Strict SDE Comparison for Cusp Coefficients and Counterexamples

Source abstract

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