Stochastic Approximation and Recursive Algorithms and Applications

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