Title
DNN Expression Rate Analysis of High-Dimensional PDEs: Application to Option Pricing
Author
Arnulf Jentzen
SAM, Department of Mathematics, ETH Zurich
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Abstract
We analyze approximation rates by deep ReLU networks of a class of multivariate solutions of Kolmogorov equations which arise in option pricing. Key technical devices are deep ReLU architectures capable of efficiently approximating tensor products. Combining this with results concerning the approximation of well-behaved (i.e., fulfilling some smoothness properties) univariate functions, this provides insights into rates of deep ReLU approximation of multivariate functions with tensor structures. We apply this in particular to the model problem given by the price of a European maximum option on a basket of d assets within the Black–Scholes model for European maximum option pricing. We prove that the solution to the d-variate option pricing problem can be approximated up to an ε-error by a deep ReLU network with depth O(ln(d)ln(ε−1)+ln(d)2) and O(d2+1nε−1n) nonzero weights, where n∈N is arbitrary (with the constant implied in O(⋅) depending on n). The techniques developed in the constructive proof are of independent interest in the analysis of the expressive power of deep neural networks for solution manifolds of PDEs in high dimension.
Keywords
Neural network approximationLow-rank approximationOption pricingHigh-dimensional PDEs
Object type
Language
English [eng]
Persistent identifier
phaidra.univie.ac.at/o:1504462
Appeared in
Title
Constructive Approximation
Volume
55
ISSN
0176-4276
Issued
2021
From page
3
To page
71
Publication
Springer Science and Business Media LLC
Version type
Date issued
2021
Access rights
Rights
Rights statement
© The Author(s) 2021
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