$\displaystyle\leq\frac{6K_{V}(p-1)}{p}\sum_{\begin{subarray}{c}\gamma\in\Pi_{n_{1}+2}^{N},~{}\hat{\mathcal{O}}(\gamma)=m\\\
\hat{\mathcal{O}}(\gamma\setminus(\gamma_{1}))=m-1\end{subarray}}\mathbb{E}\Big{[}\Big{|}X^{t,x_{\gamma_{1}},\gamma_{1},N}_{s,x_{\gamma_{2}},\dots,x_{\gamma_{n_{1}+2}}}\Big{|}^{p}\Big{]}+\frac{6K_{V}}{pN^{p}}\sum_{\begin{subarray}{c}\gamma\in\Pi_{n_{1}+2}^{N},~{}\hat{\mathcal{O}}(\gamma)=m-1\\\
\hat{\mathcal{O}}(\gamma\setminus(\gamma_{1}))=m-1\end{subarray}}\mathbb{E}\Big{[}\Big{|}X^{t,x_{\gamma_{1}},\gamma_{1},N}_{s,x_{\gamma_{2}},\dots,x_{\gamma_{n_{1}+2}}}\Big{|}^{p}\Big{]},$
(4.71)

where we used $\nabla^{2}V(x)\geq 0$ to bound (4.69) above by 0, then apply
Jensen’s inequality to (4.70) to match the acceptable weights on $N$ in
(4.72). The constant 6 arises from the fact that there are at most 6 different
values for $l\in\gamma\setminus(\gamma_{1}),m\leq n_{1}+2\leq 7$ such that
$\hat{\mathcal{O}}(\gamma\setminus(\gamma_{1})\bigcup(l))=\hat{\mathcal{O}}(\gamma\setminus(\gamma_{1}))=m-1$.

Combining case 1 and case 2: We inject the established estimates in (4.67) and
(4.71) into (4.63) and consider the summation (4.62), (noting the inclusion of
the $N^{(m-1)p-m}$ term in the summand) to obtain

$\displaystyle\sum_{m=1}^{n_{1}+2}\Bigg{(}N^{(m-1)p-m}\sum_{\gamma\in\Pi_{n_{1}+2}^{N},~{}\hat{\mathcal{O}}(\gamma)=m}\mathbb{E}\bigg{[}\frac{1}{N}\sum_{l=1}^{N}\Big{(}|X^{t,x_{\gamma_{1}},\gamma_{1},N}_{s,x_{\gamma_{2}},\dots,x_{\gamma_{n_{1}+2}}}|^{p-2}X^{t,x_{\gamma_{1}},\gamma_{1},N}_{s,x_{\gamma_{2}},\dots,x_{\gamma_{n_{1}+2}}}\Big{)}$
$\displaystyle\qquad\qquad\cdot\Big{(}\nabla^{2}V(X^{t,x_{\gamma_{1}},\gamma_{1},N}_{s}-X^{t,x_{l},l,N}_{s})(X^{t,x_{\gamma_{1}},\gamma_{1},N}_{s,x_{\gamma_{2}},\dots,x_{\gamma_{n_{1}+2}}}-X^{t,x_{l},l,N}_{s,x_{\gamma_{2}},\dots,x_{\gamma_{n_{1}+2}}})\Big{)}\bigg{]}\quad\Bigg{)}$
$\displaystyle\leq\Big{(}\frac{6K_{V}(p-1)}{p}+\frac{K_{V}}{Np}+\frac{6K_{V}}{Np}\Big{)}\sum_{m=1}^{n_{1}+2}\bigg{(}N^{(m-1)p-m}\sum_{\gamma\in\Pi_{n_{1}+2}^{N},~{}\hat{\mathcal{O}}(\gamma)=m}\mathbb{E}\Big{[}\Big{|}X^{t,x_{\gamma_{1}},\gamma_{1},N}_{s,x_{\gamma_{2}},\dots,x_{\gamma_{n_{1}+2}}}\Big{|}^{p}\Big{]}\bigg{)}$
$\displaystyle\leq
K_{V}\Big{(}6+\frac{1}{Np}\Big{)}\sum_{m=1}^{n_{1}+2}\bigg{(}N^{(m-1)p-m}\sum_{\gamma\in\Pi_{n_{1}+2}^{N},~{}\hat{\mathcal{O}}(\gamma)=m}\mathbb{E}\Big{[}\Big{|}X^{t,x_{\gamma_{1}},\gamma_{1},N}_{s,x_{\gamma_{2}},\dots,x_{\gamma_{n_{1}+2}}}\Big{|}^{p}\Big{]}\bigg{)}$
(4.72)
$\displaystyle=K_{V}\Big{(}6+\frac{1}{Np}\Big{)}e^{-\lambda_{0}^{(n_{1}+1)}(s-t)}I_{s,t}^{n_{1}+1,p}.$
(4.73)

Part 2.2: Analysis of the lower order variation term (4.61). We provide the
following details on the derivatives of the function $B_{\gamma_{1}}$: We
notice that, from Assumption H.2, for any $\bm{x}\in\mathbb{R}^{N}$,

$\displaystyle\partial_{x_{l}}B_{\gamma_{1}}=\begin{cases}-\nabla^{2}U(x_{\gamma_{1}})-\frac{1}{N}\partial_{x_{l}}\sum_{j=1}^{N}\nabla
V(x_{\gamma_{1}}-x_{j})=-\nabla^{2}U(x_{\gamma_{1}})-\frac{1}{N}\sum_{j=1}^{N}\nabla^{2}V(x_{\gamma_{1}}-x_{j}),&{\gamma_{1}}=l,\\\
-\frac{1}{N}\partial_{x_{l}}\sum_{j=1}^{N}\nabla
V(x_{\gamma_{1}}-x_{j})=\frac{1}{N}\nabla^{2}V(x_{\gamma_{1}}-x_{l}),&{\gamma_{1}}\neq
l.\end{cases}$

Thus, we have

$\displaystyle|\partial_{x_{l}}B_{\gamma_{1}}|_{\infty}\leq\begin{cases}|\nabla^{2}U|_{\infty}+\frac{1}{N}\sum_{j=1}^{N}|\nabla^{2}V|_{\infty}=\mathcal{O}(1),&{\gamma_{1}}=l,\\\
\frac{1}{N}|\nabla^{2}V|_{\infty}=\mathcal{O}(N^{-1}),&{\gamma_{1}}\neq
l,\end{cases}$ (4.74)

and

$\displaystyle|\partial^{2}_{x_{l},x_{l^{\prime}}}B_{\gamma_{1}}|_{\infty}\leq\begin{cases}|\nabla^{3}U|_{\infty}+\frac{1}{N}\sum_{j=1}^{N}|\nabla^{3}V|_{\infty}=\mathcal{O}(1),\quad&{\gamma_{1}}=l=l^{\prime},\\\
\frac{1}{N}|\nabla^{3}V|_{\infty}=\mathcal{O}(N^{-1}),\quad&{\gamma_{1}}=l\neq
l^{\prime},\\\
\frac{1}{N}|\nabla^{3}V|_{\infty}=\mathcal{O}(N^{-1}),\quad&{\gamma_{1}}=l^{\prime}\neq
l,\\\
\frac{1}{N}|\nabla^{3}V|_{\infty}=\mathcal{O}(N^{-1}),\quad&{\gamma_{1}}\neq
l=l^{\prime},\\\ 0,\quad&{\gamma_{1}}\neq l\neq l^{\prime}.\end{cases}$

Using this methodology, one can easily establish that for $n\geq
1,~{}\eta\in\Pi_{n}^{N}$

$\displaystyle|\partial^{n}_{x_{\eta_{1}},\dots,x_{\eta_{n}}}B_{\gamma_{1}}|_{\infty}$
$\displaystyle=\begin{cases}\mathcal{O}(N^{1-\hat{\mathcal{O}}(\eta\bigcup(\gamma_{1}))}),\qquad&\hat{\mathcal{O}}(\eta\bigcup(\gamma_{1}))\in\\{1,2\\},\\\
0,\qquad&\hat{\mathcal{O}}(\eta\bigcup(\gamma_{1}))\geq 3,\end{cases}$ (4.75)

with an implied constant independent of $N$, since only tuples of the form
$(\gamma_{1},\eta_{1},\ldots,\eta_{n})$, where elements take at most two
different values yield a non-zero contribution in (4.75). Proceeding exactly
as in the manner of (4.52)–(4.55), applying our induction hypothesis (4.58),
we can establish the existence of positive constants $K$ and $\alpha$ such
that

$\displaystyle\sum_{m=1}^{n_{1}+2}\bigg{(}N^{(m-1)p-m}\sum_{\gamma\in\Pi_{n_{1}+2}^{N},~{}\hat{\mathcal{O}}(\gamma)=m}\eqref{eq:
n-var - lower var }\bigg{)}\leq
K\int_{t}^{s}e^{-\alpha(u-t)}\mathrm{d}u+K\int_{t}^{s}e^{-\alpha(u-t)}I_{t,u}^{n_{1}+1,p}\mathrm{d}u.$
(4.76)

Part 3: Collecting terms and conclusion. By collecting (4.59), (4.73) and
(4.76), we have that

$\displaystyle I_{t,s}^{n_{1}+1,p}\leq
K\int_{t}^{s}e^{-\alpha(u-t)}\mathrm{d}u+p\Big{(}\lambda_{0}^{(n_{1}+1)}-\lambda+K_{V}(6+\frac{1}{Np})\Big{)}\int_{t}^{s}I_{t,u}^{n_{1}+1,p}\mathrm{d}u+K\int_{t}^{s}e^{-\alpha(u-t)}I_{t,u}^{n_{1}+1,p}\mathrm{d}u.$

Using the convexity assumption $\lambda\geq 7K_{V}$, we can conclude the
existence of $\lambda_{0}^{(n_{1}+1)}\in(0,\lambda)$ such that the term in
front of the second integral remains negative. The result follows from
Gronwall’s inequality (as was done in the case $n=2$). ∎

##  5 Decay estimates for the Kolmogorov backward equation

We establish the following estimates for the derivatives of the solution to
the Kolmogorov backward equation in terms of moments of the variation
processes. This will further help us to apply the results developed in Section
4 to the weak error analysis in Section 6.

######  Lemma 5.1.

Let $u$ be the solution to the Kolmogorov backward equation (3.4) with $g$ as
in Assumption H.2 and let $T\geq t\geq 0,N\in\mathbb{N}$. Then there exists a
constant $K>0$ (independent of $t,T,N$), such that for any
$j\in\\{1,\ldots,N\\}$ and $\bm{x}\in\mathbb{R}^{N}$

$\displaystyle|\partial_{x_{j}}u(t,\bm{x})|^{2}\leq\frac{K}{N^{2}}\mathbb{E}\Big{[}|X_{T,x_{j}}^{t,x_{j},j,N}|^{2}\Big{]}+\frac{K}{N}\sum_{i=1,~{}i\neq
j}^{N}\mathbb{E}\left[|X_{T,x_{j}}^{t,x_{i},i,N}|^{2}\right].$

###### Proof.

From the definition of $u$, using [31, Theorem 5.5 in Chapter 5] as in [33,
Proposition B.3], we deduce

$\displaystyle|\partial_{x_{j}}u(t,\bm{x})|^{2}$
$\displaystyle=\Big{|}\mathbb{E}\Big{[}~{}\sum_{i=1}^{N}\big{(}\partial_{x_{i}}g(\bm{X}_{T}^{t,\bm{x},N})\big{)}\cdot\big{(}X_{T,x_{j}}^{t,x_{i},i,N}\big{)}\Big{]}\Big{|}^{2}$
$\displaystyle\leq
2\Big{|}\mathbb{E}\Big{[}|\partial_{x_{j}}g(\bm{X}_{T}^{t,\bm{x},N})|\
|X_{T,x_{j}}^{t,x_{j},j,N}|\Big{]}\Big{|}^{2}+2\Big{|}\mathbb{E}\Big{[}~{}\sum_{i=1,~{}i\neq
j}^{N}\big{(}\partial_{x_{i}}g(\bm{X}_{T}^{t,\bm{x},N})\big{)}\cdot\big{(}X_{T,x_{j}}^{t,x_{i},i,N}\big{)}\Big{]}\Big{|}^{2}$
$\displaystyle\leq\frac{K}{N^{2}}\mathbb{E}\Big{[}|X_{T,x_{j}}^{t,x_{j},j,N}|^{2}\Big{]}+KN\sum_{i=1,~{}i\neq
j}^{N}\mathbb{E}\left[\Big{|}|\partial_{x_{i}}g(\bm{X}_{T}^{t,\bm{x},N})|\
|X_{T,x_{j}}^{t,x_{i},i,N}|\Big{|}^{2}\right]$
$\displaystyle\leq\frac{K}{N^{2}}\mathbb{E}\Big{[}|X_{T,x_{j}}^{t,x_{j},j,N}|^{2}\Big{]}+\frac{K}{N}\sum_{i=1,~{}i\neq
j}^{N}\mathbb{E}\left[|X_{T,x_{j}}^{t,x_{i},i,N}|^{2}\right],$

where we used Jensen’s inequality and the growth of derivatives of $g$ in
Assumption H.2. ∎

The next result, Lemma 5.2, generalizes Lemma 5.1 to derivatives of order
$1<n\leq 6$, whose proof is involved and relies on carefully applying Jensen’s
inequality. The next calculations aim to clarify that partitioning summations
in particular ways before the application of Jensen’s inequality can yield
sharper upper bounds.

For each $\gamma=(\gamma_{1},\ldots,\gamma_{n})\in\Pi_{n}^{N}$, let
$x_{\gamma_{1},\ldots,\gamma_{n}}$ be a real number. Then applying Jensen’s
inequality directly we have

$\displaystyle\Big{|}\sum_{\gamma\in\Pi_{1}^{N}}x_{\gamma_{1}}\Big{|}^{2}$
$\displaystyle=\Big{|}\sum_{i=1}^{N}x_{i}\Big{|}^{2}=N^{2}\Big{|}\frac{1}{N}\sum_{i=1}^{N}x_{i}\Big{|}^{2}\leq
N\sum_{i=1}^{N}|x_{i}|^{2},$
$\displaystyle\Big{|}\sum_{\gamma\in\Pi_{n}^{N}}x_{\gamma_{1},\dots,\gamma_{n}}\Big{|}^{2}$
$\displaystyle=N^{2n}\Big{|}\frac{1}{N^{n}}\sum_{\gamma\in\Pi_{n}^{N}}x_{\gamma_{1},\dots,\gamma_{n}}\Big{|}^{2}\leq\frac{N^{2n}}{N^{n}}\sum_{\gamma\in\Pi_{n}^{N}}|x_{\gamma_{1},\dots,\gamma_{n}}|^{2}=N^{n}\sum_{\gamma\in\Pi_{n}^{N}}|x_{\gamma_{1},\dots,\gamma_{n}}|^{2}.$
(5.1)

Consider now the specific two-dimensional example of
$x_{\gamma_{1},\gamma_{2}}=N^{1-\hat{\mathcal{O}}(\gamma)}$ (corresponding to
a $2\times 2$ matrix with diagonal entries $1$ and otherwise $1/N$). Using
(5.1) we have that

$\displaystyle\Big{|}\sum_{\gamma\in\Pi_{2}^{N}}x_{\gamma_{1},\gamma_{2}}\Big{|}^{2}$
$\displaystyle\leq
N^{2}\sum_{i,j=1}^{N}|x_{i,j}|^{2}=N^{2}\sum_{i=1}^{N}|x_{i,i}|^{2}+N^{2}\sum_{i,j=1,i\neq
j}^{N}|x_{i,j}|^{2}=N^{3}+N^{2}\leq 2N^{3}.$

This estimate is too naive and can be improved, as we can instead consider

$\displaystyle\Big{|}\sum_{\gamma\in\Pi_{2}^{N}}x_{\gamma_{1},\gamma_{2}}\Big{|}^{2}$
$\displaystyle\leq
2\Big{|}\sum_{i=1}^{N}x_{i,i}\Big{|}^{2}+2\Big{|}\sum_{i,j=1,i\neq
j}^{N}x_{i,j}\Big{|}^{2}\leq
2N\sum_{i=1}^{N}|x_{i,i}|^{2}+2N^{2}\sum_{i,j=1,i\neq j}^{N}|x_{i,j}|^{2}$
$\displaystyle=2N^{2}+\frac{2N^{3}(N-1)}{N^{2}}\leq 4N^{2},$

which is indeed a sharper upper bound.

This argument can be extended to the general $n$-dimensional case; that is,
take $x_{\gamma_{1},\ldots,\gamma_{n}}=N^{1-\hat{\mathcal{O}}(\gamma)}$ as the
entries of a $\bigotimes_{i=1}^{n}\mathbb{R}^{N}$-valued $n$-tensor. Observe
that

$\displaystyle|\\{\gamma\in\Pi_{n}^{N}:\hat{\mathcal{O}}(\gamma)=m\\}|=\mathcal{O}(N^{m}),$
(5.2)

a result which follows from a simple combinatorial argument: regardless of the
length $n$ of $\gamma$, we have $m$ distinct values chosen out of the range
$\\{1,\ldots,N\\}$; that is $\mathcal{O}(N^{m})$ possibilities. We have

$\displaystyle\Big{|}\sum_{\gamma\in\Pi_{n}^{N}}x_{\gamma_{1},\dots,\gamma_{n}}\Big{|}^{2}$
$\displaystyle=\Big{|}\sum_{m=1}^{n}\sum_{\begin{subarray}{c}\gamma\in\Pi_{n}^{N},\\\
\hat{\mathcal{O}}(\gamma)=m\end{subarray}}x_{\gamma_{1},\dots,\gamma_{n}}\Big{|}^{2}\leq
K\sum_{m=1}^{n}\Big{|}\sum_{\begin{subarray}{c}\gamma\in\Pi_{n}^{N},\\\
\hat{\mathcal{O}}(\gamma)=m\end{subarray}}x_{\gamma_{1},\dots,\gamma_{n}}\Big{|}^{2}$
$\displaystyle\leq
K\sum_{m=1}^{n}N^{m}\sum_{\begin{subarray}{c}\gamma\in\Pi_{n}^{N},\\\
\hat{\mathcal{O}}(\gamma)=m\end{subarray}}|x_{\gamma_{1},\dots,\gamma_{n}}|^{2}\leq
K\sum_{m=1}^{n}N^{2m}N^{2-2m}\leq KN^{2},$

with $K$ a constant, changing across the inequalities, independent of $N$ but
dependent on $n$ where $n\leq 6$. This is also a sharper upper bound than
applying Jensen’s inequality directly. We now state the generalization of
Lemma 5.1 for $1\leq n\leq 6$.

######  Lemma 5.2.

Let $u$ satisfy the Kolmogorov backward equation (3.4) with $g$ as in
Assumption H.2 and let $T\geq t\geq 0,N\in\mathbb{N}$. Then there exists a
constant $K>0$ (independent of $t,T,N$), such that for any
$n\in\mathbb{N},1\leq n\leq 6$, $\gamma\in\Pi_{n}^{N}$, and
$\bm{x}\in\mathbb{R}^{N}$

$\displaystyle|\partial$
${}_{x_{\gamma_{1}},\dots,x_{\gamma_{n}}}^{n}u(t,\bm{x})|^{2}$
$\displaystyle\leq
K\sum_{m=0}^{n}~{}~{}\sum_{\begin{subarray}{c}\ell\in\bigcup_{k=1}^{n}\Pi_{k}^{N},\\\
\hat{\mathcal{O}}(\ell\bigcup\gamma)=\hat{\mathcal{O}}(\gamma)+m\end{subarray}}~{}N^{m-2\hat{\mathcal{O}}(\ell)}~{}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sum_{\begin{subarray}{c}\alpha_{1},\dots,\alpha_{|\ell|}\in\bigcup_{k=1}^{n}\Pi_{k}^{N},\\\
\bigcup_{i=1}^{|\ell|}\alpha_{i}\simeq\gamma\end{subarray}}~{}\mathbb{E}\Big{[}~{}\prod_{i=1}^{|\ell|}\Big{|}X_{T,\alpha_{i,1},\dots,\alpha_{i,|\alpha_{i}|}}^{t,x_{\ell_{i}},\ell_{i},N}\Big{|}^{2}~{}\Big{]}},$
(5.3)

where $\alpha_{i}=(\alpha_{i,1},\dots,\alpha_{i,|\alpha_{i}|})$ and
$\alpha_{i,j}\in\\{1,\dots,N\\}$ for $j\in\\{1,\dots,|\alpha_{i}|\\}$.
Further, we have

$\displaystyle|\partial$
${}_{x_{\gamma_{1}},\dots,x_{\gamma_{n}}}^{n}u(t,\bm{x})|^{4}$
$\displaystyle\leq
K\sum_{m=0}^{n}~{}~{}\sum_{\begin{subarray}{c}\ell\in\bigcup_{k=1}^{n}\Pi_{k}^{N},\\\
\hat{\mathcal{O}}(\ell\bigcup\gamma)=\hat{\mathcal{O}}(\gamma)+m\end{subarray}}~{}N^{3m-4\hat{\mathcal{O}}(\ell)}~{}\sum_{\begin{subarray}{c}\alpha_{1},\dots,\alpha_{|\ell|}\in\bigcup_{k=1}^{n}\Pi_{k}^{N},\\\
\bigcup_{i=1}^{|\ell|}\alpha_{i}\simeq\gamma\end{subarray}}~{}\mathbb{E}\Big{[}~{}\prod_{i=1}^{|\ell|}\Big{|}X_{T,\alpha_{i,1},\dots,\alpha_{i,|\alpha_{i}|}}^{t,x_{\ell_{i}},\ell_{i},N}\Big{|}^{4}~{}\Big{]}.$
(5.4)

###### Proof.

Note that in the following proof, the positive constant $K$ is independent of
$t,T,N$ and may change line by line. Before establishing the main results
(Part 2 and Part 3), we present (Part 1) a study for the second derivatives of
$u$ (i.e., for the case $n=2$) to demonstrate the approach used and its
nuances in regards to estimate (5.3) of the lemma.

