Predictable projections and predictable dual projections of a two parameter stochastic process

2006·3.8 MB·English

by  Gray, Peter

PREDICTABLEPROJECTIONSANDPREDICTABLE
DUALPROJECTIONSOFATWOPARAMETERSTOCHASTICPROCESS
By
PETERGRAY
ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL
OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT
OFTHEREQUIREMENTSFORTHEDEGREEOF
DOCTOROFPHILOSOPHY
UNIVERSITYOFFLORIDA
2006
ACKNOWLEDGMENTS
Iamindebtedtomysupervisor,Dr.NicolaeDinculeanu,forhisinfinitepatience
withmewhileIslowlylearnedthematerial.Also,Iamgratefultothemanyexcellent
teachersthat I havehadduringmyjourneyattheUniversityofFlorida. Finally,for
thefriendlybanterandheartylaughsthat Iholdinfondmemory,IthankJulia,
Connie,andGretchen.
II
1
TABLEOFCONTENTS
ACKNOWLEDGMENTS
ii
ABSTRACT
CHAPTER
1 THECROSSSECTIONTHEOREM 1
Introduction 1
Predictable^-Algebras 2
StoppingTimes 6
Projections 7t[A] g
Sets fZs 1
TheCrossSectionTheorem 21
2 PREDICTABLEPROJECTIONS 37
Projections 37
TheUniquenessof 39
TheExistenceof /’X 43
3 PREDICTABLEDUALPROJECTIONS 50
StepFiltrations (^,) 50
PredictableDualProjections X/' 52
TheUniquenessof XF 54
PredictableDualProjectionsofMeasures 55
ProcessesAssociatedWithStochastic,K-ValuedMeasures 58
TheExistenceofX^ 64
4 VECTOR-VALUEDPREDICTABLEDUALPROJECTIONS 77
PredictableDualProjections W^ 77
TheUniquenessof W^ 81
ProcessesAssociatedWithStochastic,E-ValuedMeasures 82
TheExistenceof W^ 89
iii
5 ANEXTENSIONOFTHERADON-NIKODYMTHEOREM
TOMEASURESWITHFINITESEMIVARIATION 94
6 SUMMARYANDCONCLUSIONS 106
REFERENCELIST 107
BIOGRAPHICALSKETCH 108
IV
.
AbstractofDissertationPresentedtotheGraduateSchool
oftheUniversityofFloridainPartialFulfillmentofthe
RequirementsfortheDegreeofDoctorofPhilosophy
PREDICTABLEPROJECTIONSANDPREDICTABLE
DUALPROJECTIONSOFATWOPARAMETERSTOCHASTICPROCESS
By
PETERGRAY
August2006
Chair;NicolaeDinculeanu
MajorDepartment:Mathematics
Theframeworkofthisdissertationconsistsofaprobabilityspace (Q, afil-
tration suchthatif isafiltrationsatisfying =3^s- forevery se
thenwehave Qs = 3^s- foreverypredictablestoppingtime Sfor adouble
filtration {3^s,i)s,ie^i^ suchthat 3^s,t = 3^s- forevery >0; andaBanachspace E.
Westudyinitiallyareal-valued,twoparameterstochasticprocess X:Qx K,
andthenweextendsomeofourresultstoavector-valuedprocess Y:Qx ^
In Chapter1 westartbydefiningthepredictable o-algebra p ofsubsetsof
QX tobethe a-algebrageneratedbytheleftcontinuousprocesses X thatare
adaptedtothedoublefiltration {3^s,t)s,K'SL^.Thenweprovethemainresultofthechap-
ter,thecrosssectiontheoremforsetsin p
InChapter2 wedefinethepredictableprojectionofameasurableprocess
X:QXIR2 tobeapredictableprocess ^X; x oj suchthatforevery
V
stoppingtime Z wehave E[Xz1<z<oo>\3^z] = (^X)z1 almostsurely.Then,
usingthecrosssectiontheorem,weshowthatthepredictableprojectionisunique
uptoanevanescentset.Inaddition,wedemonstratethateverybounded,measurable
process X hasapredictableprojection.
in Chapter3 wedefinethepredictabledualprojectionofarightcontinuous,mea-
surableprocess X:Qx integrablevariationtobearightcontinuous,
predictableprocess X^:Qx k withintegrablevariationsuchthatforeach
bounded,measurableprocess (p:ClxKl R wehave E[jPep dX] = E[|(p dX^].
Thenweshowthatthepredictabledualprojectionisuniqueuptoanevanescentset.
Wealsoestablishthatthepredictabledualprojectionoftheprocess X existsifthe
filtration isastepfiltration.
In Chapters4and5 weturnourattentionfromreal-valuedprocesses X tovector-
valuedprocesses Y. Inthissetting,ourformulationsarebasednotonfinitevariation
andintegrablevariation,butonfinitesemivariationandintegrablesemivanation.
VI
CHAPTER
1
THECROSSSECTIONTHEOREM
Introduction
Thetheorysurroundingoneparameterstochasticprocesseshasapplicationsin
manyfields.Infinanceforinstance,afiltration containsinformationthatis
knownuptotime t about amarket; amartingale (X,),er^ for reflectsthe
priceofstockoptions; apredictableprocess (H/),er^ housesthenumberofshares
to beheldattime t\ andastoppingtime S for (^/),^r^ indicates whenstocks
shouldbesoldforoptimalprofit. Noteadiscrete(orstep) filtrationsufficesforgood
