Monte-Carlo Methods and Stochastic Processes: From Linear to Non-Linear, 1st Edition (Hardback) book cover

Monte-Carlo Methods and Stochastic Processes

From Linear to Non-Linear, 1st Edition

By Emmanuel Gobet

Chapman and Hall/CRC

310 pages | 30 B/W Illus.

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Description

Developed from the author’s 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.

Reviews

"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

Table of Contents

Introduction: brief overview of Monte-Carlo methods

A LITTLE HISTORY: FROM THE BUFFON NEEDLE TO NEUTRON TRANSPORT

PROBLEM 1: NUMERICAL INTEGRATION: QUADRATURE, MONTE-CARLO, AND QUASI MONTE-CARLO METHODS

PROBLEM 2: SIMULATION OF COMPLEX DISTRIBUTIONS: METROPOLIS-HASTINGS ALGORITHM, GIBBS SAMPLER

PROBLEM 3: STOCHASTIC OPTIMIZATION: SIMULATED ANNEALING AND ROBBINS-MONRO ALGORITHM

TOOLBOX FOR STOCHASTIC SIMULATION

Generating random variables

PSEUDORANDOM NUMBER GENERATOR

GENERATION OF ONE-DIMENSIONAL RANDOM VARIABLES

ACCEPTANCE-REJECTION METHODS

OTHER TECHNIQUES FOR GENERATING A RANDOM VECTOR

EXERCISES

Convergences and error estimates

LAW OF LARGE NUMBERS

CENTRAL LIMIT THEOREM AND CONSEQUENCES

OTHER ASYMPTOTIC CONTROLS

NON-ASYMPTOTIC ESTIMATES

EXERCISES

Variance reduction

ANTITHETIC SAMPLING

CONDITIONING AND STRATIFICATION

CONTROL VARIATES

IMPORTANCE SAMPLING

EXERCISES

SIMULATION OF LINEAR PROCESS

Stochastic differential equations and Feynman-Kac formulas

THE BROWNIAN MOTION

STOCHASTIC INTEGRAL AND ITÔ FORMULA

STOCHASTIC DIFFERENTIAL EQUATIONS

PROBABILISTIC REPRESENTATIONS OF PARTIAL DIFFERENTIAL EQUATIONS: FEYNMAN-KAC FORMULAS

PROBABILISTIC FORMULAS FOR THE GRADIENTS

EXERCISES

Euler scheme for stochastic differential equations

DEFINITION AND SIMULATION

STRONG CONVERGENCE

WEAK CONVERGENCE

SIMULATION OF STOPPED PROCESSES

EXERCISES

Statistical error in the simulation of stochastic differential equations

ASYMPTOTIC ANALYSIS: NUMBER OF SIMULATIONS AND TIME STEP

NON-ASYMPTOTIC ANALYSIS OF THE STATISTICAL ERROR IN EULER SCHEME

MULTI-LEVEL METHOD

UNBIASED SIMULATION USING A RANDOMIZED MULTI-LEVEL METHOD

VARIANCE REDUCTION METHODS

EXERCISES

SIMULATION OF NONLINEAR PROCESS

Backward stochastic differential equations

EXAMPLES

FEYNMAN-KAC FORMULAS

TIME DISCRETISATION AND DYNAMIC PROGRAMMING EQUATION

OTHER DYNAMIC PROGRAMMING EQUATIONS

ANOTHER PROBABILISTIC REPRESENTATION VIA BRANCHING PROCESSES

EXERCISES

Simulation by empirical regression

THE DIFFICULTIES OF A NAIVE APPROACH

APPROXIMATION OF CONDITIONAL EXPECTATIONS BY LEAST SQUARES METHODS

APPLICATION TO THE RESOLUTION OF THE DYNAMIC PROGRAMMING EQUATION BY EMPIRICAL REGRESSION

EXERCISES

Interacting particles and non-linear equations in the McKean sense

HEURISTICS

EXISTENCE AND UNIQUENESS OF NON-LINEAR DIFFUSIONS

CONVERGENCE OF THE SYSTEM OF INTERACTING DIFFUSIONS, PROPAGATION OF CHAOS, SIMULATION

Appendix: Reminders and complementary results

ABOUT CONVERGENCES

SEVERAL USEFUL INEQUALITIES

Index

About the Author

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.

Subject Categories

BISAC Subject Codes/Headings:
MAT029010
MATHEMATICS / Probability & Statistics / Bayesian Analysis