Skip to main content
SpringerLink
Log in

The Markov Additive Risk Process Under an Erlangized Dividend Barrier Strategy

  • Published:
Methodology and Computing in Applied Probability Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • 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):234–249

    Article  MathSciNet  MATH  Google Scholar 

  • 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):207–222

    Article  MathSciNet  MATH  Google Scholar 

  • Albrecher H, Boxma OJ (2005) On the discounted penalty function in a Markov-dependent risk model. Insur Math Econ 37(3):650–672

    Article  MathSciNet  MATH  Google Scholar 

  • Albrecher H, Cheung ECK, Thonhauser S (2011) Randomized observation periods for the compound Poisson risk model: Dividends. ASTIN Bull 41(2):645–672

    MathSciNet  MATH  Google Scholar 

  • 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):424–452

    Article  MathSciNet  MATH  Google Scholar 

  • Asmussen S (1989) Risk theory in a Markovian environment. Scandinavian Actuarial Journal 1989(2):69–100

    Article  MathSciNet  MATH  Google Scholar 

  • Asmussen S (2003) Applied probability and queues, 2nd Edition. Springer, New York

    MATH  Google Scholar 

  • Asmussen S, Albrecher H (2010) Ruin Probabilities, 2nd Edition. World Scientific, New Jersey

    MATH  Google Scholar 

  • Asmussen S, Avram F, Usabel M (2002) Erlangian approximations for finite-horizon ruin probabilities. ASTIN Bull 32(2):267–281

    Article  MathSciNet  MATH  Google Scholar 

  • 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):98–113

    Article  MathSciNet  MATH  Google Scholar 

  • 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):127–141

    Article  MathSciNet  MATH  Google Scholar 

  • Badescu AL, Drekic S, Landriault D (2007) Analysis of a threshold dividend strategy for a MAP risk model. Scand Actuar J 2007(4):227–247

    Article  MathSciNet  MATH  Google Scholar 

  • Borodin AN, Salminen P (2002) Handbook of Brownian motion - facts and formulae, 2nd Edition. Birkhauser-Verlag, Basel

    Book  MATH  Google Scholar 

  • Breuer L (2008) First passage times for Markov additive processes with positive jumps of phase-type. J Appl Probab 45(3):779–799

    Article  MathSciNet  MATH  Google Scholar 

  • 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):98–109

    Article  MathSciNet  MATH  Google Scholar 

  • Cheung ECK, Landriault D (2009) Perturbed MAP risk models with dividend barrier strategies. J Appl Probab 46(2):521–541

    Article  MathSciNet  MATH  Google Scholar 

  • 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):127–134

    Article  MathSciNet  MATH  Google Scholar 

  • Çinlar E (1969) Markov renewal theory. Adv Appl Probab 1(2):123–187

    Article  MathSciNet  MATH  Google Scholar 

  • Dickson DCM, Hipp C (2001) On the time to ruin for Erlang(2) risk processes. Insur Math Econ 29(3):333–344

    Article  MathSciNet  MATH  Google Scholar 

  • Feng R (2009a) On the total operating costs up to default in a renewal risk model. Insur Math Econ 45(2):305–314

    Article  MathSciNet  MATH  Google Scholar 

  • 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):71–87

    MathSciNet  MATH  Google Scholar 

  • Feng R, Shimizu Y (2014) Potential measure of spectrally negative Markov additive process with applications in ruin theory. Preprint

  • Gerber HU (1979) An Introduction to Mathematical Risk Theory. Huebner Foundation Monograph 8, Homewood, IL, Richard D. Irwin

  • 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):263–276

    Article  MathSciNet  MATH  Google Scholar 

  • Gerber HU, Shiu ESW (1998) On the time value of ruin. North Am Actuar J 2(1):48–72

    Article  MathSciNet  MATH  Google Scholar 

  • Gerber HU, Shiu ESW (2004) Optimal dividends: Analysis with Brownian motion. North Am Actuar J 8(1):1–20

    Article  MathSciNet  MATH  Google Scholar 

  • 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):73–90

    Article  MathSciNet  MATH  Google Scholar 

  • Kyprianou AE (2006) Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer-Verlag, Berlin

    MATH  Google Scholar 

  • 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):551–566

    Article  MathSciNet  MATH  Google Scholar 

  • 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):136–152

    Article  MathSciNet  Google Scholar 

  • Ramaswami V, Woolford DG, Stanford DA (2008) The Erlangization method for Markovian fluid flows. Ann Oper Res 160(1):215–225

    Article  MathSciNet  MATH  Google Scholar 

  • Salah ZB, Morales M (2012) Lévy systems and the time value of ruin for Markov additive processes. Eur Actuar J 2(2):289–317

    Article  MathSciNet  MATH  Google Scholar 

  • 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):131–144

    Article  MathSciNet  MATH  Google Scholar 

  • Stanford DA, Yu K, Ren J (2011) Erlangian approximation to finite time ruin probabilities in perturbed risk models. Scand Actuar J 2011(1):38–58

    Article  MathSciNet  MATH  Google Scholar 

  • Tsai CCL, Willmot GE (2002) A generalized defective renewal equation for the surplus process perturbed by diffusion. Insur Math Econ 30(1):51–66

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang Z (2014) On a risk model with randomized dividend-decision times. J Ind Manag Optim 10(4):1041–1058

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang Z, Yang H, Yang H (2011) On the absolute ruin in a MAP risk model with debit interest. Adv Appl Probab 43(1):77–96

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, People’s Republic of China

    Zhimin Zhang

  2. Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong, Hong Kong

    Eric C. K. Cheung

Authors
  1. Zhimin Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eric C. K. Cheung
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eric C. K. Cheung.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, Z., Cheung, E.C.K. The Markov Additive Risk Process Under an Erlangized Dividend Barrier Strategy. Methodol Comput Appl Probab 18, 275–306 (2016). https://doi.org/10.1007/s11009-014-9414-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11009-014-9414-7

Keywords

AMS 2000 Subject Classification

Search

Navigation