| Abstract: |
| We study a dynamic stochastic control problem subject to Knightian uncertainty with multi-objective (vector-valued) criteria. Assuming the preferences across expected multi-loss vectors are represented by a given preorder, we address the model uncertainty by adopting a robust or minimax perspective, minimizing expected loss across the worst-case model. 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. |
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