## On the expectations of the present values of the time of ruin perturbed by diffusion.(English)Zbl 1066.91062

The main purpose of this paper is to study the surplus process of the classical continuous time risk model containing an independent diffusion (Wiener) process. In previous papers [C.C.-L. Tsai, G.E. Wilmot (2002)], the function of the expected discount penalty is defined as depending (linearly) on a penalty scheme involving the penalty at ruin by oscillation, the penalty at ruin caused by a claim, and the corresponding expectations of the present values of the time of ruin due to oscillation and due to a claim. The expected discount penalty function is shown to depend on a compound geometric distribution tail whose properties are studied together with those of the expectations of the present values of the time of ruin due to oscillation and due to a claim (Section 2 of the paper). Recursive formulas and explicit expressions for the moments of the compound geometric distribution (tail) and for the considered expectations of the time of ruin are also derived (provided that appropriate mathematical conditions are fulfilled).
Section 3 exposes explicit analytical solutions for the same examined functions describing the present values of the time of ruin in certain specific cases: when the claim size distribution is a combination of exponentials (Example 1) or a mixture of Erlangs (Example 2).
Section 4 achieves the asymptotic formulas and the Tijms-type approximations of the compound distribution tail and of the time of ruin expectations.
The final Section 5 proposes a lower bound and an upper bound on the considered compound geometric distribution, provided that the associated claim size distribution belongs to some of the reliability-based classes.

### MSC:

 91B30 Risk theory, insurance (MSC2010) 60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 62E17 Approximations to statistical distributions (nonasymptotic) 62E20 Asymptotic distribution theory in statistics
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### References:

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