Stochastic simulation: Algorithms and analysis.

*(English)*Zbl 1126.65001
Stochastic Modelling and Applied Probability 57. New York, NY: Springer (ISBN 978-0-387-30679-7/hbk). xiv, 476 p. (2007).

The adequate statistical simulation of random quantities is one of the challenges of this century. Therefore, sampling-based computational methods have become a fundamental part of the numerical toolset of both practitioners and researchers across an enormous number of different scientific disciplines. This book provides a descriptive treatment of a variety of such sampling-based methods. Some steps to the mathematical analysis of their convergence properties and diverse applications are sketched as well.

The first half of the book focuses on general methods, whereas the second half discusses some model-specific algorithms. The book has \(14\) chapters and one appendix. Topics as random number generation, output analysis, steady state simulation, variance-reduction, rare-event simulation, derivative estimation, stochastic optimization, numerical integration, stochastic differential equations, Gaussian processes, Lévy processes, Markov chains, Monte Carlo methods, simulation of queues, branching processes and black-box algorithms are touched among many others.

The wide range of examples, exercises and applications will find interest among readers in probability, statistics, operations research, economics, finance, biology, chemistry, physics and engineering.

The following computational issues are addressed:

1. How does one generate the needed random input variables?

2. How many computer experiments should one carry out?

3. How does one compute expectations associated with stationary distributions?

4. How can one exploit the specific structure of problems to speed up the computations?

5. How does one compute efficiently probabilities or rare events?

6. How do we estimate the sensitivity of a stochastic model to changes in parameters?

7. How can we use simulation to optimize the choice of decision parameters?

As one clearly can see from this extraordinary choice of interesting topics, this book is of potential interest to many researchers, students and instructors. Due to this large variety, the book has a descriptive and introductory character and, of course, a deeper understanding of related issues requires further “digging into the current literature”.

The first half of the book focuses on general methods, whereas the second half discusses some model-specific algorithms. The book has \(14\) chapters and one appendix. Topics as random number generation, output analysis, steady state simulation, variance-reduction, rare-event simulation, derivative estimation, stochastic optimization, numerical integration, stochastic differential equations, Gaussian processes, Lévy processes, Markov chains, Monte Carlo methods, simulation of queues, branching processes and black-box algorithms are touched among many others.

The wide range of examples, exercises and applications will find interest among readers in probability, statistics, operations research, economics, finance, biology, chemistry, physics and engineering.

The following computational issues are addressed:

1. How does one generate the needed random input variables?

2. How many computer experiments should one carry out?

3. How does one compute expectations associated with stationary distributions?

4. How can one exploit the specific structure of problems to speed up the computations?

5. How does one compute efficiently probabilities or rare events?

6. How do we estimate the sensitivity of a stochastic model to changes in parameters?

7. How can we use simulation to optimize the choice of decision parameters?

As one clearly can see from this extraordinary choice of interesting topics, this book is of potential interest to many researchers, students and instructors. Due to this large variety, the book has a descriptive and introductory character and, of course, a deeper understanding of related issues requires further “digging into the current literature”.

Reviewer: Henri Schurz (Carbondale)

##### MSC:

65Cxx | Probabilistic methods, stochastic differential equations |

68U20 | Simulation (MSC2010) |

60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |