Abstract
A Markovian risk process is considered in this paper, which is generalization of the classical risk model. It is proper that a risk process with large claims is modelled as the Markovian risk model. In such a model, the occurrence of claims is described by a point process {N(t)} t≥0 with N(t) being the number of jumps during the interval (0,t] for a Markov jump process. The ruin probability ψ(u) of a company facing such a risk model is mainly studied. An integral equation satisfied by the ruin probability function ψ(u) is obtained and the bounds for the convergence rate of the ruin probability ψ(u) are given by using a generalized renewal technique developed in the paper.
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References
Grandall J. Aspects of risk theory[M]. New York: Springer-Verlag, 1991.
Gerber H U. An introduction to mathematical risk theory[M]. Philadelphia: S. S. Heubner Foundation Monograph Series 8, 1979.
Mikosch T. Heavy-tailed modelling in insurance[J]. Stochastic Models, 13(4):799–816.
Klüppelbery C, Mikosch T. Large deviations of heavy-tailed random sums with applications in insurance and finance[J]. J Appl Prob, 1997, 34(2):293–308.
Wang Hanxing, Fang Dafan, Tang Maoning. Ruin probabilities under a Markovian risk model[J]. Acta Mathematicae Applicatae Sinica, English Series, 2003, 19(4):621–630.
Asmussen S. Applied probability and queues[M]. New York: John Wiley & Sons, 1987.
Asmussen S. Risk theory in a Markovian environment[J]. Scand Actuarial J, 1989, 66–100.
Wang Y H. Bounds for the ruin probability under a Markovian modulated risk model[J]. Stochastic Models, 1999, 15(1):125–136.
Blaszczyszyn B, Rolski T. Expansions for Markov-modulated systems and approximations of ruin probability[J]. J Appl Prob, 1996, 33:57–70.
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Wang, Hx., Yan, Yz., Zhao, F. et al. Markovian risk process. Appl Math Mech 28, 955–962 (2007). https://doi.org/10.1007/s10483-007-0712-y
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DOI: https://doi.org/10.1007/s10483-007-0712-y