The Distribution of Discounted Compound PH-Renewal Processes

We continue and extend the work in Leveille et al. (2010) Scand Actuar J 3:165-184 that gives analytical formulas for the moment generating function (mgf) of some discounted compound renewal processes. Here these mgf's are used to derive and study the distribution of discounted compound Phase-type (PH) renewal sums. The approach consists in first deriving a differential equation, in the time variable, for the mgf of discounted compound sums when the inter-arrival times are PH-distributed. Then the corresponding distribution of the discounted compound PH sum is obtained by inversion of its mgf (or the equivalent Laplace transform). Analytical expressions for the asymptotic distribution of these discounted compound renewal sums are also given, to test general expressions in some limiting cases. Despite the technical difficulty, several new examples are provided where the inversion is possible, through symbolic computation in MAPLE. When the inversion is too complex, a truncated series solution method is proposed, which is not new but turns out to be very fast and accurate here, much more than other methods (such as rational approximations). The article concludes with some applications where the mgf and the distribution are used to calculate risk measures such as VaR, CTE, Esscher's, Wang's Proportional Hazard transforms and evaluate their evolution in time.