishinganotherusefulcharacterizationof therelation

≤

icx which willbeneededinthesequel. Itsanalogueintheclassicalcaseofaprobabilitymeasureis due toDana (2005)(seealso thm. 5.2.1in Dhaeneet al. 2006forarelated result). Ourproof followstheproofoftheformer.

Proposition 3.5 Let

X

∈ χ

and

Y

∈ χ

begiven. Thenthe following statementsareequivalent:

(i)

R

1

y

r

X

(t)dt ≥

R

1

y

r

Y

(t)dt, ∀y ∈ [0, 1]

(ii)

R

1

0

g(t)r

X

(t)dt ≥

R

1

0

g(t)r

Y

(t)dt, ∀g : [0, 1] → R

+

,integrable,non-decreasing.

Remark3.3 Aneconomic interpretationofthe

≤

icx

,µ

−

relation intermsof "uniform" prefer-ences is given in remark 3.1; the interpretation is basedon theinitial denition of the

≤

icx

,µ

-relation(denition3.1).

An interpretationofthe

≤

icx

,µ

−

relation intermsofambiguityissuggestedbytheequivalence establishedin proposition3.3. Indeed,letus rstconsider theinequality

µ(X > t) ≥ µ(Y > t)

where

t

∈ R

is xed. Bearing in mind that the capacity

µ

models the agent's perception of "uncertain"(or"ambiguous")events,thereadermayinterpretthepreviousinequalityashaving thefollowingmeaning: theevent

{X > t}

isperceivedbytheagentasbeinglessuncertainthan orequallyuncertaintotheevent

{Y > t}

. Then,part(ii)inproposition3.3maybelooselyread asfollows: theagent"feelslessorequallyuncertainaboutthenancialposition

X

'stakinggreat valuesonaveragethanthenancialposition

Y

's".

4 A usefultool: thegeneralized Hardy-Littlewood's

inequal-ities

Inthissectionwestateausefulresultwhichcanbeseenasa"generalization"ofthewell-known Hardy-Littlewood'sinequalitiesto thepresentsetting.

Forthestatementand theproof ofthis resultin theclassicalcaseofaprobabilitymeasure we refertotheoremA.24inFöllmerandSchied(2004);someapplicationsofthe"classical" Hardy-Littlewood's inequalities to nance can befound in the samereference. Other applications of the"classical" versionto economics and nancecanbefound in Carlier and Dana(2006); see also Carlier and Dana (2005) (and references therein) where a supermodular extension of the "classical"inequalitiesisusedininsurance.

Thegeneralizationthatwestateinthissectionwillbeneededwhiledealingwiththeoptimization problemofthefollowingsection. Thisgeneralizedversionprovestobeusefulinourongoingwork concerningsomestaticoptimizationproblemsrelatedtotheCEUtheory(cf. Grigorova2010).

Proposition 4.1 (Hardy-Littlewood'sinequalities) Let

µ

be acapacity on

(Ω, F )

. Let

X

and

Y

be two non-negative measurable functions with quantile functions (with respect to the capacity

µ

) denotedby

r

X

and

r

Y

.

1. If

µ

isconcaveandcontinuous frombelow,then

E

_µ

_{(XY ) ≤}

Z

1

0

r

X

(t)r

Y

(t)dt.

2. If

µ

isconvexandcontinuousfrombelow,then

E

_µ

_{(XY ) ≥}

Z

1

0

r

X

(1 − t)r

Y

(t)dt.

Proof: Astheproofofthisresultisrelativelylong,itis placedintheAppendix C.

5 Application to a nancial optimization problem

Thissectionisdevoted tothefollowingoptimizationproblem:

(D)

Maximize

E

µ

(ZC)

undertheconstraints

C

∈ χ

+

s.t.

C

≤

icx

,µ

X

where

χ

+

denotestheset ofnon-negativebounded measurable functions,

µ

isagivencapacity,

Z

isagivenfunctionin

χ

+

and

X

isagivenfunctionin

χ

+

.

