Lectures on Stochastic Analysis

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1、LecturesonStochasticAnalysisThomasG.KurtzDepartmentsofMathematicsandStatisticsUniversityofWisconsin-MadisonMadison,WI53706-1388RevisedSeptember7,2001MinorcorrectionsAugust23,2007Contents1Introduction.42Reviewofprobability.52.1Propertiesofexpectation...............

2、..............52.2Convergenceofrandomvariables.........................52.3Convergenceinprobability.............................62.4Norms........................................72.5Informationandindependence..........................82.6Conditionalexpectation.....

3、.........................93Continuoustimestochasticprocesses.133.1Examples......................................133.2Filtrations.....................................143.3Stoppingtimes...................................153.4Brownianmotion...........................

4、......163.5Poissonprocess..................................164Martingales.184.1OptionalsamplingtheoremandDoob’sinequalities...............184.2Localmartingales.................................194.3Quadraticvariation................................204.4Martingaleco

5、nvergencetheorem.........................215Stochasticintegrals.225.1Definitionofthestochasticintegral........................225.2Conditionsforexistence..............................235.3Semimartingales..................................285.4Changeoftimevariable...

6、...........................305.5Changeofintegrator................................315.6Localization....................................325.7Approximationofstochasticintegrals.......................335.8ConnectiontoProtter’stext............................3416Covar

7、iationandItˆo’sformula.356.1Quadraticcovariation...............................356.2Continuityofthequadraticvariation.......................366.3Ito’sformula....................................386.4Theproductruleandintegrationbyparts....................406.5Itˆo’s

8、formulaforvector-valuedsemimartingales.................417StochasticDifferentialEquations427.1Examples......................................427.2Gronwall