Precise large deviations for aggregate claims in a two-dimensional compound dependent risk model

Weiwei Ni; Kaiyong Wang; Weiwei Ni; Kaiyong Wang

doi:10.3934/math.2023456

AIMS Mathematics

2023, Volume 8, Issue 4: 9106-9117. doi: 10.3934/math.2023456

Previous Article Next Article

Research article

Precise large deviations for aggregate claims in a two-dimensional compound dependent risk model

Weiwei Ni ,
Kaiyong Wang ^,

School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215009, China

Received: 24 December 2022 Revised: 23 January 2023 Accepted: 02 February 2023 Published: 13 February 2023
MSC : 60F10, 91B05, 91G05

This paper considers a two-dimensional compound risk model. We mainly investigate the claim sizes and inter-arrival times are size-dependent. When the claim sizes have consistently varying tails, we obtain the precise large deviations for aggregate amount of claims in the above dependent compound risk model.
- precise large deviations,
- two-dimensional compound risk model,
- size-dependence,
- consistently varying distribution
Citation: Weiwei Ni, Kaiyong Wang. Precise large deviations for aggregate claims in a two-dimensional compound dependent risk model[J]. AIMS Mathematics, 2023, 8(4): 9106-9117. doi: 10.3934/math.2023456

Related Papers:

Abstract

This paper considers a two-dimensional compound risk model. We mainly investigate the claim sizes and inter-arrival times are size-dependent. When the claim sizes have consistently varying tails, we obtain the precise large deviations for aggregate amount of claims in the above dependent compound risk model.

