# Jump-Diffusion Process

The study models two competing hybrid DC pension fund managers as a stochastic differential game in which liability risk follows a jump-diffusion process with truncated exponential jump amplitudes and the exact asset-to-liability ratio formulation is used, and it derives closed-form Nash equilibrium portfolio strategies and value functions via Hamilton–Jacobi–Bellman dynamic programming, finding that equilibrium weights are independent of liability jump parameters, reduce to K=(μ−r)/σ² and are independent of both managers' risk aversion in the symmetric liability-correlation case, that competition strictly amplifies risk-taking relative to the single-agent benchmark, and that a manager's own liability jumps reduce welfare while the competitor's jumps improve it.

The study models two competing hybrid DC pension fund managers as a stochastic differential game in which liability risk follows a jump-diffusion process with truncated exponential jump amplitudes and the exact asset-to-liability ratio formulation is used, and it derives closed-form Nash equilibrium portfolio strategies and value functions via Hamilton–Jacobi–Bellman dynamic programming, finding that equilibrium weights are independent of liability jump parameters, reduce to K=(μ−r)/σ² and are independent of both managers' risk aversion in the symmetric liability-correlation case, that competition strictly amplifies risk-taking relative to the single-agent benchmark, and that a manager's own liability jumps reduce welfare while the competitor's jumps improve it.