## Abstract

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

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

Original language | English |
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Pages (from-to) | 435-477 |

Journal | ASTIN Bulletin |

Volume | 48 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2018 |

Externally published | Yes |