Stochastic simulation and Monte Carlo methods mathematical foundations of stochastic simulation
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| Format: | Buch |
| Sprache: | English |
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Berlin [u.a.]
Springer
2013
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| Schriftenreihe: | Stochastic modelling and applied probability
68 |
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| 100 | 1 | |a Graham, Carl |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Stochastic simulation and Monte Carlo methods |b mathematical foundations of stochastic simulation |c Carl Graham ; Denis Talay |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2013 | |
| 300 | |a XVI, 260 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Stochastic modelling and applied probability |v 68 | |
| 500 | |a Literaturverz. S. 253 - 255 | ||
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| 689 | 0 | 0 | |a Monte-Carlo-Simulation |0 (DE-588)4240945-7 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Talay, Denis |e Verfasser |0 (DE-588)114468907 |4 aut | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-3-642-39363-1 |
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Datensatz im Suchindex
| _version_ | 1819609914156253184 |
|---|---|
| adam_text | Contents
Part I Principles of Monte Carlo Methods
1
Introduction
.....................-,.......... 3
1.1
Why Use Probabilistic Models and Simulations?
.......... 3
1.1.1
What Are the Reasons for Probabilistic Models?
...... 4
1.1.2
What Are the Objectives of Random Simulations?
..... 6
1.2
Organization of the Monograph
................... 9
2
Strong Law of Large Numbers and Monte Carlo Methods
...... 13
2.1
Strong Law of Large Numbers, Examples of Monte Carlo Methods
13
2.1.1
Strong Law of Large Numbers, Almost Sure Convergence
. 13
2.1.2
Buffon s Needle
....................... 15
2.1.3
Neutron Transport Simulations
............... 15
2.1.4
Stochastic Numerical Methods for Partial Differential
Equations
.......................... 17
2.2
Simulation Algorithms for Simple Probability Distributions
.... 18
2.2.1
Uniform Distributions
.................... 19
2.2.2
Discrete Distributions
.................... 20
2.2.3
Gaussian Distributions
.................... 21
2.2.4
Cumulative Distribution Function Inversion, Exponential
Distributions
......................... 22
2.2.5
Rejection Method
...................... 23
2.3
Discrete-Time Martingales, Proof of the
S
LLN
........... 25
2.3.1
Reminders on Conditional Expectation
........... 25
2.3.2
Martingales and Sub-martingales, Backward Martingales
. 27
2.3.3
Proof of the Strong Law of Large Numbers
......... 30
2.4
Problems
............................... 33
3
Non-asymptotic Error Estimates for Monte Carlo Methods
..... 37
3.1
Convergence in Law and Characteristic Functions
......... 37
3.2
Central Limit Theorem
....................... 40
3.2.1
Asymptotic Confidence Intervals
.............. 41
3.3
Berry-Esseen s Theorem
...................... 42
3.4
Bikelis
Theorem
.......................... 45
3.4.1
Absolute Confidence Intervals
................ 45
3.5
Concentration Inequalities
...................... 47
3.5.1
Logarithmic Sobolev Inequalities
.............. 48
3.5.2
Concentration Inequalities, Absolute Confidence Intervals
. 50
3.6
Elementary Variance Reduction Techniques
............ 54
3.6.1
Control
Variate
........................ 54
3.6.2
Importance Sampling
.................... 55
3.7
Problems
............................... 60
Part II Exact and Approximate Simulation of Markov Processes
4
Poisson
Processes as Particular Markov Processes
........... 67
4.1
Quick Introduction to Markov Processes
.............. 67
4.1.1
Some Issues in Markovian Modeling
............ 67
4.1.2
Rudiments on Processes, Sample Paths, and Laws
..... 68
4.2
Poisson
Processes: Characterization, Properties
........... 69
4.2.1
Point Processes and
Poisson
Processes
........... 69
4.2.2
Simple and Strong Markov Property
............ 75
4.2.3
Superposition and Decomposition
.............. 77
4.3
Simulation and Approximation
................... 80
4.3.1
Simulation of Inter-arrivals
................. 80
4.3.2
Simulation of Independent
Poisson
Processes
........ 81
4.3.3
Long Time or Large Intensity Limit, Applications
..... 82
4.4
Problems
............................... 85
5
Discrete-Space Markov Processes
.................... 89
5.1
Characterization, Specification, Properties
............. 89
5.1.1
Measures, Functions, and Transition Matrices
....... 89
5.1.2
Simple and Strong Markov Property
............ 91
5.1.3
Semigroup, Infinitesimal Generator, and Evolution Law
. . 95
5.2
Constructions, Existence, Simulation, Equations
.......... 99
5.2.1
Fundamental Constructions
................. 99
5.2.2
Explosion or Existence for a Markov Process
........ 101
5.2.3
Fundamental Simulation, Fictitious Jump Method
..... 103
5.2.4
Kolmogorov Equations, Feynman—
Кас
Formula
...... 105
5.2.5
Generators and Semigroups in Bounded Operator Algebras
107
5.2.6
A Few Case Studies
..................... 112
5.3
Problems
............................... 115
6
Continuous-Space Markov Processes with Jumps
........... 121
6.1
Preliminaries
............................. 121
6.1.1
Measures, Functions, and Transition Kernels
........ 121
6.1.2
Markov Property, Finite-Dimensional Marginals
...... 123
6.1.3
Semigroup, Infinitesimal Generator
............. 125
6.2
Markov Processes Evolving Only by Isolated Jumps
........ 126
6.2.1
Semigroup, Infinitesimal Generator, and Evolution Law
. . 126
6.2.2
Construction, Simulation, Existence
............. 130
6.2.3
Kolmogorov Equations, Feynman-Kac Formula,
Bounded
Generator Case
........................ 133
6.3
Markov Processes Following an Ordinary Differential Equation