Part 1: Comparing the direct calculation with (5.3) for the example case
$n=2$. For all $j,k\in\\{1,\dots,N\\}$, $j\neq k$, using [31, Theorem 5.5 in
Chapter 5], we have

$\displaystyle|\partial_{x_{j},x_{k}}^{2}u(t,\bm{x})|^{2}$
$\displaystyle=\bigg{|}\mathbb{E}\Big{[}\sum_{i=1}^{N}\partial_{x_{i}}g(\bm{X}_{T}^{t,\bm{x},N})X^{t,x_{i},i,N}_{T,x_{j},x_{k}}\Big{]}+\mathbb{E}\Big{[}\sum_{i=1}^{N}\sum_{i^{\prime}=1}^{N}\partial^{2}_{x_{i},x_{i^{\prime}}}g(\bm{X}_{T}^{t,\bm{x},N})X^{t,x_{i},i,N}_{T,x_{j}}X^{t,x_{i^{\prime}},{i^{\prime}},N}_{T,x_{k}}\Big{]}\bigg{|}^{2}$
$\displaystyle\leq
K\Bigg{(}\bigg{|}\mathbb{E}\Big{[}\partial_{x_{j}}g(\bm{X}_{T}^{t,\bm{x},N})X^{t,x_{j},j,N}_{T,x_{j},x_{k}}+\partial_{x_{k}}g(\bm{X}_{T}^{t,\bm{x},N})X^{t,x_{k},k,N}_{T,x_{j},x_{k}}\Big{]}\bigg{|}^{2}$
(5.5) $\displaystyle\quad+\bigg{|}\mathbb{E}\Big{[}\sum_{i=1,~{}i\neq j\neq
k}^{N}\partial_{x_{i}}g(\bm{X}_{T}^{t,\bm{x},N})X^{t,x_{i},i,N}_{T,x_{j},x_{k}}~{}\Big{]}\bigg{|}^{2}$
(5.6)
$\displaystyle\quad+\bigg{|}\mathbb{E}\Big{[}\partial^{2}_{x_{j},x_{j}}g(\bm{X}_{T}^{t,\bm{x},N})X^{t,x_{j},j,N}_{T,x_{j}}X^{t,x_{j},{j},N}_{T,x_{k}}+\partial^{2}_{x_{k},x_{k}}g(\bm{X}_{T}^{t,\bm{x},N})X^{t,x_{k},k,N}_{T,x_{j}}X^{t,x_{k},{k},N}_{T,x_{k}}\Big{]}\bigg{|}^{2}$
(5.7)
$\displaystyle\quad+\bigg{|}\mathbb{E}\Big{[}\partial^{2}_{x_{j},x_{k}}g(\bm{X}_{T}^{t,\bm{x},N})X^{t,x_{j},j,N}_{T,x_{j}}X^{t,x_{k},{k},N}_{T,x_{k}}+\partial^{2}_{x_{k},x_{j}}g(\bm{X}_{T}^{t,\bm{x},N})X^{t,x_{k},k,N}_{T,x_{j}}X^{t,x_{j},{j},N}_{T,x_{k}}\Big{]}\bigg{|}^{2}$
(5.8) $\displaystyle\quad+\bigg{|}\mathbb{E}\Big{[}\sum_{i=1,~{}i\neq j\neq
k}^{N}\partial^{2}_{x_{i},x_{i}}g(\bm{X}_{T}^{t,\bm{x},N})X^{t,x_{i},i,N}_{T,x_{j}}X^{t,x_{i},{i},N}_{T,x_{k}}\Big{]}\bigg{|}^{2}$
(5.9) $\displaystyle\quad+\bigg{|}\mathbb{E}\Big{[}\sum_{i=1,~{}i\neq j\neq
k}^{N}\Big{(}\partial^{2}_{x_{j},x_{i}}g(\bm{X}_{T}^{t,\bm{x},N})X^{t,x_{j},j,N}_{T,x_{j}}X^{t,x_{i},{i},N}_{T,x_{k}}+\partial^{2}_{x_{i},x_{j}}g(\bm{X}_{T}^{t,\bm{x},N})X^{t,x_{i},i,N}_{T,x_{j}}X^{t,x_{j},{j},N}_{T,x_{k}}$
$\displaystyle\hskip
85.35826pt+\partial^{2}_{x_{k},x_{i}}g(\bm{X}_{T}^{t,\bm{x},N})X^{t,x_{k},k,N}_{T,x_{j}}X^{t,x_{i},{i},N}_{T,x_{k}}+\partial^{2}_{x_{i},x_{k}}g(\bm{X}_{T}^{t,\bm{x},N})X^{t,x_{i},i,N}_{T,x_{j}}X^{t,x_{k},{k},N}_{T,x_{k}}\Big{)}\Big{]}\bigg{|}^{2}$
(5.10)
$\displaystyle\quad+\bigg{|}\mathbb{E}\Big{[}\sum_{i,i^{\prime}=1,~{}j\neq
i\neq i^{\prime}\neq
k}^{N}\partial^{2}_{{x_{i},x_{i^{\prime}}}}g(\bm{X}_{T}^{t,\bm{x},N})X^{t,x_{i},i,N}_{T,x_{j}}X^{t,x_{i^{\prime}},{i^{\prime}},N}_{T,x_{k}}\Big{]}\bigg{|}^{2}\bigg{)},$
(5.11)

where the inequality follows simply by separating the summation terms by
number of distinct derivatives of $g$ matching also to the order of decay in
$N$, and then applying Jensen’s inequality (but still leaving the square
outside the expectations).

Under Assumption H.2 on the derivatives of $g$ and observing that (5.6), (5.9)
and (5.10) have $\mathcal{O}(N)$ terms and that (5.11) has
$\mathcal{O}(N^{2})$ terms in the summand, we apply Jensen’s inequality once
more to get

$\displaystyle\eqref{eq: lemma5.2 line 3}+\eqref{eq: lemma5.2 line
5}+\eqref{eq: lemma5.2 line 11}$
$\displaystyle\leq\frac{K}{N^{2}}\mathbb{E}\Big{[}|X^{t,x_{j},j,N}_{T,x_{j},x_{k}}|^{2}+|X^{t,x_{k},k,N}_{T,x_{j},x_{k}}|^{2}+|X^{t,x_{j},j,N}_{T,x_{j}}X^{t,x_{j},{j},N}_{T,x_{k}}|^{2}+|X^{t,x_{k},k,N}_{T,x_{j}}X^{t,x_{k},{k},N}_{T,x_{k}}|^{2}\Big{]}$
$\displaystyle\hskip 89.626pt+\sum_{i,i^{\prime}=1,~{}j\neq i\neq
i^{\prime}\neq
k}^{N}\Big{[}|X^{t,x_{i},i,N}_{T,x_{j}}X^{t,x_{i^{\prime}},{i^{\prime}},N}_{T,x_{k}}|^{2}\Big{]}$
$\displaystyle\eqref{eq: lemma5.2 line 4}+\eqref{eq: lemma5.2 line 8}$
$\displaystyle\leq\frac{K}{N}\sum_{i=1,~{}i\neq j\neq
k}^{N}\mathbb{E}\Big{[}|X^{t,x_{i},i,N}_{T,x_{j},x_{k}}|^{2}+|X^{t,x_{i},i,N}_{T,x_{j}}X^{t,x_{i},{i},N}_{T,x_{k}}|^{2}\Big{]}$
$\displaystyle\eqref{eq: lemma5.2 line 7}$
$\displaystyle\leq\frac{K}{N^{4}}\mathbb{E}\Big{[}|X^{t,x_{j},j,N}_{T,x_{j}}X^{t,x_{k},{k},N}_{T,x_{k}}|^{2}+|X^{t,x_{k},k,N}_{T,x_{j}}X^{t,x_{j},{j},N}_{T,x_{k}}|^{2}\Big{]}$
$\displaystyle\eqref{eq: lemma5.2 line 10}$
$\displaystyle\leq\frac{K}{N^{3}}\sum_{i=1,~{}i\neq j\neq
k}^{N}\mathbb{E}\Big{[}|X^{t,x_{j},j,N}_{T,x_{j}}X^{t,x_{i},{i},N}_{T,x_{k}}|^{2}+|X^{t,x_{i},i,N}_{T,x_{j}}X^{t,x_{j},{j},N}_{T,x_{k}}|^{2}$
$\displaystyle\hskip
89.626pt+|X^{t,x_{k},k,N}_{T,x_{j}}X^{t,x_{i},i,N}_{T,x_{k}}|^{2}+|X^{t,x_{i},i,N}_{T,x_{j}}X^{t,x_{k},{k},N}_{T,x_{k}}|^{2}\Big{]}.$

We now demonstrate how these estimates (5.5)–(5.11), when combined together,
can be upper bounded by the form given in (5.3) for $n=2$,
$\gamma=(j,k),~{}j\neq k$:

$\displaystyle|\partial$ ${}_{x_{j},x_{k}}^{2}u(t,\bm{x})|^{2}$
$\displaystyle\leq
K\sum_{m=0}^{2}~{}~{}\sum_{\begin{subarray}{c}\ell\in\bigcup_{k=1}^{2}\Pi_{k}^{N},\\\
\hat{\mathcal{O}}(\ell\bigcup\gamma)=\hat{\mathcal{O}}(\gamma)+m\end{subarray}}~{}N^{m-2\hat{\mathcal{O}}(\ell)}~{}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sum_{\begin{subarray}{c}\alpha_{1},\dots,\alpha_{|\ell|}\in\bigcup_{k=1}^{2}\Pi_{k}^{N},\\\
\bigcup_{i=1}^{|\ell|}\alpha_{i}\simeq\gamma\end{subarray}}~{}\mathbb{E}\Big{[}~{}\prod_{i=1}^{|\ell|}\Big{|}X_{T,\alpha_{i,1},\dots,\alpha_{i,|\alpha_{i}|}}^{t,x_{\ell_{i}},\ell_{i},N}\Big{|}^{2}~{}\Big{]}}$
$\displaystyle=K\bigg{(}\frac{1}{N^{2}}\mathbb{E}\Big{[}|X^{t,x_{j},j,N}_{T,x_{j},x_{k}}|^{2}+|X^{t,x_{k},k,N}_{T,x_{j},x_{k}}|^{2}\Big{]}+\frac{1}{N^{2}}\mathbb{E}\Big{[}|X^{t,x_{j},j,N}_{T,x_{j}}|^{2}~{}|X^{t,x_{j},j,N}_{T,x_{k}}|^{2}+|X^{t,x_{j},j,N}_{T,x_{k}}|^{2}~{}|X^{t,x_{j},j,N}_{T,x_{j}}|^{2}$
$\displaystyle\quad+|X^{t,x_{k},k,N}_{T,x_{j}}|^{2}~{}|X^{t,x_{k},k,N}_{T,x_{k}}|^{2}+|X^{t,x_{k},k,N}_{T,x_{k}}|^{2}~{}|X^{t,x_{k},k,N}_{T,x_{j}}|^{2}\Big{]}+\frac{1}{N^{4}}\mathbb{E}\Big{[}|X^{t,x_{j},j,N}_{T,x_{j}}|^{2}~{}|X^{t,x_{k},k,N}_{T,x_{k}}|^{2}$
$\displaystyle\quad+|X^{t,x_{j},j,N}_{T,x_{k}}|^{2}~{}|X^{t,x_{k},k,N}_{T,x_{j}}|^{2}+|X^{t,x_{k},k,N}_{T,x_{j}}|^{2}~{}|X^{t,x_{j},k,N}_{T,x_{k}}|^{2}+|X^{t,x_{k},k,N}_{T,x_{k}}|^{2}~{}|X^{t,x_{j},j,N}_{T,x_{j}}|^{2}\Big{]}$
$\displaystyle\quad+\frac{1}{N}\sum_{i=1,i\neq j\neq
k}^{N}\mathbb{E}\Big{[}|X^{t,x_{i},i,N}_{T,x_{j},x_{k}}|^{2}+|X^{t,x_{i},i,N}_{T,x_{k},x_{j}}|^{2}\Big{]}+\frac{1}{N^{3}}\sum_{i=1,i\neq
j\neq
k}^{N}\mathbb{E}\Big{[}|X^{t,x_{j},j,N}_{T,x_{j}}|^{2}~{}|X^{t,x_{i},i,N}_{T,x_{k}}|^{2}$
$\displaystyle\quad+|X^{t,x_{j},j,N}_{T,x_{k}}|^{2}~{}|X^{t,x_{i},i,N}_{T,x_{j}}|^{2}+|X^{t,x_{i},i,N}_{T,x_{j}}|^{2}~{}|X^{t,x_{j},j,N}_{T,x_{k}}|^{2}+|X^{t,x_{i},i,N}_{T,x_{k}}|^{2}~{}|X^{t,x_{j},j,N}_{T,x_{j}}|^{2}+|X^{t,x_{k},k,N}_{T,x_{j}}|^{2}~{}|X^{t,x_{i},i,N}_{T,x_{k}}|^{2}$
$\displaystyle\quad+|X^{t,x_{k},k,N}_{T,x_{k}}|^{2}~{}|X^{t,x_{i},i,N}_{T,x_{j}}|^{2}+|X^{t,x_{i},i,N}_{T,x_{j}}|^{2}~{}|X^{t,x_{k},k,N}_{T,x_{k}}|^{2}+|X^{t,x_{i},i,N}_{T,x_{k}}|^{2}~{}|X^{t,x_{k},k,N}_{T,x_{j}}|^{2}\Big{]}$
$\displaystyle\quad+\frac{1}{N}\sum_{i=1,i\neq j\neq
k}^{N}\mathbb{E}\Big{[}|X^{t,x_{i},i,N}_{T,x_{j}}|^{2}~{}|X^{t,x_{i},i,N}_{T,x_{k}}|^{2}+|X^{t,x_{i},i,N}_{T,x_{k}}|^{2}~{}|X^{t,x_{i},i,N}_{T,x_{j}}|^{2}\Big{]}$
$\displaystyle\quad+\frac{1}{N^{2}}\sum_{i,i^{\prime}=1,~{}j\neq i\neq
i^{\prime}\neq
k}^{N}\mathbb{E}\Big{[}|X^{t,x_{i},i,N}_{T,x_{j}}|^{2}~{}|X^{t,x_{i^{\prime}},{i^{\prime}},N}_{T,x_{k}}|^{2}+|X^{t,x_{i^{\prime}},{i^{\prime}},N}_{T,x_{j}}|^{2}~{}|X^{t,x_{i},i,N}_{T,x_{k}}|^{2}\Big{]}\bigg{)}.$
(5.12)

Comparing to the results of (5.5)–(5.11), we can see that (5.12) contains more
terms. For example, consider the term

$\frac{1}{N}\sum_{i=1,i\neq j\neq
k}^{N}\mathbb{E}\Big{[}|X^{t,x_{i},i,N}_{T,x_{j},x_{k}}|^{2}+|X^{t,x_{i},i,N}_{T,x_{k},x_{j}}|^{2}\Big{]}.$

For the case $m=1,\ell\in\Pi_{1}^{N}$, we not only take summation over
$|X^{t,x_{i},i,N}_{T,x_{j},x_{k}}|^{2}$, but also consider
$|X^{t,x_{i},i,N}_{T,x_{k},x_{j}}|^{2}$, so that the sum of the terms
(5.5)–(5.11) is bounded above by (5.12), verifying our result in the case
$n=2$, $\gamma=(j,k),~{}j\neq k$.

Part 2. The bound (5.3). From the above estimates, one can see that the idea
is to essentially partition the sums based on the relationship between indices
$i,i^{\prime},j,k$, to keep consistent orders of $N$. Having this separation
trick in mind, for any $\gamma\in\Pi_{n}^{N},~{}1\leq n\leq 6$, we consider
(recall the notation for $\hat{\mathcal{O}}(\cdot)$ and $|\cdot|$ in
Definition 3.1):

$\displaystyle|\partial^{n}_{x_{\gamma_{1}},\dots,x_{\gamma_{n}}}u(t,\bm{x})|^{2}$
$\displaystyle=\Bigg{|}\mathbb{E}\Bigg{[}~{}\sum_{\begin{subarray}{c}\alpha,\beta\in\bigcup_{k=0}^{n-1}\Pi_{k}^{N},\\\
\gamma\setminus(\gamma_{1})\in\alpha\shuffle\beta\end{subarray}}\sum_{i=1}^{N}\Big{(}\partial_{x_{i}}g(\bm{X}_{T}^{t,\bm{x},N})\Big{)}_{x_{\alpha_{1}},\dots,x_{\alpha_{|\alpha|}}}\big{(}X^{t,x_{i},i,N}_{T,x_{\gamma_{1}}}\big{)}_{x_{\beta_{1}},\dots,x_{\beta_{|\beta|}}}\Bigg{]}\Bigg{|}^{2}$
$\displaystyle\leq\Bigg{|}\mathbb{E}\Bigg{[}~{}\sum_{m=1}^{n}\sum_{\ell\in\Pi_{m}^{N}}\bigg{(}\Big{|}\partial^{m}_{{x_{\ell_{1}},\dots,x_{\ell_{m}}}}g(\bm{X}_{T}^{t,\bm{x},N})\Big{|}\sum_{\begin{subarray}{c}\alpha_{1},\dots,\alpha_{m}\in\bigcup_{k=1}^{n}\Pi_{k}^{N},\\\
\bigcup_{i=1}^{m}\alpha_{i}\simeq\gamma\end{subarray}}\prod_{i=1}^{m}\Big{|}X_{T,\alpha_{i,1},\dots,\alpha_{i,|\alpha_{i}|}}^{t,x_{\ell_{i}},\ell_{i},N}\Big{|}~{}\bigg{)}\Bigg{]}\Bigg{|}^{2}$
$\displaystyle=\Bigg{|}\mathbb{E}\Bigg{[}~{}\sum_{m=1}^{n}~{}\sum_{\begin{subarray}{c}\ell\in\bigcup_{k=1}^{n}\Pi_{k}^{N},\\\
\hat{\mathcal{O}}(\ell)=m\end{subarray}}~{}\bigg{(}\Big{|}\partial^{|\ell|}_{{x_{\ell_{1}},\dots,x_{\ell_{|\ell|}}}}g(\bm{X}_{T}^{t,\bm{x},N})\Big{|}\sum_{\begin{subarray}{c}\alpha_{1},\dots,\alpha_{|\ell|}\in\bigcup_{k=1}^{n}\Pi_{k}^{N},\\\
\bigcup_{i=1}^{|\ell|}\alpha_{i}\simeq\gamma\end{subarray}}~{}\prod_{i=1}^{|\ell|}\Big{|}X_{T,\alpha_{i,1},\dots,\alpha_{i,|\alpha_{i}|}}^{t,x_{\ell_{i}},\ell_{i},N}\Big{|}~{}\bigg{)}\Bigg{]}\Bigg{|}^{2}$
(5.13)
$\displaystyle=\Bigg{|}\mathbb{E}\Bigg{[}~{}\sum_{m=0}^{n}~{}\sum_{\begin{subarray}{c}\ell\in\bigcup_{k=1}^{n}\Pi_{k}^{N},\\\
\hat{\mathcal{O}}(\ell\bigcup\gamma)=\hat{\mathcal{O}}(\gamma)+m\end{subarray}}~{}\bigg{(}\Big{|}\partial^{|\ell|}_{{x_{\ell_{1}},\dots,x_{\ell_{|\ell|}}}}g(\bm{X}_{T}^{t,\bm{x},N})\Big{|}\sum_{\begin{subarray}{c}\alpha_{1},\dots,\alpha_{|\ell|}\in\bigcup_{k=1}^{n}\Pi_{k}^{N},\\\
\bigcup_{i=1}^{|\ell|}\alpha_{i}\simeq\gamma\end{subarray}}~{}\prod_{i=1}^{|\ell|}\Big{|}X_{T,\alpha_{i,1},\dots,\alpha_{i,|\alpha_{i}|}}^{t,x_{\ell_{i}},\ell_{i},N}\Big{|}~{}\bigg{)}\Bigg{]}\Bigg{|}^{2}$
(5.14) $\displaystyle\leq
K\sum_{m=0}^{n}N^{2m}~{}\mathbb{E}\Bigg{[}\Bigg{|}\frac{1}{N^{m}}\sum_{\begin{subarray}{c}\ell\in\bigcup_{k=1}^{n}\Pi_{k}^{N},\\\
\hat{\mathcal{O}}(\ell\bigcup\gamma)=\hat{\mathcal{O}}(\gamma)+m\end{subarray}}\bigg{(}\Big{|}\partial^{|\ell|}_{{x_{\ell_{1}},\dots,x_{\ell_{|\ell|}}}}g(\bm{X}_{T}^{t,\bm{x},N})\Big{|}$
$\displaystyle\quad\hskip
170.71652pt\cdot\sum_{\begin{subarray}{c}\alpha_{1},\dots,\alpha_{|\ell|}\in\bigcup_{k=1}^{n}\Pi_{k}^{N},\\\
\bigcup_{i=1}^{|\ell|}\alpha_{i}\simeq\gamma\end{subarray}}~{}\prod_{i=1}^{|\ell|}\Big{|}X_{T,\alpha_{i,1},\dots,\alpha_{i,|\alpha_{i}|}}^{t,x_{\ell_{i}},\ell_{i},N}\Big{|}~{}\bigg{)}\Bigg{|}^{2}~{}\Bigg{]}$
$\displaystyle\leq
K\sum_{m=0}^{n}~{}~{}\sum_{\begin{subarray}{c}\ell\in\bigcup_{k=1}^{n}\Pi_{k}^{N},\\\
\hat{\mathcal{O}}(\ell\bigcup\gamma)=\hat{\mathcal{O}}(\gamma)+m\end{subarray}}~{}N^{m}~{}\mathbb{E}\Bigg{[}\Big{|}\partial^{|\ell|}_{{x_{\ell_{1}},\dots,x_{\ell_{|\ell|}}}}g(\bm{X}_{T}^{t,\bm{x},N})\Big{|}^{2}\sum_{\begin{subarray}{c}\alpha_{1},\dots,\alpha_{|\ell|}\in\bigcup_{k=1}^{n}\Pi_{k}^{N},\\\
\bigcup_{i=1}^{|\ell|}\alpha_{i}\simeq\gamma\end{subarray}}~{}\prod_{i=1}^{|\ell|}\Big{|}X_{T,\alpha_{i,1},\dots,\alpha_{i,|\alpha_{i}|}}^{t,x_{\ell_{i}},\ell_{i},N}\Big{|}^{2}~{}\Bigg{]}$
(5.15) $\displaystyle\leq
K\sum_{m=0}^{n}~{}~{}\sum_{\begin{subarray}{c}\ell\in\bigcup_{k=1}^{n}\Pi_{k}^{N},\\\
\hat{\mathcal{O}}(\ell\bigcup\gamma)=\hat{\mathcal{O}}(\gamma)+m\end{subarray}}~{}N^{m-2\hat{\mathcal{O}}(\ell)}~{}\mathbb{E}\Bigg{[}\sum_{\begin{subarray}{c}\alpha_{1},\dots,\alpha_{|\ell|}\in\bigcup_{k=1}^{n}\Pi_{k}^{N},\\\
\bigcup_{i=1}^{|\ell|}\alpha_{i}\simeq\gamma\end{subarray}}~{}\prod_{i=1}^{|\ell|}\Big{|}X_{T,\alpha_{i,1},\dots,\alpha_{i,|\alpha_{i}|}}^{t,x_{\ell_{i}},\ell_{i},N}\Big{|}^{2}~{}\Bigg{]},$

where in (5.13) and (5.14), we regroup the summation over all
$\ell\in\bigcup_{k=1}^{n}\Pi_{k}^{N}$ based on the magnitude of
$\hat{\mathcal{O}}(\ell)$ and $\hat{\mathcal{O}}(\ell\bigcup\gamma)$. In
(5.15), we apply Jensen’s inequality to the second summation where the set
$\\{\ell\in\bigcup_{k=1}^{n}\Pi_{k}^{N}:\hat{\mathcal{O}}(\ell\bigcup\gamma)=\hat{\mathcal{O}}(\gamma)+m\\}$
has $\mathcal{O}(N^{m})$ terms ($\ell$ has $m$ degrees of freedom), thus we
end up with a factor of $N^{m}$ after calculation. For the last line, we used
Assumption H.2:
$\big{|}\partial^{|\ell|}_{x_{\ell_{1}},\dots,x_{\ell_{|\ell|}}}g\big{|}_{\infty}=\mathcal{O}(N^{-\hat{\mathcal{O}}(\ell)})$
.

