Portfolio theory, optimal transport and information geometry

量化投资策略在现代金融市场中起着举足轻重的作用。它们决定了金融机构如何处理它们频繁的交易，以及养老基金如何管理我们的储蓄。这些策略基于金融市场的理论和经验模型。许多策略是高度特定于模型的：如果模型错误，或者当市场状况发生变化时，灾难性的事件可能会发生。因此，迫切需要研究稳健的投资战略，即其成功不取决于特定模型的假设的战略。这个研究项目使用概率和几何的新工具来开发稳健的投资组合理论。同时，我们提出了在数据科学中得到重要应用的数学理论。金融思想来源于随机投资组合理论和泛投资组合理论。这些理论提供了基于金融市场持久特征的稳健投资策略和将它们结合在一起的方法；我们建议在更现实的环境下结合这些方法。从数学上讲，这些策略与最优运输有关，后者涉及以总体成本效益的方式将市场情景分配给资产配置。如果我们将不断发展的市场视为一个在高维空间中旅行的点，几何就会进入画面。我们赋予空间一个合适的几何形状，这样市场运动的方向和大小就会直接影响投资组合。这种几何可以用最优运输几何和信息几何来研究。我们建议从这个几何观点来研究市场的经验性质。通过采用多学科的方法，我们希望开发具有实际应用的新的数学理论，包括算法和软件包。特别地，我们的目的是通过研究新的代价函数来推广经典的Wasserstein几何。研究计划的结果将提供强大的工具来管理与市场集中度和波动性相关的投资组合和风险。它们还有望提高我们对最优运输和信息几何的理解，它们之间的联系已经开始引起人们的大量关注。开发的数学结果和算法有望在量化交易之外有用，并将导致统计学和机器学习方面的进一步理论发展和应用。

Quantitative investment strategies play a decisive role in modern financial markets. They determine how financial institutions deal with their frequent tradings and how pension funds manage our savings. These strategies are based on theoretical and empirical models of financial markets. Many strategies are highly model-specific: if the model is wrong, or when market conditions change, disastrous events may happen. Thus, there is a strong need to study robust investment strategies, i.e., strategies whose success do not depend on the assumptions of a specific model. This research program develops robust portfolio theory using novel tools from probability and geometry. Simultaneously, we advance the mathematical theories which have found important applications in data science. The financial ideas come from stochastic portfolio theory and universal portfolio theory. These theories provide robust investment strategies based on persistent features of financial markets and ways to combine them; we propose to combine these approach under more realistic settings. Mathematically, these strategies are related to optimal transport which is about assigning market scenarios to asset allocations in an overall cost-efficient way. Geometry enters the picture if we think of the evolving market as a point traveling in a high dimensional space. We endow the space with a suitable geometry such that the directions and magnitudes of market movements have direct impacts on the portfolio. This geometry can be studied using optimal transport and information geometry. We propose to study the empirical properties of the market from this geometric viewpoint. By adopting a multi-disciplinary approach, we hope to develop novel mathematical theories with practical applications including algorithms and software packages. In particular, we aim to generalize the classical Wasserstein geometry by studying new cost functions. The outcomes of the research program will provide robust tools to manage portfolios and risks related to market concentration and volatility. They are also expected to improve our understanding about optimal transport and information geometry whose connections have started to attract a lot of attention. The mathematical results and algorithms developed are expected to be useful beyond quantitative trading and will lead to further theoretical development and applications in statistics and machine learning.