# Weak error analysis for strong approximation schemes of SDEs with super-
linear coefficients 222This work was supported by Natural Science Foundation
of China (12071488, 11971488), Natural Science Foundation of Hunan Province
(2020JJ2040) and the innovative project of graduate students of Central South
University (2021zzts0040).
E-mail addresses<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Xiaojie Wang  School of Mathematics and Statistics, HNP-LAMA, Central South
University, Changsha, Hunan, P. R. China  Yuying Zhao corresponding author
School of Mathematics and Statistics, HNP-LAMA, Central South University,
Changsha, Hunan, P. R. China  Department of Mathematical Sciences, Worcester
Polytechnic Institute, Worcester, MA 01609 USA  Zhongqiang Zhang  Department
of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609
USA

###### Abstract

We present an error analysis of weak convergence of one-step numerical schemes
for stochastic differential equations (SDEs) with super-linearly growing
coefficients. Following Milstein’s weak error analysis on the one-step
approximation of SDEs, we prove a general conclusion on weak convergence of
the one-step discretization of the SDEs mentioned above. As applications, we
show the weak convergence rates for several numerical schemes of half-order
strong convergence, such as tamed and balanced schemes. Numerical examples are
presented to verify our theoretical analysis.

AMS subject classification: 60H35, 60H15, 65C30.

Keywords: SDEs with non-globally Lipschitz coefficients, weak convergence
theorem, modified Euler method, backward Euler method, weak convergence rate

###### Contents

  1. 1 Introduction
  2. 2 The general weak convergence theorem revisited
    1. 2.1 Notations
    2. 2.2 Stochastic differential equations
    3. 2.3 The general weak convergence theorem
  3. 3 A practical corollary on weak convergence of one-step approximation schemes
    1. 3.1 On Assumption 2.4 for one-step errors
    2. 3.2 Explicit conditions on coefficients of SDEs to fulfill Assumption 3.2
  4. 4 One-step weak errors for the Euler approximation
  5. 5 Weak convergence rates of modified Euler approximations
    1. 5.1 Boundedness of moments of modified Euler approximations
    2. 5.2 Weak convergence rates of the modified Euler approximations
    3. 5.3 Examples of modified Euler schemes
  6. 6 Weak convergence rate of the backward Euler method
  7. 7 Numerical experiments
  8. A Proof of Theorem 2.6
  9. B Proof of Lemma 3.4
  10. C Proofs of Lemmas in Section 4
  11. D Proof of Lemma 5.4
  12. E Proof of Lemma 5.5

##  1 Introduction

In the past three decades, numerical methods for stochastic differential
equations (SDEs) from various applications have been extensively explored.
When the coefficients of SDEs are globally Lipschitz continuous, numerical
methods and their analysis are well documented in [39, 27, 44], see
particularly [37, 1, 2, 45, 15, 30, 13, 50, 33, 28, 24, 43] and references
therein for further analysis of weak approximation schemes. When the
coefficients of SDEs are non-globally Lipschitz continuous with super-linear
growth, it is shown that the commonly used Euler-Maruyama method fails to
converge in the strong and weak sense (see [40], [20], [35] and [22]). Since
then many papers are focusing on SDEs with super-linearly growing
coefficients, such as [20, 22, 51, 23, 21, 47, 34, 17, 26, 8, 11, 10, 46] and
references therein.

Most works focus on strong approximations of SDEs with superlinearly growing
coefficients. There are fewer works on the weak approximation of such SDEs. In
[40], the authors introduced a new concept of discarding the approximate
trajectories that leave a sufficiently large sphere $S_{R}:=\\{x:|x|<R\\}$.
This strategy allows the authors to use any usual weak scheme for solving a
broad class of SDEs with non-globally Lipschitz coefficients, resulting in
arbitrarily small weak errors by increasing the radius of the sphere. However,
the weak convergence rates are not explicitly revealed.  In [41], the authors
devised a new kind of direction and norm decomposition methods for SDEs with
superlinearly growing coefficients and showed a linear rate of weak
convergence and one-half strong order of convergence under mild assumptions.
The authors of [9] proposed an exponential Euler scheme for one-dimensional
SDE with superlinearly growing coefficients and recovered a rate of weak
convergence of order one for the scheme.  In [10], the author analyzed the
long-time error estimates of an explicit tamed Euler scheme applied to a class
of SDEs driven by additive noise, under a one-sided Lipschitz continuity and
superlinear growth condition.

In addition, the topic of weak error analysis for SDEs with non-smooth
coefficients or non-smooth test function has and remains an active field of
research which have made tremendous progress over the past years. For example,
we observe that there are several works on weak convergence of non-globally
Lipschitz coefficients of sublinear growth (e.g, the Heston model), e.g., [36,
55, 3]. There are also many works on weak approximations of SDEs with singular
coefficients or test functions, see, e.g., [5, 6, 49, 48, 29, 42, 31, 32, 18,
7, 25, 52], where the classical scheme such as the Euler-Maruyama scheme still
provides an efficient and quantifiable numerical approximation. The common
ground of these references is roughly to establish properties of the
transition semi-groups related to the considered SDE.

In this work, we develop a weak error analysis for general one-step
approximation schemes of SDEs with non-globally Lipschitz coefficients of
superlinear growth (see Assumption 3.6). Specifically, we first revisit the
general weak convergence theorem for general one-step numerical approximations
and reformulate it under mild conditions (see Theorem 2.6) following
Milstein’s framework, see e.g., Theorem 2 of [38] and Theorem 2.1 in Chapter 2
of [39]. We then establish some explicit conditions on the coefficients of
SDEs to guarantee the aforementioned assumptions in the general convergence
theorem. These assumptions are discussed in Section 2 and we then derive a
corollary on weak convergence of numerical methods for SDEs with one-sided
Lipschitz continuous and superlinearly growing coefficients (Corollary 3.8).
As applications of the corollary, we prove a first-order (or less) weak
convergence for various modified Euler approximations (Theorem 5.6), which are
based on strong convergence, and the backward Euler method (Theorem 6.5). We
summarize our main contributions:

  * •

The weak convergence theorem is revised for general one-step numerical
approximations under mild conditions (see Theorem 2.6). Based on the theorem,
a more specific conclusion on weak convergence of numerical methods is
obtained for SDEs with one-sided Lipschitz continuous and superlinearly
growing coefficients (see Corollary 3.8)

  * •

Weak convergence rates for a class of modified Euler methods (Theorem 5.6) and
the backward Euler method (Theorem 6.5) are established, which fills some
notable gaps in literature on the weak error analysis of strong approximating
schemes.

How to apply our results. Roughly speaking, when the conditions in Assumption
3.6 are satisfied, Corollary 3.8 asserts that if moments of the numerical
solution are bounded and the scheme was proved to be convergent with order
$p+1$ in the one-step error (Assumption 3.1), then the scheme has the weak
convergence order $p$. For example, the boundedness of moments for the
modified Euler schemes is proved under Assumption 5.1 on the bounds of
modified drift and the diffusion coefficients (Theorem 5.2). Then, with
Assumption 5.3 on the errors of modifying drift and the diffusion coefficients
we can obtain from Theorem 5.6 that the modified Euler schemes (5.1) converge
with weak order $p=1$ or smaller, depending on the choices of modifications of
the coefficients; see examples in Section 5.3.

The rest of this article is structured as follows.  In the next section, we
revisit the general convergence theorem on weak convergence of one-step
schemes for SDEs with non-globally Lipschitz continuous coefficients. In
Section 3, we present explicit conditions on coefficients of SDEs, where the
assumptions are satisfied in the general convergence theorem. To prove a
first-order weak convergence, we compare the considered schemes with the
Euler-Maruyama method and its one-step error in Section 4. The one-step error
of the Euler-Maruyama method is proved in Appendix C. In Sections 5 and 6, we
apply the corollary on weak convergence obtained in Section 3 to derive weak
convergence rates for some explicit and implicit numerical methods. We present
some numerical results in Section 7.

##  2 The general weak convergence theorem revisited

In this section, we introduce notations and basic assumptions on stochastic
differential equations. We also introduce the one-step approximation and
revisit a weak convergence theorem for it.

###  2.1 Notations

Throughout this paper, the following notation is frequently used. Let
$|\cdot|$ and $\left\langle\cdot,\cdot\right\rangle$ be the Euclidean norm and
the inner product of vectors in $\mathbb{R}^{d}$, respectively. By $A^{T}$ we
denote the transpose of vector or matrix $A$. Given a matrix $A$, we use
$\|A\|:=\sqrt{\text{trace}(A^{T}A)}$ to denote the trace norm of $A$. On a
probability space $(\Omega,\mathcal{F},\mathbb{P})$, we use $\mathbb{E}$ to
mean expectation and $L^{r}(\Omega;\mathbb{R}^{d\times m})$, $r\in\mathbb{N}$,
to denote the family of $\mathbb{R}^{d\times m}$-valued variables with the
norm defined by $\|\xi\|_{L^{r}(\Omega;\mathbb{R}^{d\times
m})}=(\mathbb{E}[|\xi|^{r}])^{\frac{1}{r}}<\infty$. Next, we give some
notation for partial derivatives of functions, frequently used throughout this
paper. A vector $\alpha=\left(\alpha_{1},\ldots,\alpha_{d}\right)$ is called a
multiindex of order $|\alpha|=\alpha_{1}+\cdots+\alpha_{d},$ where each
component $\alpha_{i}\geq 0$ is a nonnegative integer. For a multiindex
$\alpha$, we define the partial derivatives of
$v\colon\mathbb{R}^{d}\rightarrow\mathbb{R}^{l}$ as

$D^{\alpha}v(x):=\tfrac{\partial^{|\alpha|}v(x)}{\partial
x_{1}^{\alpha_{1}}\cdots\partial
x_{d}^{\alpha_{d}}}=\partial_{x_{1}}^{\alpha_{1}}\cdots\partial_{x_{d}}^{\alpha_{d}}v(x).$
(2.1)

For a nonnegative integer $k$, we use
$D^{k}v(x):=\left\\{D^{\alpha}v(x):|\alpha|=k\right\\}$ to denote the set of
all partial derivatives of order $k$. Moreover, we define

$\left|D^{k}v(x)\right|=\Big{(}\sum_{|\alpha|=k}\left|D^{\alpha}v(x)\right|^{2}\Big{)}^{1/2}.$
(2.2)

We also use $Dv:=(\tfrac{\partial v}{\partial x_{1}},\tfrac{\partial
v}{\partial x_{2}},\cdots\tfrac{\partial v}{\partial x_{d}})$ (gradient
vector) and $D^{2}v:=\big{(}\frac{\partial^{2}v}{\partial x_{i}\partial
x_{j}}\big{)}_{d\times d}$ (Hessian matrix).

Next, we introduce a class of functions with superlinear growth.

######  Definition 2.1 (polynomial growth).

A function $\psi\colon\mathbb{R}^{d}\rightarrow\mathbb{R}$ is said to belong
to the class $\mathbb{H}$, written as $\psi\in\mathbb{H}$, if there exist
constants $L>0,l>0$ such that the following inequality holds

$|\psi(x)|\leq L(1+|x|^{l}).$ (2.3)

For $k\in\mathbb{N}$ we also use $\mathbb{H}^{k}$ to denote a subset of
$\mathbb{H}$, consisting of $k$-times continuously differentiable functions
which, together with its partial derivatives up to and including order $k$,
belong to $\mathbb{H}$.

The above definition is quoted from [39, Definition 1.1 in Chapter 2]. For
random functions, we introduce a concept of mean-square differentiability as
follows.

######  Definition 2.2 (mean-square differentiable).

Let $\Psi\colon\Omega\times\mathbb{R}^{d}\rightarrow\mathbb{R}$ and
$\phi_{i}\colon\Omega\times\mathbb{R}^{d}\rightarrow\mathbb{R}$ be random
functions satisfying

$\lim_{\tau\rightarrow
0}\mathbb{E}\Big{[}\Big{|}\frac{1}{\tau}\big{[}\Psi(x+\tau
e_{i})-\Psi(x)\big{]}-\phi_{i}(x)\Big{|}^{2}\Big{]}=0,\quad\forall
i\in\\{1,2,\cdots,d\\},$ (2.4)

where $e_{i}\in\mathbb{R}^{d}$ is a unit vector in $\mathbb{R}^{d}$, with the
$i$-th element being $1$. Then $\Psi$ is called to be mean-square
differentiable, with $\phi=(\phi_{1},\phi_{2},\cdots,\phi_{d})$ being the
derivative (in the mean-square differentiable sense) of $\Psi$ and we also
write $\partial_{(i)}\Psi=\phi_{i}$.

The above two definitions can be generalized to vector-valued functions in a
component-wise manner.

###  2.2 Stochastic differential equations

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a complete probability space with a
filtration $\\{\mathcal{F}_{t}\\}_{t\in[0,T]}$ ($T>0$), satisfying the usual
conditions. Let $d,m\in\mathbb{N}$, and
$W:[0,T]\times\Omega\rightarrow\mathbb{R}^{m}$ stands for standard Brownian
motions with respect to $\\{\mathcal{F}_{t}\\}_{t\in[0,T]}$. We consider the
following autonomous SDEs in the Itô form of

$\left\\{\begin{aligned}
\text{d}X(t)&=f(X(t))\,\text{d}t+g(X(t))\,\text{d}W(t),\quad t\in(0,T],\\\
X(0)&=X_{0}\in\mathbb{R}^{d},\end{aligned}\right.$ (2.5)

where
$f=(f^{1},f^{2},\cdots,f^{d})^{T}\colon\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$
is the drift coefficient and $f^{i}\colon\mathbb{R}^{d}\rightarrow\mathbb{R}$,
and $g=(g^{i,j})_{d\times m}\colon\mathbb{R}^{d}\rightarrow\mathbb{R}^{d\times
m}$ is the diffusion coefficient. The function $g$ is also written as
$g=(g^{1},g^{2},\cdots,g^{m})=(g_{1}^{T},g_{2}^{T},\cdots,g_{d}^{T})^{T}$,
where $g^{j}\colon\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ is a $d\times 1$
vector function and $g_{i}\colon\mathbb{R}^{d}\rightarrow\mathbb{R}^{m}$ is a
$1\times{m}$ vector function. The initial data $X_{0}\in\mathbb{R}^{d}$ is
assumed to be deterministic for simplicity.

