Wasserstein distributional sensitivity to model uncertainty in dynamic context

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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.