Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-29T07:51:29.443Z Has data issue: false hasContentIssue false
  • English
  • Français

ASYMPTOTICS FOR A DISCRETE-TIME RISK MODEL WITH THE EMPHASIS ON FINANCIAL RISK

Published online by Cambridge University Press:  27 June 2014

Enkelejd Hashorva  and
Jinzhu Li
Enkelejd Hashorva
Affiliation:
School of Mathematical Science and LPMC, Nankai University, Tianjin 300071, People's Republic of China E-mail: [email protected]
Jinzhu Li
Affiliation:
Department of Actuarial Science, Faculty of Business and Economics, University of Lausanne, UNIL-Dorigny 1015, Lausanne, Switzerland E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper focuses on a discrete-time risk model in which both insurance risk and financial risk are taken into account. We study the asymptotic behavior of the ruin probability and the tail probability of the aggregate risk amount. Precise asymptotic formulas are derived under weak moment conditions of involved risks. The main novelty of our results lies in the quantification of the impact of the financial risk.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 28 , Issue 4 , October 2014 , pp. 573 - 588
Copyright
Copyright © Cambridge University Press 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Bingham, N.H., Goldie, C.M. & Teugels, J.L. (1987). Regular Variation. Cambridge: Cambridge University Press.Google Scholar
2.Breiman, L. (1965). On some limit theorems similar to the arc–sin law. Theory Probab. Appl. 10(2): 323–331.CrossRefGoogle Scholar
3.Cai, J. & Tang, Q. (2004). On max-sum equivalence and convolution closure of heavy-tailed distributions and their applications. J. Appl. Probab. 41(1): 117–130.Google Scholar
4.Chen, Y. (2011). The finite-time ruin probability with dependent insurance and financial risks. J. Appl. Probab. 48(4), 1035–1048.CrossRefGoogle Scholar
5.Chen, Y. & Yuen, K.C. (2009). Sums of pairwise quasi-asymptotically independent random variables with consistent variation. Stoch. Models 25(1): 76–89.Google Scholar
6.Cline, D.B.H. (1987). Convolutions of distributions with exponential and subexponential tails. J. Aust. Math. Soc. Ser. A 43(3): 347–365.Google Scholar
7.Embrechts, P. & Goldie, C.M. (1980). On closure and factorization properties of subexponential and related distributions. J. Aust. Math. Soc. Ser. A 29(2): 243–256.Google Scholar
8.Embrechts, P., Klüpelberg, C. & Mikosch, T. (1997). Modelling Extremal Events: For Insurance and Finance. Berlin: Springer-Verlag.Google Scholar
9.Feller, W. (1971). An Introduction to Probability Theory and its Applications. Vol. II. 2nd ed.New York–London–Sydney: John Wiley & Sons.Google Scholar
10.Frolova, A., Kabanov, Y. & Pergamenshchikov, S. (2002). In the insurance business risky investments are dangerous. Financ. Stoch. 6(2), 227–235.CrossRefGoogle Scholar
11.Geluk, J. & Tang, Q. (2009). Asymptotic tail probabilities of sums of dependent subexponential random variables. J. Theor. Probab. 22(4): 871–882.Google Scholar
12.Hao, X. & Tang, Q. (2012). Asymptotic ruin probabilities for a bivariate Lévy-driven risk model with heavy-tailed claims and risky investments. J. Appl. Probab. 49(4): 939–953.CrossRefGoogle Scholar
13.Hashorva, E. & Li, J. (2013). ECOMOR and LCR reinsurance with gamma-like claims. Insur. Math. Econ. 53(1): 206–215.Google Scholar
14.Jessen, A.H. &, Mikosch, T. (2006). Regularly varying functions. Publ. Inst. Math. (Beograd) (N.S.) 80(94): 171–192.Google Scholar
15.Kalashnikov, V. & Norberg, R. (2002). Power tailed ruin probabilities in the presence of risky investments. Stoch. Process. Appl. 98(2): 211–228.Google Scholar
16.Konstantinides, D.G. & Mikosch, T. (2005). Large deviations and ruin probabilities for solutions to stochastic recurrence equations with heavy-tailed innovations. Ann. Probab. 33(5): 1992–2035.Google Scholar
17.Norberg, R. (1999). Ruin problems with assets and liabilities of diffusion type. Stoch. Process. Appl. 81(2): 255–269.Google Scholar
18.Pakes, A.G. (2004). Convolution equivalence and infinite divisibility. J. Appl. Probab. 41(2): 407–424.Google Scholar
19.Pakes, A.G. (2007). Convolution equivalence and infinite divisibility: corrections and corollaries. J. Appl. Probab. 44(2): 295–305.Google Scholar
20.Paulsen, J. (1993). Risk theory in a stochastic economic environment. Stoch. Process. Appl. 46(2): 327–361.Google Scholar
21.Paulsen, J. (2008). Ruin models with investment income. Probab. Surv. 5: 416–434.CrossRefGoogle Scholar
22.Pergamenshchikov, S. & Zeitouny, O. (2006). Ruin probability in the presence of risky investments. Stoch. Process. Appl. 116(2): 267–278.Google Scholar
23.Resnick, S.I. (1987). Extreme Values, Regular Variation, and Point Processes. New York: Springer–Verlag.Google Scholar
24.Rogozin, B.A. & Sgibnev, M.S. (1999). Banach algebras of measures on the line with given asymptotics of distributions at infinity. Siberian Math. J. 40(3): 565–576.Google Scholar
25.Rootzén, H. (1986). Extreme value theory for moving average processes. Ann. Probab. 14(2): 612–652.Google Scholar
26.Rootzén, H. (1987). A ratio limit theorem for the tails of weighted sums. Ann. Probab. 15(2): 728–747.Google Scholar
27.Samorodnitsky, G. & Taqqu, M.S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. New York: Chapman & Hall.Google Scholar
28.Tang, Q. (2006). Asymptotic ruin probabilities in finite horizon with subexponential losses and associated discount factors. Probab. Eng. Inf. Sci. 20(1): 103–113.CrossRefGoogle Scholar
29.Tang, Q. (2008). Insensitivity to negative dependence of asymptotic tail probabilities of sums and maxima of sums. Stoch. Anal. Appl. 26(3): 435–450.Google Scholar
30.Tang, Q. & Tsitsiashvili, G. (2003). Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks. Stoch. Process. Appl. 108(2): 299–325.CrossRefGoogle Scholar
31.Tang, Q. & Tsitsiashvili, G. (2004). Finite- and infinite-time ruin probabilities in the presence of stochastic returns on investments. Adv. Appl. Probab. 36(4): 1278–1299.Google Scholar
32.Yang, Y. & Wang, Y. (2013). Tail behavior of the product of two dependent random variables with applications to risk theory. Extremes 16(1): 55–74.Google Scholar
33.Zhang, Y., Shen, X. & Weng, C. (2009). Approximation of the tail probability of randomly weighted sums and applications. Stoch. Process. Appl. 119(2): 655–675.Google Scholar