Strict SDE Comparison for Cusp Coefficients and Counterexamples

Abstract

We provide a tractable sufficient condition for strict comparison for solutions of the one-dimensional stochastic differential equation \[ d X_t^x=\sigma(X_t^x)\,d B_t, \quad X_0^x=x\in I=(\ell,r), \] 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 $W^{1,p}_{\text{loc}}(I)$, $1\le p<2$. We illustrate using $\sigma(x):=1+|x|^\beta$ for $x\in\mathbb{R}$, $\beta \in (0,1)$, and show that strict comparison holds if and only if $\beta\in[\frac12,1)$. We give examples showing that neither $\sigma\in W^{1,p}_{\text{loc}}(\mathbb R)$, $p\in(1,2)$, nor $\sigma\in C^\beta(\mathbb R)$, $\beta\in[\frac12,1)$, is sufficient for strict comparison, even when combined with boundedness, uniform ellipticity, global strong existence, and pathwise uniqueness.