Generaltheoryofprocesses

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GeneralTheoryofProcessesRichardF.Bass

DepartmentofMathematicsUniversityofConnecticut

October19,1998

c1998byRichardBass.TheymaybeusedforpersonaluseorThesenotesare??

classuse,butnotforcommercialpurposes.

InordertoreadtheFrenchliteratureonstochasticintegrationandstochasticcalcu-lus,itisveryhelpfultobesomewhatfamiliarwiththegeneraltheoryofprocesses,whichincludestopicslikepredictableandoptionalprojections,sectionstheorems,Doob-Meyerdecomposition,etc.Thesenotesaremeanttoprovideabareminimumofwhatisneces-sary.Theydo,however,includeaproofofthesectiontheoremsandoftheDoob-Meyerdecomposition.

1.Predictableandoptionalσ-?elds.

Let(?,F,P)beaprobabilityspaceandlet{Ft}bea?ltrationsatisfyingtheusualconditions.ThismeansthatFtisrightcontinuous:∩ε>0Ft+ε=Ft;andcomplete:FtcontainsallsetsNinF∞forwhichP(N)=0.AprocessXtisadaptedifXtisFt-measurableforallt.Aprocessisrightcontinuousifforalmosteveryωthemapt→Xt(ω)isrightcontinuousateacht.Leftcontinuousprocessesarede?nedsimilarly.Aprocesswhosepathsarerightcontinuouswithleftlimitswillbecalledrcll(c′adl′aginFrench).

Wede?nethepredictableσ-?eldPtobetheσ-?eldon[0,∞)×?generatedbythecollectionofallleftcontinuousprocessesadaptedtoFtandtheoptionalσ-?eldOtobetheonegeneratedbythecollectionofallrightcontinuousprocessesadaptedtoFt.ThewordforpredictableinFrenchis‘pr′evisible’.Theolderliteratureuses‘well-measurable’inplaceofthewordoptional.

Astoppingtime(oroptionaltime)Tisamappingfrom?to[0,∞]suchthattheset(T≤t)isFtmeasurableforeacht.See[PTA],pp.13–14forthebasicpropertiesofstoppingtimes.GivenastoppingtimeT,wede?neFT={A∈F:A∩(T≤t)∈Ftforallt}.

AstoppingtimeTispredictableifthereexiststoppingtimesTn↑TwithTn

Let[S,T)denotethesubsetof[0,∞)×?givenby{(t,ω):S(ω)≤t

Proposition1.1.(a)Theσ-?eldOisgeneratedbysetsoftheform{[S,T):S,Tstoppingtimes}.

(b)Pisgeneratedbythesetsoftheform{(S,T]:S,Tstoppingtimes}.

(c)Pisgeneratedbythesetsoftheform{[S,T):S,Tpredictablestoppingtimes}.Proof.(a)Since1[S,T)isarightcontinuousprocess,thesetsoftheform[S,T)areoptional.Anyrightcontinuousprocesscanbeapproximatedbylinearcombinationsofprocessesoftheform1A(ω)1[a,b)(t),whereAisFsmeasurable.Suchaprocessisoftheform1[S,T)ifwede?neS(ω)=aifω∈Aand∞otherwise,andT(ω)=bifω∈Aand∞otherwise.

Theproofof(b)issimilar.

isalwaysapredictablestoppingtime(predictedbyToprove(c),notethatS+n11S+n?kfork>n).Wehave

(S,T]=∪n{∩m[S+n,T+

m)}.

Ontheotherhand,ifSandTarepredictableandarepredictedbysequencesSnandTm,resp.,then

[S,T)=∩n{∪m(Sn,Tm]}.

AconsequenceofProposition1.1(a)and(c)isthatP?O.

