Wasserstein distributional sensitivity to model uncertainty in dynamic context

Wasserstein 对动态环境中模型不确定性的分布敏感性

• 批准号: 2594682
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2594682
• 关键词: Wasserstein; distributional; sensitivity; model; uncertainty

Stochastic optimization problems in a multi-period (dynamic) setting are a staple of applied mathematics in many domains. In particular, they are the backbone of quantitative finance and financial economics, allowing us to tackle the problems from optimal investment decisions, through hedging problems to equilibrium pricing, and more. In such context, the model is usually derived from theoretical considerations, possibly combined with some calibration to market data, and typically has nice analytic representation. More recently, numerical data-driven approaches to these questions are being developed, often involving deep neural networks and machine learning techniques. In such context, the data - be it market data or generated data - is typically discrete. In both cases above, there is fundamental uncertainty about the postulated probability measure, i.e., the model. This so-called Knightian uncertainty is of fundamental importance and a subject of intense studies in mathematics and economics alike. One way to capture the model uncertainty is through the distributionally robust approach, see [1] Distributionally robust optimization (DRO) is formulated as a mini-max problem where the inner maximization is taken over a collection of probability measures (ambiguity set) and the outer minimization is taken over all the admissible controls. The ambiguity set is often given as a small perturbation of the partially observed distributional information of the reference model. The fundamental aim of this project is to understand both theoretical and numerical aspects of DRO problems when the ambiguity set is expressed using Wasserstein-like distances. These classical distances have recently been extended to the dynamic settings, under the name of adapted-Wasserstein metric. To define the adapted Wasserstein distance, we restrict ourselves to all causal couplings in the sense that the target process at time t only depends on the source process up to time t. This restriction makes the adapted Wasserstein distance essentially different from the classical Wasserstein distance. The new distances allow us to capture simultaneously the relevance of the information flow and of the geometry of the state space. Recent seminal results [2] show that the topology generated by the adapted Wasserstein distance agrees with other notions of adapted topology, e.g., the weak nested topology, Hellwig's information topology, Aldous' extended weak topology. And, indeed, it is the coarsest topology on the probability measure space that makes optimal stopping problems continuous. On the other hand, adapted Wasserstein distances allows us to treat discrete and diffuse measure at the same time. They also, crucially, allow us to capture the geometry of the state space, although the geodesic nature of the space of processes endowed with the adapted Wasserstein distance is still an open problem. The project aims to consider both discrete and continuous time, as well as limiting passage from one to the other. Likewise, the aim is both to shed understanding on the DRO problem through its analysis, including duality, as well as to develop first order approximation to the value function and optimal control. This builds on the works in a one-period setting which used regular Wasserstein distances, see [3]. The DRO setting can be potentially extended to optimal stopping problems, multi-period games, risk-averse stochastic programming, etc. Applications in machine learning, mathematical finance, and statistics will be considered. References: [1] https://doi.org/10.1287/moor.2018.0936. [2] https://doi.org/10.1007/s00440-020-00993-8. [3] https://doi.org/10.1098/rspa.2021.0176. This project falls within the EPSRC Statistics and Applied Probability research area.

多周期（动态）环境下的随机优化问题是应用数学中的一个重要问题。特别是，它们是定量金融学和金融经济学的支柱，使我们能够解决从最优投资决策到对冲问题到均衡定价等问题。在这种情况下，模型通常是从理论考虑中得出的，可能结合了一些市场数据的校准，并且通常具有很好的分析表示。最近，针对这些问题的数值数据驱动方法正在开发中，通常涉及深度神经网络和机器学习技术。在这种情况下，数据-无论是市场数据还是生成的数据-通常是离散的。在上述两种情况下，假设的概率测量存在根本的不确定性，即，该模型这种所谓的不确定性具有根本的重要性，也是数学和经济学中深入研究的一个课题。捕获模型不确定性的一种方法是通过分布鲁棒方法，参见[1]分布鲁棒优化（DRO）被公式化为一个极小-极大问题，其中内部最大化是在一组概率测度（模糊集）上进行的，外部最小化是在所有可容许控制上进行的。模糊度集通常作为参考模型的部分观测分布信息的小扰动给出。这个项目的基本目的是了解DRO问题的理论和数值方面时，模糊集表示使用Wasserstein距离。这些经典的距离最近被扩展到动态设置下，适应Wasserstein度量的名称。为了定义适应的Wasserstein距离，我们将自己限制在所有的因果耦合中，在这个意义上，时间t的目标过程只依赖于时间t之前的源过程。这个限制使得适应Wasserstein距离本质上不同于经典Wasserstein距离。新的距离使我们能够同时捕获的信息流和状态空间的几何形状的相关性。最近的开创性结果[2]表明，由自适应Wasserstein距离生成的拓扑与自适应拓扑的其他概念一致，例如，弱嵌套拓扑，Hellwig的信息拓扑，Aldous的推广弱拓扑。事实上，正是概率测度空间上的粗拓扑使得最优停时问题连续。另一方面，自适应Wasserstein距离允许我们同时处理离散和扩散测度。他们也，至关重要的是，使我们能够捕捉到的状态空间的几何形状，虽然被赋予适应Wasserstein距离的过程的空间的测地线性质仍然是一个悬而未决的问题。该项目旨在考虑离散和连续时间，以及限制从一个到另一个的通道。同样，目的是通过分析（包括对偶性）来理解DRO问题，以及开发值函数和最优控制的一阶近似。这是建立在使用常规Wasserstein距离的单周期设置中的作品上的，参见[3]。DRO设置可以潜在地扩展到最优停止问题，多期博弈，风险规避随机规划等，在机器学习，数学金融和统计学中的应用将被考虑。参考资料：[1] https://doi.org/10.1287/moor.2018.0936。[2]https://doi.org/10.1007/s00440-020-00993-8的网站。[3]https://doi.org/10.1098/rspa.2021.0176的网站。该项目属于EPSRC统计和应用概率研究领域的福尔斯。