On dynamic Knightian uncertainty models [Elektronische Ressource] : time-consistency and optimal behavior / vorgelegt von Monika Bier

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On Dynamic Knightian UncertaintyModels: Time-Consistency and OptimalBehaviorInaugural-Dissertationzur Erlangung des Grades eines Doktorsder Wirtschaftswissenschaftendurch dieFakultat fur Wirtschaftswissenschaften der Universitat Bielefeldvorgelegt vonMonika Bieraus Wheeling, West Virginia, USABielefeld, 2009Erstgutachter: Zweitgutachter:Prof. Dr. Frank Riedel JProf. Dr. Dr. Frederik HerzbergInstitut fur Mathematische Institut fur Mathematische Wirtschaftsforschung (IMW) Wirtschaftsforschung (IMW)Universitat Bielefeld Universitat BielefeldGedruckt auf alterungsbestandigem Papier nach DIN-ISO 9706to my grandmaAcknowledgmentsIn writing this thesis I have greatly bene ted from the help and encourage-ment of numerous people.First and foremost, I appreciate the guidance and support of my advisorProfessor Dr. Frank Riedel. This doctoral thesis would not have been pos-sible without his motivation, advice and ideas for further investigation. Inmany fruitful discussions he helped me with valuable comments, and sugges-tions and prevented me from getting lost in detours making my research aimsreachable again. He encouraged me to present my work on several opportu-nities for which I am thankful. In addition, I am indebted to JProfessor Dr.Dr. Frederik Herzberg for his e orts in surveying this thesis.
Publié le : jeudi 1 janvier 2009
Lecture(s) : 30
Source : NBN-RESOLVING.DE/URN:NBN:DE:HBZ:361-17704
Nombre de pages : 114
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OnDynamicKnightianUncertainty

Models:Time-ConsistencyandOptimal

iorvhaeB

Inaugural-Dissertation

zurErlangungdesGradeseinesDoktors

derWirtschaftswissenschaften

durchdie

Fakulta¨tfu¨rWirtschaftswissenschaften

derUniversit¨atBielefeld

vorgelegtvon

MonikaBier

ausWheeling,WestVirginia,USA

Bielefeld,2009

Erstgutachter:
Prof.Dr.FrankRiedel
Institutfu¨rMathematische
Wirtschaftsforschung(IMW)
Universit¨atBielefeld

Zweitgutachter:
JProf.Dr.Dr.FrederikHerzberg
Institutfu¨rMathematische
Wirtschaftsforschung(IMW)
Universita¨tBielefeld

Gedrucktaufalterungsbesta¨ndigemPapiernachDIN-ISO9706

to

ym

grandma

Acknowledgments

InwritingthisthesisIhavegreatlybenefitedfromthehelpandencourage-
mentofnumerouspeople.
Firstandforemost,Iappreciatetheguidanceandsupportofmyadvisor
ProfessorDr.FrankRiedel.Thisdoctoralthesiswouldnothavebeenpos-
siblewithouthismotivation,adviceandideasforfurtherinvestigation.In
manyfruitfuldiscussionshehelpedmewithvaluablecomments,andsugges-
tionsandpreventedmefromgettinglostindetoursmakingmyresearchaims
reachableagain.Heencouragedmetopresentmyworkonseveralopportu-
nitiesforwhichIamthankful.Inaddition,IamindebtedtoJProfessorDr.
Dr.FrederikHerzbergforhiseffortsinsurveyingthisthesis.
IacknowledgefinancialsupportfromtheGermanResearchFoundation
(DFG)andfromtheInstituteofMathematicalEconomics(IMW)atBielefeld
University.
Thanksalsotovariousseminarandconferenceparticipantswhogaveme
helpfulsuggestionsandencouragement.
Ahugecontributiontothisworkwasalsomadebyseveralcolleaguesand
friends:Mostimportantmyformerofficemate,friend,andcoauthorDaniel
EngelagewithwhomIenjoyedmanyimportantdiscussionsin-andoutsideof
theoffice.Inmyopinionwecomplementeachotherperfectlyasresearchers
buthehasalsobecomeavaluablefriend.AtthispointIalsowanttomention
hissuccessorasofficemateTatjanaChudjakowwhowasnevershortofan
economicinterpretationoramotivationalword,whateverwasneededmore.
Ontopofthatsheisanexcellenttravelcompanionandguide.Ialsowantto
thankSimon,Jan,andJo¨rgfornumeroushelpfulmathematicalremarksand
inspiringcoffeebreaks.SpecialthankstoSimonwhonevertiredoftryingto
convincemeofBielefeld’spositivesidesandmakingmefeelwelcome.
FurthermoreIwouldliketothankmycolleaguesandfriendsattheBonn
GraduateSchoolofEconomics(BGSE)formakingmefeelathome.Here
IwanttoespeciallymentionAlmiraBuzaushina,MarceloCadena,Jo¨rdis
Hengelbrock,StefanieLehmann,ChristinaMatzke,KlaasSchulze,andBernd
Schluscheforhelpfulsuggestionsandenjoyabletimes.

HugethanksalsotothemembersoftheInstituteofMathematicalEco-
nomics(IMW)forgivingmesuchaheartywelcomeandmakingmyshort
stayaspleasantasitwas.Thanksalsoforthemanyamusingeveningsin
theLounge,especiallyourfoosballtournaments.SpecialthankstoBettina
Buiwitt-Robsonforjustbeingwhosheis,toMatthiasSchleefforsolvingall
myITandbicycleproblems,andSonjaBrangewitzforallherspecialcare.
Lastbutnotleast,specialthanksgotoMarcus,myfamilyandmyfriends
fortheirpatience,theirunconditionalemotionalsupport,andencouragement
aswellastheirenduringbeliefinme.

tsentonC

1GeneralIntroduction
1.1KnightianUncertainty......................
1.2Time-Consistency.........................
1.3RiskMeasures...........................
1.4ParticularConsiderations.....................

2Time-ConsistentSetsofMeasuresonFiniteTrees
2.1Introduction............................
2.2Model...............................
2.3FromPtoA...........................
2.3.1MartingaleRepresentation...............
2.3.2Exponentialformofthedensities............
2.3.3Compact-valuednessoftheα’s..............
2.3.4StabilityunderPasting..................
2.4Necessity..............................
2.4.1ConstructionofP.....................
2.4.2Time-Consistency.....................
2.4.3Compactnessofdensities.................
2.5Examples.............................
2.5.1BinomialTree.......................
2.5.2ExponentialFamilies...................
2.5.3TrinomialTree......................

i

12356

991215167191910221121222223224

NTENTSCO

2.5.4DTV@R..........................25
2.6PossibleExtensions........................26
2.6.1Convexity.........................26
2.6.2InfiniteHorizon......................27
2.6.3LooserAssumptionsonSplittingFunction.......28
2.7Conclusions............................29

3ADualityTheoremforOptimalStoppingProblemsunder
Uncertainty31
3.1Introduction............................31
3.2Model...............................35
3.3Problem..............................38
3.4ConstructionofP.........................40
3.4.1κ-Ambiguity........................41
3.5MainPart.............................42
3.6Applications............................47
3.6.1Sub-andSupermartingales................47
3.6.2ExploitingMonotonicityintheDrift..........48
3.7Conclusion.............................52

4LearningforConvexRiskMeasureswithIncreasingInfor-
mation55
4.1Introduction............................55
4.2Model...............................59
4.3DynamicConvexRiskMeasures.................60
4.4AConstructiveApproachtoLearning..............64
4.4.1TheIntuitionofLearningviaPenalties.........64
4.4.2SpecialCase:ExplicitLearningforCoherentRisk...65
4.4.3AFirst,ParticularlyIntuitiveApproach:Simplistic
Learning..........................67

ii

5

NTENTSCO

4.4.4ASecond,MoreSophisticatedApproach:EntropicLearn-
ing.............................69
4.4.5LackofTimeConsistency................75
4.4.6ARetrospective–InBetween..............77
4.4.7LearningforagivenTime-ConsistentConvexRiskMea-
sure............................78
4.5AdaptionofBlackwell-DubinsTheorem.............80
4.6Time-ConsistentRiskMeasures.................81
4.6.1Time-ConsistentCoherentRisk.............82
4.6.2Time-ConsistentConvexRisk..............83
4.7NotNecessarilyTime-ConsistentRiskMeasures........86
4.7.1NonTime-ConsistentCoherentRisk..........87
4.7.2NonTime-ConsistentConvexRisk...........91
4.8Examples.............................92
4.8.1EntropicRisk.......................92
4.8.2Counterexample......................93
4.8.3ANonTime-ConsistentExample............94
4.9Conclusions............................95

ClosingRemarks

97

iii

1erhaptC

GeneralIntroduction

Ascanbewellobservedinthecurrentfinancialcrisischoosingthewrong
modelspecificationsformakingdecisionsorassessingriskcanhaveveryse-
vereconsequencesiftheunderlyingtheoryisnotrobusttomodeluncertainty.
Thisseemstohavebeenoneofthemajorshortcomingswhichleadtothe
presenteconomicsituation.Thereforeamajoraimineconomicsistoreduce
modelriskbydevelopingrobustapproachestodecisionmaking.
Alinkbetweenrobustcontroltheoryanddecisiontheorywasshownin
[Hansen&Sargent,01].Inrobustcontroltheorymodeluncertaintyarises
bytheperturbationofauniqueapproximatingmodel.Thiscorrespondsto
uncertaintyaboutthetruedistributioninadecisiontheoreticalansatzwhich
willbethefocusofthefollowing.
Thewellknownworksof[vonNeumann&Morgenstern,44],[Savage,54]
and[Anscombe&Aumann,63]areamongthefirsttheoreticalmodelson
decisionmaking.Onecommonpropertyofallthesemodelshoweveristhat
theyincorporateonlyonesingleknowndistributionoftheoutcomes,i.e.
whatisoftencalledapurelyriskysetting,whichinrealityisasituation
seldomfound.Thefirstcommentonthefactthatthereismorethanpure
riskcanbefoundin[Knight,21]whichiswhyuncertaintyaboutthetrue
distributionisoftenreferredtoasKnightianuncertaintybutsometimesalso

1.GENERALINTRODUCTION

ambiguity.sa

1.1KnightianUncertainty

Inhisseminalpaper[Knight,21]suggestedthatthereexistrandomoutcomes
whichcannotberepresentedbynumericalprobabilities,i.e.heestablishesa
cleardistinctionbetweenmeasurableuncertaintyhecallsriskandunmeasur-
ableuncertainty.Thisunmeasurableuncertaintycanamongotherreasons
arisewhenthedecision-makerisignorantofstatisticalfrequenciesrelevant
tohisdecisionorwhenaprioriestimationsareimpossibletoobtainorthe
decisionisuniqueinthesensethatthereisnoinformationtobuildanap-
proximationfornumericalprobabilities.Tomakethisdifferentiationabit
moreexplicitthinkoffollowingexamples.Ifonehastobetheadsortails
inacointossonewouldassumethatbothareequallylikelybasedonone’s
experience,i.e.onecansomehowmeasuretheprobabilityandfindsoneself
inapureriskysetting.Howeverifoneisaskedtobetone.g.theoutcome
ofatennismatchwithoutpossessinganyinformationabouttheplayers,it
isnotclearhowtoassignauniqueprobabilitytotheoutcomesandoneisin
anuncertainsetting.
Basedonthistheoreticalapproachistheempiricalworkof[Ellsberg,61].
FollowingtheaimofgivingevidenceforKnight’stheory,heconstructedthe
followingurn-experimentwithtwournscontaining100blackandredballs.In
thefirsturntheratioofblacktoredisunknown,itcanbeanythingbetween0
and100.Inthesecondurnthereare50redand50blackballs.Ifonedenotes
thebetofgetting$100ifaredballispulledfromurniand$0elsebyREDi,
andBLACKifortherespectivebetonablackballEllsbergclaimsthata
majorityofpeopleshowfollowingpreferences:Theyareindifferentbetween
REDiandBLACKifori=1,2butpreferRED2toRED1andBLACK2to
.CKBLA1Heshowsthatthereisabsolutelynowaytoassignprobabilitiestothe

2

1.2.TIME-CONSISTENCY

eventofredbeingpulledfromthefirsturn,toexplainthesepreferencesim-
plyingthattheclassicaldecisiontheorydealingwithoneuniquedistribution,
ase.g.introducedin[Savage,54]cannotcontainthewholetruthandthat
theredoexisttheunmeasurableuncertaintiesproposedbyKnight.
Anattempttounderpinthesefindingswithatheoreticalmodelisgiven
in[Gilboa&Schmeidler,89].Theysetupaxiomsforpreferenceswhichlead
toMaxminExpectedUtilityortheMultiplePriorsModel.Theyweakened
theIndependenceAxiomofAnscombeandAumannandintroducedanax-
iomformalizingUncertaintyAversiontothemodel.Bythisslightchange
theyaccomplishedarepresentationoftheirpreferenceswhichinsteadofone
uniquedistributioncontainedawholesetofpossibledistributions.Itleads
thedecisionmakertomaximizeinfP∈CEP[u◦f]amongallpossibleactsf,
whereCisanon-emptysetofdistributions.Heuristicallyonecaninterpret
thisashavingthedecisionmakerthinkalldistributionsinCpossibleandin
ordertobeonthesafesidehealwayslooksattheonewhichgiveshimthe
lowestutility.Iftheutilityundertheworstpossibledistributionisenough
tomakehimchooseactfsurelyitishighenoughunderallotherdistribu-
tionsaswell.Seenthiswaythesepreferencescorrespondtoanextremely
conservativedecisionmaker.

1.2Time-Consistency

Sinceuptonowthemodelswerepurelyatemporal[Epstein&Schneider,03]
generalizedtheaboveapproachtoadynamicsetting.Theyappropriately
modifiedtheaxiomsof[Gilboa&Schmeidler,89]tobenotonlystatebut
alsotimedependentandadditionallyaskedforDynamicConsistencyinthe
preferences.Withthisassumptiontheymeantiftwoactsareidenticalup
tosometimetbutoneispreferredovertheotherint+1thenthisshould
alreadybethecaseattimetimplyingthatadecisionmakerwillneverregret
hisearlierchoices.Thisrestrictiononthepreferencesyieldsaveryspecial
3

1.GENERALINTRODUCTION

propertyforthesetofdistributionsintheUtilityfunctional.Theyshowed
thatpreferencesaredynamicallyconsistentifandonlyifthecorresponding
setofdistributionsarisingintheirRecursiveMultiplePriorsRepresentation
isrectangular.Rectangularityisarestrictiononthewholesetofmeasures.
Equivalentdefinitionswereformulatedbyvariousauthors.Asurveyofthe
differentconceptsandaproofoftheirequivalencecanbefoundin[Riedel,09]
and[Delbaen,03].Amongtheseconceptsistheabovementionedrectangu-
larityintroducedin[Epstein&Schneider,03]whichisapropertyconcerning
theone-step-aheadmeasures.Theyaskedthatateverypointintimeallpos-
sibleone-step-aheadmeasurescanbeadded.Anotherconceptisstability
introducedin[Fo¨llmer&Schied,04].HerefortwomeasuresPandQinthe
setofmeasuresandeverystoppingtimeτthemeasurethattakesPupto
τandQafterwardsisalsocontainedintheset.Thelastconceptistime-
consistencywhichwasintroducedin[Delbaen,03].Thispropertydemands
thatateverystoppingtimedensityprocessescanbeconsistentlypastedto-
gether.ItisalsoequivalenttoaLawofIteratedExpectationsmeaningthat
attimesmyexpectedfuturepayoffisthesameastheexpectationinsof
myfutureexpectedpayoffint≥s.
Thesemayseemasrathertechnicalassumptionsbuttheyalsohavesome
intuitiveconsequencesforthedecisionmaker.Hecanforinstancechangehis
mindineverytimeperiodaboutwhichmeasurehethinksisthetrueoneor
theworstoneandtime-consistencyguaranteesthatthismeasureiscontained
inthesetofhispossiblemeasures.Thisimpliesthatastimepasseshewill
neverregrethispreviousdecisionssinceateverypointintimehecandecide
optimally.Anotherimplicationisthathecanusebackwardinductionfor
solvingproblemswhichmakeslargeclassesofproblemalotmoretractable.
Onefurthergeneralizationoftheseutilitymodelsisthevariationalpref-
erencemodelintroducedinastaticsetupin[Maccheronietal.,06a]and
extendedtoadynamicframeworkin[Maccheronietal.,06b].Sinceupto
nowalldistributionsinthesetwereconceivedasequallylikelytheyintro-

4

1.3.RISKMEASURES

ducedpenaltyfunctionsallowingtodifferentiatethedifferentdistributions
accordingtotheirlikelihood.

1.3RiskMeasures

Uptonowwehavefocusedonutilitymodelsthatarisefrompreferencesbut
analogousmodelscanalsobefoundbyaxiomatizingriskmeasures.
Staticcoherentriskmeasureswhichcorrespondtothemultipleprior
modelundertheassumptionofriskneutrality,adiscountfactorofone
andnointermediatepayoffswerefirstaxiomatizedin[Artzneretal.,99]
andtheirdynamicgeneralizationscaninteraliabefoundin[Riedel,04]or
[Artzneretal.,07].Therobustrepresentationofacoherentriskmeasureis
identicaltotheoneformultiplepriorpreferencesuptoaminussign,having
thedecisionmakerlookatthelargestexpectedlossasabasisforhisdecision.
Sinceforriskmeasuresitisimportanttoincorporateliquidityriskand
togivelessconservativeassessmentswhichcoherentriskmeasurescannot
provideconvexriskmeasureswereintroducedandcane.g.befoundin
[Fo¨llmer&Schied,04]forastaticsetting.Foradynamicsettingwerefer
to[Fo¨llmer&Penner,06]or[Fo¨llmeretal.,07]forriskyprojectsseenas
payoffsinthelastperiodwhileriskyprojectsseenasstochasticprocessesare
studiedin[Cheriditoetal.,06].Hereagaintheequivalencetovariational
preferencesisgivenuptoaminussign.
Time-consistencyconceptsarethesameinbothapproachesalthoughin
termsofriskmeasuresoneusuallyworkswiththeonecorrespondingtothe
lawofiteratedexpectations.
Consideringthisitmakesnodifferenceifwemakethefollowingobserva-
tionsintermsofutilityfunctionalsorriskmeasures.Eachchaptermaybe
reformulatedintermsoftheotherapproachresrectivelysinceforsimplicity
weassumeriskneutraldecisionmakerswithoutdiscountinginallchapters.
HoweverinChapter2and3wewilllookatutilityfunctionalswhileChapter
5

1.GENERALINTRODUCTION

4willbemainlyexpressedintermsofriskmeasureswherewewillgivea
furthershortoverviewofthetheoryofriskmeasuresandtheirrobustrepre-
sentation.

1.4ParticularConsiderations

Themainchaptersofthisthesis,eachofwhichisself-contained,arebased
onthreearticles.Thefirstdealswiththeconstructionoranalternativechar-
acterizationoftime-consistentsetsofmeasuresinthespecialframeworkof
finitetrees,thesecondshowsasamainresultadualitytheoremthishowever
inacontinuoustimesettingwhilethetopicofthethirdone,coauthoredby
DanielEngelageisconcernedwiththemergingofconvexriskmeasuresas
informationisgainedinthecourseoftime.
InChapter2wesolvethequestionwhattime-consistentsetsofmeasures
looklikeinfinitetrees.In[Riedel,09]time-consistentsetsofmeasuresina
discretesettingareconstructedviatheirdensityprocesseswhichgaveriseto
thequestionifalltime-consistentsetscanbeconstructedinthisway.When
restrictingtheframeworktofinitetreeswithaconstantandfinitesplitting
functionweshowthateverytime-consistentsetofmeasurescanbedescribed
viapredictableprocesses.ForeachmeasurePinatime-consistentsetPwe
getapredictableprocessαPwhosedimensionisonelessthanthesplitting
valueofthetree.ThesetofpredictableprocessesA={αP|P∈P}that
arisesviathisidentificationhasspecificfeatures.Thesefeaturesinreturn
guaranteethatatime-consistentsetofmeasurescanbecreatedfromaset
ofpredictableprocesseswiththeseproperties.Afterthischaracterizationwe
showsomeexamplesorapplicationsfortheuseofthistheorem.Additionally
weshowthatstandardgeneralizationsofthistheoremfailtogothrough,
showingthatourcharacterizationisauniversalone.
Chapter3containsaDualitytheoremwhichallowstoswitchtheor-
derofminimizationandmaximizationinordertosolveoptimalstopping

6

1.4.PARTICULARCONSIDERATIONS

problemsunderambiguity.Thistheoremisagainsetintherecursivemul-
tiplepriorsmodelof[Epstein&Schneider,03]andworksforfairlygeneral
assumptionsonourpayoffprocessXandratherstandardassumptionson
oursetofmeasuresP.Wemakestronguseofanexplicitbutgeneralcon-
structionfortime-consistentsetsofmeasuresgivenin[Delbaen,03].Wealso
applythistheoremtospecificclassesofpayoffprocesses.Itallowstodeter-
mineanoptimalstoppingtimeformultiplepriorsuper-andsubmartingales
aspayoffprocessesandinthecaseofκ-ambiguityadaptedtoourframework
andaBrownianmotionwithdriftaspayoffprocessweareabletoidentifythe
worstcasedistributionandhenceourambiguousstoppingproblemsshrinks
toaclassicalproblem.
InChapter4,coauthoredbyDanielEngelage,wetacklethequestion
ofhowinanambiguousenvironmenttheassessmentofriskandwiththis
optimalbehaviorchangesastimepassesandinformationincreases.Theve-
hicleforthisanalysiswillbeconvexriskmeasuresordynamicvariational
preferencesequivalently.Ourfirstapproachtoincorporateakindoflearning
mechanismintoconvexriskmeasuresisaconstructiveoneviathemini-
malpenaltyfunctionintherobustrepresentation.Howeverinourexplicit
approachtoconstructthepenaltyviathelikelihoodsofthedistributions
time-consistencyseemstobeamajorproblem.Thereforeinoursecondap-
proachwetakeadynamicallyconsistentsetofriskmeasuresasgivenand
showthatinthelongrunalluncertaintyisrevealed,leavingthedecision
makerbehaveasautilitymaximizerunderthetruedistribution.Formulat-
ingthisresultclosertothefundamentalBlackwell-Dubinstheoremofwhich
thisisageneralization:twodecisionmakerswhoagreeonsureandimpos-
sibleeventsbutwithdifferentopinionsoftherisktheyface,modeledhere
viadifferentpenaltyfunctions,tendtowardsagreeingonthetrueunder-
lyingdistributionintheend.Asmentionedweextendthemainresultin
[Blackwell&Dubins,62]whichholdsforprobabilitymeasurestoconvexrisk
measures.Animportantstepinthisgeneralizationisalsotheextensionto

7

1.GENERALINTRODUCTION

notnecessarilytime-consistentconvexriskmeasures.Withthisourexistence

resultforalimitingdistributionbecomesmoregeneralthantheonefound

in[Fo¨llmer&Penner,06].Tomakethingsclearerwestudyentropicrisk

measuresasanapplication.

Tothispointwehavegivenabriefoutlineofthegeneralcontextand

developmentswhichleadtothiswork.Sincethequestionsandtopicstreated

inthefollowingchaptersdifferamoredetailedscientificplacementofthis

workwillbediscussedineachchapterseparately.