2.Debutandsectiontheorems.IfA?[0,∞)×?,letπ(A)betheprojectionofAonto?.AprocessisprogressiveifforeachtthemapX:[0,t]×?→RismeasurablewithrespecttoB[0,t]×Ft,whereB[0,t]istheBorelσ-?eldon[0,t].IfFisaprogressiveset,thatis,1Fisaprogressiveprocess,thenthedebutofFisde?nedbyDF(ω)=inf{t≥0:(t,ω)∈F}.

IfXt=1A(ω)1[a,b)(t)withAmeasurablewithrespecttoFa,thenclearlyXisprogressive.Usinglinearityandtakinglimits,weseethateveryrightcontinuousadaptedprocessisprogressive,henceoptionalprocessesareprogressive.

ThedebuttheoremstatesthatTheorem2.1.IfFisprogressive,thenDFisastoppingtime.

Theoptionalsectiontheoremisthefollowing.

Theorem2.2.IfAisanoptionalsetandε>0,thereexistsastoppingtimeTsuchthatforallωwithT(ω)<∞wehave(T(ω),ω))∈AandP(T<∞)≥P(π(A))?ε.

Thepredictablesectiontheoremisthecorrespondingtheoremwithoptionalre-placedbypredictable.

Theorem2.3.IfAisapredictablesetandε>0,thereexistsapredictablestoppingtimeTsuchthatforallωwithT(ω)<∞wehave(T(ω),ω))∈AandP(T<∞)≥P(π(A))?ε.

HereisacorollaryofTheorems2.2and2.3.

Corollary2.4.(a)IfXandYareoptionalprocessessuchthatP(XT=YT)=1forevery?nitestoppingtime,thenXandYareindistinguishable:P(Xt=Ytforallt)=1.(b)IfXandYarepredictableprocesseswithP(XT=YT)=1forevery?nitepre-dictablestoppingtime,thenXandYareindistinguishable.ProofofCorollary2.4.Weprove(a),theproofof(b)beingsimilar.LetF={(t,ω):Xt(ω)=Yt(ω)}.ThenFisoptional,andifP(π(F))>0,thenthereexistsastoppingtimeTwith[T,T]?FandP(T<∞)>0.BytakingT∧Nforsu?cientlylargeN,weobtainacontradiction.??TheremainderofthissectionisdevotedtoproofsofTheorems2.1–2.3.Thismaybesafelyomittedifwished.Forsimplicity,wewillrestrictattentiontorcllreal-valuedprocesses.

RecallthatthespaceD[0,∞)withtheSkorokhod(J1)topologyiscompleteandseparable;see[Bi].(Ifonewantedtorestrictattentiontocontinuousprocesses,considerC[0,∞)instead.)

Lemma2.5.LetXtbearcllprocess.ThereexistsanincreasingsequenceofcompactsubsetsKnofD[0,∞)suchthatP(X∈∪Kn)=1.

Proof.Letε>0.LetSbetheimageof?underthemapX.SinceD[0,∞)isseparable,foreachnthereexistasequenceofopen(1/n)-ballsAniwhichcoverS.Chooseinsuch

inn∞n

thatP(X∈∪iA)>1?ε/2.Thentheset∩∪nin=1i=1i=1Aniistotallybounded.Itsclosure,Lε,isthereforecompactbecauseD[0,∞)iscomplete,andP(X∈Lε)>1?ε.NowletKn=∪n??m=1L1/m.

InviewofLemma2.5wemayconsiderourstatespacetobeR=∪nKn.

ProofofTheorem2.1.GivenaprogressivesetF,letX=1F,andde?neYt(ω)tobethe

??=elementofRwhichagreeswith1Fuptotimetandthenisconstantfromthenon.LetF

{x∈R:forsometandsomeω,x(s)=1F(s,ω)fors≤tandx(s)=1F(t,ω)fors>t}.

ThusYtisaD[0,∞)-valuedprocess;theelementsoftherangeofYtakeonlythevalues

??.0or1;thedebutofFisthesameasthe?rsttimeYentersF

IfCisaclosedsubsetofR,thenC∩KnincreasesuptoC.Sincecompactsetsareanalyticsets,thenby[PTA],PropositionII.2.3,closedsubsetsofRareanalytic,henceallBorelsubsetsofRareanalytic.

