A stochastic maximum principle for singular mean-field regime-switching optimal control
Maalvladédon Ganet Somé, Edward Korveh
TL;DR
The paper develops a stochastic maximum principle for a mean-field control problem with regime-switching and singular controls, deriving both necessary and sufficient conditions. The necessary principle uses spike variation and a second-order adjoint to handle diffusion dependence, while the sufficient principle relies on convexity/concavity of the Hamiltonian. The mean-field and regime-switching structure induces time-inconsistency, precluding standard dynamic programming, and the results are demonstrated through an inter-bank borrowing-lending model with transaction costs. Overall, the work extends existing theory to more general mean-field, regime-switching, and jump settings and provides a concrete financial application illustrating the derived optimal controls.
Abstract
In this paper, we investigate a mean-field singular stochastic optimal control problem for systems governed by mean-field regime-switching singular stochastic differential equations. The state process is assumed to depend on both a regular and a singular control, and the coefficient associated with the singular component is allowed to be regime dependent. We derive both necessary and sufficient singular stochastic maximum principles. Because the regular control domain is not assumed to be convex, we employ the spike variation technique and obtain the necessary maximum principle by introducing a second-order adjoint process. As an application, we use the main theoretical results to analyse an inter-bank borrowing and lending model with transaction costs.