Uniform Asymptotics for the Finite-Time Ruin Probability of a Dependent Risk Model with a Constant Interest Rate

This paper gives an asymptotically equivalent formula for the finite-time ruin probability of a nonstandard risk model with a constant interest rate, in which both claim sizes and inter-arrival times follow a certain dependence structure. This new dependence structure allows the underlying random va...

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Published in:	Methodology and computing in applied probability Vol. 15; no. 1; pp. 109 - 124
Main Authors:	Wang, Kaiyong, Wang, Yuebao, Gao, Qingwu
Format:	Journal Article
Language:	English
Published:	Boston Springer US 01.03.2013 Springer Nature B.V
Subjects:	Asymptotic methods Asymptotic properties Business and Management Economic models Economics Electrical Engineering Equivalence Interest rates Intervals Life Sciences Mathematical analysis Mathematical models Mathematics and Statistics Probability Random variables Risk Statistics Studies Uniform asymptotics Widely orthant dependent Finite-time ruin probability 62E10 62P05 60F05 Constant interest rate
ISSN:	1387-5841, 1573-7713
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Abstract	This paper gives an asymptotically equivalent formula for the finite-time ruin probability of a nonstandard risk model with a constant interest rate, in which both claim sizes and inter-arrival times follow a certain dependence structure. This new dependence structure allows the underlying random variables to be either positively or negatively dependent. The obtained asymptotics hold uniformly in a finite time interval. Especially, in the renewal risk model the uniform asymptotics of the finite-time ruin probability for all times have been given. The obtained results have extended and improved some corresponding results.
AbstractList	This paper gives an asymptotically equivalent formula for the finite-time ruin probability of a nonstandard risk model with a constant interest rate, in which both claim sizes and inter-arrival times follow a certain dependence structure. This new dependence structure allows the underlying random variables to be either positively or negatively dependent. The obtained asymptotics hold uniformly in a finite time interval. Especially, in the renewal risk model the uniform asymptotics of the finite-time ruin probability for all times have been given. The obtained results have extended and improved some corresponding results. This paper gives an asymptotically equivalent formula for the finite-time ruin probability of a nonstandard risk model with a constant interest rate, in which both claim sizes and inter-arrival times follow a certain dependence structure. This new dependence structure allows the underlying random variables to be either positively or negatively dependent. The obtained asymptotics hold uniformly in a finite time interval. Especially, in the renewal risk model the uniform asymptotics of the finite-time ruin probability for all times have been given. The obtained results have extended and improved some corresponding results.[PUBLICATION ABSTRACT]
Author	Gao, Qingwu Wang, Kaiyong Wang, Yuebao
Author_xml	– sequence: 1 givenname: Kaiyong surname: Wang fullname: Wang, Kaiyong organization: School of Mathematics, Soochow University, School of Mathematics and Physics, Suzhou University of Science and Technology – sequence: 2 givenname: Yuebao surname: Wang fullname: Wang, Yuebao email: ybwang@suda.edu.cn organization: School of Mathematics, Soochow University – sequence: 3 givenname: Qingwu surname: Gao fullname: Gao, Qingwu organization: School of Mathematics, Soochow University, School of Mathematics and Statistics, Nanjing Audit University
