A Decomposition Method for LQ Conditional McKean-Vlasov Control Problems with Random Coefficients

TL;DR

This paper introduces a decomposition approach for solving complex LQ McKean-Vlasov control problems with random coefficients, simplifying the problem into two classical control problems.

Contribution

It develops a novel decomposition method that transforms the original problem into two decoupled control problems, facilitating easier solutions and analysis.

Findings

The original problem's optimal control equals the sum of auxiliary problems' controls.

Auxiliary problems can be solved using classical stochastic control methods.

The approach establishes equivalence between auxiliary and original problem solutions.

Abstract

We propose a decomposition method for solving a general class of linear-quadratic (LQ) McKean-Vlasov control problems involving conditional expectations and random coefficients, where the system dynamics are driven by two independent Wiener processes. Unlike existing approaches in the literature for these problems, such as the extended stochastic maximum principle and the extended dynamic programming methods, which often involve additional technical complexities and sometimes impose restrictive conditions on control inputs, our approach decomposes the original McKean-Vlasov control problem into two decoupled stochastic optimal control problems, one of which has a constrained admissible control set. These auxiliary problems can be solved using classical methods. We establish an equivalence between the well-posedness and solvability of the auxiliary problems and those of the original problem, and show that the sum of the optimal controls of the auxiliary problems yields the optimal control of the original problem. Moreover, by applying a variational method, we characterize the optimal solution to the McKean-Vlasov control problem via two decoupled sets of (non-McKean-Vlasov) linear forward-backward stochastic differential equations, each corresponding to one of the auxiliary problems. Finally, we show that standard dynamic programming can also be applied to solve the resulting auxiliary problems.