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

Jerim Kim; Bara Kim; Hwa-Sung Kim

doi:10.3934/jimo.2012.8.909

Article Contents

2012, Volume 8, Issue 4: 909-924. Doi: 10.3934/jimo.2012.8.909

This issue Previous Article Analysis of customers' impatience in an M/M/1 queue with working vacations Next Article Stochastic decomposition in discrete-time queues with generalized vacations and applications

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
2.
School of Management, Kyung Hee University, 26 Kyunghee-daero, Dongdaemun-gu, Seoul, 130-701

Received: August 31, 2011

Revised: June 30, 2012

Published: September 2012

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Abstract

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.

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Mathematics Subject Classification: Primary: 60K25, 91B30; Secondary: 60K37.

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