######  Assumption 2.3 (Well-posedness of a strong solution).

For any $X_{0}\in\mathbb{R}^{d}$, the SDE (2.5) admits a unique pathwise
solution $X:[0,T]\times\Omega\rightarrow\mathbb{R}^{d}$, given by

$X(t)=X_{0}+\int_{0}^{t}f(X(s))\,\text{d}s+\int_{0}^{t}g(X(s))\,\text{d}W(s),\quad
a.s..$ (2.6)

Let $X(t,x;s)$ denotes the solution to (2.5) at $s$, starting from the initial
value $x$ at $t$, namely,

$X(t,x;s)=x+\int_{t}^{s}f(X(t,x;r))\,\text{d}r+\int_{t}^{s}g(X(t,x;r))\,\text{d}W(r),\quad
0\leq t\leq s\leq T.$ (2.7)

###  2.3 The general weak convergence theorem

To approximate solutions to (2.5), we construct a uniform mesh on $[0,T]$ with
$h=\frac{T}{N}$ being the step size, for any $N\in\mathbb{N}$. Here we use
$X_{t,x}(s)$ or $X(t,x;s)$ for $0\leq s\leq T$ to denote the solution to (2.5)
at the time instant $s$ with the initial condition $X_{t,x}(t)=X(t,x;t)=x$.
Also, $X(t):=X(0,X_{0};t)$. For $x\in\mathbb{R}^{d}$, $t\in[0,T]$, $h>0$,
$0<t+h\leq T$, we introduce the one-step approximation $Y(t,x;t+h)$ in the
form of

$Y(t,x;t+h)=x+\Phi(t,x,h;\xi_{t}),$ (2.8)

where $\xi_{t}$ is a random variable (in general, a vector) defined on
$(\Omega,\mathcal{F},\mathbb{P})$, having moments of a sufficiently high
order, and $\Phi$ is a function from
$[0,T]\times\mathbb{R}^{d}\times(0,T]\times\mathbb{R}^{m}$ to
$\mathbb{R}^{d}$. Similarly, by $Y(t,x;t+h)$ or $Y_{t,x}(t+h)$ we denote an
approximation of the solution at $t+h$ with initial value
$Y(t,x;t)=Y_{t,x}(t)=x$. Then we use $Y_{n+1}=Y(t_{n},Y_{n};t_{n+1})$ to
recurrently construct numerical approximations $\\{Y_{n}\\}_{0\leq n\leq N}$
on the uniform mesh grid $\\{t_{n}=nh,n=0,1,\cdots,N\\}$, given by

$Y_{0}=X_{0},\quad Y_{n+1}=Y_{n}+\Phi(t_{n},Y_{n},h;\xi_{n}),\quad
n=0,1,\ldots,N-1,$ (2.9)

where the $\xi_{n}$ for $n\geq 0$ is independent of
$Y_{{}_{0}},Y_{{}_{1}},\ldots,Y_{{}_{n}}$, $\xi_{0},\xi_{1},\ldots,\xi_{n-1}$.
Alternatively, one can write

$Y_{n+1}=Y(t_{n},Y_{n};t_{n+1})=Y(t_{0},Y_{0};t_{n+1}),\quad Y_{0}=X_{0},\quad
n=0,1,\ldots,N-1.$ (2.10)

We aim to analyze the weak error of the one-step approximation in terms of

$\big{|}\mathbb{E}\big{[}\varphi\big{(}X(t_{0},X_{0};T)\big{)}\big{]}-\mathbb{E}\big{[}\varphi\big{(}Y(t_{0},Y_{0};t_{N})\big{)}\big{]}\big{|},$
(2.11)

where $\varphi\colon\mathbb{R}^{d}\rightarrow\mathbb{R}$ is called the test
function. To carry out the weak error analysis, we introduce the function
$u\colon[0,T]\times\mathbb{R}^{d}\rightarrow\mathbb{R}$ defined by

$u(t,x)=\mathbb{E}\big{[}\varphi\big{(}X(t,x;T)\big{)}\big{]},$ (2.12)

which is the unique classical solution of the associated Kolmogorov partial
differential equation (PDE) under certain regularity assumptions on
$f,g,\varphi$ (see [14, Theorem 1.6.2] for more details). Boundedness and
differentiability properties of such solutions $u$ to the Kolmogorov PDE play
an essential role in the analysis of weak errors of numerical schemes. In [38]
(see also [39], [40]), Milstein establishes a well-known weak convergence
theorem for general one-step approximation schemes of SDEs with sufficiently
smooth, globally Lipschitz continuous coefficients and sufficiently smooth
test functions with polynomial growth. In [18], the author discusses the
classes of test functions $\varphi$ that can be allowed and also addresses how
to interpret $\mathbb{E}\big{[}\varphi\big{(}X(t,x;T)\big{)}\big{]}$ for such
cases and has to assume strong conditions on $f$ and $g$. Another approach to
prove weak convergence and, if necessary, reduce assumptions on $\varphi$ uses
Malliavin calculus, see, e.g. [28]. A further viewpoint on the topic is
provided by the concept of cubature on Wiener space, see [6], in particular
for conditions on $\varphi$. Overall, the relations among the conditions on
$\varphi$, the definition of weak convergence, and the existence and
properties of $u$ are standard practices in the literature.

In this section, we revisit the general weak convergence theorem [38, 39, 40].
Following the main idea of Milstein’s weak error analysis, we aim to
reformulate a general weak convergence theorem for one-step approximations of
SDEs without global Lipschitz condition. First, we assume on the one-step
approximation error as follows.

######  Assumption 2.4 (One-step approximation error).

For any $t\in[0,T]$ and $x\in\mathbb{R}^{d}$, $u$ given by (2.12) is well-
defined with $|u(t,x)|<\infty$. Additionally, there exists a Borel function
$\nu\colon\mathbb{R}^{d}\rightarrow\mathbb{R}^{+}$ such that

$\sup_{t\in[0,T-h]}\big{|}\mathbb{E}\big{[}u(t+h,X(t,x;t+h))\big{]}-\mathbb{E}\big{[}u(t+h,Y(t,x;t+h))\big{]}\big{|}\leq\nu(x)h^{p+1},\quad\forall
x\in\mathbb{R}^{d},$ (2.13)

where $h>0$ and $t+h\leq T$.

Under the globally Lipschitz condition, the Euler-Maruyama method satisfies
(2.13) with $p=1$ and various higher-order weak schemes were proposed by [1,
2, 30, 39, 45], which satisfy (2.13) with $p\geq 2$.

Second, we need an assumption on bounded moments of numerical solutions.

######  Assumption 2.5 (Bounded moments of numerical solutions).

The numerical approximations $\\{Y_{n}\\}_{0\leq n\leq N}$ uniquely determined
by (2.9) is well defined in $\mathbb{R}^{d}$ and there exists a function
$\Upsilon\colon\mathbb{R}^{d}\rightarrow\mathbb{R}^{+}$ such that

$\sup_{N\in\mathbb{N}}\sup_{0\leq n\leq
N}\mathbb{E}[|\nu(Y_{n})|]\leq\Upsilon(Y_{0})<\infty.$ (2.14)

Here the function $\nu$ comes from Assumption 2.4.

Following the idea of the well-known weak convergence theorem originally
proposed by Milstein ([38, Theorem 2.1], [39, Chapter 2], [40]), we obtain a
similar conclusion under Assumptions 2.3, 2.4, 2.5.  We present the proof in
Appendix A.

######  Theorem 2.6 (A general weak convergence theorem).

Let Assumptions 2.3, 2.4 and 2.5 hold. Then the numerical solution produced by
(2.9) has a weak convergence rate of order $p$ in the sense that

$\big{|}\mathbb{E}\big{[}\varphi\big{(}X(t_{0},X_{0};T)\big{)}\big{]}-\mathbb{E}\big{[}\varphi\big{(}Y(t_{0},Y_{0};t_{N})\big{)}\big{]}\big{|}\leq\color[rgb]{0,0,0}\Upsilon(X_{0})Th^{p},$
(2.15)

where the functions $\varphi\colon\mathbb{R}^{d}\rightarrow\mathbb{R}$ and
$\Upsilon\colon\mathbb{R}^{d}\rightarrow\mathbb{R}^{+}$ come from Assumptions
2.4 and 2.5, respectively.

Before closing this section, we give some remarks on the key assumption
(2.13). For the weak error analysis, boundedness and differentiability
properties of the function $u$ should be fully exploited. Essentially all
articles studying weak convergence of various numerical methods for SDEs
discuss the above properties of $u$ as a solution of the associated Kolmogorov
equation, even in the non-Lipschitz case. Examples of such articles can be
found in the introduction.

##  3 A practical corollary on weak convergence of one-step approximation
schemes

###  3.1 On Assumption 2.4 for one-step errors

In this subsection, we reveal how Assumption 2.4 can be satisfied in the case
that the test functions $\varphi$ obey certain smoothness and polynomial
growth conditions. Similar to Theorem 2.1 (b) in [39], we first make the
following assumptions on one-step errors.

######  Assumption 3.1.

There exist a measurable function $K\colon\mathbb{R}^{d}\rightarrow[0,\infty)$
such that, for any $i_{j}\in\\{1,2,\cdots,d\\}$,

$\displaystyle\bigg{|}\mathbb{E}\Big{[}\prod_{j=1}^{s}(\delta_{X,x})^{i_{j}}\Big{]}-\mathbb{E}\Big{[}\prod_{j=1}^{s}(\delta_{Y,x})^{i_{j}}\Big{]}\bigg{|}\leq{\color[rgb]{0,0,0}K(x)}h^{p+1},\,\,\,s=1,\ldots,2p+1,$
(3.1)
$\displaystyle\bigg{\|}\prod_{j=1}^{2p+2}(\delta_{X,x})^{i_{j}}\bigg{\|}_{L^{2}(\Omega;\mathbb{R})}\leq{\color[rgb]{0,0,0}K(x)}h^{p+1},\quad\bigg{\|}\prod_{j=1}^{2p+2}(\delta_{Y,x})^{i_{j}}\bigg{\|}_{L^{2}(\Omega;\mathbb{R})}\leq{\color[rgb]{0,0,0}K(x)}h^{p+1},$
(3.2)

where we denote

$(\delta_{X,x})^{i_{j}}:=X^{i_{j}}(t,x;t+h)-x^{i_{j}},\quad(\delta_{Y,x})^{i_{j}}:=Y^{i_{j}}(t,x;t+h)-x^{i_{j}},\quad
i_{j}\in\\{1,2,\cdots,d\\}.$ (3.3)

Next, we make further assumptions on the moment boundedness and mean-square
differentiability properties of the solution to SDE (2.7).

######  Assumption 3.2.

Assume that the SDE (2.7) admits a unique
$\\{\mathcal{F}_{s}\\}_{s\in[t,T]}$-adapted solution $X(t,x;s),0\leq t\leq
s\leq T$ such that for sufficiently large $q_{0}\geq 2$

$\sup_{0\leq t\leq s\leq T}\mathbb{E}[|X(t,x;s)|^{q_{0}}]\leq
C(1+|x|^{q_{0}}).$ (3.4)

Moreover, the solution $X(t,x;T)$ is $2p+2$ times mean-square differentiable
with respect to the initial data $x$ and for sufficiently large $q_{1}\geq 2$
and for any $j=1,\ldots,2p+2$ it holds that

$\sup_{x\in\mathbb{R}^{d}}\mathbb{E}[|D^{j}X(t,x;T)|^{q_{1}}]\leq
C(T,q_{1},j)<\infty,$ (3.5)

where $C(T,q_{1},j)$ is a constant depending on $T,q_{1},j$.

We mention that Assumption 3.2 is important and essentially used to obtain
boundedness and differentiability properties of the function $u(t,x)$ given by
(2.12) (see the proof of Lemma 3.4). A natural question thus arises as to how
this assumption can be satisfied in terms of conditions on coefficients $f,g$
of SDEs. In subsection 3.2, we present explicit conditions on coefficients of
SDEs to fulfill the Assumption 3.2. For simplicity, the letter $C$ is used to
denote a generic positive constant independent of time step size and may vary
for each appearance. Additionally, the moment boundedness of the numerical
solution is needed (see also Theorem 2.1 (d) in [39]).

######  Assumption 3.3.

The numerical approximations $\\{Y_{n}\\}_{0\leq n\leq N}$ produced by (2.9)
is well-defined in $\mathbb{R}^{d}$ and for sufficiently large $q_{2}\geq 0$,
$\beta\geq 1$ it holds that

$\sup_{N\in\mathbb{N}}\sup_{0\leq n\leq N}\mathbb{E}[|Y_{n}|^{q_{2}}]\leq
C(1+|Y_{0}|^{\beta q_{2}}).$ (3.6)

Given (3.4), it would be natural to require that $\beta$ in (3.6) be equal to
$1$. Indeed, (3.6) with $\beta=1$ holds for some modified Euler schemes and
the backward Euler method (2.5) (see Theorem 5.2 and Proposition 6.2).
However, this is not the case for some tamed-type methods or the balanced
method (see subsection 2.1.4 in [51]). With these assumptions as well as
smoothness and polynomial growth conditions on test functions $\varphi$, it is
straightforward to verify that Assumption 2.4 holds.

######  Lemma 3.4.

Let $\varphi\in\mathbb{H}^{2p+2}$ where, for some constants $L>0$ and
$\chi\geq 1$,

$|D^{j}\varphi(x)|\leq L(1+|x|^{\chi}),\quad j\in\\{1,2,...,2p+2\\}.$ (3.7)

Let Assumptions 3.2, 3.3 be fulfilled with $q_{0}=2\chi$, $q_{1}=4p+4$,
$q_{2}=2\chi$ and let Assumption 3.1 be satisfied. Then $u$ defined by (2.12)
fulfills Assumption 2.4 with
$\nu(x)=C(1+|x|^{\beta\chi}){\color[rgb]{0,0,0}K(x)}$.

This lemma is proved in Appendix B. The key idea originates from [38, 39].
Roughly speaking, one can use the chain rule to show that
$u(t,x)=\mathbb{E}\big{[}\varphi\big{(}X(t,x;T)\big{)}\big{]}$ is $2p+2$ times
differentiable with respect to $x$ and $u(t,\cdot)\in\mathbb{H}^{2p+2}$ for
any $t\in[0,T]$, by noting that $\varphi\in\mathbb{H}^{2p+2}$ is $2p+2$ times
differentiable and the solution $X(t,x;s)$ of the SDE (2.7) is $2p+2$ times
mean-square differentiable with respect to $x$. Then we employ the Taylor
expansion of $u$ and rely on Assumption 3.1 to obtain the one-step error
(2.13) (consult Appendix B for more details). Combining Theorem 2.6 with Lemma
3.4 gives the following corollary on weak convergence rates of one-step
approximation.

######  Corollary 3.5.

Let the assumptions of Lemma 3.4 hold and in addition let Assumption 2.5 be
satisfied with $\nu(x)=C(1+|x|^{\beta\chi}){\color[rgb]{0,0,0}K(x)}$. Then the
numerical solution produced by (2.9) has a global weak convergence rate of
order $p$, namely,

$\big{|}\mathbb{E}\big{[}\varphi\big{(}X(t_{0},X_{0};T)\big{)}\big{]}-\mathbb{E}\big{[}\varphi\big{(}Y(t_{0},Y_{0};t_{N})\big{)}\big{]}\big{|}\leq\Upsilon(X_{0})Th^{p},$
(3.8)

where the function $\Upsilon$ comes from Assumption 2.5.

###  3.2 Explicit conditions on coefficients of SDEs to fulfill Assumption 3.2

In Assumptions 3.2, it is assumed that the underlying SDE (2.7) is well-posed
with bounded moments and that the solution is mean-square differentiable with
respect to the initial data. It is a classical result that the solution
$X(t,x;s)$ to the SDE (2.7) is $2p+2$ times mean-square differentiable with
respect to the initial data $x$ and satisfies all conditions of Assumption
3.2, when the coefficients $f(x),g(x)$ of the SDE (2.7) are globally Lipschitz
continuous and $2p+2$ times differentiable with respect to $x$.  To fulfill
all conditions of Assumptions 3.2 in a non-globally Lipschitz setting, we
recall the framework of Cerrai [14] (see Hypotheses 1.1, 1.2 therein), where
the drift and diffusion coefficients are assumed to be sufficiently smooth
with polynomial growth and satisfy certain monotonicity conditions.

