A Class of Finite-Dimensional Numerically Solvable McKean-Vlasov Control Problems

TL;DR

This paper introduces a new class of finite-dimensional control problems for McKean-Vlasov models with common noise, extending quadratic cases, and compares three numerical methods through practical financial and systemic risk examples.

Contribution

It extends known linear quadratic MKV control problems to polynomial cases and demonstrates reduction to finite-dimensional control problems with a comparative analysis of numerical methods.

Findings

Polynomial MKV control problems can be reduced to finite-dimensional problems.

Quantization, regression by control randomization, and regress later are effective numerical methods.

Numerical examples include portfolio selection, liquidation, and systemic risk models.

Abstract

We address a class of McKean-Vlasov (MKV) control problems with common noise, called polynomial conditional MKV, and extending the known class of linear quadratic stochastic MKV control problems. We show how this polynomial class can be reduced by suitable Markov embedding to finite-dimensional stochastic control problems, and provide a discussion and comparison of three probabilistic numerical methods for solving the reduced control problem: quantization, regression by control randomization, and regress later methods. Our numerical results are illustrated on various examples from portfolio selection and liquidation under drift uncertainty, and a model of interbank systemic risk with partial observation.