Boykov (2002) studied a risk process where in both premia and claims were represented by different compound Poisson processes. Morales and Schoutens (2003) presented a possibility where the difference in premia and claims can be modeled by a Levy process; specifically NIG, VG, Meixner or General Hyperbolic. Recently (2013), Stanojevic and Levajkovic proposed a Cramer-Lundberg model where in the Variance Gamma (VG) process was used to model premia compound Poisson process for claims. They also used Panjers recursion with N from class (a,b,1) to obtain a recursive formula for claim size distribution.
My question is the following: Is it worthwhile to use two different Levy Processes to capture claims and premia; seeing each of these can be captured by a Levy process. I note that using for example same kind of Levy processes for premia and claim (e.g. VG, NIG) will only lead to the difference of these two processes and in some sense boil down to Boykovs result since each of these can be seen as a generalization of compound Poisson processes. As well, the dynamics may be just for medium term claims as these distributions have semi heavy tails and will be poor for extremes.
Say for example, Dufresne et al used Gamma process for claims. How does it play out using like NIG for premia and Gamma process for claims? I should say using VG for premia and Gamma process for claims may end up giving just a CGMY process of another VG process with different parameters.