在蒙特卡洛模拟中学习随机变量：用于参数偏微分方程和金融衍生品定价的机器学习
Learning the random variables in Monte Carlo simulations with stochastic gradient descent: Machine learning for parametric PDEs and financial derivative pricing
AI总结：
本文提出了一种通过机器学习在蒙特卡洛模拟中学习随机变量的方法，用于参数偏微分方程和金融衍生品定价，以克服维度灾难。
摘要：
在金融工程中，每天多次以略微不同的参数计算金融产品的价格。在许多金融模型中，这些价格可以近似通过蒙特卡洛（MC）模拟来计算。为了获得良好的近似，MC样本大小通常需要相当大，导致单次近似计算时间较长。本文介绍了一种新的近似策略，用于参数近似问题，包括上述参数金融定价问题。该策略的核心是将MC算法与机器学习技术相结合，大致来说，学习MC模拟中的随机变量（LRV）。换句话说，我们不使用随机梯度下降（SGD）优化方法来训练标准人工神经网络（ANN）的参数，而是用来学习出现在MC近似中的随机变量。我们对各种参数问题进行了数值测试，结果表明与标准MC模拟、准蒙特卡洛模拟、SGD训练的浅层ANN和SGD训练的深层ANN相比，LRV策略表现令人信服。我们的数值模拟强烈表明，LRV策略可能在某些情况下能够克服维度灾难，在标准深度学习方法无法做到的情况下。这并不与科学文献中建立的下限相矛盾，因为这种新的LRV策略不属于文献中已建立下限的算法类别。所提出的LRV策略具有普遍性，不仅限于上述参数金融定价问题，还适用于广泛的近似问题。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
