Autonomous all-moment Euler MLMC logarithmic cost

Assume the Axiom of Choice. Use the autonomous SDE and Euler scheme contract of Euler–Maruyama strong rate under Higham–Mao–Stuart hypotheses: finite-dimensional state and Brownian noise, $C^1$ drift and diffusion, a one-sided Lipschitz drift, globally Lipschitz diffusion, polynomial local-Lipschitz drift, and the stipulated uniform moments of every order for the exact solution and Euler step and interpolation processes. In particular, retain that theorem's all-moment premise; an $L^2$ initial condition alone is not a substitute.

Fix $0<T\le 1$, an integer $M\ge 2$, and for every $\mathrm{\ell }\ge 0$ set $h_{\mathrm{\ell }}=T{M}^{-\mathrm{\ell }}$ and $P_{\mathrm{\ell }}=\phi (Y_T^{h_{\mathrm{\ell }}})$, for a globally Lipschitz real payoff $\phi$ with constant $L_{\phi }$. These levels of the same SDE give the infinite compatible family required by MLMC cost complexity under bias, variance, and work rates, with target $\mathbb{E}\phi (X_T)$. At each level pair, use the same initial value and Brownian path for the fine and coarse Euler paths, summing the $M$ fine Brownian increments to form each coarse increment. Use independent sample blocks across levels as in Geometric MLMC hierarchy and coupled corrections. For $\mathrm{\ell }\ge 1$, the coupled correction variance is $O(h_{\mathrm{\ell }})$; the finest-level payoff bias is $O(h_L^{1\mathrm{/}2})$; and one coupled path-pair sample costs $O(h_{\mathrm{\ell }}^{-1})$. At level $0$, $P_0\in L^2$ by the all-moment premise and the Lipschitz payoff, and one Euler path has finite work. Since $h_0=T$ is fixed, enlarge the rate constants to include $\mathrm{Var}(P_0)=O(h_0)$ and the level-zero work $O(h_0^{-1})$. Hence $(\alpha ,\beta )=(1\mathrm{/}2,1)$ and the MLMC estimator has MSE at most ${ϵ}^2$ with work $O({ϵ}^{-2}(\log {ϵ}^{-1}{)}^2)$.