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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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S. Asmussen, "Ruin Probabilities,'', World Scientific Publishing, (2000).   Google Scholar

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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.   Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

M. F. Neuts, "Matrix Geometric Solutions in Stochastic Models: An Algorithmic Approach,'', Johns Hopkins University Press, (1981).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

M. F. Neuts, "Matrix Geometric Solutions in Stochastic Models: An Algorithmic Approach,'', Johns Hopkins University Press, (1981).   Google Scholar

[23]

M. F. Neuts, "Structured Stochastic Matrices of M/G/1 Type and Their Applications,'', Marcel Dekker, (1989).   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

S. Asmussen, "Ruin Probabilities,'', World Scientific Publishing, (2000).   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

M. F. Neuts, "Matrix Geometric Solutions in Stochastic Models: An Algorithmic Approach,'', Johns Hopkins University Press, (1981).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

M. F. Neuts, "Matrix Geometric Solutions in Stochastic Models: An Algorithmic Approach,'', Johns Hopkins University Press, (1981).   Google Scholar

[23]

M. F. Neuts, "Structured Stochastic Matrices of M/G/1 Type and Their Applications,'', Marcel Dekker, (1989).   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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