Stochastic Optimization Problems in Finance

这个项目将研究金融中的一些风险管理问题，这些问题需要有效地计算最优交易决策，这些决策旨在解决现代市场中发现的各种“摩擦”。这里调查的一些常见摩擦的例子有：不确定的波动性，资源竞争的影响，以及贸易限制。结果将以软件的形式实施，可供感兴趣的各方下载、测试和实施，这些各方可能包括大型交易机构和拥有养老基金的个人。过去25年来，衍生品市场规模的惊人增长(目前在美国的营业额达到数万亿美元)，加上最近臭名昭著(同样引人注目)的风险(MIS)管理灾难，如霸菱、奥兰治县和长期资本管理公司(Long Term Capital Management)的失败，迫切需要智能的数学和计算模型来量化此类投资的各自风险和回报。这个项目旨在建立在布莱克、斯科尔斯和默顿引入的方法的基础上，以考虑市场波动和其他摩擦的波动性质。主要的数学工具是随机控制和渐近分析领域。

This project will study a number of risk management problems in finance requiring efficient computation of optimal trading decisions that are designed to account for various "frictions" found in modern markets. Some examples of common frictions investigated here are: uncertain volatility, influence of competition for resources, and trading restrictions. The results will be implemented as software that can be downloaded, tested and implemented by interested parties, which may range from large trading institutions to individuals with pension funds.The spectacular growth in the size of the derivatives market over the last twenty-five years (currently it has a turnover of trillions of dollars in the US), plus recent infamous (and equally spectacular) risk (mis)management disasters, such as the Barings, Orange County and Long Term Capital Management fiascos, have created an urgent need for smart mathematical and computational models to quantify the respective risks and rewards of such investments. This project aims to build on the methodology introduced by Black, Scholes and Merton, to take into account the fluctuating nature of market volatility and other frictions. The main mathematical tools are in the realm of stochastic control and asymptotic analysis.