Thestudyofthisproblemhasbeeninspiredbythework ofDana(2005)intheclassicalcaseof aprobabilitymeasure(seeDana2005andreferencestherein;seealsoDanaandMeilijson2003, JouiniandKallal2001andDybvig1987). Dana(2005)considersasimilarproblemtothestated above,namely,

(

D

˜

)

Minimize

E

(ZC)

under theconstraints

C

∈ L

∞

_(P)

s.t.

X

≤

icv

C

where

E

denotes expectation with respect to

P

and

≤

icv

denotes the increasing concaveorder relationin theclassicalsense. Theproblem(

D

˜

)hasthefollowingeconomicinterpretation: the measurable function

Z

being interpretedasapricingkernel(in thecasewhere

E

(Z) = 1

),the problemistondthecontingentclaim

C

withminimalpriceamongallcontingentclaimswhich dominatetheclaim

X

inthe increasingconcaveorder,orequivalently, among alltheclaims

C

whicharepreferredto

X

byalltheinvestorswhosepreferencesaredescribedbyanon-decreasing andconcaveutilityfunction.

the agents perceive ambiguity in the same manner i.e. through the same capacity

µ

. In the casewhere thecapacity

µ

isconcave(which willbethecaselater on),theobjectivefunctional

C

7→ E

µ

(ZC)

canbeinterpretedasanon-additivepricing functional which hasthe properties

of monotonicity and convexity and the non-negative measurable function

Z

can be seen as a discount factor or, more generally, a "change of numéraire". A pricing rule of this form (in the case

Z

≡ 1

) is used in Chateauneuf et al. (2000) in order to model the selling price of a claim (its buying price being modelled by

E

µ

¯

(.)

). Thus, the problem (D) consists in nding a contingent claim

C

having the maximal price among all the non-negativecontingentclaims which are dominatedby

X

in theincreasing convexstochastic dominance(with respectto the capacity

µ

).

AdoptingtheterminologyintroducedbyJouiniandKallal(2001),wemaycallthevaluefunction

e(X, Z)

ofproblem(D)(when

Z

isxed)the"utilityprice"of

X

inthecontextofambiguity. It

willbeshownin subsection5.2that,foraxed

Z

,theutilitypriceinthecontext ofambiguity

e( . , Z)

is the smallestfunctional on

χ

+

among those which are consistentwith respect to the

≤

icx

,µ

−

relationandwhicharegreaterthanorequaltothe"marketprice"

E

µ

(Z . )

. Wehavethefollowingtheoremwhichistheanalogueoftheorem2.1in Dana(2005).

Theorem5.1 Let

µ

beaconcave andcontinuousfrombelow capacity. Foreveryfunction

X

∈

χ

+

andfor every function

Z

∈ χ

+

suchthat thedistributionfunction

G

Z

of

Z

with respectto

µ

iscontinuous, the problem

(D)

Maximize

E

µ

(ZC)

under the constraints

C

∈ χ

+

s.t.

C

≤

icx

,µ

X

has asolutionanditsvalue function

e(X, Z)

isgiven by:

e(X, Z) =

Z

1

0

r

Z

(t)r

X

(t)dt.

Proof: Wehave

e(X, Z) =

sup

0≤C,C≤

icx

,µ

X

E

_µ

_{(ZC) ≤}

_sup

0≤C,C≤

icx

,µ

X

Z

1

0

r

Z

(t)r

C

(t)dt

≤

Z

1

0

r

Z

(t)r

X

(t)dt

wherethe rstinequalityis dueto theupperboundin Hardy-Littlewood'sinequalities (propo-sition4.1),thesecondinequalityisaconsequenceofproposition3.5(applied with

g

= r

Z

).

Thusweobtainthat

e(X, Z) ≤

R

1

0

r

Z

(t)r

X

(t)dt

. Toconcludeweneedtond

C

∈ χ

+

such that

C

≤

icx

,µ

X

andsuch that

E

µ

(ZC) =

R

1

0

r

Z

(t)r

X

(t)dt

.

