 Since the function measures the actuarial present value of the penalty that occurs at formula_31, the penalty function is multiplied by the discounting factor formula_32, and then averaged over the probability distribution of the waiting time to formula_31.While Gerber and Shiu applied this function to the classical compound-Poisson model, Powers argued that an insurer’s surplus is better modeled by a family of diffusion processes.There are a great variety of ruin-related quantities that fall into the category of the expected discounted penalty function.