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, yet general, preorder, we address the model uncertainty by adopting a robust or minimax perspective, minimizing expected loss across the worst-case model. In contrast to the scalar case, major challenges for multi-loss control problems include properly defining and interpreting the notions of supremum and infimum, and addressing the non-uniqueness of these suprema and infima.  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 strong version of the dynamic programming principle (DPP) or Bellman equations by taking the value function as the collection of all the worst expected losses across all feasible actions. 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 further study a particular, but important, case of component-wise partial order of vectors, for which we additionally derive DPP under a different set-valued notion for the value function, the so-called upper image of the multi-objective problem. Finally, we provide illustrative examples motivated by financial problems.

Short bio: Igor Cialenco is a Full Professor in the Department of Applied Mathematics at Illinois Institute of Technology. His primary research interests are in mathematical finance, stochastic control and statistical analysis of SPDEs, with emphasis on environmental finance, risk and performance measures, time consistency in decision making, optimal investment and robo-advising, nonlinear pricing, counterparty risk, and reinforcement learning.

He is a member of editorial boards of numerous scientific journals, including as Managing Editor for the International Journal of Theoretical and Applied Finance (IJTAF). He served as elected Chair and Program Director for the Society for Industrial and Applied Mathematics (SIAM) Activity Group on Financial Mathematics and Engineering.

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