On the generalized Gerber–Shiu function for surplus processes with interest

In this paper, we study the generalized expected discounted penalty (Gerber–Shiu) function in a risk process with credit and debit interests. We define Tu,z to be the first time that the surplus process drops below a certain level z from the initial surplus u(>z). The time of ruin and the time of absolute ruin are special cases of this stopping time. The generalized Gerber–Shiu function is defined on three random variables: the first time that the surplus drops below z from u, Tu,z, the surplus prior to Tu,z, and the amount by which the surplus is below z.