N. V. Krylov

Controlled Diffusion Processes (eBook, PDF)

Übersetzer: Aries, A. B.

• Format: PDF

Produktbeschreibung

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

Topics include optimal stopping; one dimensional controlled diffusion; the L_p-estimates of stochastic integral distributions; the existence theorem for stochastic equations; the Itô formula for functions; and the Bellman principle, equation, and normalized equation.

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Produktdetails

• Verlag: Springer Berlin Heidelberg
• Seitenzahl: 310
• Erscheinungstermin: 26. September 2008
• Englisch
• ISBN-13: 9783540709145
• Artikelnr.: 53150765

Inhaltsangabe

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 ?-OptimalStrategies.- 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.

Rezensionen

From the reviews:
"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 stochastic processes and/or nonlinear partial differential equations of the second order. ... recommended to a wider audience of all students specializing in stochastic analysis or stochastic finance starting from MSc level." (Alexander Yu Veretennikov, Zentralblatt MATH, Vol. 1171, 2009)