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**The maximum surplus before ruin in an Erlang\((n)\) risk process and related problems.**
*(English)*
Zbl 1168.60363

Summary: We study the distribution of the maximum surplus before ruin in a Sparre Andersen risk process with the inter-claim times being Erlang(\(n\)) distributed. This distribution can be analyzed through the probability that the surplus process attains a given level from the initial surplus without first falling below zero. This probability, viewed as a function of the initial surplus and the given level, satisfies a homogeneous integro-differential equation with certain boundary conditions. Its solution can be expressed as a linear combination of \(n\) linearly independent particular solutions of the homogeneous integro-differential equation. Explicit results are obtained when the individual claim amounts are rationally distributed. When \(n=2\), all the results can be expressed explicitly in terms of the non-ruin probability. We apply our results by looking at (i) the maximum severity of ruin and (ii) the distribution of the amount of dividends under a constant dividend barrier.

### MSC:

60K10 | Applications of renewal theory (reliability, demand theory, etc.) |

60K05 | Renewal theory |

91B30 | Risk theory, insurance (MSC2010) |

### Keywords:

Sparre Andersen risk model; Erlang inter-claim times; integro-differential equation; maximum surplus before ruin; maximum severity of ruin; dividends
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\textit{S. Li} and \textit{D. C. M. Dickson}, Insur. Math. Econ. 38, No. 3, 529--539 (2006; Zbl 1168.60363)

### References:

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