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

作者：Albrecher Hansjoerg^*; Constantinescu Corina; Pirsic Gottlieb; Regen**urger Georg; Rosenkranz Markus

来源：Insurance: Mathematics and Economics , 2010, 46(1): 42-51.

DOI：10.1016/j.insmatheco.2009.02.002

We 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.