常数比例投资下延迟索赔风险模型的渐近破产概率
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摘要
研究一类带投资的延迟索赔更新风险模型的渐近破产概率,其中允许保险公司将其资产按常数比例投资于满足几何布朗运动的股票市场,其余部分投资于非负利率的债券市场,假设主索赔额和延迟索赔额序列各自为负相依同分布且属于重尾分布族L∩D族随机变量序列的情形下,根据Ito公式,给出保险公司资产的表达式,最终得到有限时间的破产概率.
The asymptotic behavior of ruin probabilities is investigated in a renewal risk model for delayed claims,in which the insurance company is allowed to invest a constant fraction of its wealth in a stock market which is described by a geometric Brownian motion and the remaining wealth in a bond with nonnegative interest force. Under the assumptions that the sequences of the main and delayed claims are negatively dependent random varies with a common distribution and that the claim sizes belong to the heavy-tailed distribution class L∩D,the expression of the wealth process is derived by the Ito formula,and the finite-time ruin probabilities are obtained.
The asymptotic behavior of ruin probabilities is investigated in a renewal risk model for delayed claims,in which the insurance company is allowed to invest a constant fraction of its wealth in a stock market which is described by a geometric Brownian motion and the remaining wealth in a bond with nonnegative interest force. Under the assumptions that the sequences of the main and delayed claims are negatively dependent random varies with a common distribution and that the claim sizes belong to the heavy-tailed distribution class L∩D,the expression of the wealth process is derived by the Ito formula,and the finite-time ruin probabilities are obtained.
引文
[1]WATERS H,PAPATRIANDAFYLOU A.Ruin probabilities allowing for delay in claims settlement[J].Insurance:Math Econ,1985,4:113-122.
[2]YUEN K C,GUO J Y.Ruin probabilities for time-correlated claims in the compound binomial model[J].Insurance:Math Econ,2001,29(1),47-57.
[3]YUEN K C,GUO J Y,NG K W.On ultimate ruin in a delayed-claims risk model[J].J Appl Probability,2005,42:163-174.
[4]XIE J H,ZOU W.Expected present value of total dividends in a delayed claims risk model under stochastic interest rates[J].Insurance:Math Econ,2010,46:415-422.
[5]ZHAO C,ZHANG C.Joint density of the number of claims until ruin and the time to ruin in the delayed renewal risk model with Erlang(n)claims[J].Compu Appl Math,2013,244(1):102-144.
[6]DENG C,ZHOU J,DENG Y.The Gerber-Shiu discounted penalty function in a delayed renewal risk model with multi-layer dividend strategy[J].Statistics and Probability Letters,2012,82:1648-1656.
[7]ZOU W,XIE J.On the Gerber-Shiu discounted penalty function in a risk model with delayed claims[J].Korean Statistical Society,2012,41:387-397.
[8]CHEN Y,HUANG Y,ZHANG W.Asymptotic ruin probabilities for proportional investment under interest force with dominatedly-varying-tailed claims[J].Korean Statistical Society,2012,41:87-95.
[9]YANG Y,YUEN K C.Finite-time and infinite-time ruin probabilities in atwo-dimensional delayed renewal risk model with Sarmanov dependent claims[J].J Math Anal Appl,2016,442(2):600-626.
[10]SO-YEUN K,GORDON E W.On the analysis of ruin-related quantities in the delayed renewal risk model[J].Insurance:Math Econ,2016,66:77-85.
[11]TANG Q.The finite time ruin probability of the compound Poisson model with constant interest force[J].Appl Probab,2005,42:608-619.
[12]WEI L.Ruin probability of the renewal model with risky investment and large claims[J].Science in China(Math),2009,A52(7):1539-1545.
[13]HIPP C,PLUM M.Optimal investment for insurers[J].Insurance:Math Econ,2000,27:215-228.
[14]GAIER J,GRANDITS P.Ruin probabilities and investment under interest force in the presence of regularly varying tails[J].Scand Actuarial J,2004,4:256-278.
[15]CHEN Y,NG K W.The ruin probability of the renewal model with constant interest force and negatively dependent heavytailed claims[J].Insurance:Math Econ,2007,40:415-423.
[16]YI L,CHEN Y,SU C.Approximation of the tail probability of randomly weighted sums of dependent random variables with dominated variation[J].J Math Anal Appl,2011,376(1):365-372.
[17]肖鸿民,何艳.常数比例投资下基于进入过程风险模型的渐进破产概率[J].河南师范大学学报(自然科学版),2015,43(2):14-19.
[18]肖鸿民,李红.重尾赔付下带常数利息力的延迟索赔风险模型的破产概率[J].西北师范大学学报(自然科学版),2011,47(6):17-20.
[19]肖鸿民,李红.负相依赔付下延迟风险模型的破产概率[J].兰州大学学报(自然科学版),2012,48(3),118-122.
