Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T06:38:12.846Z Has data issue: false hasContentIssue false
  • English
  • Français

Perturbed MAP Risk Models with Dividend Barrier Strategies

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Eric C. K. Cheung  and
David Landriault
Eric C. K. Cheung*
Affiliation:
University of Waterloo
David Landriault*
Affiliation:
University of Waterloo
*
∗Postal address: Department of Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L 3G1, Canada.
∗Postal address: Department of Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L 3G1, Canada.
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 the context of a dividend barrier strategy (see, e.g. Lin, Willmot and Drekic (2003)) we analyze the moments of the discounted dividend payments and the expected discounted penalty function for surplus processes with claims arriving according to a Markovian arrival process (MAP). We show that a relationship similar to the dividend-penalty identity of Gerber, Lin and Yang (2006) can be established for the class of perturbed MAP surplus processes, extending in the process some results of Li and Lu (2008) for the Markov-modulated risk model. Also, we revisit the same ruin-related quantities in an identical MAP risk model with the only exception that the barrier level effective at time t depends on the state of the underlying environment at this time. Similar relationships are investigated and derived. Numerical examples are also considered.

Keywords

Markovian arrivalperturbed processbarrier strategyGerber–Shiu functiondiscounted dividend payment

MSC classification

Secondary: 60J75: Jump processes 60J25: Continuous-time Markov processes on general state spaces 60J60: Diffusion processes
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 2 , June 2009 , pp. 521 - 541
Copyright
Copyright © Applied Probability Trust 2009 

References

Abate, J., Choudhury, G. L. and Whitt, W. (2000). An introduction to numerical transform inversion and its application to probability models. In Computational Probability, ed. Grassman, W. K., Kluwer, Norwell, MA, pp. 257–323.Google Scholar
Albrecher, H. and Boxma, O. (2005). On the discounted penalty function in a Markov-dependent risk model. Insurance Math. Econom. 37, 650–672.CrossRefGoogle Scholar
Ahn, S. and Ramaswami, V. (2006). Transient analysis of fluid models via elementary level-crossing arguments. Stoch. Models 22, 129–147.CrossRefGoogle Scholar
Ahn, S., Badescu, A. L. and Ramaswami, V. (2007). Time dependent analysis of finite buffer fluid flows and risk models with a dividend barrier. Queueing Systems 55, 207–222.CrossRefGoogle Scholar
Asmussen, S. (1989). Risk theory in a Markovian environment. Scand. Actuarial J. 1989, 69–100.Google Scholar
Asmussen, S. (1995). Stationary distributions via first passage times. In Advances in Queueing, CRC, Boca Raton, FL, pp. 79–102.Google Scholar
Badescu, A., Drekic, S. and Landriault, D. (2007). On the analysis of a multi-threshold Markovian risk model. Scand. Actuarial J. 2007, 248–260.CrossRefGoogle Scholar
Badescu, A. L. et al. (2005). Risk processes analyzed as fluid queues. Scand. Actuarial J. 2005, 127–141.CrossRefGoogle Scholar
Cheung, E. C. K. (2007). Discussion of ‘Moments of the dividend payments and related problems in a Markov-modulated risk model’. N. Amer. Actuarial J. 11, 145–148.CrossRefGoogle Scholar
Dickson, D. C. M. and Hipp, C. (2001). On the time to ruin for Erlang(2) risk processes. Insurance Math. Econom. 29, 333–344.CrossRefGoogle Scholar
Dufresne, D. (2001). On a general class of risk models. Austral. Actuarial J. 7, 755–791.Google Scholar
Gerber, H. U. and Shiu, E. S. W. (1998). On the time value of ruin. N. Amer. Actuarial J. 2, 48–78.Google Scholar
Gerber, H. U. and Shiu, E. S. W. (2004a). Optimal dividends: analysis with Brownian motion. N. Amer. Actuarial J. 8, 1–20.Google Scholar
Gerber, H. U. and Shiu, E. S. W. (2004b). Authors' reply: Optimal dividends: analysis with Brownian motion. N. Amer. Actuarial J. 8, 113–115.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 Bull. 36, 489–503.CrossRefGoogle Scholar
Latouche, G. and Ramaswami, V. (1999). Introduction to Matrix Analytic Methods in Stochastic Modeling. American Statistical Association, Alexandria, VA.CrossRefGoogle Scholar
Li, S. (2006). The distribution of the dividend payments in the compound Poisson risk model perturbed by diffusion. Scand. Actuarial J. 2006, 73–85.CrossRefGoogle Scholar
Li, S. and Lu, Y. (2007). Moments of the dividend payments and related problems in a Markov-modulated risk model. N. Amer. Actuarial J. 11, 65–76.Google Scholar
Li, S. and Lu, Y. (2008). The decompositions of the discounted penalty functions and dividends-penalty identity in a Markov-modulated risk model. ASTIN Bull. 38, 53–71.CrossRefGoogle Scholar
Lin, X. S., Willmot, G. E. and Drekic, S. (2003). The compound Poisson risk model with a constant dividend barrier: analysis of the Gerber–Shiu discounted penalty function. Insurance Math. Econom. 33, 551–566.Google Scholar
Lu, Y. and Tsai, C. C.-L. (2007). The expected discounted penalty at ruin for a Markov-modulated risk process perturbed by diffusion. N. Amer. Actuarial J. 11, 136–152.CrossRefGoogle Scholar
Neuts, M. F. (1989). Structured Stochastic Matrices of M/G/1 Type and Their Applications. Marcel Dekker, New York.Google Scholar
Ramaswami, V. (2006). Passage times in fluid models with application to risk processes. Methodology Comput. Appl. Prob. 8, 497–515.Google Scholar
Zhu, J. and Yang, H. (2008). Ruin theory for a Markov regime-switching model under a threshold dividend strategy. Insurance Math. Econom. 42, 311–318.Google Scholar