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Stochastic simulation and Monte Carlo methods : mathematical foundations of stochastic simulation / Carl Graham, Denis Talay.

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Author/Creator:
Graham, C. (Carl), author.
Language:
English.
Publication date:
2013
Publication:
Heidelberg : Springer, [2013]
Format:
  • Book
  • xvi, 260 pages : illustrations ; 24 cm.
Bibliography:
Includes bibliographical references (pages 253-255) 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
  • Discrete-time martingales, proof of the SLLN
  • Reminders on conditional expectation
  • Martingales and sub-martingales, backward martingales
  • Proof of the strong law of large numbers
  • Problems
  • Non-asymptotic error estimates for Monte Carlo methods
  • Convergence in law and characteristic functions
  • Central limit theorem
  • Asymptotic confidence intervals
  • Berry-Esseen'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 inter-arrivals
  • Simulation of independent poisson processes
  • Long time or large intensity limit, applications
  • Problems
  • Discrete-space 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, Feynman-Kac formula
  • Generators and semigroups in bounded operator algebras
  • A few case studies
  • Problems
  • Continuous-space Markov processes with jumps
  • Preliminaries
  • Measures, functions, and transition kernels
  • Markov property, finite-dimensional marginals
  • Semigroup, infinitesimal generator
  • Markov processes evolving only by isolated jumps
  • Semigroup, infinitesimal generator, and evolution law
  • Construction, simulation, existence
  • Kolmogorov equations, Feynman-Kac formula, bounded generator case
  • Markov processes following an ordinary differential equation between jumps: PDMP
  • Sample paths, evolution, integro-differential generator
  • Construction, simulation, existence
  • Kolmogorov equations, Feynman-Kac 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 Talay-Tubaro expansion
  • Romberg-Richardson 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
  • Non-differentiable 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
  • Long-time 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.
Note:
Also issued online
Contributor:
Talay, D. (Denis), author.
Available in another form:
9783642393631 (online)
(GyWOH)har135016798
Series:
Stochastic modelling and applied probability, 0172-4568 ; 68
Stochastic modelling and applied probability 68.
Subjects:
ISBN:
3642393624
9783642393624

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