Nowusing[PTA],TheoremII.2.8andproof,weseethatthedebutofFisthe?rstentrytimeofananalyticset,henceisastoppingtime.??Remark2.6.PropositionII.2.9of[PTA]impliesmore,thatgivenε>0thereexistcompactsubsetsMnof{(t,ω):Yt(ω)=1}withDMn↓DF,whereDMnisthe?rsttimeYentersMn.

ProofofTheorem2.2.LetAbeanoptionalset.Letε<π(A)/2andchooseMnsuchthatP(DMn<∞)>P(π(A))?ε.LetS1=TMn.Wede?neSminductivelybychoosingRmsuchthat[Rm,Rm]?Am=A∩{(t,ω):Sm(ω)=∞}andP(Rm<∞)≥P(π(Am))/2,usingtheaboveremark.WethensetSm+1=Sm∧Rm.IfweletS=infmSm,then[S,S]?AandP(S<∞)=P(π(A)).

De?neameasureμon[0,∞)×?byμ(G)=1G(S(ω),ω)1(S<∞)P(dω).ThenμiscarriedonAandhasmassP(π(A)).

NowletI0denotethecollectionofsetsoftheform[S,T),whereSandTarestoppingtimeswithS≤T,andletIdenotethecollectionof?niteunionsofsetsinI0.ThenIgeneratestheoptionalσ-?eld.

LetI+denotethecollectionofcountableunionsofsetsinIandI?thecollectionofcountableintersectionsofsetsinI.Byamonotoneclassargument(cf.theargumentthatallmeasuresinametricspaceareregular),givenA∈O,thereexistB∈I?andB??∈I+suchthatB?A?B??andμ(B???B)<ε.HencethereexistBn↓BwithBn∈I.LetHbethedebutofBandletSndenotethedebutofCn=Bn∩[H,∞).EachCnisinI,soclearly[Sn,Sn]?Cn?Bn.Bythede?nitionofCn,Sn≥H.SinceHisthedebutofB,thenB?Cn,andsoSnislessthanorequaltoH.HenceH=Snforeachn.Since∩nCn?∩nBn=Band[H,H]=[Sn,Sn]?Cn,then[H,H]?∩nCn?B?A.

Itfollowsbythede?nitionofμthatHisthedesiredstoppingtime.??ProofofTheorem2.3.WeconstructμpreciselyasintheproofofTheorem2.2.WeletI0denotethecollectionofsetsoftheform[S,T),wherenowSandTarepredictablestoppingtimeswithS≤T.WenowproceedasintheproofofTheorem2.2.ThedebutofanelementofIwillnowbepredictable,andtherestgoesthroughasbefore.??

3.Projectiontheorems.

Inthissectionwewilldiscusstheoptionalandpredictableprojectionofaprocess.InSection5wewilldiscussthedualoptionalandpredictableprojectionsofprocesseswhosepathshave?nitevariation.

Thefollowingistheoptionalprojectiontheorem.Theorem3.1.LetXbeaboundedprocessthatismeasurable.ThereexistsauniqueoptionalprocessoXsuchthat

XT1(T<∞)=E[XT1(T<∞)|FT]

(3.1)

forallstoppingtimes,includingthosetakingin?nitevalues.

XiscalledtheoptionalprojectionofX.SayingXismeasurablemeansthatX

ismeasurablewithrespecttotheproductσ-?eldB×F∞,whereBistheBorelσ-?eldon[0,∞).IfXisalreadyoptional,thenbytheuniquenessresultoX=X.

If(3.1)holds,thentakingexpectationsshowsthat

E[oXT;T<∞]=E[XT;T<∞]

(3.2)

forallstoppingtimesT.Conversely,suppose(3.2)holdsforallstoppingtimesT.IfSisastoppingtimeandA∈FS,letT=SAbede?nedasinSection1.Then(3.2)impliesthat

E[oXS1(S<∞);A]=E[XS1(S<∞);A].

