= Strict order preservation for scalar Lipschitz diffusions
Two scalar SDEs driven by the same <Brownian motion>, with the same globally Lipschitz diffusion coefficient and ordered globally Lipschitz drifts, preserve a strict initial order at every finite time. If $Z=Y-X$ starts positive, apply a reciprocal barrier to the first hit of zero. A Gronwall bound on $(Z+\epsilon)^{-1}$ makes the hitting probability at most $\epsilon e^{(L+K^2)t}/(Z_0+\epsilon)$, which tends to zero. Common noise and the Lipschitz bound on the diffusion difference are essential to this argument.
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