Robust Control

Robust optimal investment and consumption strategies with portfolio constraints and stochastic environment ArXiv ID: 2407.02831 “View on arXiv” Authors: Unknown Abstract We investigate a continuous-time investment-consumption problem with model uncertainty in a general diffusion-based market with random model coefficients. We assume that a power utility investor is ambiguity-averse, with the preference to robustness captured by the homothetic multiplier robust specification, and the investor’s investment and consumption strategies are constrained to closed convex sets. To solve this constrained robust control problem, we employ the stochastic Hamilton-Jacobi-Bellman-Isaacs equations, backward stochastic differential equations, and bounded mean oscillation martingale theory. Furthermore, we show the investor incurs (non-negative) utility loss, i.e. the loss in welfare, if model uncertainty is ignored. When the model coefficients are deterministic, we establish formally the relationship between the investor’s robustness preference and the robust optimal investment-consumption strategy and the value function, and the impact of investment and consumption constraints on the investor’s robust optimal investment-consumption strategy and value function. Extensive numerical experiments highlight the significant impact of ambiguity aversion, consumption and investment constraints, on the investor’s robust optimal investment-consumption strategy, utility loss, and value function. Key findings include: 1) short-selling restriction always reduces the investor’s utility loss when model uncertainty is ignored; 2) the effect of consumption constraints on utility loss is more delicate and relies on the investor’s risk aversion level. ...

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. ...