Wasserstein distributional sensitivity to model uncertainty in dynamic context

多周期(动态)环境下的随机优化问题是许多领域应用数学的主要内容。特别是，它们是量化金融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，该模型通常是根据理论考虑得出的，可能结合了对市场数据的一些校正，并且通常具有良好的分析表示。最近，针对这些问题的数值数据驱动的方法正在开发中，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，假设的概率度量，即模型，都存在基本的不确定性。这种所谓的奈特不确定性具有根本性的重要性，也是数学和经济学密集研究的主题。获取模型不确定性的一种方法是通过分布稳健方法，见[1]分布稳健优化(DRO)被表示为极小极大问题，其中内极大化是对概率度量集合(模糊集)进行的，而外部最小化是对所有允许的控制进行的。模糊集通常作为部分观测到的参考模型的分布信息的微小扰动给出。这个项目的基本目的是理解当模糊集用Wasserstein-like距离表示时，DRO问题的理论和数值方面。这些经典距离最近被扩展到动态环境中，命名为适应-瓦瑟斯坦度规。为了定义适应的Wasserstein距离，我们将自己限制为所有的因果耦合，即目标过程在时间t只依赖于直到时间t的源过程，这使得适应的Wasserstein距离与经典的Wasserstein距离有本质的不同。新的距离允许我们同时捕获信息流和状态空间几何形状的相关性。最近的开创性结果[2]表明，由适应Wasserstein距离生成的拓扑与适应拓扑的其他概念一致，如弱嵌套拓扑、Hellwig信息拓扑、Aldous的扩展弱拓扑等。的确，它是概率度量空间上的最粗拓扑，使得最优停止问题连续。另一方面，自适应的瓦瑟斯坦距离允许我们同时处理离散和弥散测量。至关重要的是，它们还允许我们捕捉状态空间的几何形状，尽管赋予了适应的瓦瑟斯坦距离的过程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，以及限制从一个时间到另一个时间的通行。同样，其目的既是通过分析DRO问题，包括对偶问题，也是为了发展值函数和最优控制的一阶近似。这是建立在使用常规瓦瑟斯坦距离的单周期设置中的作品的基础上的，参见[3]。DRO设置可以潜在地扩展到最优停止问题、多周期博弈、风险厌恶随机规划等。将考虑在机器学习、数学金融和统计学中的应用。参考文献：[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.该项目属于EPSRC统计和应用概率研究领域。

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.