Finite-L2 strong and conditional weak Euler MLMC cost

Assume the Axiom of Choice and the exact finite-$L^2$ SDE, initial-value, coefficient, and nested-grid hypotheses of Shared-Brownian Euler level correction rates, with $0<T\le 1$, fixed integer $M\ge 2$, and a globally Lipschitz payoff $\phi$. 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 this same SDE, with target $\mathbb{E}\phi (X_T)$. These levels give the infinite compatible family required by MLMC cost complexity under bias, variance, and work rates. The finite-$L^2$ contract here is the globally state-Lipschitz and linear-growth coefficient condition, the half-Hölder time modulus, an adapted initial value independent of future Brownian increments, and $\mathbb{E}\mathrm{\mid }{X}_0{\mathrm{\mid }}^2<\mathrm{\infty }$. With shared Brownian increments within each fine/coarse pair and independent sample blocks across levels, the strong-route rates are $(\alpha ,\beta )=(1\mathrm{/}2,1)$ and the MLMC work is $O({ϵ}^{-2}(\log {ϵ}^{-1}{)}^2)$.

If, in addition, $\phi$ is bounded and its backward value function $u(t,x)=\mathbb{E}[\phi (X_T^{t,x})]$ solves the backward equation with the mixed derivative and polynomial-growth bounds and the $p\ge 2q+8$ moment condition in Euler–Maruyama weak order one under backward regularity, then the weak bias rate is $\alpha =1$ while $\beta =1$ and the same logarithmic work bound follows. In the strong branch $P_0\in L^2$ by the finite-$L^2$ scheme moment bound and the Lipschitz payoff; in the weak branch it is bounded. The level-zero path has finite work, and because $h_0=T$ is fixed, enlarging the rate constants makes its variance and work satisfy the same bounds. This second branch is conditional on backward regularity; Lipschitz continuity of $\phi$ alone does not imply it.