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The joint distribution of the surplus prior to ruin and the deficit at ruin in some Sparre Andersen models. (English) Zbl 1043.60036
Authors’ summary: For the Sparre Andersen risk model, we derive a general expression for $$h(u,x,y)$$, the joint density function of the surplus prior to ruin and the deficit at ruin when the initial surplus is $$u$$. This density function is expressed in terms of the corresponding density function when the initial surplus is 0. We apply a known result for $$h(0,x,y)$$ in the situation when inter-claim times follow a generalised Erlang distribution to derive expressions for $$h(u,x,y)$$ when individual claims have a phase-type($$m$$) distribution, $$m \in Z^+$$. We also consider the case when inter-claim times follow a phase-type(2) distribution and derive an expression for $$h(0,x,y)$$.

##### MSC:
 60G50 Sums of independent random variables; random walks 91B30 Risk theory, insurance (MSC2010)
##### Keywords:
Sparre Andersen risk model; Joint density function; Ruin
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##### References:
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