DNN Expression Rate Analysis of High-Dimensional PDEs: Application to Option Pricing
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Date
2022-02
Publication Type
Journal Article
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yes
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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 epsilon-error by a deep ReLU network with depth O(ln(d) ln(epsilon(-1)) + ln(d)(2)) and O(d(2+1/n) epsilon(-1/n)) nonzero weights, where n is an element of N is arbitrary (with the constant implied in O(center dot) 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.
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published
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Journal / series
Volume
55 (1)
Pages / Article No.
3 - 71
Publisher
Springer
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Subject
Neural network approximation; Low-rank approximation; Option pricing; High-dimensional PDEs
Organisational unit
03435 - Schwab, Christoph / Schwab, Christoph
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Funding
175699 - Higher order numerical approximation methods for stochastic partial differential equations (SNF)