A class of finite-dimensional numerically solvable McKean-Vlasov control problems

¹ University of Leeds, Leeds, United Kingdom
e-mail: A.Balata@leeds.ac.uk
² Univ. Paris Diderot - LPSM, Paris, France
e-mail: hure@lpsm.paris
³ Operations Research and Financial Engineering, Princeton University, Princeton, USA
e-mail: lauriere@princeton.edu
⁴ Univ. Paris Diderot - LPSM and FiME Lab, Paris, France
e-mail: pham@math.univ-paris-diderot.fr
⁵ EDF and FiME Lab, Palaiseau, France
e-mail: isaque.santa-brigida-pimentel@edf.fr

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.