On the discounted aggregate claim costs until ruin in dependent Sparre Andersen risk processes

Abstract

In this paper, a dependent Sparre Andersen risk process in which the joint density of the interclaim time and the resulting claim severity satisfies the factorization as in Willmot and Woo (2012) is considered. We study a generalization of the Gerber-Shiu function (i) whose penalty function further depends on the surplus level immediately after the second last claim before ruin (Cheung et al. (2010a)); and (ii) which involves the moments of the discounted aggregate claim costs until ruin. The generalized discounted density with a moment-based component proposed in Cheung (2013) plays a key role in deriving recursive defective renewal equations. We pay special attention to the case where the marginal distribution of the interclaim times is Coxian, and the required components in the recursion are obtained. A reverse type of dependency structure where the claim severities follow a combination of exponentials is also briefly discussed, and this leads to a nice explicit expression for the expected discounted aggregate claims until ruin. Our results are applied to generate some numerical examples involving (i) the covariance of the time of ruin and the discounted aggregate claims until ruin; and (ii) the expectation, variance and third central moment of the discounted aggregate claims until ruin.

On the Joint Analysis of the Total Discounted Payments to Policyholders and Shareholders: Dividend Barrier Strategy

“…On the other hand, it also implies that no claims larger than b occur early, and this may make Z 0.01 (τ b ) smaller in the presence of discounting. Figure 4 shows that the covariance of τ b and Z 0.01 (τ b ) is positive, suggesting that the former effect dominates under our parameter setting (interested readers are referred to [16] (Section 5) for an example where the latter effect dominates and leads to negative covariance in the context of a dependent Sparre-Andersen model without dividends). In addition, we see from Figure 6 is always positive.…”

“…An exception is [16], who have recently analyzed the covariance of the aggregate discounted claims until ruin and the time of ruin (conditional on ruin occurring). In the context of a dependent Sparre-Andersen risk model where the inter-claim time and the resulting claim amount are modeled via a Farlie-Gumbel-Morgenstern copula with exponential marginals, they provided numerical illustrations showing that the above covariance could possibly take a negative value and gave some probabilistic interpretations, as well.…”

“…Specifically, under the (perturbed) compound Poisson model and the phase-type renewal models, [8,9] studied the distribution of the aggregate claims until ruin without discounting; whereas [10] (Section 6), [11] (Section 4.2) and [12] (Section 5.2) analyzed the expected aggregate discounted claims until ruin. The higher moments of the aggregate discounted claims until ruin were also considered by [13] (Section 2.1) in a risk process with Markovian claims arrival (e.g., [14], Chapter XI.1), by [15] in a renewal risk model with arbitrary inter-claim times and by [16] in a dependent Sparre-Andersen risk model (e.g., [17]). …”

A Note on a Generalized Gerber–Shiu Discounted Penalty Function for a Compound Poisson Risk Model

“…Cheung [22] extended the Gerber-Shiu discounted penalty function by introducing the penultimate claim before ruin under a Sparre-Andersen renewal risk model. Chueng [23], Cheung and Woo [24] proposed a new Gerber-Shiu type function by incorporating the total claims up to ruin. Chueng and Feng [25] studied a new kind of generalized Gerber-Shiu discounted penalty function under the Markov arrival process.…”

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

“…Given a threshold strategy, Dickson & Drekic (2006) analysed the optimal pair of threshold level and dividend rate that maximises the expectation of under a ruin probability constraint, whereas Cheung et al . (2008) derived the higher moments of and computed the optimal threshold minimising the coefficient of variation of .…”