In various scientific and industrial fields, stochastic simulations are taking on a new importance. This is due to the increasing power of computers and practitioners aim to simulate more and more complex systems, and thus use random parameters as well as random noises to model the parametric uncertainties and the lack of knowledge on the physics of these systems. The error analysis of these computations is a highly complex mathematical undertaking. Approaching these issues, the authors present stochastic numerical methods and prove accurate convergence 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 numerical methods within a theoretical framework. After briefly reviewing the basics, the authors first introduce fundamental notions in stochastic calculus and continuous-time martingale theory, then develop the analysis of pure-jump Markov processes, Poisson processes, and stochastic differential equations. In particular, they review the essential properties of Ito integrals and prove fundamental results on the probabilistic analysis of parabolic partial differential equations. These results in turn provide the basis for developing stochastic numerical methods, both from an algorithmic and theoretical point of view.
The book combines advanced mathematical tools, theoretical analysis of stochastic numerical methods, and practical issues at a high level, so as to provide optimal results on the accuracy of Monte Carlo simulations of stochastic processes. It is intended for master and Ph.D. students in the field of stochastic processes and their numerical applications, as well as for physicists, biologists, economists and other professionals working with stochastic simulations, who will benefit from the ability to reliably estimate and control the accuracy of their simulations."

This volume represents the refereed proceedings of the Sixth International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scienti?c Computing which was held in conjunction with the Second International C- ference on Monte Carlo and Probabilistic Methods for Partial Di?erential Equations at Juan-les-Pins, France, from 7-10 June 2004. The programme of this conference was arranged by a committee consisting of Henri Faure (U- versit' edeMarseille),PaulGlasserman(ColumbiaUniversity),StefanHeinrich (Universit. at Kaiserslautern), Fred J. Hickernell (Hong Kong Baptist Univ- sity), Damien Lamberton (Universit' e de Marne la Vall' ee), Bernard Lapeyre (ENPC-CERMICS), Pierre L'Ecuyer (Universit'edeMontr' eal), Pierre-Louis Lions (Coll' ege de France), Harald Niederreiter (National University of S- gapore, co-chair), Erich Novak (Universit. at Jena), Art B. Owen (Stanford University), Gilles Pag' es (Universit' e Paris 6), Philip Protter (Cornell U- versity), Ian H. Sloan (University of New South Wales), Denis Talay (INRIA Sophia Antipolis, co-chair), Simon Tavar' e (University of Southern California) and Henryk Wo' zniakowski (Columbia University and University of Warsaw). The organization of the conference was arranged by a committee consisting of Mireille Bossy and Etienne Tanr' e (INRIA Sophia Antipolis), and Madalina Deaconu(INRIALorraine). LocalarrangementswereinthehandsofMonique Simonetti and Marie-Line Ramfos (INRIA Sophia Antipolis).

The lecture courses of the CIME Summer School on Probabilistic Models for Nonlinear PDE's and their Numerical Applications (April 1995) had a three-fold emphasis: first, on the weak convergence of stochastic integrals; second, on the probabilistic interpretation and the particle approximation of equations coming from Physics (conservation laws, Boltzmann-like and Navier-Stokes equations); third, on the modelling of networks by interacting particle systems. This book, collecting the notes of these courses, will be useful to probabilists working on stochastic particle methods and on the approximation of SPDEs, in particular, to PhD students and young researchers.