######  Assumption 3.6.

  1. (A1)

The drift coefficient function $f\in C^{2p+2}(\mathbb{R}^{d};\mathbb{R}^{d})$
and there exists $r\geq 0$ such that

$\sup_{x\in\mathbb{R}^{d}}\frac{|D^{\alpha}f(x)|}{1+|x|^{2r+1-j}}<\infty,\quad|\alpha|=j,\,\,\,j\in\\{0,1,\ldots,2p+2\\};$
(3.9)

  2. (A2)

The diffusion coefficient function $g\in
C^{2p+2}(\mathbb{R}^{d};\mathbb{R}^{d\times m})$ and there exists $\rho\leq r$
such that

$\sup_{x\in\mathbb{R}^{d}}\frac{\|D^{\alpha}g(x)\|}{1+|x|^{\rho-j}}<\infty,\quad|\alpha|=j,\,\,\,j\in\\{0,1,\ldots,2p+2\\};$
(3.10)

  3. (A3)

For all $\lambda_{0}>0$, there exists $C_{\lambda_{0}}\in\mathbb{R}$ such that

$\langle Df(x)y,y\rangle+\lambda_{0}\|Dg(x)y\|^{2}\leq
C_{\lambda_{0}}|y|^{2},\quad\forall x,y\in\mathbb{R}^{d};$ (3.11)

  4. (A4)

There exist $a_{1}>0$ and $\gamma,c_{1}\geq 0$ such that for any
$x,y\in\mathbb{R}^{d}$ it holds

$\langle f(x+y)-f(x),y\rangle\leq-a_{1}|y|^{2r+2}+c_{1}(|x|^{\gamma}+1).$
(3.12)

Examples of nonlinearity satisfying Assumption 3.6 can be found in [14, Remark
1.1.1]. In the proofs which follow we will need some implications of
Assumption 3.6. The Assumption (A1) immediately implies

$|f(x)-f(y)|\leq C(1+|x|^{2r}+|y|^{2r})|x-y|,\quad\forall
x,y\in\mathbb{R}^{d},$ (3.13)

which gives the polynomial growth

$|f(x)|\leq C(1+|x|^{2r+1}),\quad\forall x\in\mathbb{R}^{d}.$ (3.14)

Similarly, it follows from Assumption (A2) that

$\|g(x)-g(y)\|\leq C(1+|x|^{\rho-1}+|y|^{\rho-1})|x-y|,\quad\forall
x,y\in\mathbb{R}^{d},$ (3.15)

and thus

$\|g(x)\|\leq C(1+|x|^{\rho}),\quad\forall x\in\mathbb{R}^{d}.$ (3.16)

In addition, Assumption 3.6 further ensures Khasminskii-type and one-sided
Lipschitz property on $f$ and $g$, which are employed in strong approximation
schemes for SDEs such as tamed and semi-implicit schemes (see, e.g., [19, 23,
51] and references therein). More precisely, Assumption (A3) implies one-sided
Lipschitz property:

$\langle x-y,f(x)-f(y)\rangle+\lambda_{0}\|g(x)-g(y)\|^{2}\leq
C_{\lambda_{0}}|x-y|^{2},\quad\forall x,y\in\mathbb{R}^{d}.$ (3.17)

Under Assumption 3.6, Theorem 1.3.6 of [14] asserts that the SDE (2.7) is
well-posed, with the solution is $2p+2$ times mean-square differentiable with
respect to the initial data $x$.

######  Theorem 3.7.

[14, Theorem 1.3.6] Let Assumption 3.6 be fulfilled. Then the SDE (2.7) admits
a unique adapted solution $X(t,x;s),0\leq t\leq s\leq T$ in $\mathbb{R}^{d}$,
which is $2p+2$ times mean-square differentiable with respect to the initial
data $x\in\mathbb{R}^{d}$ and for any $q\geq 1$ and $j=1,\ldots,2p+2$ it holds
that

$\sup_{x\in\mathbb{R}^{d}}\mathbb{E}[|D^{j}X(t,x;s)|^{q}]\leq\phi_{j,q}(s-t)\leq
C(T,q,j),$ (3.18)

for suitable continuous increasing functions $\phi_{j,q}(t)$, where $C(T,q,j)$
is a constant that depends on $T,q,j$. Moreover, for any $q\geq 1$ it holds
that

$\sup_{0\leq t\leq s\leq T}\mathbb{E}[|X(t,x;s)|^{q}]\leq C(T,q)(1+|x|^{q}),$
(3.19)

and for any $t,s_{1},s_{2}\in[0,T]$, $\gamma\geq 1$, we have

$\|X(t,x;s_{2})-X(t,x;s_{1})\|_{L^{\gamma}(\Omega;\mathbb{R}^{d})}\leq
C(1+|x|^{2r+1})|s_{2}-s_{1}|^{\frac{1}{2}}.$ (3.20)

Theorem 3.7 extends relevant results in the Lipschitz setting to a non-
Lipschitz setting. Moreover, this together with boundedness and
differentiability properties of the test function $\varphi$ (i.e.,
$\varphi\in\mathbb{H}^{2p+2}$) helps establish smoothness estimates of $u$,
i.e., $u(t,x)$ is $2p+2$ times differentiable with respect to $x$ and
$u(t,\cdot)\in\mathbb{H}^{2p+2}$ for any $t\in[0,T]$ (see the proof of Lemma
3.4 for details).  As a direct consequence of (3.13), (3.15), (3.19) and
(3.20), we have, for any $t,s_{1},s_{2}\in[0,T]$,

$\displaystyle\|f(X(t,x;s_{2}))-f(X(t,x;s_{1}))\|_{L^{\gamma}(\Omega;\mathbb{R}^{d})}$
$\displaystyle\leq C(1+|x|^{4r+1})|s_{2}-s_{1}|^{\frac{1}{2}}$ (3.21)
$\displaystyle\|g(X(t,x;s_{2}))-g(X(t,x;s_{1}))\|_{L^{\gamma}(\Omega;\mathbb{R}^{d\times
m})}$ $\displaystyle\leq C(1+|x|^{2r+\rho})|s_{2}-s_{1}|^{\frac{1}{2}}.$
(3.22)

Theorem 3.7 guarantees that all conditions in Assumption 3.2 hold, for any
$q_{0},q_{1}\geq 2$. As a direct consequence of Corollary 3.5 and Theorem 3.7,
one can immediately get the following practical corollary on weak convergence
of one-step approximation schemes.

######  Corollary 3.8 (Weak convergence of one-step approximation under
Assumption 3.6).

Let $\varphi\in\mathbb{H}^{2p+2}$ and let Assumption 3.1 be satisfied, where,
for some constants $L>0$ and $\varkappa\geq 0$,

$K(x)\leq L(1+|x|^{\varkappa}).$ (3.23)

Let Assumption 3.3 be fulfilled with $q_{2}=(2\chi)\vee(\beta\chi+\varkappa)$
and let Assumption 3.6 hold. Then the numerical solution produced by (2.9) has
a global weak convergence rate of order $p$, namely,

$\big{|}\mathbb{E}\big{[}\varphi\big{(}X(t_{0},X_{0};T)\big{)}\big{]}-\mathbb{E}\big{[}\varphi\big{(}Y(t_{0},Y_{0};t_{N})\big{)}\big{]}\big{|}\leq
C\big{(}1+|X_{0}|^{\beta(\beta\chi+\varkappa)}\big{)}h^{p}.$ (3.24)

##  4 One-step weak errors for the Euler approximation

To analyze weak convergence rates of numerical schemes considered in this
work, we introduce the following one-step Euler approximation:

$Y_{E}(t,x;t+h)=x+\int_{t}^{t+h}f(x)\,\text{d}s+\int_{t}^{t+h}g(x)\,\text{d}W(s)$
(4.1)

and denote

$(\delta_{Y_{E},x})^{i_{j}}:=Y^{i_{j}}_{E}(t,x;t+h)-x^{i_{j}},\quad
i_{j}\in\\{1,2,\ldots,d\\},$ (4.2)

where by $x^{i_{j}}$ we mean the $i_{j}$-th coordinate of the vector $x$.  The
weak error is decomposed into two parts: one being the one-step weak error of
the Euler approximation (4.1) (Lemma 4.2) and the other being the discrepancy
between the Euler approximation and the considered schemes (see Lemma 5.5 and
Lemma 6.4 for more details). We will need the following lemma to show the one-
step weak weak error of (4.1).

######  Lemma 4.1 (Moment bounds for one-step approximations).

Let Assumptions (A1) and (A2) be satisfied with $p=1$. For any $q\geq 1$, we
have

$\displaystyle\|\delta_{X,x}-\delta_{Y_{E},x}\|_{L^{2q}(\Omega;\mathbb{R}^{d})}$
$\displaystyle\leq C(1+|x|^{4r+1})h,$ (4.3)
$\displaystyle\|\delta_{Y_{E},x}\|_{L^{2q}(\Omega;\mathbb{R}^{d})}$
$\displaystyle\leq C(1+|x|^{2r+1})h^{\frac{1}{2}},$ (4.4)
$\displaystyle\|\delta_{X,x}\|_{L^{2q}(\Omega;\mathbb{R}^{d})}$
$\displaystyle\leq C(1+|x|^{2r+1})h^{\frac{1}{2}}.$ (4.5)

Here $r$ is defined in (3.9)

Lemma 4.1 leads to the desired one-step weak errors for the Euler
approximation (4.1).

######  Lemma 4.2 (One-step weak error of the Euler approximation).

Under the same condition of Lemma 4.1. Then for $i_{j}\in\\{1,2,\cdots,d\\}$
we have

$\bigg{|}\mathbb{E}\Big{[}\prod_{j=1}^{s}(\delta_{X,x})^{i_{j}}-\prod_{j=1}^{s}(\delta_{Y_{E},x})^{i_{j}}\Big{]}\bigg{|}\leq
C(1+|x|^{8r+3})h^{2},\,\,s=1,2,3.$ (4.6)

The proofs of these two lemmas can be found in Appendix C.

##  5 Weak convergence rates of modified Euler approximations

In this section, we prove weak convergence rates of several modified Euler
schemes. By Corollary 3.8, we need to prove boundedness of moments and analyze
the local error. Given a uniform mesh on $[0,T]$ with $h=\frac{T}{N}>0$
equispaced time step sizes, the following numerical scheme is defined:

$Y_{n+1}=Y_{n}+\bar{f}_{h}(Y_{n})h+\bar{g}_{h}(Y_{n})\Delta W_{n},\quad
Y_{0}=X_{0},$ (5.1)

where $\Delta W_{n}:=W(t_{n+1})-W(t_{n}),$ $n\in\\{0,1,2,\ldots,N-1\\}$ and
$\bar{f}_{h}\colon\Omega\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d},$
$\bar{g}_{h}\colon\Omega\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d\times m}$
are some measurable functions. The weak convergence rate of this scheme will
be analyzed in general. To apply the previously obtained corollary on weak
convergence, the first step is to discuss the boundedness of the moments of
the numerical approximations (5.1).

###  5.1 Boundedness of moments of modified Euler approximations

In this subsection we discuss moment boundedness for numerical solutions of
the modified Euler scheme (5.1). As noted in [51], the bounded moments for
numerical solutions are usually discussed case by case. Next, we illustrate
the moment boundedness using the conditions from [47, Lemma 2], which require
$\bar{f}_{h}$ and $\bar{g}_{h}$ to be deterministic and satisfy the following
properties.

######  Assumption 5.1.

Let $\bar{f}_{h}\colon\mathbb{R}^{d}\rightarrow\mathbb{R}^{d},$
$\bar{g}_{h}\colon\mathbb{R}^{d}\rightarrow\mathbb{R}^{d\times m}$ be
deterministic functions satisfying the following properties.
(1) There exists $\theta\in(0,\frac{1}{2}]$ such that, for $\forall
x\in\mathbb{R}^{d},$

$\displaystyle|\bar{f}_{h}(x)|$
$\displaystyle\leq\min\big{(}Ch^{-\theta}(1+|x|),|{f(x)}|\big{)},$ (5.2)
$\displaystyle\|\bar{g}_{h}(x)\|^{2}$
$\displaystyle\leq\min\big{(}Ch^{-\theta}(1+|x|^{2}),\|g(x)\|^{2}).$ (5.3)

(2) There exists a sufficiently large constant $\mu>0$ such that

$\langle x,\bar{f}_{h}(x)\rangle+\mu\|\bar{g}_{h}(x)\|^{2}\leq
C(1+|x|^{2}),\quad x\in\mathbb{R}^{d}.$ (5.4)

The moment bounds of the numerical approximations produced by (5.1) are proved
under Assumption 5.1 (see [47, Lemma 2]). Specifically, the author obtained
the moment bounds of the following continuous version of the numerical scheme
(5.1)

$Y_{t}=Y_{0}+\int_{0}^{t}\bar{f}_{h}(Y_{\lfloor
s\rfloor_{h}})\,\text{d}s+\int_{0}^{t}\bar{g}_{h}(Y_{\lfloor
s\rfloor_{h}})\,\text{d}W(s),\quad t\in[0,T].$ (5.5)

where $\lfloor t\rfloor_{h}:=t_{i}$ for
$t\in[t_{i},t_{i+1}),\,i\in\\{0,1,\ldots,N-1\\}$.

######  Theorem 5.2 ([47]).

Let Assumption (A1) be satisfied with $p=1$ and in addition assume that the
Assumption (A3) and Assumption 5.1 hold. Then for any $q\geq 2$, the
approximation process $Y_{t}$ produced by (5.5) obeys

$\sup_{t\in[0,T]}\mathbb{E}[|Y_{t}|^{q}]\leq C(1+|X_{0}|^{q}).$ (5.6)

Here $C$ depends on $T$, $q$ and is independent of $h$.

Even though the conditions in Assumption 5.1 are general, there are many
modified Euler type schemes that do not satisfy these conditions, see, e.g.,
the tamed scheme (5.20) and the balanced type schemes (5.21), (5.22). We
mention that Assumption 5.1 is only used to get moment bounds of the numerical
approximations. Once the moment bounds of numerical approximations are
available, we do not require the considered schemes to satisfy Assumption 5.1.
In Section 5.3, we present some examples of modified Euler schemes and apply
our weak convergence theorem to obtain their weak convergence rates.

###  5.2 Weak convergence rates of the modified Euler approximations

According to Corollary 3.8, we still need to verify Assumption 3.1, with the
one-step weak convergence rate in (3.1) and (3.2). To this aim, let us
consider the one-step approximation of the numerical method (5.1), given by

$Y(t,x;t+h)=x+\bar{f}_{h}(x)h+\bar{g}_{h}(x)\big{(}W(t+h)-W(t)\big{)}.$ (5.7)

For the purpose of the weak error analysis for the modified Euler schemes, we
make the following assumptions, describing the discrepancy between $f,g$ and
$\bar{f}_{h},\bar{g}_{h}$.

