MLMC work bound for bounded finite-state Poisson tau-leaping

Assume Countable Choice and Dependent Choice, and the finite-state hypotheses of Weak order one for finite-state Poisson tau-leaping with an exit bound and Order-one correction variance for coupled Poisson tau-leap levels, with bounded payoff $g$. Use levels $h_{\mathrm{\ell }}=T{2}^{-\mathrm{\ell }}$ and couple adjacent Poisson tau-leap paths by the shared-minimum construction. Under the work convention that each channel Poisson draw, propensity evaluation, and stoichiometric update has unit cost, for each $\mathrm{\ell }\ge 1$ one correction sample costs at most $C{h}_{\mathrm{\ell }}^{-1}$: there are $T\mathrm{/}{h}_{\mathrm{\ell }}$ fine half intervals, each using at most $3R$ shared and residual Poisson draws and $O(R)$ propensity and update work. The base-level sample uses one path and costs $O(R)$, which is $O(h_0^{-1})$ for fixed $T$. The level bias is $O(h_L)$ and the correction variance is $O(h_{\mathrm{\ell }})$. Therefore, for all sufficiently small $ϵ>0$, deterministic sample counts in the generic multilevel estimator can achieve mean square error at most ${ϵ}^2$ with work $O\text{\hspace{0.17em}⁣}({ϵ}^{-2}(\log {ϵ}^{-1}{)}^2)\mathrm{.}$ This conclusion applies only to the explicit Poisson leap and the bounded finite-state hypotheses above. It is not a correctness or error guarantee for the guarded binomial algorithm Critical-reaction binomial guarded tau-leaping.