Set

f

(x) := r

X

(G

Z

(x))

,then

C

:= f (Z)

isaswanted. Indeed,

C

≥ 0

. Moreover,

E

_µ

_{(ZC) = E}

_µ

_{(Zf (Z)) = E}

_µ

_{(h(Z)) =}

Z

1

0

r

h(Z)

(t)dt

where we have used proposition A.1 in the last equality and where

h

: R

+

→ R

+

is dened by

h(z) := zf (z), ∀z ≥ 0

. The function

h

being non-decreasing and the function

G

Z

being continuousbyassumption,wecanapplylemmaA.1toobtain

(5.1)

E

_µ

_{(ZC) =}

Z

1

0

h(r

Z

(t))dt =

Z

1

0

r

Z

(t)f (r

Z

(t))dt

=

Z

1

0

r

Z

(t)r

X

(G

Z

(r

Z

(t)))dt =

Z

1

0

r

Z

(t)r

X

(t)dt.

wherewehaveusedthecontinuityof

G

Z

in thelaststep. Weareleftwithestablishingthat

f

(Z) ≤

icx

,µ

X

. Wewillcheckthispropertyusingthedenition of

≤

icx

,µ

. Let

u

beanon-decreasing,convexfunction. Wehave

(5.2)

E

µ

(u(f (Z))) =

Z

1

0

r

u(f (Z))

(t)dt =

Z

1

0

u(f (r

Z

(t)))dt

wherethesecondequalityfollowsfromlemmaA.1(the function

u

◦ f

beingnon-decreasingand thefunction

G

Z

beingcontinuousbyassumption). Thisgives

(5.3)

E

_µ

_{(u(f (Z))) =}

Z

1

0

u(r

X

(G

Z

(r

Z

(t))))dt =

Z

1

0

u(r

X

(t))dt

=

Z

1

0

r

u(X)

(t)dt = E

µ

(u(X))

where the last but one equality is obtained thanks to lemma A.1 after observing that

u

is a continuousfunctionasareal-valuedconvexfunctionon

R

.

Thisconcludestheproof.



Remark5.1 The previous proof can be extended to the case where the assumption of the boundedness from aboveof

Z

is replaced by the weaker assumption that

R

1

0

|r

Z

(t)|dt < +∞

. Thisis duemainly toproposition3.5where onlythenon-negativityand theintegrabilityof

r

Z

are required. We haveneverthelesschosento present theprevious resultin the casewhere all thefunctionsarein

χ

.

Intheclassicalcasewhere

µ

isaprobabilitymeasuretheresultoftheorem5.1stillholdseven whenthecontinuityassumptiononthedistributionfunction

G

Z

of

Z

isrelaxed. Moreprecisely, wehavethefollowingresult:

Proposition 5.1 Let

µ

beaprobability measureon

(Ω, F )

. Foreveryfunction

X

∈ χ

+

andfor every function

Z

∈ χ

+

,the problem

Maximize

E

(ZC)

underthe constraints

C

∈ χ

+

s.t.

C

≤

icx

X

has asolutionanditsvalue functionisgiven by

R

1

0

r

Z

(t)r

X

(t)dt

. Thesymbol

E

denotesthe(classical) expectationwithrespectto

µ

and

≤

icx

denotesthe(classical) increasing convex stochastic dominancerelationwith respectto

µ

.

Proof: Wesketchtheprooffollowingtheproofoftheorem5.1andstressingonlyonthechanges tobemadeintheproofoftheorem5.1. NotethatapplyinglemmaA.1isstillpossiblewhenever needed in this case (even without the continuity assumption on

G

Z

) thanks to remark A.1. Nevertheless, thecontinuityof

G

Z

beingusedto obtainthelastequalityinequation (5.1),the function

f

in the proof of theorem 5.1 is now replaced by the function

f

˜

dened by

f

˜

(x) :=

r

X

(G

Z

(x))

if

x

is acontinuity point of

G

Z

and by

f

˜

(x) :=

1

G

Z

(x)−G

Z

(x−)

R

G

Z

(x)

G

Z

(x−)

r

X

(t)dt

if

x

is not a continuity point of

G

Z

. The function

f

˜

is non-decreasing and satises the property

˜

f

(r

Z

) = E

λ

(r

X

|r

Z

)

where the symbol

E

λ

(.|.)

denotes the conditional expectation with respect

totheLebesguemeasure

λ

.

