Development of Multilevel Monte Carlo Algorithms for Mathematical Finance

这项跳板奖学金将极大地帮助我在从事了20年的计算流体力学工作后，在职业生涯中期进入计算金融领域，模拟飞机燃气轮机发动机的流动。研究主题涉及使用蒙特卡罗方法对金融衍生品期权(基于股票、大宗商品、利率和汇率)的定价，该方法评估随机输入下未来可能演变的多个模拟的平均结果。随机微分方程解这个研究领域是数学中的一个主要增长领域，它支撑着伦敦各大银行的大部分日常工作，而这些银行又构成了英国经济的一大部分。6个月前，我有了一个想法，它可以显著降低实现给定精度所需的蒙特卡罗计算的计算成本。我的初步研究成果，以及对模型问题的数值试验，都非常令人鼓舞。它受到了知名学术人士的好评，并已被邀请参加三次大学演讲和四次投资银行研讨会。我提出这项奖学金的目的是在这一初步成功的基础上，进一步发展数值技术，我称之为多层蒙特卡罗方法，以提高其性能，使其与当今业界使用的领先方法相比具有竞争力。除了研究本身，该奖学金的一个主要目标是与世界各地的关键学者和伦敦的领先银行建立合作关系。我的目标是，在奖学金结束后很长一段时间内，这些都应该继续下去，银行是我后续研究的主要资金来源。此外，由于我对这一研究领域仍然是非常新的，我对支撑这一领域的随机分析理论的理解存在缺陷，我将努力解决这些问题。

This Springboard Fellowship will help me enormously in my mid-career move into computational finance after 20 years of computational fluid dynamics, simulating the flow through aircraft gas turbine engines. The research topic concerns the pricing of financial derivatives options (based on equities, commodities, interest rates and exchange rates) using Monte Carlo methods which evaluate the average outcome from multiple simulations of possible future evolution subject to random inputs. This research area, the solution of stochastic differential equations, is a major growth area in mathematics, and it underpins much of the everyday working of the major banks in London, which in turn form a large part of the UK economy.Six months ago, I had an idea which signifiantly reduces the computational cost of the Monte Carlo calculations required to achieve a given accuracy. My preliminary research results, and numerical tests on model problems, are very encouraging. It has been well received by leading academic figures and has already led to invitations for three university presentations and four seminars at investment banks. My objective with this Fellowship proposal is to build on this initial success by further developing the numerical technique, which I refer to as the multilevel Monte Carlo method, to enhance its performance and make it competitive against the leading methods used today in the industry. Alongside the research itself, a major goal of the fellowship is to build collaborations with key academics worldwide and with leading banks in London. My aim is that these should continue long after the end of the Fellowship, with the banks being my major source of funding for subsequent research. Also, as I am still very new to this field of research, there are deficiencies in my understanding of the stochastic analysis theory which underpins this field and I will work to address these.