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References	EbrahimiNGhoshMMultivariate negative dependenceCommun Stat19811030733761240010.1080/03610928108828041 TangQHeavy tails of discounted aggregrate claims in the continuous-time renewal modelJ Appl Probab20074428529423401981211.9115210.1239/jap/1183667401 KlüppelbergCSubexponential distributions and integrated tailsJ Appl Probab1988251321419295110651.6002010.2307/3214240 KonstantinidesDTangQTsitsiashviliGEstimates for the ruin probability in the classical risk model with constant interest force in the presence of heavy tailsInsur Math Econ20023144746019455431074.9102910.1016/S0167-6687(02)00189-0 WangDFinite-time ruin probability with heavy-tailed claims and constant interest rateStoch Models200824415723846891132.9150210.1080/15326340701826898 Wang Y, Cui Z, Wang K, Ma X (2009) Uniform asymptotics of the finite-time ruin probability with NLOD inter-occurrence times. (submitted) EmbrechtsPKlüppelbergCMikoschTModelling extremal events for insurance and finance1997BerlinSpringer0873.6211610.1007/978-3-642-33483-2 LiJWangKWangYFinite-time ruin probability with NQD dominated varying-tailed claims and NLOD inter-arrival timesJ Syst Sci Complex20092240741425388711186.9112410.1007/s11424-009-9173-7 LehmannELSome concepts of dependenceAnn Math Stat196637113711530146.4060110.1214/aoms/1177699260 LiuLPrecise large deviations for dependent random variables with heavy tailsStat Probab Lett200979129012981163.6001210.1016/j.spl.2009.02.001 TangQTsitsiashviliGPrecise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risksStoch Process Their Appl200310829932520190561075.91563 DenisovDFossSKorshunovDTail asymptotics for the supremum of a random walk when the mean is not finiteQueueing Syst200446153320722741056.9002810.1023/B:QUES.0000021140.87161.9c HaoXTangQA uniform asymptotic estimate for discounted aggregate claims with sunexponential tailsInsur Math Econ20084311612024420371142.6209010.1016/j.insmatheco.2008.03.009 KorshunovDLarge-deviation probabilities for maxima of sums of independent random variables with negaive mean and subexponential distributionTheory Probab Appl200246355366196869610.1137/S0040585X97979019 YangYWangYAsymptotics for ruin probability of some nagatively dependent risk models with a constant interest rate and dominatedly-varying-tailed claimsStat Probab Lett2010801431541180.6215410.1016/j.spl.2009.09.023 ChenYNgKWThe ruin probability of the renewal model with constant interest force and negatively dependent heavy-tailed claimsInsur Math Econ20074041542323109801183.6003310.1016/j.insmatheco.2006.06.004 KlüppelbergCStadtimullerURuin probability in the presence of heavy-tails and interest ratesScand Actuar J19981998149581022.6008310.1080/03461238.1998.10413991 TangQThe finite-time ruin probability of the compound Poisson model with constant interest forceJ Appl Probab20054260861921575081132.9150010.1239/jap/1127322015 KalashnikovVKonstantinidesDRuin under interest force and subexponential claims: a simple treatmentInsur Math Econ20002714514917969761056.6050110.1016/S0167-6687(00)00045-7 NelsenRBAn introduction to Copulas20062New YorkSpringer1152.62030 TangQAsymptotic ruin probabilities of the renewal model with constant interest force and regular variationScand Actuar J200520051151144.9103010.1080/03461230510006982 BinghamNHGoldieCMTeugelsJMRegular variation1987CambridgeCambridge University Press0617.26001 KongFZongGThe finite-time ruin probability for ND claims with constant interest forceStat Probab Lett2008783103310924744020537285710.1016/j.spl.2008.05.036 ClineDBHSamorodnitskyGSubexponentiality of the product of independent random variablesStoch Process Their Appl199449759812582830799.6001510.1016/0304-4149(94)90113-9 TangQAsymptotics for the finite time ruin probability in the renewal model with consistent variationStoch Models20042028129720821261130.6031210.1081/STM-200025739 BlockHWSavitsTHShakedMSome concepts of negative dependenceAnn Probab1982107657726595450501.6203710.1214/aop/1176993784 N Ebrahimi (9226_CR6) 1981; 10 X Hao (9226_CR8) 2008; 43 C Klüppelberg (9226_CR10) 1988; 25 Y Chen (9226_CR3) 2007; 40 P Embrechts (9226_CR7) 1997 D Denisov (9226_CR5) 2004; 46 F Kong (9226_CR12) 2008; 78 D Wang (9226_CR24) 2008; 24 HW Block (9226_CR2) 1982; 10 D Konstantinides (9226_CR13) 2002; 31 Q Tang (9226_CR20) 2005; 2005 Q Tang (9226_CR23) 2003; 108 Q Tang (9226_CR19) 2004; 20 Q Tang (9226_CR22) 2007; 44 Y Yang (9226_CR26) 2010; 80 L Liu (9226_CR17) 2009; 79 RB Nelsen (9226_CR18) 2006 Q Tang (9226_CR21) 2005; 42 J Li (9226_CR16) 2009; 22 NH Bingham (9226_CR1) 1987 V Kalashnikov (9226_CR9) 2000; 27 C Klüppelberg (9226_CR11) 1998; 1998 EL Lehmann (9226_CR15) 1966; 37 D Korshunov (9226_CR14) 2002; 46 9226_CR25 DBH Cline (9226_CR4) 1994; 49
