A direct approach to the discounted penalty function

This paper provides a new and accessible approach to establishing certain results concerning the discounted penalty function. The direct approach consists of two steps. In the first step, closedform expressions are obtained in the special case in which the claim amount distribution is a combination of exponential distributions. A rational function is useful in this context. For the
second step, one observes that the family of combinations of exponential distributions is dense. Hence, it suffices to reformulate the results of the first step to obtain general results. The surplus process has downward and upward jumps, modeled by two independent compound Poisson processes. If the distribution of the upward jumps is exponential, a series of new results can be
obtained with ease. Subsequently, certain results of Gerber and Shiu [H. U. Gerber and E. S. W. Shiu, North American Actuarial Journal 2(1): 48–78 (1998)] can be reproduced. The two-step approach is also applied when an independent Wiener process is added to the surplus process. Certain results are related to Zhang et al. [Z. Zhang, H. Yang, and S. Li, Journal of Computational and Applied Mathematics 233: 1773–1784 (2010)], which uses different methods.