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应用数学许多领域的主要内容。特别是，它们是定量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、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，可能结合对市场数据的一些校准，并且通常具有良好的分析表示。最近，正在开发用于解决这些问题的数值数据驱动方法，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，存在根本的不确定性。这种所谓的奈特式不确定性具有根本性的重要性，是数学和经济学领域都在深入研究的课题。捕获模型不确定性的一种方法是通过分布鲁棒方法，见[1]分布鲁棒优化（DRO）被表述为一个最小-最大问题，其中内部最大化被接管了一组概率度量（模糊集），外部最小化被接管了所有可接受的控制。模糊集通常作为参考模型部分观测到的分布信息的一个小扰动给出。这个项目的基本目的是理解当模糊集使用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设置，以适应沃瑟斯坦度规的名义。为了定义自适应Wasserstein距离，我们将自己限制为所有因果耦合，即时刻t的目标过程仅依赖于时刻t之前的源过程。这一限制使得自适应Wasserstein距离与经典Wasserstein距离本质上不同。新的距离使我们能够同时捕获信息流和状态空间几何的相关性。最近的开创性成果[2]表明，由自适应Wasserstein距离生成的拓扑与其他自适应拓扑的概念一致，例如弱嵌套拓扑、Hellwig的信息拓扑、Aldous的扩展弱拓扑。事实上，它是概率度量空间上最粗糙的拓扑，使得最优停止问题连续。另一方面，适应的Wasserstein距离允许我们同时处理离散和扩散测量。至关重要的是，它们还允许我们捕捉状态空间的几何形状，尽管被赋予适应的沃瑟斯坦距离的过程空间的测地线性质仍然是一个开放的问题。该项目旨在考虑离散时间和连续时间，以及从一个到另一个的限制通道。同样，目的是通过对DRO问题的分析，包括对偶性，以及发展一阶逼近值函数和最优控制。这建立在使用常规Wasserstein距离的单周期设置的作品上，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，风险规避随机规划等。将考虑在机器学习、数学金融和统计学方面的应用。参考资料：[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统计与应用概率研究领域。

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.