Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T08:58:42.953Z Has data issue: false hasContentIssue false
  • English
  • Français

Subexponential tails of discounted aggregate claims in a time-dependent renewal risk model

Part of: Multivariate analysis Distribution theory - Probability Applications

Published online by Cambridge University Press:  01 July 2016

Jinzhu Li ,
Qihe Tang  and
Rong Wu
Jinzhu Li*
Affiliation:
Nankai University and The University of Iowa
Qihe Tang*
Affiliation:
The University of Iowa
Rong Wu*
Affiliation:
Nankai University
*
∗ Postal address: School of Mathematical Science and LPMC, Nankai University, Tianjin 300071, P. R. China.
∗∗∗ Postal address: Department of Statistics and Actuarial Science, The University of Iowa, 241 Schaeffer Hall, Iowa City, IA 52242, USA. Email address: qihe-tang@uiowa.edu
∗ Postal address: School of Mathematical Science and LPMC, Nankai University, Tianjin 300071, P. R. China.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Consider a continuous-time renewal risk model with a constant force of interest. We assume that claim sizes and interarrival times correspondingly form a sequence of independent and identically distributed random pairs and that each pair obeys a dependence structure described via the conditional tail probability of a claim size given the interarrival time before the claim. We focus on determining the impact of this dependence structure on the asymptotic tail probability of discounted aggregate claims. Assuming that the claim size distribution is subexponential, we derive an exact locally uniform asymptotic formula, which quantitatively captures the impact of the dependence structure. When the claim size distribution is extended regularly varying tailed, we show that this asymptotic formula is globally uniform.

Keywords

Asymptoticsdependencediscounted aggregate claimextended regular variationsubexponentialityuniformity

MSC classification

Primary: 62P05: Applications to actuarial sciences and financial mathematics
Secondary: 62H20: Measures of association (correlation, canonical correlation, etc.) 60E05: Distributions
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 4 , December 2010 , pp. 1126 - 1146
Copyright
Copyright © Applied Probability Trust 2010 

References

Albrecher, H. and Boxma, O. J. (2004). A ruin model with dependence between claim sizes and claim intervals. Insurance Math. Econom. 35, 245–254.CrossRefGoogle Scholar
Albrecher, H. and Boxma, O. J. (2005). On the discounted penalty function in a Markov-dependent risk model. Insurance Math. Econom. 37, 650–672.CrossRefGoogle Scholar
Albrecher, H. and Teugels, J. L. (2006). Exponential behavior in the presence of dependence in risk theory. J. Appl. Prob. 43, 257–273.CrossRefGoogle Scholar
Asimit, A. V. and Badescu, A. L. (2010). Extremes on the discounted aggregate claims in a time dependent risk model. Scand. Actuarial. J. 2010, 93–104.CrossRefGoogle Scholar
Asmussen, S. (1998). Subexponential asymptotics for stochastic processes: extremal behavior, stationary distributions and first passage probabilities. Ann. Appl. Prob. 8, 354–374.CrossRefGoogle Scholar
Asmussen, S., Schmidli, H. and Schmidt, V. (1999). Tail probabilities for non-standard risk and queueing processes with subexponential Jumps. Adv. Appl. Prob. 31, 422–447.CrossRefGoogle Scholar
Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.CrossRefGoogle Scholar
Badescu, A. L., Cheung, E. C. K. and Landriault, D. (2009). Dependent risk models with bivariate phase-type distributions. J. Appl. Prob. 46, 113–131.CrossRefGoogle Scholar
Biard, R., Lefèvre, C. and Loisel, S. (2008). Impact of correlation crises in risk theory: asymptotics of finite-time ruin probabilities for heavy-tailed claim amounts when some independence and stationarity assumptions are relaxed. Insurance Math. Econom. 43, 412–421.CrossRefGoogle Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation. Cambridge University Press.Google Scholar
Boudreault, M., Cossette, H., Landriault, D. and Marceau, E. (2006) On a risk model with dependence between interclaim arrivals and claim sizes. Scand. Actuarial J. 2006, 265–285.CrossRefGoogle Scholar
Chen, Y. and Ng, K. W. (2007). The ruin probability of the renewal model with constant interest force and negatively dependent heavy-tailed claims. Insurance Math. Econom. 40, 415–423.CrossRefGoogle Scholar
Chen, Y. and Yuen, K. C. (2009) Sums of pairwise quasi-asymptotically independent random variables with consistent variation. Stoch. Models 25, 76–89.CrossRefGoogle Scholar
Chistyakov, V. P. (1964). A theorem on sums of independent positive random variables and its applications to branching random processes. Theory Prob. Appl. 9, 640–648.CrossRefGoogle Scholar
Cossette, H., Marceau, E. and Marri, F. (2008). On the compound Poisson risk model with dependence based on a generalized Farlie–Gumbel–Morgenstern copula. Insurance Math. Econom. 43, 444–455.CrossRefGoogle Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.CrossRefGoogle Scholar
Foss, S. and Richards, A. (2010). On sums of conditionally independent subexponential random variables. Math. Operat. Res. 35, 102–119.CrossRefGoogle Scholar
Hao, X. and Tang, Q. (2008). A uniform asymptotic estimate for discounted aggregate claims with subexponential tails. Insurance Math. Econom. 43, 116–120.CrossRefGoogle Scholar
Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman and Hall, London.Google Scholar
Kalashnikov, V. and Konstantinides, D. (2000). Ruin under interest force and subexponential claims: a simple treatment. Insurance Math. Econom. 27, 145–149.CrossRefGoogle Scholar
Klebaner, F. C. (2005). Introduction to Stochastic Calculus with Applications, 2nd edn. Imperial College Press, London.CrossRefGoogle Scholar
Klüppelberg, C. and Stadtmüller, U. (1998). Ruin probabilities in the presence of heavy-tails and interest rates. Scand. Actuarial J. 1998, 49–58.CrossRefGoogle Scholar
Konstantinides, D., Tang, Q. and Tsitsiashvili, G. (2002). Estimates for the ruin probability in the classical risk model with constant interest force in the presence of heavy tails. Insurance Math. Econom. 31, 447–460.CrossRefGoogle Scholar
Nelsen, R. B. (2006). An Introduction to Copulas, 2nd edn. Springer, New York.Google Scholar
Stein, C. (1946). A note on cumulative sums. Ann. Math. Statist. 17, 498–499.CrossRefGoogle Scholar
Tang, Q. (2005). The finite-time ruin probability of the compound Poisson model with constant interest force. J. Appl. Prob. 42, 608–619.CrossRefGoogle Scholar
Tang, Q. (2007). Heavy tails of discounted aggregate claims in the continuous-time renewal model. J. Appl. Prob. 44, 285–294.CrossRefGoogle Scholar
Tang, Q. (2008). Insensitivity to negative dependence of asymptotic tail probabilities of sums and maxima of sums. Stoch. Anal. Appl. 26, 435–450.CrossRefGoogle Scholar
Tang, Q. and Tsitsiashvili, G. (2003). Randomly weighted sums of subexponential random variables with application to ruin theory. Extremes 6, 171–188.CrossRefGoogle Scholar
Tang, Q. and Tsitsiashvili, G. (2004). Finite- and infinite-time ruin probabilities in the presence of stochastic returns on investments. Adv. Appl. Prob. 36, 1278–1299.CrossRefGoogle Scholar
Wang, D. (2008). Finite-time ruin probability with heavy-tailed claims and constant interest rate. Stoch. Models 24, 41–57.CrossRefGoogle Scholar