Deep ReLU network expression rates for option prices in high-dimensional, exponential Lévy models
研究了深度神经网络对高维指数Lévy模型下篮子期权价格的表达率，证明在特定条件下网络规模随维度多项式增长，克服维数灾难，并利用抛物平滑提出替代架构。
Abstract We study the expression rates of deep neural networks (DNNs for short) for option prices written on baskets of $d$ d risky assets whose log-returns are modelled by a multivariate Lévy process with general correlation structure of jumps. We establish sufficient conditions on the characteristic triplet of the Lévy process $X$ X that ensure $\varepsilon $ ε error of DNN expressed option prices with DNNs of size that grows polynomially with respect to ${\mathcal{O}}(\varepsilon ^{-1})$ O ( ε − 1 ) , and with constants implied in ${\mathcal{O}}(\, \cdot \, )$ O ( ⋅ ) which grow polynomially in $d$ d , thereby overcoming the curse of dimensionality (CoD) and justifying the use of DNNs in financial modelling of large baskets in markets with jumps. In addition, we exploit parabolic smoothing of Kolmogorov partial integro-differential equations for certain multivariate Lévy processes to present alternative architectures of ReLU (“rectified linear unit”) DNNs that provide $\varepsilon $ ε expression error in DNN size ${\mathcal{O}}(|\log (\varepsilon )|^{a})$ O ( | log ( ε ) | a ) with exponent $a$ a proportional to $d$ d , but with constants implied in ${\mathcal{O}}(\, \cdot \, )$ O ( ⋅ ) growing exponentially with respect to $d$ d . Under stronger, dimension-uniform non-degeneracy conditions on the Lévy symbol, we obtain algebraic expression rates of option prices in exponential Lévy models which are free from the curse of dimensionality. In this case, the ReLU DNN expression rates of prices depend on certain sparsity conditions on the characteristic Lévy triplet. We indicate several consequences and possible extensions of the presented results.