Table Of ContentSpringer Optimization and Its Applications 157
Michał Kisielewicz
Set-Valued
Stochastic
Integrals and
Applications
Springer Optimization and Its Applications
Volume 157
SeriesEditors
PanosM.Pardalos (UniversityofFlorida)
MyT.Thai (UniversityofFlorida)
HonoraryEditor
Ding-ZhuDu,UniversityofTexasatDallas
AdvisoryEditors
J.Birge(UniversityofChicago)
S.Butenko(TexasA&MUniversity)
F.Giannessi(UniversityofPisa)
S.Rebennack(KarlsruheInstituteofTechnology)
T.Terlaky(LehighUniversity)
Y.Ye(StanfordUniversity)
AimsandScope
Optimizationhascontinuedtoexpandinalldirectionsatanastonishingrate.New
algorithmicandtheoreticaltechniquesarecontinuallydevelopingandthediffusion
into other disciplines is proceeding at a rapid pace, with a spot light on machine
learning, artificial intelligence, and quantum computing. Our knowledge of all
aspects of the field has grown even more profound. At the same time, one of the
most striking trends in optimization is the constantly increasing emphasis on the
interdisciplinary nature of the field. Optimization has been a basic tool in areas
not limited to applied mathematics, engineering, medicine, economics, computer
science,operationsresearch,andothersciences.
The series Springer Optimization and Its Applications (SOIA) aims to publish
state-of-the-art expository works (monographs, contributed volumes, textbooks,
handbooks)thatfocusontheory,methods,andapplicationsofoptimization.Topics
coveredinclude,butarenotlimitedto,nonlinearoptimization,combinatorialopti-
mization,continuousoptimization,stochasticoptimization,Bayesianoptimization,
optimalcontrol,discreteoptimization,multi-objectiveoptimization,andmore.New
totheseriesportfolioincludeWorksattheintersectionofoptimizationandmachine
learning,artificialintelligence,andquantumcomputing.
Volumes from this series are indexed by Web of Science, zbMATH, Mathematical
Reviews,andSCOPUS.
Moreinformationaboutthisseriesathttp://www.springer.com/series/7393
Michał Kisielewicz
Set-Valued Stochastic
Integrals and Applications
MichałKisielewicz
FacultyofMathematics
UniversityofZielonaGóra
ZielonaGóra,Poland
ISSN1931-6828 ISSN1931-6836 (electronic)
SpringerOptimizationandItsApplications
ISBN978-3-030-40328-7 ISBN978-3-030-40329-4 (eBook)
https://doi.org/10.1007/978-3-030-40329-4
MathematicsSubjectClassification:28B20,49J21,54C60,65M75,97K60.
©SpringerNatureSwitzerlandAG2020
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional
claimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG.
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Tothememoryofmyteacher
ProfessorAndrzejAlexiewicz
Preface
The definition of set-valued integrals extending the classical Minkowski sum of
sets,wasfirstsuggestedbyR.Aumannwhodefined(see[5])theset-valuedintegral
of a measurable multifunction F : T → 2Rn as the image of its subtrajectory
integrals S(F) = {f ∈ Lp(T,F,μ,Rn) : f(t) ∈ F(t) for μ−a.(cid:2)e. t ∈ T} by
thelinearmappingJ : Lp(T,F,μ,Rn) → Rn definedbyJ(f) = f(t)μ(dt)
T
for every f ∈ Lp(T,F,μ,Rn). Later on, the above definition has been extended
toamoregeneralcase(see[4,24,25,27])dealingwithmultifunctionswithvalues
in the space 2X of all nonempty subsets of a separable Banach space (X,| · |).
It is clear that the above definition can be extended to the case where the linear
mappingJ isdefinedbyastochasticintegralandtakesitsvaluesfromtheBanach
spaceL2((cid:2),F,P,X)withagivenprobabilityspace((cid:2),F,P)andaHilbertspace
X. Such approach has been applied by F. Hiai and M. Kisielewicz (see [23, 39])
to the definition of set-valued stochastic functional integrals. In particular, in [39]
the linear mapping J, taking its values from the space L2((cid:2),F,Rn), has been
defined by Lebesgue and Itô integrals on the space L2(R+ × (cid:2),(cid:3)F,Rn) of all
squareintegrableF-non-anticipativen-dimensionalstochasticprocesses.Set-valued
stochastic functional integrals are good enough (see [42, 44–48]) to the theory of
stochasticfunctionalinclusionsxt −xs ∈ cl{Js,t(SF(F ◦x))+Js,t(SF(G◦x))},
called in the author’s monograph [48] as stochastic differential inclusions. Such
integrals are not applicable to the theory of stochastic differential inclusions and
set-valued stochastic differential equations considered in this book, because these
inclusionsandequationsaredefinedbyset-valuedstochasticintegralsthathaveto
beset-valuedrandomvariables.Therefore,wedefineset-valuedstochasticintegrals
similarlyasitwasdoneinthepaper[33].Itiseasytodefinethemforsubsetsofthe
spaceL2(R+×(cid:2),β+⊗F,Rn)withrespecttostochasticprocesseswithpathsof
boundedvariation,becauseinsuchacaseimagesofsubsetsofthisspacedefinedby
appropriatelinearmappingsaredecomposablesubsetsofthespaceL2((cid:2),F,Rn).
