Controlled Diffusion Processes

by Krylov, N. V.; Aries, A. B.

• ISBN13: 9783540709138
• ISBN10: 3540709134
• Edition: Reprint
• Format: Paperback
• Copyright: 2008-12-04
• Publisher: Springer Verlag

Summary

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.

Table of Contents

Notation	p. xi
Introduction to the Theory of Controlled Diffusion Processes	p. 1
The Statement of Problems-Bellman's Principle-Bellman's Equation	p. 2
Examples of the Bellman Equations-The Normed Bellman Equation	p. 7
Application of Optimal Control Theory-Techniques for Obtaining Some Estimates	p. 16
One-Dimensional Controlled Processes	p. 22
Optimal Stopping of a One-Dimensional Controlled Process	p. 35
Notes	p. 42
Auxiliary Propositions	p. 45
Notation and Definitions	p. 45
Estimates of the Distribution of a Stochastic Integral in a Bounded Region	p. 51
Estimates of the Distribution of a Stochastic Integral in the Whole Space	p. 61
Limit Behavior of Some Functions	p. 67
Solutions of Stochastic Integral Equations and Estimates of the Moments	p. 77
Existence of a Solution of a Stochastic Equation with Measurable Coefficients	p. 86
Some Properties of a Random Process Depending on a Parameter	p. 91
The Dependence of Solutions of a Stochastic Equation on a Parameter	p. 102
The Markov Property of Solutions of Stochastic Equations	p. 110
Ito's Formula with Generalized Derivatives	p. 121
Notes	p. 128
General Properties of a Payoff Function	p. 129
Basic Results	p. 129
Some Preliminary Considerations	p. 140
The Proof of Theorems 1.5-1.7	p. 147
The Proof of Theorems 1.8-1.11 for the Optimal Stopping Problem	p. 152
Notes	p. 161
The Bellman Equation	p. 163
Estimation of First Derivatives of Payoff Functions	p. 165
Estimation from Below of Second Derivatives of a Payoff Function	p. 173
Estimation from Above of Second Derivatives of a Payoff Function	p. 181
Estimation of a Derivative of a Payoff Function with Respect to t	p. 188
Passage to the Limit in the Bellman Equation	p. 193
The Approximation of Degenerate Controlled Processes by Nondegenerate Ones	p. 200
The Bellman Equation	p. 203
Notes	p. 211
The Construction of [epsilon]-Optimal Strategies	p. 213
[epsilon]-Optimal Markov Strategies and the Bellman Equation	p. 213
[epilson]-Optimal Markov Strategies. The Bellman Equation in the Presence of Degeneracy	p. 218
The Payoff Function and Solution of the Bellman Equation: The Uniqueness of the Solution of the Bellman Equation	p. 228
Notes	p. 243
Controlled Processes with Unbounded Coefficients: The Normed Bellman Equation	p. 245
Generalizations of the Results Obtained in Section 3.1	p. 245
General Methods for Estimating Derivatives of Payoff Functions	p. 254
The Normed Bellman Equation	p. 266
The Optimal Stopping of a Controlled Process on an Infinite Interval of Time	p. 275
Control on an Infinite Interval of Time	p. 285
Notes	p. 291
Appendices
Some Properties of Stochastic Integrals	p. 293
Some Properties of Submartingales	p. 299
Bibliography	p. 303
Index	p. 307
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