Vector-Valued Robust Stochastic Control

Abstract

We study a dynamic stochastic control problem subject to Knightian uncertainty with multiobjective (vector-valued) criteria. Assuming the preferences across expected multiloss vectors are represented by a given, yet general, preorder, we address the model uncertainty by adopting a robust or minimax perspective, minimizing expected loss across the worst-case model. For loss functions taking scalar values, there is no ambiguity in interpreting supremum and infimum. In contrast, major challenges for multi-loss control problems include properly defining and interpreting the notions of supremum and infimum, as well as addressing their non-uniqueness. To deal with these, we employ the notion of an ideal point vector-valued supremum for the robust part of the problem, while we view the control part as a multi-objective (or vector) optimization problem. Using a set-valued framework, we derive both a weak and a strong version of the dynamic programming principle (DPP) or Bellman equations for two appropriately chosen value functions: the collection of all worst expected losses across all feasible actions, and for its upper image. The weak version of Bellman's principle is proved under minimal assumptions. To establish a stronger version of DPP, we introduce the rectangularity property with respect to a general preorder. We also show that the weak minimizers obey the time consistency property. Finally, we study the important particular case of component-wise partial order of vectors, and conclude with some illustrative examples motivated by financial problems.