On the joint analysis of the total discounted payments to policyholders and shareholders: threshold dividend strategy

Abstract

In insurance risk theory, dividend and aggregate claim amount are of great research interest as they represent the insurance company's payments to its shareholders and policyholders, respectively. Since the analyses of these two quantities are performed separately in the literature, the companion paper by Cheung et al. generalised the Gerber-Shiu expected discounted penalty function by further incorporating the moments of the aggregate discounted claims until ruin and the discounted dividends until ruin. While Cheung et al. considered the compound Poisson model with a dividend barrier in which ruin occurs almost surely, the present paper looks at this generalised Gerber-Shiu function under a threshold dividend strategy where the insurer has a positive survival probability. Because the Gerber-Shiu function is only defined for sample paths leading to ruin, we will additionally study the joint moments of the aggregate discounted claims and the discounted dividends without ruin occurring. Some explicit formulas are derived when the individual claim distribution follows a combination of exponentials. Numerical illustrations involving the correlation between aggregate discounted claims and discounted dividends are given. For the case where ruin occurs, we additionally compute the correlations between the time of ruin and the above two quantities.

On weighted occupation times for refracted spectrally negative Lévy processes

“…In many insurance risk models, see for example [13,8,4] and references therein, to maximise the amount of discounted dividends paid up to time of ruin, among all admissible control strategies the optimal dividend strategy is either paying nothing or paying dividends as much as possible in the so-called solvency regions. When a constant ceiling δ is imposed for the dividend rate, under further conditions the optimal policy reduces to the so-called threshold dividend strategy and the surplus process with dividends becomes the refracted process, in which the insurance company pays nothing when the reserve is below a certain critical level, and pays dividends at the maximal rate δ when the reserve is above the level.…”

The Gerber–Shiu Discounted Penalty Function for the Bi-Seasonal Discrete Time Risk Model

“…Properties of the Gerber-Shiu function in the risk renewal models perturbed by diffusion were investigated by Chi et al (2010), Tsai (2003), Tsai and Willmot (2002), Xu et al (2014), Zhang and Cheung (2016), Zhang et al (2012Zhang et al ( , 2017bZhang et al ( , 2014. The Gerber-Shiu function of the risk models with various special strategies were considered by Avram et al (2015), Bratiichuk (2012), Cheung and Liu (2016), Cheung et al (2015), Dong et al (2009), Lin and Pavlova (2006), Lin and Sendova (2008), Liu et al (2015), Marciniak and Palmowski (2016), Shi et al (2013), Shiraishi (2016), Woo et al (2017), Zhang et al (2017a), Zhou et al (2015). This function for the risk models with various dependence structures or for risk models with investment strategies was considered by Cheung et al (2011), Cossette et al (2011), Li and Lu (2013), Mihálýko and Mihálýk (2011), Schmidli (2015), among others.…”

On the optimality of the refraction–reflection strategies for Lévy processes

“…Indeed, using the scale function, various quantities of interest related to first passage times and resolvent/occupation measures can be explicitly written; see [1,11,12,13,19,21,23]. There exist results on refracted processes driven by processes other than spectrally one-sided Lévy processes such as [3,20,24,25]. In these papers, obtained identities are more involved, without the use of the scale function and do not consider optimimization problems.…”