Convergence-rate conjecture for the randomized truncated Milstein method

Let $y(t)$ be the solution of the stochastic differential equation under Assumptions A1, A2, and A3, and let $X_\Delta(t)$ denote the continuous-time approximation produced by the randomized truncated Milstein method with step size $\Delta$. For any $p>2$, assume that Assumptions A1, A2, and A3 hold. Let $\bar{q}>0$, $\varepsilon\in(0,1/4)$, and $\Delta\in(0,1]$. Then

Convergence-rate conjecture. There is a constant $H$ independent of $\Delta$ such that

$$\sup_{t_0\leq t\leq T}\mathbb{E}\lvert y(t)-X_\Delta(t)\rvert^{\bar{q}}\leq H\left(\Delta^{\min(1-2\varepsilon,\alpha+\frac{1}{2})\bar{q}}\right).$$

The conjecture predicts that randomization improves the convergence rate from $\min(1-2\varepsilon,\alpha)$ to $\min(1-2\varepsilon,\alpha+1/2)$; the source states that a proof was still in progress and supports the claim with numerical simulations.