Gerber–Shiu Analysis in the Classical Poisson Risk Model
The central model in insurance surplus analysis is unquestionably the classical Poisson risk model. Due to its importance, this model is considered in this chapter. After presenting the underlying probabilistic structure, the fundamentally important Gerber-Shiu function is derived. The derivation is primarily algebraic, and this is done for two reasons. First, a claim size density is not assumed, thus allowing for the use of discrete models which arise naturally in applications to group life and long term disability. Second, guidance as to the direction of the derivation in more complex models is somewhat detailed in general, but can be avoided by those simply wishing to use the classical Poisson model. Finally, the Gerber-Shiu function is used to obtain explicit quantities of interest in connection with the time of ruin, the deficit at ruin, and the surplus immediately prior to ruin.