Weset

˜h(x) := x ˜

f

(x)

andwereplaceequation(5.1)bythefollowing

E

_{(ZC) =}

Z

1

0

˜

h(r

Z

(t))dt =

Z

1

0

r

Z

(t) ˜

f

(r

Z

(t))dt

=

Z

1

0

r

Z

(t)r

X

(t)dt.

wherelemma A.1and remarkA.1areusedtoobtaintherstequalityandthecharacterization oftheconditionalexpectationisused toobtainthelast.

Equation (5.2) remains unchanged, the function

f

being replaced by the function

f

˜

; we have againappliedlemmaA.1andremarkA.1toobtainit.

Equation(5.3)hastobereplacedby

E

_{(u(f (Z))) =}

Z

1

0

u( ˜

f

(r

Z

(t)))dt ≤

Z

1

0

u(r

X

(t))dt

=

Z

1

0

r

u(X)

(t)dt = E(u(X))

wherewehaveappliedJensen'sinequality.



Remark5.2 Notethatinthecasewheretheunderlyingprobabilityspace

(Ω, F , µ)

isatomless theuseoflemmaA.1(andremarkA.1)in thepreviousproofcanbereplacedbytheuseofthe following twousual arguments: the law invarianceof the functional

E

(l(.)) : χ

+

→ R

+

where

l

: R

+

→ R

+

isameasurable function andthefact that thelawof

Z

is thesameasthe lawof

r

Z

(U )

where

U

denotesauniformrandomvariableon

(0, 1)

. Then,theaboveproofisalmostthe

sameastheproofoftheorem2.1inDana(2005)(seealsoDanaandMeilijson2003andFöllmer and Schied 2004). Wenote that the useoflemma A.1 in theproof ofproposition5.1 provides analternativeargumentto the"law-invarianceargument"evenbeyondthenonatomiccase.

Remark5.3 Letusmention that,thanks toremark A.1,thecontinuityassumption on

G

Z

in theorem5.1mayberelaxedinthecaseofacapacity

µ

which,apartfromthepropertiesrequired intheorem 5.1,hastheadditionalpropertyofcontinuityfrom above.

Letusfurthernotethatforaconcavecapacity

µ

(whichisthecasein theorem5.1)theproperty ofcontinuityfromaboveof

µ

impliesthepropertyofcontinuityfrom below.

5.1 The value function of problem (D) as a risk measure

While studying the problem (

D

˜

) in the classical setting, Dana (2004) gives an interpretation of itsvaluefunction in terms ofriskmeasures. An analogous commentary canbe madein the presentsetting.

Consider the value function

e(., Z)

of problem (D) for a xed

Z

as a functional of the rst argumentandextendittothewholeset

χ

. Moreprecisely,letusconsiderthefunctional

ρ

: χ →

R

dened by

ρ(X) := e(X, Z) :=

R

1

0

r

Z

(t)r

X

(t)dt

where

Z

is a xed non-negativemeasurable function in

χ

. Fortheeasingofthepresentation,wewillassumeintherestofthis sectionthat

Z

issuchthat

R

1

positivehomogeneityof the objectivefunctional of problem (D) , wemayaswellreplace

Z

by

Z

R

1

0

r

Z,µ

(t)dt

(inthecasewhere

R

1

0

r

Z,µ

(t)dt 6= 0

)intheformulationofproblem(D).

