Convergence analysis of the Longstaff-Schwartz algorithm

Abstract

We analyse convergence properties of the Longstaff-Schwartz algorithm, a routine used in pricing American options. After a description and a numerical example of the algorithm, we will present an introduction to statistical learning theory and give a rigorous proof of Pollard's inequality. Having established the Vapnik-Chervonenkis dimension, we pass on to prove an inequality about the error that occurs within one step of the LS-Algorithm. We use all this to establish convergence theorems, even in settings where the approximation spaces are not convex, closed or linear, as long as they are uniformly bounded and have a finite Vapnik-Chervonenkis dimension. The rest of the thesis deals with applications of the convergence theorems. We are going to use polynomial approximations architectures and artificial neural networks.