Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems - Harold Kushner

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Sommaire:
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 It?'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.- 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.- 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 It? Type.- 2. Singularly Perturbed Deterministic Systems: Bounds on Paths.- 3. Singularly Perturbed It? 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 It? Processes: Weak Convergence.- 2. Stability.- References.- List of Symbols.