Thefunctional

ρ

ismonotone(

X

≤ Y

implies

ρ(X) ≤ ρ(Y )

)andtranslationinvariant(

ρ(X +b) =

ρ(X) + b, ∀b ∈ R

). Therefore, accordingto the denition given in Artzner et al. (1999),up to

aminussign

ρ

is amonetary measureof risk on

χ

(see alsoWang and Yan(2007)orEkeland et al.(2009)forthesame"sign convention"astheoneusedin thepresentpaper). Moreover,

ρ

isadditivewithrespectto comonotonicelementsof

χ

; thispropertyisdue to thecomonotonic additivityof the quantilefunction with respect to acapacity. Monetary risk measures having thepropertyofcomonotonic additivityhavealreadybeenstudiedin theliterature(cf. Föllmer and Schied 2004), the idea being that when

X

and

Y

are comonotonic,

X

cannot act as a hedge against

Y

. The riskmeasure

ρ

hastheadditional property ofbeingconsistentwith the increasing convex orderingrelation

≤

icx

,µ

which meansthat if

X

≤

icx

,µ

Y

then

ρ(X) ≤ ρ(Y )

. Thisconsistencypropertyiseasily obtainedthankstoproposition3.5 whenobservingthat the function

r

Z

which stands in the place ofthe function

g

of proposition 3.5 is non-negativeand integrable.

Furthermore, the risk measure

ρ

can be represented as a Choquet integral with respect to a certain capacity. Indeed, according to a well-known representation result for monotone and comonotoniclyadditivefunctionalson

χ

(cf. thm. 4.82. inFöllmerandSchied2004orDenneberg 1994)weknowthatthereexistsacapacity

ν

on

(Ω, F )

suchthat

ρ(X) = E

ν

(X),

forall

X

∈ χ.

Thecapacity

ν

isrelatedtotheinitialcapacity

µ

inthefollowingmanner

ν(A) = ρ(I

A

) = e(I

A

, Z) =

Z

1

0

r

Z,µ

(t)r

I

A

,µ

(t)dt =

Z

1

1−µ(A)

r

Z,µ

(t)dt, ∀A ∈ F.

Therefore,thecapacity

ν

isoftheform:

ν(A) = ψ(µ(A)), ∀A ∈ F

where

ψ(x) :=

R

1

1−x

r

Z,µ

(t)dt, ∀x ∈

[0, 1].

We verify that thefunction

ψ

is adistortionfunction inthe senseof thedenition given

in section 2; hence, the capacity

ν

= ψ ◦ µ

is a distorted capacity. Moreover, the distortion function

ψ

beingconcave,

ν

isaconcavecapacity. Thus, thefunctional

ρ

canberepresentedas aChoquetintegralwithrespecttotheconcavedistortedcapacity

ψ

◦ µ

;hence,

ρ

isapositively homogeneous,convexmonetarymeasure of risk(or equivalently,acoherentmonetary measure

Infact,riskmeasuresoftheform

E

ψ◦µ

(.)

where

µ

isaprobabilitymeasureand

ψ

isa(concave) distortionfunctionhavebeenstudiedbyWangetal.(1997)andDenneberg(1990)andarenow known under the name of distortion risk measures or distortion premium principles (see, for instance, Dhaene et al.2006 forasurveyand examples). Atthe end ofhis article, Denneberg (1990)suggestspossiblegeneralizationstothecasewheretheprobabilitymeasureisreplacedby amoregeneralsetfunction-thefunctional

ρ

that weobtaincouldbeseenassucha generaliza-tion. Adoptingthispointofview,wecouldcall

ρ

a"generalized"distortion riskmeasure. Letusnallyremarkthat thevaluefunction ofproblem(D)canbeseenalsoasananaloguein the setting ofambiguity of thenotion of maximalcorrelation risk measure (cf. Ekeland et al. 2009andthereferencestherein).

5.2 The value function of problem (D) as a premium principle

We giveanother interpretation of the value function of our problem (D) in terms of premium principlesin insurance.

