Abstract:

In this article, the expected discounted penalty function $\Phi_{\delta,\alpha}(u)$ with constant interest δ and ``discounted factor" exp(-αT_δ) is considered. As a result, the integral equation of $\Phi_{\delta,\alpha}(u)$ is derived and an exact solution for $\Phi_{\delta,\alpha}(0)$ is found. The relation between the joint density of the surplus immediately prior to ruin, and the deficit at ruin and the density of the surplus immediately prior to ruin is then obtained based on analytical methods.

Key words: Ruin, penalty function, integral equation, surplus prior to ruin, deficit at ruin