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Stochastic Modelling and Applied Probability #35: Stochastic Approximation and Recursive Algorithms and Applications

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Stochastic Modelling and Applied Probability #35: Stochastic Approximation and Recursive Algorithms and Applications Cover

 

Synopses & Reviews

Publisher Comments:

  This revised and expanded second edition presents a thorough development of the modern theory of stochastic approximation or recursive stochastic algorithms for both constrained and unconstrained problems. There is a complete development of both probability one and weak convergence methods for very general noise processes. The proofs of convergence use the ODE method, the most powerful to date. The assumptions and proof methods are designed to cover the needs of recent applications. The development proceeds from simple to complex problems, allowing the underlying ideas to be more easily understood. Rate of convergence, iterate averaging, high-dimensional problems, stability-ODE methods, two time scale, asynchronous and decentralized algorithms, state-dependent noise, stability methods for correlated noise, perturbed test function methods, and large deviations methods are covered. Many motivating examples from learning theory, ergodic cost problems for discrete event systems, wireless communications, adaptive control, signal processing, and elsewhere illustrate the applications of the theory.

Synopsis:

This book presents a thorough development of the modern theory of stochastic approximation or recursive stochastic algorithms for both constrained and unconstrained problems. This second edition is a thorough revision, although the main features and structure remain unchanged. It contains many additional applications and results as well as more detailed discussion.

Table of Contents

  Introduction   1    Review of Continuous Time Models     1.1    Martingales and Martingale Inequalities     1.2    Stochastic Integration     1.3    Stochastic Differential Equations: Diffusions     1.4    Reflected Diffusions     1.5    Processes with Jumps   2    Controlled Markov Chains     2.1    Recursive Equations for the Cost     2.2    Optimal Stopping Problems     2.3    Discounted Cost     2.4    Control to a Target Set and Contraction Mappings     2.5    Finite Time Control Problems   3    Dynamic Programming Equations     3.1    Functionals of Uncontrolled Processes     3.2    The Optimal Stopping Problem     3.3    Control Until a Target Set Is Reached     3.4    A Discounted Problem with a Target Set and Reflection     3.5    Average Cost Per Unit Time   4    Markov Chain Approximation Method: Introduction     4.1    Markov Chain Approximation     4.2    Continuous Time Interpolation     4.3    A Markov Chain Interpolation     4.4    A Random Walk Approximation     4.5    A Deterministic Discounted Problem     4.6    Deterministic Relaxed Controls   5    Construction of the Approximating Markov Chains     5.1    One Dimensional Examples     5.2    Numerical Simplifications     5.3    The General Finite Difference Method     5.4    A Direct Construction     5.5    Variable Grids     5.6    Jump Diffusion Processes     5.7    Reflecting Boundaries     5.8    Dynamic Programming Equations     5.9    Controlled and State Dependent Variance   6    Computational Methods for Controlled Markov Chains     6.1    The Problem Formulation     6.2    Classical Iterative Methods     6.3    Error Bounds     6.4    Accelerated Jacobi and Gauss-Seidel Methods     6.5    Domain Decomposition     6.6    Coarse Grid-Fine Grid Solutions     6.7    A Multigrid Method     6.8    Linear Programming   7    The Ergodic Cost Problem: Formulation and Algorithms     7.1    Formulation of the Control Problem     7.2    A Jacobi Type Iteration     7.3    Approximation in Policy Space     7.4    Numerical Methods     7.5    The Control Problem     7.6    The Interpolated Process     7.7    Computations     7.8    Boundary Costs and Controls   8    Heavy Traffic and Singular Control     8.1    Motivating Examples     8.2    The Heavy Traffic Problem     8.3    Singular Control   9    Weak Convergence and the Characterization of Processes     9.1    Weak Convergence     9.2    Criteria for Tightness in $D^{k}\left [0,\infty \right )$     9.3    Characterization of Processes     9.4    An Example     9.5    Relaxed Controls   10    Convergence Proofs     10.1    Limit Theorems     10.2    Existence of an Optimal Control     10.3    Approximating the Optimal Control     10.4    The Approximating Markov Chain     10.5    Convergence of the Costs     10.6    Optimal Stopping   11    Convergence for Reflecting Boundaries, Singular Control, and Ergodic Cost Problems     11.1    The Reflecting Boundary Problem     11.2    The Singular Control Problem     11.3    The Ergodic Cost Problem   12    Finite Time Problems and Nonlinear Filtering     12.1    Explicit Approximations: An Example     12.2    General Explicit Approximations     12.3    Implicit Approximations: An Example     12.4    General Implicit Approximations     12.5    Optimal Control Computations     12.6    Solution Methods     12.7    Nonlinear Filtering   13    Controlled Variance and Jumps     13.1    Controlled Variance: Introduction     13.2    Controlled Jumps   14    Problems from the Calculus of Variations: Finite Time Horizon     14.1    Problems with a Continuous Running Cost     14.2    Numerical Schemes and Convergence     14.3    Problems with a Discontinuous Running Cost   15    Problems from the Calculus of Variations: Infinite Time Horizon     15.1    Problems of Interest     15.2    Numerical Schemes for the Case $k(x, \alpha ) \geq k_0 > 0$     15.3    Numerical Schemes for the Case $k(x, \alpha ) \geq 0$     15.4    Remarks on Implementation and Examples   16    The Viscosity Solution Approach     16.1    Definitions and Some Properties of Viscosity Solutions     16.2    Numerical Schemes     16.3    Proof of Convergence       References       Index       List of Symbols

Product Details

ISBN:
9781441918475
Author:
Kushner, Harold J.
Publisher:
Springer
Author:
Yin, G. George
Author:
Yin, George
Author:
Kushner, Harold
Location:
New York, NY
Subject:
Statistics
Subject:
Probability Theory and Stochastic Processes
Subject:
Approximations and Expansions
Subject:
APPLICATIONS OF MATHEMATICS
Subject:
Algorithms
Subject:
Mathematics | Probability and Statistics
Subject:
Mathematics
Subject:
Language, literature and biography
Subject:
mathematics and statistics
Subject:
Distribution (Probability theory)
Copyright:
Edition Description:
Softcover reprint of hardcover 2nd ed. 2003
Series:
Stochastic Modelling and Applied Probability
Series Volume:
35
Publication Date:
20101124
Binding:
TRADE PAPER
Language:
English
Pages:
498
Dimensions:
235 x 155 mm 744 gr

Related Subjects

Science and Mathematics » Mathematics » Analysis General
Science and Mathematics » Mathematics » Applied
Science and Mathematics » Mathematics » Probability and Statistics » General
Science and Mathematics » Mathematics » Probability and Statistics » Statistics

Stochastic Modelling and Applied Probability #35: Stochastic Approximation and Recursive Algorithms and Applications New Trade Paper
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Product details 498 pages Springer - English 9781441918475 Reviews:
"Synopsis" by , This book presents a thorough development of the modern theory of stochastic approximation or recursive stochastic algorithms for both constrained and unconstrained problems. This second edition is a thorough revision, although the main features and structure remain unchanged. It contains many additional applications and results as well as more detailed discussion.
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