8

2erhaptC

Time-ConsistentSetsof

MeasuresonFiniteTrees

2.1Introduction

In1944vonNeumannandMorgensternformulatedtheirfamousaxiomsfor
preferencesoverrandompayoffs(see[vonNeumann&Morgenstern,44])and
showedthatthesepreferencesareequivalenttoanExpectedUtilityRepre-
sentationofpreferences.Aftersometimetheirmodelwascriticizedbecause
thedistributionsoftheirpayoffswereexogenouslygivenandpurelyobjec-
tive.Sincethisisaveryrestrictiveassumptiontheirmodelwasextended
in[Savage,54]andin[Anscombe&Aumann,63].Incontrasttothevon
NeumannandMorgensternmodelSavageregardedthedistributionsofthe
payoffstobepurelysubjectiveandendogenous.AnscombeandAumann
thencombinedbothmodelstakingsomeobjectivedistributionsasgivenand
havingothersarisingpurelyoutofthemodel.
Atsomepointcriticismalsoaroseagainstthesemodels.Oneofthemost
mentionedobjectionscanbefoundin[Ellsberg,61].Heconductedexperi-
mentsandempiricallyshowedthatExpectedUtilitymodelsdonotalways
mirrorreality.Onewayofexplainingthesefindingsisthatpeoplebehaveonly

2.TIME-CONSISTENTSETSONFINITETREES

boundedlyrational.Anotherwayistodistinguishbetweenuncertaintyand
risk.Whileinariskysettingthedecisionmakerissureofthedistributions
oftheoutcomesinanuncertainsettingheisunsureoftherightdistribution
andthinksmorethanonepossible.FollowingthisideaGilboaandSchmei-
dlerdevelopedtheirMultiplePriorsModelin[Gilboa&Schmeidler,89]us-
ingAnscombe’sandAumann’smodelasabasis.TheyweakenedtheIn-
dependenceAxiomandaddedanadditionalaxiomformalizingUncertainty
Aversion.ThisleadthedecisionmakertomaximizePinf∈CEP[u◦f]amongall
possibleactsf,whereCisanon-empty,closedandconvexsetofprobability
es.rsueamSincethisisapurelyatemporalmodelin[Epstein&Schneider,03]the
MultiplePriorsModelwasexpandedtoincorporatethefactortime.They
modifiedpreferencestobenotonlystatebutalsotime-dependent,adjusted
theG-S-axiomsappropriatelyandaskedforDynamicConsistencyasanad-
ditionalaxiom.Thisrestrictiononpreferencesyieldsaveryspecificproperty
ofthesetofmeasuresintheirUtilityRepresentation.Theyfoundoutthat
preferencesaredynamicallyconsistentifandonlyifthesetofmeasuresin
theirRecursiveMultiplePriorsRepresentationisrectangular.Rectangular-
ityisarestrictiononthewholesetofmeasures.Itdemandsthatitispossible
fortheone-step-aheadmeasurestobemixedarbitrarily.Sinceforsomepur-
poses(e.g.solvingconcreteoptimalstoppingproblems)thisisnotavery
easydefinitionbutneverthelessanimportantoneitisverynaturaltotry
andfindequivalentdefinitions.
Thiswasdonebyvariousauthors.In[Riedel,09]onecanfindasurvey
ofthedifferentconceptsandaproofoftheirequivalence.Amongthese
conceptsisrectangularitywhichwasintroducedin[Epstein&Schneider,03]
andisapropertyconcerningtheone-step-aheadmeasures.Theyaskedthat
ateverypointintimeallpossibleonestepaheadmeasurescanbeadded.
Anotherconceptisstability.Itwasintroducedin[Fo¨llmer&Schied,04].
HerefortwomeasuresPandQinthesetofmeasuresandeverystopping

10

2.1.INTRODUCTION

timeτthemeasurethattakesPuptoτandQafterwardsalsoliesintheset.
Thelastconceptistime-consistencywhichwasintroducedin[Delbaen,03].
Thispropertydemandsthatateverystoppingtimedensityprocessescan
beconsistentlypastedtogether.Amoreformaldefinitionofthisspecific
propertywillbegiveninthenextsection.

IntheabovecitedpaperRiedelamongotherthingsconstructedtime-
consistentsetsofmeasuresviatheirdensityprocesses.Consequentlythe
questionaroseifinthisspecialsettingalltime-consistentsetsofmeasures
canbeconstructedinthisway.Thatiswhywetookacloserlookattime-
consistentsetsofmeasuresandfoundoutthatnotquiteallsetsareofthis
kind.Howeveraslightmodificationofhisconstructiondoesthetrick.

Themaincontentofthispaperisthisalternativecharacterizationoftime-
consistentsets.Theyaredescribedviaasetofpredictableprocesseswith
specificproperties.Thiswillbeourfirstandmaintheorem.Inadditionto
showinghowthesetofmeasurescanberelatedtothissetofprocesseswe
willalsoshowthatsetsofprocesseswiththeassumedpropertiesdefinesets
oftime-consistentmeasures.Thiswillbethecontentofoursecondtheorem.
Soaltogetherwewillprovideanequivalentformulationfortime-consistent
setsofmeasures.

Thebuild-upofthispaperwillbethefollowing.Afterpinningdownthe
modelframeworkandspecifyingtheattributesofoursetsmorepreciselyin
Section2.2wewilldeductthefirsttheoreminthesucceedingSection2.3.
TheninSection2.4wewillcommitourselvestoprovingthesecondtheorem.
Inthefollowingfifthsectionwewillintroducesomeexamplesettingwhereour
resultsareapplicableanmightsimplifycalculations.Afterthatwediscuss
possibleextensionsinSection2.6andthenconcludeinthelastandseventh
.notisec

11

2.TIME-CONSISTENTSETSONFINITETREES

delMo.22TospecifythesettingwestartwithadiscretesetΩ={ω1,...,ωk}.Onthis
statespacewehaveaninformationstructure{Ft}t=0,...,TwithF0=Ωand
FT={{ω1},...,{ωk}}.ThisisasequenceofpartitionsofΩ,whichbecome
finerastimeprogresses,i.e.everysetofFt+1isasubsetofsomesetofFt
forallt.
Heuristicallythisconceptdescribestheinformationoftheprevailingstate
availableatacertaintimet.Thismeansforafixedtimetthedecisionmaker
willnotnecessarilybeabletoobservetheexactstatewhichoccursbutmerely
whichsubsetofFtisrealized.Iftheobservedsubsetconsistsofonlyasingle
statethenofcoursethedecisionmakerhasfullknowledgeoftherealization.
Ifyouwanttoexpressthisintermsofσ-fieldsandfiltrationsyoujust
takethepowersetPot(Ω)forthefiltrationFanddefinethefiltration{Ft}t
bysettingFt:=σ(Ft)i.e.FtisthesetofatomsgeneratingFt.
Forourconsiderationsweassumeourinformationstructuretohavea
constantandfinitesplittingfunctionwithsplittingvalueν.Thisimplies
thatifyoudrawthefiltrationasaninformationtreeitwillhavethesame
finitenumberofbranchesateveryvertex.Formallythesplittingfunctionf
ofaninformationstructure{Ft}tisdefinedinthefollowingway

f:Ω×[0,∞)→N,f(ω,t)={A∈Ft+1|A⊆Ft(ω)}
whereFt(ω)isthesetB∈Ftwithω∈B.Thefinitenessofthisindex
willallowustoapplythemartingalerepresentationgiveninTheorem5.15
in[Dothan,90]andtheconstancywillresultinuniqueprocessesintherep-
resentation.Wewillmakethesetwothingsmorepreciseinthefollowing
.notisecFornowwewillalsorestrictthismodeltoafinitetimehorizon[0,T].
ThefinitesplittingindexandthefinitetimehorizonresultinafiniteΩ.
21

2.2.MODEL

Tocompleteourprobabilityspacewestillneedtofixaprobabilitymea-
sureP0asareferencemeasurewhichpinsdownthesetsofmeasurezero.
Sinceweareonatreelikestructureanymeasurewhichassignsnon-zero
probabilitytoeachbranchwilldo,forsimplicityletuschoosetheuniform
distribution.
ThesetofmeasureswewanttocharacterizewillbedenotedbyP.In
thefollowingwewillmakesomeassumptionsonthissetandjustifytheir
plausibility.
Ourfirstassumptionwillbe
Assumption2.2.1.WeassumeP0∈PandforallothermeasuresP∈P
P(A)>0forallA∈FT
InthisassumptionP0’sfunctionasareferencemeasurebecomesclear.
Onecanseethatithasnoinfluenceonthestochasticstructureoftheother
measures.ItsimplyimpliesthatallmeasurescontainedinPhavethesame
nullsetswhichmeansthatweknowwhatsureandimpossibleeventsare.
In[Epstein&Marinacci,06]aneconomicinterpretationofthisassump-
tionwasgiven.Theyrelatedittoanaxiomonpreferencesfirstpostulated
in[Kreps,79].Heclaimedthatifadecisionmakerisambivalentbetween
anactxandx∪xthenheshouldalsobeambivalentbetweenx∪xand
x∪x∪x.Meaningifthepossibilityofchoosingxinadditiontoxbringsno
extrautilitycomparedtojustbeingabletochoosex,thenalsonoadditional
utilityshouldarisefrombeingabletochoosexsupplementarytox∪x.
Inoursecondassumptionweclaim
QdPdAssumption2.2.2.Pistime-consistent.Thismeansforastoppingtime
τanddensitiespt:=dP0andqt:=dP0belongingtoP,Q∈Pthatthe
ttmeasureP˜definedbythedensity
d˜Pptift≤τ
=dP0tpqττqtelse
belongstoPaswell.

31

2.TIME-CONSISTENTSETSONFINITETREES

Asmentionedintheintroductionthisassumptionalsooriginatesfrom
afeatureclaimedforpreferencesintroducedin[Epstein&Schneider,03].
TheyexpandedtheMultiplePriorsModel(cp[Gilboa&Schmeidler,89])to
adynamicsettingandaskedthedecisionmakertobedynamicallyconsistent
inhisdecisions.Withthistheymeantthatiftwoactsareidenticalupto
sometimetbutint+1theoneispreferredovertheother,thenthisshould
alreadybethecaseattimet.Thisimpliesthatadecisionmakerwillnever
regrethisearlierdecisions.IntheirpaperEpsteinandSchneiderthenshowed
thatpreferencesfulfillthisrequirementifandonlyiftheutilityfunctional
oneobtainscontainsarectangularsetofmeasures.Rectangularityisequiv-
alenttotime-consistency.Time-consistencywasintroducedin[Delbaen,03]
wherehealsoshowedtheequivalencetorectangularity.Thesetwofeatures
standforbeingabletojudgeeachperiodintimewithadifferentmeasure.
Moretechnicallytheyallowtoconsistentlypastetogetherdifferentdensities
atdifferenttimesandstillstayintheset.Theyalsomakeitpossibleto
usebackwardinductionindiscretesettingsandallowforaLawofIterated
Expectations.
ThesetusedtocharacterizePwillbedenotedbyA.Wewillshowthat
itconsistsofpredictableprocesses,iscompactandthattheprocessconstant
tozeroiscontainedinit.Furthermorewewillseethatitfulfillsaproperty
wecallstableunderpastinganddefineinthefollowingway.

Definition2.2.3.AsetofprocessesAiscalledstableunderpastingiffor
everystoppingtimeτandallprocesses(αt)t,(βt)t∈Atheprocessdefinedby
γt:=αtift≤τ
elseβt

belongstoAaswell.
LateronwewillshowifweassumethesepropertiesforasetAthenwe
canderiveasetofmeasuresPthatfeaturesouroriginalcharacteristics.
41

2.3.FROMPTOA

2.3FromPtoA
Thegoalofthissectionistoprovethemaintheoremofthispaper,which
tellsus,thateverytime-consistentsetofmeasuresinoursettingcanalsobe
describedviaasetofpredictableprocessesfulfillingcertainproperties.
Expressedmoreformallythisresultsin

Theorem2.3.1.ForeverysetofmeasuresPsatisfyingAssumptions2.2.1,
2.6.1and2.2.2thereisasetofpredictableprocessesAsuchthat
dP
Pd0tP=P=E˜t(α),α∈At,t∈{0,...,T}where
tν−1tν−1
E˜t(α)=expαhsΔmhs−lnEexpαhsΔmhs
s=1h=1s=1h=1
TheAresultingfromeachPinhabitsfollowingfeatures:

0A∈••Aiscompact.
•Aisstableunderpasting.

Inordertoprovethistheoremwewillderiveasetofpredictableprocesses
Aforeverytime-consistentsetPandthenshowthatitinhabitstherequested
features.Oneimportantstepalongthiswaywillbeamartingalerepresenta-
tiontheoremwhichwewillexplainmorethoroughlyinthenextsubsection.
Afterthatwewillshowtheconstructionoftheprocessesstartingwithan
arbitrarytime-consistentsetofmeasuressatisfyingtheaboveassumptions.
Followingthiswewillshowthattheconstructedprocessesreallyarewhat
weaskedfor.

15

2.TIME-CONSISTENTSETSONFINITETREES

2.3.1MartingaleRepresentation
Thisimportanttoolwhichwewilluseinourprooftellsusthatinoursetting
wecanfindasetofmartingaleswithwhichwecanrepresenteveryother
martingaleinoursettingwiththehelpofpredictableprocesses.Asetof
martingaleswhichhasthisrepresentationpropertyiscalledamartingale
basis.Moreformallywedefine
Definition2.3.1.Afinitesetofmartingales{m1t},...,{mkt}iscalledabasis
iffforeverymartingale{xt}therearepredictableprocesses{α1t},...,{αkt}
suchthatforevery1≤t≤T
Tkxt=x0+αhsΔmhswhereΔmhs=mhs−mh,s−1
=1s=1hIfthemartingales{m1t},...,{mkt}arepairwiseorthogonal,i.e.forevery
1≤j≤k,1≤h≤m,j=handevery0≤t≤T,mj,mht=0,then
thebasis{m1t},...,{mkt}iscalledorthogonal.
Forourpurposesitwouldbegoodtoknowinwhichcasessuchabasis
existsespeciallywithuniqueα’s.Ananswerforthisisprovidedbythe
followingproposition.Aslightlydifferentversionofthiscanbefoundin
[Dothan,90]butsincewearelookingforauniquerepresentationweneed
torestrictthesettingtoaconstantsplittingfunctionofourinformation
structure.Theproofisworksalongthesamelineastheonein[Dothan,90].
Proposition2.3.2.(MartingaleRepresentation)
GivenadiscretespaceΩ={ω1,...,ωk}whichisendowedwithaninformation
structure{Ft}t=0,...,TwithF0=ΩandFT={{ω1},...,{ωk}}andaconstant
splittingfunctionwithvalueν.Thenthereexistsanorthogonalmartingale
basism1t,...,mν−1,tforwhichthepredictableprocesses{α1xt},...,{ανx−1,t}in
therepresentationofevery{xt}areunique.
Remark2.3.3.Sinceundertheassumptionof“noarbitrage”discounted
assetsaremartingalesforamartingalemeasureP∗thismeansforabinomial

61

2.3.FROMPTOA
treesettingthatthereisoneassetMtwithwhicheveryotherassetXtcan
bereplicatedandthereforehedged.Moregeneralinann-nomialtreewecan
replicateeveryassetwithasetofn−1manyassets.

2.3.2Exponentialformofthedensities
ThenextstepwewilltakeistoshowthateverymeasureP∈Pcanbe
uniquelyrelatedtopredictableprocessesα1Pss,...,ανP−1,ss.
Remarkthatthisisexactlyoneprocesslessthanoursplittingvalueν.
TheequivalenceofthemeasuresinadditiontoP0∈P(Ass.2.2.1)givesus
thepossibilitytoidentifyeachP∈Puniquelywithitsdensitywithrespect
toP0.
FttIfyoudefineddPP0:=EddPP0foreveryt≤TandeveryP∈P
withtheexpectationtakenunderP0youobtaindensityprocesseswhichare
P0-martingales.
UsingJensen’sinequalityandDoob’sdecompositiontheoremeachofthe
abovedensitiescanbewritteninthefollowingformwhere(Mt)tisalsoa
P0-martingaleand(At)tisanon-decreasingandpredictableprocesswith
PdA0=0
dP=exp(Mt−At).
0tNowapplyingthemartingalerepresentationtheoremtoMtweobtainan
orthogonalmartingalebasis(m1s)s,...,(mν−1,s)s.Thisimpliesthatthereare
predictableprocessesα1Pss,...,ανP−1,sssuchthatourdensitiescannowbe
writteninthefollowingmannerwhereΔmhs=mhs−mh,s−1
=expαPhsΔmhs−At.
dPtν−1
dP0ts=1h=1
NowwestillhavetodeterminetheAt’s.Usingthemartingalepropertyofthe
densitiesandthemeasurabilityoftheAt’swereceivethefollowingrecursive
71

2.TIME-CONSISTENTSETSONFINITETREES

relationν−1
=1hAt+1−At=lnEexpαPh,t+1Δmh,t+1Ft.
Thisresultsin
ttν−1
s=1s=1h=1
At=(As−As−1)=lnEexpαPhsΔmhsFs−1.
Additionallythankstotheassumptionsonourinformationstructure,we
canshowthatourfiltrationisgeneratedbyourmartingalebasisandthisin
additiontothepredictabilityoftheα’sallowsustodroptheconditioning
onFs−1.
SoforourdensityddPP0twenowhavefollowingrepresentation
dP=expαPhsΔmhs−lnEexpαPhsΔmhs.
tν−1tν−1
dP0ts=1h=1s=1h=1
)12.(ThisconstructionnowallowsustonotonlyidentifyameasurePwithits
densitywithrespecttoP0andtheassociateddensityprocessbutalsowith
thepredictableprocessesintheaboverepresentationα1Pss,...,ανP−1,ss.
Consequentlyitgivesusamappingfromourdensityprocessestosetsof
predictableprocesses.
Fornotationalconvenienceandinresemblancetoastochasticexponential
wewilldenotetherighthandsideof(2.1)asE˜t(αP)seeingαP=(α1P,...,ανP−1)
asaν−1-dimensionalprocess.
Sonowifwedenotethesetofprocessesgeneratedviathisconstruction
andthedensitiesuptotimetby
At:=α1P,s,...,ανP−1,ss∈{0,...,t}|P∈Pand
Dt:=dP,...,dP|P∈P

dP01dP0t
wehaveconstructedamappingE˜t−1:Dt→At.
FromthisconstructionandfromtheassumptionthatP0∈Pwedirectly
concludethattheα’sarepredictableandthat0∈A:=AT.
81

2.3.FROMPTOA

2.3.3Compact-valuednessoftheα’s
Onefurtherthingwewanttoshowisthatthecompactnessofthedensities
resultingfromPimpliescompactnessofA.ThecompactnessonAtisdefined
viathenorm||α||t,L1:=max||αs||L1.
},...,t0∈{sThisisastraightforwardconsequenceofourassumptionsandthepre-
cedingconstruction.Intheconstructionoftheα’severystepwasunique
thankstoourassumptions.Adensitywithrespecttoadesignatedmeasure
uniquelycharacterizesameasure,thesameistruefortheconstructionof
ourdensityprocesses.Doob’sdecompositionisalsouniqueandsinceweas-
sumedafiniteandconstantsplittingfunctionthemartingalerepresentation
alsodeliversuniquepredictableprocessesoncethemartingalebasisisfixed.
Allinallthesetofα’sthatbelongstoonePisunique.Additionallyasetof
α’sprovidesexactlyonedensityandthroughthatuniquelyonemeasure.For
thisreasonourE˜tgivesusabijectivemappingfromthesetofpredictable
processesAttooursetofdensitiesDt.Thismappingisalsocontinuous
sincetheelementsofourmartingalebasisareboundedthankstothefinite
splittingindex.
Sincethisalsoimpliesacontinuousmappingbetweenthedensitiesand
thepredictableprocesses,thecompactnessononesidecarriesovertothe
r.etho

2.3.4StabilityunderPasting
Thefinalpropertyweclaimedforourprocessesisstabilityunderpasting.
ThispropertyhoweverfollowsdirectlyfromtheassumptionthatPistime-
consistent.Tomakethismoreclearfor(αtP)t,(αtQ)t∈Aandastoppingtime
τ≤Tdefine
αtPift≤τ
βt:=αQelse
t

91

2.TIME-CONSISTENTSETSONFINITETREES

∗OuraimnowistoshowthatthisprocessliesinA,i.e.thatthereexistsa
P∗∈PsuchthatddPP0=E˜t(β).IfweplugβintoEquation(2.1)anddefine
t∗ybPdP∗E˜t(αP)ift≤τ
dP:=E˜t(αQ)E˜τ(αP)else.
0tE˜τ(αQ)
wenoticethatβ∈AisequivalenttoP∗∈P.ThefactthatP∗∈Phowever
followsdirectlyfromourassumptionoftime-consistency.
IfwenowcombinetheabovepropositionswehaveshownTheorem2.3.1.

2.4Necessity

Nowletuslookattheconversionofthetheoremabove.Thegoalofthis
sectionwillbetoshowthateveryAwiththeabovepropertiesdefinesa
time-consistentsetofmeasures.SoweseethatthepropertiesofAarenot
onlysufficientbutalsonecessary.Forthispurposewewillderiveasetof
measuresPfromagivensetAofpredictableprocesseswhichareassumed
tobecompact-valuedandstableunderpasting.Additionallyweclaimthat
Acontainstheprocessconstanttozero.Ourgoalwillbetoverifythatthe
derivedPsatisfiestheassumptionsmadeinthemodelspecifications.
Formallythiswillleadtofollowingtheorem

Theorem2.4.1.ForeverysetofpredictableprocessesAthatsatisfiesthe
propertiesshowninTheorem2.3.1thereexistsasetofmeasuresP,suchthat
PdA=αdP=E˜t(α),P∈P.
0tEveryPconstructedinthiswayhasthefollowingproperties:
•P0∈PandP∼locP0forallP∈P
•Piscompact
onsistent.time-cisP•

02

2.4.NECESSITY

2.4.1ConstructionofP
Ifweusethesameidentificationasinpart2.3.2betweentheprocesses
α(αt)t∈{0,...,T}andthedensitiesweareabletoconstructadensityprocess
ddPP0foreveryα∈A.
ttFromtheconstructionitfollowsimmediatelythattheobtainedprocesses
areP0-martingaleswithexpectation1andsincethetheyareclearlystrictly
largerthanzerotheyareindeeddensityprocesses.
Letusdefineournewsetofmeasuresby
PdFt0P:=PdP=E˜t(α)forα∈A.
Sincetheprocessα≡0isassumedtobeanelementofAwegetthat
P0∈P.FromthefactthatallP∈Pareconstructedviadensityprocesses
withrespecttoP0thatarestrictpositivewecanalsodirectlyconcludethat
ourmeasuresareallequivalenttoourreferencemeasure.

2.4.2Time-Consistency
AswhenshowingthatwecanderiveAfromPtime-consistencyinoursetP
isequivalenttostabilityunderpastinginoursetAandthusthisproperty
followsinstantlyfromourassumptions.

2.4.3Compactnessofdensities
HereagainthefactthattheE˜isabijectiveandcontinuousmappingisthe
reasonwhythecompactnessoftheα’simpliescompactnessofthedensities.
Andagainsummarizingtheabovepropositionsleadsustotheproofof
Theorem2.3.1.

12

2.TIME-CONSISTENTSETSONFINITETREES

2.5Examples

Inthissectionweintroducesomeexamplesforwhichthisresultisapplicable
andmightsimplifycalculations.

