FLUCTUATION ANALYSIS FOR A CLASS OF NONLINEAR SYSTEMS WITH FAST PERIODIC SAMPLING AND SMALL STATE-DEPENDENT WHITE NOISE

The paper’s Theorem 2.4 identifies the same limiting linear SDE for Z (with drift correction −(c/2)∫0^t J(x_s)[f(x_s)+g(x_s)κ(x_s)]ds) and proves the strong error bound E[sup_{t≤T}|Z^{ε,δ}_t−Z_t|] ≤ (c+1)[ε+√δ+κ(ε)] C e^{CT} for regimes with δ/ε→c, exactly as stated in equations (16)–(17) of the paper . The model’s solution reproduces these conclusions via a slightly different route: it linearizes, isolates the sampling bias with a clean Riemann-sum lemma, and closes with BDG+Grönwall. The paper obtains the bias by a careful multi-term decomposition (equations (10)–(13) and Proposition 4.1) and then bounds all remainders (Propositions 4.2–4.5) . The end results coincide in both the limit SDE and the pathwise error rate; thus both are correct, but the proofs differ in structure.