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Moments of the Dividend Payments and Related Problems in a Markov-Modulated Risk Model

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  • Shuanming Li
  • Yi Lu

Abstract

In this paper we derive some results on the dividend payments prior to ruin in a Markovmodulated risk process in which the rate for the Poisson claim arrival process and the distribution of the claim sizes vary in time depending on the state of an underlying (external) Markov jump process {J(t); t ≥ 0}. The main feature of the model is the flexibility in modeling the arrival process in the sense that periods with very frequent arrivals and periods with very few arrivals may alternate, and that the states of {J(t); t ≥ 0} could describe, for example, epidemic types in health insurance or weather conditions in car insurance. A system of integro-differential equations with boundary conditions satisfied by the nth moment of the present value of the total dividends prior to ruin, given the initial environment state, is derived and solved. We show that the probabilities that the surplus process attains a dividend barrier from the initial surplus without first falling below zero and the Laplace transforms of the time that the surplus process first hits a barrier without ruin occurring can be expressed in terms of the solution of the above-mentioned system of integro-differential equations. In the two-state model, explicit results are obtained when both claim amounts are exponentially distributed.

Suggested Citation

  • Shuanming Li & Yi Lu, 2007. "Moments of the Dividend Payments and Related Problems in a Markov-Modulated Risk Model," North American Actuarial Journal, Taylor & Francis Journals, vol. 11(2), pages 65-76.
  • Handle: RePEc:taf:uaajxx:v:11:y:2007:i:2:p:65-76
    DOI: 10.1080/10920277.2007.10597448
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    Cited by:

    1. Lu, Yi & Li, Shuanming, 2009. "The Markovian regime-switching risk model with a threshold dividend strategy," Insurance: Mathematics and Economics, Elsevier, vol. 44(2), pages 296-303, April.
    2. Li, Shuanming & Lu, Yi, 2009. "The distribution of total dividend payments in a Sparre Andersen model," Statistics & Probability Letters, Elsevier, vol. 79(9), pages 1246-1251, May.
    3. Cheung, Eric C.K. & Landriault, David, 2010. "A generalized penalty function with the maximum surplus prior to ruin in a MAP risk model," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 127-134, February.
    4. 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.
    5. Palmowski, Zbigniew & Ramsden, Lewis & Papaioannou, Apostolos D., 2024. "Gerber-Shiu theory for discrete risk processes in a regime switching environment," Applied Mathematics and Computation, Elsevier, vol. 467(C).
    6. Ping Chen & Hailiang Yang, 2010. "Pension funding problem with regime‐switching geometric Brownian motion assets and liabilities," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 26(2), pages 125-141, March.
    7. Zhengjun Jiang & Martijn Pistorius, 2012. "Optimal dividend distribution under Markov regime switching," Finance and Stochastics, Springer, vol. 16(3), pages 449-476, July.
    8. Brill, Percy H. & Yu, Kaiqi, 2011. "Analysis of risk models using a level crossing technique," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 298-309.
    9. Chen, Ping & Yam, S.C.P., 2013. "Optimal proportional reinsurance and investment with regime-switching for mean–variance insurers," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 871-883.

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