Author | Kushner, Harold J. author |
---|---|
Title | Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems [electronic resource] / by Harold J. Kushner |
Imprint | Boston, MA : Birkhรคuser Boston, 1990 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-4482-0 |
Descript | XVIII, 234 p. online resource |
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 Itรดโs 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.โs: Introduction -- 2. Relaxed Controls: Deterministic Case -- 3. Stochastic Relaxed Controls -- 4. The Martingale Problem Revisited -- 5. Approximations, Weak Convergence and Optimality -- 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 Singularly Perturbed System -- 3. The Almost Optimality of the Averaged Filter -- 4. A Counterexample to the Averaged Filter -- 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 Itรด Type -- 2. Singularly Perturbed Deterministic Systems: Bounds on Paths -- 3. Singularly Perturbed Itรด 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 Itรด Processes: Weak Convergence -- 2. Stability -- References -- List of Symbols