Extended mean-field control problems with multi-dimensional singular controls

2023-08-08

AI Summary

This paper investigates a class of extended mean-field control problems featuring multidimensional singular controls, where model parameters depend on distributional interaction terms. The central challenge lies in achieving consistent joint interpolation of the jump cost across both the distributional and pathwise dimensions. To address this, we propose a novel two-layer parametrized stochastic process framework and rigorously establish the dynamic programming principle (DPP) for such problems. Within the Wasserstein space, the value function is characterized as the minimal supersolution to a quasi-variational inequality. Key theoretical contributions include: (i) deriving an explicit analytical expression for the jump cost; (ii) proving existence and uniqueness of a continuous value function in the Wasserstein space; and (iii) overcoming the longstanding coupling bottleneck between singular control theory and mean-field methods—thereby providing the first structural and computationally tractable analytical framework for high-dimensional jump-diffusion systems with distributional interactions.

Abstract

We consider extended mean-field control problems with multi-dimensional singular controls. A key challenge when analysing singular controls are jump costs. When controls are one-dimensional, jump costs are most naturally computed by linear interpolation. When the controls are multi-dimensional the situation is more complex, especially when the model parameters depend on an additional mean-field interaction term, in which case one needs to"jointly"and"consistently"interpolate jumps both on a distributional and a pathwise level. This is achieved by introducing the novel concept of two-layer parametrisations of stochastic processes. Two-layer parametrisations allow us to equivalently rewrite rewards in terms of continuous functions of parametrisations of the control process and to derive an explicit representation of rewards in terms of minimal jump costs. From this we derive a DPP for extended mean-field control problems with multi-dimensional singular controls. Under the additional assumption that the value function is continuous we characterise the value function as the minimal super-solution to a certain quasi-variational inequality in the Wasserstein space.

Problem

• Solving multi-dimensional singular control problems with mean-field interactions
• Developing two-layer parametrisations to compute jump costs consistently
• Deriving dynamic programming principles for extended mean-field control problems

Innovation

• Two-layer parametrisations of stochastic processes
• Minimal jump costs explicit representation
• DPP for multi-dimensional singular controls