Upper bounds for ultimate ruin probabilities in the Sparre Andersen risk model with interest and a nonlinear dividend barrier

We consider the classical risk model with constant force of interest and a nonlinear dividend barrier. Lundberg-type inequalities for the ultimate ruin probabilities are derived. The results obtained carry over those of Gerber [Gerber, H.U., 1979. An Introduction to Mathematical Risk Theory. In: Monograph Series, vol. 8. Huebner Foundation, Philadelphia], about a linear dividend barrier without interest, to the case with both interest and a nonlinear dividend barrier. More precise upper bounds for the ultimate ruin probabilities are also given for the special case of exponential claim sizes.