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

By: Contributor(s): Material type: TextTextSeries: Stochastic modelling and applied probability ; 68Publication details: Berlin : Springer-Verlag, 2013.Description: xvi, 260 pages : illustrations ; 24 cmISBN:
  • 9783642393624 (hardcover : alk. paper)
Subject(s): DDC classification:
  • 23 G738 519.23
Contents:
Part I: Principles of Monte Carlo methods -- 1. Introduction -- 1.1 Why use probabilistic models and simulations? -- 1.1.1 What are the reasons for probabilistic models? -- 1.1.2 What are the objectives of random simulations? -- 1.2 Organization of the monograph -- 2. Strong law of large numbers and Monte Carlo methods -- 2.1 Strong law of large numbers, examples of Monte Carlo methods -- 2.1.1 Strong law of large numbers, almost sure convergence -- 2.1.2 Buffon's needle -- 2.1.3 Neutron transport simulations -- 2.1.4 Stochastic numerical methods for partial differential equations -- 2.2 Simulation algorithms for simple probability distributions -- 2.2.1 Uniform distributions -- 2.2.2 Discrete distributions -- 2.2.3 Gaussian distributions -- 2.2.4 Cumulative distribution function inversion, exponential distributions -- 2.2.5 Rejection method -- 2.3 Discrete-time martingales, proof of the SLLN -- 2.3.1 Reminders on conditional expectation -- 2.3.2 Martingales and sub-martingales, backward martingales -- 2.3.3 Proof of the strong law of large numbers -- 2.4 Problems -- 3. Non-asymptotic error estimates for Monte Carlo methods -- 3.1 Convergence in law and characteristic functions -- 3.2 Central limit theorem -- 3.2.1 Asymptotic confidence intervals -- 3.3 Berry-Esseen's theorem -- 3.4 Bikelis' theorem -- 3.4.1 Absolute confidence intervals -- 3.5 Concentration inequalities -- 3.5.1 Logarithmic Sobolev inequalities -- 3.5.2 Concentration inequalities, absolute confidence intervals -- 3.6 Elementary variance reduction techniques -- 3.6.1 Control variate -- 3.6.2 Importance sampling -- 3.7 Problems -- Part II: Exact and approximate simulation of Markov processes -- 4. Poisson processes as particular Markov processes -- 4.1 Quick introduction to markov processes -- 4.1.1 Some issues in Markovian modeling -- 4.1.2 Rudiments on processes, sample paths, and laws -- 4.2 Poisson processes: characterization, properties -- 4.2.1 Point processes and poisson processes -- 4.2.2 Simple and strong markov property -- 4.2.3 Superposition and decomposition -- 4.3 Simulation and approximation -- 4.3.1 Simulation of inter-arrivals -- 4.3.2 Simulation of independent poisson processes -- 4.3.3 Long time or large intensity limit, applications -- 4.4 Problems -- 5. Discrete-space markov processes -- 5.1 Characterization, specification, properties -- 5.1.1 Measures, functions, and transition matrices -- 5.1.2 Simple and strong Markov property -- 5.1.3 Semigroup, infinitesimal generator, and evolution law -- 5.2 Constructions, existence, simulation, equations -- 5.2.1 Fundamental constructions -- 5.2.2 Explosion or existence for a Markov process -- 5.2.3 Fundamental simulation, fictitious jump method -- 5.2.4 Kolmogorov equations, Feynman-Kac formula -- 5.2.5 Generators and semigroups in bounded operator algebras -- 5.2.6 A few case studies -- 5.3 Problems -- 6. Continuous-space Markov processes with jumps -- 6.1 Preliminaries -- 6.1.1 Measures, functions, and transition kernels -- 6.1.2 Markov property, finite-dimensional marginals -- 6.1.3 Semigroup, infinitesimal generator -- 6.2 Markov processes evolving only by isolated jumps -- 6.2.1 Semigroup, infinitesimal generator, and evolution law -- 6.2.2 Construction, simulation, existence -- 6.2.3 Kolmogorov equations, Feynman-Kac formula, bounded generator case -- 6.3 Markov processes following an ordinary differential equation between jumps: PDMP -- 6.3.1 Sample paths, evolution, integro-differential generator -- 6.3.2 Construction, simulation, existence -- 6.3.3 Kolmogorov equations, Feynman-Kac formula -- 6.3.4 Application to kinetic equations -- 6.3.5 Further extensions -- 6.4 Problems -- 7. Discretization of stochastic differential equations -- 7.1 Reminders on Itô's stochastic calculus -- 7.1.1 Stochastic integrals and Itô processes -- 7.1.2 Ito's formula, existence and uniqueness of solutions of stochastic differential equations -- 7.1.3 Markov properties, martingale problems and Fokker- Planck equations -- 7.2 Euler and Milstein schemes -- 7.3 Moments of the solution and of its approximations -- 7.4 Convergence rates in Lp (... ) norm and almost surely -- 7.5 Monte Carlo methods for parabolic partial differential equations -- 7.5.1 The principle of the method -- 7.5.2 Introduction of the error analysis -- 7.6 Optimal convergence rate : the Talay-Tubaro expansion -- 7.7 Romberg-Richardson extrapolation methods -- 7.8 Probabilistic interpretation and estimates for parabolic partial differential equations -- 7.9 Problems -- Part III: Variance reduction, Girsanov's theorem, and stochastic algorithms -- 8. Variance reduction and stochastic differential equations -- 8.1 Preliminary reminders on the Girsanov theorem -- 8.2 Control variates method -- 8.3 Variance reduction for sensitivity analysis -- 8.3.1 Differentiable terminal conditions -- 8.3.2 Non-differentiable terminal conditions -- 8.4 Importance sampling method -- 8.5 Statistical romberg method -- 8.6 Problems -- 9. Stochastic algorithms -- 9.1 Introduction -- 9.2 Study in an idealized framework -- 9.2.1 Definitions -- 9.2.2 The ordinary differential equation method, martingale increments -- 9.2.3 Long-time behavior of the algorithm -- 9.3 Variance reduction for Monte Carlo methods -- 9.3.1 Searching for an importance sampling -- 9.3.2 Variance reduction and stochastic algorithms -- 9.4 Problems -- Appendix solutions to selected problems -- References -- Index.
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 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.
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Holdings
Item type Current library Call number Status Date due Barcode Item holds
Books ISI Library, Kolkata 519.23 G738 (Browse shelf(Opens below)) Available 135327
Books ISI Library, Kolkata 519.23 G738 (Browse shelf(Opens below)) Available 135326
Total holds: 0

