DNN Expression Rate Analysis of High-dimensional PDEs: Application to Option Pricing

TL;DR

This paper investigates how deep ReLU neural networks can efficiently approximate high-dimensional solutions of PDEs, specifically in option pricing, demonstrating near-optimal approximation rates with respect to dimension and accuracy.

Contribution

It provides new theoretical bounds on the approximation rates of deep ReLU networks for multi-variate PDE solutions with tensor structures, especially in financial option pricing models.

Findings

Deep ReLU networks can approximate high-dimensional PDE solutions with rates depending logarithmically on dimension.

The approximation depth scales polylogarithmically with inverse error tolerance.

The number of non-zero weights grows polynomially with dimension and inverse error.

Abstract

We analyze approximation rates by deep ReLU networks of a class of multi-variate 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 multi-variate 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 $\varepsilon$-error by a deep ReLU network with depth $\mathcal{O}\big(\ln(d)\ln(\varepsilon^{-1})+\ln(d)^2\big)$ and $\mathcal{O}\big(d^{2+\frac{1}{n}}\varepsilon^{-\frac{1}{n}}\big)$ non-zero weights, where $n\in \mathbb{N}$ is arbitrary (with the constant implied in $\mathcal{O}(\cdot)$ 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.