A note on time-inconsistent stochastic control problems with higher-order moments

In this paper, we extend the research on time-consistent stochastic control problems with higher-order moments, as formulated by [Y. Wang et al. SIAM J. Control. Optim., 63 (2025), in press]. We consider a linear controlled dynamic equation with state-dependent diffusion, and let the sum of a conventional mean-variance utility and a fairly general function of higher-order central moments be the objective functional. We obtain both the sufficiency and necessity of the equilibrium condition for an open-loop Nash equilibrium control (ONEC), under some continuity and integrability assumptions that are more relaxed and natural than those employed before. Notably, we derive an extended version of the stochastic Lebesgue differentiation theorem for necessity, because the equilibrium condition is represented by some diagonal processes generated by a flow of backward stochastic differential equations whose the data do not necessarily satisfy the usual square-integrability. Based on the derived equilibrium condition, we obtain the algebra equation for a deterministic ONEC. In particular, we find that the mean-variance equilibrium strategy is an ONEC for our higher-order moment problem if and only if the objective functional satisfies a homogeneity condition.