# American Institute of Mathematical Sciences

October  2012, 8(4): 909-924. doi: 10.3934/jimo.2012.8.909

## G/M/1 type structure of a risk model with general claim sizes in a Markovian environment

 1 Department of Mathematics, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul, 136-713, South Korea 2 School of Management, Kyung Hee University, 26 Kyunghee-daero, Dongdaemun-gu, Seoul, 130-701, South Korea

Received  September 2011 Revised  July 2012 Published  September 2012

This paper develops a discrete-time risk model with general claim sizes in a Markovian environment where both claim occurrence probabilities and the claim size distributions are dependent on the regime of the environment. We assume that the environmental regime is governed by a Markov process with a finite state space. We utilize a G/M/1 type structure in the process of the surplus level and the regime. We also employ the matrix analytic method to analyze the sojourn time of the surplus process at each level until the ruin time. Under this framework we obtain several important quantities related to ruin. First, we derive the penalty function using the results on the surplus process until the ruin time. Second, we obtain the ruin probability, the ruin time distribution and the deficit distribution at ruin. Numerical examples implement the ruin quantities that we derive.
Citation: Jerim Kim, Bara Kim, Hwa-Sung Kim. G/M/1 type structure of a risk model with general claim sizes in a Markovian environment. Journal of Industrial & Management Optimization, 2012, 8 (4) : 909-924. doi: 10.3934/jimo.2012.8.909
##### References:
 [1] I. Abate and W. Whitt, The Fourier-series method for inverting transforms of probability distributions,, Queueing Systems, 10 (1992), 5. doi: 10.1007/BF01158520. [2] S. Ahn and A. L. Badescu, On the analysis of the Gerber-Shiu discounted penalty function for risk processes with Markovian arrivals,, Insurance Mathematics & Economics, 41 (2007), 234. doi: 10.1016/j.insmatheco.2006.10.017. [3] S. Ahn, A. L. Badescu and V. Ramaswami, Time dependent analysis of finite buffer fluid flows and risk models with a dividend barrier,, Queueing System, 55 (2007), 207. doi: 10.1007/s11134-007-9017-x. [4] S. Asmussen, "Ruin Probabilities,'', World Scientific Publishing, (2000). [5] A. Badescu and D. Landriault, Applications of fluid flow matrix analytic methods in ruin theory-a review,, Revista De La Real Academia De Ciencias Exactas Fisicas Y Naturales Serie a-Matematicas, 103 (2009), 353. [6] A. L. Badescu, E. K. Cheung and D. Randriault, Dependent risk model with bivariate phase-type distributions,, J. Appl. Prob., 46 (2009), 113. doi: 10.1239/jap/1238592120. [7] S. X. Cheng, H. U. Gerber and E. S. W. Shiu, Discounted probabilities and ruin theory in the compound binomial model,, Insurance Mathematics & Economics, 26 (2000), 239. doi: 10.1016/S0167-6687(99)00053-0. [8] Y. B. Cheng, Q. H. Tang and H. L. Yang, Approximations for moments of deficit at ruin with exponential and subexponential claims,, Statistics & Probability Letters, 59 (2002), 367. doi: 10.1016/S0167-7152(02)00234-1. [9] S. N. Chiu and C. C. Yin, The time of ruin, the surplus prior to ruin and the deficit at ruin for the classical risk process perturbed by diffusion,, Insurance Mathematics & Economics, 33 (2003), 59. doi: 10.1016/S0167-6687(03)00143-4. [10] H. Cossette, D. Landriault and E. Marceau, Ruin