When do thin intermediate layers improve a two-layer dividend strategy under a ruin probability constraint?

We study a risk model whose premiums and claims are independent compound Poisson processes with phase-type jump sizes, under a multilayer strategy that pays a constant dividend rate in each layer. Starting from a two-layer strategy, we insert thin layers of total width w with intermediate rates at the threshold and adjust their position so that the ultimate ruin probability is unchanged. We prove that the expected discounted dividends change by 12χΓ⋆w2+O(w3). The coefficient of w2 factorizes into χ>0, which depends only on the dividend rates and the relative widths, and Γ⋆, which is determined solely by the two-layer strategy at the initial reserve and is computed from a phase-type boundary value problem. When Γ⋆≠0, its sign decides whether sufficiently thin layers increase dividends at a fixed ruin probability, for every choice of intermediate rates and relative widths. At an unconstrained optimal threshold where the dividend value has a negative second derivative, such layers always reduce dividends. At a locally optimal threshold under a ruin constraint, they can increase dividends, as a numerical example shows. We also express the threshold derivatives through expected discounted numbers of continuous downcrossings of the threshold.