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On differentiability of ruin functions under Markov-modulated models

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  • Zhu, Jinxia
  • Yang, Hailiang

Abstract

This paper analyzes the continuity and differentiability of several classes of ruin functions under Markov-modulated insurance risk models with a barrier and threshold dividend strategy, respectively. Many ruin related functions in the literature, such as the expectation and the Laplace transform of the Gerber-Shiu discounted penalty function at ruin, of the total discounted dividends until ruin, and of the time-integrated discounted penalty and/or reward function of the risk process, etc, are special cases of the functions considered in this paper. Continuity and differentiability of these functions in the corresponding dual models are also studied.

Suggested Citation

  • Zhu, Jinxia & Yang, Hailiang, 2009. "On differentiability of ruin functions under Markov-modulated models," Stochastic Processes and their Applications, Elsevier, vol. 119(5), pages 1673-1695, May.
    • Handle: RePEc:eee:spapps:v:119:y:2009:i:5:p:1673-1695
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    References listed on IDEAS

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    1. Stéphane Loisel, 2005. "Differentiation of some functionals of risk processes and optimal reserve allocation," Post-Print hal-00397289, HAL.
    2. Stéphane Loisel, 2005. "Differentiation of functionals of risk processes and optimal reserve allocation," Post-Print hal-00397290, HAL.
    3. Avanzi, Benjamin & U. Gerber, Hans & S.W. Shiu, Elias, 2007. "Optimal dividends in the dual model," Insurance: Mathematics and Economics, Elsevier, vol. 41(1), pages 111-123, July.
    4. Hans Gerber & Elias Shiu, 1998. "On the Time Value of Ruin," North American Actuarial Journal, Taylor & Francis Journals, vol. 2(1), pages 48-72.
    5. Bauerle, Nicole, 1996. "Some results about the expected ruin time in Markov-modulated risk models," Insurance: Mathematics and Economics, Elsevier, vol. 18(2), pages 119-127, July.
    6. Wang, Guojing & Wu, Rong, 2001. "Distributions for the risk process with a stochastic return on investments," Stochastic Processes and their Applications, Elsevier, vol. 95(2), pages 329-341, October.
    7. Stéphane Loisel, 2005. "Differentiation of some functionals of risk processes," Post-Print hal-00157739, HAL.
    8. Cai, Jun, 2004. "Ruin probabilities and penalty functions with stochastic rates of interest," Stochastic Processes and their Applications, Elsevier, vol. 112(1), pages 53-78, July.
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    Cited by:

    1. Chen, Mi & Yuen, Kam Chuen & Guo, Junyi, 2014. "Survival probabilities in a discrete semi-Markov risk model," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 205-215.
    2. Cheung, Eric C.K. & Feng, Runhuan, 2013. "A unified analysis of claim costs up to ruin in a Markovian arrival risk model," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 98-109.

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