Preface
Notation
AssumptionIndex
ProblemIndex
Chapter1.BasicStochasticCalculus
1.Probability
1.1.Probabilityspaces
1.2.Randomvariables
1.3.Conditionalexpectation
1.4.Convcrgenceofprobabilities
2.StochasticProcesses
2.1.Generalconsiderations
2.2.Brownianmotions
3.StoppingTimes
4.Martingales
5.ItS'sIntegral
5.1.NondifferentiabilityofBrownianmotion
5.2.DefinitionofItesintegralandbasicproperties
5.3.ItS'sformula
5.4.Martingalerepresentationtheorems
6.StochasticDifferentialEquations
6.1.Strongsolutions
6.2.Weaksolutions
6.3.LinearSDEs
6.4.OthertypesofSDEs
Chapter2.StochasticOptimalControlProblems
1.Introduction
2.DeterministicCasesRevisited
3.ExamplesofStochasticControlProblems
3.1.Productionplanning
3.2.Investmentvs.consumption
3.3.Reinsuranceanddividendmanagement
3.4.Technologydiffusion
3.5.Queueingsystemsinheavytraffic
4.FormulationsofStochasticOptimalControlProblems
4.1.Strongformulation
4.2.Weakformulation
5.ExistenceofOptimalControls
5.1.Adeterministicresult
5.2.Existenceunderstrongformulation
5.3.Existenceunderweakformulation
6.ReachableSetsofStochasticControlSystems
6.1.Nonconvexityofthereachablesets
6.2.Nonclnsenessofthereachablesets
7.OtherStochasticControlModels
7.1.Randomduration
7.2.Optimalstopping
7.3.Singularandimpulsecontrols
7.4.Risk-sensitivecontrols
7.5.Ergodiccontrols
7.6.Partiallyobservablesystems
8.HistoricalRemarks
Chapter3.MaximumPrincipleandStochastic
HamiitonianSystems
1.Introduction
2.TheDeterministicCaseRcvisited
3.StatementoftheStochasticMaximumPrinciple
3.1.Adjointequations
3.2.Themaximumprincipleandstochastic
Hamiltoniansystems
3.3.Aworked-outexample
4.AProofoftheMaximumPrinciple
4.1.Amomentestimate
4.2.Taylorexpansions
4.3.Dualityanalysisandcomplctionofthcproof
5.SufficientConditionsofOptimality
6.ProblemswithStatcConstraints
6.1.Formulationoftheproblemandthemaximumprinciple
6.2.Somepreliminarylemmas
6.3.AproofofTheorem6.1
7.HistoricalRemarks
Chapter4.DynamicProgrammingandHJBEquations
1.Introduction
2.TheDeterministicCascRevisited
3.TheStochasticPrincipleofOptimalityandtheHJBEquation
3.1.Astochasticframeworkfordynamicprogramming
3.2.Principlcofoptimality
3.3.TheHJBcquation
4.OtherPropertiesoftheValueFunction
4.1.Continuousdependenceonparameters
4.2.Semiconcavity
5.Viseo~itySolutions
5.1.Definitions
5.2.Someproperties
6.UniquenessofViscositySolutions
6.1.Auniquenesstheorem
6.2.ProofsofLemmas6.6and6.7
7.HistoricalRcmarks
Chapter5.TheRelationshipBetweentheMaximum
PrincipleandDynamicProgramming
1.Introduction
2.ClassicalHamilton-JacobiTheory
3.RelationshipforDeterministicSystems
3.1.Adjointvariableandvaluefunction:Smoothcase
3.2.Economicinterpretation
3.3.MethodsofcharacteristicsandtheFcynmanKacformula
3.4.Adjointvariableandvaluefunction:Nonsmoothcase
3.5.Vcrificationtheorems
4.RelationshipforStochasticSystems
4.1.Smoothcase
4.2.Nonsmoothcase:Differentialsinthespatialvariable
4.3.Nonsmoothcase:Differentialsinthetimevariable
5.StochasticVcrificationTheorems
5.1.Smoothcase
5.2.Nonsmoothcase
6.OptimalFccdbackControls
7.HistoricalRemarks
Chapter6.LinearQuadraticOptimalControlProblems
1.Introduction
2.TheDeterministicLQProblemsRevisited
2.1.Formulation
2.2.Aminimizationproblemofaquadraticfunctional
2.3.AlinearHamiltoniansystem
2.4.TheRiccatiequationandfeedbackoptimalcontrol
3.FormuLationofStochasticLQProblems
3.1.Statementoftheproblems
3.2.Examples
4.FinitenessandSolvability
5.ANecessaryConditionandaHamiltonianSystem
6.StochasticRiceatiEquations
7.GLobalSolvabilityofStochasticRiccatiEQuations
7.1.Existence:Thcstandardcase
7.2.Existence:ThecaseC=0,S=0,andQ,G>_0
7.3.Existence:Theone-dimensionalcase
8.AMean-variancePortfolioSelectionProblem
9.HistoricalRemarks
Chapter7.BackwardStochasticDifferentialEquations
1.Introduction
2.LinearBackwardStochasticDifferentialEQuations
3.NonlinearBackwardStochasticDifferentialEquations
3.1.BSDEsinfinitedeterministicdurations:Methodof
contractionmapping
3.2.BSDEsinrandomdurations:Methodofcontinuation
4.Feynman-Kac-TypeFormulae
4.1.RepresentationviaSDEs
4.2.RepresentationviaBSDEs
5.Forward-BackwardStochasticDifferentialEquations
5.1.Generalformulationandnonsolvability
5.2.Thefour-stepscheme,aheuristicderivation
5.3.SeveralsolvableclassesofFBSDEs
6.OptionPricingProblems
6.1.EuropeancalloptionsandtheBlack-Scholesformula
6.2.Otheroptions
7.HistoricalRemarks
References
Index