A discontinuous payoff with bias order one and variance order one-half

Fix an integer $M\ge 2$, put $h_{\mathrm{\ell }}=M^{-\mathrm{\ell }}$, and choose $c\in (0,1]$. Let $X$ have density $f_c(x)=\frac{1}{2}(1+cx)$ on $[-1,1]$, let $S$ be an independent Rademacher variable, and use the discontinuous payoff $\phi (x)={\mathbf{1}}_{\{x\ge 0\}}$. The target law is that of $P=\phi (X)$, and for each $\mathrm{\ell }\ge 0$ the level law is that of $P_{\mathrm{\ell }}=\phi (X+\sqrt{h_{\mathrm{\ell }}}S)$. Couple adjacent level laws using the same pair $(X,S)$ within each coupling.

For each correction sample, draw a fresh independent pair $(X,S)$ with the stated law; within each positive-level correction use that same pair for the fine and coarse payoffs, and take all samples in distinct level blocks independently as required by the definition. This gives an infinite compatible family whose finite prefixes are MLMC hierarchies. The example has bias order $\alpha =1$ and single-correction variance order $\beta =\frac{1}{2}$ for positive levels. The level-zero correction is bounded and $h_0=1$, so enlarge the variance constant to include it. One sample costs $O(1)$, so it also satisfies the abstract theorem's upper work assumption $O(h_{\mathrm{\ell }}^{-1})$. Substituting that conservative work rate into the theorem gives the guarantee $O({ϵ}^{-5\mathrm{/}2})$; the example does not claim this is a sharp cost for its constant-work evaluator.