Milstein MLMC correction variance and inverse-epsilon-squared cost

For all sufficiently fine levels, [L2] gives $\mathrm{\parallel }{Y}_{\mathrm{\ell }}(T)-X_T{\mathrm{\parallel }}_2\le C{h}_{\mathrm{\ell }}$, where $\mathrm{\parallel }Z{\mathrm{\parallel }}_2=(\mathbb{E}\mathrm{\mid }Z{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$. There are only finitely many coarser dyadic levels; each has finitely many steps and finite second moments because the coefficients and derivatives are bounded. Enlarging $C$ therefore makes the same error bound valid for every level. The same argument gives $\mathbb{E}\mathrm{\mid }\psi (Y_{\mathrm{\ell }}(T)){\mathrm{\mid }}^2<\mathrm{\infty }$, since $\mathrm{\mid }\psi (x)\mathrm{\mid }\le \mathrm{\mid }\psi (0)\mathrm{\mid }+\mathrm{Lip}(\psi )\mathrm{\mid }x\mathrm{\mid }$.

A level-$\mathrm{\ell }$ fine path has $T\mathrm{/}{h}_{\mathrm{\ell }}=2^{\mathrm{\ell }}$ steps and its coarse companion has $2^{\mathrm{\ell }-1}$ steps. Each commutative Milstein step uses a fixed finite number of coefficient/derivative evaluations and Gaussian vector samples for fixed $d,m$; [L1] requires no Lévy-area sampler. Thus one correction sample has work at most $C(2^{\mathrm{\ell }}+2^{\mathrm{\ell }-1})\le C_c{h}_{\mathrm{\ell }}^{-1}$, including the finitely many coarse initial levels after increasing $C_c$.

The fine and coarse approximations in ${\mathrm{\Delta }}_{\mathrm{\ell }}$ use the same Brownian path and hence the same exact terminal value $X_T$. By [F3], $\mathrm{\parallel }U+V{\mathrm{\parallel }}_2\le \mathrm{\parallel }U{\mathrm{\parallel }}_2+\mathrm{\parallel }V{\mathrm{\parallel }}_2$; with $h_{\mathrm{\ell }-1}=2h_{\mathrm{\ell }}$ this gives $\mathrm{\parallel }{Y}_{\mathrm{\ell }}^f(T)-Y_{\mathrm{\ell }-1}^c(T){\mathrm{\parallel }}_2\le \mathrm{\parallel }{Y}_{\mathrm{\ell }}^f(T)-X_T{\mathrm{\parallel }}_2+\mathrm{\parallel }{Y}_{\mathrm{\ell }-1}^c(T)-X_T{\mathrm{\parallel }}_2\le C(h_{\mathrm{\ell }}+h_{\mathrm{\ell }-1})\le C^{\mathrm{\prime }}{h}_{\mathrm{\ell }}\mathrm{.}$ For each $\mathrm{\ell }\ge 1$, the Lipschitz property yields $\mathrm{Var}({\mathrm{\Delta }}_{\mathrm{\ell }})\le \mathbb{E}\mathrm{\mid }{\mathrm{\Delta }}_{\mathrm{\ell }}{\mathrm{\mid }}^2\le \mathrm{Lip}(\psi {)}^2\mathbb{E}\mathrm{\mid }{Y}_{\mathrm{\ell }}^f(T)-Y_{\mathrm{\ell }-1}^c(T){\mathrm{\mid }}^2\le C_v{h}_{\mathrm{\ell }}^2$. At level zero, ${\mathrm{\Delta }}_0=P_0$ by [F1] and has finite variance by step 1.1; increasing $C_v$ to also dominate $\mathrm{Var}(P_0)\mathrm{/}{h}_0^2$ makes the same envelope hold there.

Again by the Lipschitz property and [F3], $\mathrm{\mid }\mathbb{E}{P}_{\mathrm{\ell }}-\mathbb{E}\psi (X_T)\mathrm{\mid }\le \mathbb{E}\mathrm{\mid }\psi (Y_{\mathrm{\ell }}(T))-\psi (X_T)\mathrm{\mid }\le \mathrm{Lip}(\psi )\mathrm{\parallel }{Y}_{\mathrm{\ell }}(T)-X_T{\mathrm{\parallel }}_2\le C_b{h}_{\mathrm{\ell }}\mathrm{.}$ The finite-level second-moment statement in step 1.1 verifies the square-integrability premise of [F1].

The estimates in steps 2.1, 2.2, and 1.2 verify [F2] with $\alpha =1$ and $\beta =2$. Since $2>1$ and $1\ge 1\mathrm{/}2$, applying [F2] proves the stated $O({ϵ}^{-2})$ MLMC work bound. The local variance argument uses only the strong order-one result [L2] for nested commutative-Milstein paths; AC is inherited as stated in [A1].