ON THE COMPOUND POISSON RISK MODEL WITH PERIODIC CAPITAL INJECTIONS

The analysis of capital injection strategy in the literature of insurance risk models (e.g. Pafumi, 1998; Dickson and Waters, 2004) typically assumes that whenever the surplus becomes negative, the amount of shortfall is injected so that the
company can continue its business forever. Recently, Nie et al. (2011) has proposed an alternative model in which capital is immediately injected to restore
the surplus level to a positive level b when the surplus falls between zero and b,
and the insurer is still subject to a positive ruin probability. Inspired by the idea
of randomized observations in Albrecher et al. (2011b), in this paper, we further generalize Nie et al. (2011)’s model by assuming that capital injections are
only allowed at a sequence of time points with inter-capital-injection times being Erlang distributed (so that deterministic time intervals can be approximated
using the Erlangization technique in Asmussen et al. (2002)). When the claim
amount is distributed as a combination of exponentials, explicit formulas for the
Gerber–Shiu expected discounted penalty function (Gerber and Shiu, 1998) and
the expected total discounted cost of capital injections before ruin are obtained.
The derivations rely on a resolvent density associated with an Erlang random
variable, which is shown to admit an explicit expression that is of independent
interest as well. We shall provide numerical examples, including an application
in pricing a perpetual reinsurance contract that makes the capital injections and
demonstration of how to minimize the ruin probability via reinsurance. Minimization of the expected discounted capital injections plus a penalty applied at
ruin with respect to the frequency of injections and the critical level b will also
be illustrated numerically