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 $(Z+ϵ{)}^{-1}$ makes the hitting probability at most $ϵe^{(L+K^2)t}/(Z_0+ϵ)$, which tends to zero. Common noise and the Lipschitz bound on the diffusion difference are essential to this argument.

Before the difference $Z$ hits zero, its diffusion coefficient has magnitude at most $KZ$ and its drift is at least $-LZ$. Itô's formula bounds the drift of $f_{ϵ}(Z)$ by $(L+K^2)f_{ϵ}(Z)$. Its stochastic integrand is bounded by $K/(4ϵ)$, making expectation legitimate on finite horizons. The Gronwall inequality and the value $f_{ϵ}(0)=1/ϵ$ then exclude finite-time contact from a strictly positive starting difference.