2.5.1BinomialTree
Themostbasicexampleonecanthinkofinthissettingisabinomialtree.
Ithasaconstantandfinitesplittingindexoftwo.Herethingsarestillvery
basictocalculate.Onecanforinstanceshowthataconvexsetofpriors
resultsinaconvexsetofprocessesandviceversawhichisingeneralnottrue
forahighersplittingindex.Putmoreformallywehave
Proposition2.5.1.Onabinomialtreeeveryconvexsetofmeasuresfulfill-
ingAssumptions2.2.1and2.2.2,i.e.P={(p1,...,pT)|pt∈[pt,pt]forall
t={0,...,T}},isequivalenttotherespectiveprocesseslyinginapredictable
interval[at,bt],wherept=P[Xt=up|Ft−1].
Proof.Fortheproofwewillworkourselvesthroughthetreesuccessivelyfor
everytimeperiodt.
Startingwitht=1thedensityforafixedPtakesfollowingform
dP0F1exp(αΔm1(up))+exp(αΔm1(down))
dP(up)=2p=2exp(αΔm1(up))
thiscanbetransformedto
α=ln1−p(m1(down)−m1(up))−1
pwhichisafunctionthatismonotoneandcontinuousinp.Soifp∈[p,p]
thenthisresultsinboundariesa,bwhichareF0-measurables.t.α∈[a,b].
Onecanshowtheconversionbythesameargumentationsincetheabove
formulacanbeconvertedtoafunctionp(α)whichisalsomonotoneand
continuousinα.Thereforeaconvexsetofα’sgivesusaconvexsetof
probabilities[pt,pt]wherept=Pinf∈PP[Xt=up|Ft−1].
22

2.5.EXAMPLES

Thiscaneasilybeextendedtofurthertimeperiodsbyjustlookingatthe
onestepaheadmeasuresordensitiesinananalogousway.
[Chudjakow&Vorbrink,09]presentapplicationsofthistoamericanex-
oticoptionsonabinomialtree.

2.5.2ExponentialFamilies
Afurtherexampleforexpressingtime-consistentsetsofmeasuresviapre-
dictableprocesseswasgivenin[Riedel,09].Heintroducedwhathecalls
dynamicexponentialfamilieswhichisthediscreteversionofκ-ambiguityin
[Chen&Epstein,02]butwithpredictablebounds.
Hestartswithaprobabilitystatespace(S,S,ν0)withS⊂Rd.Withthis
heconstructsaprobabilityspacewith(Ω,B,(Ft)t=1,...,T,P0),where
TS=Ω••B=tT=1Sσ-fieldgeneratedbyallprojectionst:Ω→S
•(Ft)generatedbythesequence(t)
•P0=tT=1ν0probabilitys.t.tiidwithdistributionν0
ThenbyassumingthatSeλ∙xν0(dx)<∞thelog-LaplacefunctionL(λ)=
logSeλ∙xν0(dx)iswelldefinedandwiththehelpofpredictableprocesses
(αt)thethendefinesdensitieson(Ω,B,(Ft)t,P0)via
Dtα:=exptαss−tL(αs).
=1s=1sThenforfixedpredictableprocessesa<bonegetsasetofdensitieswhich
definesatime-consistentsetofmeasuresbysetting
PdPa,b=PdP=Dtα,α∈[a,b].
0t32

32

0

2.TIME-CONSISTENTSETSONFINITETREES

2.5.3TrinomialTree
Thepurposeofthefollowingexampleistoshowthatswitchingbetweenthese
tworepresentationsdoesnotworktoowellingeneral.Startingwithatwo
periodtrinomialtreewhichmeanswehaveastatespaceΩ={s1,...,s9}and
theinformationstructureF0=Ω,F1={{s1,s2,s3},{s4,s5,s6},{s7,s8,s9}}
andF2={{s1},...,{s9}}wedefinetherathersimpletime-consistentset
111111
333333
P=+,+δ,−−δ,δ∈−,and+δ=.
Wethenconstructamartingalebasisinthistreewithrespecttotheuniform
distributionandthenshowwhatthissetlookslikeexpressedviapredictable
processesandourbasis.
Amartingalebasis{mt1},{mt2}inthiscaseisgivenby
m11m2−1
1121−2312−020−−4and−2
32−2111−11−−23−

Figure2.1:MartingaleBasis

Ifwenowcalculatetheprocessesthatbelongtoeachofthemeasures
aboveweobtain
fort=1andi=1,...,9

α11(si)=21ln1−13+δ3−3andα12(si)=31ln(1+31)(1+−3δ3−3δ)
42

andfort=2

2.5.EXAMPLES

1ln1−3−3δfori=1,2,3
√41+3δ
α21(si)=1ln(1−3−3δ)(1+3)fori=4,5,6
61+3δ
4ln1−3−3δfori=7,8,9
11+3
√1+331ln(1−3−3δ)(1+3δ)fori=1,2,3
1+31α22(si)=2ln1−3−3δ1+3δfori=4,5,6.
√δ1+331ln(1+3)(1−3−3δ)fori=7,8,9
Asonecanseeacomparablysimplesetintheonerepresentationcanbecome
relativelycomplicatedintheother.

@RDTV.4.52Anotherimportantareainwhichtime-consistentsetsofmeasureshavebeen
studiedareriskmeasures.In[Artzneretal.,99]itisshownthateverycoher-
entriskmeasureρthasarobustrepresentationinvolvingasetofmeasures
e..i,Pρt(X)=essinfEP[X|Ft].
∈PPThenin[Artzneretal.,07]itwasshownthatthefamilyofdynamicrisk
measuresρ=(ρt)tisdynamicallyconsistentiffthesetPistime-consistent.
[Roorda&Schumacher,05]introducedynamicallyconsistenttailvalueat
risk(DTV@R)asoneofthesetime-consistentriskmeasures.
AsthesetPtheytakeallmeasuresPforwhichtheonestepahead
densitieswithrespecttothereferencemeasureP0areboundedbyλ1where
λ∈(0,1]istheusualrisklevel.Ifwewanttodescribethisinourcharacter-
izationitgivesus
E˜t(α)exp(αt∙Δmt)1
=exp(αt∙Δmt−lnE[exp(αt∙Δmt)])=≤
E˜t−1(α)E[exp(αt∙Δmt)]λ
52

2.TIME-CONSISTENTSETSONFINITETREES

forallt=1,...Tandallα∈A.
ThisallowstocharacterizethesetAassoonasthemartingalebasisis
.edxfi

2.6PossibleExtensions
Inthissectionwewilldiscusspoaaibleextensionswhicharisequitenaturally.

2.6.1Convexity
Sincetime-consistentsetsareoftenusedinoptimizationproblemsconvexity
ofthesetsisoftenassumed.Itwouldbeniceifthisfeaturewouldcarryover
totheprocesses.Unfortunatelythisisnotthecaseingeneral,ascanbeseen
inthefollowingcounterexample.
Takeforexampleatrinomialtreewithstatess1,s2ands3andjustone
timeperiod.Asareferencemeasurewewillfix
111P0(s1)=,P0(s2)=andP0(s3)=.
442Asecondmeasurewillbegivenby
311Q(s1)=,Q(s2)=andQ(s3)=.
882ThedensityofQwithrespecttoP0willthenbe
dQdQ1dQ3
(s1)=1,(s2)=and(s3)=.
dP0dP02dP02
Sincewewanttoshowthatfromaconvexsetofmeasuresanon-convex
setofprocessescanarise,letusdefineoursetofmeasuresvia

P:=convH{P,Q}.
ThenletuslookatthesetofprocessesAarisingfromthisconvexset,espe-
ciallyαP0andαQ.NowifAwereaconvexset,theneveryconvexcombination
62

2.6.POSSIBLEEXTENSIONS

ofαP0andαQhastobeanelementofA.SinceαP0iszero,becausewechose
P0asourreferencemeasurewelookat21αQ.Ifwenowcalculatetheassoci-
ateddensitytothisprocess,weseethatitcanneveroriginatefromaconvex
combinationofouroriginalmeasuresandtherefore21αQ∈/AandhenceAis
notconvex.

2.6.2InfiniteHorizon
Whenextendingourstatementstoaninfinitetimehorizonletusfirstremark
thatourmodelassumptionscanallbetransferredwithoutcomplications.
Wewillhoweverneedafurtherassumptiononoursetofmeasures.This
assumptionwillbe
Assumption2.6.1.Thefamilyofdensitiesforafixedt
dP
0Dt:=dPFt|P∈P
isweaklycompactinL1(Ω,F,P0).
Technicallythisassumptionensuresthatwhenlookingatexpressionsof
thefollowingkindinfEP[Xτ]theinfimumisalwaysattainedforbounded
∈PPstoppingtimesτ.(cp.[Riedel,09])
[Arrow,71]alreadygivesaneconomicinterpretationofthispropertyby
claimingafeatureofpreferenceswhichisrelatedtothisassumptionin
[Chateauneufetal.,05].Theconditionweneedtoaskofthepreferences
toobtainthisfeatureiscalledMonotoneContinuity.Itmeansthatifanact
fispreferredoveranactgthenaconsequencexisneverthatbadthatthere
isnosmallpsuchthatxwithprobabilitypandfwithprobability(1−p)is
stillpreferredoverg.Thesameistrueforgoodconsequencesmixedwithg.
Criticstendtoobjecttothisassumptionbysayingthatiftheprobability
ofdyingisaddedtothebetteractfthensurelythepreferenceshaveto
bereversed.Howeverifwetakefforgetting100dollarsandgforgetting
72

2.TIME-CONSISTENTSETSONFINITETREES

nothingthenhavingtodrive60milestogetthe100dollarsandsoaddinga
smallprobabilityofgettingkilledwillnormallynotreversethepreferences.
Expressedformallythismeansforactsfg,aconsequencexanda
sequenceofevents{En}n∈NwithE1⊇E2⊇...and∩n∈NEn=∅thereexists
ann¯∈Nsuchthat
xifs∈En¯xifs∈En¯
f(s)ifs∈/En¯gandfg(s)ifs∈/En¯.
Theconstructionoftheprocessescanalsobemaintained,sincetheyare
alwaysconstructedforafixedtimehorizonuptoatimet.Thatisalsothe
reasonwhythemappingfromourdensitiestoourprocessesstillinhabits
thesamefeatures,i.e.itiscontinuousandbijective.Thereforeinthiscase
thecompactnessalsocarriesoverfromonesidetotheother.Itisalsoclear
thatstabilityunderpastingisequivalenttotime-consistencyforaninfinite
horizonaswell.Soaltogetherourstatementscansmoothlybeconverted
fromafinitetoaninfinitetimehorizon.

2.6.3LooserAssumptionsonSplittingFunction

Sinceourassumptionsonthefiltrationareveryrestrictive,itwouldbenice
iftheycouldberelaxedinonewayoranother.
Onewaywouldbetogiveuptheassumptionofaconstantsplitting
function.Inthiscasehoweveryourunintotheproblemthattheα’sthat
arisefromthemartingalerepresentationarenolongeruniqueandwiththat
themappingnolongerdistinctandbijective.
Asecondwayisallowingforthesplittingvaluetobecomeinfinite.This
howeverhastheconsequencethatthemartingalerepresentationwillnot
necessarilyexistanymore.
82

2.7Conclusions

2.7.CONCLUSIONS

Forourspecialsetting,i.e.discreteandwithspecialassumptionsonthe
informationstructure,wehaveconstructedanalternativecharacterization
fortime-consistentsetsofmeasures.Wehaveshownthatallsetsoftime-
consistentsetsofmeasurescanbeexpressedbypredictableprocessesand
viceversa.
Ascanbeseenintheextensionsstandardgeneralizationsfailtowork.So
asfarasIamconcernedthisisthemostgeneralizationthatcanbeformulated
inthissetting.
Forpracticalapplicationswehaveshownthatforproblemswhichcanbe
modeledintheformofdecisiontrees(withaconstantnumberofbranches
e.g.trinomialtrees)wenowknowwhatatime-consistentsetofmeasures
mustlooklikeexpressedviapredictableprocesseswhichmightsimplifycal-
culations.Sohopefullyourconstructionwillbehelpfulinthefuturee.g.for
solvingOptimalStoppingProblemswhichcanbemodeledinthisframework.

29

3erhaptC

ADualityTheoremfor
OptimalStoppingProblems
underUncertainty

3.1Introduction

ManyinvestmentproblemsarisinginEconomics,FinanceorgeneralDecision
Theorydonothaveafixedpointintimewhenthedecisionmustbemade
butatimeperiodinwhichthechoiceofinvestmentispossible.Especiallyif
thedecisionisirreversiblewhichmeanstheinvestmentcannotberecovered
withoutconsiderablelossesoptimaltimingiscrucial.Importantexamplesfor
thesekindofinvestmentsarethemarketentrytimeofafirm,theoptimal
timetoinstallanewtechnologyortheexercisestrategyofanAmerican
optionbutalsoatwhatbidtosellahouseorwhichjoboffertoaccept.This
classofproblemsiscalledoptimalstoppingproblems.
Eachoftheseproblemscanbemodeledbyoptimallytryingtostopa
stochasticprocess(Xt)twhichdescribesthefuturerandompayoffs.Clas-
sicaldecisiontheory(e.g.[vonNeumann&Morgenstern,44])proposesto

2.DUALITYTHEOREM

maximizetheexpectedpayoffunderagivendistribution,i.e.
maximizeE[Xτ]amongallstoppingtimesτ.
Butwhatifthedistributionof(Xt)tisnot(exactly)knownoroneis
unsureaboutitstrueform.Inthiscasethecurrentliteratureonwhatis
calledambiguityoruncertaintyaversionoftenrevertstothemultipleprior
modelintroducedin[Gilboa&Schmeidler,89]forthestaticcaseandin
[Epstein&Schneider,03]foradynamicsetting.Theyproposetolookat
awholesetofpossibledistributionsandtotaketheworstexpectedvalueas
afoundationofdecision-making,i.e.
PmaximizePinf∈PE[Xτ]amongallstoppingtimesτ
wherePisasetofmeasureswithspecificproperties.
Onereasonthiswayofmodelingdecisionsemergedwasthatempirical
studiese.g.[Ellsberg,61]gavesubstantialevidencethatdecisionmakersnot
onlyinhabitriskaversionbutalsouncertaintyaversion.
Whatisthedifferencebetweenriskanduncertainty?Whentalkingabout
riskonemeanstherandomnessthatisinherentinagivenandfixeddistribu-
tionwhileuncertaintyorambiguitydescribesafurthersourceofrandomness
whichspringsfromlackingknowledgeofthecorrectdistribution.Thisno-
tionwasfirstintroducedin[Knight,21],andhenceisalsooftenreferredto
asKnightianUncertainty.
Thisansatzcanfurtherbemotivatedusingtheframeworkofincomplete
marketswheretheequivalentmartingalemeasureisnolongeruniqueand
thereforeoneobtainsawholesetofmeasures.Alternativelyonecanthink
ofthesetofmeasuresbeingslightvariationsofthemeasureonethinks
therightone,thiscorrespondstotestingtherobustnessofamodel(cp.
[Hansen&Sargent,01]).
Thequestionwenowwanttostudyis,cantheorderofminimizingover
thedistributionandmaximizingviathestoppingtimebeswitched.Since
23

3.1.INTRODUCTION

forseveraloptimalstoppingproblemstheclassicalsolution,i.e.withone
fixedandknowndistribution,iswellstudiedthedualitycanbehelpfulin
solvingtheseproblemsunderuncertainty.Thereforethemainobjectiveof
thisworkistostudyunderwhatconditionsitispossibletofirstmaximize
overallstoppingtimesandthenminimizeoverthedistributions.Expressed
moreformallywewillprovethefollowingdualitytheorem
supinfEP[Xτ]=infsupEP[Xτ].
τ∈SP∈PP∈Pτ∈S
Remark3.1.1.Thisproblemcanalsobeseenasastochasticgamebetween
twoplayers,whileoneplayeristhe“maximizer”pickinganoptimalstop-
pingtimethesecondisthe“minimizer”choosingthedistribution.Whatour
theoremthenshowsisthatitisirrelevantinwhichordertheymaketheir
decisionssincethe“value”processofthisgameisalwaysthesame.
Inthispaperweproveinarathergeneralsettingaminimaxtheoremfor
optimalstoppingproblemsincontinuoustimeunderKnightianuncertainty
anddeduceanoptimalstoppingrule.Morepreciselyformildassumptionson
thepayoffprocessX(i.e.rightcontinuous,classofP-D,uppersemicontin-
uous,adaptedandana.s.finiteoptimalstoppingtime)andratherstandard
assumptionsonP(i.e.absolutecontinuity,weakcompactnessofdensities
andtime-consistency)weobtainthat
PPesssupessP∈PinfE[Xτ|Ft]=essP∈PinfesssupE[Xτ|Ft]
τ≥tτ≥t
andthatanoptimalstoppingstrategyisgivenby
P∗ρ0=inf{s≥0|esssupessP∈PinfE[Xτ|Fs]=Xs}.
s≥τWealsoshowthatthisstoppingtimeistheminimumofalloptimalstopping
rulesfortheclassicalsolutions,i.e.
ρ0∗=infρ0P,whereρ0P=inf{s≥0|esssupEP[Xτ|Fs]=Xs}.
∈PPs≥τ

33

2.DUALITYTHEOREM

Agreathelpintheproofofthemaintheoremisanexplicitbutgeneral
constructionfortime-consistentsetsofmeasuresintroducedin[Delbaen,03]
andbrieflyreviewedinSection3.4.Heshowsthateveryconvexandtime-
consistentsetofmeasurescanbeexpressedwiththehelpofaconvexvalued
correspondence.Healsoshowsthatstartingwithamartingaleandaconvex
valuedcorespondence,onecanconstructaconvexandtime-consistentsetof
es.rsueamInadditiontothetheoremwithitsimplicationsweapplytheresultsto
specialclassesofpayoffprocesses.Firstwelookatthecasewhere(Xt)tis
eitheramultiplepriorsub-orsupermartingale,whichleadstostoppingat
thelastpossibleperiodifthereisafinitetimehorizonforthesubmartingale
ortostoppingimmediatelyincaseofthesupermartingale.Afterthatwe
showhowthetheoremhelpsidentifytheworstcasedistributioninanadap-
tionofκ-ambiguity1introducedin[Chen&Epstein,02].Thisallowsforthe
ambiguousstoppingproblemtobetransformedintoaclassicalone.
Discreteversionsofthetheoremcanbefoundin[Fo¨llmer&Schied,04]
and[Riedel,09]whoalsopresentsapplications.In[Karatzas&Kou,98]one
canfindthecontinuoustimecaseforafinitetimehorizonandastrong
focusontradingconstraints.Unliketheirpaperweexplicitlyincludethe
infinitetimehorizonandembedtheexplicitconstructionforallsetsoftime-
consistentsetsofmeasuresintroducedin[Delbaen,03]intotheproof.
Thepaperisorganizedinthefollowingway.Inthenextsectionthe
modelwillbediscussedinmoredetailandtheassumptionswemakejustified.
Afterthattheproblemthatistobesolvediselaboratedmorethoroughly
inSection3.3.Section3.4containsaconstructivedescriptionofthetime-
consistentsetswelookatincludingtheadaptedversionofκ-ambiguity.The
succeedingSection3.5containstheproofofourmaintheoremwhichweapply
1Sincewearegoingtolookatproblemswithaninfinitetimehorizonandtheclassical
versionofκ-ambiguityfailstosatisfytheNovikovconditioninT=∞weneedtoadapt
thisconceptabit.

43

3.2.MODEL

todifferentstoppingproblemsinSection3.6andSection3.7concludes.

delMo.23Asafoundationforourmodelwebeginwiththefilteredprobabilityspace
Ω,F,P0,(Ft)t∈[0,∞],wherethefiltrationsatisfiestheusualassumptions,
alsoF0istrivialandFtheσ-fieldgeneratedbytheunionofallFt.Wewill
denotetheclassofallstoppingtimesτofthefiltration(Ft)twhichsatisfy
P0(τ<∞)=1bySandthosethatarelargerthanorequaltoat∈[0,∞)
bySt:={τ∈S|τ≥t}.
Furtherlet(Xt)t∈[0,∞]bearightcontinuousandadaptedprocessdescrib-
ingthepayofffromstopping.Ourdecisionmaker’staskistochooseastop-
pingtimeτofthefiltration(Ft)t.Ifhechoosesthestoppingruleτhegains
thepayoffXτ(ω)=Xτ(ω)(ω)forω∈Ω.Hisgoalistomaximizehisexpected
reward.Sinceourmodelisplacedinanambiguoussettingourdecisionmaker
isuncertainaboutthetruedistributionofX.Inordertocapturethedecision
maker’suncertaintyaversionwewillusetheRecursiveMultiplePriorModel
introducedin[Epstein&Schneider,03].Asaconsequenceheconsidersaset
ofprobabilitydistributionsPon(Ω,F)whichheallassumespossibleand
his(minimax)expectedrewardforstoppinginτisgivenby
infEP[Xτ].(3.1)
∈PPRemark3.2.1.Forsimplicitywewillonlylookatriskneutraldecisionmak-
ers.Pluswewillnotexplicitlymentiondiscountingoraspecialutilityfunc-
tion.Sincetheexpectedpayoffshouldbewell-definedforallpossiblestopping
timesweintroducefollowingnotionofclassP−Dandassumethisproperty
.XrfoDefinition3.2.2.Wesayaright-continuousprocess{Xt,Ft;0≤t<∞}is
53

2.DUALITYTHEOREM

ofclassP−Dif

PPsup∈Pτsup∈SE[|Xτ|]<∞and
Klim→∞Psup∈Pτsup∈S|Xτ|>K|Xτ|dP=0.
Remark3.2.3.Thispropertyisauniformintegrabilityconditionforam-
biguoussettings.Wenotonlyaskforuniformintegrabilityunderafixed
distributionP0butunderawholesetP.Howeverweonlylookatstopping
timesτandnotatthewholetimeset.
AfurtherpropertywewanttoassumeforthepayoffprocessXisupper
semicontinuity.
Definition3.2.4.AstochasticprocessXisuppersemicontinuousinex-
pectationfromtheleftwithrespecttotheprobabilitymeasureP0ifforany
increasingsequenceofstoppingtimes{τi}i∞=1convergingtoτ,wehave
limsupEP0[Xτi]≤EP0[Xτ].
→∞iThisensuresthatthelowerSnellenvelope
V∙=essP∈Pinfessτ∈S∙supEP[Xτ|F∙]
hasacadlagmodificationwiththeimportantconsequencethatforthestop-
pingtimesρt:=inf{s≥t|Xs=Vs}oneactuallyobtainsVρt=Xρt.A
thoroughdiscussionoftheseresultsalongwithadifferentapproachtoThe-
orem3.5.1canbefoundin[Trevi˜no,09].
ForthesetPwewillalsomakesomeassumptions.Firstofall,formainly
technicalreasonsweassume
Assumption3.2.5.P0∈PandallothermeasuresP∈Pareabsolutely
continuouswithrespecttoP0,i.e.P(A)=0ifP0(A)=0forallA∈F.
AdditionallywewillaskforPtobeconvex,i.e.forλ∈(0,1)andQ,P∈P
wehaveλQ+(1−λ)P∈P.
63

3.2.MODEL

ThisassumptionmerelyletsP0fixsomesetsofmeasurezeroandserveas
areferencemeasure.Thishasnoinfluenceonthestochasticstructureofthe
othermeasures.ItsimplyimpliesthatallmeasurescontainedinPhaveat
leastthesamenullsetsasP0whicheconomicallytranslatestothedecision
makerknowingsomesureandimpossibleevents.Technicallyitallowsus
toidentifyeachmeasureP∈PwithitsRadon-NikodymdensityddPP0with
respecttoP0.
Thesecondpartoftheassumptionassuresthatthesetsatisfyingthe
morestringentconstraintofmutualcontinuity,i.e.Pe:={Q∈P|Q∼P0}
liesdenseinPsinceeachQ∈PcanbeapproximatedbyelementsinλQ+
(1−λ)P0∈Pe.Thereforeweachievethesamebehavioralimplicationsfor
ouroptimalstoppingproblem.
Fortheassumptionofmutualcontinuityaninterpretationwasgivenin
[Epstein&Marinacci,06].Theyrelatedittoanaxiomonpreferencesfirst
postulatedin[Kreps,79].Heclaimedthatifadecisionmakerisambivalent
betweenanactxandx∪xthenheshouldalsobeambivalentbetweenx∪x
andx∪x∪x.Meaningifthepossibilityofchoosingxinadditiontox
bringsnoextrautilitycomparedtojustbeingabletochoosex,thenalsono
additionalutilityshouldarisefrombeingabletochoosexsupplementaryto
.xx∪ThesecondassumptionforoursetPwillensurethattheinfimumin
(3.1)isalwaysattainedforboundedstoppingtimesτ(cp.[Riedel,09]).We
emssuaAssumption3.2.6.Thefamilyofdensities
D:=dP|P∈P
Pd0isweaklycompactinL1(Ω,F,P0).
Aneconomicinterpretationofthispropertywasgivenby[Arrow,71]in
claimingafeatureofpreferenceswhichwasrelatedtothisassumptionin
73

2.DUALITYTHEOREM

[Chateauneufetal.,05].Theconditionweneedtoaskofthepreferencesto
obtainthisfeatureiscalledMonotoneContinuity.Itmeansthatifanactf
ispreferredoveranactgthenaconsequencexisneverthatbadthatthere
isnosmallpsuchthatxwithprobabilitypandfwithprobability(1−p)is
stillpreferredoverg.Thesameistrueforgoodconsequencesmixedwithg.
Criticstendtoobjecttothisassumptionbysayingthatiftheprobability
ofdyingisaddedtothebetteractfthensurelythepreferenceshaveto
bereversed.Howeverifwetakefforgetting100dollarsandgforgetting
nothingthenhavingtodrive60milestogetthe100dollarsandsoaddinga
smallprobabilityofgettingkilledwillnormallynotreversethepreferences.
Expressedformallythismeansforactsfg,aconsequencexanda
sequenceofevents{En}n∈NwithE1⊇E2⊇...and∩n∈NEn=∅thereexists
ann¯∈Nsuchthat
xifs∈En¯xifs∈En¯
f(s)ifs∈/En¯gandfg(s)ifs∈/En¯.
emoblPr.33Thequestionwewanttostudyintheabovesettingishowtosolveoptimal
stoppingproblemsofthefollowingform
maximizePinf∈PEP[Xτ]overallstoppingtimesτ∈S.
IfPissingletontheproblemreducestosubjectiveexpectedutilityand
thesolutioniswell-known.Weknowanoptimalsolutionisgivenby
τ∗=infs≥0|Xs=τ∈SsupsE[Xτ|Fs]
seeforexample[ElKaroui,81].Aswewillseelateronthesolutiontoour
problemincludinguncertaintyisverysimilartothisone.
Inadiscretesettingtheproblemwasstudiedin[Riedel,09].Heshows
thatwithanaddedconditiononPheattainsfollowingoptimalstopping
83

3.3.PROBLEM

timeasaresultfortheproblem
∗τ=inf{s≥0|Xs=Us},whereUs=esssupessP∈PinfE[Xτ|Fs].
∈SτsHealsoshowsthatunderthisextraassumptionandwithafinitetimehorizon
Uscanbeobtainedrecursivelybysetting
UT=XTandUs=maxXs,essinfEP[Xs+1|Fs]∀s=0,...,T−1.
∈PPThisimportantconditiononPiscalledtime-consistency.Itisacrucial
assumptionformakingdynamicallyconsistentdecisions.Inthediscreteset-
tingforinstanceitallowstheuseofbackwardinduction.Moregeneralit
impliesfollowingversionofthelawofiteratedexpectations:

essP∈PinfEessP∈PinfE[Xτ|Ft]|Fs=essP∈PinfE[Xτ|Fs]fort≥s.
Thiscanbeinterpretedinthefollowingway,ifthedecisionmakersettlesfor
thestoppingruleτthenhisexpectedreturnattimesisthesameasthe
expectationinsofwhathewouldgetifhechoseτint.
Atechnicalformulationofthispropertycanbefoundinthefollowing
finalassumptiononoursetP.
Assumption3.3.1.Pistime-consistent.Thismeansthatforeverystop-
pingtimeτandeverypairP1,P2∈Pwithdensityprocessespt1=ddPP01and
pt2,respectively,themeasureQdefinedbythedensityprocess
dQpt1ift≤τ
dP0Ft=ppτ12pt2else
τbelongstoPaswell.
Furtherimplicationsorequivalentdefinitionscaninteraliabefoundin
[Delbaen,03]or[Riedel,09].

93

2.DUALITYTHEOREM

3.4ConstructionofP
AsistypicalforthesemodelswewilluseourfirstassumptiononP,theab-
solutecontinuity,todescribeoursetofmeasuressinceitallowsustoidentify
eachmeasurewithitsdensityfunctionwithrespecttothereferencemeasure
P0.Afurtherusefulandwell-knownfactisthatforagivenmartingale(Mt)t
withrespecttoP0and(Ft)tdensitiescanbegeneratedbypredictablepro-
cessessincethestochasticexponentialE(θ∙M)describesadensityifitisa
non-negativemartingale.
Remark3.4.1.AsareminderforasemimartingaleXwithX0=0the
stochasticexponentialisdefinedinthefollowingway:

Et(X)=expXt−1[X,X]t(1+ΔXs)exp−ΔXs+1(ΔXs)2
20<s≤t2
whereΔX=Xs−Xs−andtheinfiniteproductconverges.
In[Delbaen,03]onecanfindathoroughstudyofsuchconstructions,
heshowsthatundercertainassumptionstime-consistentsetscanalwaysbe
representedinthiswayandgivesadescriptionofwhatthesetofdensities
willlooklike.Moreexplicitheprovesforeveryconvexandtime-consistent
setofdensitiesPcontainingthereferencemeasureP0,thatifthereexistsa
continuousmartingale(Mt)tsuchthatforallmeasuresP∈Pethereexists
apredictableprocessθsuchthatEP0ddPP0Ft=Et(θ∙M),thenthere
existsapredictable,convexcorrespondenceC:R+×Ω→B(Rd)suchthat
0∈C(t,ω)forall(t,ω)andsuchthat
θP=clPewithPe=Pθ|dP=E(θ∙M)withθ∈Θ
Pd0eerwhΘ={θ|θisapredictableprocesswithθ(t,ω)∈C(t,ω),
s.t.E(θ∙M)isapositiveuniformlyintegrablemartingale}

04

3.4.CONSTRUCTIONOFP
andtheclosureistakenwithrespecttotheL1normonthespaceofthe
densityprocesses.
Remark3.4.2.Theuniformintegrabilityofthestochasticexponentialguar-
anteesthatthearisingdistributionsareabsolutelycontinuouswithrespectto
P0andthepositivity,i.e.E∞(θ∙M)>0,evenguaranteesequivalence.
Remark3.4.3.Twoimportantexamplesforthesekindofsetsaretheex-
tremeconstructionswhereC(t,ω)={0}andC(t,ω)=Rd.Inthefirstcase
P={P0},i.e.singleton,andinthesecondcasePisthesetofallabsolutely
continuousprobabilitymeasureswhosedensitieshavetheappropriateform.
Remark3.4.4.Delbaenalsoshowstheconversionofthetheoremusedabove
todescribeourset.HeshowsthatforamartingaleMandapredictable
convexcorrespondenceC:R+×Ω→B(Rd)satisfying0∈C(t,ω)forall
(t,ω)andthattheprojectionofContothepredictablerangeofMisclosed2
wecanconstructatime-consistentconvexsetofmeasures.

3.4.1κ-Ambiguity
Anexplicitexamplefortheconstructionofsuchasetontheprobability
space(Ω,F,(Ft)t∈[0,∞],P0)isgiveninthefollowing.Itisstronglyrelated
toκ-ambiguityintroducedin[Chen&Epstein,02],butsincetheclassical
definitiondoesnotfulfilltheNovikovconditioninT=∞weneedtoslightly
adaptit.
FixaP0-martingale(Mt)t∈[0,∞]andthesetofpredictableprocesses
Θ:=(θt)t∈[0,∞]|θpredictableprocesswith|θt|≤κt
1t1≤whereκt=12else.
t2Thisassumptionismadeinordertodealwiththosedensitygeneratorsθthatarenot
identicallyzerobutaresuchthatθ∙Miszero.Itguaranteesthattheelementslyinginthe
closureoftheconstructedsetthatareequivalenttoP0alsohavetheformofastochastic
exponential.Formoredetailsweagainreferto[Delbaen,03].
14

2.DUALITYTHEOREM

HereC(t,ω)=[−κt,κt]isapredictableandconvexcorrespondencewith
0∈C(t,ω)forall(t,ω).
Usingtheanalysisfoundin[Czichowsky&Schweizer,09]itfollowsthat
theprojectionofContothepredictablerangeofMisclosedandsinceκt
tendstozerofastenoughtofulfilltheNovikovconditionwegettheposi-
tivityandtheuniformintegrabilityofthestochasticexponential.Therefore
Theorem1in[Delbaen,03]tellsusthattheL1-closureof
Pd0Pe=PθdP=E(θ∙M),θ∈Θ
isatime-consistentandconvexset.
ThesetP=clPeisalsoaweaklycompactset,sinceitisweaklyclosed,
uniformlyintegrable,andboundedhenceitisasetfulfillingallourassump-
.sonitRemark3.4.5.Whenconstructingtime-consistentsetsinasettingwith
aninfinitetimehorizoninthisfashiononeneedstoconsiderthatmany
martingalesknowntouswhent∈[0,∞[nolongersatisfythemartingale
conditionwhent=∞isincluded.Aprominentexampleforthisisthe
geometricBrownianmotion.

3.5MainPart
Beforewecometothemaintheoremofthepaperletusmakesurethecon-
ditionalexpectationswewillspeakaboutareproperly,i.e.P0-a.s.,defined.
Sinceoursetsofmeasuresconstructedwiththehelpof[Delbaen,03]arecon-
vexweknowthateverymeasureQ<<P0canbeapproximatedbymeasures
Qn∼P0.HencePeliesdenseinPandwedefine
essinfEP[Xτ|Ft]:=essinfEP[Xτ|Ft].
e∈PP∈PPTosimplifynotationswewilldefinethevaluefunctionofstoppingunder

24

3.5.MAINPART

theworstcasemeasureby
Ut:=esssupessinfEP[Xτ|Ft],
∈PP∈Sτtthevaluefunctionoftheinterchangedproblemby
PVt:=essP∈PinfesssupE[Xτ|Ft]
∈SτtandforfixedPwedefine
UtP:=esssupEP[Xτ|Ft].
∈SτtTheequationsarealltobeunderstoodP0-almostsurely.Additionallywe
willaskforthestoppingtimesρt:=inf{s≥t|Xs=Us}tobeP0-almost
surelyfiniteforallt<∞.
ThemainstatementofthispaperisthefollowingDualityTheoremwhich
canhelpsolveoptimalstoppingproblemsintheabovemodelsinceitallows
tointerchangetheinfimumandsupremum.
Theorem3.5.1.ForX,FandPsatisfyingtheaboveclaimsweget
Ut:=esssupessinfEP[Xτ|Ft]=essinfesssupEP[Xτ|Ft]=:Vt
τ∈StP∈PP∈Pτ∈St
Remark3.5.1.Asisalwaysthecasewithstatementsofdualityoneinequal-
ityistrivial.Soinourcaseweonlyhavetoshowthatthelefthandsideis
greaterthanorequaltotherighthandside.
Proofofthetheorem.Theproofwillconsistoftwoclaimsleadingtothe
mainstatement.Thefirstclaimis
1:Claim

Vt=essinfEP[Xρt|Ft]forall0≤t<∞,
∈PPwhereρt=inf{s∈[t,∞)|Xs=Vs}
andfromthisclaimthetheoremresultsatonce.

34

2.DUALITYTHEOREM

Inordertoprovethisclaimfirstobservethatρs∈Sandweimmediately
obtainthatthel.h.s.isgreaterthanorequaltother.h.s.Theopposite
inequalityremainstobeshown.
Todoso,wepropose
2:Claim

Vs≤EQ[Vρ|Fs]forallQ∈Pandstoppingtimess≤ρ≤ρs
Thisimmediatelyyieldsthefirstclaimbysettingρ=ρs.
Remark3.5.2.Inthefollowingtheexplicitconstructionofthesetofmea-
suresasdescribedinSection3.4willplayaveryprominentrole.Insteadof
directlyworkingwiththedistributions,wecanrestrictourselvestothedensity
ators.generWestarttheproofofthesecondclaimbyfirstfixinganarbitrarymeasure
Q∈PwhosedensityprocessregardingP0isE(θ∙M)t.Nowlookatasequence
{θk}k∈Nwhereθk∈Θforallkandallθkcoincidewithθonthestochastic
interval[[s,ρ]]andwhichsufficesfollowingconvergence
kPklim→∞essτ∈StsupE[Xτ|Ft]=Vt,
wherePkdenotesthemeasurewiththedensityE(θk∙M).
Forsuchasequencetoexistweneedtoshowthattheset

Φ:={essτ∈StsupEP[Xτ|Ft]|P∈P}
isdirecteddownwards,i.e.forallφ1,φ2∈Φwehavethatφ1∧φ2∈Φ.This
followsatoncefromLemma17of[Delbaen,03],whereheshowsthat
Φ˜:={EP[Xτ|Ft]|τ≥tisastoppingtimeandP∈P}
isalattice.
Remark3.5.3.Intherespectiveproofonecanseethattime-consistencyis
acrucialassumptionhere.

44

3.5.MAINPART

ThissequenceisdominatedbyesssupP∈Pesssupτ∈StEP[Xτ|Ft],which
isQ-integrableduetoXbeingofclassP−D.NowapplyingtheDomi-
natedConvergenceTheoremandusingthefactsthatthestoppedprocess
UPt∧ρst≥sisaP-martingaleforρsP=inf{u≥s|Xu=UuP}(cpe.g.
[Karatzas&Shreve,98])andρt≤essinfP∈PρtP(cpsubsequentlemma)we
getthesecondclaimthrough
k∈SτρEQ[Vρ|Fs]=EQklim→∞esssupEP[Xτ|Fρ]Fs
=limEQesssupEPk[Xτ|Fρ]Fs
k→∞τ∈Sρ
→∞k∈Sτ=limEPkesssupEPk[Xτ|Fρ]Fs
ρ∈PP≥essinfEPUρPFs=EP∗[UρP∧∗ρs|Fs]
∗=UsP∧ρs≥essP∈Pinfessτ∈SsupEP[Xτ|Fs]
sV=swhereP∗denotestheminimizingPintheforegoingequation.

Tocompletetheproofwestillneedtoshow
Lemma3.5.4.Defining
•ρt:=inf{s≥t|Xs=Vs}and
•ρtP:=inf{s≥t|Xs=UsP}
thenitholdsthatρt≤ρtPforallP∈P.
Proof.WeknowfollowingequationsholdalmostsurelyforallP∈P
1.Xs<VsandXρt=Vρtforallt≤s<ρt
2.Xs<UsPandXρtP=UρPtPforallt≤s<ρtP

54

2.DUALITYTHEOREM

3.Vs≤UsPforalls∈[0,∞]
Nowifweassumeρt>ρtPitfollowsfrom1and2that

PVρtP>XρtP=UρtP

whichclearlycontradicts3andthereforeρt≤ρPtforallP.
Intuitivelythismeansour“worstcase”decisionmakerhasamorepes-
simisticapprehensionofthefuturethantheotherinvestorsandhencehe
valuestheexpectedpayofflower.Soheismorelikelytoacceptwhathehas
earlierbecauseheexpectslessinthefuture.
Withthehelpofthetheoremwecannowimmediatelyshowthattheρtin
theproofgivesusasolutiontoouroptimalstoppingproblemandweobtain.

Corollary3.5.5.(i)Uisthesmallestmultiplepriorsupermartingalewith
respecttoPthatdominatesX.
(ii)Anoptimalstoppingruleisgivenbyρ0∗=inf{s≥0|Us=Xs}.
Proof.Wefirstshowthat(Ut)t∈[0,∞]isamultiplepriorsupermartingale.
Fort≥swehave
essP∈PinfEP[Ut|Fs]=essP∈PinfEPessP∈PinfEP[Xρt|Ft]|Fs
=essP∈PinfEP[Xρt|Fs]≤essτ∈SssupessP∈PinfEP[Xτ|Fs]=Us.
Remark3.5.6.Thesecondequalityisagainduetoourtime-consistency
assumption.

Nextweshowthatitisthesmallestmultiplepriorsupermartingaledomi-
natingX.ForthisassumethatWisanothermultiplepriorsupermartingale
dominatingX,thenitholdsthat

64

essinfEP[Xτ|Ft]≤essinfEP[Wτ|Ft]≤Wt∀τ∈St
∈PP∈PP

3.6.APPLICATIONS

andinparticular
PUt=essτ∈StsupessP∈PinfE[Xτ|Ft]≤Wt.
Thelastthingwewanttoshowistheoptimalityofρ0∗,thisfollowsdirectly
withthehelpofthefirstclaimintheproof,since

PPinf∈PEXρ0∗


=V0=U0=3.6Applications

supτ∈SinfP∈PEP[Xτ]≥infP∈PEP[Xτ]∀τ∈S.3.6.1Sub-andSupermartingales
Withthehelpofthistheoremwewanttosolveoptimalstoppingproblems.
Twostraightforwardexamplesareifthepayoffprocess(Xt)t∈[0,T]iseithera
multiplepriorsub-orsupermartingale.
Inthecaseofthemultiplepriorsubmartingalewhichmeans
essinfEP[Xt|Fs]≥Xsforallt≥s
∈PPwecanshowthat
PPsupPinf∈PE[Xτ]=Pinf∈PE[XT].
∈SτThismeansitisoptimaltowaituntiltheverylastperiodtostopandthe
expectedpayoffistheexpectedpayoffinthelastperiodundertheworstcase
e.rsueamToseewhythisistruefirstofallremarkthatthankstotheoptional
samplingtheorem

EP[XT]=EP[EP[XT|Fs]]≥EP[Xs]
X≥s

74

2.DUALITYTHEOREM

forallstoppingtimess≤TandfixedPsincemultiplepriorsubmartingales
aresubmartingalesforallP∈P.
Thereforesupτ∈SEP[Xτ]=EP[XT]forallPandweget
supPinf∈PEP[Xτ]=Pinf∈PsupEP[Xτ]=Pinf∈PEP[XT]
∈Sτ∈Sτandwiththattheabovestatement.
Inthecaseofamultiplepriorsupermartingaleitturnsoutthatstopping
immediatelyisoptimal.SinceXbeingamultiplepriorsupermartingale
sneamPessP∈PinfE[Xt|Fs]≤Xsforallt≥s
etgewPinf∈PEP[Xτ]≤X0∀τ∈S
andobtainthatstoppingimmediatelyisoptimal.

3.6.2ExploitingMonotonicityintheDrift
Inthefollowingletµ,σ,θ:R+×R→Rbecontinuous,boundedandadapted
functions.Forconvenienceweabbreviatethefunctionsµ(t,Xt),σ(t,Xt)and
θ(t,Xt)byµt,σtandθt.Letbθ=µ+θσfulfillaLipschitzcondition,i.e.

θθ|bt(x)−bt(y)|≤K|x−y|forapositiveconstantK.
Ontopofthisletσsatisfy|σ|≥>0and
|σt(x)−σt(y)|≤h(|x−y|)
whereh:]0,∞[→]0,∞[isastrictlyincreasingfunctionwithh(0)=0and
2−(0,)h(u)du=∞∀>0.
Fallingbackontheexampleforconstructingtime-consistentsetsviaour
adaptionofκ-ambiguityinSection3.4.1wewillinthissectionshowhowthe
theoremhelpsidentifytheworstcasemeasureinthissetinthecaseofa
84

3.6.APPLICATIONS

Brownianmotionwithdriftaspayoffprocess.Withthiswemeanthatour
payoffprocesshasfollowingdynamics
dXt=µ(t,Xt)dt+σ(t,Xt)dWt0
whereW0isaBrownianMotionwithrespecttoourunderlyingprobability
space(Ω,F,(Ft)t,P0).
Sotheproblemwewanttostudyis
supinfEP[Xτ]
∈PP∈SτfortheaboveXandthePfromSection3.4.1.Forsimplicitywewillrestrict
ourselvestoonedimensionalprocessesgivingustheadvantagethatwecan
useacomparisontheoremforourdriftslateron.
Thefirststepinouranalysiswillbetotransfertheambiguityimple-
mentedbyoursetofmeasuresontotheprocess,wewanttostop.Themain
toolforthiscanbefoundinthetheoryofweaksolutionsforstochasticdif-
ferentialequations(SDEs)andMarkovprocesses(e.g.cp[Revuz&Yor,91]
and[Shiryayev,78]).
IfaSDEhastwoweaksolutions
(Xi,Wi),(Ωi,Fi,(Fti)t,Pi)fori=1,2withX1=X2=x,
meaninginourcasethat
dXti=µ(t,Xti)dt+σ(t,Xti)dWti
whereWiisaBrownianmotionwithrespectto(Ωi,Fi,(Fti)t,Pi)fori=1,2,
thenweknowthatX1andX2havethesamelaw,i.e.
P1[X1∈Γ]=P2[X2∈Γ]forΓ∈B(C(R+)).
Inourcasethismeansifwedefineafurtherauxiliaryprocess(Xtθ)tvia
0θdXt=(µt+θtσt)dt+σtdWt
94

2.DUALITYTHEOREM

thenwiththehelpofGirsanov’stheoremandtheconstructionofourmea-
sureswegetthatforWθbeingthePθ-Brownianmotiondefinedby
tWtθ=Wt0−θsds
0whereθ∈ΘisoneofthedensitygeneratorsfromSection3.4.1that
(X,Wθ),(Ω,F,(Ft)t,Pθ)and(Xθ,W0),(Ω,F,(Ft)t,P0)
arebothweaksolutionstothesameSDEandtherefore
Pθ[X∈Γ]=P0[Xθ∈Γ]forΓ∈B(C(R+)).
Thisequalitynowallowsustoshifttheambiguityfromthesetofmeasures
tothepayoffprocesssince(Xt)ttogetherwith(Ω,F,(Ft)t,Pθ)isaMarkov
processandhence
θs(t,x)=supEP[Xτ|Xt=x]
t≥τisthesmallestexcessivemajorantofthefunctiong(z)=zandwithrespect
totheprocessX.
Remark3.6.1.Sincewearelookingfortheoptimalstoppingtimefromthe
beginning,wesett=0anddropitinthefollowing.
Wealsoknowthesmallestexcessivemajorantofg(x)withrespecttoX
canbeapproximatedby

v(x)=limnNlimQnNg(x),
whereQnNg(x)istheNthpowerofoftheoperatorQndefinedvia
Qng(x)=max{g(x),T2−ng(x)}.
isthepayoffofstoppingandT2−ng(x)=g(y)Pθ(2−n,x,dy)istheexpected
Remark3.6.2.TheoperatorQnremindsofbackwardinductionsinceg(x)
payoffint=2−n.
05

3.6.APPLICATIONS

Thisisnowwheretheuniquenessinlawfromabovecomesinsinceit
implies

Pθ(2−n,x,dy):=Pθ(X2−n∈dy|X0=x)
=P0(X2θ−n∈dy|Xθ0=x)=P0(2−n,x,dy).
Implyingsince(Xtθ)ttogetherwith(Ω,F,(Ft)t,P0)isalsoaMarkovprocess
thatthesmallestexcessivemajorantsinbothsettingsareidenticalandwe
evahθsupEP[Xτ|X0=x]=supEP0[Xτθ|X0=x]
∈Sτ∈SτwhichallowstoshifttheuncertaintyofthedistributionofXtouncertainty
ofthetruedriftofXvia
θsupθinfEP[Xτ|X0=x]=supinfEP0[Xτθ|X0θ=x]forallx∈Ω.
τ∈SP∈Pτ∈Sθ∈Θ
TheconstructionofSection3.4.1nowtellsusthat
µt+θtσt≥µt−κtσt
forallθ∈Θ,meaningthatµt−κtσtisthesmallestpossibledriftourpay-
offprocesscanhaveandwithacomparisonresultforstochasticdifferential
equationsweobtainthat
P0Xtθ≥Xt−κforallt≥0=1
andwiththat
supinfEP0Xτθ=infsupEP0Xτθ
τ∈Sθ∈Θθ∈Θτ∈S
≥infsupEP0Xτ−κ=supEP0Xτ−κ.
θ∈Θτ∈Sτ∈S
Sincetheconverseinequalityfollowsdirectlyfromthetheorem
τ∈SP∈PP∈Pτ∈Sθ∈Θτ∈Sττ∈Sτ
supinfEP[Xτ]=infsupEP[Xτ]=infsupEP0Xθ≤supEP0X−κ
15

