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Book review

Author(s) Kushner, Harold J.
Dupuis, Paul G.
Title Numerical Methods for Stochastic Control Problems in Continuous Time
Publisher Springer-Verlag
Year of publication 1992
Reviewed by Vasile Postolica

This book is a valuable synthesis concenring with numerical methods for stochastic control and optimal stochastic control problems which contains 14 chapters (40 illustrations in 43 parts and a substantial bilbliography). After a pertinent Introduction, the first chapter is conceived as a review of continuous time models, with results in the properties of the processes. In Chapter 2 one defines some of canonical control problems for the Markov chain models in order to be used afterwards as "approximating processes". The third chapter is dedicated to many standard control problems and to their numerical solutions. A rigorous introduction in the Markov chain approximation methods is described in Chapter 4, followed by the extensive constructions of approximating Markov chains contained in Chapter 5. We find important computational methods for controlled Markov chains in Chapter 6, and smart fomulations and the corresponding algorithms for the ergodic cost problem in Chapter 7. The heavy traffic and singular control problems through the agency of Markov chains approximations on interesting corresponding examples may be found in Chapter 8. Properties of convergence (weak convergence, a characterization of processes, convergence proofs and convergence for reflecting boundaries, singular control and ergodic cost problems) are indicated with important results and comments in chapters 9-11. In Chapter 12 the authors discuss several finite time problems and nonlinear filtering and they extend their study on some problems from the calculus of variations in Chapter 13. In the final chapter, the authors describe an alternative approach for proving the convergence of numerical schemes.

By our opinion, this book is a very significant contribution in the field of mathematical theories and the corresponding applications for stochastic control problems in continuous time.