######  Assumption 5.3.

For $\bar{f}_{h}\colon\Omega\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d},$
$\bar{g}_{h}\colon\Omega\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d\times
m}$, there exist $\mathsf{a},\mathsf{b},r_{1},\rho_{1}\geq 0$ such that, for
any $q\geq 1,$

$\displaystyle\|\bar{f}_{h}(x)-f(x)\|_{L^{2q}(\Omega;\mathbb{R}^{d})}$
$\displaystyle\leq Ch^{\mathsf{a}}(1+|x|^{r_{1}}),\quad x\in\mathbb{R}^{d},$
(5.8) $\displaystyle\|\bar{g}_{h}(x)-g(x)\|_{L^{2q}(\Omega;\mathbb{R}^{d\times
m})}$ $\displaystyle\leq Ch^{\mathsf{b}}(1+|x|^{\rho_{1}}),\quad
x\in\mathbb{R}^{d},$ (5.9)
$\displaystyle\big{|}\mathbb{E}\big{[}\big{(}g(x)-\bar{g}_{h}(x)\big{)}\big{(}W(t+h)-W(t)\big{)}\big{]}\big{|}$
$\displaystyle\leq Ch^{\mathsf{b}+1}(1+|x|^{\rho_{1}}),\quad
x\in\mathbb{R}^{d},h>0.$ (5.10)

Assumption 5.3 indicates that different choices of $\bar{f}_{h},\bar{g}_{h}$
would lead to different values of parameters $\mathsf{a},\mathsf{b}$, which
determine the weak convergence rates of the schemes (see Theorem 5.6 and
subsection 5.3).  Under this assumption, we can get the strong error estimates
for the one-step approximations, which are needed for the desired weak error
estimates.

######  Lemma 5.4.

Under the Assumptions (A1), (A2) and Assumption 5.3, then for any $q\geq 1$ it
holds that

$\displaystyle\|\delta_{Y_{E},x}-\delta_{Y,x}\|_{L^{2q}(\Omega;\mathbb{R}^{d})}$
$\displaystyle\leq
C(1+|x|^{r_{1}})h^{\mathsf{a}+1}+C(1+|x|^{\rho_{1}})h^{\mathsf{b}+\frac{1}{2}},$
(5.11) $\displaystyle\|\delta_{Y,x}\|_{L^{2q}(\Omega;\mathbb{R}^{d})}$
$\displaystyle\leq C(1+|x|^{2r+r_{1}+\rho_{1}+1})h^{\frac{1}{2}}.$ (5.12)

The detailed proof of Lemma 5.4 is presented in the Appendix D. With the aid
of Lemma 5.4, the next lemma gives the local error estimates of the modified
Euler scheme (5.1).

######  Lemma 5.5 (One-step weak error estimates).

Under the same conditions of Lemma 5.4, then the one-step approximation (5.7)
of the numerical method (5.1) satisfies

$\displaystyle\bigg{|}\mathbb{E}\Big{[}\prod_{j=1}^{s}(\delta_{X,x})^{i_{j}}-\prod_{j=1}^{s}(\delta_{Y,x})^{i_{j}}\Big{]}\bigg{|}$
$\displaystyle\leq
C(1+|x|^{8r+3\rho_{1}+3r_{1}+3})h^{2\land(\mathsf{a}+1)\land(\mathsf{b}+1)},\,\,s=1,2,3,$
(5.13)
$\displaystyle\bigg{\|}\prod_{j=1}^{4}(\delta_{X,x})^{i_{j}}\bigg{\|}_{L^{2}(\Omega;\mathbb{R})}$
$\displaystyle\leq C(1+|x|^{8r+4})h^{2},$ (5.14)
$\displaystyle\bigg{\|}\prod_{j=1}^{4}(\delta_{Y,x})^{i_{j}}\bigg{\|}_{L^{2}(\Omega;\mathbb{R})}$
$\displaystyle\leq C(1+|x|^{8r+4r_{1}+4\rho_{1}+4})h^{2}.$ (5.15)

The proof of this lemma is given in Appendix E. By Theorem 5.2, Lemma 5.5, and
Corollary 3.8, we are able to identify the weak convergence rates of the
modified numerical methods (5.1) as follows.

######  Theorem 5.6 (Weak convergence rate for the method (5.1)).

Let the Assumption 3.6 hold with $p=1$ and in addition assume that the
Assumptions 5.1, 5.3 hold. Then the approximation method (5.1) has a global
weak convergence rate of order $(1\land\mathsf{a}\land\mathsf{b})$: for
$\varphi\in\mathbb{H}^{4}$,

$\big{|}\mathbb{E}\big{[}\varphi\big{(}X(t_{0},X_{0};T)\big{)}\big{]}-\mathbb{E}\big{[}\varphi\big{(}Y(t_{0},Y_{0};T)\big{)}\big{]}\big{|}\leq
C\big{(}1+|X_{0}|^{(8r+4\rho_{1}+4r_{1}+\chi+4)}\big{)}h^{1\land\mathsf{a}\land\mathsf{b}},$
(5.16)

where $r$ and $\chi$ are defined in (3.9) and (3.7), respectively. If all
conditions above are satisfied except that Assumption 5.1 is replaced by
boundedness of moments of numerical solutions in Assumption 3.3, then the
approximation method (5.1) has a global weak convergence rate of order
$(1\land\mathsf{a}\land\mathsf{b})$.

###  5.3 Examples of modified Euler schemes

Here are some examples of modified Euler type schemes (5.1), where
$\bar{f}_{h}$ and $\bar{g}_{h}$ are explicitly given. Furthermore, Theorem 5.6
is applied to obtain the expected weak convergence rates of the considered
schemes.

######  Example 5.7.

(Fully tamed Euler (FTE) [47]) The author of [47] proposes two drift-diffusion
fully tamed Euler schemes (5.1), where $\bar{f}_{h},\bar{g}_{h}$ are given as
follows.

Model 1 (FTE1):

$\bar{f}_{h}(x)=\frac{f(x)}{1+h^{\alpha_{1}}|f(x)|+h^{\alpha_{2}}\|g(x)\|^{2}},\quad\bar{g}_{h}(x)=\frac{g(x)}{1+h^{\alpha_{1}}|f(x)|+h^{\alpha_{2}}\|g(x)\|^{2}},\quad
0<\alpha_{1},\alpha_{2}\leq\tfrac{1}{2}.$ (5.17)

One immediately observes that Assumption 5.1 is satisfied with
$\theta=\alpha_{1}\land\alpha_{2}\leq\tfrac{1}{2}.$ Also, by Assumption (A1),
(A2), one can easily check that Assumption 5.3 is satisfied with

$r_{1}=4r+2,\,\rho_{1}=2r+\rho+1,\,\mathsf{a}=\mathsf{b}=\alpha_{1}\land\alpha_{2}.$

From Theorem 5.6, it follows that the weak convergence rate is
$\alpha_{1}\land\alpha_{2}\leq\tfrac{1}{2}$. We also present numerical results
(Examples 7.1 and 7.2) and verify the half-order convergence by taking
$\alpha_{1}=\alpha_{2}=\frac{1}{2}.$

Model 2 (FTE2):

$\bar{f}_{h}(x)=\frac{f(x)}{1+h^{\vartheta}|x|^{2r}},\quad\bar{g}_{h}(x)=\frac{g(x)}{1+h^{\vartheta}|x|^{2r}},\quad
0<\vartheta\leq\tfrac{1}{2}.$ (5.18)

Evidently, Assumption 5.1 is fulfilled with
$\theta=\vartheta\leq\tfrac{1}{2}.$ In view of Assumptions (A1), (A2), we
observe that Assumption 5.3 is satisfied with

$r_{1}=4r+1,\,\rho_{1}=2r+\rho,\,\mathsf{a}=\mathsf{b}=\vartheta.$

By Theorem 5.6, the weak convergence rate is $\vartheta\leq\tfrac{1}{2}.$ We
also present numerical results (Examples 7.1 and 7.2) and verify the half-
order convergence by taking $\vartheta=\frac{1}{2}.$

######  Example 5.8 (Modified Euler scheme (MES)).

In this work we propose a new modified Euler scheme, where

$\bar{f}_{h}(x)=\frac{f(x)}{1+h|f(x)|^{2}},\quad\bar{g}_{h}(x)=\frac{g(x)}{1+h|f(x)|^{2}}.$
(5.19)

In a similar way, one then observes that the Assumption 5.1 is satisfied with
$\theta=\frac{1}{2}$ and furthermore that due to Assumption (A1) and (A2), the
Assumption 5.3 with

$r_{1}=6r+3,\,\rho_{1}=4r+\rho+2,\,\mathsf{a}=\mathsf{b}=1.$

According to Theorem 5.6, we derive the weak convergence rate of order $1$. We
also present numerical results (Examples 7.1 and 7.2) to verify the first-
order weak convergence.

Below, we discuss schemes which does not satisfy the Assumption 5.1 but the
moment boundedeness of numerical solutions are available in the literature
[23, 54, 51].

######  Example 5.9 (Drift-tamed Euler (DTE) [23]).

In [23], the diffusion coefficient $g$ is assumed to be globally Lipschitz
continuous. In this setting, the diffusion coefficient $g$ does not need to be
tamed and a drift-tamed Euler scheme is introduced, where

$\bar{f}_{h}(x)=\frac{f(x)}{1+h|f(x)|},\quad\bar{g}_{h}(x)=g(x).$ (5.20)

The boundedeness of moments of the numerical approximations can be found in
[23, Lemma 3.9]. Using similar arguments as above, one can show Assumption 5.3
is satisfied with $r_{1}=4r+2,\mathsf{a}=1.$ By Theorem 5.6, we get the weak
convergence rate of order $1$, which is confirmed by numerical experiment
(Example 7.2).

######  Example 5.10.

[Balanced scheme (BS) [54]] The authors of [54] propose an explicit balanced
scheme, where

$\bar{f}_{h}(x)=h^{-1}\tanh(hf(x)),\quad\bar{g}_{h}(x)=h^{-1/2}\tanh(h^{1/2}g(x)).$
(5.21)

As shown by [54, Lemma 3.1], the numerical solutions have bounded moments. By
Taylor’s formula and Assumptions (A1), (A2), one can readily verify that
Assumption 5.3 is satisfied with $r_{1}=4r+2$, $\rho_{1}=3\rho$ and
$\mathsf{a}=\mathsf{b}=1$. According to Theorem 5.6, the weak convergence rate
is one. We also present numerical results (Examples 7.1 and 7.2) and verify
the first-order convergence.

######  Example 5.11.

[Balanced-type scheme (BTS) [51]] The authors consider a balanced-type scheme,
where

$\bar{f}_{h}(x)=\frac{f(x)}{1+h|f(x)|+|g(x)[W(t+h)-W(t)]|},\>\bar{g}_{h}(x)=\frac{g(x)}{1+h|f(x)|+|g(x)[W(t+h)-W(t)]|}.$
(5.22)

Using similar arguments shows that Assumption 5.3 is satisfied with
$r_{1}=4r+2,\rho_{1}=2r+\rho+1,\mathsf{a}=\mathsf{b}=\frac{1}{2}.$ By Theorem
5.6, we obtain the weak convergence rate of order $\frac{1}{2}$, which is
confirmed by numerical result (Example 7.1).

##  6 Weak convergence rate of the backward Euler method

As another application of the corollary on weak convergence, we analyze the
weak convergence rate of the well-known backward Euler method (BEM) applied to
SDEs (2.5):

$Y_{n+1}=Y_{n}+f(Y_{n+1})h+g(Y_{n})\Delta W_{n},\quad Y_{0}=X_{0},$ (6.1)

where $\Delta W_{n}:=W(t_{n+1})-W(t_{n})$, $n\in\\{0,1,2,\ldots,N-1\\}.$ Then
the one-step approximation of (6.1) is

$Y(t,x;t+h)=x+\int_{t}^{t+h}f(Y(t,x;t+h))\,\text{d}s+\int_{t}^{t+h}g(x)\,\text{d}W(s).$
(6.2)

Denote $\delta_{X,x}:=X(t,x;t+h)-x,\,\delta_{Y,x}:=Y(t,x;t+h)-x.$ Then by
(4.1),

$\begin{split}Y(t,x;t+h)=Y_{E}(t,x;t+h)+R_{f},\quad
R_{f}:=\int_{t}^{t+h}f(Y(t,x;t+h))-f(x)\,\text{d}s.\end{split}$ (6.3)

Before proceeding further, we shall establish the well-posedness of the
implicit scheme (6.1).

######  Proposition 6.1.

Let the condition (A3) in Assumption 3.6 be satisfied and let
$hC_{\lambda_{0}}<1$. Then the implicit scheme (6.1) has a unique adapted
solution $\\{Y_{n}\\}_{n=1}^{N}$ with probability one.

The proof can be found in [4, Theorem 3.1].  As already established in [21,
Corollary 2.27], $q$-th moments of the BEM are uniformly bounded under
Assumption (A3).

######  Proposition 6.2.

Suppose Assumption (A3) holds. Then there exists $C>0,$ independent of h such
that for all $N\in\mathbb{N}$ and $n=0,1,\ldots,N,$

$\sup_{N\in\mathbb{N}}\sup_{0\leq n\leq N}\mathbb{E}[|Y_{n}|^{q}]\leq
C\big{(}1+|X_{0}|^{q}\big{)},\,\,\,\forall\,q\geq 0,$ (6.4)

where $\\{Y_{n}\\}_{0\leq n\leq N}$ is given in (6.1).

Equipped with the moment bound, we continue to analyze the weak convergence
rate of the BEM (6.1) by means of Corollary 3.8.  The local weak errors for
the BEM is similar to that for the modified Euler schemes. At first, we prove
the following lemma on one-step strong error bounds of the BEM.

######  Lemma 6.3.

Under the same condition of Proposition 6.2, letting Assumptions (A1) and (A2)
hold with $p=1,$ then for any $q\geq 1$ the following inequalities hold for
the backward Euler scheme (6.1):

$\displaystyle\|\delta_{Y,x}\|_{L^{2q}(\Omega;\mathbb{R}^{d})}$
$\displaystyle\leq C(1+|x|^{2r+1})h^{\frac{1}{2}},$ (6.5)
$\displaystyle\|R_{f}\|_{L^{2q}(\Omega;\mathbb{R}^{d})}$ $\displaystyle\leq
C(1+|x|^{4r+1})h^{\frac{3}{2}},$ (6.6)
$\displaystyle\|\delta_{X,x}-\delta_{Y,x}\|_{L^{2q}(\Omega;\mathbb{R}^{d})}$
$\displaystyle\leq C(1+|x|^{4r+1})h,$ (6.7)

where $R_{f}$ is defined in (6.3).

