Precise large deviations of aggregate claims in a risk model with regression-type size-dependence. (English) Zbl 1281.62223
Summary: We further extend the renewal risk model and introduce a more practical regression-type dependence structure between inter-arrival times and claim sizes, which is described in the framework of a web Markov skeleton process. We investigate large deviations of the aggregate claims, and, for a case of heavy-tailed claims, we obtain a precise large-deviation formula of the aggregate claims under some assumption about the regression-type size-dependence, which is consistent with existing ones in the literature.
MSC:
| 62P05 | Applications of statistics to actuarial sciences and financial mathematics |
| 60F10 | Large deviations |
| 91B30 | Risk theory, insurance (MSC2010) |
| 60J99 | Markov processes |
References:
| [1] | Albrecher, H.; Teugels, J. L., Exponential behavior in the presence of dependence in risk theory, J. Appl. Probab., 43, 1, 257-273 (2006) · Zbl 1097.62110 |
| [2] | Asimit, A. V.; Badescu, A. L., Extremes on the discounted aggregate claims in a time dependent risk model, Scand. Actuar. J., 2, 93-104 (2010) · Zbl 1224.91041 |
| [3] | Badescu, A. L.; Cheung, E. C.K.; Landriault, D., Dependent risk models with bivariate phase-type distributions, J. Appl. Probab., 46, 1, 113-131 (2009) · Zbl 1172.91009 |
| [4] | Baltrūnas, A.; Leipus, R.; S˘iaulys, J., Precise large deviation results for the total claim amount under subexponential claim sizes, Statist. Probab. Lett., 78, 1206-1214 (2008) · Zbl 1145.60018 |
| [5] | Bingham, N. H.; Goldie, C. M.; Teugels, J. L., Regular Variation (1987), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0617.26001 |
| [6] | Boudreault, M.; Cossette, H.; Landriault, D.; Marceau, E., On a risk model with dependence between interclaim arrivals and claim sizes, Scand. Actuar. J., 5, 265-285 (2006) · Zbl 1145.91030 |
| [7] | Chen, Y.; Yuen, K. C., Precise large deviations of aggregate claims in a size-dependent renewal risk model, Insurance Math. Econom., 51, 2, 457-461 (2012) · Zbl 1284.60057 |
| [8] | Cossette, H.; Marceau, E.; Marri, F., On the compound Poisson risk model with dependence based on a generalized Farlie-Gumbel-Morgenstern copula, Insurance Math. Econom., 43, 3, 444-455 (2008) · Zbl 1151.91565 |
| [9] | Kaas, R.; Tang, Q., A large deviation result for aggregate claims with dependent claim occurrences, Insurance Math. Econom., 36, 251-259 (2005) · Zbl 1110.62145 |
| [10] | Klüppelberg, C.; Mikosch, T., Large deviations of heavy-tailed random sums with applications in insurance and finance, J. Appl. Probab., 34, 293-308 (1997) · Zbl 0903.60021 |
| [11] | Koc˘etova, J.; Leipus, R.; S˘iaulys, J., A property of the renewal counting process with application to the finite-time ruin probability, Lith. Math. J., 49, 55-61 (2009) · Zbl 1185.60098 |
| [12] | Korchevsky, V. M.; Petrov, V. V., On the strong law of large numbers for sequences of dependent random variables, Vestnik St. Petersburg Univ. Math., 43, 3, 143-147 (2010) · Zbl 1253.60039 |
| [13] | Li, J.; Tang, Q.; Wu, R., Subexponential tails of discounted aggregate claims in a time-dependent renewal risk model, Adv. Appl. Probab., 42, 1126-1146 (2010) · Zbl 1205.62061 |
| [14] | Lin, J., The general principle for precise large deviations of heavy-tailed random sums, Statist. Probab. Lett., 78, 749-758 (2008) · Zbl 1140.60313 |
| [15] | Liu, L., Precise large deviations for dependent random variables with heavy tails, Statist. Probab. Lett., 79, 1290-1298 (2009) · Zbl 1163.60012 |
| [16] | Liu, Y. T.; Ma, Z. M.; Zhou, C., Web Markov skeleton processes and their applications, Tohoku Math. J., 63, 665-695 (2011) · Zbl 1236.60078 |
| [18] | Ng, K. W.; Tang, Q.; Yan, J.; Yang, H., Precise large deviations for the prospective-loss process, J. Appl. Probab., 40, 391-400 (2003) · Zbl 1028.60024 |
| [19] | Ng, K. W.; Tang, Q.; Yan, J.; Yang, H., Precise large deviations for sums of random variables with consistently varying tails, J. Appl. Probab., 41, 1, 93-107 (2004) · Zbl 1051.60032 |
| [20] | Tang, Q.; Tsitsiashvili, G., Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks, Stochastic Process. Appl., 108, 299-325 (2003) · Zbl 1075.91563 |
| [21] | Wang, S.; Wang, W., Precise large deviations for sums of random variables with consistently varying tails in multi-risk models, J. Appl. Probab., 44, 889-900 (2007) · Zbl 1134.60322 |
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.
