Stackelberg differential game for reinsurance: Mean-variance framework and random horizon

We consider a reinsurance problem for a mean-variance Stackelberg game with a random time horizon, in which an insurer and a reinsurer are the two players. The reinsurer computes its premium according to the mean-variance premium principle with parameters (θ,η)∈R₊². First, for any pair (θ,η)∈R₊², we compute the per-loss reinsurance strategy that maximizes a mean-variance functional of the insurer's surplus at the end of the random horizon. This reinsurance strategy is excess-of-loss reinsurance with constant proportional reinsurance for losses above the deductible. Then, given the information of what the insurer will choose when offered any mean-variance premium principle, we determine the optimal pair (θ^⁎,η^⁎)∈R₊² that maximizes the reinsurer's expected surplus at the end of the random horizon. We show that the equilibrium deductible and coinsurance are both independent of the risk aversion parameter of the insurer. We also show that if the claim severity is light-tailed, in the sense that its hazard rate function is non-decreasing, then the equilibrium reinsurance is pure excess-of-loss reinsurance with no loading for the variance of the loss. Moreover, we show that if the claim severity is heavy-tailed, in the sense that its hazard rate function is decreasing, then the equilibrium reinsurance has a non-trivial coinsurance, with a positive loading for the variance of the loss. Finally, we solve four examples to illustrate these latter two, important results.