Muutke küpsiste eelistusi

Stochastic Simulation and Monte Carlo Methods: Mathematical Foundations of Stochastic Simulation 2013 ed. [Kõva köide]

,
  • Formaat: Hardback, 260 pages, kõrgus x laius: 235x155 mm, kaal: 5443 g, XVI, 260 p., 1 Hardback
  • Sari: Stochastic Modelling and Applied Probability 68
  • Ilmumisaeg: 29-Jul-2013
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • ISBN-10: 3642393624
  • ISBN-13: 9783642393624
Teised raamatud teemal:
  • Kõva köide
  • Hind: 67,23 €*
  • * hind on lõplik, st. muud allahindlused enam ei rakendu
  • Tavahind: 79,09 €
  • Säästad 15%
  • Raamatu kohalejõudmiseks kirjastusest kulub orienteeruvalt 2-4 nädalat
  • Kogus:
  • Lisa ostukorvi
  • Tasuta tarne
  • Tellimisaeg 2-4 nädalat
  • Lisa soovinimekirja
Stochastic Simulation and Monte Carlo Methods: Mathematical Foundations of Stochastic Simulation 2013 ed.Suurem pilt
  • Formaat: Hardback, 260 pages, kõrgus x laius: 235x155 mm, kaal: 5443 g, XVI, 260 p., 1 Hardback
  • Sari: Stochastic Modelling and Applied Probability 68
  • Ilmumisaeg: 29-Jul-2013
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • ISBN-10: 3642393624
  • ISBN-13: 9783642393624
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9783642393624.html
Märksõnad:
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 Itō 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.
Part I Principles of Monte Carlo Methods
1 Introduction
3(10)
1.1 Why Use Probabilistic Models and Simulations?
3(6)
1.1.1 What Are the Reasons for Probabilistic Models?
4(2)
1.1.2 What Are the Objectives of Random Simulations?
6(3)
1.2 Organization of the Monograph
9(4)
2 Strong Law of Large Numbers and Monte Carlo Methods
13(24)
2.1 Strong Law of Large Numbers, Examples of Monte Carlo Methods
13(5)
2.1.1 Strong Law of Large Numbers, Almost Sure Convergence
13(2)
2.1.2 Buffon's Needle
15(1)
2.1.3 Neutron Transport Simulations
15(2)
2.1.4 Stochastic Numerical Methods for Partial Differential Equations
17(1)
2.2 Simulation Algorithms for Simple Probability Distributions
18(7)
2.2.1 Uniform Distributions
19(1)
2.2.2 Discrete Distributions
20(1)
2.2.3 Gaussian Distributions
21(1)
2.2.4 Cumulative Distribution Function Inversion, Exponential Distributions
22(1)
2.2.5 Rejection Method
23(2)
2.3 Discrete-Time Martingales, Proof of the SLLN
25(8)
2.3.1 Reminders on Conditional Expectation
25(2)
2.3.2 Martingales and Sub-martingales, Backward Martingales
27(3)
2.3.3 Proof of the Strong Law of Large Numbers
30(3)
2.4 Problems
33(4)
3 Non-asymptotic Error Estimates for Monte Carlo Methods
37(30)
3.1 Convergence in Law and Characteristic Functions
37(3)
3.2 Central Limit Theorem
40(2)
3.2.1 Asymptotic Confidence Intervals
41(1)
3.3 Berry-Esseen's Theorem
42(3)
3.4 Bikelis' Theorem
45(2)
3.4.1 Absolute Confidence Intervals
45(2)
3.5 Concentration Inequalities
47(7)
3.5.1 Logarithmic Sobolev Inequalities
48(2)
3.5.2 Concentration Inequalities, Absolute Confidence Intervals
50(4)
3.6 Elementary Variance Reduction Techniques
54(6)
3.6.1 Control Variate
54(1)
3.6.2 Importance Sampling
55(5)
3.7 Problems
60(7)
Part II Exact and Approximate Simulation of Markov Processes
4 Poisson Processes as Particular Markov Processes
67(22)
4.1 Quick Introduction to Markov Processes
67(2)
4.1.1 Some Issues in Markovian Modeling
67(1)
4.1.2 Rudiments on Processes, Sample Paths, and Laws
68(1)
4.2 Poisson Processes: Characterization, Properties
69(11)
4.2.1 Point Processes and Poisson Processes
69(6)
4.2.2 Simple and Strong Markov Property
75(2)
4.2.3 Superposition and Decomposition
77(3)
4.3 Simulation and Approximation
80(5)
4.3.1 Simulation of Inter-arrivals
80(1)
4.3.2 Simulation of Independent Poisson Processes
81(1)
4.3.3 Long Time or Large Intensity Limit, Applications