Unfortunately, it cannot be applied to set-valued stochastic integrals defined for
subsets of the space L2(R+ × (cid:2),(cid:3)F,Rn) with respect to stochastic processes
bothwithboundedandunboundedvariationpaths.Theproblemhasbeenpartially
solved in [33] by E.J. Jung and J.H. Kim. Unfortunately, the set-valued integral
vii
viii Preface
definedin[33]isstillnotapplicableinthetheoryofset-valuedstochasticdifferential
equations, because the set-valued stochastic integral, defined in [33], is not (see
[51,74]and[54])integrablybounded.
The book is devoted to the general theory of set-valued stochastic integrals,
treated as set-valued random variables and defined by images of subsets of the
spacesL2(R+×(cid:2),β+⊗F,Rd)andL2(R+×(cid:2),(cid:3)F,Rd×m)bylinearoperators
defined by both Lebesgue and Itô integrals. Such defined set-valued stochastic
integrals possess properties needed in the theories of stochastic differential inclu-
sions and set-valued stochastic differential equations. Therefore, the main part of
applications of such defined set-valued stochastic integrals deals with stochastic
differential inclusions and set-valued stochastic differential equations and some
applications of such inclusions in the stochastic optimal control theory and in
the finance mathematics. Set-valued stochastic integrals presented in the book are
connectedwithset-valuedfunctionalintegralsconsideredintheauthor’smonograph
[48].
Thecontentofthebookisdividedintonineparts.Thefirstthreearedevotedto
thebasicnotionsandtheoremsofthesettheory,thefunctionalanalysis,thetheory
ofstochasticprocesses,multifunctions,andthetheoryofdecomposablesubsetsof
thespaceLp(T,F,μ,X).Chapters4and5aredevotedtoAumann,Lebesgue,and
Itôset-valuedstochasticintegrals.Thenexttwochapterspresentsomeapplications
oftheaboveset-valuedstochasticintegralstothetheoriesofstochasticdifferential
inclusions, set-valued stochastic differential equations, and set-valued functional
inclusions. Chapters 8 and 9 contain some examples of applications of set-valued
stochastic integrals to the stochastic optimal control theory and the financial
mathematics,respectively.
The present book is intended for students, professionals in mathematics, and
those interested in applications of the theory of set-valued stochastic integrals.
Selectedfunctionalanalysisandprobabilisticmethodsandthetheoryofmultifunc-
tionsareneededforunderstandingthetext.Formulas,theorems,lemmas,remarks,
and corollaries are numbered separately in each chapter and denoted by three
numbers. The first stands for the chapter number, the second for the number of
thesection,andthelastforthenumberformula,theorem,etc.Theendsofproofs,
theorems, remarks, and corollaries are denoted by (cid:2). The main information on
bibliographical sources of the material presented in each chapter are contained in
thelastpartofthechapterentitledNotesandRemarks.
ThemanuscriptofthisbookwasreadbymycolleaguesM.MichtaandJ.Motyl
whomademanyvaluablecomments.Thelastversionofthemanuscriptwasreadby
ProfessorDiethardPallaschke.Hisremarkswereveryusefulinmylastcorrection
ofthemanuscript.Itismypleasuretothankallofthemfortheirefforts.
