Time-Consistent Portfolio Selection for Rank-Dependent Utilities in a Constrained Market

We investigate the portfolio selection problem for an agent with rank-dependent utility, for which the investment strategy is constrained to take values in a closed convex set. For a constant coefficient market and constant relative risk-aversion utilities, we characterize the deterministic strict equilibrium strategies. For the case of time-invariant probability weighting functions, we provide a comprehensive characterization of the deterministic strict equilibrium strategy. The unique nonzero equilibrium, if it exists, can be determined by solving an autonomous ordinary differential equation (ODE). In the case of time-varying probability weighting functions, we observe that there may be infinitely many nonzero deterministic strict equilibrium strategies, which are derived from the positive solutions to a nonlinear singular ODE. By specifying the maximal solution to the singular ODE, we are able to identify all the positive solutions. In addition, we address the issue of selecting an optimal strategy from the numerous equilibrium strategies available.

Funding: This research was supported by the National Key R&D Program of China [Grant 2020YFA0712700] and the National Natural Science Foundation of China [Grants 12431017, 12471447, 11971301].