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On the Moments and the Distribution of Aggregate Discounted Claims in a Markovian Environment

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  • Shuanming Li

    (Centre for Actuarial Studies, Department of Economics, The University of Melbourne, Melbourne 3010, Australia)

  • Yi Lu

    (Department of Statistics and Actuarial Science, Simon Fraser University, Burnaby, BC V5A 1S6, Canada)

Abstract

This paper studies the moments and the distribution of the aggregate discounted claims (ADCs) in a Markovian environment, where the claim arrivals, claim amounts, and forces of interest (for discounting) are influenced by an underlying Markov process. Specifically, we assume that claims occur according to a Markovian arrival process (MAP). The paper shows that the vector of joint Laplace transforms of the ADC occurring in each state of the environment process by any specific time satisfies a matrix-form first-order partial differential equation, through which a recursive formula is derived for the moments of the ADC occurring in certain states (a subset). We also study two types of covariances of the ADC occurring in any two subsets of the state space and with two different time lengths. The distribution of the ADC occurring in certain states by any specific time is also investigated. Numerical results are also presented for a two-state Markov-modulated model case.

Suggested Citation

  • Shuanming Li & Yi Lu, 2018. "On the Moments and the Distribution of Aggregate Discounted Claims in a Markovian Environment," Risks, MDPI, vol. 6(2), pages 1-16, May.
    • Handle: RePEc:gam:jrisks:v:6:y:2018:i:2:p:59-:d:148616
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    References listed on IDEAS

    as
    1. Siti Norafidah Mohd Ramli & Jiwook Jang, 2014. "Neumann Series on the Recursive Moments of Copula-Dependent Aggregate Discounted Claims," Risks, MDPI, vol. 2(2), pages 1-16, May.
    2. Jiandong Ren, 2008. "On the Laplace Transform of the Aggregate Discounted Claims with Markovian Arrivals," North American Actuarial Journal, Taylor & Francis Journals, vol. 12(2), pages 198-206.
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    5. Bargès, Mathieu & Cossette, Hélène & Loisel, Stéphane & Marceau, Étienne, 2011. "On the Moments of Aggregate Discounted Claims with Dependence Introduced by a FGM Copula," ASTIN Bulletin, Cambridge University Press, vol. 41(1), pages 215-238, May.
    6. Shuanming Li, 2008. "“On the Laplace Transform of the Aggregate Discounted Claims with Markovian Arrivals,” Jiandong Ren, April 2008," North American Actuarial Journal, Taylor & Francis Journals, vol. 12(4), pages 443-445.
    7. Ji‐Wook Jang, 2004. "Martingale Approach for Moments of Discounted Aggregate Claims," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 71(2), pages 201-211, June.
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    Cited by:

    1. Sharifah Farah Syed Yusoff Alhabshi & Zamira Hasanah Zamzuri & Siti Norafidah Mohd Ramli, 2021. "Monte Carlo Simulation of the Moments of a Copula-Dependent Risk Process with Weibull Interwaiting Time," Risks, MDPI, vol. 9(6), pages 1-21, June.
    2. Jiandong Ren & Kristina Sendova & Ričardas Zitikis, 2019. "Special Issue “Risk, Ruin and Survival: Decision Making in Insurance and Finance”," Risks, MDPI, vol. 7(3), pages 1-7, September.

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