  We consider a risk model where deficits after ruin are covered by a new type
of reinsurance contract that provides capital injections. To allow the
insurance company's survival after ruin, the reinsurer injects capital only at
ruin times caused by jumps larger than a chosen retention level. Otherwise
capital must be raised from the shareholders for small deficits. The problem
here is to determine adequate reinsurance premiums. It seems fair to base the
net reinsurance premium on the discounted expected value of any future capital
injections. Inspired by the results of Huzak et al. (2004) and Ben Salah (2014)
on successive ruin events, we show that an explicit formula for these
reinsurance premiums exists in a setting where aggregate claims are modeled by
a subordinator and a Brownian perturbation. Here ruin events are due either to
Brownian oscillations or jumps and reinsurance capital injections only apply in
the latter case. The results are illustrated explicitly for two specific risk
models and in some numerical examples.
