Monte-Carlo Methods and Stochastic Processes From Linear to Non-Linear
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.
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.
Date de parution : 09-2020
15.6x23.4 cm
Date de parution : 07-2016
15.6x23.4 cm
Thèmes de Monte-Carlo Methods and Stochastic Processes :
Mots-clés :
Backward Stochastic Differential Equation; Ordinary Differential Equation; numerical integration; Quasi Monte Carlo Methods; partial differential equations; Cumulative Distribution Function; linear and nonlinear problems; Independent Standard Gaussian Random Variables; simulation of stochastic processes; Conditional Expectation; computation of expectation; Stochastic Differential Equations; stochastic optimization; Euler Scheme; algorithm convergence; Archimedean Copula; simulation of non-linear dynamics; Acceptance Rejection Method; Dynamic Programming Equation; Non-linear Diffusion; Gaussian Vector; Stochastic Integral; Gronwall Lemma; Monte Carlo Method; Robbins Monro Algorithm; Low Discrepancy Sequence; Gaussian Copula; Local Polynomials; Non-linear PDE; Linear Backward Stochastic Differential Equation; Brownian Bridge; Average Quadratic Error; Logarithmic Sobolev Inequality