On a generalization from ruin to default in a L´ evy insurance risk model Runhuan Feng ∗ and Yasutaka Shimizu †‡ First draft: January 13, 2011 Abstract In a variety of insurance risk models, ruin-related quantities in the class of expected discounted penalty function (EDPF) were known to satisfy defective renewal equations that lead to explicit solutions. Recent development in the ruin literature has shown that similar defective renewal equations exist for a more general class of quantities than that of EDPF. This paper further extends the result on the class of functions that satisfy defective renewal equations in a spectrally negative L´ evy risk models. In particular, we present an operator-based approach as an alternative analytical tool in comparison with fluctuation theoretic methods used for similar quantities in the current literature. Key words: Time of ruin; expected discounted penalty function; defective renewal equation; compound geometric distribution; L´ evy risk model. MSC2010: 91B30; 91G80; 60G51. 1 Introduction In the recent literature of ruin theory, a class of functions, known as expected discounted penalty function (EDPF) or the Gerber-Shiu function proposed by Gerber and Shiu [13], has gained enormous research interests due to its generality in representing a variety of ruin-related quantities, including the probability of ultimate ruin, the tri-variate Laplace transform of the time to ruin, surplus prior to ruin and deficit at ruin, etc. To show its precise definition, we introduce the mathematical structure of risk models. Consider a probability space (Ω, F , (F t ) t≥0 , P) in which the evolution of an insurance company’s surplus is defined by a strong Markov process X = {X t ,t ≥ 0} together with a family of probability measures {P x ,x ≥ 0} such that P x (X 0 = x)=1. The event of ruin is hence represented by τ 0 = inf {t> 0: X t < 0}. The Gerber-Shiu function is defined by m(x) := E x [ e −δτ 0 w(X τ 0 − , |X τ 0 |)I (τ 0 < ∞) ] , x ≥ 0, ∗ Department of Mathematical Sciences, University of Wisconsin - Milwaukee, P.O. Box 413, Milwaukee, WI, USA 53202-0413; fengr@uwm.edu † Graduate School of Engineering Science, Osaka University; 1-3, Machikaneyama-cho, Toyonaka-shi, Osaka 560-8531, Japan; yasutaka@sigmath.es.osaka-u.ac.jp ‡ Japan Science and Technology Agency, CREST; Sanbancho, Chiyoda-ku, Tokyo 102-0075, Japan. 1