[20]HE S,WANG J,YAN J.Semimartingale Theory and Stochastic Calculus[M].Beijing:Science Press,1992.
[21]CLINE D B H,SAMORODNITSKY G.Subexponentiality of the product ofindependent random variables[J].Stochastic Processes Appl,1994,49(1):75-98.
[22]TANG Q,TSITSIASHVILI G.Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavytailed insurance and financial risks[J].Stochastic Processes and Their Applications,2003,108(2):299-325.
[23]EMBRECHTS P,KIUPPELBERG C,MIKOSCH T.Modeling Extremal Events for Insurance and Finance[M].Berlin:Springer-Verlag,1997.
[24]HAO X M,TANG Q H.A uniform asymptotic estimate for discounted aggregate claims with subexponential tails[J].Insurance:Math Econ,2008,43(1):116-120.
[2]YUEN K C,GUO J Y.Ruin probabilities for time-correlated claims in the compound binomial model[J].Insurance:Math Econ,2001,29(1),47-57.
[3]YUEN K C,GUO J Y,NG K W.On ultimate ruin in a delayed-claims risk model[J].J Appl Probability,2005,42:163-174.
[4]XIE J H,ZOU W.Expected present value of total dividends in a delayed claims risk model under stochastic interest rates[J].Insurance:Math Econ,2010,46:415-422.
[5]ZHAO C,ZHANG C.Joint density of the number of claims until ruin and the time to ruin in the delayed renewal risk model with Erlang(n)claims[J].Compu Appl Math,2013,244(1):102-144.
[6]DENG C,ZHOU J,DENG Y.The Gerber-Shiu discounted penalty function in a delayed renewal risk model with multi-layer dividend strategy[J].Statistics and Probability Letters,2012,82:1648-1656.
[7]ZOU W,XIE J.On the Gerber-Shiu discounted penalty function in a risk model with delayed claims[J].Korean Statistical Society,2012,41:387-397.
[8]CHEN Y,HUANG Y,ZHANG W.Asymptotic ruin probabilities for proportional investment under interest force with dominatedly-varying-tailed claims[J].Korean Statistical Society,2012,41:87-95.
[9]YANG Y,YUEN K C.Finite-time and infinite-time ruin probabilities in atwo-dimensional delayed renewal risk model with Sarmanov dependent claims[J].J Math Anal Appl,2016,442(2):600-626.
[10]SO-YEUN K,GORDON E W.On the analysis of ruin-related quantities in the delayed renewal risk model[J].Insurance:Math Econ,2016,66:77-85.
[11]TANG Q.The finite time ruin probability of the compound Poisson model with constant interest force[J].Appl Probab,2005,42:608-619.
[12]WEI L.Ruin probability of the renewal model with risky investment and large claims[J].Science in China(Math),2009,A52(7):1539-1545.
[13]HIPP C,PLUM M.Optimal investment for insurers[J].Insurance:Math Econ,2000,27:215-228.
[14]GAIER J,GRANDITS P.Ruin probabilities and investment under interest force in the presence of regularly varying tails[J].Scand Actuarial J,2004,4:256-278.
[15]CHEN Y,NG K W.The ruin probability of the renewal model with constant interest force and negatively dependent heavytailed claims[J].Insurance:Math Econ,2007,40:415-423.
[16]YI L,CHEN Y,SU C.Approximation of the tail probability of randomly weighted sums of dependent random variables with dominated variation[J].J Math Anal Appl,2011,376(1):365-372.
[17]肖鸿民,何艳.常数比例投资下基于进入过程风险模型的渐进破产概率[J].河南师范大学学报(自然科学版),2015,43(2):14-19.
[18]肖鸿民,李红.重尾赔付下带常数利息力的延迟索赔风险模型的破产概率[J].西北师范大学学报(自然科学版),2011,47(6):17-20.
[19]肖鸿民,李红.负相依赔付下延迟风险模型的破产概率[J].兰州大学学报(自然科学版),2012,48(3),118-122.
[20]HE S,WANG J,YAN J.Semimartingale Theory and Stochastic Calculus[M].Beijing:Science Press,1992.
[21]CLINE D B H,SAMORODNITSKY G.Subexponentiality of the product ofindependent random variables[J].Stochastic Processes Appl,1994,49(1):75-98.
[22]TANG Q,TSITSIASHVILI G.Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavytailed insurance and financial risks[J].Stochastic Processes and Their Applications,2003,108(2):299-325.
[23]EMBRECHTS P,KIUPPELBERG C,MIKOSCH T.Modeling Extremal Events for Insurance and Finance[M].Berlin:Springer-Verlag,1997.
[24]HAO X M,TANG Q H.A uniform asymptotic estimate for discounted aggregate claims with subexponential tails[J].Insurance:Math Econ,2008,43(1):116-120.