Theory of Stochastic Processes

ershouldbeacquainted 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.