Includes bibliographical references and index.

Part I: Principles of Monte Carlo methods --
1. Introduction --
1.1 Why use probabilistic models and simulations? --
1.1.1 What are the reasons for probabilistic models? --
1.1.2 What are the objectives of random simulations? --
1.2 Organization of the monograph --

2. Strong law of large numbers and Monte Carlo methods --
2.1 Strong law of large numbers, examples of Monte Carlo methods --
2.1.1 Strong law of large numbers, almost sure convergence --
2.1.2 Buffon's needle --
2.1.3 Neutron transport simulations --
2.1.4 Stochastic numerical methods for partial differential equations --
2.2 Simulation algorithms for simple probability distributions --
2.2.1 Uniform distributions --
2.2.2 Discrete distributions --
2.2.3 Gaussian distributions --
2.2.4 Cumulative distribution function inversion, exponential distributions --
2.2.5 Rejection method --
2.3 Discrete-time martingales, proof of the SLLN --
2.3.1 Reminders on conditional expectation --
2.3.2 Martingales and sub-martingales, backward martingales --
2.3.3 Proof of the strong law of large numbers --
2.4 Problems --

3. Non-asymptotic error estimates for Monte Carlo methods --
3.1 Convergence in law and characteristic functions --
3.2 Central limit theorem --
3.2.1 Asymptotic confidence intervals --
3.3 Berry-Esseen's theorem --
3.4 Bikelis' theorem --
3.4.1 Absolute confidence intervals --
3.5 Concentration inequalities --
3.5.1 Logarithmic Sobolev inequalities --
3.5.2 Concentration inequalities, absolute confidence intervals --
3.6 Elementary variance reduction techniques --
3.6.1 Control variate --
3.6.2 Importance sampling --
3.7 Problems --

