Vector-valued robust stochastic control

ArXiv ID: 2407.00266 “View on arXiv”

Authors: Unknown

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. For loss functions taking real (or scalar) values, there is no ambiguity in interpreting supremum and infimum. 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. 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 strong version of the dynamic programming principle (DPP) or Bellman equations by taking the value function as the collection of all 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. These results will serve as a foundation for addressing time-inconsistent problems subject to model uncertainty through the lens of a set-valued framework, as well as for studying multi-portfolio allocation problems under model uncertainty.

Keywords: Knightian Uncertainty, Multi-Objective Optimization, Dynamic Programming, Robust Control, Vector-Valued Loss

Complexity vs Empirical Score

• Math Complexity: 9.2/10
• Empirical Rigor: 2.1/10
• Quadrant: Lab Rats
• Why: The paper is highly theoretical, focusing on novel extensions of dynamic programming principles and set-valued supremum definitions under Knightian uncertainty for vector-valued criteria, with minimal computational or empirical implementation details.

  flowchart TD
    A["Research Goal<br>Multi-Objective Robust Control<br>Under Knightian Uncertainty"] --> B["Methodology Setup<br>Preorder on Vector Loss &<br>Minimax Worst-Case Perspective"]
    B --> C["Core Challenge<br>Define Sup/Inf for Vectors &<br>Handle Non-Uniqueness"]
    C --> D{"Solution Framework"}
    D --> E["General Case<br>Set-Valued Value Function"]
    D --> F["Specific Case<br>Component-Wise Partial Order"]
    E --> G["Computational Process<br>Dynamic Programming Principles (DPP)"]
    F --> G
    G --> H["Outcomes<br>Weak/Strong Bellman Equations<br>Foundations for Time-Inconsistent & Portfolio Problems"]