Stochastic Modelling and Applied Probability

Stochastic Simulation and Monte Carlo Methods

Mathematical Foundations of Stochastic Simulation

Authors: Graham, Carl, Talay, Denis

  • Combines advanced mathematical tools and theoretical analysis of stochastic numerical methods at a high level
  • Provides methods to reach optimal results on the accuracy of Monte Carlo simulations of stochastic processes
  • Contains exercises in the text and problem sets of increasing demand at the end of each chapter ​
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About this Textbook

In various scientific and industrial fields, stochastic simulations are taking on a new importance. This is due to the increasing power of computers and practitioners’ aim to simulate more and more complex systems, and thus use random parameters as well as random noises to model the parametric uncertainties and the lack of knowledge on the physics of these systems. The error analysis of these computations is a highly complex mathematical undertaking. Approaching these issues, the authors present stochastic numerical methods and prove accurate convergence rate estimates in terms of their numerical parameters (number of simulations, time discretization steps). As a result, the book is a self-contained and rigorous study of the numerical methods within a theoretical framework. After briefly reviewing the basics, the authors first introduce fundamental notions in stochastic calculus and continuous-time martingale theory, then develop the analysis of pure-jump Markov processes, Poisson processes, and stochastic differential equations. In particular, they review the essential properties of Itô integrals and prove fundamental results on the probabilistic analysis of parabolic partial differential equations. These results in turn provide the basis for developing stochastic numerical methods, both from an algorithmic and theoretical point of view.

The book combines advanced mathematical tools, theoretical analysis of stochastic numerical methods, and practical issues at a high level, so as to provide optimal results on the accuracy of Monte Carlo simulations of stochastic processes. It is intended for master and Ph.D. students in the field of stochastic processes and their numerical applications, as well as for physicists, biologists, economists and other professionals working with stochastic simulations, who will benefit from the ability to reliably estimate and control the accuracy of their simulations.

About the authors

Carl Graham is a CNRS researcher and Professeur chargé de cours (part-time associate professor) at the École Polytechnique and associate editor for Annals of Applied Probability. His main fields of research include stochastic processes, stochastic modelling and communication networks. 
Denis Talay is a senior researcher at Inria. He holds a part time research position at École Polytechnique where he had taught for 13 years. He is, or has been, an associate editor for many top journals in probability, numerical analysis, financial mathematics and scientific computing. He was the president of the French Applied Math. Society SMAI (2006-2009) and is now the Chair of its Scientific Council. His main fields of interest are stochastic modelling, numerical probability, stochastic analysis of partial differential equations and financial mathematics.

Table of contents (9 chapters)

  • Introduction

    Graham, Carl (et al.)

    Pages 3-11

  • Strong Law of Large Numbers and Monte Carlo Methods

    Graham, Carl (et al.)

    Pages 13-35

  • Non-asymptotic Error Estimates for Monte Carlo Methods

    Graham, Carl (et al.)

    Pages 37-63

  • Poisson Processes as Particular Markov Processes

    Graham, Carl (et al.)

    Pages 67-88

  • Discrete-Space Markov Processes

    Graham, Carl (et al.)

    Pages 89-119

Buy this book

eBook $54.99
price for USA (gross)
  • ISBN 978-3-642-39363-1
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $69.99
price for USA
  • ISBN 978-3-642-39362-4
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $69.99
price for USA
  • ISBN 978-3-642-43840-0
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Stochastic Simulation and Monte Carlo Methods
Book Subtitle
Mathematical Foundations of Stochastic Simulation
Authors
Series Title
Stochastic Modelling and Applied Probability
Series Volume
68
Copyright
2013
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-642-39363-1
DOI
10.1007/978-3-642-39363-1
Hardcover ISBN
978-3-642-39362-4
Softcover ISBN
978-3-642-43840-0
Series ISSN
0172-4568
Edition Number
1
Number of Pages
XVI, 260
Topics