2.DUALITYTHEOREM

weobtainthatthetheoremhelpsusidentifytheworstcasedistributionin
thiscaseandourambiguousstoppingproblemissimplifiedintoaclassical
stoppingproblemwithaknowndistribution,i.e.

supinfEP[Xτ]=supinfEP0[Xτθ]=supEP0Xτ−κ=supEP−κ[Xτ].
τ∈SP∈Pτ∈Sθ∈Θτ∈Sτ∈S
3.7Conclusion
Confrontedwithanoptimalstoppingproblemandnot(exactly)knowing
orbeingunsureofthetruedistributionofthepayoffprocessXwefind
ourselvesintheframeworkof[Epstein&Schneider,03]whoproposetosolve
thefollowingproblem

maximizeinfEP[Xτ]amongallstoppingtimesτ
∈PPwherePisthesetofallmeasureswethinkpossible.Hereweaskedifand
underwhatconditionsitispossibletointerchangetheorderofminimizing
overthedistributionsandmaximizingoverallstoppingtimes.Theresultis
aminimaxtheoremunderrathergeneralassumptionsonthepayoffprocess
XandstandardassumptionsonthesetofmeasuresP.Itincorporates
anexplicitbutstilluniversalconstructiongivenin[Delbaen,03]fortime-
consistentsetsofmeasures.
Thistheoremallowsustoidentifytheoptimalstoppingtimeforpayoff
processeswhicharemultiplepriorsub-orsupermartingalesandhaveafinite
investmenthorizon.Italsohelpsusdeterminetheworstcasedistributionin
thesettingofκ-ambiguityforoptionswithanincreasingpayoffintheun-
derlyingandwiththisturnsourambiguousdecisionproblemintoaclassical
purelyriskyone.
Itwouldbedesirabletofindfurtherapplicationsofthistheorem.Since
thecruxofapplyingthistheoremismainlyaminimizationoverthedriftit
25

3.7.CONCLUSION

wouldbepreferabletofindclassicalsolutionsthatdependonthedriftofthe

payoffprocess.Howeverthedrifthastobestochasticandsinceinclassical

solutionsthedriftiscommonlyassumedconstantforsimplicity,itmightbe

a

dogo

diae

ot

kool

at

rpscesseo

ni

a

gntiets

htwi

stochastic

estertni

es.atr

35

4erhaptC

LearningforConvexRisk

MeasureswithIncreasing

Information

4.1Introduction

Reachingdecisionsconcerningriskyprojectsinadynamicsystem,anagent
facesnewinformationconsecutivelyinfluencingherassessmentofriskinstan-
taneously.
Inthisarticle,weanswerthequestionhowanticipationofriskevolves
overtimewhenanagentgathersinformation.Weshowthat,inthelimit,all
uncertaintyisrevealedbutriskremainsiftheagentperceivesriskinterms
oftime-consistentdynamicconvexriskmeasuresand,hence,generalizethe
famousBlackwell-DubinsTheoremtoconvexriskmeasures.Wethenrelax
thetime-consistencyassumptionandshowtheresulttostillbevalid.Hereto,
afundamentalassumptionisexistenceofareferencedistributionthatfixes
impossibleandsureeventsbyvirtueofequivalenceofdistributionsunder
consideration.
Coherentriskmeasureswereintroducedbyvirtueofanaxiomaticansatz

4.INCREASINGINFORMATION&CONVEXRISK

in[Artzneretal.,99]inastaticsettingandhavebeengeneralizedtoady-
namicframeworkin[Riedel,04].Tangibleproblemsinthissetupareinter
aliadiscussedin[Riedel,09].Theequivalenttheoryofmultiplepriorprefer-
encesinastaticsetupisintroducedin[Gilboa&Schmeidler,89];adynamic
generalizationisgivenin[Epstein&Schneider,03].Applyingcoherentrisk
measuressubstantiallydecreasesmodelriskastheydonotassumeaspe-
cificprobabilitydistributiontoholdbutassumeawholesetofequallylikely
probabilitymodels.Moreover,theypossessasimplerobustrepresentation.
However,astheyassumehomogeneity,coherentriskmeasuresdonotac-
countforliquidityrisk.Thoughinfinancialapplications,theBaselIIaccord
requiresa“marginofconservatism”,coherentriskmeasuresarefartoocon-
servativewhenestimatingriskofaprojectastheyresultinaworstcase
approach.Furthermore,popularexamplesofriskmeasures,ase.g.entropic
risk,arenotcoherent.
Hence,itseemsworthwhiletoconsideramoresophisticatedaxiomatic
system:[Fo¨llmer&Schied,04]introduceconvexriskmeasuresasagener-
alizationofcoherentonesrelaxingthehomogeneityassumption.Equiva-
lently,[Maccheronietal.,06a]generalizemultiplepriorpreferencestovaria-
tionalpreferences.Convexriskmeasuresareappliedtoadynamicsetupin
[Fo¨llmer&Penner,06]forastochasticpayoffinthelastperiodor,equiva-
lently,in[Maccheronietal.,06b]intermsofdynamicvariationalpreferences.
[Cheriditoetal.,06]appliesdynamicconvexriskmeasurestostochasticpay-
offprocesses.Givenasetofpossibleprobabilisticmodels,convexriskmea-
suresarelessconservativethancoherentones.Dynamicconvexriskmeasures
aswellasdynamicvariationalpreferencespossessarobustrepresentationin
termsofminimalpenalizedexpectation.Theminimalpenalty,servingas
ameasureforuncertaintyaversion,uniquelycharacterizestheriskmeasure
or,respectively,thepreference.Conditionsontheminimaldynamicpenalty
characterizetime-consistencyofthedynamicconvexriskmeasure.
Aparametriclearningmodelinanuncertainenvironmentfordynamicco-

65

4.1.INTRODUCTION

herentriskmeasuresor,equivalently,dynamicmultiplepriorsasintroduced
in[Epstein&Schneider,03],iselaboratedin[Epstein&Schneider,07].The
mainvirtueofthisarticleistointroducelearningbasedonexperienceto
convexriskmeasuresmodels.First,wetrytointroducelearninginacon-
structiveapproach:wedesignaminimalpenaltyfunctionandplugitinto
therobustrepresentation:Sincethepenaltymightbeseenassomeinverse
likelihoodofaspecificpriordistribution,wefirstapplyaquitesimpleand
intuitivelearningmechanismtothepenalty.Wecalculatethelikelihoodof
adistributiongivenpastexperienceandusethisasupdatedpenalty.The
intuitionbehindthisapproachisquitesimple:observinggoodevents,dis-
tributionsofapayoffprocessthatare“stochasticallymoredominated”,i.e.
putmoreweightonbadevents,becomemoreunlikely,i.e.haveahigher
penalty.However,besidesitsintuitiveappeal,itturnsoutthatthisproce-
duredoesnotresultinapenaltyfunctionasitisbackwardsorientedanda
penaltyfunction,bydefinition,incorporatesprobabilitydistributionsofthe
futuremovementofthepayoffprocess.Inasecond,moresophisticatedap-
proach,wemodelapenaltyincorporatingprojectionsof“past”likelihoodson
futuredistributions.Here,wemakeuseoftheconditionalrelativeentropy
aspenaltyfunction:weachieveaproperpenaltythatpenalizesdistributions
accordingto“distance”fromthe“mostlikely”distributionservingasrefer-
encedistribution.However,theconvexriskmeasureintermsofthispenalty
turnsoutnottobetime-consistentingeneralasshownbyacounterexample.
In[Epstein&Schneider,07],time-consistencyisnotanissueasmultiplicity
ofpriorsisnotintroducedintermsofmultipleequallylikelydistributionsof
thepayoffprocessase.g.in[Riedel,09]or[Maccheronietal.,06a],butin
termsofmultipledistributionsontheparameterspace.
Ourfurtherapproachisnotconstructivebuttakestherobustrepresenta-
tionofariskmeasureintermsofminimalpenaltyforgranted.Asthemain
resultofthisarticleweachieveageneralizationofthefamousBlackwell-
DubinsTheoremin[Blackwell&Dubins,62]fromconditionalprobabilities

75

4.INCREASINGINFORMATION&CONVEXRISK

totime-consistentdynamicconvexriskmeasures.Weposeaconditionon
theminimalpenaltyintherobustrepresentation,alwayssatisfiedbycoherent
riskmeasures,forcingtheconvexriskmeasuretoconvergetotheconditional
expectedvalueunderthetrueunderlyingdistribution.Intuitively,thisre-
sultstatesthat,eventually,theuncertaindistributionisrevealedor,inother
words,uncertaintydiminishesasinformationisgatheredbutriskremains.
Theagent,asshehaslearnedabouttheunderlyingdistribution,isagain
intheframeworkofbeinganexpectedutilitymaximizerwithrespecttothe
trueunderlyingdistribution.Wehavehenceachievedlearningasanintrinsic
propertyofdynamicconvexriskmeasures.
OurgeneralizationoftheBlackwell-DubinsTheoremservesasanalterna-
tiveapproachtolimitbehavioroftime-consistentdynamicconvexriskmea-
suresastheonein[Fo¨llmer&Penner,06].Theresultparticularlystatesthe
existenceofalimitingriskmeasure.Asanexampleweconsiderdynamic
entropicriskmeasuresor,equivalently,dynamicmultiplierpreferences.We,
however,showaBlackwell-Dubinstyperesulttohold,evenifwerelaxthe
time-consistencyassumption.Again,weobtainexistenceofalimitingrisk
measurebutinamoregeneralmannerthan[Fo¨llmer&Penner,06]fornot
necessarilytime-consistentconvexandcoherentriskmeasures.
[Schnyder,02]discussesH.P.Minsky’stheoryoffinancialinstability,a
hugeportionofwhichiscausedbyherdingonfinancialmarkets.Besides,
herdingisusuallyoneofthemajorobjectionstowardsBaselII.Ourresult
howevershowsthat,inthelongrun,thereishardlyanychancetocircumvent
herdingbehavior.
Thearticleisconsideredinaparametricsetting.However,thesecondpart
canberestatedinanonparametricsetting.Itisstructuredasfollows:The
nextsectionformallyintroducestheunderlyingprobabilisticmodel.Section
4.3elaboratelydiscussesrobustrepresentationofdynamic(time-consistent)
convexriskmeasures.Constructiveapproachestolearningintermsofdy-
namicminimalpenaltyaswellastheirshortcomingsarestatedinSection

85

4.2.MODEL

4.4.Section4.5generalizestheBlackwell-DubinsTheoremtoconditionalex-
pectations.Thefollowingtwosectionsthenapplythisresulttocoherentand
convexriskmeasuresfirstinthetime-consistentcaseandtheninthecase
withouttime-consistency.Section4.8statesexamples.Thenweconclude.

delMo.24

Forourmodelwestartwithadiscretetimesett∈{0,...,T}whereTisan
infinitetimehorizon.Wewillnowconstructanunderlyingfilteredreference
space(Ω,F,(Ft)t,Pθ0)anddefineriskyprojectsX:
Wefix(S,A)asameasurablespacewhereSdescribesthepossiblestates
oftheworldatafixedpointintimetanddefineΩtobeallpossiblestates
oftheworld,formallythesetofsequencesofelementsofS.Forthislet
St=Sforallt∈{0,...,T}andthendefineΩ:=tT=0St.Onthisspace
letFbetheproductσ-fieldgeneratedbyallprojectionsπt:Ω→Stand
lettheelementsofthefiltrationFtbegeneratedbythesequenceπ1,...,πt.
AdditionallydefineallsequencesuptotimetbyΩt:=st=0Ss.Denote
genericelementsonthesespacesbyst∈St,s∈Ω,andst∈Ωt.
LetΘbeasetofparameterswhereeveryθ∈Θuniquelydefinesadistri-
butionPθon(Ω,F)withfiltration(Ft)tandfixPθ0asareferencedistribution
whichcanbeseenasthetruedistributionofthestates.Forallθ∈Θ,Pθ
isassumedtobeequivalenttoPθ0.LetMe(Pθ0)denotethesetofalldistri-
butionson(Ω,F)equivalenttoPθ0.Assumethatallthesecanbeachieved
byparametersθ∈Θ,i.e.Me(Pθ0)={Pθ|θ∈Θ}.ForPθ∈Me(Pθ0)let
Pθ(∙|Ft)denotethedistributionconditionalonFt.Duetoourassumption
toonlyconsiderdistributionsequivalenttoPθ0,thereferencedistribution
merelyfixesthenull-setsofthemodel,i.e.distinctagentsatleastagreeon
impossibleandsureevents.Thisassumptionhasnoinfluenceonthestochas-
ticstructureofthedistributionsitjusttellsthedecisionmakerswhatsure
orimpossibleeventsare.Aneconomicinterpretationofthisassumptionwas

95

4.INCREASINGINFORMATION&CONVEXRISK

givenbyEpsteinandMarinacciin[Epstein&Marinacci,06].Theyrelated
ittoanaxiomonpreferencesfirstpostulatedbyKrepsin[Kreps,79].He
claimedthatifanagentisambivalentbetweenanactxandx∪xthenhe
shouldalsobeambivalentbetweenx∪xandx∪x∪x.Meaningifthe
possibilityofchoosingxinadditiontoxbringsnoextrautilitycomparedto
justbeingabletochoosex,thenalsonoadditionalutilityshouldarisefrom
beingabletochoosexsupplementarytox∪x.
FurthermorewedefineX:Ω→RtobeanF-measurablerandomvari-
ablewhichcanbeinterpretedasapayoffatfinaltimeT.AssumeXbeing
essentiallyboundedwithesssup|X|=κ>0.Havingconstructedthefil-
teredreferencespace(Ω,F,(Ft)t≥0,Pθ0)asabove,thesetsofalmostsurely
boundedF-measurableandFt-measurablerandomvariablesaredenotedby
L∞:=L∞(Ω,F,Pθ0)andLt∞:=L∞(Ω,Ft,Pθ0),respectively.Allequations
havetobeunderstoodPθ0-almostsurely.
Remark4.2.1.Aswewillseeincourseofthearticle,theparametricset-
tingisonlyneededinthefirstpartontheconstructiveapproachtolearn-
ing.Allstatementsinthesecondpart,thegeneralizationoftheBlackwell-
Dubinstheorem,canbeposedintermsofanarbitraryunderlyingfiltered
space(Ω,F,(Ft)t≥0,P0)withdistributionsinMe(P0),whereP0denotesthe
referencedistribution,i.e.inanon-parametricsetting.Moreover,forthese
results,wedonotneedtheparticularstructureofΩintermsofaproductof
marginalspacesSt.Wehoweverfollowtheparametricapproachthroughout
toobtainaunifiedappearance.

4.3DynamicConvexRiskMeasures
Inthisarticle,weapplythetheoryofconvexriskmeasuresassetoutin
[Fo¨llmer&Penner,06]forend-periodpayoffs.Forpayoffprocesses,convex
riskmeasuresaredescribedin[Cheriditoetal.,06].Wedonotconsider
theaxiomaticapproachtoconvexriskbuttaketherobustrepresentation
06

4.3.DYNAMICCONVEXRISKMEASURES

ofdynamicconvexriskmeasuresor,equivalently,ofdynamicvariational
preferencesasgiven.

Definition4.3.1(DynamicConvexRisk&PenaltyFunctions).(a)Afamily
(ρt)tofmappingsρt:L∞→Lt∞iscalledadynamicconvexriskmeasureif
eachcomponentρtisaconditionalconvexriskmeasure,i.e.forallX∈L∞,
ρtcanberepresentedintermsof
ρt(X)=Qess∈Me(Psupθ0)EQ[−X|Ft]−αt(Q),
where(αt)tdenotesthedynamicpenaltyfunction,i.e.afamilyofmappings
αt:Me(Pθ0)→Lt∞,αt(Q)∈R+∪∞,closedandgrounded.Fortechnical
detailsonthepenaltysee[F¨ollmer&Schied,04].
(b)Equivalently,wedefinethedynamicconcavemonetaryutilityfunction
(ut)tbyvirtueofut:=−ρt,i.e.
Qut(X):=Q∈Messe(Pinfθ0)E[X|Ft]+αt(Q).
Remark4.3.2.(a)ByTheorem4.5in[F¨ollmer&Penner,06],theabove
robustrepresentationintermsofMe(Pθ0)issufficienttocapturealltime-
consistentdynamicconvexriskmeasures.
(b)Assumingriskneutralitybutuncertaintyaversion,nodiscounting,andno
intermediatepayoff,(ut)tistherobustrepresentationofdynamicvariational
preferencesasintroducedin[Maccheronietal.,06b].Inthissense,allour
resultsalsoholdequivalentlyfordynamicvariationalpreferences.However,
wehavechosentoconcentrateondynamicconvexriskmeasureshere.

Assumption4.3.3.Intherobustrepresentation,weassumethepenaltyαt
tobegivenbytheminimalpenaltyαtmin.Theminimalpenaltyisintroduced
intermsofacceptancesetsin[F¨ollmer&Penner,06],p.64:ForeveryQ∈
Me(Pθ0)
αtmin(Q):=X∈Less∞:ρt(supX)≤0EQ[−X|Ft].

61

4.INCREASINGINFORMATION&CONVEXRISK

Asstatedintherespectivereferences,everydynamicconvexriskmea-
sure(ρt)tcanbeexpressedintermsoftheaboverobustrepresentation,
uniquelybyvirtueoftheminimalpenaltyandviceversa.Thenotionof
minimalpenaltyisjustifiedbythefactthateveryotherpenaltyrepresent-
ingthesameconvexriskmeasurea.s.dominatestheminimalone,cp.
[Fo¨llmer&Penner,06]’sRemark2.7.Throughout,weassumearepresen-
tationintermsoftheminimalpenalty(αtmin)t.
Remark4.3.4(EquivalentNotation).Inourparametricset-up,adistribu-
tionPθoftheprocessisuniquelydefinedbyaparameterθ∈Θ.Hence,we
itewrθρt(X)=esssupEP[−X|Ft]−αtmin(θ).
Θ∈θFurtherassumptionsontheriskmeasureunderconsiderationwillbe
posedwhennecessary.
Remark4.3.5(OnCoherentRisk).Assetoutinthereferences,therobust
representationofcoherentriskisaspecialcaseoftherobustrepresentation
ofconvexriskwhenthepenaltyistrivial,i.e.foralltitholds
0ifPθ(∙|Ft)∈Q˜(∙|Ft),
αt(θ)=
else∞forQ˜thesetofpriordistributionsinducedbyallθinsomesetΘ˜⊂Θ.
Throughout,Q˜isassumedtobeconvexandweaklycompactor,equivalently,
Θ˜isassumedtobesuch.
Thefollowingdefinitionisamajorassumptionneededinordertosolve
tangibleeconomicproblemsunderconvexrisk.
Definition4.3.6(Time-Consistency).Adynamicconvexriskmeasure(ρt)t
iscalledtime-consistentif,forallt,s∈N,itholds
ρt=ρt(−ρt+s)
or,equivalently,ut=ut(ut+s).
26

4.3.DYNAMICCONVEXRISKMEASURES

Remark4.3.7.Forthespecialapproachhere,[Cheriditoetal.,06]show
thatitsufficestoconsiders=1intheabovedefinition.
Remark4.3.8.Asinteraliashownin[F¨ollmer&Penner,06],Theorem
4.5,time-consistencyof(ρt)tisequivalenttoaconditionontheminimal
penalty(αtmin)tcalledno-gainconditionin[Maccheronietal.,06b].
Wenowintroduceaspecialclassofdynamicconvexriskmeasuresthat
willbeusedinseveralexampleslateron:Dynamicentropicriskmeasures.
Therefore,wefirsthavetointroduce:
Definition4.3.9(RelativeConditionalEntropy).ForPQ,wedefinethe
relativeentropyofPwithrespecttoQattimet≥0as
Ht(P|Q):=EP[logZt],
where(Zt)tbyvirtueofZt:=ddQP|FtdenotesthedensityprocessofPwith
respecttoQ.Furthermore,wedefinetheconditionalrelativeentropyofP
withrespecttoQattimet≥0as

ZtZtZt
Hˆt(P|Q):=EPlogZTFt=EQZTlogZTFtI{Zt>0}.
Definition4.3.10(EntropicRiskMeasures).GivenreferencemodelQ∈
Me(P0).Letδ>0.Wesaythatdynamicconvexriskρte(X)ofarandom
variableX∈L∞,isobtainedbyadynamicentropicriskmeasuregiven
referencemodelQ∈Me(Pθ0)ifitisoftheform
ρte(X)=essesupθ0EP[−X|Ft]−δHˆt(P|Q).(4.1)
)P(∈MPEquivalently,dynamicmultiplierpreferences(ute)taredefinedbyvirtueof
ute(X)=P∈Messe(Pinfθ0)EP[X|Ft]+δHˆt(P|Q).(4.2)
Remark4.3.11.Thevariationalformulaforrelativeentropyimplies
1ρte(X)=δlog(EQ[e−δX|Ft]).

36

4.INCREASINGINFORMATION&CONVEXRISK

Intuitively,anentropicriskmeasuremeansthattheagentinanuncer-
tainsettingbelievesthereferencemodelQasmostlikelyanddistributions
“furtheraway”asmoreunlikely.Again,wecanwrite(ρte)tbyvirtueof
ρte(X)=esssupEPθ[−X|Ft]−δHˆt(θ|η),
Θ∈θwherePηdefinesthereferencemodel.

4.4AConstructiveApproachtoLearning
Inthissection,wetrytoexplicitlydevelopalearningmechanismbyvirtueof
penaltyfunctionsthatarethenusedfortherobustrepresentationofdynamic
convexriskmeasures.Wewillencounter,thatthisisnotaneligibleapproach
tomodellearningasitisstillnotclearhowtoexplicitlyformapenalty.In
alatersection,wewilljusttaketherobustrepresentationasgivenandpose
thequestionwhatcanbesaidaboutlearningwhendistinctpropertiesofthe
penaltyareassumed.

