Stochastic Simulation and Monte Carlo Methods

Abstract

In various scientific and industrial fields, stochastic simulationsare taking on a new importance. This is due to the increasing powerof computers and practitioners' aim to simulate more and more complexsystems, and thus use random parameters as well as random noisesto model the parametric uncertainties and the lack of knowledge onthe physics of these systems. The error analysis of these computationsis a highly complex mathematical undertaking. Approaching these issues,the authors present stochastic numerical methods and prove accurateconvergence rate estimates in terms of their numerical parameters(number of simulations, time discretization steps). As a result,the book is a self-contained and rigorous study of the numericalmethods within a theoretical framework. After briefly reviewing thebasics, the authors first introduce fundamental notions in stochasticcalculus and continuous-time martingale theory, then develop theanalysis of pure-jump Markov processes, Poisson processes, and stochasticdifferential equations. In particular, they review the essentialproperties of Itô integrals and prove fundamental results on theprobabilistic analysis of parabolic partial differential equations.These results in turn provide the basis for developing stochasticnumerical methods, both from an algorithmic and theoretical pointof view.The book combines advanced mathematical tools, theoretical analysisof stochastic numerical methods, and practical issues at a high level,so as to provide optimal results on the accuracy of Monte Carlo simulationsof stochastic processes. It is intended for master and Ph.D. studentsin the field of stochastic processes and their numerical applications,as well as for physicists, biologists, economists and other professionalsworking with stochastic simulations, who will benefit from the abilityto reliably estimate and control the accuracy of their simulations.