Optimal-rate conjecture for general quasilinear Skorohod SDEs

Let $X$ solve the quasilinear Skorohod SDE

$$dX_t=a(t,X_t)\,dt+\sigma(t)X_t\,dW_t,$$

where $X_0\in L^p(\Omega)$ for some $p>2$, $a$ satisfies the global Lipschitz and boundedness condition

$$|a(t,x,\omega)-a(t,y,\omega)|\leq L|x-y|,\qquad |a(t,0,\omega)|\leq L,$$

for all $t\in[0,1]$, $x,y\in\mathbb{R}$ and $\omega\in\Omega$, and $\sigma\in L^2$. Assume in addition that $X_0=F(I(f))\in\mathcal{WA}$, that $f'$ and $\sigma'$ belong to $BV$, that $a\in C^1([0,1]\times\mathbb{R})$ is nonrandom, and that $a_x$ is bounded. Optimal-rate conjecture. The optimal approximation satisfies

$$\lim_{n\rightarrow\infty}n\,\mathbb{E}\left[(X_1-\widehat{X_1}^n)^2\right]^{1/2}\in(0,\infty).$$

The preceding theorem gives existence and uniqueness of the quasilinear Skorohod SDE under the Lipschitz condition, while the conjecture predicts a finite, strictly positive first-order asymptotic error constant for its optimal approximation. The claim is presented as a belief inspired by the paper's results and is not established there.