Monte-Carlo Methods and Stochastic Processes From Linear to Non-Linear [Hardcover]

Developed from the authors course at the Ecole Polytechnique,Monte-Carlo Methods and Stochastic Processes: From Linear to Non-Linearfocuses 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.

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
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