Controlled Diffusion Processes / Edition 1

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Overview

This book deals with the optimal control of solutions of fully observable Ito-type stochastic differential equations. The validity of the Bellman differential equation for payoff functions is proved and rules for optimal control strategies are developed.
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Editorial Reviews

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“The book treats a large class of fully nonlinear parabolic PDEs via probabilistic methods. … The monograph may be strongly recommended as an excellent reading to PhD students, postdocs et al working in the area of controlled shastic processes and/or nonlinear partial differential equations of the second order. … recommended to a wider audience of all students specializing in shastic analysis or shastic finance starting from MSc level.” (Alexander Yu Veretennikov, Zentralblatt MATH, Vol. 1171, 2009)

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Product Details

Table of Contents

Notation xi

1 Introduction to the Theory of Controlled Diffusion Processes 1

1 The Statement of Problems-Bellman's Principle-Bellman's Equation 2

2 Examples of the Bellman Equations-The Normed Bellman Equation 7

3 Application of Optimal Control Theory-Techniques for Obtaining Some Estimates 16

4 One-Dimensional Controlled Processes 22

5 Optimal Stopping of a One-Dimensional Controlled Process 35

Notes 42

2 Auxiliary Propositions 45

1 Notation and Definitions 45

2 Estimates of the Distribution of a Stochastic Integral in a Bounded Region 51

3 Estimates of the Distribution of a Stochastic Integral in the Whole Space 61

4 Limit Behavior of Some Functions 67

5 Solutions of Stochastic Integral Equations and Estimates of the Moments 77

6 Existence of a Solution of a Stochastic Equation with Measurable Coefficients 86

7 Some Properties of a Random Process Depending on a Parameter 91

8 The Dependence of Solutions of a Stochastic Equation on a Parameter 102

9 The Markov Property of Solutions of Stochastic Equations 110

10 Ito's Formula with Generalized Derivatives 121

Notes 128

3 General Properties of a Payoff Function 129

1 Basic Results 129

2 Some Preliminary Considerations 140

3 The Proof of Theorems 1.5-1.7 147

4 The Proof of Theorems 1.8-1.11 for the Optimal Stopping Problem 152

Notes 161

4 The Bellman Equation 163

1 Estimation of First Derivatives of Payoff Functions 165

2 Estimation from Below of Second Derivatives of a Payoff Function 173

3 Estimation from Above of Second Derivatives of a Payoff Function 181

4 Estimation of a Derivative of a Payoff Function with Respect to t 188

5 Passage to the Limit in the Bellman Equation193

6 The Approximation of Degenerate Controlled Processes by Nondegenerate Ones 200

7 The Bellman Equation 203

Notes 211

5 The Construction of [epsilon]-Optimal Strategies 213

1 [epsilon]-Optimal Markov Strategies and the Bellman Equation 213

2 [epilson]-Optimal Markov Strategies. The Bellman Equation in the Presence of Degeneracy 218

3 The Payoff Function and Solution of the Bellman Equation: The Uniqueness of the Solution of the Bellman Equation 228

Notes 243

6 Controlled Processes with Unbounded Coefficients: The Normed Bellman Equation 245

1 Generalizations of the Results Obtained in Section 3.1 245

2 General Methods for Estimating Derivatives of Payoff Functions 254

3 The Normed Bellman Equation 266

4 The Optimal Stopping of a Controlled Process on an Infinite Interval of Time 275

5 Control on an Infinite Interval of Time 285

Notes 291

Appendices

1 Some Properties of Stochastic Integrals 293

2 Some Properties of Submartingales 299

Bibliography 303

Index 307

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