Thisimplies(3.1)holdsforthestoppingtimeS.Consequently(3.1)and(3.2)holdingforallstoppingtimesareequivalent.

ProofofTheorem3.1.TheuniquenessisimmediatefromCorollary2.4.Welookatexistence.IfXt(ω)=1F(ω)1[a,b)(t)whereF∈F∞,weletoXtbetherightcontinuousversionofE[1F|Ft]1[a,b)(t).ItiseasytoseethatoXsatis?es(3.2).Weuselinearityandlimitstode?neoXforboundedmeasurableX.??

Almostthesameproofgives

Theorem3.2.LetXbeaboundedmeasurableprocess.ThereexistsauniquepredictableprocesspX,calledthepredictableprojectionofX,suchthatE[pXT;T<∞]=E[XT;T<∞]foreverypredictablestoppingtimeT.

Proof.Uniquenessisasbefore.IfXt=1F(ω)1(a,b](t),weletpXt=1(a,b](t)Zt?(ω),whereZt?denotesthelefthandlimitofZtattimetandZtistherightcontinuousversionofE[1F|Ft].WeuselinearityandlimitstodothegeneralX.??4.Moreonpredictability.

Wecollectheresomefactsaboutpredictablestoppingtimesandpredictablepro-cesses,someofwhichwewillneedlater.IfUisarandomtime,i.e.,aF∞measur-ablemapfrom?to[0,∞],de?neFU?=σ{XU:Xispredictable}andFU=σ{XU:Xisoptional}.

We?rstneedtoshowthatthepresentde?nitionofFUagreeswiththeusualdef-initionofFU,namely,{A∈F∞:A∩(T≤t)∈Ftforallt},whenUisastoppingtime.

Lemma4.1.IfTisastoppingtime,then{A∈F∞:A∩(T≤t)∈Ftforallt}=FU=σ{XU:Xisoptional}.

Proof.Letusdenotethe?rstσ-?eldasF1,thelastasF2.SupposeA∈F1.Ifwede?neTAtobeTifω∈Aand∞otherwise,thenTAiseasilyseentobeastoppingtime,andthenX=1[TA,∞)isanoptionalprocess.SinceXT=1A,thenA∈F2.

LetXtberightcontinuousandbounded,andletTn=k/2nifk/2n≤T<(k+1)/2n.ThenXT=limXTn,anditiseasytoseethatXTn∈FTn,wherehereweareusingtheusualde?nitionofFTn.SoXT∈∩nFTn.SinceFtisrightcontinuous,thisisF1.??Lemma4.2.SupposeTisapredictablestoppingtimeandTnarestoppingtimesincreas-??∞

inguptoTwithTn

Proof.IfXisleftcontinuousandbounded,thenXT=limXTnandXTn∈FTn?????nFTn,soXT∈nFTn.

Ontheotherhand,supposeA∈FTnforsomen.De?neX=1(Un,∞),whereUn=Tnifω∈Aand∞otherwise.SinceTn

Proof.E[MT|FT?]=limnE[MT|FTn]=limnMTn=MT?ifTnisasequencethatpredictsT.??

WenowshowthateverystoppingtimeforBrownianmotionispredictable.Proposition4.4.LetFtbethe?ltrationofastandardBrownianmotion.IfTisastoppingtime,thenTispredictable.

Proof.LetTbeastoppingtimeforBrownianmotion.Letg(s)=(2/π)arctans,sothatgisacontinuousstrictlyincreasingfunctionfrom[0,∞]to[0,1].LetMtbetherightcontinuousmodi?cationofthemartingaleE[g(T)|Ft].ThepropertyofBrownianmotion

thatweuseisthateverymartingaleadaptedtothe?ltrationofaBrownianmotioniscontinuous.SoMtiscontinuous.