resultsinmanymarkets.
Thepredictableprojection(^X,),eR^andthepredictabledualprojection (XO/eR+
fortheprocess (X,),eR^ bothplayaroleinoneparameterstochastictheory.Foran
exampleofthisweretreadthefinancestagethatwassetabove.Therandomvariable
PXs, whichisthepredictableprojection (^X,),er^ evaluatedata(predictable)
stoppingtime S, mayberegardedasanupdatedversionoftheexpectedsellingprice
E[Xs] ofstockoptions,giventhemarketinformation 3^s-
Thegoalofthisdissertationistoextendthedefinitionand existence ofthepredic-
tableprojectionandthepredictabledualprojectiontoatwoparameterprocess
(Xj,/)j,,eR^. Thisextensionisdifficultbecause,whiletheset IR+ ofpositiverealnum-
bersistotallyordered,theset oforderedpairsofpositiverealnumbersisnot.
Inordertoreduceslightlythecomplexitythatweface,wewillretainaoneparameter
1
2
flavor: ourframeworkwillbebuilt around the doublefiltration = ^s-n
where isarightcontinuous,completefiltration.
Predictable a-Aigebras
Themainresultofthischapterisacrosssectiontheoremforpredictablesubsets
of QxIR2 relativetothedoublefiltration satisfying 3^s,t= 3^s- for
s,t>0.Thecrosssectiontheoremwillbederivedwithinvaluablehelpfromthe
MonotoneClasstheorem,whichmaybefoundinthetextProbabilitiesandPotential
(Dellacherie&Meyer1975,p.13-1).
Inthissectionweintroducethepredictable a-algebra p ofsubsetsofthespace
QXK+.
NotationandTerminology1.1 Thefollowingwillbeusedinthesequel.
1.1a(Q,^,P) isaprobabilityspace.
1.1bK+ isthesetofnon-negativerealnumbers.
N isthesetofnaturalnumbers.
Q+ isthesetofpositiverationalnumbers.
1R+ istheset IR+xK+, and Q+ istheset Q+xQ+.
1.1cR(lR+) istheBorel cr-algebrageneratedbytheintervals {s,t] of 0^+.
istheBorel a-algebrageneratedbytherectangles (s,r]x(«,v] of Rl.
I.ldAfunction X;Qx ^ iscalledatwoparameterprocess,andisdenoted
(X.,).
I.le^(g)B(K+) isthe <r-algebrageneratedbythesemiring B(K+).
isthe (7-algebrageneratedbythesemiring J^xB(IK2).
3
1,1f(^/),€r, isafiltrationandthereforesatisfies
• foreach r>0, isa <r-algebracontainedin 3^, and
• 3^s 3^t s<t.
Wewillwritesimply {3',),andwewillassumethatthefiltrationsatisfiestheusual
conditions:
• 3o containsallthenegligiblesets (thatis, (3,) iscomplete),and
• 3, ^ Pi every r>0 (thatis, {3,) is rightcontinuous).
S>t
See 1.1j belowforanotherassumptionabout {3,).
1.1gAfunction S:Q [0,c»] isastoppingtimeforthefiltration(3,)if{S<r}g 3,
forevery t>0.
Let S,T betwostoppingtimes.Thestochasticinterval (S,T] istheset
{(cT,r)GnXK+ I S(cj)<1<T(c7)}, while [S,T) istheset
{(c7,r)eDXIR+ I S(cj)</<T(cj)}.
Thestochasticintervals (S,T)and [S,T] aredefinedinasimilarfashion.
1.1hThe predictable cr-algebra V ofsubsetsofQxK+ isthe a-algebra
generatedbythesets Ax(^,r] and Bx{0}, where Ag3^- and Bg^o.
AstoppingtimeSispredictableifthestochasticinterval [S,oo) is apredictable
set.
1.1iLetSbeastoppingtime.
3s denotesthe cr-algebra {AeJ^j An{S<r}e^, forevery r>0}
while 3s- denotesthe cr-algebrageneratedbythesetsin 3o aswellassetsofthe
form An{S>r}, wheret>0and Ag3,.
isthefiltrationdefinedbythefollowingrules.
,
4
• ^0= ^0 and
• Qs = forevery 5’>0.
Wewillwritesimply {Q,), andwewillassumethatthefiltration (^,) issuchthat Gs
= J^s- foreverypredictablestoppingtimeSforthefiltration (Gs)- Forexample,this
isthecasewhen forevery5e1R+, where [sj isthelargestintegerless
thanorequalto5.
1.1ki3^s,t)s,eR+ isadoublefiltrationandthereforesatisfies
•• f3o^rsjea<c=h3s^,u,tv>0i,f 3s^s,<tuisaancdr-algre<bvr.acontainedin 3^, and
Wewillwritesimply andwewillassumethat = Gs forevery 5,/>0.
1.1ILet BcOx
B(c7) istheset | (t(7,j:,r)eB}.
;r[B],calledtheprojection of B,istheset
{t(7en I {fD,s,i)e B forsome s,teK+}.
1.1mThepoint (00,oo) willbedenotedby 00. Therefore,theinequality {s,t) < 00
meansthat 5 < qo and t < 00.
I.lnLet g:n^[0 00]X[0,00] beafunction.
[g] denotestheset {(c;,5,r)g x j g(gj)=(5^f)
calledthegraphof g.
Wenowcommenceourstudyofthereal-valuedtwoparameterprocess(Xsj). We
beginbydefiningthepredictable<T-algebra.
Definition1.2 Let X:Qx ^ g parameterprocess.
1.2a(X^,,) is leftcontinuous ifforevery ^oToeK+ and cjgQ, wehave