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Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems

The book deals with several closely related topics concerning approxima­ tions and perturbations of random processes and their applications to some important and fascinating classes of problems in the analysis and design of stochastic control systems and nonlinear filters. The basic mathematical met...

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Bibliographic Details
Main Author: Kushner, Harold
Format: eBook
Language:English
Published: Boston, MA Birkhäuser 1990, 1990
Edition:1st ed. 1990
Series:Systems & Control: Foundations & Applications
Subjects:
Control Theory
Systems Theory, Control
Probability Theory
System Theory
Probabilities
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 4 Controlled Singularly Perturbed Systems
  • 0. Outline of the Chapter
  • 1. Problem Formulation: Finite Time Interval
  • 2. Approximation of the Optimal Controls and Value Functions
  • 3. Discounted Cost and Optimal Stopping Problems
  • 4. Average Cost Per Unit Time
  • 5. Jump-Diffusion Processes
  • 6. Other Approaches
  • 5 Functional Occupation Measures and Average Cost Per Unit Time Problems
  • 0. Outline of the Chapter
  • 1. Measure Valued Random Variables
  • 2. Limits of Functional Occupation Measures for Diffusions
  • 3. The Control Problem
  • 4. Singularly Perturbed Control Problems
  • 5. Control of the Fast System
  • 6. Reflected Diffusions
  • 7. Discounted Cost Problem
  • 6 The Nonlinear Filtering Problem
  • 0. Outline of the Chapter
  • 1. A Representation of the Nonlinear Filter
  • 2. The Filtering Problem for the SingularlyPerturbed System
  • 3. The Almost Optimality of the Averaged Filter
  • 4. A Counterexample to the Averaged Filter
  • 3. Singularly Perturbed Itô Processes: Tightness
  • 4. The Linear Case
  • 5. Wide Bandwidth Noise
  • 6. Singularly Perturbed Wide Bandwidth Noise Driven Systems
  • 10 Parametric Singularities
  • 0. Outline of the Chapter
  • 1. Singularly Perturbed Itô Processes: Weak Convergence
  • 2. Stability
  • References
  • List of Symbols
  • 1 Weak Convergence
  • 0. Outline of the Chapter
  • 1. Basic Properties and Definitions
  • 2. Examples
  • 3. The Skorohod Representation
  • 4. The Function Space Ck [0, T]
  • 5. The Function Space Dk [0, T]
  • 6. Measure Valued Random Variables and Processes
  • 2 Stochastic Processes: Background
  • 0. Outline of the Chapter
  • 1. Martingales
  • 2. Stochastic Integrals and Itô’s Lemma
  • 3. Stochastic Differential Equations: Bounds
  • 4. Controlled Stochastic Differential Equations: Existence of Solutions
  • 5. Representing a Martingale as a Stochastic Integral
  • 6. The Martingale Problem
  • 7. Jump-Diffusion Processes
  • 8. Jump-Diffusion Processes: The Martingale Problem Formulation
  • 3 Controlled Stochastic Differential Equations
  • 0. Outline of the Chapter
  • 1. Controlled S.D.E.’s: Introduction
  • 2. Relaxed Controls: Deterministic Case
  • 3. Stochastic Relaxed Controls
  • 4. The Martingale Problem Revisited
  • 5. Approximations, Weak Convergence and Optimality
  • 5. The Near Optimality of the Averaged Filter
  • 6. A Repair and Maintainance Example
  • 7. Robustness of the Averaged Filters
  • 8. A Robust Computational Approximation to the Averaged Filter
  • 9. The Averaged Filter on the Infinite Time Interval
  • 7 Weak Convergence: The Perturbed Test Function Method
  • 0. Outline of the Chapter
  • 1. An Example
  • 2. The Perturbed Test Function Method: Introduction
  • 3. The Perturbed Test Function Method: Tightness and Weak Convergence
  • 4. Characterization of the Limits
  • 8 Singularly Perturbed Wide-Band Noise Driven Systems
  • 0. Outline of the Chapter
  • 1. The System and Noise Model
  • 2. Weak Convergence of the Fast System
  • 3. Convergence to the Averaged System
  • 4. The Optimality Theorem
  • 5. The Average Cost Per Unit Time Problem
  • 9 Stability Theory
  • 0. Outline of the Chapter
  • 1. Stability Theory for Jump-Diffusion Processes of Itô Type
  • 2. Singularly Perturbed Deterministic Systems: Bounds on Paths

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