Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems

1. Front Matter

   Pages i-xvii

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2. Weak Convergence

   • Harold J. Kushner

   Pages 1-16

3. Stochastic Processes: Background

   • Harold J. Kushner

   Pages 17-43

4. Controlled Stochastic Differential Equations

   • Harold J. Kushner

   Pages 45-60

5. Controlled Singularly Perturbed Systems

   • Harold J. Kushner

   Pages 61-91

6. Functional Occupation Measures and Average Cost per Unit Time Problems

   • Harold J. Kushner

   Pages 93-113

7. The Nonlinear Filtering Problem

   • Harold J. Kushner

   Pages 115-150

8. Weak Convergence: The Perturbed Test Function Method

   • Harold J. Kushner

   Pages 151-170

9. Singularly Perturbed Wide-Band Noise Driven Systems

   • Harold J. Kushner

   Pages 171-190

10. Stability Theory

    • Harold J. Kushner

    Pages 191-208

11. Parametric Singularities

    • Harold J. Kushner

    Pages 209-217

12. Back Matter

    Pages 219-233

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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 methods which are used and developed are those of the theory of weak con­ vergence. The techniques are quite powerful for getting weak convergence or functional limit theorems for broad classes of problems and many of the techniques are new. The original need for some of the techniques which are developed here arose in connection with our study of the particular applica­ tions in this book, and related problems of approximation in control theory, but it will be clear that they have numerous applications elsewhere in weak convergence and process approximation theory. The book is a continuation of the author's long term interest in problems of the approximation of stochastic processes and its applications to problems arising in control and communication theory and related areas. In fact, the techniques used here can be fruitfully applied to many other areas. The basic random processes of interest can be described by solutions to either (multiple time scale) Ito differential equations driven by wide band or state dependent wide band noise or which are singularly perturbed. They might be controlled or not, and their state values might be fully observable or not (e. g. , as in the nonlinear filtering problem).

• Finite
• Martingale
• Variable
• average
• diffusion process
• equation
• filtering problem
• function
• stochastic process
• theorem

• Division of Applied Mathematics, Brown University, Providence, USA

  Harold J. Kushner

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