A Decomposition Method for LQ Conditional McKean-Vlasov Control Problems with Random Coefficients
This provides a more tractable solution method for a class of stochastic control problems, but the improvement is incremental as it builds on existing decomposition ideas.
The paper proposes a decomposition method for linear-quadratic McKean-Vlasov control problems with random coefficients, decomposing the problem into two decoupled stochastic optimal control problems that can be solved using classical methods. The approach avoids technical complexities of existing methods and establishes equivalence between the auxiliary and original problems.
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.