Discrete time McKean-Vlasov control problem: a dynamic programming approach

TL;DR

This paper develops a dynamic programming approach for discrete-time McKean-Vlasov control problems, reformulating them into deterministic problems with distribution as the state, and applies it to portfolio optimization and linear-quadratic control.

Contribution

It introduces a novel dynamic programming framework for nonlinear mean-field systems in discrete time, enabling explicit solutions for complex control problems.

Findings

Explicit solutions for mean-variance portfolio selection.

Solution to multivariate linear-quadratic McKean-Vlasov control.

Validation of the dynamic programming principle in this context.

Abstract

We consider the stochastic optimal control problem of nonlinear mean-field systems in discrete time. We reformulate the problem into a deterministic control problem with marginal distribution as controlled state variable, and prove that dynamic programming principle holds in its general form. We apply our method for solving explicitly the mean-variance portfolio selection and the multivariate linear-quadratic McKean-Vlasov control problem.