Markov Chains and Monte–Carlo Simulation

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1、SITÄTVERUILNMU··SCDOINENADROUC··CODODNEMarkovChainsandMonte–CarloSimulationUlmUniversityInstituteofStochasticsLectureNotesProf.Dr.VolkerSchmidtSummer2010Ulm,July2010CONTENTS2Contents1Introduction42MarkovChains52.1SpecificationoftheModelandExamples.................................52.1

2、.1StateSpace,InitialDistributionandTransitionProbabilities.................52.1.2Examples..............................................82.1.3RecursiveRepresentation.....................................102.1.4TheMatrixofthen–StepTransitionProbabilities.......................122.2Ergo

3、dicityandStationarity........................................162.2.1BasicDefinitionsandQuasi-positiveTransitionMatrices....................162.2.2EstimatesfortheRateofConvergence;Perron–Frobenius–Theorem.............202.2.3IrreducibleandAperiodicMarkovChains..........................

4、..242.2.4StationaryInitialDistributions..................................302.2.5DirectandIterativeComputationMethods...........................352.3Reversibility;EstimatesfortheRateofConvergence..........................382.3.1DefinitionandExamples....................................

5、..382.3.2RecursiveConstructionofthe„Past”...............................422.3.3DeterminingtheRateofConvergenceunderReversibility...................432.3.4MultiplicativeReversibleVersionoftheTransitionMatrix;SpectralRepresentation.....452.3.5AlternativeEstimatefortheRateofConvergence

6、;Â2-Contrast................462.3.6Dirichlet–FormsandRayleigh–Theorem.............................502.3.7BoundsfortheEigenvalues¸2and¸`..............................533Monte–CarloSimulation583.1GenerationofPseudo-RandomNumbers.................................593.1.1SimpleApplication

7、s;Monte–CarloEstimators..........................593.1.2LinearCongruentialGenerators..................................613.1.3StatisticalTests...........................................633.2TransformationofUniformlyDistributedRandomNumbers......................683.2.1InversionMethod.

8、...................