Solving High Dimensional Partial Differential Equations in Fintech and Beyond

高盛首席财务官马蒂·查韦斯（Marty Chavez）说：“我们所做的一切都是以数学和大量软件为基础的。“在过去的几十年里，金融机构对技术的使用大大增加。金融科技，或Fintech，是一种新的趋势，彻底改变了金融业。许多传统上依赖人类技能和经验的金融活动最近已经或将被计算机系统所取代。事实上，银行和投资公司正在招聘比以往任何时候都更多的具有强大计算能力的员工。金融科技是一个广泛的主题，这项研究计划将侧重于定量分析，特别是开发高效和有效的计算技术，用于金融和其他领域的应用。一个重要的应用是金融衍生工具（如期权）的定价，这是一种与资产相关的合同，为资产所有者提供未来的特定交易权。衍生品的定价不当会导致巨大的金融损失，因此衍生品的公平定价对金融业至关重要。 本研究计划探讨解决金融及其他领域高维问题的数值方法。特别是，我们将利用机器学习（ML）技术来应对高维的挑战。ML已被证明是许多应用的强大工具，如计算机视觉，自然语言处理和医疗诊断。最近，它也被应用于金融应用，包括期权定价。然而，金融领域的许多ML应用程序将ML视为黑盒工具并提供（市场）数据来逼近期权价值函数。研究的一个方向是探索一种财务指导学习方法，将财务模型集成到ML的各个组件中，以实现更有效的方法。另一个研究方向是探索金融领域的数据分析学习方法。高维度问题超越了金融。我们注意到，上面研究的许多金融问题可以用公式表示为随机最优控制问题，这些问题出现在其他领域，如计算机图形学和医学成像。有了新的视角，我们将能够应用金融应用中的技术，并开发新技术来解决图形和成像应用中的重要问题。拟议的研究计划的成功将使金融和其他应用领域的从业者能够解决以前不可能解决的大规模和高维问题。期权的正确定价对行业的健康运行至关重要，从而改善对社会的金融服务。此外，研究扩展到其他领域，如图形仿真和医学成像，为娱乐业和医疗健康系统带来额外的好处。

A quote from Marty Chavez, CFO of Goldman Sachs, "Everything we do is underpinned by math and a lot of software." The use of technology in financial institutions has increased tremendously in the past decades. Financial technology, or Fintech, is a new trend that revolutionized the financial industry. Many of the financial activities that have been traditionally relied on human skills and experience have recently been replaced or will be replaced by computer systems. In fact, banks and investment companies are hiring more staff with strong computing skills than ever. Fintech is a broad subject and this research proposal is going to focus on quantitative analytics, and in particular on developing efficient and effective computational techniques for applications in finance and other areas. An important application is pricing of financial derivatives (e.g. options), which is a contract associated with an asset and providing their owners with specific trading rights in the future. Fair pricing of derivatives is vital to financial industry since their mispricing can result in an enormous financial loss. This research proposal investigates numerical methods for solving high dimensional problems in finance and other areas. In particular, we will exploit machine learning (ML) techniques to tackle the challenge of high dimensionality. ML has been shown to be a powerful tool for many applications such as computer vision, natural language processing, and medical diagnosis. Recently, it has also been applied to financial applications including option pricing. However, many of the ML applications in finance treat ML as a black box tool and feed (market) data to approximate the option value function. One direction of research is to explore a finance-guiding learning approach in which we integrate financial models into individual components of ML in order to achieve more effective methods. Another direction of research is to explore data analytic learning approach in finance. High dimensional problems go beyond finance. We note that many of the financial problems studied above can be formulated as stochastic optimal control problems which arise in other areas such as computer graphics and medical imaging. With the new perspective, we will be able to apply the techniques from financial applications as well as to develop new techniques for solving important problems in graphics and imaging applications. The success of the proposed research program will allow practitioners in finance and other application domains to tackle large scale and high dimensional problems that would not have been possible before. The correct pricing of options is critical to the healthy operation of the industry, which leads to improvements in financial services to the society. Furthermore, the extension of the research to other areas such as graphics simulations and medical imaging brings additional benefits to entertainment industry and medical health system.