Deep Learning Reduced Basis Method for High-Dimensional Parametric Partial Differential Equations in Finance

金融部门面临着数字革命、最大的竞争力和严重的风险以及不断提高的监管和会计要求的挑战，这导致了高度复杂的计算问题。一个突出的例子是资本储备的计算，在这种情况下，交易对手信用风险必须被纳入，作为对上次金融危机原因的直接监管回应。这对金融体系至关重要。为了开发适当的期权定价和风险管理的计算方法，我们开发了一种结合深度学习和众所周知的数值技术的新方法，对于工程中非常重视风险控制的复杂现实世界问题，这些方法已经被证明是高度可信的。更准确地说，我们发展了一种参数依赖偏微分方程组的离线-在线方法。离线后，复杂的问题在机器学习的帮助下进行处理，以准备一个高效的解算器。在线时，可以针对所有参数实时调用此求解器。这样，我们将把深度学习的效率与数值算法的可靠性和经济可解释性结合在一起。因此，我们将为机器学习在金融领域的适用性做出贡献，在金融领域，对结果的直观理解与其可靠性同样关键。高维是金融固有的，由此产生的计算问题传统上是通过蒙特卡洛模拟来解决的。对于实时风险监控、不确定性量化和信用价值调整等紧急任务，这种方法的计算成本太高。由于缺乏适当的计算工具，特别简化是当前的市场标准，产生了不可控的操作风险。参数(非线性)偏微分方程的模型降阶与深度学习的新组合使我们能够打破维度诅咒。虽然经典的离散化是不可行的，但深度神经网络允许对高维函数进行有效的逼近。虽然机器学习可以有效地评估结果，但很难解释，我们获得了经济上的可解释性。这些新技术的性能将被评估和比较，并将在现实的例子中展示它们的能力。

The financial sector is challenged by the digital revolution, utmost competitiveness, and serious risks along with advancing regulatory and accounting requirements, which leads to highly complex computational problems. A prominent example is the computation of the capital reserve where counterparty credit risks have to be incorporated as a direct regulatory response to the causes of the last financial crisis. This is of fundamental importance for the financial system. To develop adequate computational methods for option pricing and risk management we develop a new approach combining deep learning with well-understood numerical techniques, which have already proven highly trustworthy for complex real-world problems in engineering, where risk control is taken very seriously. To be more precise, we develop an offline-online method for parameter-dependent partial differential equations. Offline, the complex problem is processed with the help of machine learning to prepare an efficient solver. Online, this solver can be called in real-time for all parameters. This way, we will bring together the efficiency of deep learning with the reliability and economic interpretability of numerical algorithms. Thus we will contribute to the applicability of machine learning in the financial sector, where an intuitive understanding of the results is as crucial as their liability.High-dimensionality is intrinsic in finance and resulting computational problems are traditionally solved by Monte Carlo simulations. For urgent tasks such as real-time risk monitoring, uncertainty quantification and credit value adjustments, this approach is computationally too expensive. Lacking appropriate computational tools, ad hoc simplifications are the current market standard, yielding uncontrollable operational risk. The new combination of model order reduction for parametric (nonlinear) partial differential equations with deep learning allows us to break the curse of dimensionality. While classical discretisations are infeasible, deep neural networks allow for efficient approximations of high-dimensional functions. While with machine learning, results are efficiently evaluated, but difficult to interpret, we gain economic interpretability. The performance of the novel techniques will be evaluated and compared and their capabilities will be shown on realistic examples.