Part 3: The bound (5.4). One proves (5.4) using the same arguments as in Part
2. We have

$\displaystyle|\partial^{n}_{x_{\gamma_{1}},\dots,x_{\gamma_{n}}}u(t,\bm{x})|^{4}$
$\displaystyle\leq
K\sum_{m=0}^{n}~{}~{}\sum_{\begin{subarray}{c}\ell\in\bigcup_{k=1}^{n}\Pi_{k}^{N},\\\
\hat{\mathcal{O}}(\ell\bigcup\gamma)=\hat{\mathcal{O}}(\gamma)+m\end{subarray}}~{}N^{3m}~{}\mathbb{E}\Bigg{[}\Big{|}\partial^{|\ell|}_{{x_{\ell_{1}},\dots,x_{\ell_{|\ell|}}}}g(\bm{X}_{T}^{t,\bm{x},N})\Big{|}^{4}\sum_{\begin{subarray}{c}\alpha_{1},\dots,\alpha_{|\ell|}\in\bigcup_{k=1}^{n}\Pi_{k}^{N},\\\
\bigcup_{i=1}^{|\ell|}\alpha_{i}\simeq\gamma\end{subarray}}~{}\prod_{i=1}^{|\ell|}\Big{|}X_{T,\alpha_{i,1},\dots,\alpha_{i,|\alpha_{i}|}}^{t,x_{\ell_{i}},\ell_{i},N}\Big{|}^{4}~{}\Bigg{]}$
$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\leq
K\sum_{m=0}^{n}~{}~{}\sum_{\begin{subarray}{c}\ell\in\bigcup_{k=1}^{n}\Pi_{k}^{N},\\\
\hat{\mathcal{O}}(\ell\bigcup\gamma)=\hat{\mathcal{O}}(\gamma)+m\end{subarray}}~{}N^{3m-4\hat{\mathcal{O}}(\ell)}~{}\mathbb{E}\Bigg{[}\sum_{\begin{subarray}{c}\alpha_{1},\dots,\alpha_{|\ell|}\in\bigcup_{k=1}^{n}\Pi_{k}^{N},\\\
\bigcup_{i=1}^{|\ell|}\alpha_{i}\simeq\gamma\end{subarray}}~{}\prod_{i=1}^{|\ell|}\Big{|}X_{T,\alpha_{i,1},\dots,\alpha_{i,|\alpha_{i}|}}^{t,x_{\ell_{i}},\ell_{i},N}\Big{|}^{4}~{}\Bigg{]}.}$

∎

######  Lemma 5.3.

Let $u$ satisfy the Kolmogorov backward equation (3.4) with $g$ as in
Assumption H.2, let $T\geq t\geq 0,N\in\mathbb{N}$ and assume that the
starting points $x_{i}$ are $\mathcal{F}_{t}$-measurable random variables in
$L^{2}(\Omega,\mathbb{R})$ sampled from the same distribution for all
$i\in\\{1,\ldots,N\\}$. Then there exist positive constants $K$,
$\lambda_{0}\in(0,\lambda)$ (both are independent of $t,T,N$) such that for
any $1\leq n\leq 6$, $\gamma\in\Pi_{n}^{N}$

$\displaystyle\mathbb{E}\Big{[}\big{|}\partial^{n}_{x_{\gamma_{1}},\dots,x_{\gamma_{n}}}u(t,\bm{x})\big{|}^{2}\Big{]}$
$\displaystyle\leq Ke^{-\lambda_{0}(T-t)}N^{-2\hat{\mathcal{O}}(\gamma)},$
(5.16)
$\displaystyle\mathbb{E}\Big{[}\big{|}\partial^{n}_{x_{\gamma_{1}},\dots,x_{\gamma_{n}}}u(t,\bm{x})\big{|}^{4}\Big{]}$
$\displaystyle\leq Ke^{-\lambda_{0}(T-t)}N^{-4\hat{\mathcal{O}}(\gamma)}.$
(5.17)

###### Proof.

Note that in the following proof, the positive constants $K,\lambda_{0}$ are
independent of $t,T,N$ and may change line by line. Recall the results and
notations in Lemma 5.2, after taking the expectation, we have

$\displaystyle\mathbb{E}\Big{[}\big{|}\partial$
${}_{x_{\gamma_{1}},\dots,x_{\gamma_{n}}}^{n}u(t,\bm{x})\big{|}^{2}\Big{]}$
$\displaystyle\leq
K\sum_{m=0}^{n}~{}~{}\sum_{\begin{subarray}{c}\ell\in\bigcup_{k=1}^{n}\Pi_{k}^{N},\\\
\hat{\mathcal{O}}(\ell\bigcup\gamma)=\hat{\mathcal{O}}(\gamma)+m\end{subarray}}~{}N^{m-2\hat{\mathcal{O}}(\ell)}~{}\sum_{\begin{subarray}{c}\alpha_{1},\dots,\alpha_{|\ell|}\in\bigcup_{k=1}^{n}\Pi_{k}^{N},\\\
\bigcup_{i=1}^{|\ell|}\alpha_{i}\simeq\gamma\end{subarray}}~{}\mathbb{E}\Big{[}~{}\prod_{i=1}^{|\ell|}\Big{|}X_{T,\alpha_{i,1},\dots,\alpha_{i,|\alpha_{i}|}}^{t,x_{\ell_{i}},\ell_{i},N}\Big{|}^{2}~{}\Big{]}$
$\displaystyle\leq
K\sum_{m=0}^{n}~{}~{}\sum_{\begin{subarray}{c}\ell\in\bigcup_{k=1}^{n}\Pi_{k}^{N},\\\
\hat{\mathcal{O}}(\ell\bigcup\gamma)=\hat{\mathcal{O}}(\gamma)+m\end{subarray}}~{}N^{m-2\hat{\mathcal{O}}(\ell)}~{}\sum_{\begin{subarray}{c}\alpha_{1},\dots,\alpha_{|\ell|}\in\bigcup_{k=1}^{n}\Pi_{k}^{N},\\\
\bigcup_{i=1}^{|\ell|}\alpha_{i}\simeq\gamma\end{subarray}}\prod_{i=1}^{|\ell|}\Big{(}\mathbb{E}\Big{[}~{}\Big{|}X_{T,\alpha_{i,1},\dots,\alpha_{i,|\alpha_{i}|}}^{t,x_{\ell_{i}},\ell_{i},N}\Big{|}^{2|\ell|}~{}\Big{]}\Big{)}^{1/|\ell|},$

where we employed Hölder’s inequality. We have the following estimate for the
variation processes in the above product:

$\displaystyle\Big{(}\mathbb{E}\Big{[}~{}\Big{|}X_{T,\alpha_{i,1},\dots,\alpha_{i,|\alpha_{i}|}}^{t,x_{\ell_{i}},\ell_{i},N}\Big{|}^{2|\ell|}~{}\Big{]}\Big{)}^{1/|\ell|}$
$\displaystyle\leq\bigg{(}\frac{1}{N^{\hat{\mathcal{O}}((\ell_{i})\bigcup\alpha_{i})}}\sum_{\begin{subarray}{c}\beta\in\Pi_{|\alpha_{i}|+1}^{N},\\\
~{}\hat{\mathcal{O}}(\beta)=\hat{\mathcal{O}}((\ell_{i})\bigcup\alpha_{i})\end{subarray}}\mathbb{E}\Big{[}~{}\Big{|}X^{t,x_{\beta_{1}},\beta_{1},N}_{T,x_{\beta_{2}},\dots,x_{\beta_{|\beta|}}}\Big{|}^{2|\ell|}\Big{]}\bigg{)}^{1/|\ell|}$
$\displaystyle\leq\Big{(}\frac{K}{N^{2|\ell|(\hat{\mathcal{O}}((\ell_{i})\bigcup\alpha_{i})-1)}}e^{-\lambda_{0}2|\ell|(T-t)}\Big{)}^{1/|\ell|}$
$\displaystyle\leq
Ke^{-2\lambda_{0}(T-t)}N^{-2(\hat{\mathcal{O}}((\ell_{i})\bigcup\alpha_{i})-1)},$

where we used the second and first part of Lemma 4.5 with $p=2|\ell|$,
$m=\hat{\mathcal{O}}((\ell_{i})\bigcup\alpha_{i})$ to obtain the first two
inequalities. We now relate the orders
$(\hat{\mathcal{O}}((\ell_{i})\bigcup\alpha_{i})-1)$ from the previous
estimate and $\hat{\mathcal{O}}(\gamma)$ appearing in (5.16), by first showing
that
$\hat{\mathcal{O}}(\gamma\setminus\ell)\leq\sum_{i=1}^{|\ell|}\Big{(}\hat{\mathcal{O}}((\ell_{i})\bigcup\alpha_{i})-1\Big{)}$.
Concretely, the constraint in the second summation,
$\bigcup_{i=1}^{|\ell|}\alpha_{i}\simeq\gamma$, implies

$\displaystyle\sum_{i=1}^{|\ell|}\Big{(}\hat{\mathcal{O}}((\ell_{i})\bigcup\alpha_{i})-1\Big{)}$
$\displaystyle=\sum_{i=1}^{|\ell|}\bigg{(}\hat{\mathcal{O}}\Big{(}(\ell_{i})\bigcup\Big{(}\alpha_{i}\bigcap(\gamma\setminus\ell)\Big{)}\bigcup\Big{(}\alpha_{i}\setminus(\gamma\setminus\ell)\Big{)}\Big{)}-1\bigg{)}$
$\displaystyle=\sum_{i=1}^{|\ell|}\bigg{(}\hat{\mathcal{O}}\Big{(}\alpha_{i}\bigcap(\gamma\setminus\ell)\Big{)}+\hat{\mathcal{O}}\Big{(}(\ell_{i})\bigcup\Big{(}\alpha_{i}\setminus(\gamma\setminus\ell)\Big{)}\Big{)}-1\bigg{)}$
$\displaystyle\geq\sum_{i=1}^{|\ell|}\hat{\mathcal{O}}\Big{(}\alpha_{i}\bigcap(\gamma\setminus\ell)\Big{)}\geq\hat{\mathcal{O}}\Big{(}(\bigcup_{i=1}^{|\ell|}\alpha_{i})\bigcap(\gamma\setminus\ell)\Big{)}=\hat{\mathcal{O}}(\gamma\setminus\ell).$

Then using the constraint
$\hat{\mathcal{O}}(\ell\bigcup\gamma)=\hat{\mathcal{O}}(\gamma)+m$ of the
first summation we infer

$\sum_{i=1}^{|\ell|}\Big{(}\hat{\mathcal{O}}((\ell_{i})\bigcup\alpha_{i})-1\Big{)}\geq\hat{\mathcal{O}}(\gamma\setminus\ell)=\hat{\mathcal{O}}(\ell\bigcup\gamma)-\hat{\mathcal{O}}(\ell)=\hat{\mathcal{O}}(\gamma)+m-\hat{\mathcal{O}}(\ell).$

Hence,

$\displaystyle\mathbb{E}\Big{[}|\partial$
${}_{x_{\gamma_{1}},\dots,x_{\gamma_{n}}}^{n}u(t,\bm{x})|^{2}\Big{]}$
$\displaystyle\leq
Ke^{-2\lambda_{0}(T-t)}\sum_{m=0}^{n}~{}~{}\sum_{\begin{subarray}{c}\ell\in\bigcup_{k=1}^{n}\Pi_{k}^{N},\\\
\hat{\mathcal{O}}(\ell\bigcup\gamma)=\hat{\mathcal{O}}(\gamma)+m\end{subarray}}~{}N^{m-2\hat{\mathcal{O}}(\ell)}~{}N^{-2\sum_{i=1}^{|\ell|}\big{(}\hat{\mathcal{O}}((\ell_{i})\bigcup\alpha_{i})-1\big{)}}$
$\displaystyle\leq
Ke^{-2\lambda_{0}(T-t)}\sum_{m=0}^{n}N^{2m-2\hat{\mathcal{O}}(\ell)-2\big{(}\hat{\mathcal{O}}(\gamma)-\hat{\mathcal{O}}(\ell)+m\big{)}}\leq
Ke^{-2\lambda_{0}(T-t)}N^{-2\hat{\mathcal{O}}(\gamma)},$

which yields (5.16), as sought. Similar calculations deliver (5.17). That is,
we have

$\displaystyle\mathbb{E}\Big{[}|\partial$
${}_{x_{\gamma_{1}},\dots,x_{\gamma_{n}}}^{n}u(t,\bm{x})|^{4}\Big{]}$
$\displaystyle\leq
K\sum_{m=0}^{n}~{}~{}\sum_{\begin{subarray}{c}\ell\in\bigcup_{k=1}^{n}\Pi_{k}^{N},\\\
\hat{\mathcal{O}}(\ell\bigcup\gamma)=\hat{\mathcal{O}}(\gamma)+m\end{subarray}}~{}N^{3m-4\hat{\mathcal{O}}(\ell)}~{}\sum_{\begin{subarray}{c}\alpha_{1},\dots,\alpha_{|\ell|}\in\bigcup_{k=1}^{n}\Pi_{k}^{N},\\\
\bigcup_{i=1}^{|\ell|}\alpha_{i}\simeq\gamma\end{subarray}}\prod_{i=1}^{|\ell|}\Big{(}\mathbb{E}\Big{[}~{}\Big{|}X_{T,\alpha_{i,1},\dots,\alpha_{i,|\alpha_{i}|}}^{t,x_{\ell_{i}},\ell_{i},N}\Big{|}^{4|\ell|}~{}\Big{]}\Big{)}^{1/|\ell|}$
$\displaystyle\leq
Ke^{-2\lambda_{0}(T-t)}\sum_{m=0}^{n}~{}~{}\sum_{\begin{subarray}{c}\ell\in\bigcup_{k=1}^{n}\Pi_{k}^{N},\\\
\hat{\mathcal{O}}(\ell\bigcup\gamma)=\hat{\mathcal{O}}(\gamma)+m\end{subarray}}~{}N^{3m-4\hat{\mathcal{O}}(\ell)}~{}N^{-4\sum_{i=1}^{|\ell|}\big{(}\hat{\mathcal{O}}((\ell_{i})\bigcup\alpha_{i})-1\big{)}}$
$\displaystyle\leq
Ke^{-2\lambda_{0}(T-t)}\sum_{m=0}^{n}N^{4m-4\hat{\mathcal{O}}(\ell)-4\big{(}\hat{\mathcal{O}}(\gamma)-\hat{\mathcal{O}}(\ell)+m\big{)}}\leq
Ke^{-2\lambda_{0}(T-t)}N^{-4\hat{\mathcal{O}}(\gamma)}.$

∎

##  6 Weak error expansion and its analysis

For the convenience of the reader, we recall Lemma 3.2 which provides an
expansion for the global weak error (3.1) under Assumption H.2, for the
processes defined in (2.4) and (2.6) as follows:

$\displaystyle\mathbb{E}\Big{[}g(\bm{X}^{N}_{T})\Big{]}-\mathbb{E}\Big{[}g(\bm{X}^{N,h}_{T})\Big{]}$
$\displaystyle=h^{2}\mathbb{E}\left[\sum_{m=0}^{M-1}L(t_{m},\bm{X}^{N,h}_{t_{m}})\right]+\mathbb{E}\left[\sum_{m=0}^{M-1}R(t_{m},\bm{X}^{N,h}_{t_{m}})\right],$
(6.1)

where the map $L:\mathbb{R}_{+}\times\mathbb{R}^{N}\rightarrow\mathbb{R}$ is
defined via the maps $u$ and $(B_{i})_{i\in\\{1,\ldots,N\\}}$:

$\displaystyle L(t,\bm{x})$
$\displaystyle=\frac{1}{2}\Big{[}\sum_{i,j=1}^{N}B_{j}(\bm{x})\partial_{x_{j}}B_{i}(\bm{x})\partial_{x_{i}}u(t,\bm{x})+\frac{\sigma^{2}}{2}\sum_{i,j=1}^{N}\partial_{x_{j}}B_{i}(\bm{x})\partial_{x_{i},x_{j}}^{2}u(t,\bm{x})$
$\displaystyle\quad\quad\quad+\frac{\sigma^{2}}{2}\sum_{i,j=1}^{N}\partial^{2}_{x_{j},x_{j}}B_{i}(\bm{x})\partial_{x_{i}}u(t,\bm{x})\Big{]}.$
(6.2)

The remainder term $R(\cdot,\cdot)$ will later be written as a linear
combination of 8 remainder terms, which we will analyze in Section 6.2.

###  6.1 Estimates for the leading error term $L$

We consider the first term in (6.1) expressed as a telescoping sum:

$\displaystyle
h^{2}\mathbb{E}\Big{[}\sum_{m=0}^{M-1}L(t_{m},\bm{X}_{t_{m}}^{N,h})\Big{]}$
$\displaystyle=h\mathbb{E}\Big{[}\int_{0}^{T}L(s,\bm{X}_{s}^{N})\mathrm{d}s\Big{]}$
(6.3)
$\displaystyle\qquad+h\sum_{m=0}^{M-1}\mathbb{E}\Big{[}\int_{t_{m}}^{t_{m+1}}\Big{(}L(t_{m},\bm{X}_{t_{m}}^{N,h})-L(s,\bm{X}_{s}^{N})\Big{)}\mathrm{d}s\Big{]},$
(6.4)

and derive the following result.

######  Lemma 6.1.

Let Assumption H.2 hold and let $\xi\in L^{4}(\Omega,\mathbb{R})$. Let $L$ be
defined in (6.2). Then there exists a positive constant
$\lambda_{0}\in(0,\lambda)$ such that

$\displaystyle h^{2}\mathbb{E}\Big{[}$
$\displaystyle\sum_{m=0}^{M-1}L(t_{m},\bm{X}_{t_{m}}^{N,h})\Big{]}\leq
Kh^{3/2}+Khe^{-\lambda_{0}T}.$ (6.5)

###### Proof.

Note that in the following proof, the positive constants $K,\lambda_{0}$ are
independent of $t,T,N$ and may change line by line. The proof is carried out
by analysing (6.3) and (6.4) separately to reach (6.5).

Part 1: Estimating (6.3). Let
$(X_{*}^{1,N},\ldots,X_{*}^{N,N})=\bm{X}_{*}^{N}\sim\mu^{N,*}$, where
$\mu^{N,*}$ is the stationary distribution of the IPS (viewed as a
$\mathbb{R}^{N}$-valued SDE). Using integration by parts, we have

$\displaystyle\mathbb{E}\Big{[}L(t,\bm{X}_{*}^{N})\Big{]}=\int_{\mathbb{R}^{N}}L(t,\bm{x})\mu^{N,*}(\mathrm{d}\bm{x})=0.$

Hence, we may write

$\displaystyle\mathbb{E}\Big{[}\int_{0}^{T}L(s,\bm{X}_{s}^{N})\mathrm{d}s\Big{]}$
$\displaystyle=\int_{0}^{T}\mathbb{E}\Big{[}L(s,\bm{X}_{s}^{N})-L(s,\bm{X}_{*}^{N})\Big{]}\mathrm{d}s$
$\displaystyle=\int_{0}^{T}\int_{0}^{1}\mathbb{E}\Big{[}\big{\langle}\partial_{\bm{x}}L(s,\rho\bm{X}_{s}^{N}+(1-\rho)\bm{X}_{*}^{N},~{}\bm{X}_{s}^{N}-\bm{X}_{*}^{N}\big{\rangle}\Big{]}\mathrm{d}\rho~{}\mathrm{d}s.$

Let $\bm{X}_{s,\rho}^{N}\coloneqq\rho\bm{X}_{s}^{N}+(1-\rho)\bm{X}_{*}^{N}$.
Using the chain rule, we deduce the following:

