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1.2.4 Backward Stochastic Differential Equations1.3 Asset Pricing Problems; 1.3.1 Formulation; 1.3.2 Backward SDE for the Selling Price; 1.3.3 Parabolic Equation Associated with (1.84); 2 Optimal Control for Diffusion Processes ; 2.1 Introduction; 2.1.1 Formulations; 2.1.2 Value Functions: Basic Properties; 2.2 Dynamic Programming Principle (DPP); 2.2.1 Discrete-Time Dynamic Programming Principle; 2.2.2 Approximation Theorem; 2.2.3 Dynamic Programming Principle; 2.2.4 Brownian Adapted Controls; 2.2.5 Characterization of the Semigroup (V[theta]t, [less than or equal to] t); 2.3 Verification Theorems and Optimal Controls.
Preface; Acknowledgement; Contents; Notations; Abbreviations; 1 Stochastic Differential Equations ; 1.1 Review of Stochastic Processes; 1.1.1 Random Variables; Monotone Convergence Theorem; Fatou's Lemma; Dominated Convergence Theorem; Convergence Theorem; 1.1.2 Stochastic Processes; Burkholder-Davis-Gundy Inequality; 1.1.3 Itô Integrals; 1.1.4 Itô's Formula; Itô's Formula; Itô-Krylov Formula (Kr09, 2.10, Theorem 1); 1.2 Stochastic Differential Equations; 1.2.1 Lipschitz Continuous SDEs with Random Coefficients; 1.2.2 Girsanov Transformations; 1.2.3 SDEs with Deterministic Borel Coefficients.
2.3.1 Verification Theorems2.3.2 Examples of Optimal Control; 2.4 Optimal Investment Models; 2.4.1 Formulations; 2.4.2 Investment Problems for Power Utility Function; 2.4.3 Optimal Investment Strategy; 3 Viscosity Solutions for HJB Equations; 3.1 Formulations; 3.1.1 Definition of Viscosity Solution Based on Parabolic Differentials; 3.1.2 Equivalent Definitions; 3.1.3 Viscosity Solutions via Semigroups; 3.2 Uniqueness of Viscosity Solutions; 3.2.1 Crandall-Ishii Lemma; Ishii's Lemma; Crandall-Ishii Lemma CI90; 3.2.2 Structural Condition; Structural Condition; 3.2.3 Comparison Principle.
3.3 HJB Equations for Control-Stopping Problems3.3.1 Formulations; 3.3.2 DPP; 3.3.3 Semigroups Associated with DPP; 3.3.4 American Option Price; 4 Stochastic Differential Games ; 4.1 Formulations; 4.1.1 Admissible Controls and Strategies; 4.1.2 Formulation of Stochastic Differential Games; 4.2 DPP; 4.2.1 D-Lower and D-Upper Value Functions; 4.2.2 DPP for Lower- and Upper Value Functions; 4.3 Isaacs Equations; 4.3.1 Semigroups Related to the DPP; 4.3.2 Viscosity Solutions of the Isaacs Equations; 4.4 Risk Sensitive Stochastic Controls and Differential Games; 4.4.1 Logarithmic Transformation.
4.4.2 Small Noise Limit4.4.3 Note on Control with Infinite Time Horizon; 5 Stochastic Parabolic Equations; 5.1 Preliminaries; 5.1.1 H-Random Variables; Expectation of X; Conditional Expectation; 5.1.2 Continuous Martingales; 5.1.3 Correlation Operators; 5.2 Stochastic Integrals; 5.2.1 Definitions and Basic Properties; 5.2.2 Martingale Inequalities; 5.3 Stochastic Parabolic Equations with Colored Wiener Noises; 5.3.1 Preliminaries; 5.3.2 Linear Stochastic Parabolic Equations; 5.3.3 Regularities of Solutions; 5.3.4 Semilinear Stochastic Parabolic Equations with Lipschitz Nonlinearity.
Preface; Acknowledgement; Contents; Notations; Abbreviations; 1 Stochastic Differential Equations ; 1.1 Review of Stochastic Processes; 1.1.1 Random Variables; Monotone Convergence Theorem; Fatou's Lemma; Dominated Convergence Theorem; Convergence Theorem; 1.1.2 Stochastic Processes; Burkholder-Davis-Gundy Inequality; 1.1.3 Itô Integrals; 1.1.4 Itô's Formula; Itô's Formula; Itô-Krylov Formula (Kr09, 2.10, Theorem 1); 1.2 Stochastic Differential Equations; 1.2.1 Lipschitz Continuous SDEs with Random Coefficients; 1.2.2 Girsanov Transformations; 1.2.3 SDEs with Deterministic Borel Coefficients.
2.3.1 Verification Theorems2.3.2 Examples of Optimal Control; 2.4 Optimal Investment Models; 2.4.1 Formulations; 2.4.2 Investment Problems for Power Utility Function; 2.4.3 Optimal Investment Strategy; 3 Viscosity Solutions for HJB Equations; 3.1 Formulations; 3.1.1 Definition of Viscosity Solution Based on Parabolic Differentials; 3.1.2 Equivalent Definitions; 3.1.3 Viscosity Solutions via Semigroups; 3.2 Uniqueness of Viscosity Solutions; 3.2.1 Crandall-Ishii Lemma; Ishii's Lemma; Crandall-Ishii Lemma CI90; 3.2.2 Structural Condition; Structural Condition; 3.2.3 Comparison Principle.
3.3 HJB Equations for Control-Stopping Problems3.3.1 Formulations; 3.3.2 DPP; 3.3.3 Semigroups Associated with DPP; 3.3.4 American Option Price; 4 Stochastic Differential Games ; 4.1 Formulations; 4.1.1 Admissible Controls and Strategies; 4.1.2 Formulation of Stochastic Differential Games; 4.2 DPP; 4.2.1 D-Lower and D-Upper Value Functions; 4.2.2 DPP for Lower- and Upper Value Functions; 4.3 Isaacs Equations; 4.3.1 Semigroups Related to the DPP; 4.3.2 Viscosity Solutions of the Isaacs Equations; 4.4 Risk Sensitive Stochastic Controls and Differential Games; 4.4.1 Logarithmic Transformation.
4.4.2 Small Noise Limit4.4.3 Note on Control with Infinite Time Horizon; 5 Stochastic Parabolic Equations; 5.1 Preliminaries; 5.1.1 H-Random Variables; Expectation of X; Conditional Expectation; 5.1.2 Continuous Martingales; 5.1.3 Correlation Operators; 5.2 Stochastic Integrals; 5.2.1 Definitions and Basic Properties; 5.2.2 Martingale Inequalities; 5.3 Stochastic Parabolic Equations with Colored Wiener Noises; 5.3.1 Preliminaries; 5.3.2 Linear Stochastic Parabolic Equations; 5.3.3 Regularities of Solutions; 5.3.4 Semilinear Stochastic Parabolic Equations with Lipschitz Nonlinearity.