References

[1]	A. Aleškevičienė, R. Leipus, J. Šiaulys, A probabilistic look at tail behavior of random sums under consistent variation with applications to the compound renewal risk, Extremes, 11 (2008), 261–279. https://doi.org/10.1007/s10687-008-0057-3 doi: 10.1007/s10687-008-0057-3
[2]	N. H. Bingham, C. M. Goldie, J. L. Teugels, Regular variation, Cambridge: Cambridge University Press, 1987.
[3]	X. Bi, S. Zhang, Precise large deviation of aggregate claims in a risk model with regression-type size-dependence, Stat. Probabil. Lett., 83 (2013), 2248–2255. https://doi.org/10.1016/j.spl.2013.06.009 doi: 10.1016/j.spl.2013.06.009
[4]	J. Cai, H. Li, Dependence properties and bounds for ruin probabilities in multivariate compound risk models, J. Multivariate Anal., 98 (2007), 757–773. https://doi.org/10.1016/j.jmva.2006.06.004 doi: 10.1016/j.jmva.2006.06.004
[5]	D. B. H. Cline, G. Samorodnitsky, Subexponentiality of the product of independent random variables, Stoch. Proc. Appl., 49 (1994), 75–98. https://doi.org/10.1016/0304-4149(94)90113-9 doi: 10.1016/0304-4149(94)90113-9
[6]	G. A. Delsing, M. R. H. Mandjes, P. J. C. Spreij, E. M. M. Winands, An optimization approach to adaptive multi-dimensional capital management, Insur. Math. Econ., 84 (2019), 87–97. https://doi.org/10.1016/j.insmatheco.2018.10.001 doi: 10.1016/j.insmatheco.2018.10.001
[7]	P. Embrechts, C. Klüppelberg, T. Mikosch, Modelling extremal events for insurance and finance, Berlin: Springer, 1997.
[8]	K. Fu, X. Shen, H. Li, Precise large deviations for sums of claim-size vectors in a two-dimensional size-dependent renewal risk model, Acta Math. Appl. Sin. Engl. Ser., 37 (2021), 539–547. https://doi.org/10.1007/s10255-021-1030-z doi: 10.1007/s10255-021-1030-z
[9]	H. Guo, S. Wang, C. Zhang, Precise large deviations of aggregate claims in a compound size-dependent renewal risk model, Commun. Stat. Theor. Method., 46 (2017), 1107–1116. https://doi.org/10.1080/03610926.2015.1010011 doi: 10.1080/03610926.2015.1010011
[10]	J. Kočetova, R. Leipus, J. Šiaulys, A property of the renewal counting process with application to the finite-time probability, Lith. Math. J., 49 (2009), 55–61. https://doi.org/10.1007/s10986-009-9032-1 doi: 10.1007/s10986-009-9032-1
[11]	D. G. Konstantinides, F. Loukissas, Precise large deviations for consistently varying-tailed distribution in the compound renewal risk model, Lith. Math. J., 50 (2010), 391–400. https://doi.org/10.1007/s10986-010-9094-0 doi: 10.1007/s10986-010-9094-0
[12]	D. Lu, Lower bounds of large deviation for sums of long-tailed claims in a multi-risk model, Stat. Probabil. Lett., 82 (2012), 1242–1250. https://doi.org/10.1016/j.spl.2012.03.020 doi: 10.1016/j.spl.2012.03.020
[13]	R. B. Nelsen, An introduction to copulas, New York: Springer, 2006.
[14]	X. Shen, H. Tian, Precise large deviations for sums of two-dimensional random vectors and dependent components with extended regularly varying tails, Commun. Stat. Theor. Method., 45 (2016), 6357–6368. https://doi.org/10.1080/03610926.2013.839794 doi: 10.1080/03610926.2013.839794
[15]	Q. Tang, C. Su, T. Jiang, J. Zhang, Large deviations for heavy-tailed random sums in compound renewal model, Stat. Probabil. Lett., 52 (2001), 91–100. https://doi.org/10.1016/S0167-7152(00)00231-5 doi: 10.1016/S0167-7152(00)00231-5
[16]	Q. Tang, Insensitivity to negative dependence of the asymptotic behavior of precise large deviations, Electron. J. Probab., 11 (2006), 107–120. https://doi.org/10.1214/EJP.v11-304 doi: 10.1214/EJP.v11-304
[17]	K. Wang, Y. Cui, Y. Mao, Estimates for the finite-time ruin probability of a time-dependent risk model with a Brownian perturbation, Math. Probl. Eng., 2020 (2020), 7130243. https://doi.org/10.1155/2020/7130243 doi: 10.1155/2020/7130243
[18]	K. Wang, L. Chen, Precise large deviations for the aggregate claims in a dependent compound renewal risk model, J. Inequal. Appl., 257 (2019), 1–25. https://doi.org/10.1186/s13660-019-2209-1 doi: 10.1186/s13660-019-2209-1
[19]	S. Wang, W. Wang, Precise large deviations for sums of random variables with consistently varying tails in multi-risk mode, J. Appl. Probab., 44 (2007), 889–900. https://doi.org/10.1239/jap/1197908812 doi: 10.1239/jap/1197908812
[20]	S. Wang, W. Wang, Precise large deviations for sums of random variables with consistent variation in dependent multi-risk models, Commun. Stat. Theor. Method., 42, (2013), 4444–4459. https://doi.org/10.1080/03610926.2011.648792
[21]	B. Xun, K. C. Yuen, K. Wang, The finite-time ruin probability of a risk model with a general counting process and stochastic return, J. Ind. Manag. Optim., 18 (2022), 1541–1556. https://doi.org/10.3934/jimo.2021032 doi: 10.3934/jimo.2021032
[22]	Y. Yang, R. Leipus, J. Šiaulys, Precise large deviations for compound random sums in the presence of dependence structures, Comput. Math. Appl., 64 (2012), 2074–2083. https://doi.org/10.1016/j.camwa.2012.04.003 doi: 10.1016/j.camwa.2012.04.003
[23]	Y. Yang, K. Wang, J. Liu, Z. Zhang, Asymptotics for a bidimensional risk model with two geometric Lévy price processes, J. Ind. Manag. Optim., 15 (2019), 481–505. http://dx.doi.org/10.3934/jimo.2018053 doi: 10.3934/jimo.2018053

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)