Stochastic simulation and Monte Carlo methods : mathematical foundations of stochastic simulation
 Author/Creator
 Graham, C. (Carl), author.
 Language
 English.
 Publication
 Heidelberg : Springer, [2013]
 Physical description
 xvi, 260 pages : illustrations ; 24 cm.
 Series
 Stochastic modelling and applied probability 68.
Access
Available online

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QA274 .G725 2013

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QA274 .G725 2013
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Contributors
 Contributor
 Talay, D. (Denis), author.
Contents/Summary
 Bibliography
 Includes bibliographical references (pages 253255) and index.
 Contents

 Principles of Monte Carlo methods
 Introduction
 Why use probabilistic models and simulations?
 What are the reasons for probabilistic models?
 What are the objectives of random simulations?
 Organization of the monograph
 Strong law of large numbers and Monte Carlo methods
 Strong law of large numbers, examples of Monte Carlo methods
 Strong law of large numbers, almost sure convergence
 Buffon's needle
 Neutron transport simulations
 Stochastic numerical methods for partial differential equations
 Simulation algorithms for simple probability distributions
 Uniform distributions
 Discrete distributions
 Gaussian distributions
 Cumulative distribution function inversion, exponential distributions
 Rejection method
 Discretetime martingales, proof of the SLLN
 Reminders on conditional expectation
 Martingales and submartingales, backward martingales
 Proof of the strong law of large numbers
 Problems
 Nonasymptotic error estimates for Monte Carlo methods
 Convergence in law and characteristic functions
 Central limit theorem
 Asymptotic confidence intervals
 BerryEsseen's theorem
 Bikelis' theorem
 Absolute confidence intervals
 Concentration inequalities
 Logarithmic Sobolev inequalities
 Concentration inequalities, absolute confidence intervals
 Elementary variance reduction techniques
 Control variate
 Importance sampling
 Problems
 Exact and approximate simulation of Markov processes
 Poisson processes as particular Markov processes
 Quick introduction to markov processes
 Some issues in Markovian modeling
 Rudiments on processes, sample paths, and laws
 Poisson processes: characterization, properties
 Point processes and poisson processes
 Simple and strong markov property
 Superposition and decomposition
 Simulation and approximation
 Simulation of interarrivals
 Simulation of independent poisson processes
 Long time or large intensity limit, applications
 Problems
 Discretespace markov processes
 Characterization, specification, properties
 Measures, functions, and transition matrices
 Simple and strong Markov property
 Semigroup, infinitesimal generator, and evolution law
 Constructions, existence, simulation, equations
 Fundamental constructions
 Explosion or existence for a Markov process
 Fundamental simulation, fictitious jump method
 Kolmogorov equations, FeynmanKac formula
 Generators and semigroups in bounded operator algebras
 A few case studies
 Problems
 Continuousspace Markov processes with jumps
 Preliminaries
 Measures, functions, and transition kernels
 Markov property, finitedimensional marginals
 Semigroup, infinitesimal generator
 Markov processes evolving only by isolated jumps
 Semigroup, infinitesimal generator, and evolution law
 Construction, simulation, existence
 Kolmogorov equations, FeynmanKac formula, bounded generator case
 Markov processes following an ordinary differential equation between jumps: PDMP
 Sample paths, evolution, integrodifferential generator
 Construction, simulation, existence
 Kolmogorov equations, FeynmanKac formula
 Application to kinetic equations
 Further extensions
 Problems
 Discretization of stochastic differential equations
 Reminders on Itô's stochastic calculus
 Stochastic integrals and Itô processes
 Itô's formula, existence and uniqueness of solutions of stochastic differential equations
 Markov properties, martingale problems and Fokker Planck equations
 Euler and Milstein schemes
 Moments of the solution and of its approximations
 Convergence rates in Lp (... ) norm and almost surely
 Monte Carlo methods for parabolic partial differential equations
 The principle of the method
 Introduction of the error analysis
 Optimal convergence rate : the TalayTubaro expansion
 RombergRichardson extrapolation methods
 Probabilistic interpretation and estimates for parabolic partial differential equations
 Problems
 Variance reduction, Girsanov's theorem, and stochastic algorithms
 Variance reduction and stochastic differential equations
 Preliminary reminders on the Girsanov theorem
 Control variates method
 Variance reduction for sensitivity analysis
 Differentiable terminal conditions
 Nondifferentiable terminal conditions
 Importance sampling method
 Statistical romberg method
 Problems
 Stochastic algorithms
 Introduction
 Study in an idealized framework
 Definitions
 The ordinary differential equation method, martingale increments
 Longtime behavior of the algorithm
 Variance reduction for Monte Carlo methods
 Searching for an importance sampling
 Variance reduction and stochastic algorithms
 Problems
 Appendix solutions to selected problems
 References
 Index.
 Publisher's Summary
 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 selfcontained 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 continuoustime martingale theory, then develop the analysis of purejump Markov processes, Poisson processes, and stochastic differential equations. In particular, they review the essential properties of Ito 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.
(source: Nielsen Book Data)
Subjects
Bibliographic information
 Publication date
 2013
 Responsibility
 Carl Graham, Denis Talay.
 Series
 Stochastic modelling and applied probability, 01724568 ; 68
 Note
 Also issued online
 Available in another form
 9783642393631 (online)
 (GyWOH)har135016798
 ISBN
 3642393624
 9783642393624