Robust Equilibrium Strategy for Mean–Variance–Skewness Portfolio Selection Problem with Long Memory

This paper considers a robust time-consistent mean–variance–skewness portfolio selection problem for an ambiguity-averse investor by taking into account wealth-dependent risk aversion, wealth-dependent skewness preference, long memory and model uncertainty. The robust equilibrium investment strategy and the corresponding equilibrium value function are characterized for such a problem by employing an extended Hamilton–Jacobi–Bellman–Isaacs (HJBI) system via a game theoretic approach. Furthermore, for a special robust time-consistent mean–variance–skewness portfolio selection problem, the robust equilibrium investment strategy and the corresponding equilibrium value function are respectively obtained in semi-closed form. Finally, some numerical experiments are provided to indicate several new findings including (i) the robust equilibrium investment strategy displays long memory; (ii) in most cases, the mean–variance–skewness investor would invest more in the risky asset than the mean–variance investor; (iii) the skewness preference has no impact on the robust equilibrium investment strategy when the risk aversion coefficient is large enough; (iv) the skewness preference could slow down the reduction of investment in the risky asset due to the ambiguity aversion.