Stochastic maximum principle for mean-field type singular optimal control problem with discounted cost

In this article, mean-field type stochastic singular optimal control problem with discounted cost is studied over an infinite time interval. The discounted cost makes the cost functional is bounded, which guarantees the existence of optimal control. The control variable has two components as classical and singular control. Moreover the singular control satisfies bounded variation, non-decreasing continuous on the left with right limits. The proposed system is investigated in two different cases, such as without and with delay. In addition, infinite horizon version of stochastic maximum principle is established by using the convex control domain in each case. The obtained theoretical results are applied to optimal harvesting problem and optimal consumption problem.