Moving-Boundary Fluctuation Analysis: Premium Drift, Ladder Structure, and Ruin in Phase-Type Cumulative Shock Models

We extend the phase-tagged fluctuation framework for cumulative shock models from a fixed failure threshold to a linearly moving boundary u0+cτn, the premium drift regime that underlies insurance ruin theory. The moving boundary turns the first-passage problem from a fixed-level crossing into a crossing of a drifting walk, for which the partial sum truncation of the fixed-threshold theory no longer applies. We resolve this with a drift-aware ladder apparatus: a rank-one drift ladder matrixG↑(c)=g(c)eα whose scalar g(c) solves a discrete Lundberg equation a(z)=zc, a closed-form ascending ladder-height law obtained from the roots of the Lundberg polynomial, and a compound-geometric (Pollaczek–Khinchine) representation of the ruin probability. Building on these, we derive in closed form the full five-variable moving-boundary reliability functional Φνruin(ξ,u,v,ϑ,θ), the drift analogue of the fixed-threshold phase-tagged functional, and show it lies in the same two-dimensional matrix subspace span{H(θ),H(ω)}: the premium drift deforms the scalar coefficients through a first-passage transform while leaving the matrix structure invariant. As the principal application, we obtain the phase-resolved Gerber–Shiu expected-discounted-penalty function, with the joint transform of the time of ruin, the deficit at ruin, and the surplus prior to ruin. The classical scalar Gerber–Shiu function and the fixed-threshold functional are recovered as projections and as the c→0 limit, respectively. All closed forms reduce to root-finding on a single polynomial for rational model primitives, and every structural result is verified against Monte Carlo simulation and exact recursion.