LetVt=Mt?g(T∧t)=E[g(T)?g(T∧t)|Ft].ThenVtisacontinuousnonnegativesupermartingale.ClearlyVT=0.IfSisthe?rsttimethatVtis0,thenS≤T.Also,0=EVS=E[g(T)?g(T∧S)],soS≥T.

WenowletTn=inf{t:Vt=1/n}.BythecontinuityofV,itisclearthateachTn

isstrictlylessthanTifT<∞andtheTnincreaseuptoT.HenceTispredictable.??Finally,letussupposethatAtisarightcontinuousadaptedprocesswhosepathsarenondecreasing.Wecallsuchaprocessanincreasingprocess.?AtdenotesthejumpofAattimet.

Proposition4.5.SupposeAtisanincreasingprocessand?AT=0wheneverTisatotallyinaccessiblestoppingtimeand?ATisFT?measurablewheneverTisapredictablestoppingtime.ThenAispredictable.

Proof.LetUmibetheithtime?At∈(2?m,2?m+1].Fixm,i.SinceAdoesnotjumpattotallyinaccessibletimes,thereexistsapredictabletimeS1suchthatP(S1=Umi)>0.LookingatthestoppingtimewhichequalsUmiifUmi(ω)=S1(ω)andwhichisin?niteotherwise,thereexistsS2predictablesuchthatP(S2=Umi,Umi=S1)>0becauseAdoesnotjumpattotallyinaccessibletimes.LookingatthestoppingtimewhichequalsUmiifUmi(ω)isnotequaltoeitherS1(ω)orS2(ω)andin?nityotherwise,weselectS3,etc.ThusthereisacountablesequenceofpredictablestoppingtimestheunionofwhosegraphscontainsthegraphofUmi.

Wedothisforeachmandiandobtainacountablecollectionofpredictablestoppingtimes,theunionofwhosegraphscontainsallthejumptimesofA.WeordertheminsomewayasR1,R2,....De?neT1=R1,de?neT2byrequiring[T2,T2]=[R2,R2]?[R1,R1]andingeneralTnby[Tn,Tn]=[Rn,Rn]?∪m

??c

ItistheneasytoseethatwecanwriteAt=Ac+ti(?ATn)1[Tn,∞),whereAisacontinuousincreasingprocess.Sotheproofwillbecompleteonceweshow(?ATn)1[Tn,∞)isapredictableprocess.WriteBfor?ATnandTforTn.

LetCbeaBorelsubsetofR?{0}andletD=B?1(C)={ω∈?:B(ω)∈C}.SinceBisFT?measurable,thereexistsapredictableprocessXsuchthat1D=X(T).Then[TD,TD]=[T,T]∩X?1(1),orTDispredictable.Since(B1[T,∞))?1(C)=[TD,∞),thenB1[T,∞)ispredictable,andwearedone.??5.Dualprojectiontheorems.

InthissectionAtisarightcontinuousincreasingprocess.Wedonotnecessarily

??∞

assumethatAtisadapted.De?neμAon[0,∞)×?byμA(B)=E01B(t,ω)dAt(ω).

??∞

Wede?neμA(X)byE0XtdAtisXisboundedandmeasurable.NotethatifX=0,thenμA(X)=0.

Theorem5.1.Supposeμisaboundedmeasuresuchthatμ(X)=0wheneverX=0.ThenthereexistsauniqueincreasingprocessAsuchthatμ=μA.

Proof.Ifμ=μA=μB,lett>0andletCbethesetwhereAt>Bt+ε.ThenμA([0,t]×C)≥μB([0,t]×C)+εP(C),whichimpliesP(C)=0.Sinceεisarbitrary,thenAt=Bt,a.s.SinceAandBarerightcontinuous,weconcludeA=B.

