Overview
- 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 ?
- Includes supplementary material: sn.pub/extras
Part of the book series: Stochastic Modelling and Applied Probability (SMAP, volume 68)
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Table of contents (9 chapters)
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Front Matter
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Principles of Monte Carlo Methods
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Front Matter
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Exact and Approximate Simulation of Markov Processes
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Front Matter
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Variance Reduction, Girsanov’s Theorem, and Stochastic Algorithms
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Front Matter
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Back Matter
Keywords
About this book
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.
Authors and Affiliations
About the authors
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.
Bibliographic Information
Book Title: Stochastic Simulation and Monte Carlo Methods
Book Subtitle: Mathematical Foundations of Stochastic Simulation
Authors: Carl Graham, Denis Talay
Series Title: Stochastic Modelling and Applied Probability
DOI: https://doi.org/10.1007/978-3-642-39363-1
Publisher: Springer Berlin, Heidelberg
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer-Verlag Berlin Heidelberg 2013
Hardcover ISBN: 978-3-642-39362-4Published: 29 July 2013
Softcover ISBN: 978-3-642-43840-0Published: 06 August 2015
eBook ISBN: 978-3-642-39363-1Published: 16 July 2013
Series ISSN: 0172-4568
Series E-ISSN: 2197-439X
Edition Number: 1
Number of Pages: XVI, 260
Topics: Probability Theory and Stochastic Processes, Numerical Analysis, Quantitative Finance