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Risks 2018, 6(2), 59; https://doi.org/10.3390/risks6020059

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

1
Centre for Actuarial Studies, Department of Economics, The University of Melbourne, Melbourne 3010, Australia
2
Department of Statistics and Actuarial Science, Simon Fraser University, Burnaby, BC V5A 1S6, Canada
*
Author to whom correspondence should be addressed.
Received: 7 May 2018 / Revised: 17 May 2018 / Accepted: 21 May 2018 / Published: 23 May 2018
(This article belongs to the Special Issue Risk, Ruin and Survival: Decision Making in Insurance and Finance)
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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. View Full-Text
Keywords: aggregate discounted claims; Markovian arrival process; partial integro-differential equation; covariance aggregate discounted claims; Markovian arrival process; partial integro-differential equation; covariance
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Li, S.; Lu, Y. On the Moments and the Distribution of Aggregate Discounted Claims in a Markovian Environment. Risks 2018, 6, 59.

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