© 2008

Numerical Methods for Controlled Stochastic Delay Systems


Part of the Systems & Control: Foundations & Applications book series (SCFA)

About this book


The Markov chain approximation methods are widely used for the numerical solution of nonlinear stochastic control problems in continuous time. This book extends the methods to stochastic systems with delays. Because such problems are infinite-dimensional, many new issues arise in getting good numerical approximations and in the convergence proofs. Useful forms of numerical algorithms and system approximations are developed in this work, and the convergence proofs are given. All of the usual cost functions are treated as well as singular and impulsive controls. A major concern is on representations and approximations that use minimal memory.

Features and topics include:

* Surveys properties of the most important stochastic dynamical models, including singular control, and those for diffusion and reflected diffusion models.

* Gives approximations to the dynamical models that simplify the numerical problem, but have only small effects on the behavior.

* Develops an ergodic theory for reflected diffusions with delays, as well as model simplifications useful for numerical approximations for average cost per unit time problems.

* Provides numerical algorithms for models with delays in the path, or path and control, with reduced memory requirements.

* Develops transformations of the problem that yield more efficient approximations when the control, driving Wiener process, and/or reflection processes might be delayed, as well as the path.

* Presents examples with applications to control and modern communications systems.

The book is the first on the subject and will be of interest to all those who work with stochastic delay equations and whose main interest is in either the use of the algorithms or the underlying mathematics. An excellent resource for graduate students, researchers, and practitioners, the work may be used as a graduate-level textbook for a special topics course or seminar on numerical methods in stochastic control.


Ergodic theory Markov Markov chain approximation methods Minimum Transformation algorithms convergence proofs differential equation numerical algorithms numerical methods reflected or stopped diffusion stochastic control stochastic delay systems wave equation weak convergence theory

Authors and affiliations

  1. 1.Div. Applied MathematicsBrown UniversityProvidenceU.S.A.

Bibliographic information


From the reviews:

"Overall this is a book entirely devoted to numerics of controlled stochastic systems with delays. In addition to analysis, it contains many numerical and simulation results. The book should be beneficial to both people working in the numerical methods of stochastic controls and people working in various applications who need to use numerical algorithms. It is perhaps the only comprehensive numerical study of controlled diffusions with delays to date...[I]t is conceivable that this book will become a standard reference in the stochastic control literature."   —Mathematical Reviews

“This book extends the Markov chain approximation methods from nonlinear stochastic control problems in continuous time to stochastic systems with delays which arise in the realistic modeling of many physical, mechanical, biological or medical systems. … An Index, a Symbol Index and a 95 titles reference list are also provided. … used as a graduate level textbook or as a reference for researchers or practitioners dealing with stochastic delay equations, and interested in either the use of algorithms or of the underlying mathematics.” (Adriana-Ioana Lefter, Memoirs of the Scientific Sections, 2009)

“The book deals with numerical methods for nonlinear continuous time stochastic control systems with delays. … The book appears of particular importance for the practical solution to stochastic control problems for systems with delays. … it does not contain only results, but also ideas and suggestions so that it can be considered as an important source of inspiration for further work in this field.” (Giovanni Di Masi, Zentralblatt MATH, Vol. 1219, 2011)