Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-24T02:00:49.253Z Has data issue: false hasContentIssue false
  • English
  • Français

The Decompositions of the Discounted Penalty Functions and Dividends-Penalty Identity in a Markov-Modulated Risk Model

Published online by Cambridge University Press:  17 April 2015

Shuanming Li  and
Yi Lu
Shuanming Li
Affiliation:
Centre for Actuarial Studies – Department of Economics, The University of Melbourne, Victoria 3010 – Australia, E-mail: shli@unimelb.edu.au Fax: 61-3-83446899
Yi Lu
Affiliation:
Department of Statistics & Actuarial Science, Simon Fraser University, 8888 University Drive, Burnaby, BC, Canada V5A 1S6, E-mail: yilu@sfu.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we study the expected discounted penalty functions and their decompositions in a Markov-modulated risk process in which the rate for the Poisson claim arrivals and the distribution of the claim amounts vary in time depending on the state of an underlying (external) Markov jump process. The main feature of the model is the flexibility modeling the arrival process in the sense that periods with very frequent arrivals and periods with very few arrivals may alternate. Explicit formulas for the expected discounted penalty function at ruin, given the initial surplus, and the initial and terminal environment states, are obtained when the initial surplus is zero or when all the claim amount distributions are from the rational family. We also investigate the distributions of the maximum surplus before ruin and the maximum severity of ruin. The dividends-penalty identity is derived when the model is modified by applying a barrier dividend strategy.

Keywords

Markov-modulated risk modelexpected discounted penalty functionmaximum surplus before ruinmaximum severity of ruindividends-penalty identity
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 38 , Issue 1 , May 2008 , pp. 53 - 71
Copyright
Copyright © ASTIN Bulletin 2008

References

Albrecher, H. and Boxma, O.J. (2005) On the discounted penalty function in a Markov-dependent risk model. Insurance: Mathematics and Economics 37, 650–672.Google Scholar
Asmussen, S. (1989) Risk theory in a Markovian environment. Scandinavian Actuarial Journal, 69–100.Google Scholar
Asmussen, S., Frey, A., Rolski, T. and Schmidt, V. (1995) Does Markov-modulation increase the risk? ASTIN Bulletin 25, 49–66.CrossRefGoogle Scholar
Bäuerle, N. (1996) Some results about the expected ruin time in Markov-modulated risk models. Insurance: Mathematics and Economics 18, 119–127.Google Scholar
Dickson, D.C.M. and Gray, J. (1984) Approximations to ruin probability in the presence of an upper absorbing barrier. Scandinavian Actuarial Journal, 105–115.Google Scholar
Dickson, D.C.M. and Hipp, C. (2001) On the time to ruin for Erlang(2) risk process,Insurance: Mathematics and Economics 29, 333–344.Google Scholar
Gerber, H.U., Lin, X.S., and Yang, H. (2006) A note on the dividends-penalty identity and the optimal dividend barrier. ASTIN Bulletin 36, 489–503.CrossRefGoogle Scholar
Gerber, H.U. and Shiu, E.S.W. (2005) The time value of ruin in a Sparre Andersen model. North American Actuarial Journal 9(2), 49–69.CrossRefGoogle Scholar
Li, S. and Garrido, J. (2004) On a class of renewal risk models with a constant dividend barrier. Insurance: Mathematics and Economics 35, 691–701.Google Scholar
Li, S. and Dickson, D.C.M. (2006) The maximum surplus before ruin in an Erlang(n) risk process and related problems. Insurance: Mathematics and Economics 38, 529–539.Google Scholar
Li, S. and Lu, Y. (2007) Moments of the dividend payments and related problems in a Markov-modulated risk model. North American Actuarial Journal 11(2), 65–76.CrossRefGoogle Scholar
Lu, Y. (2006) On the severity of ruin in a Markov-modulated risk model. Scandinavian Actuarial Journal, 183–202.Google Scholar
Lu, Y. and Li, S. (2005) On the probability of ruin in a Markov-modulated risk model. Insurance: Mathematics and Economics 37, 522–532.Google Scholar
Ng, A.C.Y. and Yang, H. (2006) On the joint distribution of surplus before and after ruin under a Markovian regime switching model. Stochastic Processes and their Applications 116, 244–266.CrossRefGoogle Scholar
Picard, P. (1994) On some measures of the severity of ruin in the classical Poisson model. Insurance: Mathematics and Economics 14, 107–115.Google Scholar
Reinhard, J.M. (1984) On a class of semi-Markov risk models obtained as classical risk models in a Markovian environment. ASTIN Bulletin 14, 23–43.CrossRefGoogle Scholar
Schmidli, H. (1997) Estimation of the Lundberg coefficient for a Markov modulated risk model. Scandinavian Actuarial Journal, 48–57.CrossRefGoogle Scholar
Snoussi, M. (2002) The severity of ruin in Markov-modulated risk models. Bulletin of the Swiss Association of Actuaries, 31–43.Google Scholar
Wu, Y. (1999) Bounds for the ruin probability under a Markovian modulated risk model. Commun. Statist.-Stochastic Models, 15(1), 125–136.CrossRefGoogle Scholar
Zhu, J. and Yang, H. (2007) Ruin theory for a Markov regime-switching model under a threshold dividend strategy. Insurance: Mathematics and economics, doi:10.1016/j.insmatheco.2007.03.004CrossRefGoogle Scholar