Some Remarks on Delayed Renewal Risk Models

Some extensions to the delayed renewal risk models are considered. In particular, the independence assumption between the interclaim time and the subsequent claim size is relaxed, and the classical Gerber-Shiu penalty function is generalized by incorporating more variables. As a result, general structures regarding various joint densities of ruin related quantities as well as their probabalistic interpretations are provided. The numerical example in case of time-dependent claim sizes is provided, and also the usual delayed model with time-independent claim sizes is discussed including a special case with exponential claim sizes. Furthermore, asymptotic formulas for the associated compound geometric tail for the present model are derived using two alternative methods.

Keywords: Delayed renewal risk model; Gerber-Shiu function; Last ladder height; Last interclaim time; Cramer's asymptotic ruin formula.