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 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 ITO 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
(0 Comentarios)