4.4.1TheIntuitionofLearningviaPenalties
Inafirst,intuitiveapproach,weexplicitlyintroducealearningmechanism
tothepenalty(αt)tintermsofalikelihoodfunction.Thefundamentalidea
isthatthepenaltymightbeviewedasameasureforthelikelihoodofa
distribution.Intheextremecaseofcoherentrisk,thismeans
•αt(θ)=∞:Pθisnotpossible,
•αt(θ)=0:Pθisamongthemostlikely.
Ingeneral,thelargerαt,thelesslikelytherespectivedistribution.Statedin
otherterms,(αt)tisameasureforuncertaintyaversion:giventwopenalties
(αt1)tand(αt2)t,thea.s.largeronecorrespondstothelessuncertaintyaverse
46

4.4.ACONSTRUCTIVEAPPROACHTOLEARNING

¯agent.Intheentropiccase,αt(θ)=Ht(Pθ|Pθ),theconditionalrelativeen-
tropyofPθwithrespecttoPθ¯attimet,theagentconsidersPθ¯mostlikelyas
Ht(Pθ¯|Pθ¯)=0anddistributions“furtheraway”asmoreandmoreunlikely.
Inthecoherentcasecharacterizedbyatrivialpenalty,learningmeansto
alternatethesetsQ˜t:={P∈Q˜|P(∙|Ft)∈Q˜(∙|Ft)},t=0,...,Tofcondi-
tionalpriorsonwhichthepenaltyhasvaluezero:whenmoreinformationis
availableandhence,moremightbeknownaboutthedistributionthatrules
theworld,Q˜t⊃Qt˜+1,i.e.penaltyisincreasingint.Forsomecutoffvalue
β,anintuitiveapproachwouldbeintermsofsomelikelihoodfunctionl:
θα(θ)=0ifl(P|θ,Ft)≥β,
t.seel∞Asadirectgeneralizationtoconvexriskmeasures,onemightconsiderthe
log-likelihood−log(l(Q|θ,Ft))aspenalty.Itwillturnoutthatthisapproach
isnoteligiblesinceapenaltydefinedintermsoflikelihoodfunctionsisnot
feasible.Hence,wecomeupwithadistinctansatzinwhichpenaltyisgiven
byrelativeconditionalentropy.Wethenachieveadynamicconvexrisk
measurebutrunintotroubleregardingtime-consistency.Amodeldefined
asaboveservesasameasuretheoreticfundamentofH.P.Minsky’stheory
offinancialinstability:Asequenceof“good”eventscausesthepenaltyto
besmallerfordistributionsthatstochasticallydominateforthepayoffunder
consideration.Uponobservingfavorableevents,theagentthinksthatnature
hasbecomekinder.Thismighthelptounderstandunderestimationofrisk
leadingtobubblesandfinancialinstabilityintimesofgrowthandfinancial
ess.ccsu

4.4.2SpecialCase:ExplicitLearningforCoherentRisk

[Epstein&Schneider,07]introducelearningforcoherentriskintermsof
likelihoodratiotests.Aswewillseelater,theydonotconsiderthesetsof
priors(Qt)tasforexamplein[Riedel,04]buttheprocessPt(Ft)ofone-step
56

4.INCREASINGINFORMATION&CONVEXRISK

aheadconditionalbeliefs,formallyintroducedbelow,astheseimmediately
representthelearningprocess.Moreover,[Epstein&Schneider,07]distin-
guishbetweeninformationthatcanbelearnedandinformationthatcannot:
Informationthatcanbelearnedisincorporatedinathesetofpriorsnot
beingsingleton,informationthatcannotbelearnedisincorporatedinthe
setoflikelihoodfunctionsnotbeingsingleton.
Formally,letthestatespacebegivenbyST:=⊗tT=1St,St=S,Θasinthe
generalmodel.Thespaceofparameterswillbeslightlymodified,i.e.every
θ∈ΘuniquelycharacterizesadistributiononSandnotonΩ;however,
thismodificationisrestrictedtothecurrentsubsection.LetQ0⊂M(Θ)be
thesetofpriorsonΘandLthesetoflikelihoods,i.e.everyl∈Lsatisfies
l(∙|θ)∈M(S)andl(st|∙)isFt-measurableforst∈St.Setst=(s1,...,st),
si∈Si.Everyµ0∈Q0togetherwithafamilyoflikelihoods(l1,l2,...)∈L∞
inducesapriorP∈Me(P0)ofthepayoffprocessor,equivalently,theprocess
(pt)tofone-step-aheadconditionals
pt(∙|st)=l(∙|θ)dµt(θ|st)∈M(St+1),
Θwhereµtisderivedfromµ0asdescribedbelowandµt(∙|st)∈Qt(st),theset
ofposteriorbeliefsonΘgivenhistoryst.Hence,multiplicityofbeliefsis
describedby
ΘPt(st)=pt(∙|st)=l(∙|θ)dµt(θ)µt∈Qtα(st),l∈L
:=L(∙|θ)dQtα(θ).
ΘTocompletethemodel,itleavestoshowhow(µ0;l1,...)induceµtor,equiv-
alently,howQt(st)isobtained.For(µ0;l1,...),theposteriorsareobtained
byBayesianupdating:
dµt(∙,st,µ0,lt)
lt(st|∙)t−1t−1
Θ=lt(st|θ˜)dµt−1(θ˜,st−1,µ0,lt−1)dµt−1(∙,s,µ0,l).
66

4.4.ACONSTRUCTIVEAPPROACHTOLEARNING

Then,theposteriorsareachievedbyvirtueofalikelihoodratiotestinterms
oftheunconditionaldatadensity:
t
Qtα(st):=µt(st,µ0,lt)µ0∈Q0,lt∈Lt,lj(sj|θ)dµ0(θ)
=1jt≥βµ¯∈Qmax,l¯t∈Ltl¯j(sj|θ)dµ¯0(θ)
00=1j

forsomeboundβ∈R+.
Remark4.4.1.Conceptually,thereisahugedifferencebetweentheap-
proachin[Epstein&Schneider,07]and[Gilboa&Schmeidler,89]:Inthe
latter,theterm“multiplepriors”meansmultipledistributionsofthepayoff
stream,allbeingequallylikely,intheformer,itmeansmultipledistribu-
tionsoftheparameter,i.e.multipledistributionsonthedistributionsof
thepayoffstream.Hence,[Epstein&Schneider,07]isageneralizationof
[Gilboa&Schmeidler,89]asthelatterframeworkisachievedwithQ0={µ0}
withµ0theuniformdistributiononsomesubsetofΘ.Inthatcasewehave
atrivialαandhenceacoherentriskmeasure.Intuitively,auniformdistri-
butiononasubsetofΘcorrespondstotheagentbelievingalldistributionsin
thatsubsetbeingequallylikelyandtheothersimpossible.

Nevertheless,fruitfulinsightsfrom[Epstein&Schneider,07]canbegained
forourapproachinparticulartheincorporationofalikelihoodratiotest.We
goastepcloserto[Gilboa&Schmeidler,89]andintroduceasingledistribu-
tiononΘinducingauniquepenaltyforadynamicconvexriskmeasure.

4.4.3AFirst,ParticularlyIntuitiveApproach:Sim-
plisticLearning
Asstatedabove,multiplepriorpreferencesmeantheagenthasauniform
distributiononasubsetofΘ:Sheissureaboutwhichparametersarepossible
andwhichnot,buthasnotendencytowardstheirlikeliness.Inaway,this

76

4.INCREASINGINFORMATION&CONVEXRISK

correspondstoanon-informativeweightingoratrivialpenaltyfunctionα0.
Weactonthisnon-informativeapproachandassumethefollowingpenalty
attimezero:LetΘ˜⊂Θ.Thepenaltycorrespondingtothisdistributionis
givenby:
˜αt(θ)=0ifθ∈Θ,
.seel∞Hence,initiallytheconvexriskmeasureisactuallycoherent:
θθρ0(X):=esssupEP[−X]−α0(θ)=esssupEP[−X].
θ∈Θθ∈Θ˜
Wenowcomeupwithasimplelearningmechanismdirectlydefiningthe
dynamicpenaltyfunction(αt)tintermsoflikelihoods.Att=0,wehave
alreadycharacterizedthepenalty.Furthermore,weset
l(s1|θ)Qθ(s1)
α1(θ):=−lnsup¯l(s|θ¯)=−lnsup¯Qθ¯(s),
11θθwheres1=s1and
l(s2|θ)Qθ(s1)Qθ(s2|θ,s1)
α2(θ)=−ln2¯=−lnγ,
supθ¯l(s|θ)2
whereγ2:=supθ∈ΘQθ(s1)Qθ(s2|θ,s1).
Definition4.4.2.Wesaythatthepenalty(αt)tintherobustrepresentation
oftheconvexdynamicriskmeasure(ρt)tisachievedbysimplisticlearning,
ifitisoftheform:
α(θ):=−lni=1Q(si|θ,s),
tθi−1
tγttwhereγt:=supθ∈Θi=1Qθ(si|θ,si−1).
Remark4.4.3(Onimpropernessofsimplisticlearning).(αt)tachievedby
simplisticleaningisnotafeasiblepenaltyfunction.
Proof.Apenaltyattshouldincludetheconditionaldistributionsfromt
onwardsasseeninthedefinition.Inourlikelihoodapproachαtonlydepends
ondistributionsuptotimet,i.e.alreadyrealizedentitiesofthedensity
ess.corp

86

4.4.ACONSTRUCTIVEAPPROACHTOLEARNING

4.4.4ASecond,MoreSophisticatedApproach:En-
tropicLearning
Wenowincorporatethelikelihoodfunctionintherelativeentropyinorder
toachieveariskmeasurebasedonthewellknownandelegantentropicrisk
es.rsueamHere,weassumeθ=(θt)t∈Θ;everyentityθtcharacterizesadistribution
inM(St)possiblydependenton(θi)i<t.Thefamilyθ=(θt)tthendefinesa
priorPθ∈Me(Pθ0).Setθt:=(θ1,...,θt)analogoustost.
Intheforegoingsection,wehaveseenthemajorproblemtobethatour
“penalty”wasonlycontingentonthepastevolutionofthedensityprocess.
Thereishoweverawholebunchofpossibilitiestoestimatethefuturebyuse
ofpastinformation.Aprominentrouteisbyvirtueofmaximumlikelihood
estimator.

Definition4.4.4(ExperienceBasedLearning).(a)Givenlikelihoodl.Being
attimet,learningissaidtobenaiveiftheestimatorθˆtforθtisachieved
solelybytakingintoaccountmaximumlikelihoodfortheobservationstat
.ttime(b)Learningiscalledintermediateorexperiencebasedatlevelm,ifθˆtis
themaximumlikelihoodestimatorofthelastmobservations(st−m,...,st)
MLE−m∈argmaxl(st−m,...,st|θt,θˆt−1,st−m−1).
Θ∈θt(c)Learningissaidtobeofmaximumlikelihoodtype,if,atanyt,θˆtisthe
maximumlikelihoodestimatorofthewholehistory.

Notethatthenaiveestimatorisjusttheintermediateoneatlevelzero.
Furthermore,notethatourdefinitionofexperiencebasedmaximumlike-
lihood.Inthenextdefinition,wecharacterizehowlearningresultsina
distributionforthepayoff.

96

4.INCREASINGINFORMATION&CONVEXRISK

Definition4.4.5(LearningDistributions).Beingattimet,havingobtained
θˆtandtheforegoingestimators(θˆi)i<t,thereferencefamilyθˆofparameters
bydachieveis

θˆi=θˆii≤t,
θˆti>t.
Havingseenhowagentslearnaboutthebestfittingdistribution,wenow
formallyintroduceentropiclearningforwhichdynamicentropicriskmea-
suresinDefinition4.3.10serveasavehicle:Wechoosethebestfittingdis-
tributionasreferencedistributionintheconditionalrelativeentropy.
Theagent’svariationalutilityincorporatinglearningisinoursetupgiven
byaconvexriskmeasurewithanentropicpenaltyfunction:
Definition4.4.6(ExperienceBasedEntropicRisk).Apenalty(αˆt)tissaid
tobeachievedbyexperiencebasedentropiclearningifgivenas
αˆt(η):=δHˆt(Pη|Pθˆ)
forδ>0andθˆ=(θˆt)tachievedasinDefinition4.4.5,η=(ηt)t∈Θ.The
resultingconvexriskmeasure(ρˆt)tincorporatingthisverypenaltyfunctionis
thencalledexperiencebasedentropicrisk.
Remark4.4.7.(αˆtθ)tiswelldefinedaspenalty;thisisinteraliashownin
[F¨ollmer&Schied,04].Duetoourconstruction,thepenaltynowincorpo-
ratesconditionaldistributionsoffuturemovements.
Remark4.4.8.Whentheparameterisalsotherealizationofanentityin
thedensityprocess,e.g.inatree(cp.theexamplebelow),relativeentropy
candirectlybewrittenas
θdPθdPθˆ
αˆt(θ)=EPlndPθ0dPθ0Ft.
Remark4.4.9.Naiveentropiclearningreflectsthetendencyoftheagentto
forget(orignore)aboutthedistantpastandjustassumethepresenttobethe
07

4.4.ACONSTRUCTIVEAPPROACHTOLEARNING

bestestimatoroftheunderlyingmodel.Thislearningmechanismisthenof
courseparticularlyadjuvantinexplainingabubbleasitishardertoseethat
thefinancialsystemmovesawayfromthefundamentals.
Despite[Epstein&Schneider,07]wedonotconsidermultiplicityoflikeli-
hoodshere.Hence,wedonotincorporateinformationthatcannotbelearned
uponinourmodel.Thoughrealworldapplicationswithseveraltrueparam-
eters,e.g.inincompletefinancialmarketswithamultiplicityofequivalent
martingalemeasures,wouldbemodeledintermsofmultiplelikelihoods.
However,ourmainresultinthissectionon“time-inconsistency”ofexpe-
riencebasedentropicriskwouldnotchangewhenextendingthemodelto
multiplelikelihoods.
Proposition4.4.10.Themodeliswelldefined,i.e.foreveryt,ρˆtisa
conditionalconvexriskmeasure.
Proof.Ascaneasilybeseen,themodelsatisfiestheaxiomsofconvexrisk
measures:ρˆt:L∞→Lt∞and
•ρˆtismonotone,i.e.ρˆt(X)≤ρˆt(Y)forX≥Ya.s.
•ρˆtiscash-invariant,i.e.ρˆt(X+m)=ρˆt(X)−m∀m∈Lt,X∈LT
•ρˆtisconvexasafunctiononLT

Asinteraliashown[Fo¨llmer&Penner,06],Proposition4.4,dynamicen-
tropicriskmeasuresaretime-consistentwhenthereferencedistributionis
notlearnedbutfixedatthebeginning.However,nowthatthereference
distributionisalsostochastic,weachieve:
Proposition4.4.11.Experiencebasedentropicriskisingeneralnottime-
onsistent.cProof.Asproofweconstructthefollowingcounterexampleshowinganex-
periencebasedentropicriskmeasurewhichisnottime-consistent.

17

4.INCREASINGINFORMATION&CONVEXRISK

Example4.4.12(EntropicRiskinaTree).Sinceourexampleismainlyfor
demonstrationpurposeswerestrictourselvestoasimpleCox-Ross-Rubinstein
modelwith3timeperiods.Eachtimeperiodisindependentofthosebefore.
Onecouldimaginethatineverytimeperiodadifferentcoinisthrownand
theresultofthecointossdeterminestherealizationinthetree,e.g.from
headsresultsupandfromtailsdown.Thepayoffsofourrandomvariable
Xarelimitedtothelasttime-periodandareasshowninthefigurebelow.
Fortractabilityreasonswealsoconfineourselvestoasinglelikelihoodfunc-
tionl(∙|θ).Forthesamereasonwewillalsousetheextremecaseofnaive
updatingwhichmeansourreferencedistributionwillmerelydependonthe
lastobservedeventinourtree.Theprobabilityforgoingupinthistreewill
alwaysbeassumedtolieintheinterval[a,b]where0<a≤b<1.
3

p∈[a,b]1
0-1

2

0-2

Figure4.1:Cox-Ross-RubinsteinModel

1

-1

-3

Time-period2:Sincewewanttoshowacontradictiontotime-consistency
wewillshowthattherecursiveformula

ρˆt(X)=ρˆt(−ρˆt+s(X))forallt∈[0,T]ands∈N
27

4.4.ACONSTRUCTIVEAPPROACHTOLEARNING

isviolated.Sowestartwiththecalculationofρ2(X)forthedifferentsetsin
2F

θρˆ2(X)(up,up)
=esssupE[−X|F2](up,up)−Elnθ∗2|F2(up,up)
p∈[a,b]2
=sup−3p−1+p−plnp−(1−p)ln1−p
p∈[a,b]b1−b
=lnbe−3+(1−b)e−1,
wherethereferencedistributionPθ∗inducedbyθ∗isdeterminedbythefol-
maximization:lowingθ∗=(θ0∗,θ1∗,θ2∗),θ2∗∈argmaxl(up|θ2)
θ2∈[a,b]
givingusthemaximum-likelihoodestimatorforwhathappenedinthelast
time-periodwhichwealsothinkistherightdistributionforthenexttime-
d.ioerpTheresultofthiscomputationcanalsobeobtainedbyusingavariational
formwhichcanforexamplebefoundin[F¨ollmer&Penner,06]andtakes
mforlowingfolthe∗ρˆt(X)=lnEPθ[exp(−X)|Ft],
wherePθ∗isagainthereferencedistributionthedecisionmakerestablishes
bylookingatthepast,which,aswelookatnaivelearning,willagainonly
bewhathappenedinthelastperiod.Sincethisgiveswayforaneasierand
quickercomputationwewillusethisformforthefollowingcalculations:
∗ρˆ2(X)(down,up)=lnEPθ[exp(−X)|F2](down,up)
=lnbe−1+(1−b)e1,
ˆρ2(X)(up,down)=lnEPθ∗[exp(−X)|F2](up,down)
=lnae−1+(1−a)e1.
37

37

4.INCREASINGINFORMATION&CONVEXRISK

Hereonecannicelyobservetheextremenessofthenaivelearningapproach.
Eventhoughthedecisionmakerinthesetwocalculationsislocatedatthe
samevertexinthetreehehasverydifferentbeliefsabouttheprobabilityof
goingupordownwhichcausesstrongshiftsinhisriskconception.
Inthecaseofgoingfirstdownthenupheclearlybelievesupwillbemore
prθ∗obableinthenextstep.Thisisvisibleinhischoiceofreferencemeasure
Pinthepenaltyfunctionwhichhesetsbforgoingupand1−bforgoing
down.Incontrasttothisthedecisionmakerwhohasobservedupandthendown
willputmoreweightontheprobabilityofgoingdowninthenextstepand
thereforesetshisreferencemeasureaforupand1−afordown.
Forthelastpossibleeventintime2ourrisk-measuretakesthefollowing
value:

∗ρˆ2(X)(down,down)=lnEPθ[exp(−X)|F2](down,down)
=lnae1+(1−a)e3.
Time-period1:Ifforthenexttime-periodwemaintaintheassumptionof
time-consistencyandmakeuseoftherecursiveformula,usingthevariational
formaswedidabovewillyield
∗ρˆ1(X)(up)=ρˆ1(−ρˆ2(X))(up)=lnEPθ[exp(ρˆ2(X))|F1](up)
=lnbbe−3+(1−b)e−1+(1−b)ae−1+(1−a)e1
=lnb2e−3+(a+b)(1−b)e−1+(1−a)e1.
Nowifwecalculateρˆ1(X)(up)withoutthetime-consistencyassumptionmean-
ingwecannotusetherecursiveformulaweobtainthefollowingequation:
p,qp,qθ1θ2
ρˆ1(X)(up)=p,qess∈[a,bsup]E[−X|F1](up)−Elnθ1∗θ2∗|F1(up)
=lnb2e−3+2b(1−b)e−1+(1−b)2e1.
Thisclearlyisnotthesameasweobtainedundertheassumptionoftime-
consistency.Howeverifourdynamicexperiencebasedentropicriskmeasure
47

4.4.ACONSTRUCTIVEAPPROACHTOLEARNING

weretime-consistentthesecalculationsshouldgiveusthesameresults.Hence
thisexampleclearlyshowsusthattheassumptionofourriskmeasurebeing
time-consistentonlyleadsuptocontradictionsandcanthereforenotbetrue.
FtToemphasizethereasonfortheseinconsistenciessetZt:=ddPPθθ21,where
Pθiisthereferencedistributiontheagentobtainsattimeiwhenlookingat
pastrealizationsandthenmaximizingtherespectivelikelihoodfunction.Then
forinstancefort=1andω=upweobtain:
ρˆ1(−ρˆ2(X−lnZT))(up)
Z2
=lnEPθ1expρ2X3−lnZ3|F1(up)
Z2=lnbEPθ2e−XZ|F2(up,up)
Z2+(1−b)EPθ2e−XZ|F2(up,down)
Z2=lnbbe−3bbbbb+(1−b)e−1bb(1−b)bb
bbbbbbb(1−b)bb
+(1−b)ae−1b(1−b)bb(1−b)
b(1−b)ab(1−b)
+(1−a)e1b(1−b)(1−b)b(1−b)
b(1−b)(1−a)b(1−b)
=lnb2e−3+2b(1−b)e−1+(1−b)2e1=ρ1(X)(up),
which,ifZZTi=1(generallytrue),clearlycontradictstime-consistency.
InthisspecialcaseforexamplethemeasurePθ1correspondstothemea-
sureassigningtheprobabilitybtoupineverytimeperiod,whereasPθ2isthe
measureassigningbtoupinthefirst2timeperiodsandainthelast.That
iswhye.g.Z3(up,down,up)=bb(1(1−−bb))abandZZ23(up,down,up)=ab.

4.4.5LackofTimeConsistency
Aswehaveseenintheforegoingparagraphourdefinitionofexperiencebased
entropicriskdoesnotresultinatime-consistentdynamicconvexriskmea-
57

4.INCREASINGINFORMATION&CONVEXRISK

sure.Thisinsightissomewhatdisappointingastimeconsistencyisapros-
perousvehicletosolvetangibleproblems.Ontheotherhand,[Schied,07]
showsthatameaningfultheoryofconvexriskcanevenbeachievedinanot
generallytime-consistentsetting.
Wehavetoposethefollowingquestion:Doesthereexistanylearning
modelforthereferencedistributionsuchthatdynamicentropicriskbecomes
time-consistent?

Remark4.4.13.Themajorissuethatmightcomeintomindistheinde-
pendenceofthereferencedistributionoffuturehistories.Aswewillsee,
thisisbasicallythereasonforthegeneralimpossibilityresultbelow.Fur-
thermore,theworst-casedistributionchosenbynatureisheavilydependent
onthereferencedistribution.Asthelatteronemaychangeinabroadva-
rietyofmanners,thereisnogoodreasontoexpectnaturetochooseina
way.onsistenttime-c

Next,weposethemostgeneraldefinitionoflearninginentropicset-ups.

Definition4.4.14.AreferencedistributionPθ˜forexperiencebasedentropic
riskissaidtobeobtainedbygenerallearningifthefamily(θ˜t)tisafamily
ofrandomvariables,i.e.notdeterministicallyfixedfromscratch.Wecall
ggtheresultingdynamicconvexriskmeasure(ρ˜t)tdefinedbyvirtueofα˜t:=
Hˆt(∙|(θ˜t)t)intherobustrepresentationgeneralexperiencebasedentropicrisk.
Weseethatourdefinitionofexperiencebasedentropicrisksatisfiesthe
abovedefinitionasinthatcontextlearningtakesplaceintermsofmaximum
likelihood.
Usingthisgeneraldefinitionoflearning,wecanshowanimpossibility
resultfortime-consistencyofgeneralexperiencebasedentropicrisk.

Proposition4.4.15.Generalexperiencebasedentropicrisk(ρ˜tg)tisingen-
time-cnotaleronsistent.

76

4.4.ACONSTRUCTIVEAPPROACHTOLEARNING

Proof.Letθ˜=(θ˜1,...)beobtainedbygenerallearningandtθ˜suchthat
Ft˜˜tθ˜
+1tPθ=Pθ(∙|Ft).LetZt+1:=dQdtQ+1θ˜.Then,wehave
tθ˜
ρ˜gt(X)=lnEQe−XFt
˜t=lnEQtθ˜elnEQθ[e−X|Ft+1]Ft
t+1
Qtθ˜lnEQθ˜ZZTe−XFt+1
=lnEet+1Ft
Qtθ˜−(−ρt+1(X−ln(ZtZT+1)))
=lnEeFt
ZT=ρ˜tg(−ρ˜tg+1(X−ln()))
Z+1t=ρ˜tg(−ρ˜tg+1(X)),
ifZZT=1a.s.,i.e.if,intuitivelyspeaking,learningactuallytakesplaceand,
+1thence,thereferencedistributionsatdistincttimeperiodsdiffer.

Theforegoingresultimmediatelyimpliesourmainintuitionforexpe-
riencebasedentropicrisknotbeingtime-consistentthoughquitepuzzling
asentropicriskmeasuresarebroadlyusedasstandardexamplefortime-
consistentconvexrisk.

Remark4.4.16(MainIntuition).Theminimalpenaltyfunctionuniquely
definesariskmeasure.Changingthereferencedistributionduetolearning
resultsinadifferentminimalpenaltyandhence,adistinctriskmeasure.
Hence,anexperiencebasedentropicriskmeasureisactuallyafamilyofdy-
namicentropicriskmeasuresandourdefinitionoftime-consistencyisnot
able.appliceven

4.4.6ARetrospective–InBetween
Inthissection,wehavestatedaconstructiveapproachtolearningforconvex
riskmeasures.Wehaveencounteredseveralproblemsindoingthat:
77

4.INCREASINGINFORMATION&CONVEXRISK

•Inourfirstintuitiveapproach,weranintoproblemsindefininga
penaltyfunctionnotentirelycontingentonthepastevolutionofthe
densityprocess.