Toproveexistence,foreachrationalqde?neνq(C)=μ([0,q]×C).Clearlyνqis

??qbetheRadon-NikodymderivativeofνqabsolutelycontinuouswithrespecttoP.LetA

??isincreasinginq.LetAt=limsupq↓t,q>tA??q.ItwithrespecttoP.Sinceμispositive,A

iseasytocheckthatμA=μ.??Theorem5.2.SupposeμA(X)=μA(oX)foreveryboundedmeasurableX.ThenAtisoptional.

Proof.SinceAtisrightcontinuous,weneedonlyshowthatAtisadapted.LetYbeaboundedF∞measurablerandomvariable,Z=Y?E[Y|Ft],andXs(ω)=1[0,t](s)Z(ω).IfTisastoppingtime,then(T≤t)∈Ft,andsoE[XT;T<∞]=E[Z;T≤t]=0,

??∞

Proof.SinceμA(oX)=μA(p(oX))=μA(pX)=μA(X),thenbyTheorem5.2Atisoptional.WeneedtoshowthatAdoesnotjumpattotallyinaccessibletimes,that?ATisFT?measurableatpredictabletimesT,andthenuseProposition4.5.

LetTbeatotallyinaccessibletimeandletB=(?AT>0).LetX=1[TB,TB].ThenpX=0,so0=μA(pX)=μA(X)=E[?ATB]=E[?AT;?AT>0].

NowsupposeTispredictable.LetYbeaboundedmeasurablerandomvariable,Z=Y?E[Y|FT?]andX=Z1[T,T].IfSispredictable,E[XS;S<∞]=E[Z1(S=T);S<∞]=0since1(S=T<∞)=1[S,S](T),whichisFT?measurable.ThuspX=0,and0=μA(pX)=μA(X)=E[Z?AT].AsintheproofofTheorem5.2,wecanconcludethat?ATisFT?measurable.??

Wenowde?nethedualoptionalprojectionandthedualpredictableprojectionofanincreasingprocess.GivenarightcontinuousincreasingnotnecessarilyadaptedprocessAt,de?neμobyμo(X)=μA(oX)forboundedmeasurableX.IfX=0,thenoX=0,soμ(oX)=0.Clearlyμo(oX)=μA(o(oX))=μA(oX)=μo(X).ByourconstructionofoX,weseethatoX≥0ifX≥0,henceμoisapositivemeasure.ThereforebyTheorem5.2μocorrespondstoanoptionalincreasingprocessAo,thedualoptionalprojectionofA.

Thedualoptionalprojectionisusedinexcursiontheory.Morecommonlyusedisthedualpredictableprojection,whichisde?nedinasimilarway.De?neμp(X)=μA(pX),andletApbethepredictableincreasingprocessassociatedtoμp.WeoftendenoteApby??andcallitthecompensatorofA.ThereasonforthisterminologyisthefollowingA

proposition.

??tisamartingale.Proposition5.4.LetAtbeanadaptedincreasingprocess.ThenAt?A

Proof.Lets

ifω∈B,in?nityotherwise.LetX=1(S,T].ThenE[At?As;B]=μA(X)=μA(pX)=

μAp(X)=E[Ap??t?As;B],whichdoesit.6.TheDoob-Meyerdecomposition.

WesayaprocessXisofclassDifthefamily{XT:Tastoppingtime}isuniformlyintegrable.

Proposition6.1.IfMisapredictablelocalmartingale,thenMiscontinuous.Proof.Bylocalization,itsu?cestoconsideruniformlyintegrablemartingales.ByCorol-lary4.3,E[MT|FT?]=MT?.SinceMispredictable,MTisFT?measurable,andsoE[MT|FT?]=MT.HenceMT=MT?atallpredictablestoppingtimes,andinparticularatthe?rsttimethepredictableprocessMtjumpsmorethanε.??

TheDoob-Meyerdecompositionisthefollowing.

