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In financial engineering, prices of financial products are computedapproximately many times each trading day with (slightly) different parametersin each calculation. In many financial models such prices can be approximatedby means of Monte Carlo (MC) simulations. To obtain a good approximation the MCsample size usually needs to be considerably large resulting in a longcomputing time to obtain a single approximation. In this paper we introduce anew approximation strategy for parametric approximation problems including theparametric financial pricing problems described above. A central aspect of theapproximation strategy proposed in this article is to combine MC algorithmswith machine learning techniques to, roughly speaking, learn the randomvariables (LRV) in MC simulations. In other words, we employ stochasticgradient descent (SGD) optimization methods not to train parameters of standardartificial neural networks (ANNs) but to learn random variables appearing in MCapproximations. We numerically test the LRV strategy on various parametricproblems with convincing results when compared with standard MC simulations,Quasi-Monte Carlo simulations, SGD-trained shallow ANNs, and SGD-trained deepANNs. Our numerical simulations strongly indicate that the LRV strategy mightbe capable to overcome the curse of dimensionality in the $L^\infty$-norm inseveral cases where the standard deep learning approach has been proven not tobe able to do so. This is not a contradiction to lower bounds established inthe scientific literature because this new LRV strategy is outside of the classof algorithms for which lower bounds have been established in the scientificliterature. The proposed LRV strategy is of general nature and not onlyrestricted to the parametric financial pricing problems described above, butapplicable to a large class of approximation problems.

Learning the random variables in Monte Carlo simulations with stochastic  gradient descent: Machine learning for parametric PDEs and financial  derivative pricing