Consideraninsurancecompanywhich usesagivenpremium principleasareferencebutwhich isnowwillingtotakeintoaccountothercriteriaof"riskiness"modelledthroughthestochastic dominancerelation

≤

icx

,µ

. Ininsurance,elementsof

χ

+

areusuallyconsideredaspaymentswhich thecompanyhastomake(orlossesithastoface)andpremiumprinciplesarefunctionalson

χ

+

takingvaluesin

R

;thesefunctionalsareusuallynon-decreasing.Inthisframework,theobjective functional ofproblem (D) , namelythefunctional

ρ

0

: χ

+

7→ R

+

dened by

ρ

0

(X) := E

µ

(ZX)

, can be seen as the reference premium principle used by the company. We remark that the premium principle

ρ

0

may be perceived as akind of a generalizationof the Esscher premium principlewhichiswell-knownininsurance(seeforinstanceYoung2004foradenitionandother examples). In this context, the value function

e(., Z)

of problem (D) is interpreted asa new premium principlewhich has(among other "desirable"properties) theproperty ofconsistency withrespectto therelation

≤

icx

,µ

andwhich isgreaterthanorequaltothereferencepremium principle

ρ

0

(i.e.

e(X, Z) ≥ E

µ

(ZX), ∀X ∈ χ

+

). The latter property is due to the fact that

e(., Z)

isthevaluefunctionofproblem (D)andtothereexivityoftherelation

≤

icx

,µ

. Moreover,wehavethefollowingproperty:

which satisesthe property ofconsistencywithrespecttotherelation

≤

icx

,µ

andwhichisgreater thanorequal to

ρ

0

where

ρ

0

isgiven by

ρ

0

(X) := E

µ

(ZX), ∀X ∈ χ

+

.

Proof: Let

F

: χ

+

7→ R

be afunctional which is consistent with

≤

icx

,µ

and which is greater thanorequalto

ρ

0

. Forall

X

∈ χ

+

and forall

C

∈ χ

+

suchthat

C

≤

icx

,µ

X

,thepropertyof consistencywith respect to therelation

≤

icx

,µ

impliesthat

F

(X) ≥ F (C)

. Moreover,

F

(C) ≥

E

_µ

_(ZC).

So,bytaking thesupremumovertheset

{C ∈ χ

+

s.t.

C

≤

icx

,µ

X

}

,wehave

F(X) ≥

e(X, Z)

.



We concludethat the valuefunction

e(., Z)

of problem (D) is the smallestpremium principle amongthosewhichareconsistentwithrespecttotheincreasingconvexdominancerelation

≤

icx

,µ

andwhich aregreaterthanorequaltotheinitialpremiumprinciple

ρ

0

.

Thanks to the above considerationsthe insurance company may use problem (D) asa way of dening anew premium principle

e(., Z)

on

χ

+

(which premium principleinduces atotal pre-order on

χ

+

unlikethe stochasticdominance relation

≤

icx

,µ

which is onlyapartial pre-order). Loosely speaking, the newly obtained premium principle is "richer" than the initial premium principle

ρ

0

because other criteria of "riskiness" and the "change of numéraire"

Z

have been takenintoaccountthroughproblem(D) .

6 Future perspectives

Asseenin theprevioussection,acloselyrelatedquestiontotheconceptsstudied inthisarticle is the problem of risk measures respecting stochastic dominance relations. We are studying, in particular, thequestionofquantile-based riskmeasures withrespectto agivencapacityi.e. risk measures based on the quantilefunction

r

X,µ

where

µ

is agiven capacityand where the measurablefunction

X

modelsanancialposition,andweareexploringtheirconsistencywith respecttothestochasticdominancerelationsdenedabove(cf. Grigorova2010).

A Appendix: Some basic results about capacities and

Cho-quet integrals

The results of this appendix A can be found in the book by Föllmer and Schied (2004) (cf. section 4.7 of this reference)and/or in the oneby Denneberg (1994)and are recalled herefor reader'sconvenience.

A.1 Choquet integrals and quantile functions

Wehavethefollowingwell-knownresultwherewemaketheconventionthattheassertionisvalid providedtheexpressionsmakesense. Theresultcanbefoundin FöllmerandSchied(2004)for thebounded caseordeduced fromDenneberg(1994)(cf. pages61-62in chapter5ofthelatter reference).

Proposition A.1 Let

X

beareal-valuedmeasurablefunction andlet

r

X

beaquantile function of

X

with respecttoacapacity

µ

,then

E

_µ

_{(X) =}

Z

1

0

r

X

(t)dt.

