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E-raamat: Monte-Carlo Methods and Stochastic Processes: From Linear to Non-Linear

(Ecole Polytechnique - CMAP, Palaiseau Cedex, France)
  • Formaat: PDF+DRM
  • Ilmumisaeg: 15-Sep-2016
  • Kirjastus: Chapman & Hall/CRC
  • Keel: eng
  • ISBN-13: 9781498746236
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Monte-Carlo Methods and Stochastic Processes: From Linear to Non-LinearSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 15-Sep-2016
  • Kirjastus: Chapman & Hall/CRC
  • Keel: eng
  • ISBN-13: 9781498746236
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Developed from the authors course at the Ecole Polytechnique, Monte-Carlo Methods and Stochastic Processes: From Linear to Non-Linear focuses on the simulation of stochastic processes in continuous time and their link with partial differential equations (PDEs). It covers linear and nonlinear problems in biology, finance, geophysics, mechanics, chemistry, and other application areas. The text also thoroughly develops the problem of numerical integration and computation of expectation by the Monte-Carlo method.

The book begins with a history of Monte-Carlo methods and an overview of three typical Monte-Carlo problems: numerical integration and computation of expectation, simulation of complex distributions, and stochastic optimization. The remainder of the text is organized in three parts of progressive difficulty. The first part presents basic tools for stochastic simulation and analysis of algorithm convergence. The second part describes Monte-Carlo methods for the simulation of stochastic differential equations. The final part discusses the simulation of non-linear dynamics.

Arvustused

"Emmanuel Gobet has successfully put together the modern tools for Monte Carlo simulations of continuous-time stochastic processes. He takes us from classical methods to new challenging nonlinear situations from various fields of applications, and rightly explains that naive approaches can be misleading. The book is self-contained, rigorous and definitely a must-have for anyone performing simulations and worrying about quantifying statistical errors."

- Jean-Pierre Fouque, Director of the Center for Financial Mathematics and Actuarial Research, University of California, Santa Barbara

"This book is a modern and broad presentation of Monte Carlo techniques related to the simulation of several types of continuous-time stochastic processes. The discussion is pedagogical (the book originates from a course on Monte Carlo methods); in particular, each chapter contains exercises. Nevertheless, detailed and rigorous proofs of difficult results are provided; generalizations, which often deal with current research questions, are mentioned. Both theoretical and practical aspects are considered.

The book is divided into three parts. The third one, which treats the simulation of some non-linear processes in connexion with non-linear PDEs, certainly provides a nice and original contribution, and concerns topics which have been investigated only very recently."

- Charles-Edouard Brehier, Mathematical Reviews, June 2017 "This book is a modern and broad presentation of Monte Carlo techniques related to the simulation of several types of continuous-time stochastic processes. The discussion is pedagogical (the book originates from a course on Monte Carlo methods); in particular, each chapter contains exercises. Nevertheless, detailed and rigorous proofs of difficult results are provided; generalizations, which often deal with current research questions, are mentioned. Both theoretical and practical aspects are considered.

The book is divided into three parts. The third one, which treats the simulation of some non-linear processes in connexion with non-linear PDEs, certainly provides a nice and original contribution, and concerns topics which have been investigated only very recently."

