The mean field optimal switching problem: Variational inequality approach

This paper is concerned with the optimal switching problem, the dynamics of which is McKean-Vlasov diffusions and the criterion is a function of the law of the state process. A remarkable new feature in this setting is that the switching time also impacts the dynamics of the state process through the dependence of the coefficients on the law. The mean field switching problem is introduced in weak formulation in terms of the joint marginal law of the state process and switching process. This specification satisfies a dynamic programming principle, and the corresponding variational inequality is an obstacle problem on the Wasserstein space. Our verification result characterizes the nature of optimal switching policies, highlighting the crucial need to randomized switching. To illuminate the idea, we take two-mode switching as an example for simplicity.