Mean-field neural networks-based algorithms for McKean-Vlasov control problems *

Paper: arXiv 2212.11518

Authors: Huyên Pham, Xavier Warin

Abstract

This paper is devoted to the numerical resolution of McKean-Vlasov control problems via the class of mean-field neural networks introduced in our companion paper [25] in order to learn the solution on the Wasserstein space. We propose several algorithms either based on dynamic programming with control learning by policy or value iteration, or backward SDE from stochastic maximum principle with global or local loss functions. Extensive numerical results on different examples are presented to illustrate the accuracy of each of our eight algorithms. We discuss and compare the pros and cons of all the tested methods.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 4.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper demonstrates high mathematical sophistication by leveraging mean-field neural networks and Wasserstein space formulations to solve McKean-Vlasov control problems. However, empirical rigor is limited to synthetic numerical examples rather than real-world quantitative financial datasets or market backtests.

Research Flowchart

  flowchart TD
    A[Research Goal: Solve McKean-Vlasov Control Problems via Mean-Field Neural Networks] --> B[Inputs: Dynamics on Wasserstein Space & Loss Functions]
    B --> C[Methodology: Develop 8 Neural Network Algorithms]
    C --> D[Approach 1: Dynamic Programming<br>Policy & Value Iteration]
    C --> E[Approach 2: Stochastic Maximum Principle<br>BSDE with Global & Local Losses]
    D & E --> F[Computational Execution: Train Mean-Field NNs across Example Cases]
    F --> G[Outcomes: Demonstrated High Accuracy & Comparative Trade-offs of all 8 Methods]