In financial engineering, prices of financial products are computed approximately many times each trading day with (slightly) different parameters in each calculation. In many financial models such prices can be approximated by means of Monte Carlo (MC) simulations. To obtain a good approximation the MC sample size usually needs to be considerably large resulting in a long computing time to obtain a single approximation. In this paper we introduce a new approximation strategy for parametric approximation problems including the parametric financial pricing problems described above. A central aspect of the approximation strategy proposed in this article is to combine MC algorithms with machine learning techniques to, roughly speaking, learn the random variables (LRV) in MC simulations. In other words, we employ stochastic gradient descent (SGD) optimization methods not to train parameters of standard artificial neural networks (ANNs) but to learn random variables appearing in MC approximations. We numerically test the LRV strategy on various parametric problems with convincing results when compared with standard MC simulations, Quasi-Monte Carlo simulations, SGD-trained shallow ANNs, and SGD-trained deep ANNs. Our numerical simulations strongly indicate that the LRV strategy might be capable to overcome the curse of dimensionality in the $L^\infty$-norm in several cases where the standard deep learning approach has been proven not to be able to do so. This is not a contradiction to lower bounds established in the scientific literature because this new LRV strategy is outside of the class of algorithms for which lower bounds have been established in the scientific literature. The proposed LRV strategy is of general nature and not only restricted to the parametric financial pricing problems described above, but applicable to a large class of approximation problems.

翻译：在金融工程中，金融产品价格每天需多次近似计算，每次计算采用（略有）不同的参数。在许多金融模型中，此类价格可通过蒙特卡罗（MC）模拟进行近似。为获得良好的近似效果，MC样本量通常需要相当大，导致单次近似的计算时间较长。本文针对包括上述参数化金融定价问题在内的参数化近似问题，提出一种新的近似策略。该策略的核心思想在于将MC算法与机器学习技术相结合，简而言之，即学习MC模拟中的随机变量（LRV）。换言之，本文采用随机梯度下降（SGD）优化方法，其目的并非训练标准人工神经网络（ANN）的参数，而是学习MC近似过程中出现的随机变量。我们在多种参数化问题上对LRV策略进行了数值测试，并与标准MC模拟、拟蒙特卡罗模拟、SGD训练的浅层ANN及SGD训练的深层ANN进行对比，结果令人信服。数值模拟强烈表明，在若干标准深度学习方法已被证明无法克服维度灾难的情况下，LRV策略可能具备在$L^\infty$范数下克服该问题的能力。这与科学文献中建立的下界并不矛盾，因为该新LRV策略不属于已建立下界的算法类别。本文提出的LRV策略具有通用性，不仅局限于上述参数化金融定价问题，还可应用于一大类近似问题。