•Inoursecondone,weranintotime-consistencyproblems.
Inaway,inthenextsection,weputthecartbeforethehorse:We
justtaketherobustrepresentationintermsofminimalpenaltyoftime-
consistentdynamicconvexriskmeasuresasgivenandaskourselveswhatcan
besaidabout“learning”inthatrespect.Wewillshowanequivalenttothe
fundamentalBlackwell-DubinsTheoremforconvexriskmeasures.Aswillbe
seen,thisresultwillbeequivalentlysatisfiedwheneverthetrueparameter
iseventuallylearneduponasdefinedinthesubsequentsubsection.Our
resultstatessomekindofherdingbehavioraseverymarketparticipantwill
eventuallyperceiveriskinthesamemanner.

4.4.7LearningforagivenTime-ConsistentConvexRisk
eursMeaWenowwanttoencounter,whetherweactuallyhavetoconstructalearning
mechanismoriflearningisnotalreadyincorporatedinsomesenseinthe
conceptofatime-consistentconvexriskmeasure.

Remark4.4.17.Wehavestatedthatthetime-consistencyproblemencoun-
teredsofarinlearningmodelsisduetothefactthatpenaltiesarenotjust
randomvariablesbutrandomitself,i.e.alsothefunctionalformdependson
theobservations.Thisassumptioningeneralcontradictstime-consistencyas
weactuallymayachievedistinctriskmeasuresataparticularpointintime.
However,thebasisforlearningisalreadyincorporatedinconvexriskasthe
domainofpenaltyconsistsofBayesianupdateddistributionsoftheprocess.

Letushenceassumeatrueunderlyingparameterθ0∈Θandtheagent
evaluatesriskintermsofrobustrepresentationoftime-consistentdynamic

87

4.4.ACONSTRUCTIVEAPPROACHTOLEARNING

convexrisk(ρt)twithminimalpenalty(αtmin)t.Wethenstatethefollowing
definition:

Definition4.4.18.Wesaythatθ0iseventuallylearneduponif
ρt(X)−EPθ0[−X|Ft]→0Pθ0−a.e.
.tfor∞→Proposition4.4.19.Theabovedefinitionissatisfiedifandonlyif
→∞tS+1tlimρt(X)−−ρt+1(X)Pθ0(dst+1|Ft)=0Pθ0−a.e.
Proof.cp.[Klibanoffetal.,08],Proposition5.

Inthetime-consistentcase,thefollowingassertionisequivalenttoDefi-
nition4.4.18:

Proposition4.4.20.Givenatime-consistentdynamicconvexriskmeasure
(ρt)t,thenθ0iseventuallylearneduponifandonlyif
αtmin(θ)t−→∞→0Pθ0−a.e
forallθsuchthatα0min(θ)<∞.
Proof.As(ρt)tisassumedtobetime-consistent,itholdsforallt

ρt=ρt(−ρt+1)
or,moreelaborately,forallX

ρt(X)
=supEPθ[−X|Ft]−αtmin(θ)
Θ∈θ=supEPθ[−ρt+1(X)|Ft]−αtmin(θ).
θΘ∈

79

4.INCREASINGINFORMATION&CONVEXRISK

AsfurtherforallX

−ρt+1(X)Pθ0(dst+1|Ft)
S+1t=EPθ0[−ρt+1(X)|Ft]
θ=supEP[−ρt+1(X)|Ft]−α¯tmin(θ),
Θ∈θwhere(α¯tmin)tisdefinedas
α¯min(θ):=0ifθ=θ0
t,seel∞→∞ttheprooffollowsreadily:αtmin(θ)−→α¯tmin(θ)byProposition4.4.19.Theo-
rem5.4.(4)in[Fo¨llmer&Penner,06]thenshowsequivalencetoavanishing
limitgiventime-consistency.

Inthesubsequentsections,weshowthenotionofbeingeventuallylearned
upontobesatisfiedbyconvexriskmeasuresincaseoftime-consistencyand
underlessstringentassumptionsintermsofBlackwell&Dubins.

4.5AdaptionofBlackwell-DubinsTheorem

Asacornerstoneforourmainresultonconvergenceofdynamicconvex
riskmeasures,wefirstgeneralizethefamousBlackwell-Dubinstheorem,cp.
[Blackwell&Dubins,62],fromconditionalprobabilitiestoconditionalexpec-
tationsofriskyprojects.Assetoutinthemodelsection,weassumeexistence
ofareferencedistributionPθ0,θ0∈Θ,asin[Blackwell&Dubins,62].This
referencehastobeinterpersonallybeingagreedupon.

Proposition4.5.1.LetPθbeabsolutelycontinuouswithrespecttoPθ0for

08

4.6.TIME-CONSISTENTRISKMEASURES

someθ∈Θ,1Xasinthedefinitionofthemodel,then
EPθ[X|Ft]−EPθ0[X|Ft]→0Pθ0-almostsurelyfort→∞.
Proof.ForimprovingreadabilitydenotePθ0byPandPθbyQ.
GivenPandQ,Qbeingassumedabsolutelycontinuouswithrespectto
P,i.e.ddPQ=q,andforeveryn,ddPQ((∙|F∙|Ftt))=q(∙|Ft).Then,thefollowinglineof
equationsholds:

EQ[X|Ft]=EQ(∙|Ft)[X]
=EP(∙|Ft)[q(∙|Ft)X]

andhence
EQ[X|Ft]−EP[X|Ft]=EP(∙|Ft)[(q(∙|Ft)−1)X]
≤κEP(∙|Ft)[(q(∙|Ft)−1)]
=κ(q(∙|Ft)−1)P(d∙|Ft),
whichconvergestozeroP-a.s.byBlackwell-Dubinstheoremas(Ft)tisas-
sumedtobeafiltrationand,hence,anincreasingfamilyofσ-fields.
Remark4.5.2.Asweseeintheproof,theparametricsettingisnotneeded.
Theassertioncanbeshowninthesamefashioninanon-parametricsetting.
Thesameholdstrueforsubsequentresults.

4.6Time-ConsistentRiskMeasures

WewillnowshowaBlackwell-Dubinstyperesultforcoherentaswellas
convexriskmeasuresincasetime-consistencyisassumed.Weseethatthe
riskmeasureeventuallyequalstheexpectedvalueunderthetrueparameter;
inthissense,uncertaintyvanishesbutriskremains.
1Notethatwehaveassumedalldistributionsinducedbyparametersθ∈Θtobe
equivalent.Inparticular,allthoseareabsolutelycontinuouswithrespecttoeachother
andthisassumptionisnorestrictionwithinoursetup.Alsonotethattherespectiveθ
doesnothavetobeθ0.

18

4.INCREASINGINFORMATION&CONVEXRISK

4.6.1Time-ConsistentCoherentRisk

Let(ρt)tbeatime-consistentcoherentriskmeasurepossessingrobustrepre-
sentation

θρt(X)=supEP[−X|Ft],
˜Θ∈θwithΘ˜⊂Θassumedtobeaconvexandcompactsetofparametersinducing
aweaklycompactandconvexsetofpriorsQ˜⊂Me(Pθ0).
Proposition4.6.1.ForeveryessentiallyboundedF-measurablerandom
variableXandtime-consistentcoherentriskmeasure(ρt)twehave
ρt(X)−EPθ0[−X|Ft]→0Pθ0-almostsurelyfort→∞.
Proof.Thankstotheassumptionoftime-consistencyandcompactnessthere
∗θexistsaparameterθ∗∈Θ˜suchthatρt(X)=EP[−X|Ft]forallt∈
{0,...,T}resultinginthefollowingequation
θθ∗θ
ρt(X)−EP0[−X|Ft]=EP[−X|Ft]−EP0[−X|Ft]
∗andthisconvergestozeroastincreasesandPθ∼Pθ0byProposition4.5.1.

Remark4.6.2.Notethatwehavenotassumedθ0∈Θ˜.
Remark4.6.3.TheassumptionthatΘ˜isweaklycompactisaverycrucial
assumption,asitassuresthatthesupremumisactuallyattained.Addition-
allyitisanecessarypropertyforourresulttohold,whichisshowninthe
4.6.4.ositionopPr

Proposition4.6.4.Weakcompactnessoftheset{Pθ|θ∈Θ˜}ofpriorsisa
necessaryconditionforourresultinProposition4.6.1tohold.

Proof.Fortheproof,seethecounterexampleinSection4.8.2.

82

4.6.TIME-CONSISTENTRISKMEASURES

4.6.2Time-ConsistentConvexRisk
Let(ρt)tbeatime-consistentdynamicconvexriskmeasure,hence,possessing
thefollowingrobustrepresentation:
ρt(X)=esssupEPθ[−X|Ft]−αtmin(θ)
Θ∈θwithdynamicminimalpenalty(αtmin)t.
Assumption4.6.5.Weassume(ρt)ttobecontinuousfrombelowforallt,
i.e.foreverysequenceofrandomvariables(Xj)j,Xj∈L∞forallj,with
XjX∈L∞wehavelimj→∞ρt(Xj)=ρt(X).
Remark4.6.6.Inthecoherentcase,continuityfrombelowisequivalentto
weakcompactnessoftheset{Pθ|(αt(θ))t=0}={Pθ|θ∈Θ˜}ofpriorsas
interaliashownin[Riedel,09].

Thisassumptionhastechnicaladvantagesasitensuresthesupremumto
beachievedintherobustrepresentationofρt.AproofisgiveninTheorem
1.2of[Fo¨llmeretal.,07].Itisalsoshownthatcontinuityfrombelowimplies
continuityfromabove.Tosumup:continuityfromaboveisequivalenttothe
existenceofarobustrepresentation.Continuityfrombelow(whichgeneral-
izesthecompactnessassumptioninthecoherentcase)isequivalenttothe
existenceofarobustrepresentationintermsofadistinctpriordistribution,
thesocalledworstcasedistribution.
Fromaneconomicpointofview,continuityfrombelowresultsfroma
featureofpreferencesalreadyclaimedin[Arrow,71]andrelatedtothisas-
sumptionby[Chateauneufetal.,05].Theconditiononpreferencesweneed
toaskforinordertoobtainthisfeatureiscalledMonotoneContinuity:If
anactfispreferredoveranactgthenaconsequencexisneverthatbad
thatthereisnosmallpsuchthatxwithprobabilitypandfwithprobability
(1−p)isstillpreferredoverg.Thesameistrueforgoodconsequencesmixed
.ghtwi

38

4.INCREASINGINFORMATION&CONVEXRISK

Formallythismeans,foractsfg,aconsequencexandasequenceof
events{En}n∈NwithE1⊇E2⊇...and∩n∈NEn=∅thereexistsann¯∈N
suchthat
xifs∈En¯xifs∈En¯
f(s)ifs∈/En¯gandfg(s)ifs∈/En¯.
NowwiththehelpofthisassumptionwecanshowtheBlackwell-Dubins
resultfortime-consistentconvexriskmeasures:
Proposition4.6.7.ForeveryessentiallyboundedF-measurablerandom
variableXandtime-consistentconvexriskmeasure(ρt)t,continuousfrom
holdsitelow,bρt(X)−EPθ0[−X|Ft]→0Pθ0-almostsurelyfort→∞
ifthereexistsθ∈Θsuchthatαtmin(θ)→0Pθ0-almostsurelyandα0min(θ)<
.∞Remark4.6.8(OntheAssumption).BythemainassumptioninProposi-
tion4.6.7thereoughttobesomeθsuchthatthepenaltyvanishesinthelong
run.Thisintuitivelymeansthat,eventually,natureatleasthastopretend
somedistributiontobethecorrectone.Weseethatthisissatisfiede.g.in
thecoherentorintheentropiccase.
Theassertionthenstatesthatitdoesnotmatterwhichriskmeasurewas
chosenaslongasthepenaltyisfiniteinthebeginning.Inthetime-consistent
case,thepenaltythenvanishesforallthoseparametersandtheconvexrisk
eventuallywillbecoherent.
Aswewillseelater,inthenon-time-consistentcase,naturehastopaya
pricefornotchoosingadistributiontime-consistentlyasinthatcasepenalty
hastovanishforthetrueunderlyingparameter.Toconclude:whennature
choosestheworstcasedistributiontime-consistently,shemerelyhastopre-
tendsomedistributiontobetheunderlyingone.Ifshedoesnotchoosethe
worstcasemeasuresatanystagetime-consistently,shehastorevealthetrue
underlyingdistributioninthelongrun.

84

4.6.TIME-CONSISTENTRISKMEASURES

Remark4.6.9.ByTheorem5.4in[F¨ollmer&Penner,06]duetotime-
consistencytheassumptionαtmin(θ)→0Pθ0-almostsurelyforsomeθ∈Θis
equivalenttoαtmin(θ)→0Pθ0-almostsurelyforallθ∈Θwithα0(θ)<∞.
Proofoftheproposition.Byourassumptionson(ρt)tthereexistsθ∗∈Θ
suchthattheassertionbecomes
∗tEPθ[−X|Ft]−αmin(θ∗)−EPθ0[−X|Ft]→0Pθ0-a.s.
Bytheforegoingpropositiononcoherentrisk,weknowthatthisassertion
holdsifandonlyif
αtmin(θ∗)→0Pθ0-a.s.
Theorem5.4in[Fo¨llmer&Penner,06]impliesthisconvergencebeingequiv-
totenlaαtmin(θ)→0Pθ0-a.s.
forsomeθ∈Θsuchthatα0(θ)<∞asassumedtoholdintheassertion.
Corollary4.6.10.ByProposition4.4.20undertheconditionsofProposition
4.6.7,θ0iseventuallylearnedupon.
Again,notethatwehavenotassumedθ0suchthatα0(θ0)<∞.
Corollary4.6.11.Everydynamictime-consistentconvexriskmeasure(ρt)t
satisfyingtheassumptionsoftheProposition4.6.7isasymptoticallyprecise
asinthesenseof[F¨ollmer&Penner,06],i.e.ρt(X)→ρ∞(X)=−X,and
viceversa.Inparticular,thisholdsforthecoherentcaseast→∞.
Proof.Bytheassumptionofcontinuityfrombelow,weknowthataworst
casemeasureintherobustrepresentationof(ρt)tisactuallyachieved.By
Theorem5.4(5)in[Fo¨llmer&Penner,06]wehavethatρt(X)→ρ∞(X)≥
−Xaswehaveassumedαtmin(θ0)→0.Thentheassertionisshownby
Proposition5.11in[Fo¨llmer&Penner,06].

58

4.INCREASINGINFORMATION&CONVEXRISK

Remark4.6.12.In[F¨ollmer&Penner,06]time-consistencyisdirectlyused
toshowtheexistenceofthelimitρ∞:=limt→∞ρt.As,byassumptionsonX
inthemodel,limt→∞(EPθ0[−X|Ft])existsweachieveexistenceofρ∞from
ourresultnotdirectlyfromtime-consistency.Inourpropositionthecon-
vergenceoftheαcorrespondstoasymptoticprecision,howeverstartingata
differentpointofview.Thequestionnowisiftime-consistencyisanecessary
conditionforourresulttohold.Ifso,wehavegainednothing,ifnot,wehave
amoregeneralexistenceresultforρ∞than[F¨ollmer&Penner,06].Wewill
tackletheproblemofnecessityoftime-consistencyforourresultwithinthe
ction.senextProposition4.6.13.(ρt)tbeingcontinuousfrombelowisanecessarycon-
ditionfortheresultinTheorem4.6.7tohold.
Proof.InProposition4.6.4weshownecessityofweakcompactnessoftheset
ofpriorsforcoherentriskmeasures.However,weakcompactnessisequivalent
tocontinuityfrombelowandcoherentriskmeasuresareparticularexamples
forconvexones.Thisproofstheassertion.
Remark4.6.14.InProposition4.6.7,iftheredoesnotexistθsuchthat
αtmin(θ)→0butαtmin(θ∗)≤c∈R+forallt≥n0forsomen0∈Nthenthere
isatleastanupperboundontheremaininguncertainty:
|ρt(X)−EPθ0[−X|Ft]|≤c

.tas∞→

4.7NotNecessarilyTime-ConsistentRiskMea-
sures

WewillnowachieveaBlackwell-Dubinstyperesultfordynamiccoherentand
convexriskmeasuresforwhichwedonotposethetime-consistencyassump-
tion.However,westillassumethedynamicriskmeasuretobecontinuous

68

4.7.NOTNECESSARILYTIME-CONSISTENTRISKMEASURES

frombelow,i.e.inthecoherentcasethesetofpriorstobeweaklycompact.
Wecanstillshowthatanticipationofriskconvergestotheexpectedvalue
ofariskyprojectXasdefinedinthemodelwithrespecttotheunderlying
parameterθ0.

4.7.1NonTime-ConsistentCoherentRisk
Wewillnowrestatetheresultinamannerthattime-consistencyisnot
needed.Wehoweverneedtoassumethatlearningtakesplace;whichisa
moreliberalassumptionthantime-consistencyasseeninSection4.8.3.
Definition4.7.1.(a)Givenadynamicconvexriskmeasure(ρt)t,continu-
∗ousfrombelowbutnotnecessarilytime-consistent,wecalladistributionPθt
instantaneousworstcasedistributionattifitsatisfies2
∗θρt(X)=EPt[−X|Ft]−αtmin(θt∗).
(b)Wesaylearningtakesplaceifthereexistsaθ∈Θ,Pθ∼Pθ0,suchthat
theinstantaneousworstcasemeasuresPθt∗→Pθweaklyfort→∞.Inthe
coherentcaseweneedθ∈Θ˜asthepenaltyisinfiniteotherwise.
Inthisverydefinition,weseehowever,thattheagentdoesnothaveto
learnthetrueunderlyingparameterθ0.Inthissense,naturemightmislead
hertoawrongparameter.
Wecannowrelaxthetime-consistencyassumptioninthemainresultof
thisarticle.Notethattime-consistencyisaspecialcaseofDefinition4.7.1
givencontinuityfrombelowasinthatcasethesequenceofinstantaneous
worstcaseparametersisconstant.Hence,weachievethemoregeneralresult:
Proposition4.7.2.Let(ρt)tbeanotnecessarilytime-consistentdynamic
coherentriskmeasureforwhichlearningtakesplace.Then
θρt(X)−EP0[−X|Ft]→0Pθ0-almostsurelyfort→∞.
2Note,thatexistenceislocallyguaranteedbycontinuityfrombelow.Aswehowever
havenotassumedtime-consistency,theinstantaneousworstcasedistributionsateachtime
periodmaydiffer,henceglobalexistenceisnotnecessarilyfulfilled.

87

4.INCREASINGINFORMATION&CONVEXRISK

Proof.Tomakethingsclearerwewillwritetheproofintermsofpenalty
functionsandnotintermsofpriors.Weknowthatacoherentriskmeasure
hasarobustrepresentationofaconvexriskmeasurewithapenalty
min0ifPθ(∙|Ft)∈Q˜(∙|Ft),
αt(θ)=
seel∞whereQ˜isthesetofpriors,i.e.Q˜={Pθ|(αtmin(θ))t=0}uniquelydefining
thecoherentriskmeasure.Asweareinthecaseofacoherentriskmeasure,
weparticularlyhaveαtmin(θt∗)=0.
First,notethatincaseαtmin(θ)→∞forallθ∈Θ˜3,ourconvergence
θresultcannothold,aslimt→∞EP0[−X|Ft]existsandisfinitebyassumption.
Secondly,inthetime-consistent(coherentaswellasconvex)case,itsuf-
ficestoassumeαtmin(θ¯)→0forsomeθ¯∈Θ.Thisassumptioninthetime-
consistentcaseisequivalenttoαtmin(θ)→0forallθforwhichα0min(θ)<∞
byTheorem5.4in[Fo¨llmer&Penner,06].
Letusnowturntotheproofitself:AsQ˜isassumedtobeweaklycompact
andnon-empty,i.e.thereexistsadistributionthathaspenaltyzero,we
achieveaninstantaneousworstcasedistributionateachtimestep,i.e.at
anyt,thereexistsθt∗∈Θs.t.
∗∗θθρt(X)=EPt[−X|Ft]−αtmin(θt∗)=EPt[−X|Ft].
Ofcourse,dueto“time-inconsistency”,wemighthaveθi∗=θj∗fori=j.
Theproofiscompletedbyshowingthefollowingconvergence4

∗EPθn[−X|Ft]→EPθ0[−X|F∞]forn,t→∞.
3Ofcourse,convergenceistrivialinthiscaseduetotrivialityofthepenaltyfunction.
4Byourassumptionsweknow:
∗θθ•EPn[−X|Ft]→EP[−X|Ft]forn→∞asθn∗→θbyPortemonteau’sTheorem.
•EPθ∗n[−X|Ft]→EPθn∗[−X|F∞]fort→∞byProposition4.5.1.
Thequestionnowis,whethertheresultalsoholdswhenlettingn,t→∞atonce.
Inthetime-consistentcase,whereθ∗i=θj∗foralli,j,thisisimmediatebyProposition
1.5.4.

88

4.7.NOTNECESSARILYTIME-CONSISTENTRISKMEASURES

Inordertodothiswelookatthefollowingequationforn≥twhichusesthe
projectivityofthedensity,i.e.oftheRadon-Nikodymderivative:
θ∗θdPθn∗
0PdFnEPn[−X|Ft]=EP0[−Xθ|Ft].
∗θFnDefinethefollowingsequenceofrandomvariablesYn:=−XddPPθ0n.These
havefiniteexpectationandthankstoourassumptionthatlearningtakes
placeandtheoriginalBlackwell-Dubinsresultwehave
∗θ∞n→∞dPθ0F∞
Pθ0[limYn=−X]=Pθ0[−XdP=−X]=1.
Then,byLemma4.7.4,theassertionfollows.
Remark4.7.3.Again,notethatwehavenotassumedθ0∈Θ˜.
Intheforegoingproof,weneedageneralmartingaleconvergenceresult
asstatedin[Blackwell&Dubins,62],Theorem2.WeknowfromDoob’s
famousmartingaleconvergenceresultthat
θθEP[X|Ft]=limEP[X|F∞]a.s.
→∞tundersuitableassumptions.Thequestionis:IfXnnXinsomesense,is
ittruethat
EPθ[X|F∞]=limEPθ[Xn|Ft]a.s.?
→∞n,tApositiveanswerisgiveninthefollowinglemma.
Lemma4.7.4.Fixθ.Let(Yn)nbeasequenceofF-measurablerandom
variablessuchthatEPθ[supn|Yn|]<∞.AssumeYn→n→∞Yalmostsurely
forsomeF-measurablerandomvariableY.Then,itholds5
θθlimEP[Yn|Ft]=EP[Y|F].
→∞n,t5TheconvergenceintheassertionofthelemmacanalsobeshowninL1.

