Deep Kusuoka Approximation: High-Order Spatial Approximation for Solving High-Dimensional Kolmogorov Equations and Its Application to Finance

The paper introduces a new deep learning-based high-order spatial approximation for a solution of a high-dimensional Kolmogorov equation where the initial condition is only assumed to be a continuous function and the condition on the vector fields associated with the differential operator is very ge...

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Bibliographic Details
Published in:Computational economics Vol. 64; no. 3; pp. 1443 - 1461
Main Authors: Naito, Riu, Yamada, Toshihiro
Format: Journal Article
Language:English
Published: New York Springer US 01.09.2024
Springer
Springer Nature B.V
Subjects:
Approximation
Behavioral/Experimental Economics
Computation
Computational mathematics
Computer Appl. in Social and Behavioral Sciences
Deep learning
Differential equations
Differentiation
Economic Theory/Quantitative Economics/Mathematical Methods
Economics
Economics and Finance
Fields (mathematics)
Learning
Math Applications in Computer Science
Mathematical analysis
Neural networks
Numerical differentiation
Operations Research/Decision Theory
Operators (mathematics)
Optimization techniques
Partial differential equations
Random variables
Viscosity
Deep learning
Financial diffusions
Kolmogorov equations
Kusuoka approximation
Delta computing
ISSN:0927-7099, 1572-9974
Online Access:Get full text
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Summary:The paper introduces a new deep learning-based high-order spatial approximation for a solution of a high-dimensional Kolmogorov equation where the initial condition is only assumed to be a continuous function and the condition on the vector fields associated with the differential operator is very general, i.e. weaker than Hörmander’s hypoelliptic condition. In particular, the deep learning-based method is constructed based on the Kusuoka approximation. Numerical results for high-dimensional partial differential equations up to 500-dimension cases appearing in option pricing problems show the validity of the method. As an application, a computation scheme for the delta is shown using “deep” numerical differentiation.
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ISSN:0927-7099
1572-9974
DOI:10.1007/s10614-023-10476-2