Complete monotonicity for the ruin-time density in the Cramér–Lundberg model

In this article, we study complete monotonicity of the defective ruin-time density in the classical Cramér–Lundberg model for exponential, Erlang, and two-phase hyperexponential claim-size distributions. Specifically, we ask how the claim-size structure and initial capital determine whether this density is a mixture of exponentials. This is actuarially relevant because solvency management concerns not only whether ruin occurs but also when; a mixture representation makes finite-horizon ruin timing transparent and tractable for capital monitoring and timely intervention. We characterize complete monotonicity through a Stieltjes criterion for the discounted ruin transform. For exponential claims, it holds within a specific initial-capital regime. For integer-shape Erlang claims of order at least two, it fails for every initial capital, making the exponential law the only positive Erlang case. For two-phase hyperexponential claims, it is governed by an initial-capital threshold under a structural ordering that is also necessary in the generic case.