×

zbMATH — the first resource for mathematics

On the distribution of surplus immediately after ruin under interest force. (English) Zbl 1012.91027
A compound Poisson model is considrered \(dU_\delta(t)= pdt+ U_\delta(t) \delta dt-dX(t)\). Here \(U_\delta(t)\) is the value of the reserve at time \(t\), \(p\) is the premium rate that the insurance company receives, \(\delta\) is the interest force, and \(X(t)= \sum^{N(t)}_{j=1} Y_j\), where \(N(t)\) is a homogeneous Poisson process counting the number of claims in the time interval \((0,t]\), and \(Y_j\) is the amount of the \(j\)th claim. Using the techniques of B. Sundt and J. L. Tengels [Isur. Math. Econ. 16, 7-22 (1995; Zbl 0838.62098)], the equations are obtained for the probability of ruin \(G_\delta (u,y)\) beginning with initial reserve \(u\) and the deficit at most \(y> 0\) immediately after the claim causing ruin. The bounds for \(G_\delta (0,y)\) are derived. In the case \(\delta=0\), the asymptotic result for \(G_\delta (u,y)\), as \(u\to +\infty\), is obtained, which generalizes the result in J. Grandell [Aspects of Risk Theory. Springer, Berlin (1991; Zbl 0717.62100)] to the case where the severity of ruin is taken into account.

MSC:
91B30 Risk theory, insurance (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Boogaert, P.; Crijns, V., Upper bound on ruin probabilities in case of negative loadings and positive interest rates, Insurance: mathematics and economics, 6, 221-232, (1987) · Zbl 0642.62058
[2] Di Lorenzo, E.; Tessitore, G., Approximation solutions of severity of ruin, Blatter deutsche gesellschaft fur versicherungsmathematik, XXII, 705-709, (1996) · Zbl 0860.62078
[3] 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
[4] Dufresne, F.; Gerber, H.U., The probability and severity of ruin for combinations of exponential claim amount distributions and their translations, Insurance: mathematics and economics, 7, 75-80, (1988) · Zbl 0637.62101
[5] Gerber, H.V.; Shiu, E.S.W., The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin, Insurance: mathematics and economics, 21, 129-137, (1997) · Zbl 0894.90047
[6] Gerber, H.V.; Shiu, E.S.W., On the time value of ruin, North American actuarial journal, 2, 1, 48-78, (1998)
[7] Gerber, H.V.; Goovaerts, M.J.; Kaas, R., On the probability and severity of ruin, Astin bulletin, 17, 2, 151-153, (1987)
[8] Grandell, J., 1991. Aspects of Risk Theory. Springer, Berlin. · Zbl 0717.62100
[9] Paulsen, J., Ruin theory with compounding assets: a survey, Insurance: mathematics and economics, 22, 3-16, (1998) · Zbl 0909.90115
[10] Paulsen, J.; Gjessing, H.K., Ruin theory with stochastic return on investments, Advances in applied probability, 29, 965-985, (1997) · Zbl 0892.90046
[11] Sundt, B.; Teugels, J.L., Ruin estimates under interest force, Insurance: mathematics and economics, 16, 7-22, (1995) · Zbl 0838.62098
[12] Willmot, G.E.; Lin, X., Exact and approximate properties of the distribution of surplus before and after ruin, Insurance: mathematics and economics, 23, 91-110, (1997) · Zbl 0914.90074
[13] Yang, H., 1999. Non-exponential bounds for ruin probability with interest effect included. Scandinavian Actuarial Journal, 66-79. · Zbl 0922.62113
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.