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

金融行业面临着数字革命、激烈竞争、严重风险以及不断推进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准备金的计算，其中必须将交易对手信用风险纳入其中，作为对上次金融危机起因的直接监管反应。这对金融体系至关重要。为了开发足够的期权定价和风险管理的计算方法，我们开发了一种将深度学习与易于理解的数值技术相结合的新方法，这种方法已经被证明对于工程中复杂的现实问题非常值得信赖，其中风险控制非常重视。为了更精确，我们开发了一种离线-在线方法来求解参数相关的偏微分方程。在离线状态下，复杂的问题在机器学习的帮助下进行处理，以准备一个有效的求解器。在线时，可以对所有参数实时调用此求解器。通过这种方式，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起来。因此，我们将为机器学习在金融领域的适用性做出贡献，在这个领域，对结果的直观理解与它们的责任一样重要。高维性在金融中是固有的，由此产生的计算问题传统上是通过蒙特卡罗模拟来解决的。对于诸如实时风险监控、不确定性量化和信用价值调整等紧急任务，这种方法在计算上过于昂贵。由于缺乏适当的计算工具，临时简化是目前的市场标准，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数的诅咒。虽然经典离散是不可行的，但深度神经网络允许对高维函数进行有效的近似。而通过机器学习，结果被有效地评估，但难以解释，我们获得了经济可解释性。本文将对这些新技术的性能进行评估和比较，并通过实际实例展示它们的能力。

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.