Probability methods for approximations in stochastic control and for elliptic equations / Harold J. Kushner.

Format:

Book

Author/Creator:

Kushner, Harold J. (Harold Joseph), 1933-

Series:

Mathematics in science and engineering ; v. 129.

Mathematics in science and engineering ; v. 129

Language:

English

Subjects (All):

Stochastic control theory.

Differential equations, Elliptic.

Approximation theory.

Probabilities.

Physical Description:

1 online resource (263 p.)

Place of Publication:

New York : Academic Press, 1977.

Language Note:

English

Summary:

Probability methods for approximations in stochastic control and for elliptic equations

Contents:

Front Cover; Probability Methods for Approximations in Stochastic Control and for Elliptic Equations; Copyright Page; Contents; Preface; Acknowledgments; Chapter 1. Probability Background; 1.1 The Wiener Processes; 1.2 Martingales; 1.3 Markov Processes; 1.4 Stochastic Integrals; 1.5 Stochastic Differential Equations; Chapter 2. Weak Convergence of Probability Measures; 2.1 Probability Measures on the Real Line. Real-Valued Random Variables; 2.2 Probability Measures on Metric Spaces; 2.3 The Spaces Cm[a, ß] of Continuous Functions; 2.4 The Space Dm[a, ß]; 2.5 Weak Convergence on Other Spaces

Chapter 3. Markov Chains and Control Problems with Markov Chain Models3.1 Equations Satisfied by Functionals of Markov Chains; 3.2 Optimal Stopping Problems; 3.3 Controlled Markov Chains: Families of Controlled Strategies; 3.4 Optimal Control until a Boundary Is Reached; 3.5 Optimal Discounted Cost; 3.6 Optimal Stopping and Control; 3.7 Impulsive Control Systems; 3.8 Control over a Fixed Time Interval; 3.9 Linear Programming Formulation of the Markov Chain Control Problems; Chapter 4. Elliptic and Parabolic Equations and Functionals of Diffusions

4.1 Assumptions and Uniqueness Results: No Control4.2 Functionals of Uncontrolled Diffusions; 4.3 Partial Differential Equations Associated with Functionals of Diffusions. a( . ) Uniformly Positive Definite; 4.4 a( . ) Degenerate; 4.5 Partial Differential Equations Formally Satisfied by Path Functionals; 4.6 The Characteristic Operator of the Diffusion; 4.7 Optimal Control Problems and Nonlinear Partial Differential Equations; Chapter 5. A Simple Application of the Invariance Theorems; 5.1 A Functional Limit Theorem; 5.2 An Application to Numerical Analysis

Chapter 6. Elliptic Equations and Uncontrolled Diffusions6.1 Problem Formulation; 6.2 The Finite Difference Method and an Approximating Markov Chain; 6.3 Convergence of the Approximations to a Diffusion Process; 6.4 Convergence of the Cost Functionals Rh ( . ); 6.5 The Discounted Cost Problem; 6.6 An Alternative Representation for ßhn and W( . ); 6.7 Monte Carlo; 6.8 Approximation of Invariant Measures; 6.9 Remarks and Extensions; 6.10 Numerical Data; Chapter 7. Approximations for Parabolic Equations and Nonlinear Filtering Problems; 7.1 Problem Statement

7.2 The Finite Difference Approximation and Weak Convergence7.3 Implicit Approximations; 7.4 Discounted Cost: Explicit Method; 7.5 Nonlinear Filtering; 7.6 Numerical Data: Estimation of an Invariant Measure; Chapter 8. Optimal Stopping and Impulsive Control Problems; 8.1 Discretization of the Optimal Stopping Problem; 8.2 Optimality of the Limiting Stopping Time p; 8.3 Constrained Optimal Stopping Problems; 8.4 Discretization of the Impulsive Control Problem; 8.5 Optimality of the Limits ( pi, vi} and R(x, {pi, vi}); 8.6 Numerical Data for the Optimal Stopping Problem

Chapter 9. Approximations to Optimal Controls and Nonlinear Partial Differential Equations

Notes:

Description based upon print version of record.

Includes bibliographical references (p. 237-239) and indexes.

ISBN:

1-282-75543-9

9786612755439

0-08-095638-6

OCLC:

700919069