Theorem6.1.IfZtisarcllsubmartingaleofclassDthatis0at0,Zt=Mt+At,whereMtisauniformlyintegrablemartingalethatis0at0andAtisapredictableincreasingprocessthatis0at0.Thedecompositionisunique.

Proof.IfZ=M+A=N+B,thenM?N=B?A,andsoM?Nisapredictablemartingale.ByProposition6.1,M?Nisacontinuousmartingale.SinceM?N=B?A,thenM?Nisacontinuousmartingaleofboundedvariation,henceM?N=0by[PTA],PropositionI.4.19.Thisprovesuniqueness.

LetIdenotethe?niteunionsofsubsetsof[0,∞)×?oftheform(S,T]whereS≤Tarestoppingtimes.De?neμ((S,T])=E[ZT?ZS].SinceZisasubmartingale,

thenμisnonnegative.ItisnothardtoseethatIisanalgebraandthatμis?nitelyadditiveonI.

IfK=(S1,T1]∪···∪(Sn,Tn]withS1≤T1≤S2≤···≤Tn,letK=[S1,T1]∪···∪[Sn,Tn].

IfH=(S,T]andε>0,letSn=S+(1/n)andTn=TifS+(1/n)μ(H)?ε.

WenowprovethatμiscountablyadditiveonI.SupposeHi∈IwithHi↓?.Weneedtoshowthatμ(Hn)↓0.

Letε>0andchooseKi∈IsuchthatKn?Hnwithμ(Kn)>μ(Hn)?ε/2n.LetLn=K1∩···Kn.Thenforeachnwehaveμ(Hn)≤μ(Ln)+ε.SinceLn?Kn?Hn,wehaveLn↓?.

LetRnbethedebutofLn.LetFn={t:(t,ω)∈Ln}.Thisisaclosedsubsetof[0,∞),andRn(ω)∈Fmwhenevern≥mandRn(ω)<∞.ThestoppingtimesRnincrease;letRbethelimit.IfR(ω)<∞,thenR(ω)∈Fnforeachm,whichcontradicts∩Lm=?.ThereforeR=∞.SinceZisofclassD,thenZRnconvergesa.s.andinL1.Thusμ(Ln)≤E[Z∞?ZRn]→0.Hencelimsupμ(Hn)<ε,andsinceεisarbitrary,μ(Hn)→0.

ThisprovesthatμiscountablyadditiveonI.BytheCarath′eodoryextensiontheorem,μmaybeextendedtoameasureonP.

De?neμ??(X)=μ(pX).Thenμ??(pX)=μ??(X),andsothereexistsapredictableincreasingprocessAtsuchthatμ??=μA.IfSisanystoppingtime,E[A∞?AS]=μ??((S,∞))=E[Z∞?ZS].Lettingt>0andB∈Ft,de?neS=tifω∈Bandin?nityotherwise.ThenE[A∞?At;B]=E[Z∞?Zt;B],orMt=Zt?Atisauniformlyintegrablemartingale.??Proposition6.2.AtiscontinuousifandonlyifEZTn→EZTwheneverTn↑TandTn

Proof.SinceE[A∞?ATn]=E[Z∞?ZTn],thenE[A∞?AT?;T<∞]=E[Z∞?ZT?;T<∞],usingthefactthatZisofclassD.AlsoE[A∞?AT]=E[Z∞?ZT].SoE[AT?AT?]=E[ZT?ZT?].ThisiszeroifandonlyifEZT=EZT?.??References.

[PTA]R.Bass,ProbabilisticTechniquesinAnalysis.NewYork,Springer-Verlag,1995.[Bi]P.Billingsley,ConvergenceofProbabilityMeasures.NewYork,Wiley,1968.

[DM]C.DellacherieandP.-A.Meyer,ProbabilitiesandPotential,vol.A.Amsterdam,North-Holland,1978.

[RW]L.C.G.RogersandD.Williams,Di?usions,MarkovProcesses,andMartingales:vol.2,It?oCalculus.NewYork,Wiley,1987.