(Carlsson)Stochastic Differential Equations

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1、StochasticDierentialEquations:ModelsandNumerics1JesperCarlssonKyoung-SookMoonAndersSzepessyRaulTemponeGeorgiosZourarisSeptember7,20091Thisisadraft.Commentsandimprovementsarewelcome.Contents1IntroductiontoMathematicalModelsandtheirAnalysis41.1NoisyEvolutionofStockValu

2、es.......................51.2MolecularDynamics.............................61.3OptimalControlofInvestments........................71.4CalibrationoftheVolatility.........................81.5TheCoarse-grainingandDiscretizationAnalysis.............82StochasticIntegrals112

3、.1ProbabilityBackground............................112.2BrownianMotion...............................122.3ApproximationandDenitionofStochasticIntegrals...........133StochasticDierentialEquations233.1ApproximationandDenitionofSDE...................233.2It^o'sFormula.

4、................................303.3StratonovichIntegrals............................353.4SystemsofSDE................................364TheFeynman-KacFormulaandtheBlack-ScholesEquation384.1TheFeynman-KacFormula.........................384.2Black-ScholesEquation....

5、.......................395TheMonte-CarloMethod445.1StatisticalError.................................445.2TimeDiscretizationError..........................486FiniteDierenceMethods546.1AmericanOptions...............................546.2LaxEquivalenceTheorem.............

6、.............567TheFiniteElementMethodandLax-Milgram'sTheorem627.1TheFiniteElementMethod.........................637.2ErrorEstimatesandAdaptivity........................677.2.1AnAPrioriErrorEstimate......................6717.2.2AnAPosterioriErrorEstimate...............

7、....697.2.3AnAdaptiveAlgorithm.......................707.3Lax-Milgram'sTheorem............................718MarkovChains,DualityandDynamicProgramming768.1Introduction..................................768.2MarkovChains.................................778.3ExpectedValue

8、s...............................798.4DualityandQualitativeProperties......................818.5DynamicProgramming....