目录：
1ExamplesandDynamicProgramming1
1.1OptimalControl..........................1
1.2Examples.............................2
1.2.1Deterministicminimaltimeproblem..........3
1.2.2MertonΓfisoptimalinvestment-consumptionproblem..
1.2.3Finitetimeutilitymaximization.............5
1.2.4Mertonproblemwithtransactioncosts.........5
1.2.5Super-replicationwithportfolioconstraints.......7
1.2.6BuyerΓfispriceandtheno-arbitrageinterval.......
1.2.7Super-replicationwithgammaconstraints.......8
1.3DynamicProgrammingPrinciple................9
1.3.1FormalProofofDPP...................10
1.3.2ExamplesfortheDPP..................11
1.4DynamicProgrammingEquation................13
1.4.1FormalDerivationoftheDPE..............14
1.4.2In?nitehorizon......................16
1.5ExamplesfortheDPE......................16
1.5.1MertonProblem.....................16
1.5.2MinimalTimeProblem..................19
1.5.3Transactioncosts.....................20
1.5.4Super-replicationwithportfolioconstraints.......22
1.5.5TargetReachabilityProblem...............22
2Super-Replicationunderportfolioconstraints25
2.1SolutionbyDuality........................25
2.1.1Black-ScholesCase....................26
2.1.2GeneralCase.......................27
2.2DirectSolution..........................29
2.2.1ViscositySolutions....................29
2.2.2Supersolution.......................30
2.2.3Subsolution........................32
2.2.4TerminalConditionorΓflFace-LiftingΓ?..........
Super-ReplicationwithGammaConstraints38
3.1PureUpperBoundCase.....................39
3.1.1Supersolution.......................39
3.1.2Subsolution........................41
3.1.3TerminalCondition....................41
3.2DoubleStochasticIntegrals...................44
3.3GeneralGammaConstraint...................51
3.4Examples.............................53
3.4.1EuropeanCallOption..................53
3.4.2EuropeanPutOption:..................54
3.4.3EuropeanDigitalOption.................55
3.4.4UpandOutEuropeanCallOption...........56
3.5GuessforTheDualFormulation.................58