Controlled Diffusion Processes

Cover
Title page
Preface
Notation
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 ε-Optimal Strategies
1. ε-Optimal Markov Strategies and the Hellman Equation
2. ε-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. Generalizations 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
Bibliography
Index