On joint ruin probabilities of a two-dimensional risk model with constant interest rate. (English) Zbl 1266.91034
Summary: We consider the two-dimensional risk model introduced in [F. Avram, Z. Palmowski and M. Pistorius, Insur. Math. Econ. 42, No. 1, 227–234 (2008; Zbl 1141.91482)] with constant interest rate. We derive the integral-differential equations of the Laplace transforms, and asymptotic expressions for the finite-time ruin probabilities with respect to the joint ruin times \(T_{\max}(u_{1},u_{2})\) and \(T_{\min}(u_{1},u_{2})\) respectively.
MSC:
| 91B30 | Risk theory, insurance (MSC2010) |
| 60J25 | Continuous-time Markov processes on general state spaces |
| 60H30 | Applications of stochastic analysis (to PDEs, etc.) |
Keywords:
two-dimensional risk model; constant interest rate; joint ruin probability; integral-differential equation; asymptotic expressionCitations:
Zbl 1141.91482References:
| [1] | Asmussen, S. (2000). Ruin Probabilities . World Scientific, River Edge, NJ. · Zbl 0960.60003 |
| [2] | Avram, F., Palmowski, Z. and Pistorius, M. (2008). A two-dismensional ruin problem on the positive quadrant. Insurance Math. Econom. 42 , 227-234. · Zbl 1141.91482 · doi:10.1016/j.insmatheco.2007.02.004 |
| [3] | Cai, J. and Li, H. (2005). Multivariate risk model of phase type. Insurance Math. Econom. 36 , 137-152. · Zbl 1122.60049 · doi:10.1016/j.insmatheco.2004.11.004 |
| [4] | Cai, J. and Li, H. (2007). Dependence properties and bounds for ruin probabilities in multivariate compound risk models. J. Multivariate Anal. 98 , 757-773. · Zbl 1280.91090 · doi:10.1016/j.jmva.2006.06.004 |
| [5] | Chan, W.-S., Yang, H. and Zhang, L. (2003). Some results on ruin probabilities in a two-dimensional risk model. Insurance Math. Econom. 32 , 345-358. · Zbl 1055.91041 · doi:10.1016/S0167-6687(03)00115-X |
| [6] | Gerber, H. U. (1981). On the probability of ruin in the presence of a linear dividend barrier. Scand. Actuarial J. 1981 , 105-115. · Zbl 0455.62086 · doi:10.1080/03461238.1981.10413735 |
| [7] | Li, J., Liu, Z. and Tang, Q. (2007). On the ruin probabilities of a bidimensional perturbed risk model. Insurance Math. Econom. 41 , 185-195. · Zbl 1119.91056 · doi:10.1016/j.insmatheco.2006.10.012 |
| [8] | Paulsen, J. (2008). Ruin models with investment income. Prob. Surveys 5 , 416-434. · Zbl 1189.91077 · doi:10.1214/08-PS134 |
| [9] | Protter, P. E. (1992). Stochastic Integration and Differential Equations: A New Approach . Springer, Berlin. · Zbl 0694.60047 |
| [10] | Rolski, T., Schmidli, H., Schimidt, V. and Teugels, J. (1999). Stochastic Processes for Finance and Insurance . John Wiley, Chichester. · Zbl 0940.60005 |
| [11] | Yuen, K. C., Guo, J. and Wu, X. (2006). On the first time of ruin in the bivariate compound Poisson model. Insurance Math. Econom. 38 , 298-308. · Zbl 1095.62120 · doi:10.1016/j.insmatheco.2005.08.011 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