Thefollowinglemmaistheanalogueoflemma A.23. in FöllmerandSchied(2004)and can befoundin Denneberg(1994).

LemmaA.1 Let

X

= f (Y )

where

f

isanon-decreasing functionandlet

r

Y

beaquantile func-tionof

Y

with respecttoacapacity

µ

. Supposethat

f

and

G

Y

havenocommondiscontinuities, then

f

◦ r

Y

isaquantilefunction of

X

with respectto

µ

. Inparticular,

r

X

(t) = r

f(Y )

(t) = f (r

Y

(t))

for almost every

t

∈ (0, 1),

where

r

X

denotesaquantile function of

X

with respectto

µ

.

RemarkA.1 Ifthecapacity

µ

satisestheadditionalpropertiesofcontinuityfrombelowand from above, the assumption of no common discontinuities of the functions

f

and

G

Y

can be droppedinthepreviouslemma. Theproofisthenanalogoustotheproofintheclassicalcaseof aprobability measure(cf. lemmaA.23. inFöllmerand Schied 2004foraproof inthe classical

A.2 A monotone convergence theorem for Choquet integrals

We recall a monotone convergence theorem for Choquet integrals with respect to a capacity which is continuous from below; we refer the reader to Denneberg (1994) for a proof of this result.

TheoremA.1 (monotoneconvergence) Let

µ

be acapacity on

(Ω, F )

which is continuous frombelow. Foranon-decreasingsequence

(X

n

)

ofnon-negativemeasurable functions,the limit function

X

:= lim

n→∞

X

n

ismeasurable and

lim

n→∞

E

µ

(X

n

) = E

µ

(X).

A.3 Comonotonic functions

Wehavethefollowingcharacterizationofcomonotonic functions which correspondsto proposi-tion4.5inDenneberg(1994)(seealso FöllmerandSchied2004)

Proposition A.2 For tworeal-valuedmeasurable functions

X

,

Y

on

(Ω, F )

the following con-ditionsare equivalent:

(i)

X

and

Y

arecomonotonic.

(ii) There existsameasurable function

Z

on

(Ω, F )

and twonon-decreasing functions

f

and

g

on

R

suchthat

X

= f (Z)

and

Y

= g(Z)

.

The notion of comonotonic functions proves to be veryuseful while dealing with Choquet integralsthanksto thefollowingresult(cf. lemma 4.84in Föllmerand Schied 2004, aswell as corollary4.6inDenneberg1994).

LemmaA.2 If

X, Y

: Ω → R

is a pair of comonotonic functions and if

r

X

, r

Y

, r

X+Y

are quantilefunctions(with respect toacapacity

µ

)of

X, Y, X

+ Y

respectively,then

r

X+Y

= r

X

+ r

Y

,

for almost every

t.

B Appendix: The proofs of Lemma 3.1 and Proposition 3.5

Proofof lemma3.1:

Throughout this proof we set

φ(x) := G

(2)

denoteby

φ

∗

theconjugatefunctionof

φ

i.e.

φ

∗

_{(y) := sup}

x∈R

(xy − φ(x))

. Letusrstremarkthat

φ(x) =

Z

+∞

x

µ(X > u)du = E

µ

((X − x)

+

) =

Z

1

0

(r

X

(t) − x)

+

dt,

the second equality is thestraightforward transformation used in the proof of proposition 3.3 andthethirdisdueto propositionA.1andtolemmaA.1.

Therefore,for

y

= 0

,wehave

φ

∗

(0) = − inf

x

∈R

Z

1

0

(r

X

(t) − x)

+

dt

= − lim

x→+∞

Z

1

0

(r

X

(t) − x)

+

dt

= 0,

wherewehaveusedthenon-increasingnessofthefunction

x

7→

R

1

0

(r

X

(t)−x)

+

dt

andtheLebesgue convergencetheorem. For

y

= −1

,wehave

φ

∗

(−1) = sup

x

∈R



−x −

Z

1

0

(r

X

(t) − x)

+

dt



= − lim

x→−∞

Z

1

0

max(r

X

(t), x)dt = −

Z

1

0

r

X

(t)dt.

