The Markov Additive Risk Process Under an Erlangized Dividend Barrier Strategy

  • Zhimin Zhang
  • Eric C. K. Cheung


In this paper, we consider a Markov additive insurance risk process under a randomized dividend strategy in the spirit of Albrecher et al. (2011). Decisions on whether to pay dividends are only made at a sequence of dividend decision time points whose intervals are Erlang(n) distributed. At a dividend decision time, if the surplus level is larger than a predetermined dividend barrier, then the excess is paid as a dividend as long as ruin has not occurred. In contrast to Albrecher et al. (2011), it is assumed that the event of ruin is monitored continuously (Avanzi et al. (2013) and Zhang (2014)), i.e. the surplus process is stopped immediately once it drops below zero. The quantities of our interest include the Gerber-Shiu expected discounted penalty function and the expected present value of dividends paid until ruin. Solutions are derived with the use of Markov renewal equations. Numerical examples are given, and the optimal dividend barrier is identified in some cases.


Markov additive process Barrier strategy Inter-dividend-decision times Gerber-Shiu function Dividends Markov renewal equation Erlangization 

AMS 2000 Subject Classification

91B30 97M30 60J27 60J75 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ahn S, Badescu AL (2007) On the analysis of the Gerber-Shiu discounted penalty function for risk processes with Markovian arrivals. Insur Math Econ 41(2):234–249MathSciNetCrossRefMATHGoogle Scholar
  2. Ahn S, Badescu AL, Ramaswami V (2007) Time dependent analysis of finite buffer fluid flows and risk models with a dividend barrier. Queueing Syst 55(4):207–222MathSciNetCrossRefMATHGoogle Scholar
  3. Albrecher H, Boxma OJ (2005) On the discounted penalty function in a Markov-dependent risk model. Insur Math Econ 37(3):650–672MathSciNetCrossRefMATHGoogle Scholar
  4. Albrecher H, Cheung ECK, Thonhauser S (2011) Randomized observation periods for the compound Poisson risk model: Dividends. ASTIN Bull 41(2):645–672MathSciNetMATHGoogle Scholar
  5. Albrecher H, Cheung ECK, Thonhauser S (2013) Randomized observation periods for the compound Poisson risk model: The discounted penalty function. Scand Actuar J 2013(6):424–452MathSciNetCrossRefMATHGoogle Scholar
  6. Asmussen S (1989) Risk theory in a Markovian environment. Scandinavian Actuarial Journal 1989(2):69–100MathSciNetCrossRefMATHGoogle Scholar
  7. Asmussen S (2003) Applied probability and queues, 2nd Edition. Springer, New YorkMATHGoogle Scholar
  8. Asmussen S, Albrecher H (2010) Ruin Probabilities, 2nd Edition. World Scientific, New JerseyMATHGoogle Scholar
  9. Asmussen S, Avram F, Usabel M (2002) Erlangian approximations for finite-horizon ruin probabilities. ASTIN Bull 32(2):267–281MathSciNetCrossRefMATHGoogle Scholar
  10. Avanzi B, Cheung ECK, Wong B, Woo J-K (2013) On a periodic dividend barrier strategy in the dual model with continuous monitoring of solvency. Insur Math Econ 52(1):98–113MathSciNetCrossRefMATHGoogle Scholar
  11. Badescu AL, Breuer L, Da Silva Soares A, Latouche G, Remiche M-A, Stanford A (2005) Risk processes analyzed as fluid queues. Scand Actuar J 2005(2):127–141MathSciNetCrossRefMATHGoogle Scholar
  12. Badescu AL, Drekic S, Landriault D (2007) Analysis of a threshold dividend strategy for a MAP risk model. Scand Actuar J 2007(4):227–247MathSciNetCrossRefMATHGoogle Scholar
  13. Borodin AN, Salminen P (2002) Handbook of Brownian motion - facts and formulae, 2nd Edition. Birkhauser-Verlag, BaselCrossRefMATHGoogle Scholar
  14. Breuer L (2008) First passage times for Markov additive processes with positive jumps of phase-type. J Appl Probab 45(3):779–799MathSciNetCrossRefMATHGoogle Scholar
  15. Cheung ECK, Feng R (2013) A unified analysis of claim costs up to ruin in a Markovian arrival risk model. Insur Math Econ 53(1):98–109MathSciNetCrossRefMATHGoogle Scholar
