Supplementary file 1_A stochastic differential game between two hybrid pension fund managers under a jump-diffusion liability process.pdf

Background<p>Hybrid defined contribution (DC) pension funds, combining fixed contributions with stochastic liability dynamics, face competitive pressures when members reallocate contributions to higher-performing funds.</p>Methods<p>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.</p>Results<p>The Nash equilibrium portfolio weights are independent of liability jump parameters. In the symmetric liability-correlation case, they reduce to K = (μ−r)/σ², independent of both managers' risk aversion. In the asymmetric case, risk aversion enters through a cross-liability correction term.