A stochastic differential game between two hybrid pension fund managers under a jump-diffusion liability process

Abstract

Background Hybrid defined contribution (DC) pension funds, combining fixed contributions with stochastic liability dynamics, face competitive pressures when members reallocate contributions to higher-performing funds. Methods We model two competing hybrid DC pension fund managers in a stochastic differential game. Liability risk is modelled as a jump-diffusion process with truncated exponential jump amplitudes, using the exact asset-to-liability ratio formulation. The Hamilton–Jacobi–Bellman dynamic programming approach yields Nash equilibrium portfolio strategies and value functions in closed form. Results The Nash equilibrium portfolio weights are independent of liability jump parameters. In the symmetric liability-correlation case, they reduce to K = (μ− r )/σ 2 , independent of both managers' risk aversion. In the asymmetric case, risk aversion enters through a cross-liability correction term. Competition strictly amplifies risk-taking relative to the single-agent benchmark. Liability jumps significantly affect welfare: a manager's own jumps reduce welfare while the competitor's jumps improve it. Discussion These results extend Nash equilibrium analysis for DC pension funds to jump-diffusion liabilities, showing that while jump parameters do not alter equilibrium portfolio strategies, they materially affect welfare through two loading functions capturing own-jump welfare costs and cross-player spillovers. Numerical illustrations confirm all theoretical predictions.