Between Jumps: PDMP
....................... 136
6.3.1
Sample Paths, Evolution, Integro-Differential Generator
. . 136
6.3.2
Construction, Simulation, Existence
............. 141
6.3.3
Kolmogorov Equations, Feynman—
Кас
Formula
...... 144
6.3.4
Application to Kinetic Equations
.............. 146
6.3.5
Further Extensions
...................... 149
6.4
Problems
............................... 151
7
Discretization of Stochastic Differential Equations
.......... 155
7.1
Reminders on
Itô s
Stochastic Calculus
............... 155
7.1.1
Stochastic Integrals and
Itô
Processes
............ 155
7.1.2
Itô s
Formula, Existence and Uniqueness of Solutions
of Stochastic Differential Equations
............. 160
7.1.3
Markov Properties, Martingale Problems and Fokker-
Planck Equations
...................... 162
7.2
Euler
and Milstein Schemes
..................... 165
7.3
Moments of the Solution and of Its Approximations
........ 168
7.4
Convergence Rates in LP(Q) Norm and Almost Surely
...... 173
7.5
Monte Carlo Methods for Parabolic Partial Differential Equations
. 176
7.5.1
The Principle of the Method
................. 176
7.5.2
Introduction of the Error Analysis
.............. 177
7.6
Optimal Convergence Rate: The Talay—Tubaro Expansion
..... 180
7.7
Romberg-Richardson Extrapolation Methods
............ 185
7.8
Probabilistic Interpretation and Estimates for Parabolic Partial
Differential Equations
........................ 186
7.9
Problems
............................... 191
Part III Variance Reduction, Girsanov s Theorem, and Stochastic
Algorithms
8
Variance Reduction and Stochastic Differential Equations
...... 199
8.1
Preliminary Reminders on the Girsanov Theorem
......... 199
8.2
Control
Variâtes
Method
....................... 200
8.3
Variance Reduction for Sensitivity Analysis
............ 202
8.3.1
Differentiable Terminal Conditions
............. 202
8.3.2
Non-differentiable Terminal Conditions
........... 204
8.4
Importance Sampling Method
.................... 206
8.5
Statistical Romberg Method
..................... 209
8.6
Problems
............................... 210
9
Stochastic Algorithms
.......................... 213
9.1
Introduction
............................. 213
9.2
Study in an Idealized Framework
.................. 214
9.2.1
Definitions
.......................... 214
9.2.2
The Ordinary Differential Equation Method, Martingale
Increments
.......................... 216
9.2.3
Long-Time Behavior of the Algorithm
........... 217
9.3
Variance Reduction for Monte Carlo Methods
........... 221
9.3.1
Searching for an Importance Sampling
.......... . 221
9.3.2
Variance Reduction and Stochastic Algorithms
....... 223
9.4
Problems
............................... 225
Appendix Solutions to Selected Problems
.................231
References
...................................253
Index
......................................257
Stochastic Modelling and Applied Probability
Carl Graham -DenisTalay
Stochastic Simulation and Monte Carlo Methods
Mathematical Foundations of Stochastic Simulation
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
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 appli¬
cations, 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.
|
| any_adam_object | 1 |
| author | Graham, Carl Talay, Denis |
| author_GND | (DE-588)114468907 |
| author_facet | Graham, Carl Talay, Denis |
| author_role | aut aut |
| author_sort | Graham, Carl |
| author_variant | c g cg d t dt |
| building | Verbundindex |
| bvnumber | BV041108154 |
| classification_rvk | SK 820 |
| ctrlnum | (OCoLC)864489387 (DE-599)DNB1035564386 |
| dewey-full | 518.282 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 518 - Numerical analysis |
| dewey-raw | 518.282 |
| dewey-search | 518.282 |
| dewey-sort | 3518.282 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV041108154 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-24T03:23:04Z |
| institution | BVB |
| isbn | 9783642393624 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-026084419 |
| oclc_num | 864489387 |
| open_access_boolean | |
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| physical | XVI, 260 S. |
| publishDate | 2013 |
| publishDateSearch | 2013 |
| publishDateSort | 2013 |
| publisher | Springer |
| record_format | marc |
| series | Stochastic modelling and applied probability |
| series2 | Stochastic modelling and applied probability |
| spellingShingle | Graham, Carl Talay, Denis Stochastic simulation and Monte Carlo methods mathematical foundations of stochastic simulation Stochastic modelling and applied probability Monte-Carlo-Simulation (DE-588)4240945-7 gnd |
| subject_GND | (DE-588)4240945-7 |
| title | Stochastic simulation and Monte Carlo methods mathematical foundations of stochastic simulation |
| title_auth | Stochastic simulation and Monte Carlo methods mathematical foundations of stochastic simulation |
| title_exact_search | Stochastic simulation and Monte Carlo methods mathematical foundations of stochastic simulation |
| title_full | Stochastic simulation and Monte Carlo methods mathematical foundations of stochastic simulation Carl Graham ; Denis Talay |
| title_fullStr | Stochastic simulation and Monte Carlo methods mathematical foundations of stochastic simulation Carl Graham ; Denis Talay |
| title_full_unstemmed | Stochastic simulation and Monte Carlo methods mathematical foundations of stochastic simulation Carl Graham ; Denis Talay |
| title_short | Stochastic simulation and Monte Carlo methods |
| title_sort | stochastic simulation and monte carlo methods mathematical foundations of stochastic simulation |
| title_sub | mathematical foundations of stochastic simulation |
| topic | Monte-Carlo-Simulation (DE-588)4240945-7 gnd |
| topic_facet | Monte-Carlo-Simulation |
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