Numerical Methods for Stochastic Control Problems in Continuous Time

1. Front Matter

   Pages i-ix

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2. Introduction

   • Harold J. Kushner, Paul G. Dupuis

   Pages 1-5

3. Review of Continuous Time Models

   • Harold J. Kushner, Paul G. Dupuis

   Pages 7-33

4. Controlled Markov Chains

   • Harold J. Kushner, Paul G. Dupuis

   Pages 35-51

5. Dynamic Programming Equations

   • Harold J. Kushner, Paul G. Dupuis

   Pages 53-65

6. The Markov Chain Approximation Method: Introduction

   • Harold J. Kushner, Paul G. Dupuis

   Pages 67-88

7. Construction of the Approximating Markov Chain

   • Harold J. Kushner, Paul G. Dupuis

   Pages 89-149

8. Computational Methods for Controlled Markov Chains

   • Harold J. Kushner, Paul G. Dupuis

   Pages 151-192

9. The Ergodic Cost Problem: Formulation and Algorithms

   • Harold J. Kushner, Paul G. Dupuis

   Pages 193-216

10. Heavy Traffic and Singular Control Problems: Examples and Markov Chain Approximations

    • Harold J. Kushner, Paul G. Dupuis

    Pages 217-245

11. Weak Convergence and the Characterization of Processes

    • Harold J. Kushner, Paul G. Dupuis

    Pages 247-267

12. Convergence Proofs

    • Harold J. Kushner, Paul G. Dupuis

    Pages 269-301

13. Convergence for Reflecting Boundaries, Singular Control and Ergodic Cost Problems

    • Harold J. Kushner, Paul G. Dupuis

    Pages 303-323

14. Finite Time Problems and Nonlinear Filtering

    • Harold J. Kushner, Paul G. Dupuis

    Pages 325-345

15. Problems from the Calculus of Variations

    • Harold J. Kushner, Paul G. Dupuis

    Pages 347-410

16. The Viscosity Solution Approach to Proving Convergence of Numerical Schemes

    • Harold J. Kushner, Paul G. Dupuis

    Pages 411-421

17. Back Matter

    Pages 423-439

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This book is concerned with numerical methods for stochastic control and optimal stochastic control problems. The random process models of the controlled or uncontrolled stochastic systems are either diffusions or jump diffusions. Stochastic control is a very active area of research and new prob­ lem formulations and sometimes surprising applications appear regularly. We have chosen forms of the models which cover the great bulk of the for­ mulations of the continuous time stochastic control problems which have appeared to date. The standard formats are covered, but much emphasis is given to the newer and less well known formulations. The controlled process might be either stopped or absorbed on leaving a constraint set or upon first hitting a target set, or it might be reflected or "projected" from the boundary of a constraining set. In some of the more recent applications of the reflecting boundary problem, for example the so-called heavy traffic approximation problems, the directions of reflection are actually discontin­ uous. In general, the control might be representable as a bounded function or it might be of the so-called impulsive or singular control types. Both the "drift" and the "variance" might be controlled. The cost functions might be any of the standard types: Discounted, stopped on first exit from a set, finite time, optimal stopping, average cost per unit time over the infinite time interval, and so forth.

• Markov chain
• Variation
• algorithms
• numerical analysis
• stochastic processes

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

  Harold J. Kushner, Paul G. Dupuis

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