The Gerber-Shiu function and the generalized Cramér-Lundberg model.

*(English)*Zbl 1239.91081Summary: We extend some results in Cramér by considering the expected discounted penalty function as a generalization of the infinite time ruin probability. We consider his ruin theory model that allows the claim sizes to take positive as well as negative values. Depending on the sign of these amounts, they are interpreted either as claims made by insureds or as income from deceased annuitants, respectively. We then demonstrate that when the events’ arrival process is a renewal process, the Gerber – Shiu function satisfies a defective renewal equation. Subsequently, we consider some special cases such as when claims have exponential distribution or the arrival process is a compound Poisson process and annuity-related income has Erlang \((n, \beta )\) distribution. We are then able to specify the parameter and the functions involved in the above-mentioned defective renewal equation.

##### MSC:

91B30 | Risk theory, insurance (MSC2010) |

##### Keywords:

defective renewal equation; Erlang \((n, \beta )\) annuity-related income distribution; expected discounted penalty function; probability of ruin; Laplace transform of the time to ruin; two-sided jumps
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\textit{C. Labbé} et al., Appl. Math. Comput. 218, No. 7, 3035--3056 (2011; Zbl 1239.91081)

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