Solving High Dimensional Partial Differential Equations in Fintech and Beyond

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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.