References_xml	– reference: ClineDBHSamorodnitskyGSubexponentiality of the product of independent random variablesStoch Process Their Appl199449759812582830799.6001510.1016/0304-4149(94)90113-9 – reference: BlockHWSavitsTHShakedMSome concepts of negative dependenceAnn Probab1982107657726595450501.6203710.1214/aop/1176993784 – reference: EmbrechtsPKlüppelbergCMikoschTModelling extremal events for insurance and finance1997BerlinSpringer0873.6211610.1007/978-3-642-33483-2 – reference: KalashnikovVKonstantinidesDRuin under interest force and subexponential claims: a simple treatmentInsur Math Econ20002714514917969761056.6050110.1016/S0167-6687(00)00045-7 – reference: KlüppelbergCStadtimullerURuin probability in the presence of heavy-tails and interest ratesScand Actuar J19981998149581022.6008310.1080/03461238.1998.10413991 – reference: KorshunovDLarge-deviation probabilities for maxima of sums of independent random variables with negaive mean and subexponential distributionTheory Probab Appl200246355366196869610.1137/S0040585X97979019 – reference: LiuLPrecise large deviations for dependent random variables with heavy tailsStat Probab Lett200979129012981163.6001210.1016/j.spl.2009.02.001 – reference: TangQHeavy tails of discounted aggregrate claims in the continuous-time renewal modelJ Appl Probab20074428529423401981211.9115210.1239/jap/1183667401 – reference: LehmannELSome concepts of dependenceAnn Math Stat196637113711530146.4060110.1214/aoms/1177699260 – reference: TangQAsymptotics for the finite time ruin probability in the renewal model with consistent variationStoch Models20042028129720821261130.6031210.1081/STM-200025739 – reference: HaoXTangQA uniform asymptotic estimate for discounted aggregate claims with sunexponential tailsInsur Math Econ20084311612024420371142.6209010.1016/j.insmatheco.2008.03.009 – reference: YangYWangYAsymptotics for ruin probability of some nagatively dependent risk models with a constant interest rate and dominatedly-varying-tailed claimsStat Probab Lett2010801431541180.6215410.1016/j.spl.2009.09.023 – reference: ChenYNgKWThe ruin probability of the renewal model with constant interest force and negatively dependent heavy-tailed claimsInsur Math Econ20074041542323109801183.6003310.1016/j.insmatheco.2006.06.004 – reference: NelsenRBAn introduction to Copulas20062New YorkSpringer1152.62030 – reference: TangQThe finite-time ruin probability of the compound Poisson model with constant interest forceJ Appl Probab20054260861921575081132.9150010.1239/jap/1127322015 – reference: LiJWangKWangYFinite-time ruin probability with NQD dominated varying-tailed claims and NLOD inter-arrival timesJ Syst Sci Complex20092240741425388711186.9112410.1007/s11424-009-9173-7 – reference: TangQTsitsiashviliGPrecise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risksStoch Process Their Appl200310829932520190561075.91563 – reference: BinghamNHGoldieCMTeugelsJMRegular variation1987CambridgeCambridge University Press0617.26001 – reference: KlüppelbergCSubexponential distributions and integrated tailsJ Appl Probab1988251321419295110651.6002010.2307/3214240 – reference: Wang Y, Cui Z, Wang K, Ma X (2009) Uniform asymptotics of the finite-time ruin probability with NLOD inter-occurrence times. (submitted) – reference: KonstantinidesDTangQTsitsiashviliGEstimates for the ruin probability in the classical risk model with constant interest force in the presence of heavy tailsInsur Math Econ20023144746019455431074.9102910.1016/S0167-6687(02)00189-0 – reference: WangDFinite-time ruin probability with heavy-tailed claims and constant interest rateStoch Models200824415723846891132.9150210.1080/15326340701826898 – reference: KongFZongGThe finite-time ruin probability for ND claims with constant interest forceStat Probab Lett2008783103310924744020537285710.1016/j.spl.2008.05.036 – reference: TangQAsymptotic ruin probabilities of the renewal model with constant interest force and regular variationScand Actuar