Equilibrium Moment Analysis of Itô SDEs

The paper derives the Euler–Maruyama (EM) fixed-point identity for the second moment at equilibrium (its Eq. (5)) and, for the cubic example F(x) = −x^3 − 3σ^2 x and G(x)^2 = 15σ^6 + 36σ^4 x^2 + 9σ^2 x^4, obtains the leading-order relation (its Eq. (7)) and the exact Δt-level relation (its Eq. (8)). It then concludes that for σ > √2/3 the right-hand side is strictly positive, contradicting the equilibrium identity and forcing the second moment to diverge. The candidate solution reproduces exactly this chain: the same EM moment identity, the same coefficients for μ2, μ4, μ6, and the same contradiction argument for σ > √2/3. The small caveat noted in the paper—that deriving Eq. (5) implicitly assumes μ2 is finite and swaps integrals—is also acknowledged. Overall, the reasoning and formulas match the paper’s results and logic closely. See the paper’s presentation of the EM kernel and equilibrium map, the second-moment identity (Eq. (5)), and its specialization to this F, G producing Eqs. (7) and (8) and the σ-threshold conclusion.