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The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. (English) Zbl 0894.90047
Summary: We examine the joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. The time of ruin is analyzed in terms of its Laplace transform, which can naturally be interpreted as discounting. We show that, as a function of the initial surplus, the joint density satisfies a certain renewal equation. We generalize Dickson’s formula (see [D. C. M. Dickson, Insurance: Mathematics and Economics 11, 191–207 (1992)] which expresses the joint distribution of the surplus immediately before ruin and the deficit at ruin in terms of the probability of ultimate ruin.

##### MSC:
 91B30 Risk theory, insurance (MSC2010) 91G50 Corporate finance (dividends, real options, etc.) 60K05 Renewal theory
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