98

4.INCREASINGINFORMATION&CONVEXRISK

Proof.Were-sampletheproofin[Blackwell&Dubins,62]:Fork∈N,set
Gk:=sup{Yn|n≥k}.Ifn≥k,wehencehaveYn≤Gkandthus
θθEP[Yn|Ft]≤EP[Gk|Ft](4.3)
forallt.TogetherwithDoob’smartingaleconvergenceresultandLebesgue’s
theorem,weachieve
θz:=limsupEP[Yn|Ft]
→∞jj≥n,t3)(4.θP≤jlim→∞supE[Gk|Ft]
j≥tθ=limEP[Gk|Ft]
→∞tDo=obEPθ[G|F]
k

dnaz≤limEPθ[Gk|F]Leb=esgueEPθ[Y|F].
→∞kInthesametoken,
θθx:=jlim→∞t,ninf≥jEP[Yn|Ft]≥EP[Y|F],
whichcompletestheproofsince
θθx=liminfEP[Yn|Ft]≤limsupEP[Yn|Ft]=z.
j→∞t,n≥jj→∞n,t≥j

Remark4.7.5(OnBlackwell-DubinsTypeLearning).Blackwell-Dubinsap-
pliesforlearningmodelsbutdoesnotnecessarilyresultintime-consistency
asthisnotionisnowmotivatedasaspecialcaseofournotionofθ0tobe
eventuallylearnedupon.
Wehavebuiltabridgebetweenthefirstandthesecondpartofthisarticle:
inthefirstpartwehaveachieveddynamicconvexriskmeasuresbyvirtueof
learningthatdidnotturnouttobetime-consistent.Hence,wehaveshown,
thatourresultevenholdsforthosemodels,e.g.entropiclearning.
09

4.7.NOTNECESSARILYTIME-CONSISTENTRISKMEASURES

Remark4.7.6.Note,thattheabovenewversionofthefundamentalresult
particularlyholdsfortime-consistentdynamiccoherentriskmeasuresasthen
suchalimitingθasintheDefinition4.7.1(b)alwaysexists,theworstcase
one.However,weparticularlyhaveanexistenceresultforthelimitρ∞:=
limt→∞ρtinthenontime-consistentcaseandthusamoregeneralexistence
resultthanin[F¨ollmer&Penner,06].

4.7.2NonTime-ConsistentConvexRisk
Asinthecaseofcoherentriskmeasures,wenowstateourgeneralization
oftheBlackwell-Dubinstheoremwhenthedynamicconvexriskmeasureis
notassumedtobetime-consistent.Asinthecoherentcase,weassumethat
learningtakesplace,i.e.thereexistsθ∈Θsuchthattheinstantaneousworst
caseθ∗t→θast→∞.Furthermore,wehavetoassumeαtmin(θt∗)→0as
n→∞:6Asintheforegoingproof,weachieveconvergenceoftheconditional
expectationsunderthefamilyofinstantaneousworstcasedistributionstothe
conditionalexpectationunderθ0.

Proposition4.7.7.ForeveryriskyprojectXassetoutinthemodeland
dynamicconvexriskmeasure(ρt)t,continuousfrombelowbutnotnecessarily
haveweonsistent,time-cρt(X)−EPθ0[−X|Ft]→0Pθ0-almostsurelyfort→∞
iflearningtakesplaceforaninstantaneousworstcasesequence(θt∗)ttoward
someθ∈Θandwehave
αtmin(θt∗)→0.
Proof.ApplyingtheprocedureusedintheproofofProposition4.7.2tothe
proofofProposition4.6.7showstheassertion.
6Note,again,wedonothavetoassumeαtmin(θ0)→0.

19

4.INCREASINGINFORMATION&CONVEXRISK

4.8Examples

Inthissection,wefirstconsiderdynamicentropicriskmeasuresasapromi-
nenteconomicexampleoftime-consistentdynamicconvexriskmeasures.In
thesecondpartwestateacounterexampleservingasproofforProposition
4.6.4and4.6.13.Asalastpoint,weconsideradynamicriskmeasurethatis
nottime-consistent.

4.8.1EntropicRisk
Here,wewillhavealookattime-consistentdynamicentropicriskmeasure
(ρte)t.RecallitsDefinition4.3.10intermsof
ρte(X):=δlogEe−γXFt
forsomemodelparameterδ>0.Afundamentalresultshowsthatthe
robustrepresentationofdynamicentropicriskisgivenintermsofconditional
relativeentropyaspenaltyfunction,i.e.foralln,wehave
11θZ
Zγγtαtmin(θ)=Hˆt(Pθ|Pη):=EPlnTFt,
θFtwhereZt:=ddPPη,theRadon-NikodymderivativeofPθwithrespecttoPη
conditionalonFt.
ThefundamentalBlackwell-DubinsTheoremimmediatelyshowsthat
Pθ(∙|Ft)−Pη(∙|Ft)→0
foreveryθ,η.Hence,wehavethatZT→1Pθ0-a.s.fort→∞andhence
Ztαtmin(θ)→0
showingProposition4.6.7tohold.Thisisanalternativewaytoshowthe
lastassertioninTheorem6.3in[Fo¨llmer&Penner,06]directly.
29

4.8.EXAMPLES

4.8.2Counterexample
ToshownecessityofcontinuityfrombelowinProposition4.6.7weconsider
thefollowingexampleintroducedin[Fo¨llmer&Penner,06]:
TheunderlyingprobabilityspaceconsistsofthestatespaceΩ=(0,1]
endowedwiththeLebesguemeasurePθ0andafiltration(Ft)tgeneratedby
thedyadicpartitionsofΩ.ThismeansFtisgeneratedbythesetsJt,k:=
(k2−t,(k+1)2−t]fork=0,...,2t−1.Inthissetting[Fo¨llmer&Penner,06]
constructatime-consistentcoherentandthereforeconvexriskmeasureswith
αtmin(θ0)→0Pθ0-a.s.ofthefollowingform:
∞ρt(X)=−esssup{m∈Lt|m≤X}.
ThatthissequencefromallpropertiesassumedinProposition4.6.7isonly
missingcontinuityfrombelow(hereequivalenttoweakcompactnessofpriors)
canbeseeninthefollowingway:LettbearbitrarybutfixedandXdefined
byvirtueof
0forω∈(0,(2t−1)2−t],
X(ω)=1else.
Thenwecanconstructasequence(Xn)n,XnX,suchthatρt(Xn)=0
forallnbutρt(X)=−X=0.Thisshows(ρt)tnotbeingcontinuousfrom
w.oelbNowwestillhavetoshowthatforthisconstructionthestatementofour
propositionisnotfulfilled.ToverifythislookatasetAassumedtobe
F:=σ(t≥0Ft)-measurablesuchthatPθ0[A]>0andPθ0[Ac∩Jt,k]=0for
alltandk.Forthisset,itholds
→∞t→∞tlimρt(1A)−EPθ0[−1A|Ft]=lim0+Pθ0[A|Ft]=Pθ0[A]>0
andhencenecessityofthecontinuityassumptionisshown.
TheskepticalreadermightnowobjectthatsuchasetAmightnotexist.
ForsakeofcompletenesswebrieflyquoteasetAfrom[Fo¨llmer&Penner,06]
39

4.INCREASINGINFORMATION&CONVEXRISK

thatsatisfiesourassumptions:LetAbedefinedbyvirtueofitscomplement
A:=Ut(k2−t),
∞2t−1c
=1k=1twhereUtdenotesthet-neighborhoodandt∈]0,2−2t].

4.8.3ANonTime-ConsistentExample
Here,weconsidertheentropiclearningmodelintroducedinDefinition4.4.6
explicitlyintermsofΩ=⊗tSt.LetPθdenotethedistributioninducedby
θ=(θt)t,θtinducingamarginaldistributioninM(St).Thoughthemodel
looksquitesimilartodynamicentropicriskmeasures,webrieflyrecallit:Let
therobustrepresentationofadynamicconvexriskmeasure(ρˆt)tbegivenby
virtueofthepenalty
αˆtmin(θ):=δHˆt(Pθ|Pθˆ),
δ>0andθˆ=(θˆt)tbeachievedasinDefinition4.4.5:fort∈N,θˆtisthe
maximumlikelihoodestimatoroftheforegoingobservationsandθˆi:=θˆtfor
i>t.Restrictingourselvestotheiidcase,weknowthatweachieveθˆt→θ¯0,
Pθ0-a.s.,whereθ0=(θ¯0)tforsomeθ¯0inducingamarginaldistributionin
M(St).Bydefinition,(ρˆt)tisadynamicconvexriskmeasure.Asshown
inProposition4.4.15,(ρˆt)tisnottime-consistent.Bystandardresultson
conditionalentropicriskmeasures,(ρˆt)tiscontinuousfrombelow.
Furthermore,Proposition4.7.7isapplicableandhence,ourgeneraliza-
tionofBlackwell-Dubins’theoremholdsforexperiencebasedentropicrisk.
Indeed:BydefinitionofthepenaltyandourconsiderationsinSection4.8.1,
αˆtmin(θ)→0ast→∞forallθ∈Θ.Secondly,asthemaximumlikelihood
estimatorisasymptoticallystable,i.e.θˆt→θ¯0,theconditionalreference
ˆdistributionsPθ(∙|Ft)converge.Thus,theworstcaseinstantaneousdistribu-
∗tionsPθtconvergeasinDefinition4.7.1duetocontinuityoftheentropyand
astheeffectivedomainofthepenaltyisgivenbyconditionaldistributions,a
49

4.9.CONCLUSIONS

factthatismadeparticularlyprecisein[Maccheronietal.,06b].7

4.9Conclusions

ThemajorcontributionofourresultsistocarryoverthefamousBlackwell-
Dubinstheoremfromprobabilitydistributionstoconvexriskmeasures.Itis
particularlystrikingthattheresultsstillholdwhentime-consistencyisnot
posedasanassumption.
Hereto,thepresentarticleistwofold:Inthefirstpart,weshowthat
explicitlyconstructingdynamicconvexriskmeasuresbyvirtueofapenalty
emergingfromalearningmechanismandinsertedintherobustrepresen-
tationofconvexriskmeasuresleadstotime-consistencyproblems.Inthe
secondpart,wehavethenassumedatime-consistentdynamicconvexrisk
measureforgrantedandaskedthequestionoflimitbehavior;moreelabo-
ratelyitsconvergencetotheexpectedvalueunderthetrueunderlyingdis-
tribution.
WethereforeintroducedageneralizationofthefamousBlackwell-Dubins
theoremon“MergingofOpinions”toconditionalexpectedvalues.Existence
ofaworstcasedistributionduetocontinuityfrombelowandtime-consistency
thenallowedforafurthergeneralizationtocoherentandconvexriskmea-
sures.Inparticular,wehaveobtainedtheexistenceofthelimitingrisk
measureρ∞inthatcase.
Byvirtueofacounterexample,wehaveshownnecessityofcontinuity
frombelowforourresult.However,wehaveshownthattime-consistency
isnotnecessaryfortheresulttohold.Inparticular,wehaveobtained
amoregeneralexistenceresultforthelimitingriskmeasureρ∞thanin
[Fo¨llmer&Penner,06].OurgeneralizationoftheBlackwell-Dubinstheorem
7Thenotationisquitemisleadingatthispoint:theworstcaseinstantaneousdistribu-
∗tionsPθt∈Me(Pθ0)asinDefinition4.7.1isadistributionon(Ω,F)asθt∗isanelement
ofΘandnota“marginal”parameterastheaboveθts.
59

4.INCREASINGINFORMATION&CONVEXRISK

wasshowntobeequivalenttothenotionoftheparameterbeingeventually
learneduponandthenotionofasymptoticprecisionin[Fo¨llmer&Penner,06]
inthetime-consistentcase.
Furtherresearchshouldbeconductedinthedirectionofourresults.
First,ofcourse,theriddleofexplicitlyconstructingconvexriskmeasures
byvirtueofthepenaltyfunctionisstilltosolve;inparticular,howalearn-
ingmechanismmightbeintroducedwithoutdestroyingtheassumptionof
time-consistency.Weakernotionsoftime-consistencythataresatisfiedin
a“learning”environmentshouldbeintroducedalongwithacomprehensive
theoryallowingforsolutionsoftangibleeconomicandsocialproblems.
Inthearticleathand,wehaveconsideredriskyprojectswithfinalpayoffs,
i.e.randomvariablesoftheformX∈F.Wehaveshownconvergenceofcon-
vexriskmeasurestotheconditionalexpectedvaluewithrespecttothetrue
underlyingdistribution:ageneralizationoftheBlackwell-Dubinstheoremto
(notnecessarilytime-consistent)convexriskmeasuresforfinalpayoffs.To
usitseemsbeinganinteresting,yetchallenging,tasktogeneralizeourresult
tothecaseofconvexriskmeasuresforstochasticpayoffprocesses(Xt)twith
respecttosomefiltration(Ft)t,whereeachXtdenotesthestochasticpayoff
inperiodt.[Cheriditoetal.,06]introducedynamicconvexriskmeasuresfor
thesestochasticprocessesandelaboratelydiscusstime-consistencyissuesbut
donotinspectlimitingbehavior.Amajordifficultyinthecaseofstochastic
processesisthattheassumptionofequivalentdistributionsshouldbere-
placedbylocalequivalence,cp.[Riedel,09].Hence,themainquestionturns
outtobeiftheresultstillholdsassuminglocalinsteadofglobalequivalence
asdonehere.

69

C5erhapt

ClosingRemarks

Withinthethreechaptersofthisthesiswehavestudiedseveralproblems
arisinginthecontextofdynamicdecisionproblemsunderKnightianUncer-
tainty.Eachchapterdiscussesitsrespectivetopicindetailandendswith
aconclusionsummarizingitsresults.Neverthelessforcompletenesswewill
brieflyrestateourachievementsatthispoint:
First,wepresentedanalternativecharacterizationfortime-consistentsets
ofmeasuresonfinitetrees.Itallowstoexpressoursetofmeasuresthrougha
setofpredictableprocesseswhichinreturnagaindefinesatime-consistentset
ofmeasures.Thisrepresentationisuniqueuptothechoiceofamartingale
basis.Tryingtogeneralizeourassumptionsinstandardwaysinorderto
achieveamoreuniversalrepresentationfailed,showingthescopeofthis
characterization.
Inthethirdchapterwestudiedifandunderwhatconditionsaduality
theoremforoptimalstoppingproblemsholdsinthemultiplepriorsframework
of[Epstein&Schneider,03].Theresultisaminimaxtheoremforrather
generalassumptionsonthepayoffprocessandstandardassumptionsonthe
setofmeasures.Weusethistheoremtoidentifytheworstcasemeasurein
thesettingofκ-ambiguityadaptedtoourframeworkandapplyittomultiple
priorsuper-andsubmartingalesdeterminingtheoptimalstoppingtimes.

5.CLOSINGREMARKS

Finally,wehaveconsidereddynamicconvexriskmeasureswheninfor-

mationisgatheredincourseoftime.Wehavegeneralizedthefundamental

Blackwell-Dubinstheoremfrom[Blackwell&Dubins,62]tonotnecessarily

time-consistentdynamicconvexriskmeasuresandhavethusshowntheircon-

vergencetoconditionalexpectedvalueswithrespecttothetrueunderlying

distribution:

es.ruden

89

Intuitively

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tlsure

wsosh

ttah

uncertainty

vanishes

tub

irsk

Bibliography

[Anscombe&Aumann,63]ANSCOMBE,F.J.&AUMANN,R.J.(1963):A
DefinitionofSubjectiveProbability,AnnalsofMathematicalStatistics,
Vol.34,pp.199-205.

[Arrow,71]ARROW,K.(1971):EssaysintheTheoryofRiskBearing,
MarkhamPublishingCo.,Chicago.

[Artzneretal.,99]ARTZNER,P.;DELBAEN,F.;EBER,J.-M.&HEATH,
D.(1999):CoherentMeasuresofRisk,MathematicalFinance,Vol.9,
No.3,pp.203-228.

[Artzneretal.,07]ARTZNER,P.;DELBAEN,F.;EBER,J.-M.;HEATH,
D.&KU,H.(2007):CoherentMultiperiodRiskAdjustedValuesand
Bellman’sPrinciple,AnnalsofOperationsResearch,Vol.152,No1,pp.5-
2.2

[Blackwell&Dubins,62]BLACKWELL,D.&DUBINS,L.(1962):Merging
ofOpinionswithIncreasingInformation,Ann.Math.Statistics,Vol.33,
No.3,pp.882-886.

[Chateauneufetal.,05]CHATEAUNEUF,A.;MACCHERONI,F.;MARI-
NACCI,M.&TALLON,J.-M.(2005):MonotoneContinuousMultiple
Priors,EconomicTheory,Vol.26,No.4,pp.973-982.

99

[Chen&Epstein,02]CHEN,Z.&EPSTEIN,L.(2002):Ambiguity,Risk,
andAssetReturnsinContinuousTime,EconometricaVol.70,No.4,
pp.1403-1443

[Cheriditoetal.,06]CHERIDITO,P.;DELBAEN,F.&KUPPER,M.
(2006):DynamicMonetaryRiskMeasuresforBoundedDiscrete-Time
Processes,ElectronicJournalofProbability,Vol.11,No.3,pp.57-106.

[Cheridito&Stadje,07]CHERIDITO,P.&STADJE,M.(2009):Time-
inconsistencyofVaRandtime-consistentalternatives,FinanceResearch
Letters,Vol.6,No.1,pp.40-46.

[Chudjakow&Vorbrink,09]CHUDJAKOW,T.&VORBRINK,J.(2009):
ExerciseStrategiesforAmericanExoticOptionsunderAmbiguity,
IMW-Workingpaper

[Czichowsky&Schweizer,09]CZICHOWSKY,C.&SCHWEIZER,M.
(2009):ClosednessintheSemimartingaleTopologyforSpacesof
StochasticIntegralswithConstrainedIntegrands,FINRISKWorking
.erpap

[Delbaen,03]DELBAEN,F.(2003):TheStructureofm-StableSetsandin
ParticularoftheSetofRiskNeutralMeasures,InYor,M.,Emery,
M.(eds.):InMemoriamPaul-Andre´Meyer-S´eminairedeProbabilit´es
XXXIX.BerlinHeidelbergNewYork:Springer,pp.215-258.

[Dothan,90]DOTHAN,M.(1990):PricesinFinancialMarkets,1.edition,
OxfordUniversityPress.

[ElKaroui,81]ELKAROUI,N.(1981):LesAspectesProbabilistesduCon-
troˆleStochastique,LectureNotesinMathematics,Vol.876,pp.73-238.
Springer,Berlin.

[Ellsberg,61]ELLSBERG,D.(1961):Risk,Ambiguity,andtheSavageAx-
ioms,QuarterlyJournalofEconomics,Vol.75,pp.643-669.
[Engelage,09]ENGELAGE,D.(2009):OptimalStoppingunderConvex
Risk,inEssaysonCoherentandConvexRiskMeasures,Ph.D.The-
sis,BonnGraduateSchoolofEconomics.
[Epstein&Marinacci,06]EPSTEIN,L.&MARINACCI,M.(2006):Mu-
tualAbsoluteContinuityofMultiplePriors,JournalofEconomicThe-
ory,Vol.137,No.1,pp.716-720.
[Epstein&Schneider,03]EPSTEIN,L.&SCHNEIDER,M.(2003):Recur-
siveMultiplePriors,JournalofEconomicTheory,Vol.113,pp.1-13.
[Epstein&Schneider,07]EPSTEIN,L.&SCHNEIDER,M.(2007):Learn-
ingunderAmbiguity,WorkingPaper,UniversityofRochester.
[Fo¨llmer&Schied,04]FO¨LLMER,H.&SCHIED,A.(2004):Stochastic
Finance,AnIntroductioninDiscreteTime,2ndedition,WalterDe-
Gruyter.
[Fo¨llmer&Penner,06]FO¨LLMER,H.&PENNER,I.(2006):ConvexRisk
MeasuresandtheDynamicsoftheirPenaltyFunctions,Statisticsand
Decisions,Vol.24,pp.61-96.
[Fo¨llmeretal.,07]FO¨LLMER,H.;SCHIED,A.&WEBER,S.(2007):Ro-
bustPreferencesandRobustPortfolioChoice,ORIE,CornellUniversity.
[Gilboa&Schmeidler,89]GILBOA,I.&SCHMEIDLER,D.(1989):Max-
iminExpectedUtilitywithnon-uniquePrior,JournalofMathematical
Economics,Vol.18,pp.141-153.
[Hansen&Sargent,01]HANSEN,L.&SARGENT,T.(2001):Robustcon-
trolandModelUncertainty,AmericanEconomicReviewPapersand
Proceedings,Vol.91,pp.60-66.

[Karatzas&Kou,98]KARATZAS,I.&KOU,S.(1998):HedgingAmerican
ContingentClaimswithConstrainedPortfolios,FinanceandStochas-
tics,Vol.2,pp.215-258.

[Karatzas&Shreve,98]KARATZAS,I.&SHREVE,S.(1998):Methodsof
MathematicalFinance,NewYork:Springer

[Klibanoffetal.,08]KLIBANOFF,P.;MARINACCI,M.&MUKERJI,S.
(2008):RecursiveSmoothAmbiguityPreferences,JournalofEconomic
Theory,forthcoming.

[Knight,21]KNIGHT,F.H.(1921):Risk,UncertaintyandProfit,Boston,
MA:Hart,Schaffner&Marx;HoughtonMifflinCompany,1921.

[Kreps,79]KREPS,D.(1979):ARepresentationTheoremfor‘Preference
forFlexibility’,Econometrica,Vol.47,No.3,pp.565-577.

[Maccheronietal.,06a]MACCHERONI,F.;MARINACCI,M.&RUSTI-
CHINI,A.(2006):AmbiguityAversion,Robustness,andtheVariational
RepresentationofPreferences,Econometrica,Vol.74,No.6,pp.1447-
8.491

[Maccheronietal.,06b]MACCHERONI,F.;MARINACCI,M.&RUSTI-
CHINI,A.(2006):DynamicVariationalPreferences,JournalofEco-
nomicTheory,Vol.128,pp.4-44.

[McNeiletal.,05]MCNEIL,A.;FREY,R.&EMBRECHTS,P.(2005):
Quantitativeriskmanagement:concepts,techniques,andtools,Prince-
tonUniversityPress.

[vonNeumann&Morgenstern,44]VONNEUMANN,J.&MORGEN-
STERN,O.(1944):TheoryofGamesandEconomicBehavior,Second
Edition,PrincetonUniversityPress,Princeton,NJ,1947.

[Revuz&Yor,91]REVUZ,D.&YOR,M.(1991):ContinuousMartingales
andBrownianMotion,Springer-Verlag,Berlin.
[Riedel,04]RIEDEL,F.(2004):DynamicCoherentRiskMeasures,Stochas-
ticProcessesandtheirApplications,Vol.112,pp.185-200.
[Riedel,09]RIEDEL,F.(2009):OptimalStoppingforMultiplePriors,
Econometrica,Vol.77(3),pp.857-908.
[Roorda&Schumacher,05]ROORDA,B.&SCHUMACHER,H.(2005):
TimeConsistencyConditionsforAcceptabilityMeasures,withanAp-
plicationtoTailValueatRisk,Insurance:MathematicsandEconomics,
Vol.40(2),pp.209-230.
[Savage,54]SAVAGE,L.J.(1954):TheFoundationofStatistics,JohnWi-
leyandSons,NewYork.Revisedandenlargededition,Dover,NewYork,
2971[Schied,07]SCHIED,A.(2007):OptimalInvestmentforRisk-and
Ambiguity-AversePreferences:ADualityApproach,Financeand
Stochastics,Vol.11,pp.107-129.
[Schnyder,02]SCHNYDER,M.(2002):DieHypothesefinanziellerInstabil-
ita¨tvonHymanP.Minsky–EinVersuchdertheoretischenAbgrenzung
undErweiterung.
[Shiryayev,78]SHIRYAYEV,A.N.(1978):OptimalStoppingRules,
Springer-Verlag,NewYork.
[Trevi˜no,09]TREVI˜NO-AGUILAR,E.(2009)T-Systemsandthelower
SnellEnvelope,arXiv:‘QuantativeFinancePapers’,No0902.4245v1.

MonikaBier

KurzerLebenlauf

Geborenam19.Ma¨rz1980

Abitur1999amGymnasiumamStadtpark,Krefeld

StudiumderMathematikander
Rheinischen-Friedrich-Wilhelms-Universita¨tBonn

Diplom-Mathematikerin12/2005

PromotionsstudiumanderRheinischen-Friedrich-Wilhelms-Universita¨t
BonnundderUniversta¨tBielefeldab3/2006

PromotionzumDr.rer.pol.anderWirtschaftswissenschaftlichen
Fakulta¨t,Universta¨tBielefeld,9/2010

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