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

Emmanuel Gobet



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.

Features

• Covers a broad spectrum of advanced and modern tools of probability, statistics, and PDEs, along with systematic computational concerns regarding numerical efficiency
• Emphasizes the main algorithms and most important convergence phenomena
• Encourages students to implement the algorithms to improve their own computational intuition
• Presents simple proofs of results

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


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