arXiv:2203.10680v1 [q-fin.RM] 21 Mar 2022 The Gerber-Shiu discounted penalty function: From practical perspectives YUE HE * ,REIICHIRO KAWAI † ,Y ASUTAKA SHIMIZU ‡ AND KAZUTOSHI Y AMAZAKI § Abstract The Gerber-Shiu function provides a unified framework for the evaluation of a variety of risk quantities. Ever since its establishment, it has attracted constantly increasing interests in actuarial science, whereas the conventional research has been focused on finding analytical or semi-analytical solutions, either of which is rarely available, except for limited classes of penalty functions on rather simple risk models. In contrast to its great generality, the Gerber-Shiu function does not seem sufficiently prevalent in practice, largely due to a variety of difficulties in numerical approximation and statistical inference. To enhance research activities on such implementation aspects, we provide a full review of existing formulations and underlying surplus processes, as well as an extensive survey of analytical, semi-analytical and asymptotic methods for the Gerber-Shiu function, which altogether shed fresh light on its numerical methods and statistical inference for further developments. On the basis of an exhaustive collection of 207 references, the present survey can serve as an insightful guidebook to model and method selection from practical perspectives as well. Keywords: Gerber-Shiu function; Laplace transform; integro-differential equations; series expansions; scale function. JEL classification: C14, C63, G22. 2020 Mathematics Subject Classification: 91G05, 60G40, 91G60, 91G70. Contents 1 Introduction 2 2 The Gerber-Shiu discounted penalty function 3 2.1 Standard formulation ...................................................... 3 2.2 Generalization .......................................................... 5 2.2.1 Penalty function .................................................... 5 2.2.2 Discount factor ..................................................... 5 2.2.3 Finite time horizon ................................................... 5 2.3 Surplus processes ........................................................ 6 2.3.1 Cram´ er-Lundberg models ............................................... 6 2.3.2 L´ evy risk model .................................................... 6 2.3.3 Diffusion perturbation ................................................. 7 2.3.4 Sparre-Andersen models ................................................ 7 2.3.5 Markov additive processes ............................................... 8 2.3.6 Surplus processes with additional components ..................................... 8 2.4 Specialized formulations for applications ........................................... 8 2.4.1 Tax, dividends and interests .............................................. 8 2.4.2 Credit default swaps .................................................. 9 2.4.3 Capital injection .................................................... 9 2.4.4 Risk measures ..................................................... 10 3 Analytical, semi-analytical and asymptotic methods 10 3.1 Laplace transform of the Gerber-Shiu function ......................................... 10 3.1.1 Cram´ er-Lundberg and L´ evy models .......................................... 11 3.1.2 Advanced models .................................................... 11 3.2 Defective renewal equations .................................................. 11 3.2.1 Cram´ er-Lundberg model ................................................ 12 This work was partially supported by JSPS Grants-in-Aid for Scientific Research 19H01791, 20K03758, 20K22301, 21K03347 and 21K03358. * Email address: y.he@maths.usyd.edu.au. Postal Address: School of Mathematics and Statistics, The University of Sydney, Australia. † Email address: raykawai@g.ecc.u-tokyo.ac.jp. Postal address: Graduate School of Arts and Sciences / Mathematics and Informatics Center, The University of Tokyo, Japan, and School of Mathematics and Statistics, The University of Sydney, Australia. ‡ Email address: shimizu@waseda.jp. Postal address: Department of Applied Mathematics, Waseda University, Japan. § Email address: kyamazak@kansai-u.ac.jp. Postal address: Department of Mathematics, Kansai University, Japan. 1