- Charles-Edouard Brehier, Mathematical Reviews, June 2017

Introduction: Brief Overview Of Monte-Carlo Methods 1(28)
A Little History: From The Buffon Needle To Neutron Transport
3(7)
Three Typical Problems In Random Simulation
10(1)
Problem 1 Numerical Integration: Quadrature, Monte-Carlo And Quasi Monte-Carlo Methods
10(8)
Problem 2 Simulation Of Complex Distributions: Metropolis-Hastings Algorithm, Gibbs Sampler
18(5)
Problem 3 Stochastic Optimization: Simulated Annealing And The Robbinsmonro Algorithm
23(6)
PART A TOOLBOX FOR STOCHASTIC SIMULATION
29(86)
Chapter 1 Generating random variables
31(18)
1.1 Pseudorandom Number Generator
31(1)
1.2 Generation Of One-Dimensional Random Variables
32(5)
1.2.1 Inversion method
32(4)
1.2.2 Gaussian variables
36(1)
1.3 Acceptance-Rejection Methods
37(5)
1.3.1 Generation of conditional distribution
37(1)
1.3.2 Generation of (non-conditional) distributions by the acceptance-rejection method
38(2)
1.3.3 Ratio-of-uniforms method
40(2)
1.4 Other Techniques For Generating A Random Vecto
42(5)
1.4.1 The Gaussian vector
43(1)
1.4.2 Modeling of dependence using copulas
44(3)
1.5 Exercises
47(2)
Chapter 2 Convergences and error estimates
49(40)
2.1 Law Of Large Numbers
49(3)
2.2 Central Limit Theorem And Consequences
52(13)
2.2.1 Central limit theorem in dimension 1 and beyond
52(2)
2.2.2 Asymptotic confidence regions and intervals
54(2)
2.2.3 Application to the evaluation of a function of E(X)
56(5)
2.2.4 Applications in the evaluation of sensitivity of expectations
61(4)
2.3 Other Asymptotic Controls
65(2)
2.3.1 Berry-Essen bounds and Edgeworth expansions
65(1)
2.3.2 Law of iterated logarithm
66(1)
2.3.3 "Almost sure" central limit theorem
66(1)
2.4 Non-Asymptotic Estimates
67(18)
2.4.1 About exponential inequalities
67(2)
2.4.2 Concentration inequalities in the case of bounded random variables
69(1)
2.4.3 Uniform concentration inequalities
70(7)
2.4.4 Concentration inequalities in the case of Gaussian noise
77(8)
2.5 Exercises
85(4)
Chapter 3 Variance reduction
89(26)
3.1 Antithetic Sampling
89(3)
3.2 Conditioning And Stratification
92(2)
3.2.1 Conditioning technique
92(1)
3.2.2 Stratification technique
92(2)
3.3 Control Variates
94(3)
3.3.1 Concept
94(1)
3.3.2 Optimal choice
95(2)
3.4 Importance Sampling
97(15)
3.4.1 Changes of probability measure: basic notions and applications to Monte-Carlo methods
97(6)
3.4.2 Changes of probability measure by affine transformations
103(4)
3.4.3 Change of probability measure by Esscher transform
107(3)
3.4.4 Adaptive methods
110(2)
3.5 Exercises
112(3)
PART B SIMULATION OF LINEAR PROCESSES
115(96)
Chapter 4 Stochastic differential equations and Feynman-Kac formulas
117(46)
4.1 Brownian Motion
119(13)
4.1.1 A brief history
119(1)
4.1.2 Definition
119(5)
4.1.3 Simulation
124(4)
4.1.4 Heat equation
128(3)
4.1.5 Quadratic variation
131(1)
4.2 Stochastic Integral And Ito Formula
132(6)
4.2.1 Filtration and stopping times
133(1)
4.2.2 Stochastic integral and its properties
134(3)
4.2.3 Ito process and Ito formula
137(1)
4.3 Stochastic Differential Equations
138(4)
4.3.1 Definition, existence, uniqueness
138(1)
4.3.2 Flow property and Markov property
139(1)
4.3.3 Examples
139(3)
4.4 Probabilistic Representations Of Partial Differential Equations: Feynman-Kac Formulas
142(11)
4.4.1 Infinitesimal generator
142(2)
4.4.2 Linear parabolic partial differential equation with Cauchy condition
144(4)
4.4.3 Linear elliptic partial differential equation
148(1)
4.4.4 Linear parabolic partial differential equation with Cauchy-Dirichlet condition
149(4)
4.4.5 Linear elliptic partial differential equation with Dirichlet condition
153(1)
4.5 Probabilistic Formulas For The Gradients
153(3)
