Asymptotic Expansion and Weak Approximation: Applications of Malliavin Calculus and Deep Learning

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Akihiko Takahashi Toshihiro Yamada

Asymptotic Expansion and Weak Approximation

Applications of Malliavin Calculus and Deep Learning

Akihiko Takahashi, Toshihiro Yamada

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English

Springer Nature Switzerland AG
03 October 2025

Differential calculus & equations; Probability & statistics; Mathematical & statistical software; Neural networks & fuzzy systems

Series: SpringerBriefs in Statistics

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.

By: Akihiko Takahashi, Toshihiro Yamada
Imprint: Springer Nature Switzerland AG
Country of Publication: Switzerland
Dimensions: Height: 235mm, Width: 155mm,
ISBN: 9789819682799
ISBN 10: 9819682797
Series: SpringerBriefs in Statistics
Pages: 97
Publication Date: 03 October 2025
Audience: Professional and scholarly , College/higher education , Undergraduate , Further / Higher Education
Format: Paperback
Publisher's Status: Active

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.

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