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The density of the time to ruin for a Sparre Andersen process with Erlang arrivals and exponential claims. (English) Zbl 1144.91025
The Sparre Andersen surplus process \(\{U(t)\}_{t\geq0}\) is defined by \(U(t)=u+ct-\sum_{i=1}^{N(t)}X_{i}\), where \(u\) is the initial surplus, \(c\) is the rate of premium income per unit time, \(\{X_{i}\}_{i=1}^{\infty}\) is a sequence of i.i.d. random variables, and \(\{N(t)\}_{t\geq0}\) is a counting process. The sequence of i.i.d. random variables \(\{W_{i}\}_{i=1}^{\infty}\) represents the claim inter-arrival times, with \(W_1\) being the time until the first claim, and it is assumed that claim amounts are independent of claim inter-arrival times. The authors derive expressions for the density of the time to ruin given that ruin occurs in the considered Sparre Andersen model in which individual claim amounts are exponentially distributed and inter-arrival times are Erlang distributed.

MSC:
91B30 Risk theory, insurance (MSC2010)
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[1] Gerber, H.U. and Shiu, E.S.W. (1998) On the time value of ruin <i>North American Actuarial Journal</i>, 2, pp. 48 - 78. · Zbl 1081.60550
[2] Seal, H.L.(1978) <i> Survival probabilities: the goal of risk theory</i>. New York : John Wiley & Sons. · Zbl 0386.62088
[3] Garcia, J.M.A., 2002, Explicit solutions for survival probabilities in a finite time horizon. Presented to the 6th International IME Congress, Lisbon. http://pascal.iseg.utl.pt/cemapre/ime2002/index.html(No. 49).
[4] Drekic, S. and Willmot, G.E. (2003) On the density and moments of the time of ruin with exponential claims <i>ASTIN Bulletin</i>, 33, pp. 11 - 21. · Zbl 1062.60007
[5] Dickson, D.C.M. and Willmot, G.E., 2005, The density of the time to ruin in the classical Poisson risk model. <i>ASTIN Bulletin</i> 35, 45–60. · Zbl 1097.62113
[6] Dickson, D.C.M. and Hipp, C. (2001) On the time to ruin for Erlang(2) risk processes <i>Insurance: Mathematics & Economics</i>, 29, pp. 333 - 344. · Zbl 1074.91549
[7] Li, S. and Garrido, J. (2004) On ruin for the Erlang(n) risk process <i>Insurance: Mathematics & Economics</i>, 34, pp. 391 - 408. · Zbl 1188.91089
[8] Gerber, H.U.andShiu, E.S.W., 2005, The time value of ruin in a Sparre Andersen model. <i>North American Actuarial Journal</i> , 9, 2, 49–84. · Zbl 1085.62508
[9] Grandell, J.(1991) <i> Aspects of Risk Theory</i>. New York : Springer-Verlag. · Zbl 0717.62100
[10] Abramowitz, M. and Stegun, I.A.(1965) <i> Handbook of Mathematical Functions</i>. New York : Dover. · Zbl 0171.38503
[11] Sneddon, I.N.(1972) <i> The Use of Integral Transforms</i>. New York : McGraw-Hill. · Zbl 0237.44001
[12] Henrici, P.(1977) <i> Applied and Computational Complex Analysis, Volume 2</i>. New York : John Wiley & Sons. · Zbl 0363.30001
[13] Marsden, J.E. and Hoffman, M.J.(1999) <i> Basic Complex Analysis 3rd edition</i>. New York : W.H. Freeman.
[14] Olver, F.W.J.(1974) <i> Asymptotics and Special Functions</i>. New York : Academic Press. · Zbl 0303.41035
[15] Henrici, P.(1986) <i> Applied and Computational Complex Analysis, Volume 3</i>. New York : John Wiley & Sons. · Zbl 0578.30001
[16] Whittaker, E.T. and Watson, G.N.(1927) <i> A Course of Modern Analysis 2nd edition</i>. Cambridge University Press. · Zbl 0108.26903
[17] Dickson, D.C.M. and Waters, H.R. (2002) The distribution of the time to ruin in the classical risk model <i>ASTIN Bulletin</i>, 32, pp. 299 - 313. · Zbl 1098.62136
[18] Segerdahl, C.-O. (1955) When does ruin occur in the collective theory of risk? <i>Skandinavisk Aktuarietidskrift</i>, XXXVIII, pp. 22 - 36. · Zbl 0067.12105
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