Proof of Lemma 6.3. Repeating the same lines in the proof of (C.2) and (6.4)
results in

$\displaystyle\mathbb{E}\big{[}|(\delta_{Y,x})|^{2q}\big{]}$ $\displaystyle=$
$\displaystyle\mathbb{E}\Big{[}\Big{|}\int_{t}^{t+h}f\big{(}Y(t,x;t+h)\big{)}\,\text{d}s+\int_{t}^{t+h}g(x)\,\text{d}W(s)\Big{|}^{2q}\Big{]}$
(6.8) $\displaystyle\leq$ $\displaystyle
Ch^{2q-1}\int_{t}^{t+h}\big{\|}f\big{(}Y(t,x;t+h)\big{)}\big{\|}_{L^{2q}(\Omega,\mathbb{R}^{d})}^{2q}\,\text{d}s+Ch^{q-1}\int_{t}^{t+h}\big{\|}g(x)\big{\|}_{L^{2q}(\Omega,\mathbb{R}^{d\times
m})}^{2q}\,\text{d}s$ $\displaystyle\leq$ $\displaystyle
C(1+|x|^{2q(2r+1)})h^{q},$

which confirms (6.5). To prove (6.6), we use the Hölder inequality, (3.21) and
(6.5) to derive that

$\displaystyle\mathbb{E}\big{[}\big{|}R_{f}\big{|}^{2q}\big{]}$
$\displaystyle\leq$ $\displaystyle
Ch^{2q-1}\int_{t}^{t+h}\big{\|}f(Y(t,x;t+h))-f(x)\big{\|}_{L^{2q}(\Omega,\mathbb{R}^{d})}^{2q}\,\text{d}s$
(6.9) $\displaystyle\leq$ $\displaystyle C(1+|x|^{2q(4r+1)})h^{3q}.$

Also, one can follow the same lines in the proof of (4.3) to show that

$\displaystyle\mathbb{E}\big{[}|\delta_{X,x}-\delta_{Y,x}|^{2q}\big{]}$ (6.10)
$\displaystyle=$
$\displaystyle\mathbb{E}\Big{[}\Big{|}\int_{t}^{t+h}f\big{(}X(s)\big{)}-f\big{(}Y(t,x;t+h)\big{)}\,\text{d}s+\int_{t}^{t+h}g\big{(}X(s)\big{)}-g(x)\,\text{d}W(s)\Big{|}^{2q}\Big{]}$
$\displaystyle\leq$ $\displaystyle
Ch^{2q-1}\int_{t}^{t+h}\big{\|}f(X(s))-f(x)\big{\|}_{L^{2q}(\Omega,\mathbb{R}^{d})}^{2q}\,\text{d}s+Ch^{2q-1}\int_{t}^{t+h}\big{\|}f(x)-f(Y(t,x;t+h))\big{\|}_{L^{2q}(\Omega,\mathbb{R}^{d})}^{2q}\,\text{d}s$
$\displaystyle\quad+Ch^{q-1}\int_{t}^{t+h}\big{\|}g(X(s))-g(x)\big{\|}_{L^{2q}(\Omega,\mathbb{R}^{d\times
m})}^{2q}\,\text{d}s$ $\displaystyle\leq$ $\displaystyle
C(1+|x|^{2q(4r+1)})h^{2q},$

as required. This thus finishes the proof of the lemma. ∎

Based on Taylor’s expansion and Lemma 6.3, we are able to obtain the following
local errors of the BEM (6.1).

######  Lemma 6.4.

Under the same conditions of Lemma 6.3, one can obtain the local errors of the
BEM (6.2)

$\displaystyle\bigg{|}\mathbb{E}\Big{[}\prod_{j=1}^{s}(\delta_{X,x})^{i_{j}}-\prod_{j=1}^{s}(\delta_{Y,x})^{i_{j}}\Big{]}\bigg{|}$
$\displaystyle\leq C(1+|x|^{8r+3})h^{2},\,\,s=1,2,3,$ (6.11)
$\displaystyle\bigg{\|}\prod_{j=1}^{4}(\delta_{Y,x})^{i_{j}}\bigg{\|}_{L^{2}(\Omega;\mathbb{R})}$
$\displaystyle\leq C(1+|x|^{8r+4})h^{2}.$ (6.12)

Proof of Lemma 6.4. For $s=1,$ we first note that

$\displaystyle\big{|}\mathbb{E}\big{[}(\delta_{X,x})^{i_{1}}-(\delta_{Y,x})^{i_{1}}\big{]}\big{|}=\big{|}\mathbb{E}\big{[}(\delta_{X,x})^{i_{1}}-(\delta_{Y_{E},x})^{i_{1}}-R_{f}^{i_{1}}\big{]}$
(6.13) $\displaystyle\leq$
$\displaystyle\big{|}\mathbb{E}\big{[}(\delta_{X,x})^{i_{1}}-\delta_{Y_{E},x})^{i_{1}}\big{]}\big{|}+\bigg{|}\mathbb{E}\bigg{[}\int_{t}^{t+h}f^{i_{1}}\big{(}Y(t,x;t+h)\big{)}-f^{i_{1}}(x)\,\text{d}s\bigg{]}\bigg{|}.$

The estimate for the first term has been done in (C.3), namely,

$\big{|}\mathbb{E}\big{[}(\delta_{X,x})^{i_{1}}-(\delta_{Y_{E},x})^{i_{1}}\big{]}\big{|}\leq
C(1+|x|^{4r+1})h^{2}.$ (6.14)

Next we bound the second term. Employing Taylor’s expansion in the form of

$\displaystyle f^{i_{1}}\big{(}Y(t,x;t+h)\big{)}$ $\displaystyle=$
$\displaystyle f^{i_{1}}(x)+\sum^{d}_{j_{1}=1}\frac{\partial
f^{i_{1}}(x)}{\partial x_{j_{1}}}\big{(}Y^{j_{1}}(t,x;t+h)-x^{j_{1}}\big{)}$
(6.15)
$\displaystyle\quad+\sum^{d}_{j_{1},j_{2}=1}\int_{0}^{1}\frac{\partial^{2}f^{i_{1}}\big{(}x+\theta\big{(}Y(t,x;t+h)-x\big{)}\big{)}}{\partial
x_{j_{1}}\partial x_{j_{2}}}(1-\theta)\,\text{d}\theta$
$\displaystyle\quad\cdot\big{(}Y^{j_{1}}(t,x;t+h)-x^{j_{1}}\big{)}\big{(}Y^{j_{2}}(t,x;t+h)-x^{j_{2}}\big{)},$

and using the assumption on the function $f$ in (3.9) and the Hölder
inequality, one can infer

$\displaystyle\Big{|}\mathbb{E}\Big{[}\int_{t}^{t+h}f^{i_{1}}\big{(}Y(t,x;t+h)\big{)}-f^{i_{1}}(x)\,\text{d}s\Big{]}\Big{|}$
(6.16) $\displaystyle\leq$
$\displaystyle\sum^{d}_{j_{1}=1}\Big{|}\mathbb{E}\Big{[}\int_{t}^{t+h}\frac{\partial
f^{i_{1}}(x)}{\partial
x_{j_{1}}}\big{(}Y^{j_{1}}(t,x;t+h)-x^{j_{1}}\big{)}\,\text{d}s\Big{]}\Big{|}$
$\displaystyle\quad+\sum^{d}_{j_{1},j_{2}=1}\Big{|}\mathbb{E}\Big{[}\int_{t}^{t+h}\int_{0}^{1}\frac{\partial^{2}f^{i_{1}}(x+\theta(Y(t,x;t+h)-x))}{\partial
x_{j_{1}}\partial x_{j_{2}}}(1-\theta)\,\text{d}\theta\,\text{d}s$
$\displaystyle\quad\cdot\big{(}Y^{j_{1}}(t,x;t+h)-x^{j_{1}}\big{)}\big{(}Y^{j_{2}}(t,x;t+h)-x^{j_{2}}\big{)}\Big{]}\Big{|}$
$\displaystyle\leq$ $\displaystyle
C(1+|x|^{2r})\Big{|}\mathbb{E}\Big{[}\int_{t}^{t+h}\big{(}Y^{j_{1}}(t,x;t+h)-x^{j_{1}}\big{)}\,\text{d}s\Big{]}\Big{|}$
$\displaystyle\quad+C(1+|x|^{2r-1})h\,\|(\delta_{Y,x})^{j_{1}}\|_{L^{2}(\Omega;\mathbb{R})}\|(\delta_{Y,x})^{j_{2}}\|_{L^{2}(\Omega;\mathbb{R})}$
$\displaystyle\leq$ $\displaystyle C(1+|x|^{6r+1})h^{2}.$

This together with (6.14) implies

$\big{|}\mathbb{E}\big{[}(\delta_{X,x})^{i_{1}}-(\delta_{Y,x})^{i_{1}}\big{]}\big{|}\leq
C(1+|x|^{6r+1})h^{2}.$ (6.17)

For $s=2$, we first note that

$\displaystyle\big{|}\mathbb{E}\big{[}(\delta_{X,x})^{i_{1}}(\delta_{X,x})^{i_{2}}-(\delta_{Y,x})^{i_{1}}(\delta_{Y,x})^{i_{2}}\big{]}\big{|}$
(6.18) $\displaystyle\leq$
$\displaystyle\big{|}\mathbb{E}\big{[}(\delta_{X,x})^{i_{1}}((\delta_{X,x})^{i_{2}}-(\delta_{Y,x})^{i_{2}})\big{]}\big{|}+\big{|}\mathbb{E}\big{[}((\delta_{X,x})^{i_{1}}-(\delta_{Y,x})^{i_{1}})(\delta_{Y,x})^{i_{2}}\big{]}\big{|}$
$\displaystyle=$
$\displaystyle\big{|}\mathbb{E}\big{[}(\delta_{X,x})^{i_{1}}((\delta_{X,x})^{i_{2}}-(\delta_{Y_{E},x})^{i_{2}}-R_{f}^{i_{2}})\big{]}\big{|}+\big{|}\mathbb{E}\big{[}((\delta_{X,x})^{i_{1}}-(\delta_{Y_{E},x})^{i_{1}}-R_{f}^{i_{1}})(\delta_{Y,x})^{i_{2}}\big{]}\big{|}$
$\displaystyle\leq$
$\displaystyle\big{|}\mathbb{E}\big{[}(\delta_{X,x})^{i_{1}}((\delta_{X,x})^{i_{2}}-(\delta_{Y_{E},x})^{i_{2}})\big{]}\big{|}+\big{|}\mathbb{E}\big{[}((\delta_{X,x})^{i_{1}}-(\delta_{Y_{E},x})^{i_{1}})(\delta_{Y,x})^{i_{2}}\big{]}\big{|}$
$\displaystyle\quad+\big{|}\mathbb{E}\big{[}(\delta_{X,x})^{i_{1}}R_{f}^{i_{2}}\big{]}\big{|}+\big{|}\mathbb{E}\big{[}R_{f}^{i_{1}}(\delta_{Y,x})^{i_{2}}\big{]}\big{|}.$

By (C.22), it follows that

$\big{|}\mathbb{E}\big{[}(\delta_{X,x})^{i_{1}}((\delta_{X,x})^{i_{2}}-(\delta_{Y_{E},x})^{i_{2}})\big{]}\big{|}\leq
C(1+|x|^{6r+2})h^{2}.$ (6.19)

The estimate for the second is obtained in the same way. Utilizing the Hölder
inequality, (4.5) and (6.6) yields

$\big{|}\mathbb{E}\big{[}(\delta_{X,x})^{i_{1}}R_{f}^{i_{2}}\big{]}\big{|}\leq\|(\delta_{X,x})^{i_{1}}\|_{L^{2}(\Omega;\mathbb{R})}\,\|R^{i_{2}}_{f}\|_{L^{2}(\Omega;\mathbb{R})}\leq
C(1+|x|^{6r+2})h^{2}.$ (6.20)

Following a similar way as (6.20), we acquire

$\big{|}\mathbb{E}\big{[}R_{f}^{i_{1}}(\delta_{Y,x})^{i_{2}}\big{]}\big{|}\leq
C(1+|x|^{6r+2})h^{2}.$ (6.21)

Collecting (6.19), (6.20), and (6.21) enables us to arrive at

$\big{|}\mathbb{E}\big{[}(\delta_{X,x})^{i_{1}}(\delta_{X,x})^{i_{2}}-(\delta_{Y,x})^{i_{1}}(\delta_{Y,x})^{i_{2}}\big{]}\big{|}\leq
C(1+|x|^{6r+2})h^{2}.$ (6.22)

Finally, for $s=3,$ we make a decomposition as follows:

$\displaystyle\big{|}\mathbb{E}\big{[}(\delta_{X,x})^{i_{1}}(\delta_{X,x})^{i_{2}}(\delta_{X,x})^{i_{3}}-(\delta_{Y,x})^{i_{1}}(\delta_{Y,x})^{i_{2}}(\delta_{Y,x})^{i_{3}}\big{]}\big{|}$
(6.23) $\displaystyle\leq$
$\displaystyle\big{|}\mathbb{E}\big{[}(\delta_{X,x})^{i_{1}}((\delta_{X,x})^{i_{2}}(\delta_{X,x})^{i_{3}}-(\delta_{Y,x})^{i_{2}}(\delta_{Y,x})^{i_{3}})\big{]}\big{|}+\big{|}\mathbb{E}\big{[}((\delta_{X,x})^{i_{1}}-(\delta_{Y,x})^{i_{1}})(\delta_{Y,x})^{i_{2}}(\delta_{Y,x})^{i_{3}}\big{]}\big{|}$
$\displaystyle\leq$
$\displaystyle\big{|}\mathbb{E}\big{[}(\delta_{X,x})^{i_{1}}(\delta_{X,x})^{i_{2}}((\delta_{X,x})^{i_{3}}-(\delta_{Y,x})^{i_{3}})\big{]}\big{|}+\big{|}\mathbb{E}\big{[}(\delta_{X,x})^{i_{1}}(\delta_{Y,x})^{i_{3}}((\delta_{X,x})^{i_{2}}-(\delta_{Y,x})^{i_{2}})\big{]}\big{|}$
$\displaystyle\quad+\big{|}\mathbb{E}\big{[}((\delta_{X,x})^{i_{1}}-(\delta_{Y,x})^{i_{1}})(\delta_{Y,x})^{i_{2}}(\delta_{Y,x})^{i_{3}}\big{]}\big{|}.$

Using (4.5), (6.5), (6.7) and the Hölder inequality yields

$\big{|}\mathbb{E}\big{[}(\delta_{X,x})^{i_{1}}(\delta_{X,x})^{i_{2}}((\delta_{X,x})^{i_{3}}-(\delta_{Y,x})^{i_{3}})\big{]}\big{|}\leq
C(1+|x|^{8r+3})h^{2},$ (6.24)
$\big{|}\mathbb{E}\big{[}(\delta_{X,x})^{i_{1}}(\delta_{Y,x})^{i_{3}}((\delta_{X,x})^{i_{2}}-(\delta_{Y,x})^{i_{2}})\big{]}\big{|}\leq
C(1+|x|^{8r+3})h^{2},$ (6.25)

and

$\big{|}\mathbb{E}\big{[}((\delta_{X,x})^{i_{1}}-(\delta_{Y,x})^{i_{1}})(\delta_{Y,x})^{i_{2}}(\delta_{Y,x})^{i_{3}}\big{]}\big{|}\leq
C(1+|x|^{8r+3})h^{2}.$ (6.26)

Putting (6.17), (6.22) and (6.24)- (6.26) together, we obtain

$\Big{|}\mathbb{E}\Big{[}\prod_{j=1}^{s}(\delta_{X,x})^{i_{j}}\Big{]}-\mathbb{E}\Big{[}\prod_{j=1}^{s}(\delta_{Y,x})^{i_{j}}\Big{]}\Big{|}\leq
C(1+|x|^{8r+3})h^{2},\,\,s=1,2,3.$ (6.27)

Similar to the proof of (5.14), one can employ the Hölder inequality and
(6.5), to deduce

$\bigg{\|}\prod_{j=1}^{4}(\delta_{Y,x})^{i_{j}}\bigg{\|}_{L^{2}(\Omega;\mathbb{R})}\leq\prod_{j=1}^{4}\Big{\|}(\delta_{Y,x})^{i_{j}}\Big{\|}_{L^{8}(\Omega;\mathbb{R})}\leq
C(1+|x|^{8r+4})h^{2},$ (6.28)

which finishes the proof of Lemma 6.4. ∎

Based on the local error and Corollary 3.8, we are able to obtain the global
weak convergence rate of the BEM (6.1).