probabilities in the discrete time renewal risk model,, Insurance Mathematics & Economics, 38 (2006), 309. doi: 10.1016/j.insmatheco.2005.09.005. [11] H. Cossette, E. Marceau and F. Toureille, Risk models based on time series for count random variables,, Insurance Mathematics & Economics, 48 (2011), 19. doi: 10.1016/j.insmatheco.2010.08.007. [12] H. U. Gerber and E. S. W. Shiu, The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin,, Insurance Mathematics & Economics, 21 (1997), 129. doi: 10.1016/S0167-6687(97)00027-9. [13] B. Kim, H. S. Kim and J. Kim, A risk model with paying dividends and random environment,, Insurance Mathematics & Economics, 42 (2008), 717. doi: 10.1016/j.insmatheco.2007.08.001. [14] M. F. Neuts, "Matrix Geometric Solutions in Stochastic Models: An Algorithmic Approach,'', Johns Hopkins University Press, (1981). [15] G. Latouche and V. Ramaswami, "Introduction to Matrix Analytic Methods in Stochastic Modeling,'', American Statistic Association and the Society for Industrial and Applied Mathematics, (1999). doi: 10.1137/1.9780898719734. [16] D. Landriault, On a generalization of the expected discounted penalty function in a discrete-time insurance risk model,, Applied Stochastic Models in Business and Industry, 24 (2008), 525. doi: 10.1002/asmb.713. [17] S. M. Li, Y. Lu and J. Garrido, A review of discrete-time risk models,, Revista De La Real Academia De Ciencias Exactas Fisicas Y Naturales Serie a-Matematicas, 103 (2009), 321. [18] X. S. Lin and G. E. Willmot, Analysis of a defective renewal equation arising in ruin theory,, Insurance Mathematics & Economics, 25 (1999), 63. doi: 10.1016/S0167-6687(99)00026-8. [19] X. S. Lin and K. P. Pavlova, The compound Poisson risk model witha threshold dividend strategy,, Insurance Mathematics & Economics, 38 (2006), 57. doi: 10.1016/j.insmatheco.2005.08.001. [20] X. S. Lin and G. E. Willmot, The moments of the time of ruin, the surplus before ruin, and the deficit at ruin,, Insurance Mathematics & Economics, 27 (2000), 19. doi: 10.1016/S0167-6687(00)00038-X. [21] Y. Lu and S. M. Li, The Markovian regime-switching risk model with a threshold dividend strategy,, Insurance Mathematics & Economics, 44 (2009), 296. doi: 10.1016/j.insmatheco.2008.04.004. [22] M. F. Neuts, "Matrix Geometric Solutions in Stochastic Models: An Algorithmic Approach,'', Johns Hopkins University Press, (1981). [23] M. F. Neuts, "Structured Stochastic Matrices of M/G/1 Type and Their Applications,'', Marcel Dekker, (1989). [24] K. P. Pavlova and G. E. Willmot, The discrete stationary renewal risk model and the Gerber-Shiu discounted penalty function,, Insurance Mathematics & Economics, 33 (2003), 440. [25] M. S. Sgibnev, The matrix analogue of the Blackwell renewal theorem on the real line,, Sbornik: Mathematics, 197 (2006), 69. doi: 10.1070/SM2006v197n03ABEH003762. [26] H. Yang, Z. M. Zhang and C. M. Lan, Ruin problems in a discrete Markov risk model,, Statistics & Probability Letters, 79 (2009), 21. doi: 10.1016/j.spl.2008.07.009. [27] K. C. Yuen and J. Y. Guo, Ruin probabilities for time-correlated claims in the compound binomial model,, Insurance Mathematics & Economics, 29 (2001), 47. doi: 10.1016/S0167-6687(01)00071-3. [28] J. Zhu and H. Yang, Ruin theory for a Markov regime-switching model under a threshold dividend strategy,, Insurance Mathematics & Economics, 42 (2008), 311. doi: 10.1016/j.insmatheco.2007.03.004.

