Lundberg-Type Bounds for the Joint Distribution of Surplus Immediately Before and at Ruin Under the Sparre Andersen Model

Tail bounds for the joint distribution of the surplus prior to and at ruin

“…The main purpose of the present paper is to obtain new, lower and upper, bounds for the function H (u, x, y) in (2) recently been obtained by Ng and Yang (2005a), see also Ng and Yang (2005b). However, in these papers it is assumed that the adjustment coefficient of the risk process exists, so that in particular, these bounds are not available if claim sizes are heavy tailed, which is often the case in practice.…”

Asymptotic aspects of the Gerber–Shiu function in the renewal risk model using Wiener–Hopf factorization and convolution equivalence

“…Very often it is impossible to get closed-form formulas, or the established integro-differential equations are not solvable at all, or the calculation is rather involved. For bounds, we refer to Ng and Yang (2005), Psarrakos and Politis (2008), and Psarrakos (2008).…”

A connection between the discounted and non-discounted expected penalty functions in the Sparre Andersen risk model

“…As a result, for an arbitrary interclaim time distribution and phase-type claim sizes, we derive simple explicit formulas for the discounted probability of ruin and the discounted distribution of the deficit at ruin. These formulas generalize those for the corresponding nondiscounted quantities in Asmussen (2000), Drekic et al (2004) and Ng and Yang (2005). In addition, the formulas can be evaluated efficiently by the iteration scheme provided and thus give a way to calculate the expected discounted penalty without resorting to finding roots of high-order polynomials.…”

“…Landriault and Willmot (2008) considered a Coxian claim size distribution and derived explicit expressions for the expected discounted penalty function, subject to some restrictions on its form. One the other hand, in the non-discounted paradigm where typical quantities of interest are the ultimate non-discounted distribution of the deficit (see also Ng and Yang (2005)). In this paper, as a continuation of Ren (2007), and of the subsequent discussions by Ko (2007), Li (2008) and Badescu (2008), we study the expected discounted penalty function in a probabilistic manner, similar to that of Asmussen (2000).…”