Mean-field control problems with multi-dimensional singular controls

Paper: arXiv 2308.04378

Authors: Robert Denkert, Ulrich Horst

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.

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a highly sophisticated theoretical framework introducing two-layer parametrisations to solve mean-field control problems with multi-dimensional singular controls. While mathematically rigorous and novel in its continuous-time stochastic control approach, it is purely theoretical with no empirical backtesting or real-world data application.

Research Flowchart

  flowchart TD
    A[Research Goal: Solve Extended Mean-Field Control<br/>with Multi-Dimensional Singular Controls] --> B[Input: Model with Jump Costs &<br/>Mean-Field Interaction Terms]
    B --> C[Method: Develop Two-Layer Parametrisations<br/>of Stochastic Processes]
    C --> D[Jointly & Consistently Interpolate Jumps<br/>Pathwise and Distributionally]
    D --> E[Derive Explicit Reward Representation<br/>via Minimal Jump Costs]
    E --> F[Establish Dynamic Programming<br/>Principle DPP]
    F --> G[Outcome: Characterize Value Function as Minimal<br/>Super-Solution to QVI in Wasserstein Space]