On a general class of renewal risk process: analysis of the Gerber-Shiu function

Abstract

We consider a compound renewal (Sparre Andersen) risk process with interclaim times that have a K n distribution (i.e. the Laplace transform of their density function is a ratio of two polynomials of degree at most n ∈ N). The Laplace transform of the expected discounted penalty function at ruin is derived. This leads to a generalization of the defective renewal equations given by Willmot (1999) and Gerber and Shiu (2005). Finally, explicit results are given for rationally distributed claim severities.

On the discounted penalty function in a Markov-dependent risk model

“…which is Equation (20) of Li and Garrido [21]. For instance, let w(x 1 , x 2 ) be the Dirac delta function with respect to x 1 = y 1 , x 2 = y 2 (i.e.ω(s) = e −s y 1 b(y 1 + y 2 )).…”

“…Gerber and Shiu [16], Li and Garrido [20]) as well as phase-type interclaim distributions (see Avram and Usabel [6] and Li and Garrido [21]) as special cases (just choose appropriate transition probabilities and let B j be degenerate at 0 for all but one state among {1, . .…”

An Insurance Risk Model with Parisian Implementation Delays

“…Historically, in actuarial risk theory, a lot of attention has been given to the analysis of events related to the time of default which is assumed to occur if and when the surplus process falls below a certain threshold level for the first time (see, e.g., Gerber and Shiu (1998), Li and Garrido (2005) and Willmot (2007)). Without loss of generality, which is due to the spatial homogeneity of most risk processes, this threshold level has commonly been assumed to be the artificial level 0.…”

Time dependent analysis of finite buffer fluid flows and risk models with a dividend barrier

“…Finally, our work can also be considered as deriving a special form of the discounted penalty function introduced by Gerber and Shiu [26,27] where u is the initial surplus, T represents the time of ruin, R(T − ) the surplus prior to ruin, |R(T )| the deficit at ruin and χ(·) an indicator function. Some papers dealing with the general case for specific models are: Li and Garrido [31,32], Albrecher and Boxma [9].…”