Stochastic Simulation and Monte Carlo Methods: Mathematical Foundations of Stochastic Simulation
In various scientific and industrial fields, shastic 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 shastic 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 shastic calculus and continuous-time martingale theory, then develop the analysis of pure-jump Markov processes, Poisson processes, and shastic 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 shastic numerical methods, both from an algorithmic and theoretical point of view.

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

1115844165
Stochastic Simulation and Monte Carlo Methods: Mathematical Foundations of Stochastic Simulation
In various scientific and industrial fields, shastic 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 shastic 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 shastic calculus and continuous-time martingale theory, then develop the analysis of pure-jump Markov processes, Poisson processes, and shastic 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 shastic numerical methods, both from an algorithmic and theoretical point of view.

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

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Stochastic Simulation and Monte Carlo Methods: Mathematical Foundations of Stochastic Simulation

Stochastic Simulation and Monte Carlo Methods: Mathematical Foundations of Stochastic Simulation

Stochastic Simulation and Monte Carlo Methods: Mathematical Foundations of Stochastic Simulation

Stochastic Simulation and Monte Carlo Methods: Mathematical Foundations of Stochastic Simulation

Overview

In various scientific and industrial fields, shastic 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 shastic 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 shastic calculus and continuous-time martingale theory, then develop the analysis of pure-jump Markov processes, Poisson processes, and shastic 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 shastic numerical methods, both from an algorithmic and theoretical point of view.

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

Product Details

ISBN-13: 9783642393624
Publisher: Springer Berlin Heidelberg
Publication date: 07/17/2013
Series: Stochastic Modelling and Applied Probability , #68
Edition description: 2013
Pages: 260
Product dimensions: 6.10(w) x 9.25(h) x 0.03(d)

About the Author

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 shastic processes, shastic 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 shastic modelling, numerical probability, shastic analysis of partial differential equations and financial mathematics.

Table of Contents

Part I:Principles of Monte Carlo Methods.- 1.Introduction.- 2.Strong Law of Large Numbers and Monte Carlo Methods.- 3.Non Asymptotic Error Estimates for Monte Carlo Methods.- Part II:Exact and Approximate Simulation of Markov Processes.- 4.Poisson Processes.- 5.Discrete-Space Markov Processes.- 6.Continuous-Space Markov Processes with Jumps.- 7.Discretization of Shastic Differential Equations.- Part III:Variance Reduction, Girsanov’s Theorem, and Shastic Algorithms.- 8.Variance Reduction and Shastic Differential Equations.- 9.Shastic Algorithms.- References.- Index.​
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