Controlled Diffusion Processes

1 Introduction to the Theory of Controlled Diffusion Processes.- 1. The Statement of Problems-Bellman's Principle-Bellman's Equation.- 2. Examples of the Bellman Equations-The Normed Bellman Equation.- 3. Application of Optimal Control Theory-Techniques for Obtaining Some Estimates.- 4. One-Dimensional Controlled Processes.- 5. Optimal Stopping of a One-Dimensional Controlled Process.- Notes.- 2 Auxiliary Propositions.- 1. Notation and Definitions.- 2. Estimates of the Distribution of a Stochastic Integral in a Bounded Region.- 3. Estimates of the Distribution of a Stochastic Integral in the Whole Space.- 4. Limit Behavior of Some Functions.- 5. Solutions of Stochastic Integral Equations and Estimates of the Moments.- 6. Existence of a Solution of a Stochastic Equation with Measurable Coefficients.- 7. Some Properties of a Random Process Depending on a Parameter.- 8. The Dependence of Solutions of a Stochastic Equation on a Parameter.- 9. The Markov Property of Solutions of Stochastic Equations.- 10. Ito's Formula with Generalized Derivatives.- Notes.- 3 General Properties of a Payoff Function.- 1. Basic Results.- 2. Some Preliminary Considerations.- 3. The Proof of Theorems 1.5-1.7.- 4. The Proof of Theorems 1.8-1.11 for the Optimal Stopping Problem.- Notes.- 4 The Bellman Equation.- 1. Estimation of First Derivatives of Payoff Functions.- 2. Estimation from Below of Second Derivatives of a Payoff Function.- 3. Estimation from Above of Second Derivatives of a Payoff Function.- 4. Estimation of a Derivative of a Payoff Function with Respect to t.- 5. Passage to the Limit in the Bellman Equation.- 6. The Approximation of Degenerate Controlled Processes by Nondegenerate Ones.- 7. The Bellman Equation.- Notes.- 5 The Construction of ?-Optimal Strategies.- 1. ?-Optimal Markov Strategies and the Bellman Equation.- 2. ?-Optimal Markov Strategies. The Bellman Equation in the Presence of Degeneracy.- 3. The Payoff Function and Solution of the Bellman Equation: The Uniqueness of the Solution of the Bellman Equation.- Notes.- 6 Controlled Processes with Unbounded Coefficients: The Normed Bellman Equation.- 1. Generalization of the Results Obtained in Section 3.1.- 2. General Methods for Estimating Derivatives of Payoff Functions.- 3. The Normed Bellman Equation.- 4. The Optimal Stopping of a Controlled Process on an Infinite Interval of Time.- 5. Control on an Infinite Interval of Time.- Notes.- Appendices.- 1. Some Properties of Stochastic Integrals.- 2. Some Properties of Submartingales.