Some results on ruin probabilities in a two-dimensional risk model.

*(English)*Zbl 1055.91041Summary: 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) |

##### Keywords:

Adjustment coefficient; Asymptotic good bound; Integral-differential equation; Phase-type distribution
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\textit{W.-S. Chan} et al., Insur. Math. Econ. 32, No. 3, 345--358 (2003; Zbl 1055.91041)

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