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.An explicit expression for the Gerber-Shiu function when u=z is obtained when the credit and debit interest rates are equal, and explicit results for the Gerber-Shiu function under exponential claims are then obtained. Using these results, we investigate the probability that the surplus reaches an upper level without dropping below a lower level and the distribution of the maximum severity of ruin.

► The generalized Gerber-Shiu function is defined on three random variables. ► The probability of reaching an upper level before dropping below a lower level is studied. ► The distribution of the maximum severity of ruin is derived.