This book provides a self-contained lecture on a Malliavin calculus approach to asymptotic expansion and weak approximation of stochastic differential equations (SDEs), along with numerical methods for computing parabolic partial differential equations (PDEs).
Constructions of weak approximation and asymptotic expansion are given in detail using Malliavin's integration by parts with theoretical convergence analysis.
Weak approximation algorithms and Python codes are available with numerical examples.
Moreover, the weak approximation scheme is effectively applied to high-dimensional nonlinear problems without suffering from the curse of dimensionality
through combining with a deep learning method.
Readers including graduate-level students, researchers, and practitioners can understand both theoretical and applied aspects of recent developments of asymptotic expansion and weak approximation.

Highlights arbitrary order asymptotic expansion and weak approximation of stochastic differential equations Is written in self-contained form, and computation schemes, algorithms, and Python codes are available Provides a deep learning method with Malliavin calculus for solving high-dimensional nonlinear problems

Autorentext

Akihiko Takahashi is at Graduate School of Economics, The University of Tokyo

Toshihiro Yamada is at Graduate School of Economics, Hitotsubashi University

Inhalt

Chapter 1. Introduction.- Chapter 2. Itô calculus.- Chapter 3. Malliavin calculus.- Chapter 4. Asymptotic expansion.- Chapter 5. Weak approximation.- Chapter 6. Application: Deep learning-based weak approximation.