# zbMATH — the first resource for mathematics

Some results on ruin probabilities in a two-dimensional risk model. (English) Zbl 1055.91041
Summary: Ruin theory under multi-dimensional risk models is very complex. Even in the two-dimensional case, the problem is challenging. In this paper, we consider a bivariate risk model. Three different types of ruin probabilities are defined. Using some results of one-dimensional risk processes, simple bounds for the two-dimensional ruin probabilities are obtained. Numerical examples and simulation experiments are given to illustrate the tightness of the bounds. A partial integral–differential equation satisfied by the two-dimensional ruin probabilities is derived. Although special cases and examples in this paper provide some exciting results, the problem of ruin probability in a multi-dimensional risk model is still far from solved. We hope that this paper stimulates more research by actuaries in this area.

##### MSC:
 91B30 Risk theory, insurance (MSC2010)
Full Text:
##### References:
 [1] Ambagaspitiya, R.S., On the distributions of two classes of correlated aggregate claims, Insurance: mathematics and economics, 24, 301-308, (1999) · Zbl 0945.62110 [2] Asmussen, S.; Rolski, T., Computational methods in risk theory: a matrix-algorithmic approach, Insurance: mathematics and economics, 10, 259-274, (1991) · Zbl 0748.62058 [3] Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A., Nesbitt, C.J., 1997. Actuarial Mathematics, 2nd ed. The Society of Actuaries. · Zbl 0634.62107 [4] Denuit, M.; Genest, C.; Marceau, E., Stochastic bounds on sums of dependent risks, Insurance: mathematics and economics, 25, 85-104, (1999) · Zbl 1028.91553 [5] Dhaene, J.; Denuit, M., The safest dependence structure among risks, Insurance: mathematics and economics, 25, 11-21, (1999) · Zbl 1072.62651 [6] Dhaene, J.; Goovaerts, M.J., Dependency of risks and stop-loss order, ASTIN bulletin, 26, 201-212, (1996) [7] Dhaene, J.; Goovaerts, M.J., On the dependency of risks in the individual life model, Insurance: mathematics and economics, 19, 243-253, (1997) · Zbl 0931.62089 [8] Dickson, D.C.M., Hipp, C., 1998. Ruin problems for phase-type(2) risk processes. Working Paper, Centre of Actuarial Studies, the University of Melbourbe, Australia. http://www.ecom.unimelb.edu.au/actwww. · Zbl 0907.90097 [9] Ditkin, V.A., Prudnikov, A.P., 1962. Operational Calculus in Two Variables and its Applications. Pergamon Press, Oxford (English Translation Version). · Zbl 0116.30902 [10] Goovaerts, M.J.; Dhaene, J., The compound Poisson approximation for a portfolio of dependent risks, Insurance: mathematics and economics, 18, 81-85, (1996) · Zbl 0853.62079 [11] Halmos, P., 1978. Measure Theory, 2nd ed. Springer, New York. [12] Hu, T.; Wu, Z., On the dependence of risks and the stop-loss premiums, Insurance: mathematics and economics, 24, 323-332, (1999) · Zbl 0945.62109 [13] Müller, A., Stop-loss order for portfolios of dependent risks, Insurance: mathematics and economics, 21, 219-223, (1997) · Zbl 0894.90022 [14] Müller, A., Stochastic orderings generated by integrals: a unified study, Advances in applied probability, 29, 414-428, (1997) · Zbl 0890.60015 [15] Panjer, H.H., Willmot, G.E., 1992. Insurance Risk Models. The Society of Actuaries. [16] Rolski, T., Schmidli, H., Schmidt, V., Teugels, J., 1999. Stochastic Processes For Finance and Insurance. Wiley, New York. · Zbl 0940.60005 [17] Sundt, B., On multivariate panjer recursions, ASTIN bulletin, 29, 1, 29-45, (1999)
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.