This book is concerned with numerical methods for stochastic control and optimal stochastic control problems. The random process models of the controlled or uncontrolled stochastic systems are either diffusions or jump diffusions. Stochastic control is a very active area of research and new prob lem formulations and sometimes surprising applications appear regularly. We have chosen forms of the models which cover the great bulk of the for mulations of the continuous time stochastic control problems which have appeared to date. The standard formats are covered, but much emphasis is given to the newer and less well known formulations. The controlled process might be either stopped or absorbed on leaving a constraint set or upon first hitting a target set, or it might be reflected or "projected" from the boundary of a constraining set. In some of the more recent applications of the reflecting boundary problem, for example the so-called heavy traffic approximation problems, the directions of reflection are actually discontin uous. In general, the control might be representable as a bounded function or it might be of the so-called impulsive or singular control types. Both the "drift" and the "variance" might be controlled. The cost functions might be any of the standard types: Discounted, stopped on first exit from a set, finite time, optimal stopping, average cost per unit time over the infinite time interval, and so forth.
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ISBN-13
978-1-4684-0441-8 (9781468404418)
DOI
10.1007/978-1-4684-0441-8
Schweitzer Klassifikation
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 The Markov Chain Approximation Method: Introduction.- 4.1 The Markov Chain Approximation Method.- 4.2 Continuous Time Interpolation and Approximating Cost Function.- 4.3 A Continuous Time Markov Chain Interpolation.- 4.4 A Random Walk Approximation to the Wiener Process.- 4.5 A Deterministic Discounted Problem.- 4.6 Deterministic Relaxed Controls.- 5 Construction of the Approximating Markov Chain.- 5.1 Finite Difference Type Approximations: One Dimensional Examples.- 5.2 Numerical Simplifications and Alternatives for Example 4.- 5.3 The General Finite Difference Method.- 5.4 A Direct Construction of the Approximating Markov Chain.- 5.5 Variable Grids.- 5.6 Jump Diffusion Processes.- 5.7 Approximations for Reflecting Boundaries.- 5.8 Dynamic Programming Equations.- 6 Computational Methods for Controlled Markov Chains.- 6.1 The Problem Formulation.- 6.2 Classical Iterative Methods: Approximation in Policy and Value Space.- 6.3 Error Bounds for Discounted Problems.- 6.4 Accelerated Jacobi and Gauss-Seidel Methods.- 6.5 Domain Decomposition and Implementation on Parallel Processors.- 6.6 A State Aggregation Method.- 6.7 Coarse Grid-Fine Grid Solutions.- 6.8 A Multigrid Method.- 6.9 Linear Programming Formulations and Constraints.- 7 The Ergodic Cost Problem: Formulations and Algorithms.- 7.1 The Control Problem for the Markov Chain: Formulation.- 7.2 A Jacobi Type Iteration.- 7.3 Approximation in Policy Space.- 7.4 Numerical Methods for the Solution of (3.4).- 7.5 The Control Problem for the Approximating Markov Chain.- 7.6 The Continuous Parameter Markov Chain Interpolation.- 7.7 Computations for the Approximating Markov Chain.- 7.8 Boundary Costs and Controls.- 8 Heavy Traffic and Singular Control Problems: Examples and Markov Chain Approximations.- 8.1 Motivating Examples.- 8.2 The Heavy Traffic Problem: A Markov Chain Approximation.- 8.3 Singular Control: A Markov Chain Approximation.- 9 Weak Convergence and the Characterization of Processes.- 9.1 Weak Convergence.- 9.2 Criteria for Tightness in Dk [0, ?).- 9.3 Characterization of Processes.- 9.4 An Example.- 9.5 Relaxed Controls.- 10 Convergence Proofs.- 10.1 Limit Theorems and Approximations of Relaxed Controls.- 10.2 Existence of an Optimal Control: Absorbing Boundary.- 10.3 Approximating the Optimal Control.- 10.4 The Approximating Markov Chain: Weak Convergence.- 10.5 Convergence of the Costs: Discounted Cost and Absorbing Boundary.- 10.6 The Optimal Stopping Problem.- 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 The Explicit Approximation Method: An Example.- 12.2 The General Explicit Approximation Method.- 12.3 The Implicit Approximation Method: An Example.- 12.4 The General Implicit Approximation Method.-12.5 The Optimal Control Problem: Approximations and Dynamic Programming Equations.- 12.6 Methods of Solution, Decomposition and Convergence.- 12.7 Nonlinear Filtering.- 13 Problems from the Calculus of Variations.- 13.1 Problems of Interest.- 13.2 Numerical Schemes and Convergence for the Finite Time Problem.- 13.3 Problems with a Controlled Stopping Time.- 13.4 Problems with a Discontinuous Running Cost.- 14 The Viscosity Solution Approach to Proving Convergence of Numerical Schemes.- 14.1 Definitions and Some Properties of Viscosity Solutions.- 14.2 Numerical Schemes.- 14.3 Proof of Convergence.- References.- List of Symbols.
