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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)
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