Equilibrium strategies for stochastic control problems with higher-order moments and applications to portfolio selection

In this paper we derive a novel characterization result for time-consistent stochastic control problems with higher-order moments, originally formulated by Wang et al. [SIAM J. Control. Optim., 63 (2025), 1560--1589], and newly explore many solvable instances including a mean-variance-excess kurtosis portfolio selection problem. By improving an asymptotic result of the variational process for the uniform boundedness and integrability properties, we obtain both the sufficiency and necessity of an equilibrium condition for an open-loop Nash equilibrium control (ONEC). This condition is simply formulated by the diagonal processes of a flow of backward stochastic differential equations (BSDEs) whose data do not necessarily satisfy the usual square-integrability condition. In particular, for linear controlled dynamics with deterministic parameters, we show that the ONEC can be derived by solving a polynomial algebraic equation under a class of nonlinear objective functions. Interestingly, the mean-variance equilibrium strategy is an ONEC for our general higher-order moment problem if and only if a homogeneity condition holds. Additionally, in the case with random parameters, we characterize the ONEC by finitely many BSDEs with a recurrence relation. As an intuitive illustration, the solution to the mean-variance-skewness problems is given by a quadratic BSDE.