The Markov Additive Risk Process Under an Erlangized Dividend Barrier Strategy

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.

Lévy insurance risk process with Poissonian taxation

“…the model dynamics of the taxed surplus process still follows X θ with the time of ruin now defined by τ c θ = inf {t > 0 : X θ t < 0}. Such a model assumption can be viewed as a complement to risk processes with periodic dividend barrier strategy and continuous monitoring of solvency, which were studied by Avanzi et al (2013Avanzi et al ( , 2014, Zhang (2014) and Zhang & Cheung (2014a, 2014b. In addition, one can also treat the present model as the reverse case of Albrecher & Ivanovs (2014, Section 6), which is concerned with Poissonian monitoring of ruin but continuous checking for tax payments.…”

“…A taxation system with delayed tax payment is also studied. In Section 4, we consider the situation where solvency is monitored continuously, but tax is still payable at Poissonian time points (see Avanzi et al (2013Avanzi et al ( , 2014, Zhang (2014) and Zhang & Cheung (2014a, 2014b for similar assumptions in dividend problems), and results analogous to those in Section 3 can readily be obtained. Section 5 ends the paper with some numerical illustrations.…”

A note on a Lévy insurance risk model under periodic dividend decisions

“…Since then, such a variant has been analyzed under perturbed compound Poisson and Markov additive insurance risk models by [44] and [45]. We remark that the reverse case with continuous dividend decisions and Poissonian monitoring of ruin was also studied by [4].…”

“…In this paper, we shall consider the modifications as in [9] and [45] in the context of a spectrally negative Lévy risk process X. The dynamics of the modified surplus process, namely X b = {X b t } t≥0 , under a barrier-type strategy are described below.…”

“…[8,41,42,38]), and it has recently gained popularity in risk processes with periodic decisions as well (e.g. [2,9,45,47]). We also refer interested readers to e.g.…”