Strong equilibrium strategy for the higher-order moment control problem

Determine whether a strong equilibrium strategy exists for the time-inconsistent stochastic control problem defined by the controlled stochastic differential equation dX_s = b(s, X_s, u_s) ds + σ(s, X_s, u_s) dW_s with objective J(t, x, u) = E_t[Ψ(t, x, X^{t,x,u}_T, E_t[X^{t,x,u}_T], E_t[(X^{t,x,u}_T − E_t[X^{t,x,u}_T])^2], …, E_t[(X^{t,x,u}_T − E_t[X^{t,x,u}_T])^n])] (for fixed integer n ≥ 2), and, if such a strategy exists, characterize or construct it under smooth perturbations in the sense of strong equilibrium (as in Huang–Zhou 2021; He–Jiang 2021).

Background

The paper adopts the closed-loop Nash equilibrium control (CNEC) notion as a time-consistent solution, using spike variations with constant perturbations. A stronger notion—strong equilibrium strategy—requires optimality against smooth functional perturbations and has been shown to fail to exist for many classical time-inconsistent problems under general perturbations.

The authors explicitly state that they do not address the existence or construction of strong equilibrium strategies for their higher-order moment objective, highlighting this as a direction left for future research.