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
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.
Review :
"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