82(3)
4.4 Problems
85(4)
5 Discrete-Space Markov Processes
89(32)
5.1 Characterization, Specification, Properties
89(10)
5.1.1 Measures, Functions, and Transition Matrices
89(2)
5.1.2 Simple and Strong Markov Property
91(4)
5.1.3 Semigroup, Infinitesimal Generator, and Evolution Law
95(4)
5.2 Constructions, Existence, Simulation, Equations
99(16)
5.2.1 Fundamental Constructions
99(2)
5.2.2 Explosion or Existence for a Markov Process
101(2)
5.2.3 Fundamental Simulation, Fictitious Jump Method
103(2)
5.2.4 Kolmogorov Equations, Feynman-Kac Formula
105(2)
5.2.5 Generators and Semigroups in Bounded Operator Algebras
107(5)
5.2.6 A Few Case Studies
112(3)
5.3 Problems
115(6)
6 Continuous-Space Markov Processes with Jumps
121(34)
6.1 Preliminaries
121(5)
6.1.1 Measures, Functions, and Transition Kernels
121(2)
6.1.2 Markov Property, Finite-Dimensional Marginals
123(2)
6.1.3 Semigroup, Infinitesimal Generator
125(1)
6.2 Markov Processes Evolving Only by Isolated Jumps
126(10)
6.2.1 Semigroup, Infinitesimal Generator, and Evolution Law
126(4)
6.2.2 Construction, Simulation, Existence
130(3)
6.2.3 Kolmogorov Equations, Feynman-Kac Formula, Bounded Generator Case
133(3)
6.3 Markov Processes Following an Ordinary Differential Equation Between Jumps: PDMP
136(15)
6.3.1 Sample Paths, Evolution, Integro-Differential Generator
136(5)
6.3.2 Construction, Simulation, Existence
141(3)
6.3.3 Kolmogorov Equations, Feynman-Kac Formula
144(2)
6.3.4 Application to Kinetic Equations
146(3)
6.3.5 Further Extensions
149(2)
6.4 Problems
151(4)
7 Discretization of Stochastic Differential Equations
155(44)
7.1 Reminders on Ito's Stochastic Calculus
155(10)
7.1.1 Stochastic Integrals and Ito Processes
155(5)
7.1.2 Ito's Formula, Existence and Uniqueness of Solutions of Stochastic Differential Equations
160(2)
7.1.3 Markov Properties, Martingale Problems and Fokker-Planck Equations
162(3)
7.2 Euler and Milstein Schemes
165(3)
7.3 Moments of the Solution and of Its Approximations
168(5)
7.4 Convergence Rates in Lp(Ω) Norm and Almost Surely
173(3)
7.5 Monte Carlo Methods for Parabolic Partial Differential Equations
176(4)
7.5.1 The Principle of the Method
176(1)
7.5.2 Introduction of the Error Analysis
177(3)
7.6 Optimal Convergence Rate: The Talay-Tubaro Expansion
180(5)
7.7 Romberg-Richardson Extrapolation Methods
185(1)
7.8 Probabilistic Interpretation and Estimates for Parabolic Partial Differential Equations
186(5)
7.9 Problems
191(8)
Part III Variance Reduction, Girsanov's Theorem, and Stochastic Algorithms
8 Variance Reduction and Stochastic Differential Equations
199(14)
8.1 Preliminary Reminders on the Girsanov Theorem
199(1)
8.2 Control Variates Method
200(2)
8.3 Variance Reduction for Sensitivity Analysis
202(4)
8.3.1 Differentiable Terminal Conditions
202(2)
8.3.2 Non-differentiable Terminal Conditions
204(2)
8.4 Importance Sampling Method
206(3)
8.5 Statistical Romberg Method
209(1)
8.6 Problems
210(3)
9 Stochastic Algorithms
213(18)
9.1 Introduction
213(1)
9.2 Study in an Idealized Framework
214(7)
9.2.1 Definitions
214(2)
9.2.2 The Ordinary Differential Equation Method, Martingale Increments
216(1)
9.2.3 Long-Time Behavior of the Algorithm
217(4)
9.3 Variance Reduction for Monte Carlo Methods
221(4)
9.3.1 Searching for an Importance Sampling
221(2)
9.3.2 Variance Reduction and Stochastic Algorithms
223(2)
9.4 Problems
225(6)
Appendix Solutions to Selected Problems 231(22)
References 253(4)
Index 257
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 stochastic processes, stochastic 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 stochastic modelling, numerical probability, stochastic analysis of partial differential equations and financial mathematics.