This volume represents the proceedings of the Workshop on Numerical Methods and Stochastics held at The Fields Institute in April 1999. The goal of the workshop was to identify emerging ideas in probability theory that influence future work in both probability and numerical computation. The book focuses on new results and gives novel approaches to computational problems based on the latest techniques from the theory of probability and stochastic processes.Three papers discuss particle system approximations to solutions of the stochastic filtering problem. Two papers treat particle system equations. The paper on 'rough paths' describes how to generate good approximations to stochastic integrals. An expository paper discusses a long-standing conjecture: the stochastic fast dynamo effect. A final paper gives an analysis of the error in binomial and trinomial approximations to solutions of the Black-Scholes stochastic differential equations. The book is intended for graduate students and research mathematicians interested in probability theory.
Numerical methods for solving the stochastic filtering problem by D. Crisan Optimal filtering on discrete sets by D. Crisan and T. Lyons The Monte-Carlo method for filtering with discrete-time observations: Central limit theorems by P. Del Moral and J. Jacod Approximations of Markovian non linear partial differential equations by particle systems by A. Guionnet Non-Markovian limit diffusions and spin glasses by A. Guionnet Towards pathwise stochastic fast dynamo in magneto-hydrodynamics by S. B. Hazra and F. G. Viens System control and rough paths by T. J. Lyons Embedding and the convergence of the binomial and trinomial tree schemes by J. B. Walsh and O. D. Walsh.