show all references

##### References:
 [1] I. Abate and W. Whitt, The Fourier-series method for inverting transforms of probability distributions,, Queueing Systems, 10 (1992), 5. doi: 10.1007/BF01158520. [2] S. Ahn and A. L. Badescu, On the analysis of the Gerber-Shiu discounted penalty function for risk processes with Markovian arrivals,, Insurance Mathematics & Economics, 41 (2007), 234. doi: 10.1016/j.insmatheco.2006.10.017. [3] S. Ahn, A. L. Badescu and V. Ramaswami, Time dependent analysis of finite buffer fluid flows and risk models with a dividend barrier,, Queueing System, 55 (2007), 207. doi: 10.1007/s11134-007-9017-x. [4] S. Asmussen, "Ruin Probabilities,'', World Scientific Publishing, (2000). [5] A. Badescu and D. Landriault, Applications of fluid flow matrix analytic methods in ruin theory-a review,, Revista De La Real Academia De Ciencias Exactas Fisicas Y Naturales Serie a-Matematicas, 103 (2009), 353. [6] A. L. Badescu, E. K. Cheung and D. Randriault, Dependent risk model with bivariate phase-type distributions,, J. Appl. Prob., 46 (2009), 113. doi: 10.1239/jap/1238592120. [7] S. X. Cheng, H. U. Gerber and E. S. W. Shiu, Discounted probabilities and ruin theory in the compound binomial model,, Insurance Mathematics & Economics, 26 (2000), 239. doi: 10.1016/S0167-6687(99)00053-0. [8] Y. B. Cheng, Q. H. Tang and H. L. Yang, Approximations for moments of deficit at ruin with exponential and subexponential claims,, Statistics & Probability Letters, 59 (2002), 367. doi: 10.1016/S0167-7152(02)00234-1. [9] S. N. Chiu and C. C. Yin, The time of ruin, the surplus prior to ruin and the deficit at ruin for the classical risk process perturbed by diffusion,, Insurance Mathematics & Economics, 33 (2003), 59. doi: 10.1016/S0167-6687(03)00143-4. [10] H. Cossette, D. Landriault and E. Marceau, Ruin probabilities in the discrete time renewal risk model,, Insurance Mathematics & Economics, 38 (2006), 309. doi: 10.1016/j.insmatheco.2005.09.005. [11] H. Cossette, E. Marceau and F. Toureille, Risk models based on time series for count random variables,, Insurance Mathematics & Economics, 48 (2011), 19. doi: 10.1016/j.insmatheco.2010.08.007. [12] H. U. Gerber and E. S. W. Shiu, The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin,, Insurance Mathematics & Economics, 21 (1997), 129. doi: 10.1016/S0167-6687(97)00027-9. [13] B. Kim, H. S. Kim and J. Kim, A risk model with paying dividends and random environment,, Insurance Mathematics & Economics, 42 (2008), 717. doi: 10.1016/j.insmatheco.2007.08.001. [14] M. F. Neuts, "Matrix Geometric Solutions in Stochastic Models: An Algorithmic Approach,'', Johns Hopkins University Press, (1981). [15] G. Latouche and V. Ramaswami, "Introduction to Matrix Analytic Methods in Stochastic Modeling,'', American Statistic Association and the Society for Industrial and Applied Mathematics, (1999). doi: 10.1137/1.9780898719734. [16] D. Landriault, On a generalization of the expected discounted penalty function in a discrete-time insurance risk model,, Applied Stochastic Models in Business and Industry, 24 (2008), 525. doi: 10.1002/asmb.713. [17] S. M. Li, Y. Lu and J. Garrido, A review of discrete-time risk models,, Revista De La Real Academia De Ciencias Exactas Fisicas Y Naturales Serie a-Matematicas, 103 (2009), 321. [18] X. S. Lin and G. E. Willmot, Analysis of a defective renewal equation arising in ruin theory,, Insurance Mathematics & Economics, 25 (1999), 63. doi: 10.1016/S0167-6687(99)00026-8. [19] X. S. Lin and K. P. Pavlova, The compound Poisson risk model witha threshold dividend strategy,, Insurance Mathematics & Economics, 38 (2006), 57. doi: 10.1016/j.insmatheco.2005.08.001. [20] X. S. Lin and G. E. Willmot, The moments of the time of ruin, the surplus before ruin, and the deficit at ruin,, Insurance Mathematics & Economics, 27 (2000), 19. doi: 10.1016/S0167-6687(00)00038-X. [21] Y. Lu and S. M. Li, The Markovian regime-switching risk model with a threshold dividend strategy,, Insurance Mathematics & Economics, 44 (2009), 296. doi: 10.1016/j.insmatheco.2008.04.004. [22] M. F. Neuts, "Matrix Geometric Solutions in Stochastic Models: An Algorithmic Approach,'', Johns Hopkins University Press, (1981). [23] M. F. Neuts, "Structured Stochastic Matrices of M/G/1 Type and Their Applications,'', Marcel Dekker, (1989). [24] K. P. Pavlova and G. E. Willmot, The discrete stationary renewal risk model and the Gerber-Shiu discounted penalty function,, Insurance Mathematics & Economics, 33 (2003), 440. [25] M. S. Sgibnev, The matrix analogue of the Blackwell renewal theorem on the real line,, Sbornik: Mathematics, 197 (2006), 69. doi: 10.1070/SM2006v197n03ABEH003762. [26] H. Yang, Z. M. Zhang and C. M. Lan, Ruin problems in a discrete Markov risk model,, Statistics & Probability Letters, 79 (2009), 21. doi: 10.1016/j.spl.2008.07.009. [27] K. C. Yuen and J. Y. Guo, Ruin probabilities for time-correlated claims in the compound binomial model,, Insurance Mathematics & Economics, 29 (2001), 47. doi: 10.1016/S0167-6687(01)00071-3. [28] J. Zhu and H. Yang, Ruin theory for a Markov regime-switching model under a threshold dividend strategy,, Insurance Mathematics & Economics, 42 (2008), 311. doi: 10.1016/j.insmatheco.2007.03.004.
