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1.3 The Stochastic Integral1.3.1 Definition of the Stochastic Integral; 1.3.2 Basic Properties and Estimates; 1.4 Stochastic Differential Equations; 1.4.1 Mild and Strong Solutions; 1.4.2 Existence and Uniqueness of Solutions; 1.4.3 Properties of Solutions; 1.4.4 Uniqueness in Law; 1.5 Further Existence and Uniqueness Results in Special Cases; 1.5.1 SDEs Coming from Boundary Control Problems; 1.5.2 Semilinear SDEs with Additive Noise; 1.5.3 Semilinear SDEs with Multiplicative Noise; 1.6 Transition Semigroups; 1.7 Itô's and Dynkin's Formulae; 1.8 Bibliographical Notes
2 Optimal Control Problems and Examples2.1 Stochastic Optimal Control Problems: General Formulation; 2.1.1 Strong Formulation; 2.1.2 Weak Formulation; 2.2 The Dynamic Programming Principle: Setup and Assumptions; 2.2.1 The Setup; 2.2.2 The General Assumptions; 2.2.3 The Assumptions in the Case of Control Problems for Mild Solutions; 2.3 The Dynamic Programming Principle: Statement and Proof; 2.3.1 Pullback to the Canonical Reference Probability Space; 2.3.2 Independence of Reference Probability Spaces; 2.3.3 The Proof of the Abstract Principle of Optimality; 2.4 Infinite Horizon Problems
2.5 The HJB Equation and Optimal Synthesis in the Smooth Case2.5.1 The Finite Horizon Problem: Parabolic HJB Equation; 2.5.2 The Infinite Horizon Problem: Elliptic HJB Equation; 2.6 Some Motivating Examples; 2.6.1 Stochastic Controlled Heat Equation: Distributed Control; 2.6.2 Stochastic Controlled Heat Equation: Boundary Control; 2.6.3 Stochastic Controlled Heat Equation: Boundary Control and Boundary Noise; 2.6.4 Optimal Control of the Stochastic Burgers Equation; 2.6.5 Optimal Control of the Stochastic Navier
Stokes Equations
2.6.6 Optimal Control of the Duncan
Mortensen
Zakai Equation2.6.7 Super-Hedging of Forward Rates; 2.6.8 Optimal Control of Stochastic Delay Equations; 2.7 Bibliographical Notes; 3 Viscosity Solutions; 3.1 Preliminary Results; 3.1.1 B-Continuity and Weak and Strong B-Conditions; 3.1.2 Estimates for Solutions of Stochastic Differential Equations; 3.1.3 Perturbed Optimization; 3.2 A Maximum Principle; 3.3 Viscosity Solutions; 3.3.1 Bounded Equations; 3.4 Consistency of Viscosity Solutions; 3.5 Comparison Theorems; 3.5.1 Degenerate Parabolic Equations; 3.5.2 Degenerate Elliptic Equations
2 Optimal Control Problems and Examples2.1 Stochastic Optimal Control Problems: General Formulation; 2.1.1 Strong Formulation; 2.1.2 Weak Formulation; 2.2 The Dynamic Programming Principle: Setup and Assumptions; 2.2.1 The Setup; 2.2.2 The General Assumptions; 2.2.3 The Assumptions in the Case of Control Problems for Mild Solutions; 2.3 The Dynamic Programming Principle: Statement and Proof; 2.3.1 Pullback to the Canonical Reference Probability Space; 2.3.2 Independence of Reference Probability Spaces; 2.3.3 The Proof of the Abstract Principle of Optimality; 2.4 Infinite Horizon Problems
2.5 The HJB Equation and Optimal Synthesis in the Smooth Case2.5.1 The Finite Horizon Problem: Parabolic HJB Equation; 2.5.2 The Infinite Horizon Problem: Elliptic HJB Equation; 2.6 Some Motivating Examples; 2.6.1 Stochastic Controlled Heat Equation: Distributed Control; 2.6.2 Stochastic Controlled Heat Equation: Boundary Control; 2.6.3 Stochastic Controlled Heat Equation: Boundary Control and Boundary Noise; 2.6.4 Optimal Control of the Stochastic Burgers Equation; 2.6.5 Optimal Control of the Stochastic Navier
Stokes Equations
2.6.6 Optimal Control of the Duncan
Mortensen
Zakai Equation2.6.7 Super-Hedging of Forward Rates; 2.6.8 Optimal Control of Stochastic Delay Equations; 2.7 Bibliographical Notes; 3 Viscosity Solutions; 3.1 Preliminary Results; 3.1.1 B-Continuity and Weak and Strong B-Conditions; 3.1.2 Estimates for Solutions of Stochastic Differential Equations; 3.1.3 Perturbed Optimization; 3.2 A Maximum Principle; 3.3 Viscosity Solutions; 3.3.1 Bounded Equations; 3.4 Consistency of Viscosity Solutions; 3.5 Comparison Theorems; 3.5.1 Degenerate Parabolic Equations; 3.5.2 Degenerate Elliptic Equations