Part II: Exact and approximate simulation of Markov processes --
4. Poisson processes as particular Markov processes --
4.1 Quick introduction to markov processes --
4.1.1 Some issues in Markovian modeling --
4.1.2 Rudiments on processes, sample paths, and laws --
4.2 Poisson processes: characterization, properties --
4.2.1 Point processes and poisson processes --
4.2.2 Simple and strong markov property --
4.2.3 Superposition and decomposition --
4.3 Simulation and approximation --
4.3.1 Simulation of inter-arrivals --
4.3.2 Simulation of independent poisson processes --
4.3.3 Long time or large intensity limit, applications --
4.4 Problems --

5. Discrete-space markov processes --
5.1 Characterization, specification, properties --
5.1.1 Measures, functions, and transition matrices --
5.1.2 Simple and strong Markov property --
5.1.3 Semigroup, infinitesimal generator, and evolution law --
5.2 Constructions, existence, simulation, equations --
5.2.1 Fundamental constructions --
5.2.2 Explosion or existence for a Markov process --
5.2.3 Fundamental simulation, fictitious jump method --
5.2.4 Kolmogorov equations, Feynman-Kac formula --
5.2.5 Generators and semigroups in bounded operator algebras --
5.2.6 A few case studies --
5.3 Problems --

6. Continuous-space Markov processes with jumps --
6.1 Preliminaries --
6.1.1 Measures, functions, and transition kernels --
6.1.2 Markov property, finite-dimensional marginals --
6.1.3 Semigroup, infinitesimal generator --
6.2 Markov processes evolving only by isolated jumps --
6.2.1 Semigroup, infinitesimal generator, and evolution law --
6.2.2 Construction, simulation, existence --
6.2.3 Kolmogorov equations, Feynman-Kac formula, bounded generator case --
6.3 Markov processes following an ordinary differential equation between jumps: PDMP --
6.3.1 Sample paths, evolution, integro-differential generator --
6.3.2 Construction, simulation, existence --
6.3.3 Kolmogorov equations, Feynman-Kac formula --
6.3.4 Application to kinetic equations --
6.3.5 Further extensions --
6.4 Problems --

7. Discretization of stochastic differential equations --
7.1 Reminders on Itô's stochastic calculus --
7.1.1 Stochastic integrals and Itô processes --
7.1.2 Ito's formula, existence and uniqueness of solutions of stochastic differential equations --
7.1.3 Markov properties, martingale problems and Fokker- Planck equations --
7.2 Euler and Milstein schemes --
7.3 Moments of the solution and of its approximations --
7.4 Convergence rates in Lp (... ) norm and almost surely --
7.5 Monte Carlo methods for parabolic partial differential equations --
7.5.1 The principle of the method --
7.5.2 Introduction of the error analysis --
7.6 Optimal convergence rate : the Talay-Tubaro expansion --
7.7 Romberg-Richardson extrapolation methods --
7.8 Probabilistic interpretation and estimates for parabolic partial differential equations --
7.9 Problems --

Part III: Variance reduction, Girsanov's theorem, and stochastic algorithms --
8. Variance reduction and stochastic differential equations --
8.1 Preliminary reminders on the Girsanov theorem --
8.2 Control variates method --
8.3 Variance reduction for sensitivity analysis --
8.3.1 Differentiable terminal conditions --
8.3.2 Non-differentiable terminal conditions --
8.4 Importance sampling method --
8.5 Statistical romberg method --
8.6 Problems --

9. Stochastic algorithms --
9.1 Introduction --
9.2 Study in an idealized framework --
9.2.1 Definitions --
9.2.2 The ordinary differential equation method, martingale increments --
9.2.3 Long-time behavior of the algorithm --
9.3 Variance reduction for Monte Carlo methods --
9.3.1 Searching for an importance sampling --
9.3.2 Variance reduction and stochastic algorithms --
9.4 Problems --
Appendix solutions to selected problems --
References --
Index.

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.

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