Byanalogouscomputations, weobtainthat

φ

∗

_{(y) = +∞}

for

y >

0

, aswellas

φ

∗

_{(y) = +∞}

for

y <

−1

.

Finally,letusconsider thecasewhere

y

∈ (−1, 0)

.

Thefunction

f

denedby

f

(x) := xy − φ(x)

isconcave(thefunction

φ

beingconvex). Noticing that

f

(x) = xy −

R

+∞

x

(1 − G

X

(u))du

,weseethat theright-handandleft-hand derivativesof

f

at

x

are given by

f

0

+

(x) = y + (1 − G

X

(x+))

and

f

0

−

(x) = y + (1 − G

X

(x−))

. Apoint

x

is a

maximumpointforthefunction

f

if











f

0

+

(x) ≤ 0

f

0

−

(x) ≥ 0

whichisequivalentto











G

X

(x+) ≥ y + 1

G

X

(x−) ≤ y + 1

which, inturn,isequivalentto

x

beinga

(y + 1)

-quantileof

X

. So,bytaking

x

= r

X

(y + 1)

,we have

φ

∗

(y) = yr

X

(y + 1) −

Z

1

0

(r

X

(t) − r

X

(y + 1))

+

dt

= −

Z

1

y+1

r

X

(t)dt

which concludestheproof.



Proofof Proposition3.5Theimplication(ii)

⇒

(i)isobviousbytaking

g(t) := I

[y,1]

(t)

which isnon-negative,non-decreasingandintegrable.

Letus nowturnto the converse implication. Supposethat (i)holds true. Theassertion(ii) is trueforanyfunction

g

oftheform

g(t) := I

[y,1]

(t)

.

Letnow

g

beanon-negative,non-decreasing stepfunction. Then gcanbewritten asfollows:

g(t) =

P

k

i=1

a

i

I

[bi,1]

where

a

i

≥ 0

and

0 = b

1

< b

2

· · · < b

k

<

1

. Thus,wehave

Z

1

0

g(t)r

X

(t)dt =

k

X

i=1

a

i

Z

1

b

i

r

X

(t)dt ≥

k

X

i=1

a

i

Z

1

b

i

r

Y

(t)dt =

Z

1

0

g(t)r

Y

(t)dt.

Letnow

g

beanon-negative,non-decreasingfunction. Then

g

canbeapproximatedfrombelow by asequence

(g

n

)

of non-negative, non-decreasingstepfunctions. Due to the previous obser-vation, we then have

R

1

0

g

n

(t)r

X

(t)dt ≥

R

1

0

g

n

(t)r

Y

(t)dt.

The function

g

being integrable and thefunctions

r

X

and

r

Y

beingbounded (since

X

and

Y

are in

χ

), we canapply theLebesgue convergencetheoremtopassto thelimitin thepreviousinequalitywhichconcludestheproof.



C Appendix: The generalized Hardy-Littlewood's

inequal-ities

Wegivetheproofofproposition4.1.

Let us rst prove the upper bound part in proposition 4.1. Before we proceed, we need the followingtwolemmas:

LemmaC.1 Let

µ

beacapacityon

(Ω, F )

whichiscontinuousfrombelow. Let

(X

n

)

bea non-decreasing sequence of non-negative measurable functions and let

X

denote the limit function. Thenthesequenceofdistributionfunctions(withrespectto

µ

)of

X

n

convergestothedistribution function (withrespectto

µ

)of

X

i.e.

lim

n→∞

G

X

n

(x) = G

X

(x), ∀x ∈ ¯

R

₊

_.

Proof: The proofof this lemmais containedin theproof of theorem8.1 in Denneberg(1994) andisomitted.

RemarkC.1 WenotethatlemmaC.1remainsvalidevenwhenthenon-negativityassumption onthefunctions ofthesequence

(X

n

)

isrelaxed.