Robust equilibrium strategies in a defined benefit pension plan game

This paper investigates the robust non-zero-sum games in an aggregated overfunded defined benefit pension plan. The sponsoring firm is concerned with the investment performance of the fund surplus, while the participants act like a union to claim a share of the fund surplus. The financial market consists of one risk-free asset and n risky assets. The firm and the union are ambiguous about the financial market and care about the robust strategies under the worst-case scenario. The union's objective is to maximize the expected discounted utility of the additional benefits. The firm's two objectives are to maximize the expected discounted utility of the fund surplus and the probability of the fund surplus reaching an upper level before hitting a lower level in the worst-case scenario. We present a general robust non-zero-sum game with stopping times, which contains the two objectives as special cases. Hamilton-Jacobi-Bellman-Isaacs equations and verification theorem are presented for the robust optimization problem. We obtain explicit solutions in the related two robust non-zero-sum games for the firm and the union. Numerical results are illustrated to depict the economic behaviors of the robust equilibrium strategies in these two different games.