######  Theorem 6.5.

Suppose Assumption 3.6 hold with $p=1$. Then the backward Euler method (6.1)
has a global weak convergence rate of order $1$, namely, for
$\varphi\in\mathbb{H}^{4}$

$\big{|}\mathbb{E}\big{[}\varphi\big{(}X(t_{0},X_{0};T)\big{)}\big{]}-\mathbb{E}\big{[}\varphi\big{(}Y(t_{0},Y_{0};T)\big{)}\big{]}\big{|}\leq
C\big{(}1+|X_{0}|^{(8r+4+\chi)}\big{)}h.$ (6.29)

##  7 Numerical experiments

In this section, we present some numerical experiments to illustrate the
previous theoretical findings. In the following simulations, we used the
Mersenne Twister random generator with seed $100$.  Newton’s method is used to
solve the nonlinear algebraic equations at each step of the implicit schemes
with the two successive iterative solutions smaller than $10^{-6}$ in the
$l^{2}$-norm.

We test several numerical schemes for two model problems. The first one
satisfies Assumption 3.6 (non-globally Lipschitz both drift and diffusion).
The second example has non-globally Lipschitz drift and global Lipschitz
diffusion. The aim of the tests is to compare the weak convergence rates of
the numerical methods proposed in subsection 5.3.

Along this section, we consider four different test functions:

$\varphi(x)=x,\,\,x^{2},\,\,\cos(x),\,\,e^{-x^{2}}.$ (7.1)

The reference values of the models are computed based on a Monte Carlo method
combined with the scheme (6.1) and $h_{ref}=2^{-14}$:

$\mathbb{E}\big{[}\varphi(X_{T})\big{]}\thickapprox\frac{1}{M}\sum_{i=1}^{M}\varphi\big{(}X_{T}(\omega_{i},h_{ref})\big{)}.$
(7.2)

######  Example 7.1.

As the first test model, we consider the following model

$\left\\{\begin{aligned}
\text{d}X(t)&=\big{(}1-X^{5}(t)+X^{3}(t)\big{)}\,\text{d}t+\big{(}\tfrac{1}{10}X^{2}(t)+2\big{)}\,\text{d}W(t),\quad
t\in(0,T],\\\ X(0)&=x.\end{aligned}\right.$ (7.3)

The conditions in Assumption 3.6 are fulfilled with $r=2$ and $\rho=2$.

In this example, we test the weak errors of the backward Euler method (BEM)
(6.1), the fully tamed Euler (FTE1) (5.17), the fully tamed Euler (FTE2)
(5.18), the modified Euler scheme (MES) (5.19), the balanced scheme (BS)
(5.21) and the balanced-type scheme (BTS) (5.22).

For the first presented numerical experiments, we consider a unit terminal
time $T=1$, the initial condition $x=2,8$, and the time step sizes
$h=2^{-k},k=6,7,8,9,10.$ In addition, the quantity
$\mathbb{E}\big{[}\varphi(Y_{T})\big{]}$ is estimated by a Monte Carlo
approximation, involving $M=3\times 10^{6}$ independent trajectories. The
statistical errors for the Monte Carlo approximations are computed with $95\%$
confidence [53, subsection 2.5.1] and the resulting statistical errors are 10
times smaller than the reported weak errors.

In Figure 1 and Figure 2, we take $x=2$ and present the weak errors of six
different numerical methods with the test functions (7.1) against five
different stepsizes on a log-log scale. As expected, the weak convergence
rates of the MES (5.19), the BEM (6.1), and the BS (5.21) are close to $1$,
whereas the FTE1 (5.17), the FTE2 (5.18), and the BTS (5.22) give convergence
rates close to $\frac{1}{2}$.  When $x=8$, our numerical experiments show that
the tamed-type methods including the FTE1 (5.17), the FTE2 (5.18), and the MES
(5.19) can not produce satisfactory accuracy with the considered time step
sizes $h=2^{-k},k=6,7,8,9,10$, while the BS (5.21), the BTS (5.22) and the BEM
(6.1) always give high accuracy approximations. In this case, we also
implement the Euler-Maruyama (EM) method for the test problem and find that
all ($100\%$) the EM trajectories explode, even for very small step-size
$h=2^{-10}\approx 0.001$.

Figure 1: Weak convergence rates for the stochastic differential equation
(7.3) for $\varphi(x)=x$ (Left) and $\varphi(x)=x^{2}$ (Right).

Figure 2: Weak convergence rates for the the stochastic differential equation
(7.3) for $\varphi(x)=\cos(x)$ (Left) and $\varphi(x)=e^{-x^{2}}$ (Right).

######  Example 7.2.

Consider the stochastic FitzHugh-Nagumo (FHN) model [12] in the form of:

$\begin{split}&\left(\begin{array}[]{c}\text{d}X_{1}(t)\\\
\text{d}X_{2}(t)\end{array}\right)=\left(\\!\begin{array}[]{c}X_{1}(t)-X_{1}^{3}(t)-X_{2}(t)\\\
X_{1}(t)-X_{2}(t)+1\end{array}\\!\right)\text{d}t+\left(\\!\begin{array}[]{cc}X_{1}(t)+1&0\\\
0&X_{2}(t)+1\end{array}\\!\right)\text{d}W(t)\end{split}$ (7.4)

for $t\in(0,T]$ and $X(0)=[0,0]$, with solution
$X(t):=\big{(}X_{1}(t),X_{2}(t)\big{)}^{T}$ for $t\in[0,T]$, where
$W(t):=\big{(}W_{1}(t),W_{2}(t)\big{)}^{T}$ is a two dimensional Brownian
motion.

In this example, conditions in Assumption 3.6 are fulfilled with $r=1$ and
$\rho=1$. Here we test weak convergent rates of the BEM (6.1), the FTE1
(5.17), the FTE2 (5.18), the MES (5.19), the DTE scheme (5.20), and the BS
(5.21).

For the second numerical experiment, we consider a unit terminal time $T=1$
and the time step $h=2^{-k},k=7,8,9,10,11$, and $M=10^{6}$ independent
trajectories in the Monte Carlo approximation. The statistical errors for the
Monte Carlo approximations turn out to be negligible as the Monte Carlo error
computed with 95% confidence is at least 10 times smaller than the weak
errors.

From Figure 3 and Figure 4, we observe that weak convergence rates of (5.19),
(5.20), (5.21), (6.1) are $1$. For the schemes (5.17) and (5.18), we observe a
weak convergence of order $\frac{1}{2}$.

Figure 3: Weak rates for stochastic FHN model with $\varphi(x)=x$ (Left) and
$\varphi(x)=x^{2}$ (Right).

Figure 4: Weak rates for stochastic FHN model with $\varphi(x)=\cos(x)$ (Left)
and $\varphi(x)=e^{-x^{2}}$ (Right).

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##  Appendix A Proof of Theorem 2.6

The proof exploits the idea of the proof of [38, Theorem 2] and [39, Theorem
2.1] in the global Lipschitz case. Recall $Y_{0}=X_{0}$ and

$X\big{(}t,x;T\big{)}=X\big{(}s,X_{t,x}(s);T\big{)},\quad 0\leq t\leq s\leq
T.$ (A.1)

Noting $t_{N}=T$ and recalling the definition (2.12) yield

$\mathbb{E}\big{[}\varphi\big{(}X(t_{0},X_{0};T)\big{)}\big{]}-\mathbb{E}\big{[}\varphi\big{(}Y(t_{0},Y_{0};t_{N})\big{)}\big{]}=\mathbb{E}[u(t_{0},X_{0})]-\mathbb{E}[u(t_{N},Y_{N})].$
(A.2)

Again, by the definition (2.12) and (A.1),

$\displaystyle u(t,x)$ $\displaystyle=$
$\displaystyle\mathbb{E}\big{[}\varphi(X(t,x;T))\big{]}=\mathbb{E}\big{[}\varphi(X(s,X_{t,x}(s);T))\big{]}$
(A.3) $\displaystyle=$
$\displaystyle\mathbb{E}\big{[}\mathbb{E}\big{(}\varphi(X(s,X_{t,x}(s);T))|\sigma(X_{t,x}(s))\big{)}\big{]}=\mathbb{E}[u(s,X_{t,x}(s))],\quad\text{
for }0\leq t\leq s\leq T.$

By taking $t=t_{i},$ $s=t_{i+1},$ a conditional version of such an equality
$u(t_{i},Y_{i})=\mathbb{E}\big{[}u\big{(}t_{i+1},X_{t_{i},Y_{i}}(t_{i+1}\big{)}|\sigma(Y_{i})\big{]},$
implies

$\mathbb{E}\big{[}u(t_{i},Y_{i})\big{]}=\mathbb{E}\big{[}u(t_{i+1},X(t_{i},Y_{i};t_{i+1}))\big{]},\quad
i=0,1,...,N-1.$ (A.4)

Using this repeatedly, we obtain

$\displaystyle\mathbb{E}\big{[}u(t_{0},X_{0})\big{]}$ $\displaystyle=$
$\displaystyle\mathbb{E}\big{[}u(t_{1},X(t_{1}))\big{]}-\mathbb{E}\big{[}u(t_{1},Y_{1})\big{]}+\mathbb{E}\big{[}u(t_{2},X(t_{1},Y_{1};t_{2}))\big{]}$
(A.5) $\displaystyle=$
$\displaystyle\mathbb{E}\big{[}u(t_{1},X(t_{0},X_{0};t_{1}))\big{]}-\mathbb{E}\big{[}u(t_{1},Y_{1})\big{]}$
$\displaystyle\quad+\mathbb{E}\big{[}u(t_{2},X(t_{1},Y_{1};t_{2}))\big{]}-\mathbb{E}\big{[}u(t_{2},Y_{2})\big{]}+\mathbb{E}\big{[}u(t_{3},X(t_{2},Y_{2};t_{3}))\big{]}$
$\displaystyle=$ $\displaystyle\cdots$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{N-2}\Big{(}\mathbb{E}\big{[}u(t_{i+1},X(t_{i},Y_{i};t_{i+1}))\big{]}-\mathbb{E}\big{[}u(t_{i+1},Y_{i+1})\big{]}\Big{)}+\mathbb{E}\big{[}u(t_{N-1},Y_{N-1})\big{]}.$

By (A.4), we get

$\mathbb{E}\big{[}u(t_{N-1},Y_{N-1})\big{]}-\mathbb{E}\big{[}u(t_{N},Y_{N})\big{]}=\mathbb{E}\big{[}u(t_{N},X(t_{N-1},Y_{N-1};t_{N}))\big{]}-\mathbb{E}\big{[}u(t_{N},Y_{N})\big{]},$
(A.6)

and thus one can derive from (A.5) that

$\displaystyle\mathbb{E}\big{[}u(t_{0},X_{0})\big{]}-\mathbb{E}\big{[}u(t_{N},Y_{N})\big{]}$
(A.7) $\displaystyle=$
$\displaystyle\sum_{i=0}^{N-1}\Big{(}\mathbb{E}\big{[}u(t_{i+1},X(t_{i},Y_{i};t_{i+1}))\big{]}-\mathbb{E}\big{[}u(t_{i+1},Y_{i+1})\big{]}\Big{)}$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{N-1}\Big{(}\mathbb{E}\big{[}u(t_{i+1},X(t_{i},Y_{i};t_{i+1}))\big{]}-\mathbb{E}\big{[}u(t_{i+1},Y(t_{i},Y_{i};t_{i+1}))\big{]}\Big{)}.$

Finally, combining a conditional version of (2.13) with (A.7) yields

$\displaystyle\big{|}\mathbb{E}\big{[}\varphi\big{(}X(t_{0},X_{0};T)\big{)}\big{]}-\mathbb{E}\big{[}\varphi\big{(}Y(t_{0},Y_{0};t_{N})\big{)}\big{]}\big{|}$
(A.8) $\displaystyle\leq$
$\displaystyle\sum_{i=0}^{N-1}\Big{|}\mathbb{E}\big{[}u(t_{i+1},X(t_{i},Y_{i};t_{i+1}))\big{]}-\mathbb{E}\big{[}u(t_{i+1},Y(t_{i},Y_{i};t_{i+1}))\big{]}\Big{|}$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{N-1}\Big{|}\mathbb{E}\Big{[}\mathbb{E}\big{(}u(t_{i+1},X(t_{i},Y_{i};t_{i+1}))-u(t_{i+1},Y(t_{i},Y_{i};t_{i+1}))|\sigma(Y_{i})\big{)}\Big{]}\Big{|}$
$\displaystyle\leq$
$\displaystyle\sum_{i=0}^{N-1}\mathbb{E}[|\nu(Y_{i})|]h^{p+1}\leq\Upsilon(X_{0})Th^{p},$

which validates the desired assertion (2.15). ∎

##  Appendix B Proof of Lemma 3.4

Proof. Since $\varphi(x)$ is differentiable and the solution $X(t,x;T)$ of the
SDE (2.7) is mean-square differentiable with respect to the initial data $x$
by assumption, $u(t,x)$ is differentiable with respect to $x$ and

$\tfrac{\partial u(t,x)}{\partial
x_{i}}=\tfrac{\partial\mathbb{E}[\varphi(X(t,x;T))]}{\partial
x_{i}}=\mathbb{E}[D\varphi(X(t,x;T))\tfrac{\partial X(t,x;T)}{\partial
x_{i}}]=\mathbb{E}\Big{[}\sum_{j=1}^{d}\partial_{x_{j}}\varphi(X(t,x;T))\tfrac{\partial
X^{j}(t,x;T)}{\partial x_{i}}\Big{]}.$ (B.1)

Noting further that the partial derivative of $\varphi$ belongs to
$\mathbb{H}$ by assumption, one can use (3.5) and the Hölder inequality to
deduce

$|\tfrac{\partial u(t,x)}{\partial
x_{i}}|\leq\|D\varphi(X(t,x;T))\|_{L^{2}(\Omega,\mathbb{R}^{d})}\|\tfrac{\partial
X(t,x;T)}{\partial x_{i}}\|_{L^{2}(\Omega,\mathbb{R}^{d})}\leq
C(1+|x|^{\chi}).$ (B.2)

For higher order derivatives, we recall that $\varphi$’s partial derivatives
of order up to $2p+2$ inclusively, belong to $\mathbb{H}$ and the solution of
the SDE is $2p+2$ times mean-square differentiable with respect to the initial
data $x$. These facts together with the chain rule enable us to similarly show
that $u(t,x)$ is $2p+2$ times differentiable with respect to $x$ and

$\big{|}\tfrac{\partial^{k}u(t+h,x)}{\partial x_{i_{1}}\ldots\partial
x_{i_{k}}}\big{|}\leq C(1+|x|^{\chi}),\quad k=2,3,\cdots,2p+2.$ (B.3)

