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

We analyze approximation rates by deep ReLU networks of a class multi-variate solutions Kolmogorov equations which arise in option pricing. Key technical devices are architectures capable efficiently approximating tensor products. Combining this with results concerning the well behaved (i.e. fulfilling some smoothness properties) univariate functions, provides insights into functions structures. apply particular to model problem given price European maximum on basket $d$ assets within Black-Scholes for prove that solution $d$-variate pricing can be approximated up an $\varepsilon$-error network 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 constant implied $\mathcal{O}(\cdot)$ depending $n$). The techniques developed constructive proof independent interest analysis expressive power neural manifolds PDEs high dimension.