An algebraic operator approach to the analysis of Gerber–Shiu functions

Abstract

a b s t r a c tWe introduce an algebraic operator framework to study discounted penalty functions in renewal risk models. For inter-arrival and claim size distributions with rational Laplace transform, the usual integral equation is transformed into a boundary value problem, which is solved by symbolic techniques. The factorization of the differential operator can be lifted to the level of boundary value problems, amounting to iteratively solving first-order problems. This leads to an explicit expression for the Gerber-Shiu function in terms of the penalty function.

An operator-based approach to the analysis of ruin-related quantities in jump diffusion risk models

“…However, one would reasonably expect tractable analytical solutions in the family of Lévy processes, which is indeed the case in our analysis of H. Various operator-based approaches have appeared in Sparre-Andersen models such as in Gerber and Shiu (2005), Albrecher et al (2010) and Li and Garrido (2005). To the best knowledge of the author, this is the first time operators have been employed to solve the Poisson equation in a jump diffusion model.…”

“…The operator T is known as the Dickson-Hipp operator in ruin literature and operators T and E also known as Green operators are used extensively in the analysis of the Gerber-Shiu function in the context of a renewal risk model in Albrecher et al (2010). The interactions among the three operators are summarized in the list of operator identities included in Appendix B.…”

“…In a forthcoming paper by Feng and Shimizu (2010), the operator-based approach is further extended to analyze a more general Lévy risk model. It should be pointed out that a subset of the operator identities shown in Appendix B has been used in Albrecher et al (2010) as a basis for automated computer algebra systems, such as Mathematica, for solving boundary value problems in the context of a renewal risk model. The approach in the present paper continues in this direction and shows the potential of further broadening the spectrum of problems solvable by computer algebra systems.…”

Investigation of operation of intermediate storages applying probability density functions satisfying linear differential equation

“…We investigate the case when the density function of the inter-arrival times satisfies a linear differential equation with constant coefficients subject to some general initial conditions. This set of density functions is dense in the set of continuous density functions [8] and is investigated in risk models as well [9]. Furthermore, it contains such famous distributions as Coxian distributions, K n -type distributions, Erlang(n) distributions, combination of exponential distributions.…”

An elementary approach to discrete models of dividend strategies

“…InAlbrecher et al (2010), R(ρ; D) is denoted as B ρ . The operator T r inLi (2005) andLi et al (2009) is…”