Strict SDE Comparison for Cusp Coefficients and Counterexamples

We provide a tractable sufficient condition for strict comparison for solutions of the one-dimensional stochastic differential equation $$d{X}_t^x=\sigma (X_t^x)d{B}_t,X_0^x=x\in I=(\mathrm{\ell },r),$$ for $\sigma >0$ and continuous. Our proof is based on a two-dimensional Lyapunov argument, which allows us to prove strict comparison for some coefficients in $W_{\text{loc}}^{1,p}(I)$, $1\le p<2$. We illustrate using $\sigma (x):=1+\mathrm{\mid }x{\mathrm{\mid }}^{\beta }$ for $x\in \mathbb{R}$, $\beta \in (0,1)$, and show that strict comparison holds if and only if $\beta \in [\frac{1}{2},1)$. We give examples showing that neither $\sigma \in W_{\text{loc}}^{1,p}(\mathbb{R})$, $p\in (1,2)$, nor $\sigma \in C^{\beta }(\mathbb{R})$, $\beta \in [\frac{1}{2},1)$, is sufficient for strict comparison, even when combined with boundedness, uniform ellipticity, global strong existence, and pathwise uniqueness.