Speedy convolution algorithms and Panjer recursions for phase-type distributions

For phase-type claim size distributions, the Panjer algorithm can be simplified in the sense that the non local term in the recursion can be replaced by a local one, which yields a considerable reduction of computation time. This is done for the discrete case with arithmetic claim size, for the continuous case with a density for the claim size, and finally for the mixed case in which the claim size distribution is a mixture of two distributions, the first arithmetic, the second continuous. In the last two cases, numerical integration can be replaced by the solution of an interacting system of differential equations.