Probability Methods for Approximations in Stochastic Control and for Elliptic Equations
Published on 14. April 1977
242 pages
978-0-08-095638-1 (ISBN)
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Probability Methods for Approximations in Stochastic Control and for Elliptic Equations
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Harold J. Kushner
Probability Methods for Approximation in Stochastic Control and for Elliptic Equations
Book
01/1977
Academic Press
€95.32
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Content
- Front Cover
- Probability Methods for Approximations in Stochastic Control and for Elliptic Equations
- Copyright Page
- Contents
- Preface
- Acknowledgments
- Chapter 1. Probability Background
- 1.1 The Wiener Processes
- 1.2 Martingales
- 1.3 Markov Processes
- 1.4 Stochastic Integrals
- 1.5 Stochastic Differential Equations
- Chapter 2. Weak Convergence of Probability Measures
- 2.1 Probability Measures on the Real Line. Real-Valued Random Variables
- 2.2 Probability Measures on Metric Spaces
- 2.3 The Spaces Cm[a, ß] of Continuous Functions
- 2.4 The Space Dm[a, ß]
- 2.5 Weak Convergence on Other Spaces
- Chapter 3. Markov Chains and Control Problems with Markov Chain Models
- 3.1 Equations Satisfied by Functionals of Markov Chains
- 3.2 Optimal Stopping Problems
- 3.3 Controlled Markov Chains: Families of Controlled Strategies
- 3.4 Optimal Control until a Boundary Is Reached
- 3.5 Optimal Discounted Cost
- 3.6 Optimal Stopping and Control
- 3.7 Impulsive Control Systems
- 3.8 Control over a Fixed Time Interval
- 3.9 Linear Programming Formulation of the Markov Chain Control Problems
- Chapter 4. Elliptic and Parabolic Equations and Functionals of Diffusions
- 4.1 Assumptions and Uniqueness Results: No Control
- 4.2 Functionals of Uncontrolled Diffusions
- 4.3 Partial Differential Equations Associated with Functionals of Diffusions. a( . ) Uniformly Positive Definite
- 4.4 a( . ) Degenerate
- 4.5 Partial Differential Equations Formally Satisfied by Path Functionals
- 4.6 The Characteristic Operator of the Diffusion
- 4.7 Optimal Control Problems and Nonlinear Partial Differential Equations
- Chapter 5. A Simple Application of the Invariance Theorems
- 5.1 A Functional Limit Theorem
- 5.2 An Application to Numerical Analysis
- Chapter 6. Elliptic Equations and Uncontrolled Diffusions
- 6.1 Problem Formulation
- 6.2 The Finite Difference Method and an Approximating Markov Chain
- 6.3 Convergence of the Approximations to a Diffusion Process
- 6.4 Convergence of the Cost Functionals Rh ( . )
- 6.5 The Discounted Cost Problem
- 6.6 An Alternative Representation for ßhn and W( . )
- 6.7 Monte Carlo
- 6.8 Approximation of Invariant Measures
- 6.9 Remarks and Extensions
- 6.10 Numerical Data
- Chapter 7. Approximations for Parabolic Equations and Nonlinear Filtering Problems
- 7.1 Problem Statement
- 7.2 The Finite Difference Approximation and Weak Convergence
- 7.3 Implicit Approximations
- 7.4 Discounted Cost: Explicit Method
- 7.5 Nonlinear Filtering
- 7.6 Numerical Data: Estimation of an Invariant Measure
- Chapter 8. Optimal Stopping and Impulsive Control Problems
- 8.1 Discretization of the Optimal Stopping Problem
- 8.2 Optimality of the Limiting Stopping Time p
- 8.3 Constrained Optimal Stopping Problems
- 8.4 Discretization of the Impulsive Control Problem
- 8.5 Optimality of the Limits ( pi, vi} and R(x, {pi, vi})
- 8.6 Numerical Data for the Optimal Stopping Problem
- Chapter 9. Approximations to Optimal Controls and Nonlinear Partial Differential Equations
- 9.1 Optimal Stopping and Continuously Acting Control: Formulation and Approximations
- 9.2 The Limit Is a Controlled, Stopped Diffusion
- 9.3 Optimality of the Limit
- 9.4 Discounted Cost
- 9.5 Control until a Boundary Is Reached: Discounted Case
- 9.6 Control until a Boundary Is Reached: No Discounting
- 9.7 The Impulsive Control Problem
- 9.8 Numerical Results
- Chapter 10. Approximations to Stochastic Delay Equations and to Diffusions and Partial Differential Equations with Reflecting Boundaries
- 10.1 Approximations to Stochastic Differential Delay Equations. Introduction
- 10.2 Approximations to Elliptic and Parabolic Equations with Neumann Boundary Conditions. Formulation
- 10.3 The Finite Difference Approximations
- 10.4 Continuous Time Interpolations and Convergence
- 10.5 Extensions of the Reflection Problem
- Chapter 11. The Separation Theorem of Optimal Stochastic Control Theory
- 11.1 Assumptions and the System Model
- 11.2 The Separation and Optimality Theorems
- References
- Index of Selected Symbols
- Index