By the Taylor expansion, we get

$\displaystyle\quad\mathbb{E}\big{[}u(t+h,X(t,x;t+h))\big{]}-\mathbb{E}\big{[}u(t+h,Y(t,x;t+h))\big{]}$
(B.4) $\displaystyle=$
$\displaystyle\sum^{2p+1}_{k=1}\sum^{d}_{i_{1},\ldots,i_{k}=1}\frac{1}{k!}\frac{\partial^{k}u(t+h,x)}{\partial
x_{i_{1}}\ldots\partial
x_{i_{k}}}\mathbb{E}\Big{[}\prod^{k}_{j=1}(\delta_{X,x})^{i_{j}}-\prod^{k}_{j=1}(\delta_{Y,x})^{i_{j}}\Big{]}+\mathbb{E}[\mathcal{R}_{2p+2}],$

where we used the notation (3.3) and denote

$\big{(}\delta_{X,x}\big{)}^{\alpha}:=\prod_{j=1}^{2p+2}(\delta_{X,x})^{i_{j}},\quad\big{(}\delta_{Y,x}\big{)}^{\alpha}:=\prod_{j=1}^{2p+2}(\delta_{Y,x})^{i_{j}},\quad\alpha=(i_{1},i_{2},...,i_{2p+2})$

and

$\displaystyle\mathcal{R}_{2p+2}$ $\displaystyle:=$
$\displaystyle\sum_{|\alpha|=2p+2}\frac{|\alpha|}{\alpha!}\bigg{(}\int_{0}^{1}(1-s)^{|\alpha|-1}D^{\alpha}u\big{(}t+h,x+s\delta_{X,x}\big{)}\text{d}s\,(\delta_{X,x})^{\alpha}$
(B.5)
$\displaystyle\qquad-\int_{0}^{1}(1-s)^{|\alpha|-1}D^{\alpha}u\big{(}t+h,x+s\delta_{Y,x}\big{)}\text{d}s\,(\delta_{Y,x})^{\alpha}\bigg{)}.$

Combining the assumption (3.1) with (B.2), (B.3) yields

$\sum^{2p+1}_{k=1}\sum^{d}_{i_{1},\ldots,i_{k}=1}\frac{1}{k!}\Big{|}\frac{\partial^{k}u(t+h,x)}{\partial
x_{i_{1}}\ldots\partial
x_{i_{k}}}\Big{|}\cdot\Big{|}\mathbb{E}\Big{[}\prod^{k}_{j=1}(\delta_{X,x})^{i_{j}}-\prod^{k}_{j=1}(\delta_{Y,x})^{i_{j}}\Big{]}\Big{|}\leq
C(1+|x|^{\chi})K(x)h^{p+1}.$ (B.6)

Using the Hölder inequality, (3.4), (3.6) and (B.3) one can derive

$\displaystyle\big{|}\mathbb{E}[\mathcal{R}_{2p+2}]\big{|}$
$\displaystyle\leq$
$\displaystyle\sum_{|\alpha|=2p+2}C\int_{0}^{1}\|D^{\alpha}u\big{(}t+h,(1-s)x+sX(t,x;t+h)\big{)}\|_{L^{2}(\Omega;\mathbb{R})}\text{d}s\|(\delta_{X,x})^{\alpha}\|_{L^{2}(\Omega;\mathbb{R})}$
(B.7)
$\displaystyle\quad+\sum_{|\alpha|=2p+2}C\int_{0}^{1}\|D^{\alpha}u\big{(}t+h,(1-s)x+sY(t,x;t+h)\big{)}\|_{L^{2}(\Omega;\mathbb{R})}\text{d}s\|(\delta_{Y,x})^{\alpha}\|_{L^{2}(\Omega;\mathbb{R})}$
$\displaystyle\leq$
$\displaystyle\sum_{|\alpha|=2p+2}C\big{(}1+|x|^{\chi}+\|X(t,x;t+h)\|^{\chi}_{L^{2\chi}(\Omega;\mathbb{R})}\big{)}\,K(x)h^{p+1}$
$\displaystyle\quad+\sum_{|\alpha|=2p+2}C\big{(}1+|x|^{\chi}+\|Y(t,x;t+h)\|^{\chi}_{L^{2\chi}(\Omega;\mathbb{R})}\big{)}\,K(x)h^{p+1}$
$\displaystyle\leq$ $\displaystyle C(1+|x|^{\beta\chi})K(x)h^{p+1},$

where we used the assumption $q_{0}=2{\chi}$, $q_{1}=4p+4$, and
$q_{2}=2{\chi},\beta\geq 1.$ Plugging (B.6) and (B.7) into (B.4) enables us to
get

$\big{|}\mathbb{E}\big{[}\varphi\big{(}X(t,x;t+h)\big{)}\big{]}-\mathbb{E}\big{[}\varphi\big{(}Y(t,x;t+h)\big{)}\big{]}\big{|}\leq
C(1+|x|^{\beta\chi})K(x)h^{p+1}.$ (B.8)

The proof of the lemma is thus complete. ∎

##  Appendix C Proofs of Lemmas in Section 4

We first prove Lemma 4.1. According to (2.6) and (4.1), one can use the Hölder
inequality, the moment inequality [16, 53], (3.21) and (3.22), to acquire

$\displaystyle\mathbb{E}[|\delta_{X,x}-\delta_{Y_{E},x}|^{2q}]$ (C.1)
$\displaystyle=$
$\displaystyle\mathbb{E}\Big{[}\Big{|}\int^{t+h}_{t}f\big{(}X(t,x;s)\big{)}-f(x)\,\text{d}s+\int^{t+h}_{t}g\big{(}X(t,x;s)\big{)}-g(x)\,\text{d}W(s)\Big{|}^{2q}\Big{]}$
$\displaystyle\leq$ $\displaystyle
C\mathbb{E}\Big{[}\Big{|}\int^{t+h}_{t}f\big{(}X(t,x;s)\big{)}-f(x)\,\text{d}s\Big{|}^{2q}\Big{]}+C\mathbb{E}\Big{[}\Big{|}\int^{t+h}_{t}g\big{(}X(t,x;s)\big{)}-g(x)\,\text{d}W(s)\Big{|}^{2q}\Big{]}$
$\displaystyle\leq$ $\displaystyle
Ch^{2q-1}\int^{t+h}_{t}\big{\|}f\big{(}X(t,x;s)\big{)}-f(x)\big{\|}_{L^{2q}(\Omega,\mathbb{R}^{d})}^{2q}\,\text{d}s+Ch^{q-1}\int^{t+h}_{t}\big{\|}g\big{(}X(t,x;s)\big{)}-g(x)\big{\|}_{L^{2q}(\Omega,\mathbb{R}^{d\times
m})}^{2q}\,\text{d}s$ $\displaystyle\leq$ $\displaystyle
C(1+|x|^{2q(4r+1)})h^{2q},$

which implies (4.3), as required. Similar arguments, together with (3.9) and
(3.10) yield that

$\displaystyle\mathbb{E}\big{[}|\delta_{Y_{E},x}|^{2q}\big{]}$
$\displaystyle=$
$\displaystyle\mathbb{E}\Big{[}\Big{|}\int^{t+h}_{t}f(x)\,\text{d}s+\int^{t+h}_{t}g(x)\,\text{d}W(s)\Big{|}^{2q}\Big{]}$
(C.2) $\displaystyle\leq$ $\displaystyle
Ch^{2q}\big{|}f(x)\big{|}^{2q}+Ch^{q}\|g(x)\|^{2q}$ $\displaystyle\leq$
$\displaystyle C(1+|x|^{2q(2r+1)})h^{q},$

so that (4.4) is validated. The estimate of $\delta_{X,x}$ as (4.5) can be
done similarly. ∎We then prove Lemma 4.2. We treat the case $s=1$ first. Using
the multi-dimensional Itô formula, the Cauchy-Schwarz inequality, (3.9) and
(3.10) shows

$\displaystyle\Big{|}\mathbb{E}\big{[}(\delta_{X,x})^{i_{1}}-(\delta_{Y_{E},x})^{i_{1}}\big{]}\Big{|}=\Big{|}\mathbb{E}\Big{[}\int^{t+h}_{t}f^{i_{1}}\big{(}X(s)\big{)}-f^{i_{1}}(x)\,\text{d}s\Big{]}\Big{|}$
(C.3) $\displaystyle=$
$\displaystyle\Big{|}\mathbb{E}\Big{[}\int_{t}^{t+h}\int^{s}_{t}\left\langle
Df^{i_{1}}\big{(}X(r)\big{)},f\big{(}X(r)\big{)}\right\rangle\,\text{d}r\,\text{d}s\Big{]}+\mathbb{E}\Big{[}\int_{t}^{t+h}\int^{s}_{t}\left\langle
Df^{i_{1}}\big{(}X(r)\big{)},g\big{(}X(r)\big{)}\,\text{d}W(r)\right\rangle\,\text{d}s\Big{]}$
$\displaystyle\quad+\frac{1}{2}\mathbb{E}\Big{[}\int_{t}^{t+h}\int^{s}_{t}\text{trace}\big{(}g^{T}(X(r))D^{2}f^{i_{1}}\big{(}X(r)\big{)}g(X(r))\big{)}\,\text{d}r\,\text{d}s\Big{]}\Big{|}$
$\displaystyle\leq$
$\displaystyle\Big{|}\mathbb{E}\Big{[}\int_{t}^{t+h}\int^{s}_{t}\big{|}Df^{i_{1}}\big{(}X(r)\big{)}\big{|}\big{|}f\big{(}X(r)\big{)}\big{|}\,\text{d}r\,\text{d}s\Big{]}\Big{|}$
$\displaystyle\quad+C\Big{|}\mathbb{E}\Big{[}\int_{t}^{t+h}\int_{t}^{s}\big{\|}D^{2}f^{i_{1}}\big{(}X(r)\big{)}\big{\|}\big{\|}g\big{(}X(r)\big{)}\big{\|}^{2}\,\text{d}r\,\text{d}s\Big{]}\Big{|}$
$\displaystyle\leq$ $\displaystyle
C(1+|x|^{4r+1})h^{2}+C(1+|x|^{2r+2\rho-1})h^{2}$ $\displaystyle\leq$
$\displaystyle C(1+|x|^{4r+1})h^{2}.$

For $s=2,$ we first derive

$\displaystyle\big{|}\mathbb{E}\big{[}(\delta_{X,x})^{i_{1}}(\delta_{X,x})^{i_{2}}-(\delta_{Y_{E},x})^{i_{1}}(\delta_{Y_{E},x})^{i_{2}}\big{]}\big{|}$
(C.4) $\displaystyle\leq$
$\displaystyle\big{|}\mathbb{E}\big{[}((\delta_{X,x})^{i_{1}}-(\delta_{Y_{E},x})^{i_{1}})(\delta_{Y_{E},x})^{i_{2}}\big{]}\big{|}+\big{|}\mathbb{E}\big{[}(\delta_{X,x})^{i_{1}}((\delta_{X,x})^{i_{2}}-(\delta_{Y_{E},x})^{i_{2}})\big{]}\big{|}=:I_{1}+I_{2}.$

Before proceeding further with the estimate of $I_{1},$ we decompose it as
follows:

$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\Big{|}\mathbb{E}\Big{[}\Big{(}\int_{t}^{t+h}f^{i_{1}}\big{(}X(s)\big{)}-f^{i_{1}}(x)\,\text{d}s+\int_{t}^{t+h}g_{i_{1}}\big{(}X(s)\big{)}-g_{i_{1}}(x)\,\text{d}W(s)\Big{)}$
(C.5)
$\displaystyle\qquad\times\Big{(}\int_{t}^{t+h}f^{i_{2}}(x)\,\text{d}s+\int_{t}^{t+h}g_{i_{2}}(x)\,\text{d}W(s)\Big{)}\Big{]}\Big{|}$
$\displaystyle\leq$
$\displaystyle\Big{|}\mathbb{E}\Big{[}\Big{(}\int_{t}^{t+h}f^{i_{1}}\big{(}X(s)\big{)}-f^{i_{1}}(x)\,\text{d}s\Big{)}\cdot\int_{t}^{t+h}f^{i_{2}}(x)\,\text{d}s\Big{]}\Big{|}$
$\displaystyle\quad+\Big{|}\mathbb{E}\Big{[}\Big{(}\int_{t}^{t+h}g_{i_{1}}\big{(}X(s)\big{)}-g_{i_{1}}(x)\,\text{d}W(s)\Big{)}\cdot\int_{t}^{t+h}f^{i_{2}}(x)\,\text{d}s\Big{]}\Big{|}$
$\displaystyle\quad+\Big{|}\mathbb{E}\Big{[}\Big{(}\int_{t}^{t+h}f^{i_{1}}\big{(}X(s)\big{)}-f^{i_{1}}(x)\,\text{d}s\Big{)}\cdot\int_{t}^{t+h}g_{i_{2}}(x)\,\text{d}W(s)\Big{]}\Big{|}$
$\displaystyle\quad+\Big{|}\mathbb{E}\Big{[}\Big{(}\int_{t}^{t+h}g_{i_{1}}\big{(}X(s)\big{)}-g_{i_{1}}(x)\,\text{d}W(s)\Big{)}\cdot\int_{t}^{t+h}g_{i_{2}}(x)\,\text{d}W(s)\Big{]}\Big{|}$
$\displaystyle=$
$\displaystyle\Big{|}\mathbb{E}\Big{[}\Big{(}\int_{t}^{t+h}f^{i_{1}}\big{(}X(s)\big{)}-f^{i_{1}}(x)\,\text{d}s\Big{)}\cdot\int_{t}^{t+h}f^{i_{2}}(x)\,\text{d}s\Big{]}\Big{|}$
$\displaystyle\quad+\Big{|}\mathbb{E}\Big{[}\Big{(}\int_{t}^{t+h}f^{i_{1}}\big{(}X(s)\big{)}-f^{i_{1}}(x)\,\text{d}s\Big{)}\cdot\int_{t}^{t+h}g_{i_{2}}(x)\,\text{d}W(s)\Big{]}\Big{|}$
$\displaystyle\quad+\Big{|}\mathbb{E}\Big{[}\Big{(}\int_{t}^{t+h}g_{i_{1}}\big{(}X(s)\big{)}-g_{i_{1}}(x)\,\text{d}W(s)\Big{)}\cdot\int_{t}^{t+h}g_{i_{2}}(x)\,\text{d}W(s)\Big{]}\Big{|}$
$\displaystyle=:$ $\displaystyle I_{1,1}+I_{1,2}+I_{1,3}.$

Since $I_{1,1}$ is an easy term, we treat it first. In view of (3.14) and
(3.21), we get

$I_{1,1}\leq
h|f^{i_{2}}(x)|\int_{t}^{t+h}\big{|}\mathbb{E}[f^{i_{1}}(X(s))-f^{i_{1}}(x)]\big{|}\,\text{d}s\leq
C(1+|x|^{6r+2})h^{\frac{5}{2}}.$ (C.6)

Regarding $I_{1,2},$ with the aid of the Hölder inequality and the Itô
isometry, one employs (3.10) and (3.21) to obtain

$\displaystyle I_{1,2}$ $\displaystyle\leq$
$\displaystyle\int_{t}^{t+h}\|f^{i_{1}}\big{(}X(s)\big{)}-f^{i_{1}}(x)\|_{L^{2}(\Omega,\mathbb{R})}\,\text{d}s\cdot\Big{\|}\int_{t}^{t+h}g_{i_{2}}(x)\,\text{d}W(s)\Big{\|}_{L^{2}(\Omega,\mathbb{R})}$
(C.7) $\displaystyle\leq$ $\displaystyle
C(1+|x|^{4r+1})h^{\frac{3}{2}}\Big{(}\int_{t}^{t+h}\mathbb{E}\big{[}|g_{i_{2}}(x)|^{2}\big{]}\,\text{d}s\Big{)}^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle C(1+|x|^{4r+\rho+1})h^{2}.$

Now we come to the estimate of $I_{1,3}$, which requires more careful
arguments. In light of the multi-dimensional Itô formula, one can separate the
considered term $I_{1,3}$ as follows:

