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
This is a preview of subscription content, log in to check access.
Albrecher H, Cheung ECK, Thonhauser S (2011) Randomized observation periods for the compound Poisson risk model: Dividends. ASTIN Bull 41(2):645–672MathSciNetMATHGoogle 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):424–452MathSciNetCrossRefMATHGoogle 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):98–113MathSciNetCrossRefMATHGoogle 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):127–141MathSciNetCrossRefMATHGoogle 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):73–90MathSciNetCrossRefMATHGoogle Scholar
Kyprianou AE (2006) Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer-Verlag, BerlinMATHGoogle 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):551–566MathSciNetCrossRefMATHGoogle 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):136–152MathSciNetCrossRefGoogle 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):131–144MathSciNetCrossRefMATHGoogle Scholar
Stanford DA, Yu K, Ren J (2011) Erlangian approximation to finite time ruin probabilities in perturbed risk models. Scand Actuar J 2011(1):38–58MathSciNetCrossRefMATHGoogle Scholar
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