Stat219_Notes - Stochastic Processes Amir Dembo(Revised by...

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Stochastic Processes Amir Dembo (Revised by Kevin Ross) September 25, 2008 E-mail address : [email protected], [email protected] Department of Statistics, Stanford University, Stanford, CA 94305.
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Contents Preface 5 Chapter1. Probability,measureandintegration 7 1.1. Probabilityspacesand σ -fields 7 1.2. Randomvariablesandtheirexpectation 10 1.3. Convergenceofrandomvariables 19 1.4. Independence,weakconvergenceanduniformintegrability 25 Chapter2. ConditionalexpectationandHilbertspaces 35 2.1. Conditionalexpectation: existenceanduniqueness 35 2.2. Hilbertspaces 39 2.3. Propertiesoftheconditionalexpectation 43 2.4. Regularconditionalprobability 46 Chapter3. StochasticProcesses: generaltheory 49 3.1. Definition,distributionandversions 49 3.2. Characteristicfunctions,Gaussianvariablesandprocesses 55 3.3. Samplepathcontinuity 62 Chapter4. Martingalesandstoppingtimes 67 4.1. Discretetimemartingalesandfiltrations 67 4.2. Continuoustimemartingalesandrightcontinuousfiltrations 73 4.3. Stoppingtimesandtheoptionalstoppingtheorem 76 4.4. Martingalerepresentationsandinequalities 82 4.5. Martingaleconvergencetheorems 88 4.6. Branchingprocesses: extinctionprobabilities 90 Chapter5. TheBrownianmotion 95 5.1. Brownianmotion: definitionandconstruction 95 5.2. ThereflectionprincipleandBrownianhittingtimes 101 5.3. SmoothnessandvariationoftheBrowniansamplepath 104 Chapter6. Markov,PoissonandJumpprocesses 111 6.1. Markovchainsandprocesses 111 6.2. Poissonprocess,Exponentialinter-arrivalsandorderstatistics 119 6.3. Markovjumpprocesses,compoundPoissonprocesses 125 HomeworkProblems 127 Bibliography 129 Index 131 3
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Preface These arethe lecturenotesfor aonequartergraduatecourseinStochasticPro- cessesthatItaughtatStanfordUniversityin2002and2003. Thiscourseisintended forincomingmasterstudentsinStanford’sFinancialMathematicsprogram,forad- vanced undergraduates majoring in mathematics and for graduate students from Engineering,Economics,StatisticsortheBusinessschool. Onepurposeofthistext istopreparestudentstoarigorousstudyofStochasticDifferentialEquations. More broadly,itsgoalistohelpthereaderunderstandthebasicconceptsofmeasurethe- orythatarerelevanttothemathematicaltheoryofprobabilityandhowtheyapply totherigorousconstructionofthemostfundamentalclassesofstochasticprocesses. Towardsthisgoal,weintroduceinChapter1therelevantelementsfrommeasure and integration theory, namely, the probability space and the σ -fields of events in it, random variables viewed as measurable functions, their expectation as the correspondingLebesgueintegral,independence,distributionandvariousnotionsof convergence. This is supplemented in Chapter 2 by the study of the conditional expectation, viewed as a random variable defined via the theory of orthogonal projectionsinHilbertspaces. AfterthisexplorationofthefoundationsofProbabilityTheory,weturninChapter 3 to the general theory of Stochastic Processes, with an eye towards processes indexed by continuous time parameter such as the Brownian motion of Chapter 5 and the Markov jump processes of Chapter 6. Having this in mind, Chapter
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