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 \(\varepsilon \)-error by a deep ReLU network with depth \({\mathcal {O}}\big (\ln (d)\ln (\varepsilon ^{-1})+\ln (d)^2\big )\) and \({\mathcal {O}}\big (d^{2+\frac{1}{n}}\varepsilon ^{-\frac{1}{n}}\big )\) nonzero 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.

我们通过期权定价中出现的一类Kolmogorov方程多元解决方案的深层ReLU网络分析近似率。关键技术设备是能够有效逼近张量积的深层ReLU架构。将其与有关行为良好（即，满足某些平滑特性）的单变量函数逼近的结果相结合，可以深入了解具有张量结构的多元函数的深ReLU逼近率。我们将其尤其应用于由Black–Scholes模型中用于欧洲最大期权定价的一揽子d资产上的欧洲最大期权价格给出的模型问题。我们证明d变量期权定价问题的解决方案可以近似到\（\ varepsilon \） -深度为\（{\ mathcal {O}} \ big（\ ln（d）\ ln（\ varepsilon ^ {-1}）+ \ ln（d）^的深度ReLU网络的错误2 \ big）\）和\（{\ mathcal {O}} \ big（d ^ {2+ \ frac {1} {n}} \ varepsilon ^ {-\ frac {1} {n}} \ big） \）非零权重，其中\（n在{\ mathbb {N}} \）中是任意的（根据n隐含在\（{\ mathcal {O}}（\ cdot）\）中的常数）。在构造性证明中开发的技术对于分析高维PDE的求解流形的深层神经网络的表达能力具有独立的兴趣。