Insurance: Mathematics and Economics 46 (2010) 1–2 Contents lists available at ScienceDirect Insurance: Mathematics and Economics journal homepage: www.elsevier.com/locate/ime Editorial for the special issue on Gerber–Shiu functions Ruin theory is one of the classical fields of actuarial mathemat- ics, with methodological links to various other areas of applied probability. While in the early days the probability of ruin was the main object of study, more recently a number of other ruin quan- tities that measure the risk related to the event of default in a non- life insurance portfolio have been investigated, in particular the time of ruin, the deficit at ruin and the surplus prior to ruin. In two now classical papers, Gerber and Shiu (1997, 1998) identified a very elegant way to study these quantities simultaneously us- ing a so-called discounted penalty function. The resulting expected discounted penalty function is now usually referred to as the Ger- ber–Shiu function. This extension of the ruin probability turned out to be a very fruitful concept that triggered intensive research. The behavior of the Gerber–Shiu function has been studied in increas- ingly complex models for the underlying surplus process, as well as its applications in insurance and finance. The diversity of techniques and perspectives under which this function was studied in the last few years has already led to the organization of two international workshops on Gerber–Shiu functions with about 50 participants each. The first one took place at Concordia University in Montreal (August 7–8, 2006) and the second one at the Radon Institute of the Austrian Academy of Sciences in Linz, Austria (August 27–29, 2008). As organizers of these two conferences, we witnessed the rapid increase in activity in the field, so it seemed natural to initiate a thematic issue of Insurance: Mathematics and Economics on the topic of Gerber–Shiu functions. It features research papers on recent methodological advances, as well as papers that establish further connections of the theory to other fields of mathematics and applied probability. The contributions in the issue are grouped according to their underlying model assumptions. In the first paper, Hanspeter Schmidli shows how change of measure techniques can simplify and unify the analysis of Gerber– Shiu functions and gives examples in several models. David Dick- son and Shuanming Li then identify expressions for the joint den- sity of the time to ruin and the deficit at ruin in an Erlang(2) risk model using certain decomposition properties in connection with Laplace transforms. In the next paper, Qihe Tang and Li Wei extend the Wiener–Hopf factorization in the continuous-time renewal risk model to include the times of ascending and de- scending ladders, which then leads to asymptotic formulas for the Gerber–Shiu function. A renewal risk model with K n distributions for the interclaim time is then studied by Gordon Willmot and Jae-Kyung Woo. They derive a defective renewal equation and its solution for a generalization of the Gerber–Shiu function that also involves the last interclaim time before ruin. Another ap- proach to analyze Gerber–Shiu functions in renewal models based on symbolic computation techniques is introduced in the paper of Hansjörg Albrecher, Corina Constantinescu, Gottlieb Pirsic, Georg Regensburger and Markus Rosenkranz. In particular, it is shown that if the differential operator of the associated boundary value problem can be factorized, then explicit expressions for the Ger- ber–Shiu function can be obtained by iteratively solving first-order problems. Yichun Chi, Sebastian Jaimungal and Sheldon Lin con- sider a risk model perturbed by diffusion with stochastic volatility driven by an underlying Ornstein–Uhlenbeck process. They apply singular perturbation theory to obtain an asymptotic expansion of the associated Gerber–Shiu function and give results on the ac- curacy of the first two terms of this expansion. An expected dis- counted penalty at absolute ruin in the presence of tax payments and interest rates in a compound Poisson framework is discussed by Ruixing Ming, Wenyuan Wang and Liqun Xiao. Exit probabilities and exit times of spectrally negative Lévy processes, of which the compound Poisson process and the Brownian motion are special cases, have been analyzed in terms of fluctuation theory in other contexts. Enrico Biffis and Andreas Kyprianou give an explicit characterization of a generalized version of the Gerber–Shiu function that includes the last minimum before ruin, via scale functions in the general setup of spectrally negative Lévy processes. They discuss further links between classical tools of probability theory and the risk theory context. In another paper, Enrico Biffis and Manuel Morales represent this generalized Gerber–Shiu function in the Lévy context in terms of convolutions and derive its defective renewal equation. Jean-Francois Renaud and Ronnie Loeffen then give criteria on the Lévy measure under which a barrier strategy optimizes the expected value of discounted dividend payments minus a penalty term that depends on the deficit at ruin. As a by-product, they also manage to relax previously established sufficient conditions for the barrier strategy to be optimal for the classical problem of maximizing expected dividend payments until ruin. For a fixed horizontal barrier level, Hans Gerber, Elias Shiu and Hailiang Yang consider a discrete-time model for the surplus of an insurance company in terms of a time- homogeneous Markov chain and show in an elegant elementary way that in this case the dividends–penalty identity holds. This identity expresses the expected discounted penalty at ruin through the expected discounted dividends until ruin and the expected discounted penalty at ruin without dividends. The determination of optimal dividend strategies (typically band strategies) is also discussed. Finally, Eric Cheung, David Landriault, Gordon Willmot and Jae- Kyung Woo investigate the structure of Gerber–Shiu functions in renewal models that allow for dependence between claim sizes and interclaim times. In particular, the minimum surplus before ruin and the surplus immediately after the second last claim before ruin can be included in this approach and several properties of the independent model are shown to hold in this dependent setup as well. Eric Cheung and David Landriault then show in another contribution that an extension of the Gerber–Shiu 0167-6687/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.insmatheco.2009.10.004