The expected discounted penalty function under a risk model with stochastic income.

*(English)*Zbl 1181.91100The paper deals with a ruin model framed within the general scenario of the expected discounted penalty function, where premiums and claims follow compound Poisson processes.

In this framework the authors deduce a defective renewal equation and an integral equation involving the expected penalty function; in particular explicit expressions of such functions are given in some special cases.

Moreover, assuming that premiums follow the Erlang\((n, \beta)\) distribution, the authors specialize the defective renewal equation for the expected discounted penalty function.

The theoretical results are applied to specific examples under particular hypotheses on claims and premiums.

In this framework the authors deduce a defective renewal equation and an integral equation involving the expected penalty function; in particular explicit expressions of such functions are given in some special cases.

Moreover, assuming that premiums follow the Erlang\((n, \beta)\) distribution, the authors specialize the defective renewal equation for the expected discounted penalty function.

The theoretical results are applied to specific examples under particular hypotheses on claims and premiums.

Reviewer: Emilia Di Lorenzo (Napoli)

##### MSC:

91B30 | Risk theory, insurance (MSC2010) |

60K05 | Renewal theory |

##### Keywords:

defective renewal equation; Erlang\((n, \beta)\) premium distribution; expected discounted penalty function; probability of ruin; Laplace transform of the time to ruin
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\textit{C. Labbé} and \textit{K. P. Sendova}, Appl. Math. Comput. 215, No. 5, 1852--1867 (2009; Zbl 1181.91100)

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