Data-driven Distributionally Robust Stochastic Optimization via Wasserstein Distance with Applications to Portfolio Risk Management and Inventory Control

Abstract

The central theme of this dissertation is stochastic optimization under distributional ambiguity. One canthink of this as a two player game between a decision maker, who tries to minimize some loss or maximize some reward, and an adversarial agent that chooses the worst case, or least favorable, distribution (to the decision maker) from some ambiguity set. The Wasserstein distance metric is used to specify the ambiguity set which is known as a Wasserstein ball of some finite radius d. At the center of this ball, is the empirical distribution, which serves as a proxy for the true underlying distribution. In that sense, this line of research has been called data-driven robust optimization in the academic literature. The primal problem is infinite dimensional since the Wasserstein ball contains all finite and discrete distributions within distance d of the empirical distribution. As such, it would appear more difficult to solve the stochastic optimization problem in this setting. This research makes use of (recent) Lagrangian duality results in distributional robustness and (classical) moments duality results to formulate and solve the simpler finite dimensional dual problem. Different problem formulations are considered, both with and without moment constraints on the ambiguity set. Some interesting practical applications of these results include single stage and multistage problems in portfolio risk management and inventory control. We also investigate the notion of time consistency between the static and dynamic (multi-period) problem formulations. Time consistency is a desirable property in that the decision maker knows that the optimal policy determined at time zero will not change as realizations of the data process and corresponding system state are observed. In particular, this dissertation considers optimal decision making for portfolio problems in counterparty credit risk, funding risk, statistical arbitrage, option exercise, asset purchasing/selling, and quantification of certain profit and risk metrics. In addition, we consider the classical newsvendor model (both with and without moment constraints) in the single period and multi-period settings. We conclude with some commentary on our findings throughout this work and provide some suggestions for further research.