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How long is the surplus below zero? (English) Zbl 0777.62096

For the classical compound Poisson continuous-time surplus process the following evaluations are considered: duration of the first negative surplus, duration of any other negative surplus, total duration of negative surplus.
The author develops the Gerber model [H. U. Gerber, Insur. Math. Econ. 9, No. 2/3, 115-119 (1990; Zbl 0731.62153)], using his martingale method. The symmetry between the distributions of time of ruin and duration of a negative surplus is discussed for the zero initial surplus. Finally, the author presents two examples, considering exponential and gamma \((2,\beta)\) distributions.
Reviewer: L.S.Ioffe (Haifa)

MSC:

62P05 Applications of statistics to actuarial sciences and financial mathematics

Citations:

Zbl 0731.62153

Software:

Mathematica
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Full Text: DOI

References:

[1] Bowers, N. L.; Gerber, H. U.; Hickman, C. J.; Jones, D. A.; Nesbitt, C. J., Actuarial Mathematics (1987), Society of Actuaries: Society of Actuaries Itasca, IL · Zbl 0634.62107
[2] Dickson, D. C.M., On the distribution of the surplus prior to ruin, Insurance: Mathematics & Economics, 11, 3 (1992) · Zbl 0770.62090
[3] Dufresne, F.; Gerber, H. U., The surpluses immediately before and at ruin, and the amount of the claim causing ruin, Insurance: Mathematics & Economics, 7, 193-199 (1988) · Zbl 0674.62072
[4] Gerber, H. U., An Introduction to Mathematical Risk Theory (1979), University of Pennsylvania: University of Pennsylvania Philadelphia, PA, S.S. Huebner Foundation for Insurance Education · Zbl 0431.62066
[5] Gerber, H. U., Mathematical fun with run theory, Insurance: Mathematics & Economics, 7, 15-23 (1988) · Zbl 0657.62121
[6] Gerber, H. U., When does the surplus reach a given target?, Insurance: Mathematics & Economics, 9, 115-119 (1990) · Zbl 0731.62153
[7] Gerber, H. U.; Goovaerts, M. J.; Kaas, R., On the probability and severity of ruin, Astin Bulletin, 17, 151-163 (1987)
[8] Wolfram, S., The Advanced Book Program, (Mathematica - A system for doing mathematics by computer (1991), Addison-Wesley: Addison-Wesley Redwood City, CA)
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