# 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
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