Mean-field control problems with multi-dimensional singular controls

TL;DR

This paper develops a novel framework for analyzing extended mean-field control problems with multi-dimensional singular controls, introducing two-layer parametrizations to handle complex jump costs and deriving a dynamic programming principle.

Contribution

It introduces two-layer parametrizations for multi-dimensional singular controls, enabling explicit reward representation and a new dynamic programming principle in mean-field control.

Findings

Explicit representation of rewards in terms of minimal jump costs

Derivation of a dynamic programming principle for complex controls

Characterization of the value function as a minimal super-solution

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.