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 and 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), 5-87. 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), 234-249. 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), 207-222. 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), 353-372.

[6]

A. L. Badescu, E. K. Cheung and D. Randriault, Dependent risk model with bivariate phase-type distributions, J. Appl. Prob., 46 (2009), 113-131. 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), 239-250. 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), 367-378. 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), 59-66. 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), 309-323. 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), 19-28. 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), 129-137. 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), 717-726. 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, Baltimore, 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), 525-539. 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), 321-337.

[18]

X. S. Lin and G. E. Willmot, Analysis of a defective renewal equation arising in ruin theory, Insurance Mathematics & Economics, 25 (1999), 63-84. 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), 57-80. 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), 19-44. 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), 296-303. 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, Baltimore, 1981.

[23]

M. F. Neuts, "Structured Stochastic Matrices of M/G/1 Type and Their Applications,'' Marcel Dekker, New York, NY, 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), 440-440.

[25]

M. S. Sgibnev, The matrix analogue of the Blackwell renewal theorem on the real line, Sbornik: Mathematics, 197 (2006), 69-86. 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), 21-28. 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), 47-57. 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), 311-318. 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), 5-87. 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), 234-249. 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), 207-222. 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), 353-372.

[6]

A. L. Badescu, E. K. Cheung and D. Randriault, Dependent risk model with bivariate phase-type distributions, J. Appl. Prob., 46 (2009), 113-131. 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), 239-250. 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), 367-378. 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), 59-66. 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), 309-323. 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), 19-28. 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), 129-137. 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), 717-726. 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, Baltimore, 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), 525-539. 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), 321-337.

[18]

X. S. Lin and G. E. Willmot, Analysis of a defective renewal equation arising in ruin theory, Insurance Mathematics & Economics, 25 (1999), 63-84. 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), 57-80. 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), 19-44. 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), 296-303. 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, Baltimore, 1981.

[23]

M. F. Neuts, "Structured Stochastic Matrices of M/G/1 Type and Their Applications,'' Marcel Dekker, New York, NY, 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), 440-440.

[25]

M. S. Sgibnev, The matrix analogue of the Blackwell renewal theorem on the real line, Sbornik: Mathematics, 197 (2006), 69-86. 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), 21-28. 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), 47-57. 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), 311-318. doi: 10.1016/j.insmatheco.2007.03.004.

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