$\displaystyle\mathbb{E}\Big{[}$
$\displaystyle\Big{\langle}\partial_{\bm{x}}L(s,\bm{X}_{s,\rho}^{N}),~{}\bm{X}_{s}^{N}-\bm{X}_{*}^{N}\Big{\rangle}\Big{]}=\frac{1}{2}\mathbb{E}\bigg{[}\sum_{i,j,k=1}^{N}\Big{(}\partial_{x_{k}}B_{j}(\bm{X}_{s,\rho}^{N})\partial_{x_{j}}B_{i}(\bm{X}_{s,\rho}^{N})\partial_{x_{i}}u(s,\bm{X}_{s,\rho}^{N})$
$\displaystyle\quad\quad+B_{j}(\bm{X}_{s,\rho}^{N})\partial^{2}_{x_{j},x_{k}}B_{i}(\bm{X}_{s,\rho}^{N})\partial_{x_{i}}u(s,\bm{X}_{s,\rho}^{N})+B_{j}(\bm{X}_{s,\rho}^{N})\partial_{x_{j}}B_{i}(\bm{X}_{s,\rho}^{N})\partial_{x_{i},x_{k}}^{2}u(s,\bm{X}_{s,\rho}^{N})\Big{)}\cdot\Big{(}X_{s}^{k,N}-X_{*}^{k,N}\Big{)}$
$\displaystyle\quad\quad+\frac{\sigma^{2}}{2}\sum_{i,j,k=1}^{N}\Big{(}\partial^{2}_{x_{j},x_{k}}B_{i}(\bm{X}_{s,\rho}^{N})\partial^{2}_{x_{i},x_{j}}u({s},\bm{X}_{s,\rho}^{N})+\partial_{x_{j}}B_{i}(\bm{X}_{s,\rho}^{N})\partial^{3}_{x_{i},x_{j},x_{k}}u({s},\bm{X}_{s,\rho}^{N})\Big{)}\cdot\Big{(}X_{s}^{k,N}-X_{*}^{k,N}\Big{)}$
$\displaystyle\quad\quad+\frac{\sigma^{2}}{2}\sum_{i,j,k=1}^{N}\Big{(}\partial^{3}_{x_{j},x_{j},x_{k}}B_{i}(\bm{X}_{s,\rho}^{N})\partial_{x_{i}}u({s},\bm{X}_{s,\rho}^{N})+\partial^{2}_{x_{j},x_{j}}B_{i}(\bm{X}_{s,\rho}^{N})\partial^{2}_{x_{i},x_{k}}u({s},\bm{X}_{s,\rho}^{N})\Big{)}\cdot\Big{(}X_{s}^{k,N}-X_{*}^{k,N}\Big{)}\bigg{]}$
$\displaystyle\leq
K\sum_{i,j,k=1}^{N}\sqrt{\mathbb{E}\Big{[}\big{|}X_{s}^{k,N}-X_{*}^{k,N}\big{|}^{2}\Big{]}}\cdot\bigg{\\{}\sqrt{\mathbb{E}\Big{[}\big{|}\partial_{x_{k}}B_{j}(\bm{X}_{s,\rho}^{N})\partial_{x_{j}}B_{i}(\bm{X}_{s,\rho}^{N})\partial_{x_{i}}u(s,\bm{X}_{s,\rho}^{N})\big{|}^{2}\Big{]}}$
$\displaystyle\quad\quad+\sqrt{\mathbb{E}\Big{[}\big{|}B_{j}(\bm{X}_{s,\rho}^{N})\partial^{2}_{x_{j},x_{k}}B_{i}(\bm{X}_{s,\rho}^{N})\partial_{x_{i}}u(s,\bm{X}_{s,\rho}^{N})\big{|}^{2}\Big{]}}+\sqrt{\mathbb{E}\Big{[}\big{|}B_{j}(\bm{X}_{s,\rho}^{N})\partial_{x_{j}}B_{i}(\bm{X}_{s,\rho}^{N})\partial^{2}_{x_{i},x_{k}}u(s,\bm{X}_{s,\rho}^{N})\big{|}^{2}\Big{]}}$
$\displaystyle\quad\quad+\sqrt{\mathbb{E}\Big{[}\big{|}\partial^{2}_{x_{j},x_{k}}B_{i}(\bm{X}_{s,\rho}^{N})\partial_{x_{i},x_{j}}^{2}u({s},\bm{X}_{s,\rho}^{N})\big{|}^{2}\Big{]}}+\sqrt{\mathbb{E}\Big{[}\big{|}\partial_{x_{j}}B_{i}(\bm{X}_{s,\rho}^{N})\partial_{x_{i},x_{j},x_{k}}^{3}u({s},\bm{X}_{s,\rho}^{N})\big{|}^{2}\Big{]}}$
$\displaystyle\quad\quad+\sqrt{\mathbb{E}\Big{[}\big{|}\partial^{3}_{x_{j},x_{j},x_{k}}B_{i}(\bm{X}_{s,\rho}^{N})\partial_{x_{i}}u({s},\bm{X}_{s,\rho}^{N})\big{|}^{2}\Big{]}}+\sqrt{\mathbb{E}\Big{[}\big{|}\partial^{2}_{x_{j},x_{j}}B_{i}(\bm{X}_{s,\rho}^{N})\partial_{x_{i},x_{k}}^{2}u({s},\bm{X}_{s,\rho}^{N})\big{|}^{2}\Big{]}}~{}\bigg{\\}},$
(6.6)

where we used the Cauchy–Schwarz inequality. We now work through (6.6). As for
the $L^{2}$-distance to the invariant distribution (the first expectation in
the sum), note that (A.11) in Lemma A.4 implies that for any $s\geq 0$ we have
$\mathbb{E}[|X_{s}^{k,N}-X_{*}^{k,N}|^{2}]\leq Ke^{-2\lambda s}$. The approach
to deal with the remaining seven terms is more or less identical. We inject
the estimate (5.16) of Lemma 5.3 for the derivatives of $u$ and the bounds for
the derivatives of $B$ established in (4.75). The inequality below preserves
the exact ordering of (6.6), and we highlight that obtaining (6.7) requires
the additional use of the linear growth of $B_{j}$, the Cauchy–Schwarz
inequality, the $L^{4}$-estimates of $\bm{X}^{N}$ (in Proposition 2.1) and the
$L^{4}$-estimate (5.17) for the derivatives of $u$ (recalling that $\xi\in
L^{4}(\Omega,\mathbb{R})$); for some positive constant
$\lambda_{0}\in(0,\lambda)$ chosen small enough we have

$\displaystyle\mathbb{E}\Big{[}$
$\displaystyle\Big{\langle}\partial_{\bm{x}}L(s,\bm{X}_{s,\rho}^{N}),~{}\bm{X}_{s}^{N}-\bm{X}_{*}^{N}\Big{\rangle}\Big{]}$
$\displaystyle\leq K\sqrt{e^{-2\lambda
s}}\cdot\sqrt{e^{-\lambda_{0}(T-s)}}\sum_{\gamma\in\Pi^{N}_{3}}\Big{(}\frac{1}{N^{(\hat{\mathcal{O}}((\gamma_{2},\gamma_{3}))-1)+(\hat{\mathcal{O}}((\gamma_{1},\gamma_{2}))-1)+1}}$
$\displaystyle\hskip
142.26378pt+\frac{1}{N^{(\hat{\mathcal{O}}(\gamma)-1)+1}}+\frac{1}{N^{(\hat{\mathcal{O}}((\gamma_{1},\gamma_{2}))-1)+\hat{\mathcal{O}}((\gamma_{1},\gamma_{3}))}}$
(6.7) $\displaystyle\hskip
142.26378pt+\frac{1}{N^{(\hat{\mathcal{O}}(\gamma)-1)+\hat{\mathcal{O}}((\gamma_{1},\gamma_{2}))}}+\frac{1}{N^{(\hat{\mathcal{O}}((\gamma_{1},\gamma_{2}))-1)+\hat{\mathcal{O}}(\gamma)}}$
$\displaystyle\hskip
142.26378pt+\frac{1}{N^{(\hat{\mathcal{O}}(\gamma)-1)+1}}+\frac{1}{N^{(\hat{\mathcal{O}}((\gamma_{1},\gamma_{2}))-1)+\hat{\mathcal{O}}((\gamma_{1},\gamma_{3}))}}\Big{)}$
$\displaystyle\leq K\sqrt{e^{-2\lambda
s}e^{-\lambda_{0}(T-s)}}\sum_{\gamma\in\Pi^{N}_{3}}\frac{1}{N^{\hat{\mathcal{O}}(\gamma)}}$
(6.8) $\displaystyle\leq K\sqrt{e^{-2\lambda s}e^{-\lambda_{0}(T-s)}},$ (6.9)

where the inequality in (6.8) follows from the fact that
$\hat{\mathcal{O}}((\gamma_{1},\gamma_{2}))+\hat{\mathcal{O}}((\gamma_{1},\gamma_{3}))-1\geq\hat{\mathcal{O}}(\gamma)$
for any $\gamma\in\Pi_{3}^{N}$ (seen by checking the cases). The final result
(6.9) follows from recalling (5.2), in turn implying that the summation term
in (6.8) is indeed $\mathcal{O}(1)$.

To conclude this first part of the proof, we gather our estimates and obtain

$\displaystyle\mathbb{E}\Big{[}\int_{0}^{T}L(s,\bm{X}_{s}^{N})\mathrm{d}s\Big{]}\leq
K\int_{0}^{T}\int_{0}^{1}\sqrt{e^{-\lambda_{0}(T-s)}e^{-2\lambda
s}}~{}\mathrm{d}\rho\mathrm{d}s=Ke^{-\frac{\lambda_{0}}{2}T}\int_{0}^{T}e^{\frac{\lambda_{0}-2\lambda}{2}s}\mathrm{d}s\leq
Ke^{-\frac{\lambda_{0}}{2}T}.$

Part 2: Estimating (6.4). For the term (6.4), we have

$\displaystyle\sum_{m=0}^{M-1}\mathbb{E}\Big{[}\int_{t_{m}}^{t_{m+1}}\Big{(}L(t_{m},\bm{X}_{t_{m}}^{N,h})-L(s,\bm{X}_{s}^{N})\Big{)}\mathrm{d}s\Big{]}$
$\displaystyle=\sum_{m=0}^{M-1}\int_{t_{m}}^{t_{m+1}}\mathbb{E}\Big{[}L(t_{m},\bm{X}_{t_{m}}^{N,h})-L(t_{m},\bm{X}_{t_{m}}^{N})\Big{]}\mathrm{d}s$
(6.10)
$\displaystyle\quad+\sum_{m=0}^{M-1}\int_{t_{m}}^{t_{m+1}}\mathbb{E}\Big{[}L(t_{m},\bm{X}_{t_{m}}^{N})-L(s,\bm{X}_{t_{m}}^{N})\Big{]}\mathrm{d}s$
(6.11)
$\displaystyle\quad+\sum_{m=0}^{M-1}\int_{t_{m}}^{t_{m+1}}\mathbb{E}\Big{[}L(s,\bm{X}_{t_{m}}^{N})-L(s,\bm{X}_{s}^{N})\Big{]}\mathrm{d}s.$
(6.12)

Part 2.1: Estimating (6.10) and (6.12). For (6.10), similar to the
calculations (6.6)–(6.9) (in the previous part of the proof), we derive

$\displaystyle\mathbb{E}\Big{[}$ $\displaystyle
L(t_{m},\bm{X}_{t_{m}}^{N,h})-L(t_{m},\bm{X}_{t_{m}}^{N})\Big{]}$
$\displaystyle=\int_{0}^{1}\mathbb{E}\Big{[}\Big{\langle}\partial_{\bm{x}}L\big{(}t_{m},\rho\bm{X}_{t_{m}}^{N,h}+(1-\rho)\bm{X}_{t_{m}}^{N}\big{)},~{}\bm{X}_{t_{m}}^{N,h}-\bm{X}_{t_{m}}^{N}\Big{\rangle}\Big{]}\mathrm{d}\rho$
$\displaystyle\leq\frac{K}{N}\sum_{i=1}^{N}~{}\sqrt{e^{-\lambda_{0}(T-t_{m})}\mathbb{E}\Big{[}\big{|}X_{t_{m}}^{i,N,h}-X_{t_{m}}^{i,N}\big{|}^{2}\Big{]}~{}}\leq
Kh^{1/2}e^{-\lambda_{0}(T-t_{m+1})/2},$ (6.13)

where we used Proposition 2.2 for the strong error rate. Similarly, for
(6.12), we have for $s\in[t_{m},t_{m+1}]$,

$\displaystyle\mathbb{E}\Big{[}$ $\displaystyle
L(s,\bm{X}_{t_{m}}^{N})-L(s,\bm{X}_{s}^{N})\Big{]}=\int_{0}^{1}\mathbb{E}\Big{[}\Big{\langle}\partial_{\bm{x}}L\big{(}s,\rho\bm{X}_{t_{m}}^{N}+(1-\rho)\bm{X}_{s}^{N}\big{)},~{}\bm{X}_{s}^{N}-\bm{X}_{t_{m}}^{N}\Big{\rangle}\Big{]}\mathrm{d}\rho$
$\displaystyle\leq\frac{K}{N}\sum_{i=1}^{N}~{}\sqrt{e^{-\lambda_{0}(T-s)}\mathbb{E}\Big{[}\big{|}X_{s}^{i,N}-X_{t_{m}}^{i,N}\big{|}^{2}\Big{]}~{}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\leq
K\sqrt{s-t_{m}}e^{-\lambda_{0}(T-s)/2}\leq
Kh^{1/2}e^{-\lambda_{0}(T-t_{m+1})/2},}$ (6.14)

where we used Proposition A.1 and that $h\geq s-t_{m}$.

Part 2.2: Estimating (6.11). For the term (6.11), we get for all $m$ and
$s\in[t_{m},t_{m+1}]$

$\displaystyle\mathbb{E}\Big{[}L(t_{m},$
$\displaystyle\bm{X}_{t_{m}}^{N})-L(s,\bm{X}_{t_{m}}^{N})\Big{]}$
$\displaystyle=\frac{1}{2}\mathbb{E}\bigg{[}~{}\sum_{i,j=1}^{N}\bigg{(}B_{j}(\bm{X}_{t_{m}}^{N})\partial_{x_{j}}B_{i}(\bm{X}_{t_{m}}^{N})\Big{(}\partial_{x_{i}}u(t_{m},\bm{X}_{t_{m}}^{N})-\partial_{x_{i}}u(s,\bm{X}_{t_{m}}^{N})\Big{)}$
$\displaystyle\qquad+\frac{\sigma^{2}}{2}\partial_{x_{j}}B_{i}(\bm{X}_{t_{m}}^{N})\Big{(}\partial_{x_{i},x_{j}}^{2}u(t_{m},\bm{X}_{t_{m}}^{N})-\partial^{2}_{x_{i},x_{j}}u(s,\bm{X}_{t_{m}}^{N})\Big{)}$
$\displaystyle\qquad+\frac{\sigma^{2}}{2}\partial^{2}_{x_{j},x_{j}}B_{i}(\bm{X}_{t_{m}}^{N})\Big{(}\partial_{x_{i}}u(t_{m},\bm{X}_{t_{m}}^{N})-\partial_{x_{i}}u(s,\bm{X}_{t_{m}}^{N})\Big{)}\bigg{]}$
$\displaystyle\leq
K\sum_{i,j=1}^{N}\sqrt{\mathbb{E}\Big{[}\Big{|}B_{j}(\bm{X}_{t_{m}}^{N})\partial_{x_{j}}B_{i}(\bm{X}_{t_{m}}^{N})\Big{|}^{2}\Big{]}\mathbb{E}\Big{[}\Big{|}\partial_{x_{i}}u(t_{m},\bm{X}_{t_{m}}^{N})-\partial_{x_{i}}u(s,\bm{X}_{t_{m}}^{N})\Big{|}^{2}\Big{]}}$
(6.15)
$\displaystyle\qquad+K\sum_{i,j=1}^{N}\sqrt{\mathbb{E}\Big{[}\Big{|}\partial_{x_{j}}B_{i}(\bm{X}_{t_{m}}^{N})\Big{|}^{2}\Big{]}\mathbb{E}\Big{[}\Big{|}\partial_{x_{i},x_{j}}^{2}u(t_{m},\bm{X}_{t_{m}}^{N})-\partial^{2}_{x_{i},x_{j}}u(s,\bm{X}_{t_{m}}^{N})\Big{|}^{2}\Big{]}}$
(6.16)
$\displaystyle\qquad+K\sum_{i,j=1}^{N}\sqrt{\mathbb{E}\Big{[}\Big{|}\partial^{2}_{x_{j},x_{j}}B_{i}(\bm{X}_{t_{m}}^{N})\Big{|}^{2}\Big{]}\mathbb{E}\Big{[}\Big{|}\partial_{x_{i}}u(t_{m},\bm{X}_{t_{m}}^{N})-\partial_{x_{i}}u(s,\bm{X}_{t_{m}}^{N})\Big{|}^{2}\Big{]}}.$
(6.17)

We first study the differences of first order derivatives of $u$ in (6.15) and
(6.17), and then study the difference of second order derivatives of $u$
(6.16). Collecting these estimates and using the bounds on the derivatives of
$B$ yields the upper bound on (6.11). Then we will be in a position to
conclude the main result.

Part 2.2.1: The first order derivative terms. We first derive the following
estimate:

$\displaystyle\mathbb{E}\Big{[}$
$\displaystyle\big{|}\partial_{x_{i}}u(t_{m},\bm{X}_{t_{m}}^{N})-\partial_{x_{i}}u(s,\bm{X}_{t_{m}}^{N})\big{|}^{2}\Big{]}$
$\displaystyle=\mathbb{E}\bigg{[}\Big{|}~{}~{}\sum_{j=1}^{N}\mathbb{E}\Big{[}\partial_{x_{j}}g(\bm{X}_{T}^{t_{m},\bm{x},N})\cdot
X_{T,x_{i}}^{t_{m},x_{j},j,N}-\partial_{x_{j}}g(\bm{X}_{T}^{s,\bm{x},N})\cdot
X_{T,x_{i}}^{s,x_{j},j,N}\Big{]}~{}\bigg{|}_{\begin{subarray}{c}\bm{x}=\bm{X}_{t_{m}}^{N}\end{subarray}}~{}\Big{|}^{2}\bigg{]}$
$\displaystyle\leq
K\mathbb{E}\bigg{[}~{}\Big{|}~{}\mathbb{E}\Big{[}~{}\sum_{j=1,j\neq
i}^{N}\Big{(}\partial_{x_{j}}g(\bm{X}_{T}^{t_{m},\bm{x},N})\cdot
X_{T,x_{i}}^{t_{m},x_{j},j,N}-\partial_{x_{j}}g(\bm{X}_{T}^{s,\bm{x},N})\cdot
X_{T,x_{i}}^{s,x_{j},j,N}\Big{)}\Big{]}~{}\bigg{|}_{\begin{subarray}{c}\bm{x}=\bm{X}_{t_{m}}^{N}\end{subarray}}~{}\Big{|}^{2}\bigg{]}$
$\displaystyle\qquad\qquad+K\mathbb{E}\bigg{[}~{}\Big{|}~{}\mathbb{E}\Big{[}\partial_{x_{i}}g(\bm{X}_{T}^{t_{m},\bm{x},N})\cdot
X_{T,x_{i}}^{t_{m},x_{i},i,N}-\partial_{x_{i}}g(\bm{X}_{T}^{s,\bm{x},N})\cdot
X_{T,x_{i}}^{s,x_{i},i,N}\Big{]}~{}\bigg{|}_{\begin{subarray}{c}\bm{x}=\bm{X}_{t_{m}}^{N}\end{subarray}}~{}\Big{|}^{2}\bigg{]}$
$\displaystyle\leq K\mathbb{E}\bigg{[}~{}\Big{|}~{}\sum_{j=1,j\neq
i}^{N}\Big{(}\partial_{x_{j}}g(\bm{X}_{T}^{t_{m},\bm{X}_{t_{m}}^{N},N})\cdot
X_{T,x_{i}}^{t_{m},X_{t_{m}}^{j,N},j,N}-\partial_{x_{j}}g(\bm{X}_{T}^{s,\bm{X}_{t_{m}}^{N},N})\cdot
X_{T,x_{i}}^{s,X_{t_{m}}^{j,N},j,N}\Big{)}~{}\Big{|}^{2}~{}\bigg{]}$
$\displaystyle\qquad\qquad+K\mathbb{E}\bigg{[}~{}\Big{|}~{}\partial_{x_{i}}g(\bm{X}_{T}^{t_{m},\bm{X}_{t_{m}}^{N},N})\cdot
X_{T,x_{i}}^{t_{m},X_{t_{m}}^{i,N},i,N}-\partial_{x_{i}}g(\bm{X}_{T}^{s,\bm{X}_{t_{m}}^{N},N})\cdot
X_{T,x_{i}}^{s,X_{t_{m}}^{i,N},i,N}~{}\Big{|}^{2}~{}\bigg{]}$ (6.18)
$\displaystyle\leq KN\sum_{j=1,j\neq
i}^{N}\mathbb{E}\bigg{[}~{}\Big{|}~{}\Big{(}\partial_{x_{j}}g(\bm{X}_{T}^{t_{m},\bm{X}_{t_{m}}^{N},N})-\partial_{x_{j}}g(\bm{X}_{T}^{s,\bm{X}_{t_{m}}^{N},N})\Big{)}\cdot
X_{T,x_{i}}^{t_{m},X_{t_{m}}^{j,N},j,N}\Big{|}^{2}\bigg{]}$
$\displaystyle\qquad\qquad+KN\sum_{j=1,j\neq
i}^{N}\mathbb{E}\bigg{[}~{}\Big{|}\partial_{x_{j}}g(\bm{X}_{T}^{s,\bm{X}_{t_{m}}^{N},N})\cdot\Big{(}X_{T,x_{i}}^{t_{m},X_{t_{m}}^{j,N},j,N}-X_{T,x_{i}}^{s,X_{t_{m}}^{j,N},j,N}\Big{)}~{}\Big{|}^{2}~{}\bigg{]}$
$\displaystyle\qquad\qquad+K\mathbb{E}\bigg{[}~{}\Big{|}~{}\Big{(}\partial_{x_{i}}g(\bm{X}_{T}^{t_{m},\bm{X}_{t_{m}}^{N},N})-\partial_{x_{i}}g(\bm{X}_{T}^{s,\bm{X}_{t_{m}}^{N},N})\Big{)}\cdot
X_{T,x_{i}}^{t_{m},X_{t_{m}}^{i,N},i,N}~{}\Big{|}^{2}~{}\bigg{]}$
$\displaystyle\qquad\qquad+K\mathbb{E}\bigg{[}~{}\Big{|}~{}\partial_{x_{i}}g(\bm{X}_{T}^{s,\bm{X}_{t_{m}}^{N},N})\cdot\Big{(}X_{T,x_{i}}^{t_{m},X_{t_{m}}^{i,N},i,N}-X_{T,x_{i}}^{s,X_{t_{m}}^{i,N},i,N}\Big{)}~{}\Big{|}^{2}~{}\bigg{]},$
(6.19)

where we employed the Cauchy–Schwarz and Jensen inequalities, as well as the
tower property for conditional expectations to obtain (6.18). Inequality
(6.19) follows from a standard re-arrangement of (6.18).

An application of Hölder’s inequality and Assumption H.2 on the function $g$
further yields

$\displaystyle\eqref{eq-aux-diff of 1st order derives time}$
$\displaystyle\leq\frac{K}{N}\sum_{j=1,j\neq
i}^{N}\sqrt{\mathbb{E}\Big{[}\big{|}X_{T}^{t_{m},X_{t_{m}}^{j,N},j,N}-X_{T}^{s,X_{t_{m}}^{j,N},j,N}\big{|}^{4}\Big{]}~{}\mathbb{E}\Big{[}\big{|}X_{T,x_{i}}^{t_{m},X_{t_{m}}^{j,N},j,N}\big{|}^{4}\Big{]}}$
$\displaystyle\qquad\qquad+\frac{K}{N}\sum_{j=1,j\neq
i}^{N}\mathbb{E}\Big{[}\big{|}X_{T,x_{i}}^{t_{m},X_{t_{m}}^{j,N},j,N}-X_{T,x_{i}}^{s,X_{t_{m}}^{j,N},j,N}\big{|}^{2}\Big{]}$
$\displaystyle\qquad\qquad+\frac{K}{N^{2}}\sqrt{\mathbb{E}\Big{[}\big{|}X_{T}^{t_{m},X_{t_{m}}^{i,N},i,N}-X_{T}^{s,X_{t_{m}}^{i,N},i,N}\big{|}^{4}\Big{]}~{}\mathbb{E}\Big{[}\big{|}X_{T,x_{i}}^{t_{m},X_{t_{m}}^{i,N},i,N}\big{|}^{4}\Big{]}}$
$\displaystyle\qquad\qquad+\frac{K}{N^{2}}\mathbb{E}\Big{[}\big{|}X_{T,x_{i}}^{t_{m},X_{t_{m}}^{i,N},i,N}-X_{T,x_{i}}^{s,X_{t_{m}}^{i,N},i,N}\big{|}^{2}\Big{]}$
$\displaystyle\leq\frac{K}{N^{2}}(s-t_{m})e^{-2\lambda_{2}(T-s)},$ (6.20)

where we used Lemma 4.5 with $n=1$ (note that $\lambda_{0}$ can be replaced by
$\lambda_{1}$ in this case), Lemma A.2 and Proposition 4.3 with
$\lambda_{2}\in(0,\min\\{\lambda-2K_{V},\lambda_{1}\\})$ for the final
inequality.