4.5.1 Pathwise differentiation method
154(1)
4.5.2 Likelihood method
155(1)
4.6 Exercises
156(7)
Chapter 5 Euler scheme for stochastic differential equations
163(28)
5.1 Definition And Simulation
164(6)
5.1.1 Definition as an Ito process, quadratic moments
164(2)
5.1.2 Simulation
166(2)
5.1.3 Application to computation of diffusion expectation: discretization error and statistical error
168(2)
5.2 Strong Convergence
170(3)
5.3 Weak Convergence
173(5)
5.3.1 Convergence at order 1
173(3)
5.3.2 Extensions
176(2)
5.4 Simulation Of Stopped Processes
178(8)
5.4.1 Discrete approximation of exit time
179(2)
5.4.2 Brownian bridge method
181(3)
5.4.3 Boundary shifting method
184(2)
5.5 Exercises
186(5)
Chapter 6 Statistical error in the simulation of stochastic differential equations
191(20)
6.1 Asymptotic Analysis: Number Of Simulations And Time Step
191(3)
6.2 Non-Asymptotic Analysis Of The Statistical Error In The Euler Scheme
194(3)
6.3 Multi-Level Method
197(5)
6.4 Unbiased Simulation Using A Randomized Multi-Level Method
202(4)
6.5 Variance Reduction Methods
206(2)
6.5.1 Control variates
206(1)
6.5.2 Importance sampling
207(1)
6.6 Exercises
208(3)
PART C SIMULATION OF NON-LINEAR PROCESSES
211(74)
Chapter 7 Backward stochastic differential equations
213(28)
7.1 Examples
214(7)
7.1.1 Examples coming from reaction-diffusion equations
214(3)
7.1.2 Examples coming from stochastic modeling
217(4)
7.2 Feynman-Kac Formulas
221(6)
7.2.1 A general result
221(3)
7.2.2 Toy model
224(3)
7.3 Time Discretization And Dynamic Programming Equation
227(4)
7.3.1 Discretization of the problem
227(1)
7.3.2 Error analysis
228(3)
7.4 Other Dynamic Programming Equations
231(2)
7.5 Another Probabilistic Representation Via Branching Processes
233(3)
7.6 Exercises
236(5)
Chapter 8 Simulation by empirical regression
241(32)
8.1 The Difficulties Of A Naive Approach
241(3)
8.2 Approximation Of Conditional Expectations By Least Squares Methods
244(13)
8.2.1 Empirical regression
245(1)
8.2.2 SVD method
246(3)
8.2.3 Example of approximation space: the local polynomials
249(1)
8.2.4 Error estimations, robust with respect to the model
250(2)
8.2.5 Adjustment of the parameters in the case of local polynomials
252(2)
8.2.6 Proof of the error estimations
254(3)
8.3 Application To The Resolution Of The Dynamic Programming Equation By Empirical Regression
257(10)
8.3.1 Learning sample and approximation space
257(1)
8.3.2 Calculation of the empirical regression functions
258(2)
8.3.3 Equation of the error propagation
260(6)
8.3.4 Optimal adjustment of the convergence parameters in the case of local polynomials
266(1)
8.4 Exercises
267(6)
Chapter 9 Interacting particles and non-linear equations in the McKean sense
273(12)
9.1 Heuristics
273(5)
9.1.1 Macroscopic scale versus microscopic scale
273(2)
9.1.2 Examples and applications
275(3)
9.2 Existence And Uniqueness Of Non-Linear Diffusions
278(1)
9.3 Convergence Of The System Of Interacting Diffusions, Propagation Of Chaos And Simulation
279(6)
APPENDIX A Reminders and complementary results
285(8)
A.1 About Convergences
285(1)
A.1.1 Convergence a.s., in probability and in L1
285(1)
A.1.2 Convergence in distribution
286(1)
A.2 Several Useful Inequalities
287(1)
A.2.1 Inequalities for moments
287(3)
A.2.2 Inequalities in the deviation probabilities
290(3)
Bibliography 293(14)
Index 307
Emmanuel Gobet is a professor of applied mathematics at Ecole Polytechnique. His research interests include algorithms of probabilistic type and stochastic approximations, financial mathematics, Malliavin calculus and stochastic analysis, Monte Carlo simulations, statistics for stochastic processes, and statistical learning.
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