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“On the Laplace Transform of the Aggregate Discounted Claims with Markovian Arrivals,” Jiandong Ren, April 2008

SHUANMING LI* In this paper Dr. Ren has obtained some very elegant matrix expressions for the Laplace transform and the first two moments of the aggregate discounted claims in a risk process in a Marokovian environment (MAP). The purpose of this discussion is to show how to calculate the nth moment of the aggregate discounted claims recursively when the force of interest is constant 0, that is, v(t) e t. We slightly change the notation and use D0(i, j) and D1(i, j) to denote the (i, j)–th element of D0 and D1, respectively. We extend the assumption in the paper under discussion by assuming that claim amounts that occur at a transition of the underlying Markov process from state i to state j depend on both i and j. We denote by pi,j and (s) e pi,j(x) dx the density function and the Laplace  p ˆi,j 0 transform, respectively, of claim amounts that occur at a transition. First, we shall show that L( , t) satisfies the following first-order linear partial differential equation: