  In this work, we study the approximation of expected values of functional
quantities on the solution of a stochastic differential equation (SDE), where
we replace the Monte Carlo estimation with the evaluation of a deep neural
network. Once the neural network training is done, the evaluation of the
resulting approximating function is computationally highly efficient so that
using deep neural networks to replace costly Monte Carlo integration is
appealing, e.g., for near real-time computations in quantitative finance.
However, the drawback of these nowadays widespread ideas lies in the fact that
training a suitable neural network is likely to be prohibitive in terms of
computational cost. We address this drawback here by introducing a multilevel
approach to the training of deep neural networks. More precisely, we combine
the deep learning algorithm introduced by Beck et al. with the idea of
multilevel Monte Carlo path simulation of Giles. The idea is to train several
neural networks, each having a certain approximation quality and computational
complexity, with training data computed from so-called level estimators,
introduced by Giles. We show that under certain assumptions, the variance in
the training process can be reduced by shifting most of the computational
workload to training neural nets at coarse levels where producing the training
data sets is comparably cheap, whereas training the neural nets corresponding
to the fine levels requires only a limited number of training data sets. We
formulate a complexity theorem showing that the multilevel idea can indeed
reduce computational complexity.
