Relaxed stochastic maximum principle for nonzero sum stochastic differential games driven by McKean–Vlasov SDEs with regime-switching

This paper investigates a nonzero sum stochastic differential game where the state dynamics are governed by McKean–Vlasov type stochastic differential equations with regime-switching. By employing the convex perturbation technique and 𝐿 -derivatives, we establish the relaxed stochastic maximum principle that provides necessary optimality conditions for Nash equilibria. Moreover, we prove two verification theorems: one for convex control domains and the other for non-convex control domains where the coefficients satisfy additional structural conditions. As an application, a linear-quadratic problem is solved.