Closed-loop strong equilibria for time-inconsistent control problems with higher-order moments

In this paper, we study closed-loop strong equilibrium strategies for the time-inconsistent control problem with higher-order moments formulated by [Wang et al. SIAM J. Control. Optim., 63 (2025), 1560--1589]. Since time-inconsistency makes the dynamic programming principle inapplicable, the problem is treated as a game between the decision maker and her future selves. As a time-consistent solution, the previously proposed Nash equilibrium control is merely a stationary point and does not necessarily reach a maximum in the game-theoretical prospective. To address this issue, we consider the strong equilibrium strategy, from which any spike deviation will be worse off. We derive sufficient conditions for strong equilibrium strategies by expanding the objective function corresponding to the spike deviation with respect to the variational factor up to higher orders, and simplify the result by exploiting the partial differential equations that a Nash equilibrium control satisfies. Next, we examine whether the derived Nash equilibrium control is a strong equilibrium strategy. In particular, we provide a sufficient condition for the case with linear controlled equations. When the model parameters are constant, we find that the derived Nash equilibrium control for the mean-variance problem must be a strong equilibrium strategy, while that for the mean-variance-skewness-kurtosis problem is not necessarily a strong equilibrium strategy.