*Article* **A Note on Gerber–Shiu Function with Delayed Claim Reporting under Constant Force of Interest**

**Kokou Essiomle and Franck Adekambi \***

School of Economics, University of Johannesburg, Johannesburg 2006, South Africa; 201900490@student.uj.ac.za **\*** Correspondence: fadekambi@uj.ac.za

**Abstract:** In this paper, we analyze the Gerber–Shiu discounted penalty function for a constant interest rate in delayed claim reporting times. Using the Poisson claim arrival scenario, we derive the differential equation of the Laplace transform of the generalized Gerber–Shiu function and show that the differential equation can be transformed to a Volterra equation of the second kind with a degenerated kernel. In the case of an exponential claim distribution, a closed-expression for the Gerber–Shiu function is obtained via sequence expansion. This result allows us to calculate the absolute (relative) ruin probability. Additionally, we discuss a method of solving the Volterra equation numerically and provide an illustration of the ruin's probability to support the finding.

**Keywords:** Gerber–Shiu function; constant force of interest; Volterra equation; absolute ruin; delayed reporting times
