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

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