 [1] Rongfei Liu, Dingcheng Wang, Jiangyan Peng. Infinite-time ruin probability of a renewal risk model with exponential Levy process investment and dependent claims and inter-arrival times. Journal of Industrial & Management Optimization, 2017, 13 (2) : 995-1007. doi: 10.3934/jimo.2016058 [2] Emilija Bernackaitė, Jonas Šiaulys. The finite-time ruin probability for an inhomogeneous renewal risk model. Journal of Industrial & Management Optimization, 2017, 13 (1) : 207-222. doi: 10.3934/jimo.2016012 [3] Qingwu Gao, Zhongquan Huang, Houcai Shen, Juan Zheng. Asymptotics for random-time ruin probability in a time-dependent renewal risk model with subexponential claims. Journal of Industrial & Management Optimization, 2016, 12 (1) : 31-43. doi: 10.3934/jimo.2016.12.31 [4] Wuyuan Jiang. The maximum surplus before ruin in a jump-diffusion insurance risk process with dependence. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3037-3050. doi: 10.3934/dcdsb.2018298 [5] Yinghui Dong, Guojing Wang. Ruin probability for renewal risk model with negative risk sums. Journal of Industrial & Management Optimization, 2006, 2 (2) : 229-236. doi: 10.3934/jimo.2006.2.229 [6] Yuebao Wang, Qingwu Gao, Kaiyong Wang, Xijun Liu. Random time ruin probability for the renewal risk model with heavy-tailed claims. Journal of Industrial & Management Optimization, 2009, 5 (4) : 719-736. doi: 10.3934/jimo.2009.5.719 [7] Massimiliano Tamborrino. Approximation of the first passage time density of a Wiener process to an exponentially decaying boundary by two-piecewise linear threshold. Application to neuronal spiking activity. Mathematical Biosciences & Engineering, 2016, 13 (3) : 613-629. doi: 10.3934/mbe.2016011 [8] Jiangyan Peng, Dingcheng Wang. Asymptotics for ruin probabilities of a non-standard renewal risk model with dependence structures and exponential Lévy process investment returns. Journal of Industrial & Management Optimization, 2017, 13 (1) : 155-185. doi: 10.3934/jimo.2016010 [9] Qiuying Li, Lifang Huang, Jianshe Yu. Modulation of first-passage time for bursty gene expression via random signals. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1261-1277. doi: 10.3934/mbe.2017065 [10] Yinghua Dong, Yuebao Wang. Uniform estimates for ruin probabilities in the renewal risk model with upper-tail independent claims and premiums. Journal of Industrial & Management Optimization, 2011, 7 (4) : 849-874. doi: 10.3934/jimo.2011.7.849 [11] Lin Xu, Rongming Wang. Upper bounds for ruin probabilities in an autoregressive risk model with a Markov chain interest rate. Journal of Industrial & Management Optimization, 2006, 2 (2) : 165-175. doi: 10.3934/jimo.2006.2.165 [12] Jan Lorenz, Stefano Battiston. Systemic risk in a network fragility model analyzed with probability density evolution of persistent random walks. Networks & Heterogeneous Media, 2008, 3 (2) : 185-200. doi: 10.3934/nhm.2008.3.185 [13] Regina S. Burachik, C. Yalçın Kaya. An update rule and a convergence result for a penalty function method. Journal of Industrial & Management Optimization, 2007, 3 (2) : 381-398. doi: 10.3934/jimo.2007.3.381 [14] Cheng Ma, Xun Li, Ka-Fai Cedric Yiu, Yongjian Yang, Liansheng Zhang. On an exact penalty function method for semi-infinite programming problems. Journal of Industrial & Management Optimization, 2012, 8 (3) : 705-726. doi: 10.3934/jimo.2012.8.705 [15] Zhiqing Meng, Qiying Hu, Chuangyin Dang. A penalty function algorithm with objective parameters for nonlinear mathematical programming. Journal of Industrial & Management Optimization, 2009, 5 (3) : 585-601. doi: 10.3934/jimo.2009.5.585 [16] Changjun Yu, Kok Lay Teo, Liansheng Zhang, Yanqin Bai. A new exact penalty function method for continuous inequality constrained optimization problems. Journal of Industrial & Management Optimization, 2010, 6 (4) : 895-910. doi: 10.3934/jimo.2010.6.895 [17] Yang Yang, Kam C. Yuen, Jun-Feng Liu. Asymptotics for ruin probabilities in Lévy-driven risk models with heavy-tailed claims. Journal of Industrial & Management Optimization, 2018, 14 (1) : 231-247. doi: 10.3934/jimo.2017044 [18] Xiaoqing Liang, Lihua Bai. Minimizing expected time to reach a given capital level before ruin. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1771-1791. doi: 10.3934/jimo.2017018 [19] Steve Drekic, Jae-Kyung Woo, Ran Xu. A threshold-based risk process with a waiting period to pay dividends. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1179-1201. doi: 10.3934/jimo.2018005 [20] Vladimir Gaitsgory, Tanya Tarnopolskaya. Threshold value of the penalty parameter in the minimization of $L_1$-penalized conditional value-at-risk. Journal of Industrial & Management Optimization, 2013, 9 (1) : 191-204. doi: 10.3934/jimo.2013.9.191

2018 Impact Factor: 1.025