  16. Cheung ECK, Landriault D (2009) Perturbed MAP risk models with dividend barrier strategies. J Appl Probab 46(2):521–541MathSciNetCrossRefMATHGoogle Scholar
  17. Cheung ECK, Landriault D (2010) A generalized penalty function with the maximum surplus prior to ruin in a MAP risk model. Insur Math Econ 46(1):127–134MathSciNetCrossRefMATHGoogle Scholar
  18. Çinlar E (1969) Markov renewal theory. Adv Appl Probab 1(2):123–187MathSciNetCrossRefMATHGoogle Scholar
  19. Dickson DCM, Hipp C (2001) On the time to ruin for Erlang(2) risk processes. Insur Math Econ 29(3):333–344MathSciNetCrossRefMATHGoogle Scholar
  20. Feng R (2009a) On the total operating costs up to default in a renewal risk model. Insur Math Econ 45(2):305–314MathSciNetCrossRefMATHGoogle Scholar
  21. Feng R (2009b) A matrix operator approach to the analysis of ruin-related quantities in the phase-type renewal risk model. Bull Swiss Assoc Actuaries 2009(1&2):71–87MathSciNetMATHGoogle Scholar
  22. Feng R, Shimizu Y (2014) Potential measure of spectrally negative Markov additive process with applications in ruin theory. PreprintGoogle Scholar
  23. Gerber HU (1979) An Introduction to Mathematical Risk Theory. Huebner Foundation Monograph 8, Homewood, IL, Richard D. IrwinGoogle Scholar
  24. Gerber HU, Landry B (1998) On the discounted penalty at ruin in a jump-diffusion and the perpetual put option. Insur Math Econ 22(3):263–276MathSciNetCrossRefMATHGoogle Scholar
  25. Gerber HU, Shiu ESW (1998) On the time value of ruin. North Am Actuar J 2(1):48–72MathSciNetCrossRefMATHGoogle Scholar
  26. Gerber HU, Shiu ESW (2004) Optimal dividends: Analysis with Brownian motion. North Am Actuar J 8(1):1–20MathSciNetCrossRefMATHGoogle Scholar
  27. Ji L, Zhang C (2012) Analysis of the multiple roots of the Lundberg fundamental equation in the PH(n) risk model. Appl Stochast Model Bus Ind 28(1):73–90MathSciNetCrossRefMATHGoogle Scholar
  28. Kyprianou AE (2006) Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer-Verlag, BerlinMATHGoogle Scholar
  29. Lin XS, Willmot GE, Drekic S (2003) The compound poisson risk model with a constant dividend barrier: analysis of the Gerber-Shiu discounted penalty function. Insur Math Econ 33(3):551–566MathSciNetCrossRefMATHGoogle Scholar
  30. Lu Y, Tsai CCL (2007) The expected discounted penalty at ruin for a Markov-modulated risk process perturbed by diffusion. North Am Actuar J 11(2):136–152MathSciNetCrossRefGoogle Scholar
  31. Ramaswami V, Woolford DG, Stanford DA (2008) The Erlangization method for Markovian fluid flows. Ann Oper Res 160(1):215–225MathSciNetCrossRefMATHGoogle Scholar
  32. Salah ZB, Morales M (2012) Lévy systems and the time value of ruin for Markov additive processes. Eur Actuar J 2(2):289–317MathSciNetCrossRefMATHGoogle Scholar
  33. Stanford DA, Avram F, Badescu AL, Breuer L, Da Silva Soares A, Latouche G (2005) Phase-type approximations to finite-time ruin probabilities in the Sparre-Anderson and stationary renewal risk models. ASTIN Bull 35(1):131–144MathSciNetCrossRefMATHGoogle Scholar
  34. Stanford DA, Yu K, Ren J (2011) Erlangian approximation to finite time ruin probabilities in perturbed risk models. Scand Actuar J 2011(1):38–58MathSciNetCrossRefMATHGoogle Scholar
  35. Tsai CCL, Willmot GE (2002) A generalized defective renewal equation for the surplus process perturbed by diffusion. Insur Math Econ 30(1):51–66MathSciNetCrossRefMATHGoogle Scholar
  36. Zhang Z (2014) On a risk model with randomized dividend-decision times. J Ind Manag Optim 10(4):1041–1058MathSciNetCrossRefMATHGoogle Scholar
  37. Zhang Z, Yang H, Yang H (2011) On the absolute ruin in a MAP risk model with debit interest. Adv Appl Probab 43(1):77–96MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.College of Mathematics and StatisticsChongqing UniversityChongqingPeople’s Republic of China
  2. 2.Department of Statistics and Actuarial ScienceThe University of Hong KongHong KongHong Kong

Personalised recommendations