J200520051151144.9103010.1080/03461230510006982 – reference: DenisovDFossSKorshunovDTail asymptotics for the supremum of a random walk when the mean is not finiteQueueing Syst200446153320722741056.9002810.1023/B:QUES.0000021140.87161.9c – reference: EbrahimiNGhoshMMultivariate negative dependenceCommun Stat19811030733761240010.1080/03610928108828041 – volume: 79 start-page: 1290 year: 2009 ident: 9226_CR17 publication-title: Stat Probab Lett doi: 10.1016/j.spl.2009.02.001 – volume: 31 start-page: 447 year: 2002 ident: 9226_CR13 publication-title: Insur Math Econ doi: 10.1016/S0167-6687(02)00189-0 – volume: 27 start-page: 145 year: 2000 ident: 9226_CR9 publication-title: Insur Math Econ doi: 10.1016/S0167-6687(00)00045-7 – volume: 46 start-page: 355 year: 2002 ident: 9226_CR14 publication-title: Theory Probab Appl doi: 10.1137/S0040585X97979019 – volume: 46 start-page: 15 year: 2004 ident: 9226_CR5 publication-title: Queueing Syst doi: 10.1023/B:QUES.0000021140.87161.9c – volume: 2005 start-page: 1 issue: 1 year: 2005 ident: 9226_CR20 publication-title: Scand Actuar J doi: 10.1080/03461230510006982 – volume: 24 start-page: 41 year: 2008 ident: 9226_CR24 publication-title: Stoch Models doi: 10.1080/15326340701826898 – volume: 80 start-page: 143 year: 2010 ident: 9226_CR26 publication-title: Stat Probab Lett doi: 10.1016/j.spl.2009.09.023 – volume: 25 start-page: 132 year: 1988 ident: 9226_CR10 publication-title: J Appl Probab doi: 10.2307/3214240 – volume: 108 start-page: 299 year: 2003 ident: 9226_CR23 publication-title: Stoch Process Their Appl doi: 10.1016/j.spa.2003.07.001 – volume: 1998 start-page: 49 issue: 1 year: 1998 ident: 9226_CR11 publication-title: Scand Actuar J doi: 10.1080/03461238.1998.10413991 – volume: 49 start-page: 75 year: 1994 ident: 9226_CR4 publication-title: Stoch Process Their Appl doi: 10.1016/0304-4149(94)90113-9 – volume: 20 start-page: 281 year: 2004 ident: 9226_CR19 publication-title: Stoch Models doi: 10.1081/STM-200025739 – volume: 40 start-page: 415 year: 2007 ident: 9226_CR3 publication-title: Insur Math Econ doi: 10.1016/j.insmatheco.2006.06.004 – volume: 78 start-page: 3103 year: 2008 ident: 9226_CR12 publication-title: Stat Probab Lett doi: 10.1016/j.spl.2008.05.036 – volume: 10 start-page: 765 year: 1982 ident: 9226_CR2 publication-title: Ann Probab doi: 10.1214/aop/1176993784 – volume-title: An introduction to Copulas year: 2006 ident: 9226_CR18 – volume: 42 start-page: 608 year: 2005 ident: 9226_CR21 publication-title: J Appl Probab doi: 10.1239/jap/1127322015 – volume: 10 start-page: 307 year: 1981 ident: 9226_CR6 publication-title: Commun Stat doi: 10.1080/03610928108828041 – volume: 37 start-page: 1137 year: 1966 ident: 9226_CR15 publication-title: Ann Math Stat doi: 10.1214/aoms/1177699260 – volume: 44 start-page: 285 year: 2007 ident: 9226_CR22 publication-title: J Appl Probab doi: 10.1239/jap/1183667401 – volume: 22 start-page: 407 year: 2009 ident: 9226_CR16 publication-title: J Syst Sci Complex doi: 10.1007/s11424-009-9173-7 – volume-title: Modelling extremal events for insurance and finance year: 1997 ident: 9226_CR7 doi: 10.1007/978-3-642-33483-2 – ident: 9226_CR25 – volume-title: Regular variation year: 1987 ident: 9226_CR1 doi: 10.1017/CBO9780511721434 – volume: 43 start-page: 116 year: 2008 ident: 9226_CR8 publication-title: Insur Math Econ doi: 10.1016/j.insmatheco.2008.03.009
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Snippet	This paper gives an asymptotically equivalent formula for the finite-time ruin probability of a nonstandard risk model with a constant interest rate, in which...
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SubjectTerms	Asymptotic methods Asymptotic properties Business and Management Economic models Economics Electrical Engineering Equivalence Interest rates Intervals Life Sciences Mathematical analysis Mathematical models Mathematics and Statistics Probability Random variables Risk Statistics Studies
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Title	Uniform Asymptotics for the Finite-Time Ruin Probability of a Dependent Risk Model with a Constant Interest Rate
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