Geometric MLMC hierarchy and coupled corrections

Fix a refinement factor $M\in \mathbb{N}$ with $M\ge 2$, a base scale $T>0$, and a finite level $L\in \mathbb{N}\cup \{0\}$. The geometric mesh widths are $h_{\mathrm{\ell }}=T{M}^{-\mathrm{\ell }}$ for $0\le \mathrm{\ell }\le L$. Let $P$ be a real random variable with $\mathbb{E}\mathrm{\mid }P\mathrm{\mid }<\mathrm{\infty }$, and for each $\mathrm{\ell }$ let $P_{\mathrm{\ell }}$ be a real random variable with $\mathbb{E}[P_{\mathrm{\ell }}^2]<\mathrm{\infty }$.

At level $0$, set ${\mathrm{\Delta }}_0=P_0$. At each level $\mathrm{\ell }\ge 1$, choose a joint law for a pair $(P_{\mathrm{\ell }}^{\mathrm{f}},P_{\mathrm{\ell }-1}^{\mathrm{c}})$ whose marginals are the laws of $P_{\mathrm{\ell }}$ and $P_{\mathrm{\ell }-1}$, respectively. Such a joint law is a coupling of the two levels. Define the correction ${\mathrm{\Delta }}_{\mathrm{\ell }}=P_{\mathrm{\ell }}^{\mathrm{f}}-P_{\mathrm{\ell }-1}^{\mathrm{c}}$. This construction gives $\mathbb{E}{\mathrm{\Delta }}_{\mathrm{\ell }}=\mathbb{E}{P}_{\mathrm{\ell }}-\mathbb{E}{P}_{\mathrm{\ell }-1}$.

For each level choose a positive integer $N_{\mathrm{\ell }}$ and take ${\mathrm{\Delta }}_{\mathrm{\ell },1},\dots ,{\mathrm{\Delta }}_{\mathrm{\ell },N_{\mathrm{\ell }}}$ to be independent copies of ${\mathrm{\Delta }}_{\mathrm{\ell }}$. Require the complete sample blocks from distinct levels to be independent. Write ${\bar{\mathrm{\Delta }}}_{\mathrm{\ell }}=N_{\mathrm{\ell }}^{-1}{\sum }_{i=1}^{N_{\mathrm{\ell }}}{\mathrm{\Delta }}_{\mathrm{\ell },i}$. The finite-level multilevel Monte Carlo estimator is ${\widehat{\mu }}_{L,\mathbf{N}}={\sum }_{\mathrm{\ell }=0}^L{\bar{\mathrm{\Delta }}}_{\mathrm{\ell }}\mathrm{.}$ All corrections have finite second moments because each level approximation does. The shared randomness inside one fine/coarse pair is what can reduce the correction variance; the estimator still uses independent sample blocks across levels.

The definition makes no claim that a coupling is optimal, that the bias or correction variance has a particular rate, or that a data-dependent stopping rule reliably estimates the remaining bias.