Asymptotic Expansion and Weak Approximation – Application of Malliavin Calculus and Deep Learning –(Forthcoming in JSS Research Series in Statistics, a subseries of SpringerBriefs in Statistics.)

Abstract

The book presents recent developments in asymptotic methods on Wiener space and introduces a type of higher-order weak approximation for stochastic differential equations using certain Brownian polynomials based on asymptotic expansions. Furthermore, it offers a weak approximation scheme with a deep learning method, providing broad applications for high-dimensional nonlinear problems. In this context, we demonstrate how to combine our asymptotic expansion-based weak approximation with a neural network approximation, applicable to high-dimensional nonlinear models such as those in finance.