Part 2.2.2: The second order derivative terms. Similarly, for the differences
of the second order derivatives of $u$ in (6.16), applying the tower property
of the conditional expectation once more we have

$\displaystyle\mathbb{E}$
$\displaystyle\Big{[}\big{|}\partial_{x_{i},x_{j}}^{2}u(t_{m},\bm{X}_{t_{m}}^{N})-\partial_{x_{i},x_{j}}^{2}u(s,\bm{X}_{t_{m}}^{N})\big{|}^{2}\Big{]}$
$\displaystyle=\mathbb{E}\bigg{[}~{}\bigg{|}~{}~{}\sum_{k=1}^{N}\mathbb{E}\Big{[}\partial_{x_{k}}g(\bm{X}_{T}^{t_{m},\bm{x},N})\cdot
X_{T,x_{i},x_{j}}^{t_{m},x_{k},k,N}-\partial_{x_{k}}g(\bm{X}_{T}^{s,\bm{x},N})\cdot
X_{T,x_{i},x_{j}}^{s,x_{k},k,N}\Big{]}$
$\displaystyle\qquad+\sum_{k,k^{\prime}=1}^{N}\mathbb{E}\Big{[}\partial_{x_{k},x_{k^{\prime}}}g(\bm{X}_{T}^{t_{m},\bm{x},N})\cdot
X_{T,x_{i}}^{t_{m},x_{k},k,N}X_{T,x_{j}}^{t_{m},x_{k^{\prime}},k^{\prime},N}$
$\displaystyle\hskip
170.71652pt-\partial_{x_{k},x_{k^{\prime}}}g(\bm{X}_{T}^{s,\bm{x},N})\cdot
X_{T,x_{i}}^{s,x_{k},k,N}X_{T,x_{j}}^{s,x_{k^{\prime}},k^{\prime},N}\Big{]}~{}\bigg{|}_{\begin{subarray}{c}\bm{x}=\bm{X}_{t_{m}}^{N}\end{subarray}}~{}\bigg{|}^{2}\bigg{]}$
$\displaystyle\leq
K\mathbb{E}\Big{[}\Big{|}~{}~{}\sum_{k=1}^{N}\Big{(}\partial_{x_{k}}g(\bm{X}_{T}^{t_{m},\bm{X}_{t_{m}}^{N},N})-\partial_{x_{k}}g(\bm{X}_{T}^{s,\bm{X}_{t_{m}}^{N},N})\Big{)}\cdot
X_{T,x_{i},x_{j}}^{t_{m},X_{t_{m}}^{k,N},k,N}\Big{|}^{2}\Big{]}$ (6.21)
$\displaystyle\qquad+K\mathbb{E}\Big{[}\Big{|}\sum_{k=1}^{N}\Big{(}X_{T,x_{i},x_{j}}^{t_{m},X_{t_{m}}^{k,N},k,N}-X_{T,x_{i},x_{j}}^{s,X_{t_{m}}^{k,N},k,N}\Big{)}\cdot\partial_{x_{k}}g(\bm{X}_{T}^{s,\bm{X}_{t_{m}}^{N},N})\Big{|}^{2}\Big{]}$
(6.22)
$\displaystyle\qquad+K\mathbb{E}\Big{[}\Big{|}~{}~{}\sum_{k,k^{\prime}=1}^{N}\Big{(}\partial_{x_{k},x_{k^{\prime}}}g(\bm{X}_{T}^{t_{m},\bm{X}_{t_{m}}^{N},N})-\partial_{x_{k},x_{k^{\prime}}}g(\bm{X}_{T}^{s,\bm{X}_{t_{m}}^{N},N})\Big{)}\cdot
X_{T,x_{i}}^{t_{m},X_{t_{m}}^{k,N},k,N}X_{T,x_{j}}^{t_{m},X_{t_{m}}^{k^{\prime},N},k^{\prime},N}\Big{|}^{2}\Big{]}$
(6.23)
$\displaystyle\qquad+K\mathbb{E}\Big{[}\Big{|}~{}~{}\sum_{k,k^{\prime}=1}^{N}\Big{(}X_{T,x_{i}}^{t_{m},X_{t_{m}}^{k,N},k,N}-X_{T,x_{i}}^{s,X_{t_{m}}^{k,N},k,N}\Big{)}\cdot\partial_{x_{k},x_{k^{\prime}}}g(\bm{X}_{T}^{s,\bm{X}_{t_{m}}^{N},N})\cdot
X_{T,x_{j}}^{t_{m},X_{t_{m}}^{k^{\prime},N},k^{\prime},N}\Big{|}^{2}\Big{]}$
(6.24)
$\displaystyle\qquad+K\mathbb{E}\Big{[}\Big{|}~{}~{}\sum_{k,k^{\prime}=1}^{N}\Big{(}X_{T,x_{j}}^{t_{m},X_{t_{m}}^{k^{\prime},N},k^{\prime},N}-X_{T,x_{j}}^{s,X_{t_{m}}^{k^{\prime},N},k^{\prime},N}\Big{)}\cdot\partial_{x_{k},x_{k^{\prime}}}g(\bm{X}_{T}^{s,\bm{X}_{t_{m}}^{N},N})\cdot
X_{T,x_{i}}^{s,X_{t_{m}}^{k,N},k,N}\Big{|}^{2}\Big{]}.$ (6.25)

We are required to analyze each of the terms (6.21)–(6.25) separately. By
further applying Jensen’s inequality and Hölder’s inequality, we derive the
following estimates for the first two terms (6.21) and (6.22) based on the
values of $\hat{\mathcal{O}}((i,j,k))$

$\displaystyle\eqref{eq: uxx term1}$ $\displaystyle\leq
K\sqrt{\mathbb{E}\Big{[}\Big{|}~{}\partial_{x_{i}}g(\bm{X}_{T}^{t_{m},\bm{X}_{t_{m}}^{N},N})-\partial_{x_{i}}g(\bm{X}_{T}^{s,\bm{X}_{t_{m}}^{N},N})\Big{|}^{4}\Big{]}\mathbb{E}\Big{[}\Big{|}X_{T,x_{i},x_{j}}^{t_{m},X_{t_{m}}^{i,N},i,N}\Big{|}^{4}~{}\Big{]}}$
$\displaystyle\qquad+K\sqrt{\mathbb{E}\Big{[}\Big{|}~{}\partial_{x_{j}}g(\bm{X}_{T}^{t_{m},\bm{X}_{t_{m}}^{N},N})-\partial_{x_{j}}g(\bm{X}_{T}^{s,\bm{X}_{t_{m}}^{N},N})\Big{|}^{4}\Big{]}\mathbb{E}\Big{[}\Big{|}X_{T,x_{i},x_{j}}^{t_{m},X_{t_{m}}^{j,N},j,N}\Big{|}^{4}~{}\Big{]}}$
$\displaystyle\qquad+NK\sum_{k=1,k\notin\\{i,j\\}}^{N}\sqrt{\mathbb{E}\Big{[}\Big{|}~{}\partial_{x_{k}}g(\bm{X}_{T}^{t_{m},\bm{X}_{t_{m}}^{N},N})-\partial_{x_{k}}g(\bm{X}_{T}^{s,\bm{X}_{t_{m}}^{N},N})\Big{|}^{4}\Big{]}\mathbb{E}\Big{[}\Big{|}X_{T,x_{i},x_{j}}^{t_{m},X_{t_{m}}^{k,N},k,N}\Big{|}^{4}~{}\Big{]}}.$
(6.26)

Now, similar to (6.20), an application of Assumption H.2 for the function $g$
and then injecting the bounds from Lemma 4.5 with $n=2$ (note that
$\lambda_{0}$ can be replaced by $\lambda_{4}$ in this case) and Lemma A.2
yield

$\displaystyle\eqref{eq-aux:wrap-up to eq: uxx term1}$
$\displaystyle\leq\frac{K}{N^{2}}\sqrt{\mathbb{E}\Big{[}\big{|}X_{T}^{t_{m},X_{t_{m}}^{i,N},i,N}-X_{T}^{s,X_{t_{m}}^{i,N},i,N}\big{|}^{4}\Big{]}~{}\mathbb{E}\Big{[}\big{|}X_{T,x_{i},x_{j}}^{t_{m},X_{t_{m}}^{i,N},i,N}\big{|}^{4}\Big{]}}$
$\displaystyle\qquad+\frac{K}{N^{2}}\sqrt{\mathbb{E}\Big{[}\big{|}X_{T}^{t_{m},X_{t_{m}}^{j,N},j,N}-X_{T}^{s,X_{t_{m}}^{j,N},j,N}\big{|}^{4}\Big{]}~{}\mathbb{E}\Big{[}\big{|}X_{T,x_{i},x_{j}}^{t_{m},X_{t_{m}}^{j,N},j,N}\big{|}^{4}\Big{]}}$
$\displaystyle\qquad+\frac{K}{N}\sum_{k=1,k\notin\\{i,j\\}}^{N}\sqrt{\mathbb{E}\Big{[}\big{|}X_{T}^{t_{m},X_{t_{m}}^{k,N},k,N}-X_{T}^{s,X_{t_{m}}^{k,N},k,N}\big{|}^{4}\Big{]}~{}\mathbb{E}\Big{[}\big{|}X_{T,x_{i},x_{j}}^{t_{m},X_{t_{m}}^{k,N},k,N}\big{|}^{4}\Big{]}}$
$\displaystyle\leq
K(s-t_{m})e^{-2\lambda_{2}(T-s)}e^{-2\lambda_{4}(T-t_{m})}\bigg{(}\frac{1}{N^{2}\cdot
N^{2\hat{\mathcal{O}}((i,j))-2}}+\frac{1}{N}\sum_{k=1,k\notin\\{i,j\\}}^{N}\frac{1}{N^{2\hat{\mathcal{O}}((i,j,k))-2}}\bigg{)}.$
(6.27)

Similarly, for (6.22), using Proposition A.3, we have

$\displaystyle\eqref{eq: uxx term2}$ $\displaystyle\leq
K\sqrt{\mathbb{E}\Big{[}\Big{|}\Big{(}X_{T,x_{i},x_{j}}^{t_{m},X_{t_{m}}^{i,N},i,N}-X_{T,x_{i},x_{j}}^{s,X_{t_{m}}^{i,N},i,N}\Big{)}\Big{|}^{4}\Big{]}\mathbb{E}\Big{[}\Big{|}\partial_{x_{i}}g(\bm{X}_{T}^{s,\bm{X}_{t_{m}}^{N},N})\Big{|}^{4}\Big{]}}$
$\displaystyle\quad+K\sqrt{\mathbb{E}\Big{[}\Big{|}\Big{(}X_{T,x_{i},x_{j}}^{t_{m},X_{t_{m}}^{j,N},j,N}-X_{T,x_{i},x_{j}}^{s,X_{t_{m}}^{j,N},j,N}\Big{)}\Big{|}^{4}\Big{]}\mathbb{E}\Big{[}\Big{|}\partial_{x_{j}}g(\bm{X}_{T}^{s,\bm{X}_{t_{m}}^{N},N})\Big{|}^{4}\Big{]}}$
$\displaystyle\quad+NK\sum_{k=1,k\notin\\{i,j\\}}^{N}\sqrt{\mathbb{E}\Big{[}\Big{|}\Big{(}X_{T,x_{i},x_{j}}^{t_{m},X_{t_{m}}^{k,N},k,N}-X_{T,x_{i},x_{j}}^{s,X_{t_{m}}^{k,N},k,N}\Big{)}\Big{|}^{4}\Big{]}\mathbb{E}\Big{[}\Big{|}\partial_{x_{k}}g(\bm{X}_{T}^{s,\bm{X}_{t_{m}}^{N},N})\Big{|}^{4}\Big{]}}$
$\displaystyle\leq
K(s-t_{m})e^{-2\lambda_{4}(T-t_{m})}\bigg{(}\frac{1}{N^{2}\cdot
N^{2\hat{\mathcal{O}}((i,j))-2}}+\frac{1}{N}\sum_{k=1,k\notin\\{i,j\\}}^{N}\frac{1}{N^{2\hat{\mathcal{O}}((i,j,k))-2}}\bigg{)}.$
(6.28)

For (6.23)–(6.25), we will make use of the following bound, which follows from
exploiting the Lipschitz property of $\partial_{x_{k},x_{k}^{\prime}}g$ from
Assumption H.2 and applying Lemma A.2

$\displaystyle\mathbb{E}\Big{[}\Big{|}\partial_{x_{k},x_{k^{\prime}}}g(\bm{X}_{T}^{t_{m},\bm{X}_{t_{m}}^{N},N})-\partial_{x_{k},x_{k^{\prime}}}g(\bm{X}_{T}^{s,\bm{X}_{t_{m}}^{N},N})\Big{|}^{4}\Big{]}\leq\frac{K}{N^{4\hat{\mathcal{O}}((k,k^{\prime}))}}e^{-4\lambda_{2}(T-s)}.$

Hence, similarly to (6.27)–(6.28), partitioning the sum based on values taken
by $\hat{\mathcal{O}}((i,j,k,k^{\prime}))$

$\displaystyle\eqref{eq: uxx term3}$ $\displaystyle\leq
K\bigg{(}\sum_{\begin{subarray}{c}k,k^{\prime}\in\\{1,\dots,N\\},\\\
\hat{\mathcal{O}}((i,j,k,k^{\prime}))=\hat{\mathcal{O}}((i,j))\end{subarray}}+N\sum_{\begin{subarray}{c}k,k^{\prime}\in\\{1,\dots,N\\},\\\
\hat{\mathcal{O}}((i,j,k,k^{\prime}))=\hat{\mathcal{O}}((i,j))+1\end{subarray}}+N^{2}\sum_{\begin{subarray}{c}k,k^{\prime}\in\\{1,\dots,N\\},\\\
\hat{\mathcal{O}}((i,j,k,k^{\prime}))=\hat{\mathcal{O}}((i,j))+2\end{subarray}}\bigg{)}$
$\displaystyle\quad\cdot\sqrt{\mathbb{E}\Big{[}\Big{|}\partial_{x_{k},x_{k^{\prime}}}g(\bm{X}_{T}^{t_{m},\bm{X}_{t_{m}}^{N},N})-\partial_{x_{k},x_{k^{\prime}}}g(\bm{X}_{T}^{s,\bm{X}_{t_{m}}^{N},N})\Big{|}^{4}\Big{]}\mathbb{E}\Big{[}\Big{|}X_{T,x_{i}}^{t_{m},X_{t_{m}}^{k,N},k,N}X_{T,x_{j}}^{t_{m},X_{t_{m}}^{k^{\prime},N},k^{\prime},N}\Big{|}^{4}\Big{]}}$
$\displaystyle\leq K(s-t_{m})e^{-2\lambda_{2}(T-s)}e^{-4\lambda_{1}(T-t_{m})}$
$\displaystyle\qquad\qquad\cdot\Bigg{(}\sum_{\begin{subarray}{c}k,k^{\prime}\in\\{1,\dots,N\\},\\\
\hat{\mathcal{O}}((i,j,k,k^{\prime}))=\hat{\mathcal{O}}((i,j))\end{subarray}}\frac{1}{N^{2\hat{\mathcal{O}}((k,k^{\prime}))}}\frac{1}{N^{2(\hat{\mathcal{O}}((i,k))+\hat{\mathcal{O}}((j,k^{\prime}))-2))}}$
$\displaystyle\qquad\qquad\qquad\qquad+N\sum_{\begin{subarray}{c}k,k^{\prime}\in\\{1,\dots,N\\},\\\
\hat{\mathcal{O}}((i,j,k,k^{\prime}))=\hat{\mathcal{O}}((i,j))+1\end{subarray}}\frac{1}{N^{2\hat{\mathcal{O}}((k,k^{\prime}))}}\frac{1}{N^{2(\hat{\mathcal{O}}((i,k))+\hat{\mathcal{O}}((j,k^{\prime}))-2))}}$
$\displaystyle\qquad\qquad\qquad\qquad+N^{2}\sum_{\begin{subarray}{c}k,k^{\prime}\in\\{1,\dots,N\\},\\\
\hat{\mathcal{O}}((i,j,k,k^{\prime}))=\hat{\mathcal{O}}((i,j))+2\end{subarray}}\frac{1}{N^{2\hat{\mathcal{O}}((k,k^{\prime}))}}\frac{1}{N^{2(\hat{\mathcal{O}}((i,k))+\hat{\mathcal{O}}((j,k^{\prime}))-2))}}\Bigg{)}$
(6.29)
$\displaystyle\leq\frac{K(s-t_{m})e^{-2\lambda_{2}(T-s)}e^{-4\lambda_{1}(T-t_{m})}}{N^{2\hat{\mathcal{O}}((i,j))}}\cdot\sum_{m=0}^{2}N^{m}\sum_{\begin{subarray}{c}k,k^{\prime}\in\\{1,\dots,N\\},\\\
\hat{\mathcal{O}}((i,j,k,k^{\prime}))=\hat{\mathcal{O}}((i,j))+m\end{subarray}}\frac{1}{N^{2m}}$
(6.30)
$\displaystyle\leq\frac{K(s-t_{m})e^{-2\lambda_{2}(T-s)}e^{-4\lambda_{1}(T-t_{m})}}{N^{2\hat{\mathcal{O}}((i,j))}}.$
(6.31)

The term (6.29) is derived by applying the Cauchy–Schwarz inequality, which
then requires the $L^{8}$-moments of $X_{T,x_{i}}^{t_{m},X_{t_{m}}^{k,N},k,N}$
and $X_{T,x_{j}}^{t_{m},X_{t_{m}}^{k^{\prime},N},k^{\prime},N}$ obtained from
Lemma 4.2 (that holds without integrability requirements on
$X_{t_{m}}^{k,N},X_{t_{m}}^{k^{\prime},N}$ since (4.1) is a linear ODE with
bounded coefficients starting from either $1$ or $0$), and noting we can unify
the bounds in Lemma 4.2 as (note that $\lambda_{1}<\lambda$ and
$\hat{\mathcal{O}}((i,k))$ is either 0 or 1)

$\displaystyle\mathbb{E}\big{[}\,|X_{T,x_{i}}^{t_{m},X_{t_{m}}^{k,N},k,N}|^{8}\big{]}\leq\frac{K}{N^{8(\hat{\mathcal{O}}((i,k))-1)}}e^{-8\lambda_{1}(T-t_{m})}.$

To obtain (6.30), we used the following bounds, which can be confirmed by
checking cases

$\displaystyle 2\hat{\mathcal{O}}\big{(}(k,k^{\prime})\big{)}$
$\displaystyle+2\big{(}\hat{\mathcal{O}}((i,k))+\hat{\mathcal{O}}((j,k^{\prime}))-2\big{)}$
$\displaystyle\geq\begin{cases}2+2(1+\hat{\mathcal{O}}((i,j))-2)=2\hat{\mathcal{O}}((i,j)),&\hat{\mathcal{O}}((i,j,k,k^{\prime}))=\hat{\mathcal{O}}((i,j)),\\\
4+2(1+2-2)=6\geq
2+2\hat{\mathcal{O}}((i,j)),&\hat{\mathcal{O}}((i,j,k,k^{\prime}))=\hat{\mathcal{O}}((i,j))+1,~{}k\neq
k^{\prime},\\\ 2+2(2+2-2)=6\geq
2+2\hat{\mathcal{O}}((i,j)),&\hat{\mathcal{O}}((i,j,k,k^{\prime}))=\hat{\mathcal{O}}((i,j))+1,~{}k=k^{\prime},\\\
4+2(2+2-2)=8\geq
4+2\hat{\mathcal{O}}((i,j)),&\hat{\mathcal{O}}((i,j,k,k^{\prime}))=\hat{\mathcal{O}}((i,j))+2.\end{cases}$

In (6.31), we used (5.2) to ensure the summation term in (6.30) is
$\mathcal{O}(1)$. We establish bounds for (6.24) and (6.25) in a similar
fashion and obtain

$\displaystyle\eqref{eq: uxx term4}+\eqref{eq: uxx term5}$
$\displaystyle\leq\frac{K(s-t_{m})e^{-2\lambda_{4}(T-s)}}{N^{2\hat{\mathcal{O}}((i,j))}}.$
(6.32)

Substituting (6.27)–(6.32) into (6.21)–(6.25) yields

$\displaystyle\mathbb{E}\Big{[}\big{|}\partial_{x_{i},x_{j}}^{2}u(t_{m},\bm{X}_{t_{m}}^{N})-\partial_{x_{i},x_{j}}^{2}u(s,\bm{X}_{t_{m}}^{N})\big{|}^{2}\Big{]}\leq~{}\frac{K(s-t_{m})e^{-2\lambda_{4}(T-s)}}{N^{2\hat{\mathcal{O}}((i,j))}}.$
(6.33)

Part 2.3: Collecting the estimates for (6.15)–(6.17). Consequently,
substituting (6.20) and (6.33) into (6.15)–(6.17) and using the bounds for the
derivatives of the function $B$ in (4.74), we conclude that there exists some
constant $\lambda_{0}\in(0,\lambda)$ such that for all $m$ and
$s\in[t_{m},t_{m+1}]$,

$\displaystyle\mathbb{E}\Big{[}$ $\displaystyle
L(t_{m},\bm{X}_{t_{m}}^{N})-L(s,\bm{X}_{t_{m}}^{N})\Big{]}$ $\displaystyle\leq
K\sum_{i,j=1}^{N}\Bigg{(}\frac{\sqrt{s-t_{m}}}{N^{\hat{\mathcal{O}}((i,j))-1}\cdot
N}e^{-\lambda_{2}(T-s)}+\frac{\sqrt{s-t_{m}}}{N^{\hat{\mathcal{O}}((i,j))-1}N^{\hat{\mathcal{O}}((i,j))}}e^{-\lambda_{4}(T-s)}+\frac{\sqrt{s-t_{m}}}{N^{\hat{\mathcal{O}}((i,j))-1}\cdot
N}e^{-\lambda_{2}(T-s)}\Bigg{)}$ $\displaystyle\leq
Kh^{1/2}e^{-\lambda_{0}(T-s)}\sum_{i,j=1}^{N}\frac{1}{N^{\hat{\mathcal{O}}((i,j))}}\leq
Kh^{1/2}e^{-\lambda_{0}(T-t_{m+1})},$ (6.34)

where once again the final inequality arises from recalling (5.2).

