×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Beekman, J.A., Two stochastic processes, (1974), Almqvist & Wiksell Stockholm · Zbl 0137.35601
[2] Cox, D.R.; Miller, H.D., The theory of stochastic processes, (1965), Methuen London · Zbl 0149.12902
[3] Dickson, D.C.M., On the distribution of surplus prior to ruin, Insurance: mathematics and economics, 11, 191-207, (1992) · Zbl 0770.62090
[4] Dickson, D.C.M.; Egídio dos Reis, A.D., Ruin problems and dual events, Insurance: mathematics and economics, 14, 51-60, (1994) · Zbl 0803.62091
[5] Dufresne, F.; Gerber, H.U., The surpluses immediately before and at ruin, and the amount of the claim causing ruin, Insurance: mathematics and economics, 7, 193-199, (1988) · Zbl 0674.62072
[6] Feller, W., ()
[7] Gerber, H.U., An introduction to mathematical risk theory, () · Zbl 0431.62066
[8] Gerber, H.U., When does the surplus reach a given target?, Insurance: mathematics and economics, 9, 115-119, (1990) · Zbl 0731.62153
[9] Gerber, H.U.; Shiu, E.S.W., On the time value of ruin, North American actuarial journal, 2, 1, (1998), to appear · Zbl 1333.91027
[10] Kendall, D.G., Some problems in the theory of dams, Journal of the royal statistical society series B, 19, 207-212, (1957) · Zbl 0118.35502
[11] Lundberg, F., Some supplementary researches on the collective risk theory, Skandinavisk aktuarietidskrift, 15, 137-158, (1932) · Zbl 0004.36006
[12] Panjer, H.H.; Willmot, G.E., Insurance risk models, (1992), Society of Actuaries Schaumburg, IL
[13] Prabhu, N.U., On the ruin problem of collective risk theory, Annals of mathematical statistics, 32, 757-764, (1961) · Zbl 0103.13302
[14] Prabhu, N.U., Stochastic storage processes: queues, insurance risk, and dams, (1980), Springer New York · Zbl 0453.60094
[15] Resnick, S.I., Adventures in stochastic processes, (1992), Birkhäuser Boston · Zbl 0762.60002
[16] Seal, H.L., Stochastic theory of a risk business, (1969), Wiley New York · Zbl 0196.23501
[17] Takács, L., Combinatorial methods in the theory of stochastic processes, (1967), Wiley New York, Reprinted by Krieger, Huntington, NY, 1977 · Zbl 0189.17602
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.