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

Jerim Kim, Bara Kim, Hwa Sung Kim

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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.

Original languageEnglish
Pages (from-to)909-924
Number of pages16
JournalJournal of Industrial and Management Optimization
Issue number4
Publication statusPublished - 2012 Nov 1



  • First passage time
  • Penalty function
  • Risk model
  • Ruin probability
  • Surplus process

ASJC Scopus subject areas

  • Business and International Management
  • Strategy and Management
  • Control and Optimization
  • Applied Mathematics

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