## The density of the time to ruin for a Sparre Andersen process with Erlang arrivals and exponential claims.(English)Zbl 1144.91025

The Sparre Andersen surplus process $$\{U(t)\}_{t\geq0}$$ is defined by $$U(t)=u+ct-\sum_{i=1}^{N(t)}X_{i}$$, where $$u$$ is the initial surplus, $$c$$ is the rate of premium income per unit time, $$\{X_{i}\}_{i=1}^{\infty}$$ is a sequence of i.i.d. random variables, and $$\{N(t)\}_{t\geq0}$$ is a counting process. The sequence of i.i.d. random variables $$\{W_{i}\}_{i=1}^{\infty}$$ represents the claim inter-arrival times, with $$W_1$$ being the time until the first claim, and it is assumed that claim amounts are independent of claim inter-arrival times. The authors derive expressions for the density of the time to ruin given that ruin occurs in the considered Sparre Andersen model in which individual claim amounts are exponentially distributed and inter-arrival times are Erlang distributed.

### MSC:

 91B30 Risk theory, insurance (MSC2010)
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### References:

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