Part 3: Concluding the proof. Substituting the results of (6.13), (6.14),
(6.34) to (6.10)–(6.12), we conclude that

$\displaystyle\sum_{m=0}^{M-1}$
$\displaystyle\mathbb{E}\Big{[}\int_{t_{m}}^{t_{m+1}}\Big{(}L(t_{m},\bm{X}_{t_{m}}^{N,h})-L(s,\bm{X}_{s}^{N})\Big{)}\mathrm{d}s\Big{]}\leq
Kh^{3/2}\sum_{m=0}^{M-1}e^{-\lambda_{0}(T-t_{m+1})}\leq Kh^{1/2}.$

Therefore, for the left-hand side of (6.5), there exists some positive
constants $\lambda_{0}\in(0,\lambda),K$ such that

$\displaystyle h^{2}\mathbb{E}\Big{[}$
$\displaystyle\sum_{m=0}^{M-1}L(t_{m},\bm{X}_{t_{m}}^{N,h})\Big{]}\leq
Kh^{3/2}+Khe^{-\lambda_{0}T}.$

∎

######  Remark 6.1 (On losing $1/2$ in the rate of convergence).

Letting $T\rightarrow\infty$ in (6.5), and temporarily ignoring higher-order
remainder terms, we have that the weak error is of order $3/2$. In [40] this
term satisfies the bound (with $K$ depending on the dimension $N$)

$h^{2}\mathbb{E}\Big{[}\sum_{m=0}^{M-1}L(t_{m},\bm{X}_{t_{m}}^{N,h})\Big{]}\leq
Kh^{2}+Khe^{-\lambda_{0}T},$

and thus a weak order of $2$ is attained.

The loss of $1/2$ in the convergence rate in our estimates occurs in the final
step of (6.13). In [40] this term is estimated with the weak error (starting
on (6.10)), whilst here, we are not able to do so. In fact, to recover the
missing $1/2$-rate by following using the arguments [40] one would need to
show that $L$ in (3.7) satisfies condition (3) of Assumption H.2 – see
additionally our Remark 3.1. At present, this is an open question.

###  6.2 Analysis of residual terms

A close inspection of the proof of the main result in [40] shows that there
are 8 remainder terms which need to be analyzed and we do so in the next
lemma. We will make use of the following helpful abbreviation: for
$m\in\\{0,\ldots,M-1\\}$, we define using (2.3) and (2.6)

$\displaystyle\Delta X_{t_{m}}^{i,N,h}:=$ $\displaystyle
X^{i,N,h}_{t_{m+1}}-X^{i,N,h}_{t_{m}}$ $\displaystyle=$
$\displaystyle-\Big{(}\nabla
U(X^{i,N,h}_{t_{m}})+\frac{1}{N}\sum_{j=1}^{N}\nabla
V(X_{t_{m}}^{i,N,h}-X_{t_{m}}^{j,N,h})\Big{)}h+\frac{\sigma}{2}(\Delta
W_{m}^{i}+\Delta W_{m+1}^{i})$ (6.35) $\displaystyle=$ $\displaystyle
B_{i}(\bm{X}^{N,h}_{t_{m}})h+\frac{\sigma}{2}(\Delta W_{m}^{i}+\Delta
W_{m+1}^{i}).$

Further, we define the continuous extension of $\bm{X}^{N,h}_{t_{m}}$: for all
$s\in[0,h]$

$\displaystyle X_{t_{m}+s}^{i,N,h}:=$ $\displaystyle
X^{i,N,h}_{t_{m}}+\Big{(}\nabla
U(X^{i,N,h}_{t_{m}})+\frac{1}{N}\sum_{j=1}^{N}\nabla
V(X_{t_{m}}^{i,N,h}-X_{t_{m}}^{j,N,h})+\frac{\sigma}{2h}\Delta
W_{m}^{i}\Big{)}s+\frac{\sigma}{2}\Delta W_{m+1,s}^{i}$ (6.36)
$\displaystyle=$ $\displaystyle
X_{t_{m}}^{i,N,h}+\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{(}B_{i}(\bm{X}^{N,h}_{t_{m}})+\frac{\sigma\Delta
W_{m}^{i}}{2h}\Big{)}s+\frac{\sigma}{2}\Delta W_{m+1,s}^{i},$
$\displaystyle\Delta W^{i}_{m,s}$
$\displaystyle={W}^{i}_{t_{m-1}+s}-{W}^{i}_{t_{m-1}},\text{ for }m>0.$ (6.37)

Equivalently, we could write

$\displaystyle\Delta\bm{X}_{t_{m}}^{N,h}$
$\displaystyle:=\bm{X}^{N,h}_{t_{m+1}}-\bm{X}^{N,h}_{t_{m}}=B(\bm{X}^{N,h}_{t_{m}})h+\frac{\sigma}{2}(\Delta\bm{W}_{m}+\Delta\bm{W}_{m+1}),$
$\displaystyle\bm{X}_{t_{m}+s}^{N,h}$
$\displaystyle:=\bm{X}^{N,h}_{t_{m}}+\Big{(}B(\bm{X}^{N,h}_{t_{m}})+\frac{\sigma\Delta\bm{W}_{m,h}}{2h}\Big{)}s+\frac{\sigma}{2}\Delta\bm{W}_{m+1,s},$

where $\Delta\bm{X}_{t_{m}}^{N,h}\coloneqq(\Delta
X_{t_{m}}^{1,N,h},\ldots,\Delta X_{t_{m}}^{N,N,h})$,
$\Delta\bm{W}_{m}\coloneqq(\Delta W^{1}_{m},\ldots,\Delta W^{N}_{m})$ and
$\Delta\bm{W}_{m,s}\coloneqq(\Delta W^{1}_{m,s},\ldots,\Delta W^{N}_{m,s})$
(for $m>0$).  Instead of dealing with the expression above, we rewrite the
scheme in a different way (see [40, p7]). For $m\in\\{0,\ldots,M-1\\}$ and
$i\in\\{1,\dots,N\\}$, with $\hat{X}^{i,N,h}_{t_{0}}=X^{i,N}_{t_{0}}$, define

$\displaystyle\hat{X}^{i,N,h}_{t_{m+1}}$
$\displaystyle=\hat{X}^{i,N,h}_{t_{m}}+\sigma\Delta W_{m}^{i}$ (6.38)
$\displaystyle\qquad-\Big{(}\nabla
U(\hat{X}^{i,N,h}_{t_{m}}+\frac{\sigma}{2}\Delta
W_{m}^{i})+\frac{1}{N}\sum_{j=1}^{N}\nabla
V\big{(}\hat{X}_{t_{m}}^{i,N,h}+\frac{\sigma}{2}\Delta
W_{m}^{i}-\hat{X}_{t_{m}}^{j,N,h}-\frac{\sigma}{2}\Delta
W_{m}^{j}\big{)}\Big{)}h,$ $\displaystyle\hat{\bm{X}}^{N,h}_{t_{m+1}}$
$\displaystyle\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}=\hat{\bm{X}}^{N,h}_{t_{m}}+\sigma\Delta\bm{W}_{m}+B\big{(}\hat{\bm{X}}^{N,h}_{t_{m}}+\frac{\sigma}{2}\Delta\bm{W}_{m}\big{)}h,$

so that $X^{i,N,h}_{t_{m}}=\hat{X}^{i,N,h}_{t_{m}}+\sigma\Delta W_{m}^{i}/2$,
for all $i\in\\{1,\ldots,N\\}$ (or
$\bm{X}^{N,h}_{t_{m}}=\hat{\bm{X}}^{N,h}_{t_{m}}+\sigma\Delta\bm{W}_{m}/2$).

Now, for $s\in[0,h)$ and $m\in\\{1,\ldots,M\\}$, we define the following
auxiliary process

$\displaystyle\overline{X}^{i,N,h}_{t_{m-1}+s}$
$\displaystyle=\hat{X}^{i,N,h}_{t_{m}}+\frac{\sigma}{2}\Delta
W_{m,s}^{i},\quad\overline{X}^{i,N,h}_{t_{m-1}}=\hat{X}^{i,N,h}_{t_{m}},\quad$
(6.39) $\displaystyle\overline{\bm{X}}_{t_{m-1}+s}^{N,h}$
$\displaystyle=\hat{\bm{X}}_{{t_{m}}}^{N,h}+\frac{\sigma}{2}\Delta\bm{W}_{m,s},\quad\overline{\bm{X}}_{t_{m-1}}=\hat{\bm{X}}_{{t_{m}}}^{N,h},$

where we remark that this form is used in the proof of Lemma 6.2 (e.g., in
(6.46)). The moment stability of these auxiliary schemes is discussed in the
appendix, see Lemma A.5. We refer the reader to [53] for different versions of
such schemes, coined there ‘postprocessed schemes’, achieving higher-order
weak convergence in the ergodic setting.

######  Lemma 6.2.

Let Assumption H.2 hold and let $\xi\in L^{10}(\Omega,\mathbb{R})$. Then for
the remainder term $R$ in (6.1), there exists a positive constant $K$
independent of $h,T,M$ and $N$, such that

$\displaystyle\mathbb{E}\left[\sum_{m=0}^{M-1}R(t_{m},\bm{X}^{N,h}_{t_{m}})\right]\leq
Kh^{3/2}.$

###### Proof.

Recall that the remainder term $r(t_{m},\cdot)$ (corresponding to our
$R(t_{m},\cdot)$ in (6.1)) in [40, Equation (3.17)], itself is given as a
linear combination of remainders denoted by $h^{3}r_{i}(t_{m},\cdot)$ for
$i\in\\{1,\ldots,8\\}$ appearing in Equations (3.8), (3.10)–(3.16) in [40,
p7-9]. In our case, we derive bounds for
$\mathbb{E}[R(t_{m},\bm{X}^{N,h}_{t_{m}})]\leq\sum_{i=1}^{8}|\mathbb{E}[R^{i}_{t_{m}}]|$,
where each $R^{i}_{t_{m}}$ term corresponds to the $h^{3}r_{i}(t_{m},\cdot)$
terms ($i\in\\{1,\ldots,8\\}$) of [40, p7-9].

It may not be immediately transparent how the $h^{3}r_{i}(t_{m},\cdot)$ terms
correspond to our $R_{t_{m}}^{i}$ terms. We explicitly present the derivations
of $\mathbb{E}[R_{t_{m}}^{1}]$, $\mathbb{E}[R_{t_{m}}^{4}]$ and
$\mathbb{E}[R_{t_{m}}^{6}]$, which we feel encapsulate the techniques and
proof methodology. The derivation of the residual terms
$R_{t_{m}}^{2},\,R_{t_{m}}^{3},\,R_{t_{m}}^{5},\,R_{t_{m}}^{7},\,R_{t_{m}}^{8}$,
can be found in Appendix A.4. In this proof, let $K$ be a constant independent
of $m,M,N,T,h$ which may change from line to line.

Part 1: Recall the $\Delta X$ notation introduced in (6.35). The remainder
term $h^{3}r_{1}$ in Equation (3.8) of [40, p7] is established from the Taylor
expansion with Lagrange’s form of the remainder term. That is, for all $m$,
there exists some $\rho_{m}\in(0,1)$ such that

$\displaystyle\mathbb{E}\big{[}R^{1}_{t_{m}}\big{]}$
$\displaystyle=K\mathbb{E}\bigg{[}\sum_{\gamma\in\Pi_{5}^{N}}\Delta
X^{\gamma_{1},N,h}_{t_{m}}\dots\Delta
X^{\gamma_{5},N,h}_{t_{m}}~{}\partial^{5}_{\gamma_{1},\dots,\gamma_{5}}u(t_{m+1},\bm{X}_{m,{\rho_{m}}}^{N,h})\bigg{]}$

where
$\bm{X}_{m,{\rho_{m}}}^{N,h}:={\rho_{m}}\bm{X}_{t_{m}}^{N,h}+(1-{\rho_{m}})\bm{X}_{t_{m+1}}^{N,h}$.

We deal with $|\mathbb{E}[R^{1}_{t_{m}}]|$ via Hölder’s inequality to isolate
the $\Delta X$ terms from the derivatives of $u$ term,

$\displaystyle\big{|}\mathbb{E}\big{[}R^{1}_{t_{m}}\big{]}\big{|}$
$\displaystyle\leq
K\sum_{\gamma\in\Pi_{5}^{N}}\sqrt{\mathbb{E}\Big{[}\big{|}\Delta
X^{\gamma_{1},N,h}_{t_{m}}\dots\Delta
X^{\gamma_{5},N,h}_{t_{m}}\big{|}^{2}\Big{]}\mathbb{E}\Big{[}~{}\Big{|}\partial^{5}_{x_{\gamma_{1}},\dots,x_{\gamma_{5}}}~{}u(t_{m+1},\bm{X}_{m,{\rho_{m}}}^{N,h})\Big{|}^{2}\Big{]}}$
$\displaystyle\leq
Kh^{5/2}\sum_{\gamma\in\Pi_{5}^{N}}e^{-\lambda_{0}(T-t_{m+1})}N^{-\hat{\mathcal{O}}(\gamma)}\leq
Kh^{5/2}e^{-\lambda_{0}(T-t_{m+1})},$ (6.40)

which holds for some $\lambda_{0}\in(0,\lambda)$. We derive (6.40) using Lemma
5.3 to treat the derivatives of $u$, while for the $\Delta X$ terms, we use
their explicit forms (6.35) and the fact that $\nabla U,\nabla V$ are of
linear growth. In particular, the assumption $\xi\in
L^{10}(\Omega,\mathbb{R})$, allows to use Proposition 2.1 and Proposition 2.2
to control the $L^{10}$-moments of the processes. The moment properties of the
increments extract the leading order $h$ terms; providing the $h^{5/2}$
leading order. Note that for $R^{1}_{t_{m}}$ we incur a loss of $h^{1/2}$ in
the leading term while in [40] there is no such loss – this issue has already
appeared in the proof of Lemma 6.1 and is discussed in Remark 6.1 (and Remark
3.1). Critically, [40] has a $|\cdot|_{\infty}$\- norm bound for the 5-th
derivative of $u$ while here we are only able to bound it in expectation. We
are forced to use Hölder’s inequality.

Similar to [40, Equation (3.12)], we show how the $R_{t_{m}}^{4}$
(corresponding to $r_{4}(t_{m},\cdot)h^{3}$ in [40, Equation (3.12)]) is
generated and give the exact form of $\mathbb{E}[R_{t_{m}}^{4}]$. Expanding
out the increments $\Delta X^{\gamma_{i},N,h}_{t_{m}}$, $i\in\\{1,2,3\\}$, and
recall the definition of $\Delta W_{m,2h}$ in (6.37), we have

$\displaystyle\sum_{\gamma\in\Pi_{3}^{N}}$
$\displaystyle\mathbb{E}\Big{[}\Delta X^{\gamma_{1},N,h}_{t_{m}}\Delta
X^{\gamma_{2},N,h}_{t_{m}}\Delta
X^{\gamma_{3},N,h}_{t_{m}}~{}\partial^{3}_{x_{\gamma_{1}},x_{\gamma_{2}},x_{\gamma_{3}}}u(t_{m+1},\bm{X}_{t_{m}}^{N,h})\Big{]}$
$\displaystyle=$ $\displaystyle
K\sum_{\gamma\in\Pi_{3}^{N}}\mathbb{E}\Big{[}\Delta
W_{m,2h}^{\gamma_{1}}\Delta W_{m,2h}^{\gamma_{2}}\Delta
W_{m,2h}^{\gamma_{3}}~{}\partial^{3}_{x_{\gamma_{1}},x_{\gamma_{2}},x_{\gamma_{3}}}u(t_{m+1},\bm{X}_{t_{m}}^{N,h})\Big{]}$
(6.41)
$\displaystyle+Kh\sum_{\gamma\in\Pi_{3}^{N}}\mathbb{E}\Big{[}B_{\gamma_{1}}({\bm{X}}_{t_{m}}^{N,h})\Delta
W_{m,2h}^{\gamma_{2}}\Delta
W_{m,2h}^{\gamma_{3}}~{}\partial^{3}_{x_{\gamma_{1}},x_{\gamma_{2}},x_{\gamma_{3}}}u(t_{m+1},\bm{X}_{t_{m}}^{N,h})\Big{]}$
(6.42)
$\displaystyle+Kh^{2}\sum_{\gamma\in\Pi_{3}^{N}}\mathbb{E}\Big{[}B_{\gamma_{1}}({\bm{X}}_{t_{m}}^{N,h})B_{\gamma_{2}}({\bm{X}}_{t_{m}}^{N,h})\Delta
W_{m,2h}^{\gamma_{3}}~{}\partial^{3}_{x_{\gamma_{1}},x_{\gamma_{2}},x_{\gamma_{3}}}u(t_{m+1},\bm{X}_{t_{m}}^{N,h})\Big{]}$
(6.43)
$\displaystyle+Kh^{3}\sum_{\gamma\in\Pi_{3}^{N}}\mathbb{E}\Big{[}B_{\gamma_{1}}({\bm{X}}_{t_{m}}^{N,h})B_{\gamma_{2}}({\bm{X}}_{t_{m}}^{N,h})B_{\gamma_{3}}({\bm{X}}_{t_{m}}^{N,h})~{}\partial^{3}_{x_{\gamma_{1}},x_{\gamma_{2}},x_{\gamma_{3}}}u(t_{m+1},\bm{X}_{t_{m}}^{N,h})\Big{]}.$
(6.44)

We first deal with the term (6.41), by applying an Itô-Taylor expansion (see
[35, Section 5.1, p.163–164]) around $\hat{\bm{X}}_{{t_{m}}}^{N,h}$ to obtain

$\displaystyle\eqref{eq: r4 term1}$
$\displaystyle=K\sum_{\gamma\in\Pi_{3}^{N}}\mathbb{E}\Big{[}\Delta
W_{m,2h}^{\gamma_{1}}\Delta W_{m,2h}^{\gamma_{2}}\Delta
W_{m,2h}^{\gamma_{3}}~{}\partial^{3}_{x_{\gamma_{1}},x_{\gamma_{2}},x_{\gamma_{3}}}u(t_{m+1},\hat{\bm{X}}_{{t_{m}}}^{N,h})\Big{]}$
(6.45)
$\displaystyle+K\sum_{\gamma\in\Pi_{4}^{N}}\mathbb{E}\bigg{[}\int_{t_{m}}^{t_{m}+h}\partial_{x_{\gamma_{1}}}\bigg{(}\Delta
W_{m,2h}^{\gamma_{2}}\Delta W_{m,2h}^{\gamma_{3}}\Delta
W_{m,2h}^{\gamma_{4}}~{}\partial^{3}_{x_{\gamma_{2}},x_{\gamma_{3}},x_{\gamma_{4}}}u(t_{m+1},\overline{\bm{X}}_{q_{1}}^{N,h})\bigg{)}\mathrm{d}W^{\gamma_{1}}_{q_{1}}\bigg{]}$
(6.46)
$\displaystyle+K\sum_{\gamma\in\Pi_{4}^{N}}\mathbb{E}\bigg{[}\int_{t_{m}}^{t_{m}+h}\partial^{2}_{x_{\gamma_{1}},x_{\gamma_{1}}}\bigg{(}\Delta
W_{m,2h}^{\gamma_{2}}\Delta W_{m,2h}^{\gamma_{3}}\Delta
W_{m,2h}^{\gamma_{4}}~{}\partial^{3}_{x_{\gamma_{2}},x_{\gamma_{3}},x_{\gamma_{4}}}u(t_{m+1},\overline{\bm{X}}_{q_{1}}^{N,h})\bigg{)}\mathrm{d}{q_{1}}\bigg{]}.$
(6.47)

One observes that $\eqref{eq: r4 term 1.1}=0$ since for
$\gamma\in\Pi_{3}^{N}$, regardless of the value of
$\hat{\mathcal{O}}(\gamma)$, there will always be an odd power of a Brownian
increment $\Delta W_{m,2h}$ presented in (6.37), independent of the Brownian
increment contained in the $\hat{\bm{X}}_{{t_{m}}}^{N,h}$ term. The term
(6.47) however, does not vanish since the $\Delta W_{m,2h}$ term is not
independent of $\overline{\bm{X}}_{q_{1}}^{N,h}$, because the latter contains
the Brownian increment $W_{q_{1}}-W_{t_{m}}$.

Applying a further Itô-Taylor expansion around $\hat{\bm{X}}_{t_{m}}^{N,h}$,
and splitting the zeroth order term into the cases
$\hat{\mathcal{O}}(\gamma)\in\\{1,2\\}$ yields

$\displaystyle\eqref{eq: r4 term 1.2}=$ $\displaystyle
Kh^{2}\sum_{i=1}^{N}\mathbb{E}\bigg{[}\partial^{4}_{x_{i},x_{i},x_{i},x_{i}}u(t_{m+1},\hat{\bm{X}}_{t_{m}}^{N,h})\bigg{]}+Kh^{2}\sum_{i,j=1,~{}i\neq
j}^{N}\mathbb{E}\bigg{[}\partial^{4}_{x_{i},x_{i},x_{j},x_{j}}u(t_{m+1},\hat{\bm{X}}_{t_{m}}^{N,h})\bigg{]}$
(6.48)
$\displaystyle+K\sum_{\gamma\in\Pi_{5}^{N}}\mathbb{E}\bigg{[}\int_{t_{m}}^{t_{m}+h}\int_{t_{m}}^{q_{1}}\partial_{x_{\gamma_{1}},x_{\gamma_{2}}}\bigg{(}\Delta
W_{m,2h}^{\gamma_{3}}\Delta W_{m,2h}^{\gamma_{4}}\Delta
W_{m,2h}^{\gamma_{5}}~{}\partial^{3}_{x_{\gamma_{4}},x_{\gamma_{4}},x_{\gamma_{5}}}u(t_{m+1},\overline{\bm{X}}_{q_{2}}^{N,h})\bigg{)}\mathrm{d}W^{\gamma_{1}}_{q_{2}}\
\mathrm{d}W^{\gamma_{2}}_{q_{1}}\bigg{]}$ (6.49)
$\displaystyle+K\sum_{\gamma\in\Pi_{5}^{N}}\mathbb{E}\bigg{[}\int_{t_{m}}^{t_{m}+h}\int_{t_{m}}^{q_{1}}\partial^{3}_{x_{\gamma_{1}},x_{\gamma_{1}},x_{\gamma_{2}}}\bigg{(}\Delta
W_{m,2h}^{\gamma_{3}}\Delta W_{m,2h}^{\gamma_{4}}\Delta
W_{m,2h}^{\gamma_{5}}~{}\partial^{3}_{x_{\gamma_{4}},x_{\gamma_{4}},x_{\gamma_{5}}}u(t_{m+1},\overline{\bm{X}}_{q_{2}}^{N,h})\bigg{)}\mathrm{d}{q_{2}}\
\mathrm{d}W^{\gamma_{2}}_{q_{1}}\bigg{]},$ (6.50)

where the two summation terms in (6.2) correspond to the second and third
summation in [40, (3.12)] and they do not contribute to the remainder term
$R^{4}_{t_{m}}$. Similarly for (6.42), we have

$\displaystyle\eqref{eq: r4
term2}=Kh\sum_{i,j=1}^{N}\mathbb{E}\Big{[}B_{i}(\hat{\bm{X}}_{{t_{m}}}^{N,h})\partial^{3}_{x_{i},x_{j},x_{j}}u(t_{m+1},\hat{\bm{X}}_{{t_{m}}}^{N,h})\Big{]}$
(6.51)
$\displaystyle+Kh\sum_{\gamma\in\Pi_{4}^{N}}\mathbb{E}\bigg{[}\int_{t_{m}}^{t_{m}+h}\partial_{x_{\gamma_{1}}}\bigg{(}B_{\gamma_{2}}(\overline{\bm{X}}_{q_{1}}^{N,h})\Delta
W_{m,2h}^{\gamma_{3}}\Delta
W_{m,2h}^{\gamma_{4}}~{}\partial^{3}_{x_{\gamma_{2}},x_{\gamma_{3}},x_{\gamma_{4}}}u(t_{m+1},\overline{\bm{X}}_{q_{1}}^{N,h})\bigg{)}\mathrm{d}W^{\gamma_{1}}_{q_{1}}\bigg{]}$
(6.52)
$\displaystyle+Kh\sum_{\gamma\in\Pi_{4}^{N}}\mathbb{E}\bigg{[}\int_{t_{m}}^{t_{m}+h}\partial^{2}_{x_{\gamma_{1}},x_{\gamma_{1}}}\bigg{(}B_{\gamma_{2}}(\overline{\bm{X}}_{q_{1}}^{N,h})\Delta
W_{m,2h}^{\gamma_{3}}\Delta
W_{m,2h}^{\gamma_{4}}~{}\partial^{3}_{x_{\gamma_{2}},x_{\gamma_{3}},x_{\gamma_{4}}}u(t_{m+1},\overline{\bm{X}}_{q_{1}}^{N,h})\bigg{)}\mathrm{d}{q_{1}}\bigg{]},$
(6.53)

where (6.51) corresponds to the first summation in [40, (3.12)], thus also
does not contribute to $R_{t_{m}}^{4}$. Leimkuhler et al. isolates and
separates the terms (6.2) and (6.51) from $h^{3}r_{4}$, since these terms
cleverly cancel with other corresponding lower order terms in the expansion of
other $r_{i}$’s when everything is summed over.

