Continuous-time optimal investment in DC pension plans with path-dependent reference points

This paper investigates the optimal investment strategies of defined contribution (DC) pension plans under an S-shaped utility function with a path-dependent reference point. The reference point is assumed to be the initial wealth adjusted by subsequent paper gains and losses, whose dynamic is given by a differential equation. A new approach is proposed to convert the problem into an equivalent one with a single state variable and a fixed reference point, enabling explicit solutions via the martingale method in a complete market. In the incomplete market with short-selling prohibition, a new proof of the equivalence between the primal and concavified problems is provided, and the explicit solution is derived by the dual control method. Numerical tests are carried out to assess the effects of the model parameters and short-selling prohibition, as well as to compare path-dependent and constant reference points in terms of strategies, gains and risks.