Global stochastic maximum principle of regime-switching systems with time-varying delay under non-convex control domains

This paper establishes a global stochastic maximum principle for regime-switching systems with time-varying delays under non-convex control domains. The proposed approach relies on the spike variation and duality methods. One contribution of this paper is the establishment of the well-posedness of anticipated backward stochastic differential equations featuring both time-varying delays and regime-switching, which arise as the first- and the second-order adjoint equations required to handle the non-convexity of the control domain. Additionally, the classical duality relation is extended to accommodate the time-varying delay. Finally, a linear–quadratic problem is presented as an example to demonstrate the applicability of the theoretical results.