As for (6.43), we have

$\displaystyle\eqref{eq: r4
term3}=Kh^{2}\sum_{\gamma\in\Pi_{3}^{N}}\mathbb{E}\Big{[}B_{\gamma_{1}}(\hat{\bm{X}}_{{t_{m}}}^{N,h})B_{\gamma_{2}}(\hat{\bm{X}}_{{t_{m}}}^{N,h})\Delta
W_{m,2h}^{\gamma_{3}}~{}\partial^{3}_{x_{\gamma_{1}},x_{\gamma_{2}},x_{\gamma_{3}}}u(t_{m+1},\hat{\bm{X}}_{{t_{m}}}^{N,h})\Big{]}$
(6.54)
$\displaystyle+Kh^{2}\sum_{\gamma\in\Pi_{4}^{N}}\mathbb{E}\bigg{[}\int_{t_{m}}^{t_{m}+h}\partial_{x_{\gamma_{1}}}\bigg{(}B_{\gamma_{2}}(\overline{\bm{X}}_{q_{1}}^{N,h})B_{\gamma_{3}}(\overline{\bm{X}}_{q_{1}}^{N,h})\Delta
W_{m,2h}^{\gamma_{4}}~{}\partial^{3}_{x_{\gamma_{2}},x_{\gamma_{3}},x_{\gamma_{4}}}u(t_{m+1},\overline{\bm{X}}_{q_{1}}^{N,h})\bigg{)}\mathrm{d}W^{\gamma_{1}}_{q_{1}}\bigg{]}$
(6.55)
$\displaystyle+Kh^{2}\sum_{\gamma\in\Pi_{4}^{N}}\mathbb{E}\bigg{[}\int_{t_{m}}^{t_{m}+h}\partial^{2}_{x_{\gamma_{1}},x_{\gamma_{1}}}\bigg{(}B_{\gamma_{2}}(\overline{\bm{X}}_{q_{1}}^{N,h})B_{\gamma_{3}}(\overline{\bm{X}}_{q_{1}}^{N,h})\Delta
W_{m,2h}^{\gamma_{4}}~{}\partial^{3}_{x_{\gamma_{2}},x_{\gamma_{3}},x_{\gamma_{4}}}u(t_{m+1},\overline{\bm{X}}_{q_{1}}^{N,h})\bigg{)}\mathrm{d}{q_{1}}\bigg{]},$
(6.56)

where the first term in this expansion (6.54) is zero, following the same
reasoning as for (6.45).

For the residual term $R^{4}_{t_{m}}$, we have that via direct application of
Hölder’s inequality, Itô’s isometry and Fubini’s Theorem, we have

$\displaystyle\mathbb{E}$ $\displaystyle\big{[}R^{4}_{t_{m}}\big{]}=\eqref{eq:
r4 term4}+\eqref{eq: r4 term 1.3}+\eqref{eq: R4 comb1}+\eqref{eq: R4
comb2}+\eqref{eq: R4 comb4}+\eqref{eq: R4 comb5}+\eqref{eq: R4
comb6}+\eqref{eq: R4 comb7}$ $\displaystyle\leq
Kh^{3}\sum_{\gamma\in\Pi_{3}^{N}}\mathbb{E}\Big{[}B_{\gamma_{1}}({\bm{X}}_{t_{m}}^{N,h})B_{\gamma_{2}}({\bm{X}}_{t_{m}}^{N,h})B_{\gamma_{3}}({\bm{X}}_{t_{m}}^{N,h})~{}\partial^{3}_{x_{\gamma_{1}},x_{\gamma_{2}},x_{\gamma_{3}}}u(t_{m+1},\bm{X}_{t_{m}}^{N,h})\Big{]}$
$\displaystyle\quad+Kh^{3/2}~{}\int_{t_{m}}^{t_{m}+h}~{}\bigg{(}\mathbb{E}\bigg{[}~{}\bigg{|}\sum_{\gamma\in\Pi_{4}^{N}}\partial^{5}_{x_{\gamma_{1}},x_{\gamma_{1}},x_{\gamma_{2}},x_{\gamma_{3}},x_{\gamma_{4}}}u(t_{m+1},\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})~{}\bigg{|}^{2}~{}~{}\bigg{]}\bigg{)}^{1/2}~{}\mathrm{d}q_{1}$
$\displaystyle\quad+Kh^{2}\bigg{(}\int_{t_{m}}^{t_{m}+h}\mathbb{E}\bigg{[}~{}\bigg{|}\sum_{\gamma\in\Pi_{5}^{N}}\partial^{5}_{x_{\gamma_{1}},\dots,x_{\gamma_{5}}}u(t_{m+1},\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})~{}\bigg{|}^{2}~{}~{}\bigg{]}~{}\mathrm{d}q_{1}\bigg{)}^{1/2}$
$\displaystyle\quad+Kh^{2}\bigg{(}\int_{t_{m}}^{t_{m}+h}\mathbb{E}\bigg{[}~{}\bigg{|}\sum_{\gamma\in\Pi_{5}^{N}}\partial^{6}_{x_{\gamma_{1}},x_{\gamma_{1}},x_{\gamma_{2}},x_{\gamma_{4}},x_{\gamma_{4}},x_{\gamma_{5}}}u(t_{m+1},\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})~{}\bigg{|}^{2}~{}~{}\bigg{]}~{}~{}\mathrm{d}q_{1}\bigg{)}^{1/2}$
$\displaystyle\quad+Kh^{2}\bigg{(}\int_{t_{m}}^{t_{m}+h}\mathbb{E}\bigg{[}~{}\bigg{|}\sum_{\gamma\in\Pi_{4}^{N}}\partial_{x_{\gamma_{1}}}\bigg{(}B_{\gamma_{2}}(\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\partial^{3}_{x_{\gamma_{2}},x_{\gamma_{3}},x_{\gamma_{4}}}u(t_{m+1},\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\bigg{)}~{}\bigg{|}^{2}~{}~{}\bigg{]}~{}\mathrm{d}q_{1}\bigg{)}^{1/2}$
$\displaystyle\quad+Kh^{3/2}\int_{t_{m}}^{t_{m}+h}\bigg{(}\mathbb{E}\bigg{[}~{}\bigg{|}\sum_{\gamma\in\Pi_{4}^{N}}\partial^{2}_{x_{\gamma_{1}},x_{\gamma_{1}}}\bigg{(}B_{\gamma_{2}}(\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\partial^{3}_{x_{\gamma_{2}},x_{\gamma_{3}},x_{\gamma_{4}}}u(t_{m+1},\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\bigg{)}~{}\bigg{|}^{2}~{}~{}\bigg{]}\bigg{)}^{1/2}~{}\mathrm{d}q_{1}$
$\displaystyle\quad+Kh^{5/2}\bigg{(}\int_{t_{m}}^{t_{m}+h}\mathbb{E}\bigg{[}~{}\bigg{|}\sum_{\gamma\in\Pi_{4}^{N}}\partial_{x_{\gamma_{1}}}\bigg{(}B_{\gamma_{2}}(\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})B_{\gamma_{3}}(\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\partial^{3}_{x_{\gamma_{2}},x_{\gamma_{3}},x_{\gamma_{4}}}u(t_{m+1},\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\bigg{)}~{}\bigg{|}^{2}~{}~{}\bigg{]}~{}\mathrm{d}q_{1}\bigg{)}^{1/2}$
$\displaystyle\quad+Kh^{5/2}\int_{t_{m}}^{t_{m}+h}\bigg{(}\mathbb{E}\bigg{[}~{}\bigg{|}\sum_{\gamma\in\Pi_{4}^{N}}\partial^{2}_{x_{\gamma_{1}},x_{\gamma_{1}}}\bigg{(}B_{\gamma_{2}}(\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})B_{\gamma_{3}}(\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\partial^{3}_{x_{\gamma_{2}},x_{\gamma_{3}},x_{\gamma_{4}}}u(t_{m+1},\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\bigg{)}~{}\bigg{|}^{2}~{}~{}\bigg{]}~{}\bigg{)}^{1/2}\mathrm{d}q_{1}.$

As for $R_{t_{m}}^{6}$, recalling the definition of the operator
$\mathcal{L}_{N}$

$\displaystyle\mathcal{L}_{N}u$
$\displaystyle=\sum_{i=1}^{N}\Big{(}B_{i}\partial_{x_{i}}u+\frac{1}{2}\sigma^{2}\partial^{2}_{x_{i},x_{i}}u\Big{)},$

the remainder term $r_{6}(t_{m},\cdot)h^{3}$ in [40, Equation (3.14)] (apply
$\mathcal{L}_{N}$ three times) corresponds to

$\displaystyle\mathbb{E}\big{[}R^{6}_{t_{m}}\big{]}$
$\displaystyle=\mathbb{E}\bigg{[}\int_{t_{m}}^{t_{m}+h}\int_{t_{m}}^{q_{1}}\int_{t_{m}}^{q_{2}}(\mathcal{L}_{N})^{3}u(t_{m+1},{\bm{X}}_{q_{3}}^{N,h})\mathrm{d}q_{3}~{}\mathrm{d}q_{2}~{}\mathrm{d}q_{1}~{}\bigg{]}$
$\displaystyle=\mathbb{E}\bigg{[}\int_{t_{m}}^{t_{m}+h}\int_{t_{m}}^{q_{1}}\int_{t_{m}}^{q_{2}}\Bigg{(}K\sum_{\gamma\in\Pi_{3}^{N}}B_{\gamma_{1}}({\bm{X}}_{q_{3}}^{N,h})B_{\gamma_{2}}({\bm{X}}_{q_{3}}^{N,h})B_{\gamma_{3}}({\bm{X}}_{q_{3}}^{N,h})\partial^{3}_{x_{\gamma_{1}},x_{\gamma_{2}},x_{\gamma_{3}}}u(t_{m+1},{\bm{X}}_{q_{3}}^{N,h})$
$\displaystyle\qquad\qquad+K\sum_{\gamma\in\Pi_{3}^{N}}\partial^{2}_{x_{\gamma_{3}},x_{\gamma_{3}}}\bigg{(}B_{\gamma_{1}}({\bm{X}}_{q_{3}}^{N,h})B_{\gamma_{2}}({\bm{X}}_{q_{3}}^{N,h})\partial^{2}_{x_{\gamma_{1}},x_{\gamma_{2}}}u(t_{m+1},{\bm{X}}_{q_{3}}^{N,h})\bigg{)}$
$\displaystyle\qquad\qquad+K\sum_{\gamma\in\Pi_{3}^{N}}\partial^{4}_{x_{\gamma_{2}},x_{\gamma_{2}},x_{\gamma_{3}},x_{\gamma_{3}}}\bigg{(}B_{\gamma_{1}}({\bm{X}}_{q_{3}}^{N,h})\partial_{x_{\gamma_{1}}}u(t_{m+1},{\bm{X}}_{q_{3}}^{N,h})\bigg{)}$
$\displaystyle\qquad\qquad+K\sum_{\gamma\in\Pi_{6}^{N},\hat{\mathcal{O}}(\gamma)\leq
3}\partial^{6}_{x_{\gamma_{1}},\dots,x_{\gamma_{6}}}u(t_{m+1},{\bm{X}}_{q_{3}}^{N,h})\Bigg{)}\mathrm{d}q_{3}~{}\mathrm{d}q_{2}~{}\mathrm{d}q_{1}~{}\bigg{]}.$

Once again, a direct application of Hölder’s inequality and Fubini’s Theorem

$\displaystyle\big{|}\mathbb{E}\big{[}R^{6}_{t_{m}}\big{]}\big{|}$
$\displaystyle\leq
Kh^{2}\int_{t_{m}}^{t_{m}+h}\bigg{(}\mathbb{E}\bigg{[}~{}\bigg{|}~{}\sum_{\gamma\in\Pi_{3}^{N}}B_{\gamma_{1}}(\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})B_{\gamma_{2}}(\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})B_{\gamma_{3}}(\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\partial^{3}_{x_{\gamma_{1}},x_{\gamma_{2}},x_{\gamma_{3}}}u(t_{m+1},\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})$
$\displaystyle\qquad\qquad+\sum_{\gamma\in\Pi_{3}^{N}}\partial^{2}_{x_{\gamma_{3}},x_{\gamma_{3}}}\bigg{(}B_{\gamma_{1}}(\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})B_{\gamma_{2}}(\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\partial^{2}_{x_{\gamma_{1}},x_{\gamma_{2}}}u(t_{m+1},\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\bigg{)}$
$\displaystyle\qquad\qquad+\sum_{\gamma\in\Pi_{3}^{N}}\partial^{4}_{x_{\gamma_{2}},x_{\gamma_{2}},x_{\gamma_{3}},x_{\gamma_{3}}}\bigg{(}B_{\gamma_{1}}(\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\partial_{x_{\gamma_{1}}}u(t_{m+1},\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\bigg{)}$
$\displaystyle\qquad\qquad+\sum_{\gamma\in\Pi_{6}^{N},\hat{\mathcal{O}}(\gamma)\leq
3}\partial^{6}_{x_{\gamma_{1}},\dots,x_{\gamma_{6}}}u(t_{m+1},\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})~{}\bigg{|}^{2}\bigg{]}\bigg{)}^{1/2}~{}\mathrm{d}q_{1}.$

Part 2: Other remainder terms. We now present the dominations of the other
residual terms, whose explicit expressions can be found in Appendix A.4.
These, once again, are a result of applying Hölder’s inequality, Itô’s
isometry and Fubini’s Theorem. We have

$\displaystyle\big{|}\mathbb{E}\big{[}R^{2}_{t_{m}}\big{]}\big{|}$
$\displaystyle\leq
Kh^{2}\bigg{(}~{}\int_{t_{m}}^{t_{m}+h}\mathbb{E}\bigg{[}~{}\bigg{|}\sum_{\gamma\in\Pi_{5}^{N}}\partial^{5}_{x_{\gamma_{1}},\dots,x_{\gamma_{5}}}u(t_{m+1},\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})~{}\bigg{|}^{2}~{}\bigg{]}~{}\mathrm{d}q_{1}~{}\bigg{)}^{1/2}$
$\displaystyle\quad+Kh^{5/2}\bigg{(}~{}\int_{t_{m}}^{t_{m}+h}\mathbb{E}\bigg{[}~{}\bigg{|}\sum_{\gamma\in\Pi_{6}^{N}}\partial^{6}_{x_{\gamma_{1}},\dots,x_{\gamma_{6}}}u(t_{m+1},\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})~{}\bigg{|}^{2}~{}\bigg{]}~{}\mathrm{d}q_{1}~{}\bigg{)}^{1/2},$
$\displaystyle\big{|}\mathbb{E}\big{[}R^{3}_{t_{m}}\big{]}\big{|}$
$\displaystyle\leq
Kh^{2}\bigg{(}~{}\int_{t_{m}}^{t_{m}+h}\mathbb{E}\bigg{[}~{}\bigg{|}\sum_{\gamma\in\Pi_{3}^{N}}\partial_{x_{\gamma_{1}}}\bigg{(}\partial_{x_{\gamma_{3}}}B_{\gamma_{2}}(\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\partial^{2}_{x_{\gamma_{2}},x_{\gamma_{3}}}u(t_{m+1},\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\bigg{)}~{}\bigg{|}^{2}~{}\bigg{]}~{}\mathrm{d}q_{1}~{}\bigg{)}^{1/2}$
$\displaystyle\quad+Kh^{2}\bigg{(}~{}\int_{t_{m}}^{t_{m}+h}\mathbb{E}\bigg{[}~{}\bigg{|}\sum_{\gamma\in\Pi_{3}^{N}}\partial^{2}_{x_{\gamma_{1}},x_{\gamma_{1}}}\bigg{(}\partial_{x_{\gamma_{3}}}B_{\gamma_{2}}(\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\partial^{2}_{x_{\gamma_{2}},x_{\gamma_{3}}}u(t_{m+1},\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\bigg{)}~{}\bigg{|}^{2}~{}\bigg{]}~{}\mathrm{d}q_{1}~{}\bigg{)}^{1/2}$
$\displaystyle\quad+Kh^{2}\bigg{(}~{}\int_{t_{m}}^{t_{m}+h}\mathbb{E}\bigg{[}~{}\bigg{|}\sum_{\gamma\in\Pi_{3}^{N}}\partial_{x_{\gamma_{1}}}\bigg{(}B_{\gamma_{2}}(\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\partial^{2}_{x_{\gamma_{2}},x_{\gamma_{3}},x_{\gamma_{3}}}u(t_{m+1},\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\bigg{)}~{}\bigg{|}^{2}~{}\bigg{]}~{}\mathrm{d}q_{1}~{}\bigg{)}^{1/2}$
$\displaystyle\quad+Kh^{2}\bigg{(}~{}\int_{t_{m}}^{t_{m}+h}\mathbb{E}\bigg{[}~{}\bigg{|}\sum_{\gamma\in\Pi_{3}^{N}}\partial^{2}_{x_{\gamma_{1}},x_{\gamma_{1}}}\bigg{(}B_{\gamma_{2}}(\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\partial^{2}_{x_{\gamma_{2}},x_{\gamma_{3}},x_{\gamma_{3}}}u(t_{m+1},\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\bigg{)}~{}\bigg{|}^{2}~{}\bigg{]}~{}\mathrm{d}q_{1}~{}\bigg{)}^{1/2}$
$\displaystyle\quad+Kh^{2}\bigg{(}~{}\int_{t_{m}}^{t_{m}+h}\mathbb{E}\bigg{[}~{}\bigg{|}\sum_{\gamma\in\Pi_{5}^{N}}\partial^{5}_{x_{\gamma_{1}},\dots,x_{\gamma_{5}}}u(t_{m+1},\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})~{}\bigg{|}^{2}~{}\bigg{]}~{}\mathrm{d}q_{1}~{}\bigg{)}^{1/2}$
$\displaystyle\quad+Kh^{5/2}\bigg{(}~{}\int_{t_{m}}^{t_{m}+h}\mathbb{E}\bigg{[}~{}\bigg{|}\sum_{\gamma\in\Pi_{6}^{N}}\partial^{6}_{x_{\gamma_{1}},\dots,x_{\gamma_{6}}}u(t_{m+1},\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})~{}\bigg{|}^{2}~{}\bigg{]}~{}\mathrm{d}q_{1}~{}\bigg{)}^{1/2},$
$\displaystyle\big{|}\mathbb{E}\big{[}R^{5}_{t_{m}}\big{]}\big{|}$
$\displaystyle\leq
Kh^{5/2}\sum_{\gamma\in\Pi_{4}^{N}}\bigg{(}\mathbb{E}\bigg{[}~{}\bigg{|}\partial^{4}_{x_{\gamma_{1}},x_{\gamma_{2}},x_{\gamma_{3}},x_{\gamma_{4}}}u(t_{m+1},\bm{X}_{t_{m}}^{N,h})~{}\bigg{|}^{2}~{}\bigg{]}~{}\bigg{)}^{1/2}$
$\displaystyle\quad+Kh^{2}\bigg{(}~{}\int_{t_{m}}^{t_{m}+h}\mathbb{E}\bigg{[}~{}\bigg{|}\sum_{\gamma\in\Pi_{5}^{N}}\partial^{5}_{x_{\gamma_{1}},\dots,x_{\gamma_{5}}}u(t_{m+1},\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})~{}\bigg{|}^{2}~{}\bigg{]}~{}\mathrm{d}q_{1}~{}\bigg{)}^{1/2}$
$\displaystyle\quad+Kh^{2}\bigg{(}\int_{t_{m}}^{t_{m}+h}~{}\mathbb{E}\bigg{[}~{}\bigg{|}\sum_{\gamma\in\Pi_{5}^{N}}\partial^{6}_{x_{\gamma_{1}},x_{\gamma_{1}},x_{\gamma_{2}},x_{\gamma_{3}},x_{\gamma_{4}},x_{\gamma_{5}}}u(t_{m+1},\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})~{}\bigg{|}^{2}~{}\bigg{]}~{}\mathrm{d}q_{1}~{}\bigg{)}^{1/2},$
$\displaystyle\big{|}\mathbb{E}\big{[}R^{7}_{t_{m}}\big{]}\big{|}$
$\displaystyle\leq
Kh^{2}\bigg{(}~{}\int_{t_{m}}^{t_{m}+h}\mathbb{E}\bigg{[}~{}\bigg{|}~{}\sum_{\gamma\in\Pi_{6}^{N},\hat{\mathcal{O}}(\gamma)\in\\{4,5\\}}\partial^{6}_{x_{\gamma_{1}},\dots,x_{\gamma_{6}}}u(t_{m+1},\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})~{}\bigg{|}^{2}~{}\bigg{]}~{}\mathrm{d}q_{1}~{}\bigg{)}^{1/2}$
$\displaystyle+Kh^{2}\bigg{(}~{}\int_{t_{m}}^{t_{m}+h}\mathbb{E}\bigg{[}~{}\bigg{|}~{}\sum_{\gamma\in\Pi_{5}^{N},\hat{\mathcal{O}}(\gamma)=4}\partial^{6}_{x_{\gamma_{1}},\dots,x_{\gamma_{6}}}u(t_{m+1},\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})~{}\bigg{|}^{2}~{}\bigg{]}~{}\mathrm{d}q_{1}~{}\bigg{)}^{1/2},$
$\displaystyle\big{|}\mathbb{E}\big{[}R^{8}_{t_{m}}\big{]}\big{|}$
$\displaystyle\leq
Kh^{2}\bigg{(}~{}\int_{t_{m}}^{t_{m}+h}\mathbb{E}\bigg{[}~{}\bigg{|}~{}\sum_{\gamma\in\Pi_{5}^{N},\hat{\mathcal{O}}(\gamma)\leq
3}\partial^{5}_{x_{\gamma_{1}},\dots,x_{\gamma_{5}}}u(t_{m+1},\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})~{}\bigg{|}^{2}~{}\bigg{]}\mathrm{d}q_{1}~{}\bigg{)}^{1/2}$
$\displaystyle+Kh^{2}\int_{t_{m}}^{t_{m}+h}~{}\bigg{(}\mathbb{E}\bigg{[}~{}\bigg{|}~{}\sum_{\gamma\in\Pi_{6}^{N},\hat{\mathcal{O}}(\gamma)\leq
3}\partial^{6}_{x_{\gamma_{1}},\dots,x_{\gamma_{6}}}u(t_{m+1},\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})~{}\bigg{|}^{2}~{}\bigg{]}\bigg{)}^{1/2}\mathrm{d}q_{1}~{}$
$\displaystyle+Kh^{2}\bigg{(}~{}\int_{t_{m}}^{t_{m}+h}$
$\displaystyle~{}\mathbb{E}\bigg{[}~{}\bigg{|}~{}\sum_{\begin{subarray}{c}\alpha,\beta\in\bigcup_{k=0}^{5}\Pi_{k}^{N},~{}\hat{\mathcal{O}}(\alpha\bigcup\beta)=3\\\
|\alpha|+|\beta|=5,~{}|\beta|\geq
2\end{subarray}}\partial^{|\alpha|}_{x_{\alpha_{1}},\dots,x_{\alpha_{|\alpha|}}}B_{\beta_{1}}(\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})~{}\partial^{|\beta|}_{x_{\beta_{1}},\dots,x_{\beta_{|\beta|}}}u(t_{m+1},\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\bm{X}}_{q_{1}}^{N,h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\bigg{)}~{}~{}\bigg{|}^{2}~{}\bigg{]}\mathrm{d}q_{1}~{}\bigg{)}^{1/2}.$

Similar to (6.40) and the proof of Lemma 6.1, we can use the linear growth of
$B_{j}$ in combination with the assumption $\xi\in L^{10}(\Omega,\mathbb{R})$
and the Cauchy–Schwarz inequality to establish $L^{10}$-estimates of the
$B_{j}$ (see, Proposition 2.1 and Lemma A.5) which, for instance, are employed
in the estimation of $R^{3}_{t_{m}},R^{5}_{t_{m}},R^{8}_{t_{m}}$. An
application of Lemma 5.3 allows to control the moments of derivatives of the
solution, $u$, to the Kolmogorov backward equation. Hence, all the above
expectations are well-defined and bounded by
$K(h^{5/2}+h^{3})e^{-\lambda_{0}(T-t_{m+1})}$.

