Thiscollectionofproblemsisplannedasatextbookforuniversitycoursesinthe
theoryofstochasticprocessesandrelatedspecialcourses.
Theproblemsinthebook haveawidespectrumofthelevelofdif
cultyandcanbeusefulforreaderswith
variouslevelsofmasteringinthetheoryofstochasticprocesses.
Togetherwithte-
nicalandillustrativeproblemsintendedforbeginners,thebookcontainsanumber
ofproblemsoftheoreticalnaturethatcanbeusefulforstudentsandundergraduate
studentsthatpursueadvancedstudiesinthetheoryofstochasticprocessesandits-
plications.
Amongothers,theimportantaimofthebookistoprovideateachingstaff anef
cienttoolforpreparingseminarstudies,tests,andexamsconcerninguniversity
coursesinthetheoryofstochasticprocessesandrelatedtopics.
Whilecomposingthe
book,theauthorshavepartiallyusedthecollectionsofproblemsinprobabilityt-
ory[16,65,75,83]. Also,someexercisesandproblemsfromthemonographsand
textbooks[4,9,19,22,82]wereused.
Atthesametime,alargepartofourproblem bookcontainsoriginalmaterial.
Thebookisorganizedasfollows.
Theproblemsarecollectedintochapters,each
chapterbeingdevotedtoacertaintopic.
Atthebeginningofeachchapter,theth-
reticalgroundsforthecorrespondingtopicaregivenbrie
ytogetherwiththelistof
bibliography,whichthereadercanuseinordertostudythistopicinmoredetail.
For
themostoftheproblems,eitherhintsorcompletesolutions(oranswers)aregiven,
andsomeoftheproblemsareprovidedwithbothhintsandsolutions(answers).
H-
ever,theauthorsdonotrecommendthatareaderusethehintssystematically,because
solvingaproblemwithoutassistanceismuchmoreusefulthanusingaready-made
idea.
Somestatementsthathaveaparticulartheoreticalinterestareformulatedon
theoreticalgrounds,andtheirproofsareformulatedasproblemsforthereader.
Such problemsaresuppliedwitheithercompletesolutionsordetailedhints.
Inordertoworkwiththeproblembookef ciently,areadershouldbeacquainted
withprobabilitytheory,calculus,andmeasuretheorywithinthescopeofresp-
tiveuniversity courses. Standard notions, suchas random variable,
measurability, independence, Lebesgue measure and integral, and so
on are used without ad- tionaldiscussion.
Allthenewnotionsandstatementsrequiredforsolvingthepr-
lemsaregiveneitherontheoreticalgroundsorintheformulationsoftheproblems
vii viii Preface straightforwardly.
However,sometimesanotionisusedinthetextbeforeitsformal de nition.
Forinstance,theWienerandPoissonprocessesareprocesseswithin-
pendentincrementsandthusareformallyintroducedinaTheoreticalgroundsfor
Chapter5,buttheseprocessesareusedwidelyintheproblemsofChapters2to4.
Theauthorsrecommendthatareaderwhocomestoanunknownnotionorobject
usetheIndexinorderto ndthecorrespondingformalde nition. Thesamerec-
mendationconcernssomestandardabbreviationsandsymbolslistedattheendofthe
book.
Someproblemsinthebookformcycles:solutionstooneofthemaregrounded
onstatementsofothersoronauxiliaryconstructionsdescribedinsomepreceding
solutions.
Sometimes,onthecontrary,itisproposedtoprovethesamestatement
withindifferentproblemsusingessentiallydifferenttechniques.
Theauthorsrec- mendareaderpayspeci
cattentiontothesefruitfulinternallinksbetweenvarious
topicsofthetheoryofstochasticprocesses.
Everypartofthebookwascomposedsubstantiallybyoneauthor. Chapters1-6,
and16arecomposedbyA. Kulik,Chapters7,12-15,18,and19byYu. Mishura,
Chapters 8-10 by A. Pilipenko, Chapter 17 by A. Kukush, and Chapter
20 by D. Gusak. Chapter11waspreparedjointlybyD. GusakandA.
Pilipenko. Atthe
sametime,everyauthorhasmadeacontributiontootherpartsofthebookbyprop-
ingseparateproblemsorcyclesofproblems,improvingpreliminaryversionsoft-
oreticalgrounds,andeditingthe naltext. The authors would like to
express their deep gratitude to M. Portenko and A.
Ivanovfortheircarefulreadingofapreliminaryversionofthebookandva-
ablecommentsthatledtosigni cantimprovementofthetext.
Theauthorsarealso gratefultoT. Yakovenko,G. Shevchenko,O.
Soloveyko, Yu. Kartashov, Yu. K- menko,A. Malenko,andN.
Ryabovafortheirassistanceintranslation,preparing
lesandpictures,andcomposingthesubjectindexandreferences.
Thetheoryofstochasticprocessesisanextendeddiscipline,andtheauthors-
derstandthattheproblembookinitscurrentformmaycausecriticalremarksfrom
readers,concerningeitherthestructureofthebookorthecontentofseparatech-
ters.
Whilepublishingtheproblembookinitscurrentform,theauthorsareopenfor
remarks,comments,andpropositions,andexpressinadvancetheirgratitudetoall
theircorrespondents. Kyiv DmytroGusak December2008 AlexanderKukush
AlexeyKulik YuliyaMishura AndreyPilipenko Contents 1 De?nition of
stochastic process. Cylinder?-algebra, ?nite-dimensional
distributions, the Kolmogorov theorem...1 Theoreticalgrounds ...1
Bibliography...3 Problems...3 Hints...7 AnswersandSolutions...9 2
Characteristics of a stochastic process. Mean and covariance
functions. Characteristic functions...11 Theoreticalgrounds ...11
Bibliography...13 Problems...13 Hints...16 AnswersandSolutions...17
3 Trajectories. Modi?cations. Filtrations...21 Theoreticalgrounds
...21 Bibliography...24 Problems...24 Hints...29
AnswersandSolutions...31 4 Continuity. Differentiability.
Integrability...33 Theoreticalgrounds ...33 Bibliography...34
Problems...34 Hints...38 AnswersandSolutions...40 ix x Contents 5
Stochastic processes with independent increments. Wiener and
Poisson processes. Poisson point measures...
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