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Some characteristics of a surplus process in the presence of an upper barrier. (English) Zbl 1055.91058
Summary: For the classical model of risk theory, we consider the time until the insurer’s surplus reaches an upper barrier \(a\). We consider some properties of hitting times, first unconditionally and then conditional on the event that ruin does not occur before the surplus exceeds \(a\). For the latter case, we obtain an upper bound for the expected time to reach \(a\) given that ruin has not occurred. We also study the more general case where the insurer’s target profit is not constant but varies with time. By considering the random walk associated with the surplus process, we show that the distribution of the overshoot over a general barrier is always exponential and we obtain expressions for the moments of first passage times for a non-constant barrier.

91B30 Risk theory, insurance (MSC2010)
62P05 Applications of statistics to actuarial sciences and financial mathematics
Full Text: DOI
[1] Dickson, D.C.M., Gray, G.R., 1984. Approximations to the probability of ruin in the presence of an upper absorbing barrier. Scandnavian Actuarial Journal, 105-115. · Zbl 0584.62174
[2] Feller, W., 1971. An Introduction to Probability Theory and Its Applications, Vol. II, 2nd Edition. Wiley, New York. · Zbl 0219.60003
[3] Gerber, H.U., When does a surplus reach a given target?, Insurance: mathematics and economics, 9, 115-119, (1990) · Zbl 0731.62153
[4] Gerber, H.U.; Goovaerts, M.J.; Kaas, R., On the probability and severity of ruin, ASTIN bulletin, 17, 151-163, (1987)
[5] Gut, A., On the moments and limit distributions of some first passage times, Annals of probability, 2, 277-308, (1974) · Zbl 0278.60031
[6] Gut, A., 1988. Stopped Random Walks: Limit Theorems and Applications. Springer, New York. · Zbl 0634.60061
[7] Lin, X.S.; Willmot, G.E., The moments of the time of ruin the surplus before ruin, and the deficit at ruin, Insurance: mathematics and economics, 27, 19-44, (2000) · Zbl 0971.91031
[8] Picard, P., On some measures of the severity of ruin in the classical Poisson model, Insurance: mathematics and economics, 14, 107-115, (1994) · Zbl 0813.62093
[9] Picard, P.; Lefevre, C., On the first crossing of the surplus process with a given upper barrier, Insurance: mathematics and economics, 14, 163-179, (1994) · Zbl 0806.62089
[10] Woodroofe, M., A renewal theorem for curved boundaries and moments of first passage times, Annals of probability, 4, 67-80, (1976) · Zbl 0368.60099
[11] Woodroofe, M., 1982. Nonlinear Renewal Theory in Sequential Analysis. SIAM, Philadelphia, PA. · Zbl 0487.62062
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