Part 3: Collecting the estimates. For the eight residual terms
$R^{1}_{t_{m}},\ldots,R^{8}_{t_{m}}$, we have

$\displaystyle\sum_{i=1}^{8}\big{|}\mathbb{E}\big{[}R^{i}_{t_{m}}\big{]}\big{|}\leq
K(h^{5/2}+h^{3})e^{-\lambda_{0}(T-t_{m+1})}.$

Combining all of the above estimates, we conclude

$\displaystyle\sum_{m=0}^{M-1}\mathbb{E}\big{[}R(t_{m},\bm{X}^{N,h}_{t_{m}})\big{]}\leq\sum_{m=0}^{M-1}\sum_{i=1}^{8}\big{|}\mathbb{E}\big{[}R^{i}_{t_{m}}\big{]}\big{|}\leq\sum_{m=0}^{M-1}K(h^{5/2}+h^{3})e^{-\lambda_{0}(T-t_{m+1})}\leq
Kh^{3/2}.$

∎

##  7 Numerical illustration

We illustrate the performance of the non-Markovian Euler scheme (2.6) with a
simple linear MV-SDE example. Consider the following mean-field equation:

$\displaystyle\mathrm{d}X_{t}=\Big{(}\alpha\big{(}X_{t}-\mathbb{E}[X_{t}]\big{)}-X_{t}\Big{)}\mathrm{d}t+\sigma\mathrm{d}W_{t},\quad
X_{0}\in L^{10}(\Omega,\mathbb{R}),$ (7.1)

where $\alpha,\sigma>0$. Explicit calculations yield
$\mathbb{E}[X_{t}]=\mathbb{E}[X_{0}]e^{-t}$ and thus this process admits the
following stationary distribution

$\displaystyle\mu^{*}(x)=Z\exp\Big{(}-\frac{\alpha+1}{\sigma^{2}}x^{2}\Big{)},$
(7.2)

where $Z$ is the renormalization constant such that
$\int_{\mathbb{R}}\mu^{*}(x)\mathrm{d}x=1$. We use the following recipe to
compute errors in the numerical experiments.

  1. 1.

Choose a large enough domain $[a,b]$ such that
$\big{(}F_{\textrm{cdf}}(b)-F_{\textrm{cdf}}(a)\big{)}>1-10^{-6}$, where
$F_{\textrm{cdf}}$ denotes the cumulative density function (CDF) of the
invariant distribution $\mu$ of (7.2).

  2. 2.

Split the domain $[a,b]$ into $N_{\textrm{bins}}$ equally spaced bins and
compute the true density in each bin. These values are denoted
$(\mu_{i}^{\textrm{true}})_{i\in\\{1,\dots,N_{\textrm{bins}}\\}}$ and obtained
using numerical integration. Additionally, the first bin and the value
$\mu_{1}^{\textrm{true}}$ takes the interval $(-\infty,a]$ into account while
the last bin (and the value $\mu_{N_{\textrm{bins}}}^{\textrm{true}}$) takes
the interval $[b,\infty)$. In this way, one has
$\sum_{i=1}^{N_{\textrm{bins}}}\mu_{i}^{\textrm{true}}=1$.

  3. 3.

We simulate $N$ particles up to time $T$ using different time-stepping schemes
and compute an approximation of the density (using a histogram approach)
denoted by $(\mu_{i}^{\textrm{proxy}})_{i\in\\{1,\dots,N_{\textrm{bins}}\\}}$
based on the simulated paths.

  4. 4.

As in [40], we compute the relative entropy error and the $L_{2}$-Error as

$\displaystyle\textrm{ Relative Entropy Error
}=\sum_{i=1}^{N_{\textrm{bins}}}\mu_{i}^{\textrm{true}}\ln\Big{(}\frac{\mu_{i}^{\textrm{true}}}{\mu_{i}^{\textrm{proxy}}}\Big{)}\quad\textrm{and}\quad\textrm{
$L_{2}$-Error
}=\sqrt{\sum_{i=1}^{N_{\textrm{bins}}}|\mu_{i}^{\textrm{true}}-\mu_{i}^{\textrm{proxy}}|^{2}}.$

  5. 5.

We also show a PoC result by computing the $L_{2}$-Error for different sizes
$N$ of particles with fixed time $T$ and timestep $h$.

The goal of this simple example is to simulate the IPS associated with (7.1)
up to $T=9$ using the classical Euler method and the non-Markovian Euler
method, respectively, and compare the results to (7.2). Following the recipe
above, we set the domain as $[a,b]=[-1.8,1.8]$ and split it into $72$ bins.

(a) Relative Entropy Error

(b) $L_{2}$-Error

(c) PoC $L_{2}$-Error (log-scale), $T=9$

Figure 1: Simulation of the linear MV-SDE (7.1) with $\alpha=0.5,\sigma=0.8$,
$N=10^{7},h=0.16$, and $X_{0}\sim\mathcal{N}(\pi,1)$ (a normal distribution
with mean value $\pi$ and variance $1$). Both schemes run on the exact same
samples of the initial condition and Brownian increments. (a) Entropy Error of
the Euler method and non-Markovian method in log-scale over time. (b)
$L_{2}$-Error of the Euler method and non-Markovian method in log-scale over
time. (c) $L_{2}$-Error in particle size $N$ of the Euler method and non-
Markovian method in log-scale with respect to different number of particles
$N$ at $T=9$.

Figure 1 (a) and (b) show that the non-Markovian method (uniformly)
outperforms the Euler method in the approximation of the stationary
distribution $\mu$ in both error metrics and the difference in performance
becomes more evident as time increases. This is expected from the result in
Theorem 3.3, since the first order term decays exponentially as $T$ increases.

Figure 1 (c) shows (for a fixed timestep $h=0.04$) that the PoC $L_{2}$-Error
of the non-Markovian method decays consistently as $N$ increases with a rate
of approximately $0.5$ (the expected strong PoC $\mathcal{O}(1/\sqrt{N})$
rate). The error of the Euler method plateaus for $N>10^{5}$ as the error from
the time-discretization dominates the particle error; see also Table 2 below
for more information.

$\alpha$ | $\sigma$ | $a$ | $b$ | $N_{\textrm{bins}}$ | timestep | Entropy Error |  $L_{2}$-error
---|---|---|---|---|---|---|---
Euler | NM | Euler | NM
$0.5$ | $0.8$ | $-1.8$ | $1.8$ | $72$ | $0.04$ | 2.33E-04 | 4.71E-06 | 2.37E-03 | 3.56E-04
0.16 | 3.84E-03 | 4.33E-06 | 9.47E-03 | 3.37E-04
0.24 | 9.26E-03 | 4.40E-06 | 1.47E-02 | 3.25E-04
0.48 | 4.31E-02 | 3.25E-06 | 3.18E-02 | 2.92E-04
$0.3$ | $1.5$ | $-3.0$ | $3.0$ | $120$ | $0.04$ | 1.84E-04 | 7.85E-06 | 1.44E-03 | 3.81E-04
0.16 | 2.98E-03 | 5.94E-06 | 5.88E-03 | 2.82E-04
0.24 | 6.84E-03 | 6.07E-06 | 9.00E-03 | 3.19E-04
0.48 | 3.08E-02 | 5.72E-06 | 1.95E-02 | 3.09E-04
Table 1: Simulation results for MV-SDE (7.1) with $N=10^{7},T=8.64$ and
different choices of parameters using the non-Markovian (NM) Euler and
standard Euler method, respectively. (As for Fig. 1:
$X_{0}\sim\mathcal{N}(\pi,1)$ and both schemes run on the exact same samples
of the initial condition and Brownian increments.)

Table 1 shows that the non-Markovian method has a significantly better
approximation accuracy compared to the Euler method under different choices of
timesteps and model parameters. The Euler method produces larger errors as the
timestep increases and the non-Markovian method yields stable results across
all choices for the timestep.

The results in Table 2 show (at fixed timestep $h=0.04$) the entropy error and
the $L_{2}$-Error of the non-Markovian method decaying as the number of
particles $N$ increases. However, the error of the Euler method remains stable
for $N>10^{5}$ (i.e., there is a plateau). Due to computational limitations,
we are not able to show results beyond $N>10^{7}$. The terminal time $T=8.64$
is chosen for convenience only (due to the smaller timestep $h=0.04$).

$\alpha$ | $\sigma$ | $a$ | $b$ | $N_{\textrm{bins}}$ | $N$ | Entropy Error |  $L_{2}$-Error
---|---|---|---|---|---|---|---
Euler | NM | Euler | NM
$0.5$ | $0.8$ | $-1.8$ | $1.8$ | $72$ | $10^{3}$ | - | - | 2.89E-02 | 3.28E-02
$10^{4}$ | - | - | 1.01E-02 | 1.04E-02
$10^{5}$ | 8.21E-04 | 4.83E-04 | 4.29E-03 | 3.10E-03
$10^{6}$ | 2.74E-04 | 4.66E-05 | 2.31E-03 | 1.26E-03
$10^{7}$ | 2.33E-04 | 4.71E-06 | 2.37E-03 | 3.56E-04
Table 2:  Simulation results for MV-SDE (7.1) with $h=0.04$ and $T=8.64$ for
increasing numbers of particles $N$. (As for Fig. 1:
$X_{0}\sim\mathcal{N}(\pi,1)$ and both schemes run on the exact same samples
of the initial condition and Brownian increments.)

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##  Appendix A Appendix

###  A.1 Proof of Proposition 2.1

###### Proof of Proposition 2.1.

Assumption H.1 is sufficient to guarantee the existence of the unique
stationary distribution; see [13] (under their Assumption (A’)). The uniform
PoC result follows from [42, Theorem 1.2]. This addresses the proposition’s
last two statements.

The system’s wellposedness (as an SDE in $\mathbb{R}^{N}$) follows [42]. For
completeness, we present a short proof for the moment stability of the IPS,
highlighting that the constant $K$ appearing in the RHS of the inequality is
independent of $t$ and $N$.

Let $p\geq 2$. Performing similar calculations as in [20, Appendix A] and
applying Gronwall’s inequality, we deduce that there exists some positive
constant $K$ independent of $N,t\geq 0$, such that for any $i\neq j$

$\displaystyle\mathbb{E}[\,|X^{i,N}_{t}-X^{j,N}_{t}|^{p}]\leq K<\infty.$ (A.1)

Employing Itô’s formula, we deduce for any $t\geq 0$ , $p\geq 2$ and
$\kappa\in[0,\lambda)$

$\displaystyle e^{p\kappa t}|X^{i,N}_{t}|^{p}-|\xi^{i}|^{p}$
$\displaystyle\leq\int_{0}^{t}p\bigg{(}\kappa|X_{s}^{i,N}|^{2}+\big{(}X_{s}^{i,N}\big{)}\cdot\Big{(}-\nabla
U(X_{s}^{i,N})-\frac{1}{N}\sum_{j=1}^{N}\nabla
V(X_{s}^{i,N}-X_{s}^{j,N})\Big{)}\bigg{)}e^{p\kappa
s}|X^{i,N}_{s}|^{p-2}\mathrm{d}s$
$\displaystyle\quad+\sigma^{2}\frac{p(p-1)}{2}\int_{0}^{t}e^{p\kappa
s}|X_{s}^{i,N}|^{p-2}\mathrm{d}s+\sigma p\int_{0}^{t}e^{p\kappa
s}|X_{s}^{i,N}|^{p-2}X_{s}^{i,N}\mathrm{d}W_{s}^{i}.$

Taking expectations on both sides, using the inequality: for $q\in\\{1,2\\}$

$\displaystyle a^{p-q}b^{q}\leq\frac{p-q}{p}\varepsilon
a^{p}+\frac{q}{p\varepsilon^{(p-q)/q}}b^{p},\qquad\textrm{for any
$\varepsilon>0$ and $a,b>0$},$

assumption H.1 and (A.1) yields

$\displaystyle\mathbb{E}$ $\displaystyle\left[e^{p\kappa
t}|X^{i,N}_{t}|^{p}\right]-\mathbb{E}\left[|\xi^{i}|^{p}\right]$
$\displaystyle\leq p\int_{0}^{t}(\kappa-\lambda)\mathbb{E}\left[e^{p\kappa
s}|X^{i,N}_{s}|^{p}\right]\mathrm{d}s$
$\displaystyle\qquad\qquad+\int_{0}^{t}e^{p\kappa
s}\Big{(}\frac{K}{N}\sum_{j=1}^{N}\mathbb{E}\Big{[}|X^{i,N}_{s}|^{p-1}~{}|X^{i,N}_{s}-X^{j,N}_{s}|\Big{]}\Big{)}\mathrm{d}s+K\int_{0}^{t}e^{p\kappa
s}\mathbb{E}\left[|X_{s}^{i,N}|^{p-2}\right]\mathrm{d}s$ $\displaystyle\leq
p\int_{0}^{t}(\kappa-\lambda+\varepsilon)\mathbb{E}\left[e^{\kappa
ps}|X^{i,N}_{s}|^{p}\right]\mathrm{d}s+K\int_{0}^{t}e^{p\kappa
s}\mathrm{d}s\leq Ke^{p\kappa t},$

for some positive constant $K$ and $\varepsilon>0$ arbitrarily small. ∎

We provide the following auxiliary results:

######  Proposition A.1.

Let the assumptions and set up of Proposition 2.1 hold and let the IPS be
given as in (2.4). Then there exists $K\geq 0$, independent of $t,s$ and $N$,
such that for any $s\geq t\geq 0,~{}s-t<1$

$\displaystyle\max_{i\in\\{1,\ldots,N\\}}\mathbb{E}\Big{[}\big{|}X_{s}^{i,N}-X_{t}^{i,N}\big{|}^{2}\Big{]}\leq
K(s-t).$

###### Proof.

By Itô’s formula and an application of the Cauchy–Schwarz and Jensen
inequalities, we have that for any $s\geq t\geq 0,~{}(s-t)<1$,
$i\in\\{1,\dots,N\\}$,

$\displaystyle\mathbb{E}\Big{[}$
$\displaystyle\big{|}X_{s}^{i,N}-X_{t}^{i,N}\big{|}^{2}\Big{]}=\mathbb{E}\bigg{[}\bigg{|}\int_{t}^{s}-\nabla
U(X_{u}^{i,N})-\frac{1}{N}\sum_{j=1}^{N}\nabla
V(X_{u}^{i,N}-X_{u}^{j,N})\mathrm{d}u+\sigma\int_{t}^{s}\mathrm{d}W_{u}^{i}\bigg{|}^{2}\bigg{]}$
$\displaystyle\leq K(s-t)\int_{t}^{s}\mathbb{E}\Big{[}\big{|}\nabla
U(X_{u}^{i,N})\big{|}^{2}\Big{]}+\frac{1}{N}\sum_{j=1}^{N}\mathbb{E}\Big{[}\big{|}\nabla
V(X_{u}^{i,N}-X_{u}^{j,N})\big{|}^{2}\Big{]}\mathrm{d}u+K(s-t)$
$\displaystyle\leq
K(s-t)\Big{(}1+\int_{t}^{s}\mathbb{E}\Big{[}\big{|}X_{u}^{i,N}\big{|}^{2}\Big{]}+\frac{1}{N}\sum_{j=1}^{N}\mathbb{E}\Big{[}\big{|}X_{u}^{j,N}\big{|}^{2}\Big{]}\mathrm{d}u\Big{)}\leq
K(s-t),$

where we used Proposition 2.1 in the last estimate. ∎

###  A.2 Proof of Proposition 2.2

For clarity, we prove each of the proposition’s statements separately. The
main argument requires care as an analysis by cases is needed, but within each
case the estimation procedure is standard.

###### Proof of Proposition 2.2 – Statement (1): Moment estimates.

For the scheme defined in (2.6), we introduce the following notations for any
$m\in\\{0,\ldots,M\\}$,

$\displaystyle b_{m}^{i,N,h}\coloneqq-\nabla
U(X^{i,N,h}_{t_{m}})-\frac{1}{N}\sum_{j=1}^{N}\nabla
V(X_{t_{m}}^{i,N,h}-X_{t_{m}}^{j,N,h}),\quad\Delta\overline{W}_{m}^{i,N,h}\coloneqq\frac{\Delta
W_{m}^{i}+\Delta W_{m+1}^{i}}{2}.$ (A.2)

Assumption H.1 implies that

$\displaystyle\big{(}X_{t_{m}}^{i,N,h}-X_{t_{m}}^{j,N,h}\big{)}\cdot$
$\displaystyle\big{(}~{}b_{m}^{i,N,h}-b_{m}^{j,N,h}\big{)}\leq-\lambda\big{|}X_{t_{m}}^{i,N,h}-X_{t_{m}}^{j,N,h}\big{|}^{2},$
$\displaystyle\big{|}~{}b_{m}^{i,N,h}-b_{m}^{j,N,h}\big{|}^{2}$
$\displaystyle\leq 2\big{|}\nabla U(X^{i,N,h}_{t_{m}})-\nabla
U(X^{j,N,h}_{t_{m}})\big{|}^{2}$
$\displaystyle\quad+\frac{2}{N}\sum_{k=1}^{N}\big{|}\nabla
V(X_{t_{m}}^{i,N,h}-X_{t_{m}}^{k,N,h})-\nabla
V(X_{t_{m}}^{j,N,h}-X_{t_{m}}^{k,N,h})\big{|}^{2}$ $\displaystyle\leq
2(\lambda^{2}+K_{V}^{2})|X_{t_{m}}^{i,N,h}-X_{t_{m}}^{j,N,h}|^{2}\leq
K|X_{t_{m}}^{i,N,h}-X_{t_{m}}^{j,N,h}|^{2}.$

An inspection of the above inequalities indicates that in order to establish
the $L^{p}$-moments for $X_{t_{m}}^{i,N,h}$, we need $L^{p}$-estimates on the
local differences $X_{t_{m}}^{i,N,h}-X_{t_{m}}^{j,N,h}$ (a term that appears
from the interaction kernel). This proof is split accordingly; Part 1 deals
with the latter while Part 2 with the former.

Part 1: Moments of local differences uniformly bounded in time. We first prove
that for all $i,j\in\\{1,\ldots,N\\}$, $m\in\\{0,\ldots,M\\},p\geq 2$, with
$p$ an even number, that
$\mathbb{E}\big{[}|X_{t_{m}}^{i,N,h}-X_{t_{m}}^{j,N,h}|^{p}\big{]}$ is
uniformly bounded in time. Note that due to the nature of the scheme,
$X_{t_{m}}^{i,N,h}$ is not independent of $\Delta\overline{W}_{m}^{i,N,h}$,
and thus we analyze the different cases as time evolves, i.e.,
$m\in\\{0,1,2\\}$ and $m\geq 3$ below. (This same procedure will be used in
Part 2 of the proof.)

Case: $m=0$. For any $i,j\in\\{1,\ldots,N\\}$, we have

$\displaystyle\mathbb{E}\big{[}|X_{t_{0}}^{i,N,h}-X_{t_{0}}^{j,N,h}|^{p}\big{]}$
$\displaystyle\leq
K\mathbb{E}\big{[}|X_{t_{0}}^{i,N,h}|^{p}+|X_{t_{0}}^{j,N,h}|^{p}\big{]}=K\mathbb{E}\big{[}|\xi|^{p}\big{]}.$
(A.3)

Case: $m=1$. We get

$\displaystyle|X_{t_{1}}^{i,N,h}-X_{t_{1}}^{j,N,h}|^{2}$
$\displaystyle=|X_{t_{0}}^{i,N,h}+b_{0}^{i,N,h}h-X_{t_{0}}^{j,N,h}-b_{0}^{j,N,h}h|^{2}+|\sigma\Delta\overline{W}_{0}^{i,N,h}-\sigma\Delta\overline{W}_{0}^{j,N,h}|^{2}$