Stochastic differential reinsurance games with capital injections

This paper investigates a class of reinsurance game problems between two insurance companies under the framework of non-zero-sum stochastic differential games. Both insurers can purchase proportional reinsurance contracts from reinsurance markets and have the option of conducting capital injections. We assume the reinsurance premium is calculated under the generalized variance premium principle. The objective of each insurer is to maximize the expected value that synthesizes the discounted utility of his surplus relative to a reference point, the penalties caused by his own capital injection interventions, and the gains brought by capital injections of his competitor. We prove the verification theorem and derive explicit expressions of the Nash equilibrium strategy by solving the corresponding quasi-variational inequalities. Numerical examples are also conducted to illustrate our results.