ZielonaGóra,Poland MichałKisielewicz
Contents
1 Preliminaries ................................................................. 1
1.1 SetTheoryandTopologicalPreliminaries............................. 1
1.2 FunctionalAnalysisPreliminaries ..................................... 4
1.3 SpaceofSubsetsofMetricSpace...................................... 12
1.4 LebesgueandBochnerIntegrals ....................................... 17
1.5 RandomVariables....................................................... 24
1.6 StochasticProcesses .................................................... 37
1.7 PropertiesofExitTimesofContinuousProcesses.................... 42
1.8 StochasticIntegrals ..................................................... 43
1.9 NotesandRemarks ..................................................... 58
2 Multifunctions................................................................ 61
2.1 ContinuityofMultifunctions........................................... 61
2.2 MeasurabilityofMultifunctions........................................ 69
2.3 SubtrajectoryIntegrals.................................................. 73
2.4 NotesandRemarks ..................................................... 79
3 DecomposableSubsetsofLp(T,F,µ,X)................................ 81
3.1 TheSpaceLp(T,F,μ,X)............................................. 81
3.2 DecomposableSubsetsofLp(T,F,μ,X)............................ 86
3.3 DecomposableHullsofSubsetsofLp(T,F,μ,X) .................. 91
3.4 ConditionalExpectationofSubsetsofLp(T,F,μ,X) .............. 100
3.5 Set-ValuedMartingalesandMartingaleSelectors..................... 103
3.6 NotesandRemarks ..................................................... 105
4 AumannStochasticIntegrals ............................................... 107
4.1 AumannIntegralsofSubsetsofLp(T,F,μ,X)...................... 107
4.2 AumannStochasticIntegrals........................................... 117
4.3 LebesgueSet-ValuedStochasticIntegrals ............................. 121
4.4 ApproximationofAumannStochasticIntegrals....................... 122
4.5 SelectionTheoremsforAumannStochasticIntegrals ................ 134
4.6 IndefiniteAumannStochasticIntegrals................................ 137
4.7 NotesandRemarks ..................................................... 138
ix
x Contents
5 ItôSet-ValuedIntegrals..................................................... 141
5.1 ItôSet-ValuedFunctionalIntegrals .................................... 141
5.2 ItôSet-ValuedIntegrals................................................. 150
5.3 UnboundednessofItôSet-ValuedIntegrals ........................... 155
5.4 BoundednessofItôSet-ValuedIntegrals .............................. 163
5.5 IndefiniteItôSet-ValuedIntegrals...................................... 167
5.6 IntegralRepresentationofSet-ValuedMartingales ................... 172
5.7 ApproximationofItôSet-ValuedIntegrals ............................ 174
5.8 SelectionTheoremsforItôSet-ValuedIntegrals...................... 187
5.9 NotesandRemarks ..................................................... 193
6 StochasticDifferentialInclusions .......................................... 195
6.1 ExistenceofSolutionsofSDI(F,G).................................. 195
6.2 PropertiesofStrongSolutionsSetsofSDI(F,G).................... 198
6.3 WeakCompactnessofWeakSolutionsSetsofSDI(F,G)........... 200
6.4 NotesandRemarks ..................................................... 209
7 Set-ValuedStochasticEquationsandInclusions ......................... 211
7.1 ExistenceofStrongSolutionsofSDE(F,G)......................... 211
7.2 ExistenceofWeakSolutionsofSDE(F,G) .......................... 217
7.3 WeakCompactnessofWeakSolutionsSetsofSDE(F,G).......... 225
7.4 Set-ValuedFunctionalInclusions ...................................... 232
7.5 AttainableSetsofStochasticFunctionalInclusions .................. 237
7.6 NotesandRemarks ..................................................... 247
8 StochasticOptimalControlProblems..................................... 249
8.1 OptimalControlProblemsforSystemsDescribedbySDE(f,g)... 249
8.2 OptimalControlProblemsforSystemsDescribedbySDI(F,G)... 255
8.3 NotesandRemarks ..................................................... 257
9 MathematicalFinanceProblems........................................... 259
9.1 Market,Portfolio,andArbitrage ....................................... 259
9.2 OptionPricingandConsumptionProcesses........................... 261
9.3 FinanceOptimalControlProblems .................................... 262
9.4 RecursiveUtilityOptimizationProblem............................... 270
9.5 NotesandRemarks ..................................................... 273
References......................................................................... 275
Index............................................................................... 279
List of Symbols
N Setofpositiveintegers,1
R Setofrealnumbers,1
∅ Emptyset,1
a ∈A aisanelementofasetA,1
A⊂B AisasubsetofB(setinclusion),1
A\B ComplementofBwithrespecttoA,1
limsup A Upperlimitofasequenceofsets,1
n
liminf A Lowerlimitofasequenceofsets,1
n
(X,T) Topologicalspace,2
cl[S],S ClosureofasetSof(X,T),2
(X,ρ) Metricspace,4
(X,(cid:8)·(cid:8)) Normedspace,5
∗
X Normedconjugate(dual)ofX,7
σ(X,Y) Weaktopologywithrespecttoduality,7
∗
σ(X,X ) WeaktopologyofanormedspaceX,7
s(·,A) SupportfunctionofasetA,7
coS ConvexhullofasetS,8
coS ClosedconvexhullofasetS,8
∗ ∗ ∗
σ(X ,X) Weak topologyofX ,11
Cl(X) Familyofnonemptyclosedsubsetsof(X,ρ),12
Comp(X) Familyofnonemptycompactsubsetsof(X,ρ),12
Conv(X) Familyofnonemptycompactconvexsubsetsof(X,|·|),12
¯
h(A,B) HausdorffsubdistancebetweensetsAandB,12
h(A,B) HausdorffdistancebetweensetsAandB,12
P(X) Spaceofnonemptysubsetsofthemetricspace(X,ρ),11
LiA Topologicallowerlimitofaset-valuedsequence,16
n
LsA Topologicalupperlimitofaset-valuedsequence,16
n
LimA Topologicallimitofaset-valuedsequence,16
n
A(cid:10)B SymmetricdifferenceofsetsAandB,17
RNP Radon–Nikodymproperty,23
P ⇒P Weakconvergenceofsequencesofprobabilitymeasures,28
n
xi
