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.
Emmanuel Gobet (Ecole Polytechnique - CMAP Palaiseau Cedex France)
Country of Publication:
20 July 2016
Further / Higher Education
A / AS level
Introduction: brief overview of Monte-Carlo methodsA LITTLE HISTORY: FROM THE BUFFON NEEDLE TO NEUTRON TRANSPORT PROBLEM 1: NUMERICAL INTEGRATION: QUADRATURE, MONTE-CARLO, AND QUASI MONTE-CARLO METHODSPROBLEM 2: SIMULATION OF COMPLEX DISTRIBUTIONS: METROPOLIS-HASTINGS ALGORITHM, GIBBS SAMPLER PROBLEM 3: STOCHASTIC OPTIMIZATION: SIMULATED ANNEALING AND ROBBINS-MONRO ALGORITHM TOOLBOX FOR STOCHASTIC SIMULATIONGenerating 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 CONSEQUENCESOTHER ASYMPTOTIC CONTROLS NON-ASYMPTOTIC ESTIMATES EXERCISES Variance reduction ANTITHETIC SAMPLING CONDITIONING AND STRATIFICATION CONTROL VARIATES IMPORTANCE SAMPLING EXERCISES SIMULATION OF LINEAR PROCESSStochastic differential equations and Feynman-Kac formulas THE BROWNIAN MOTION STOCHASTIC INTEGRAL AND ITO FORMULA STOCHASTIC DIFFERENTIAL EQUATIONS PROBABILISTIC REPRESENTATIONS OF PARTIAL DIFFERENTIAL EQUATIONS: FEYNMAN-KAC FORMULAS PROBABILISTIC FORMULAS FOR THE GRADIENTSEXERCISES Euler scheme for stochastic differential equationsDEFINITION 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 PROCESSBackward 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
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.
Reviews for Monte-Carlo Methods and Stochastic Processes: From Linear to Non-Linear
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