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.
Table of Contents
Introduction: brief overview of Monte-Carlo methods. TOOLBOX FOR STOCHASTIC SIMULATION: Generating random variables. Convergences and error estimates. Variance reduction. SIMULATION OF LINEAR PROCESS: Stochastic differential equations and Feynman-Kac formulas. Euler scheme for stochastic differential equations. Statistical error in the simulation of stochastic differential equations. SIMULATION OF NONLINEAR PROCESS: Backward stochastic differential equations. Simulation by empirical regression. Interacting particles and non-linear equations in the McKean sense. Appendix. 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.
"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