Title: Vector-valued robust stochastic control Show Chinese title

Title: 向量值鲁棒随机控制

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. Show Chinese abstract

Abstract: 我们研究了一类在Knightian不确定性条件下具有多目标（向量值）准则的动态随机控制问题。 假设预期多损失向量之间的偏好由一个给定但一般的前序关系表示，我们通过采用鲁棒或minimax视角来处理模型不确定性，即最小化最差情况下的期望损失。 对于取实数值（或标量值）的损失函数，对上确界和下确界的解释没有歧义。 与标量情形不同，多损失控制问题的主要挑战包括正确定义和解释上确界和下确界的概念，并解决这些上确界和下确界的非唯一性。 为此，我们在问题的鲁棒部分使用了理想点向量值上确界的概念，同时将控制部分视为一个多目标（或向量）优化问题。 利用集值框架，我们将值函数视为所有可行行动下最差期望损失的集合，并由此推导出动态规划原理（DPP）或Bellman方程的弱形式和强形式。 Bellman原理的弱形式在最小假设下被证明。为了建立更强的DPP版本，我们引入了一般前序关系下的矩形性性质。 我们也进一步研究了一个特殊的但重要的情形，即向量分量的部分序关系，对于这种情况，我们基于多目标问题的所谓上图像的集值概念，额外推导出DPP。 最后，我们提供了几个受金融问题启发的示例。 这些结果将作为通过集值框架解决受模型不确定性影响的时间不一致问题的基础，同时也将用于研究在模型不确定性下的多投资组合分配问题。