$\displaystyle I_{1,3}$ $\displaystyle\leq$
$\displaystyle\Big{|}\mathbb{E}\Big{[}\Big{(}\int_{t}^{t+h}\Big{\langle}\int_{t}^{s}Dg_{i_{1}}(X(r))\cdot
g(X(r))\text{d}W(r),\text{d}W(s)\Big{\rangle}\Big{)}\int_{t}^{t+h}g_{i_{2}}(x)\,\text{d}W(s)\Big{]}\Big{|}$
(C.8)
$\displaystyle\quad+\Big{|}\mathbb{E}\Big{[}\Big{(}\int_{t}^{t+h}\Big{\langle}\int_{t}^{s}Dg_{i_{1}}\big{(}X(r)\big{)}\cdot
f(X(r))\text{d}r,\text{d}W(s)\Big{\rangle}\Big{)}\int_{t}^{t+h}g_{i_{2}}(x)\,\text{d}W(s)\Big{]}\Big{|}$
$\displaystyle\quad+\frac{1}{2}\sum_{l=1}^{m}\Big{|}\mathbb{E}\Big{[}\int_{t}^{t+h}\int_{t}^{s}\text{trace}\big{(}g^{T}(X(r))D^{2}g^{i_{1},l}(X(r))g(X(r))\,\text{d}r\,\text{d}W^{l}(s)\big{)}\\!\int_{t}^{t+h}g_{i_{2}}(x)\,\text{d}W(s)\Big{]}\Big{|}$
$\displaystyle=:$ $\displaystyle I^{(1)}_{1,3}+I^{(2)}_{1,3}+I^{(3)}_{1,3}.$

The three terms $I^{(1)}_{1,3},$ $I^{(2)}_{1,3},$ $I^{(3)}_{1,3}$ will be
estimated separately. The generalized Itô isometry yields that

$\displaystyle I^{(1)}_{1,3}$ $\displaystyle=$
$\displaystyle\Big{|}\mathbb{E}\Big{[}\int_{t}^{t+h}\Big{(}\int_{t}^{s}Dg_{i_{1}}\big{(}X(r)\big{)}\cdot
g\big{(}X(r)\big{)}\text{d}W(r)\Big{)}g_{i_{2}}(x)\,\text{d}s\Big{]}\Big{|}$
(C.9) $\displaystyle=$
$\displaystyle\Big{|}\int_{t}^{t+h}\mathbb{E}\Big{[}\int_{t}^{s}Dg_{i_{1}}\big{(}X(r)\big{)}g\big{(}X(r)\big{)}\text{d}W(r)\Big{]}g_{i_{2}}(x)\text{d}s\Big{|}$
$\displaystyle=$ $\displaystyle 0.$

In order to properly handle $I^{(2)}_{1,3},$ we utilize the Hölder inequality,
(3.9) and (3.10) to obtain

$\displaystyle I^{(2)}_{1,3}$ $\displaystyle\leq$
$\displaystyle\Big{\|}\int_{t}^{t+h}\Big{\langle}\int_{t}^{s}Dg_{i_{1}}\big{(}X(r)\big{)}\cdot
f(X(r))\text{d}r,\text{d}W(s)\Big{\rangle}\Big{\|}_{L^{2}(\Omega,\mathbb{R})}\Big{\|}\int_{t}^{t+h}g_{i_{2}}(x)\,\text{d}W(s)\Big{\|}_{L^{2}(\Omega,\mathbb{R})}$
(C.10) $\displaystyle\leq$ $\displaystyle
C(1+|x|^{2r+\rho})h^{\frac{3}{2}}\cdot(1+|x|^{\rho})h^{\frac{1}{2}}\leq
C(1+|x|^{2r+2\rho})h^{2}.$

For the term $I^{(3)}_{1,3},$ similar techniques used in (C.10) help us to
show

$I^{(3)}_{1,3}\leq C(1+|x|^{4\rho-2})h^{2}.$ (C.11)

Substituting the above estimates for $I^{(1)}_{1,3},$ $I^{(2)}_{1,3}$ and
$I^{(3)}_{1,3}$ into (C.8) gives

$I_{1,3}\leq C(1+|x|^{2r+2\rho})h^{2}.$ (C.12)

Collecting the estimates of the three parts of the $I_{1},$ we thus arrive at

$I_{1}\leq C(1+|x|^{6r+2})h^{2}.$ (C.13)

In the sequel, we focus on the estimate of $I_{2}.$ Likewise, we decompose it
as follows:

$\displaystyle I_{2}$ $\displaystyle=$
$\displaystyle\Big{|}\mathbb{E}\Big{[}\Big{(}\int_{t}^{t+h}f^{i_{2}}\big{(}X(s)\big{)}-f^{i_{2}}(x)\,\text{d}s+\int_{t}^{t+h}g_{i_{2}}\big{(}X(s)\big{)}-g_{i_{2}}(x)\,\text{d}W(s)\Big{)}$
(C.14)
$\displaystyle\qquad\times\Big{(}\int_{t}^{t+h}f^{i_{1}}\big{(}X(s)\big{)}\,\text{d}s+\int_{t}^{t+h}g_{i_{1}}\big{(}X(s)\big{)}\,\text{d}W(s)\Big{)}\Big{]}\Big{|}$
$\displaystyle\leq$
$\displaystyle\Big{|}\mathbb{E}\Big{[}\Big{(}\int_{t}^{t+h}f^{i_{2}}\big{(}X(s)\big{)}-f^{i_{2}}(x)\,\text{d}s\Big{)}\cdot\int_{t}^{t+h}f^{i_{1}}\big{(}X(s)\big{)}\,\text{d}s\Big{]}\Big{|}$
$\displaystyle\quad+\Big{|}\mathbb{E}\Big{[}\Big{(}\int_{t}^{t+h}g_{i_{2}}\big{(}X(s)\big{)}-g_{i_{2}}(x)\,\text{d}W(s)\Big{)}\cdot\int_{t}^{t+h}f^{i_{1}}\big{(}X(s)\big{)}\,\text{d}s\Big{]}\Big{|}$
$\displaystyle\quad+\Big{|}\mathbb{E}\Big{[}\Big{(}\int_{t}^{t+h}f^{i_{2}}\big{(}X(s)\big{)}-f^{i_{2}}(x)\,\text{d}s\Big{)}\cdot\int_{t}^{t+h}g_{i_{1}}\big{(}X(s)\big{)}\,\text{d}W(s)\Big{]}\Big{|}$
$\displaystyle\quad+\Big{|}\mathbb{E}\Big{[}\Big{(}\int_{t}^{t+h}g_{i_{2}}\big{(}X(s)\big{)}-g_{i_{2}}(x)\,\text{d}W(s)\Big{)}\cdot\int_{t}^{t+h}g_{i_{1}}\big{(}X(s)\big{)}\,\text{d}W(s)\Big{]}\Big{|}$
$\displaystyle=:$ $\displaystyle I_{2,1}+I_{2,2}+I_{2,3}+I_{2,4}.$

Next we focus on the estimations of $I_{2,1}$-$I_{2,4}$. Similar arguments in
the estimate of $I_{1,2}$ lead to

$\displaystyle I_{2,1}$ $\displaystyle\leq C(1+|x|^{6r+2})h^{2}.$ (C.15)
$\displaystyle I_{2,2}$ $\displaystyle\leq C(1+|x|^{4r+\rho+1})h^{2}.$ (C.16)
$\displaystyle I_{2,3}$ $\displaystyle\leq C(1+|x|^{4r+\rho+1})h^{2}.$ (C.17)

Now we are in a position to estimate $I_{2,4}.$ Using the multi-dimensional
Itô formula, we can get

$\displaystyle I_{2,4}$ $\displaystyle\leq$
$\displaystyle\Big{|}\mathbb{E}\Big{[}\Big{(}\int_{t}^{t+h}\Big{\langle}\int_{t}^{s}Dg_{i_{2}}\big{(}X(r)\big{)}\cdot
g\big{(}X(r)\big{)}\text{d}W(r),\text{d}W(s)\Big{\rangle}\Big{)}\\!\int_{t}^{t+h}g_{i_{1}}\big{(}X(s)\big{)}\,\text{d}W(s)\Big{]}\Big{|}$
(C.18)
$\displaystyle\quad+\Big{|}\mathbb{E}\Big{[}\Big{(}\int_{t}^{t+h}\Big{\langle}\int_{t}^{s}Dg_{i_{2}}\big{(}X(r)\big{)}\cdot
f(X(r))\text{d}r,\text{d}W(s)\Big{\rangle}\Big{)}\\!\int_{t}^{t+h}g_{i_{1}}\big{(}X(s)\big{)}\,\text{d}W(s)\Big{]}\Big{|}$
$\displaystyle\quad+\frac{1}{2}\sum_{l=1}^{m}\Big{|}\mathbb{E}\Big{[}\int_{t}^{t+h}\int_{t}^{s}\text{trace}\big{(}g^{T}(X(r))D^{2}g^{i_{2},l}\big{(}X(r)\big{)}g\big{(}X(r)\big{)}\,\text{d}r\,\text{d}W^{l}(s)\big{)}\\!\int_{t}^{t+h}g_{i_{1}}\big{(}X(s)\big{)}\,\text{d}W(s)\Big{]}\Big{|}$
$\displaystyle=:$ $\displaystyle I^{(1)}_{2,4}+I^{(2)}_{2,4}+I^{(3)}_{2,4}.$

Next we come to the estimate of $I^{(1)}_{2,4},$ which requires more careful
arguments. According to the generalized Itô isometry, the Hölder inequality,
(3.10) and (3.22) yields that

$\displaystyle I^{(1)}_{2,4}$ $\displaystyle=$
$\displaystyle\Big{|}\mathbb{E}\Big{[}\Big{(}\int_{t}^{t+h}\Big{\langle}\int_{t}^{s}Dg_{i_{2}}\big{(}X(r)\big{)}\cdot
g\big{(}X(r)\big{)}\text{d}W(r),\text{d}W(s)\Big{\rangle}\Big{)}\\!\int_{t}^{t+h}g_{i_{1}}\big{(}X(s)\big{)}-g_{i_{1}}(x)\,\text{d}W(s)\Big{]}\Big{|}$
(C.19)
$\displaystyle\quad+\Big{|}\mathbb{E}\Big{[}\Big{(}\int_{t}^{t+h}\Big{\langle}\int_{t}^{s}Dg_{i_{2}}\big{(}X(r)\big{)}\cdot
g\big{(}X(r)\big{)}\text{d}W(r),\text{d}W(s)\Big{\rangle}\Big{)}\\!\int_{t}^{t+h}g_{i_{1}}(x)\,\text{d}W(s)\Big{]}\Big{|}$
$\displaystyle=$
$\displaystyle\Big{|}\int_{t}^{t+h}\mathbb{E}\Big{[}\big{(}\int_{t}^{s}Dg_{i_{2}}\big{(}X(r)\big{)}g\big{(}X(r)\big{)}\text{d}W(r)\big{)}\\!\big{(}g_{i_{1}}\big{(}X(s)\big{)}-g_{i_{1}}(x)\big{)}\,\text{d}s\Big{]}\Big{|}$
$\displaystyle\leq$
$\displaystyle\int_{t}^{t+h}\big{\|}\int_{t}^{s}Dg_{i_{2}}\big{(}X(r)\big{)}g\big{(}X(r)\big{)}\text{d}W(r)\big{\|}_{L^{2}(\Omega,\mathbb{R})}\cdot\big{\|}g_{i_{1}}\big{(}X(s)\big{)}-g_{i_{1}}(x)\big{\|}_{L^{2}(\Omega,\mathbb{R})}\text{d}s$
$\displaystyle\leq$ $\displaystyle C(1+|x|^{2r+3\rho-1})h^{2}.$

Similar arguments in the estimate of $I^{(2)}_{1,3}$ lead to

$\displaystyle I^{(2)}_{2,4}$ $\displaystyle\leq C(1+|x|^{2r+2\rho})h^{2},$
(C.20) $\displaystyle I^{(3)}_{2,4}$ $\displaystyle\leq
C(1+|x|^{4\rho-2})h^{2}.$ (C.21)

From the above estimations, it follows that

$I_{2}\leq C(1+|x|^{6r+2})h^{2}.$ (C.22)

This together with (C.13) helps us to derive from (C.4) that

$\big{|}\mathbb{E}\big{[}(\delta_{X,x})^{i_{1}}(\delta_{X,x})^{i_{2}}-(\delta_{Y_{E},x})^{i_{1}}(\delta_{Y_{E},x})^{i_{2}}\
\big{]}\big{|}\leq C(1+|x|^{6r+2})h^{2}.$ (C.23)

To handle the case $s=3,$ we first infer that

$\displaystyle\big{|}\mathbb{E}\big{[}(\delta_{X,x})^{i_{1}}(\delta_{X,x})^{i_{2}}(\delta_{X,x})^{i_{3}}-(\delta_{Y_{E},x})^{i_{1}}(\delta_{Y_{E},x})^{i_{2}}(\delta_{Y_{E},x})^{i_{3}}\big{]}\big{|}$
(C.24) $\displaystyle\leq$
$\displaystyle\big{|}\mathbb{E}\big{[}(\delta_{X,x})^{i_{1}}((\delta_{X,x})^{i_{2}}(\delta_{X,x})^{i_{3}}-(\delta_{Y_{E},x})^{i_{2}}(\delta_{Y_{E},x})^{i_{3}})\big{]}\big{|}+\big{|}\mathbb{E}\big{[}((\delta_{X,x})^{i_{1}}-(\delta_{Y_{E},x})^{i_{1}})(\delta_{Y_{E},x})^{i_{2}}(\delta_{Y_{E},x})^{i_{3}}\big{]}\big{|}$
$\displaystyle\leq$
$\displaystyle\big{|}\mathbb{E}\big{[}(\delta_{X,x})^{i_{1}}(\delta_{X,x})^{i_{2}}((\delta_{X,x})^{i_{3}}-(\delta_{Y_{E},x})^{i_{3}})\big{]}\big{|}+\big{|}\mathbb{E}\big{[}(\delta_{X,x})^{i_{1}}(\delta_{Y_{E},x})^{i_{3}}((\delta_{X,x})^{i_{2}}-(\delta_{Y_{E},x})^{i_{2}})\big{]}\big{|}$
$\displaystyle\quad+\big{|}\mathbb{E}\big{[}((\delta_{X,x})^{i_{1}}-(\delta_{Y_{E},x})^{i_{1}})(\delta_{Y_{E},x})^{i_{2}}(\delta_{Y_{E},x})^{i_{3}}\big{]}\big{|}.$

Combining the Hölder inequality, (4.3), (4.4), and (4.5) shows

$\big{|}\mathbb{E}\big{[}(\delta_{X,x})^{i_{1}}(\delta_{X,x})^{i_{2}}(\delta_{X,x})^{i_{3}}-(\delta_{Y_{E},x})^{i_{1}}(\delta_{Y_{E},x})^{i_{2}}(\delta_{Y_{E},x})^{i_{3}}\big{]}\big{|}\leq
C(1+|x|^{8r+3})h^{2}.$ (C.25)

Putting these estimates with $s=1,2,3$ together gives the desired assertion. ∎

##  Appendix D Proof of Lemma 5.4

Using the Hölder inequality, (5.8) and (5.9) gives

$\displaystyle\mathbb{E}[|\delta_{Y_{E},x}-\delta_{Y,x}|^{2q}]$
$\displaystyle=$
$\displaystyle\mathbb{E}\Big{[}\Big{|}\int_{t}^{t+h}f(x)-\bar{f}_{h}(x)\,\text{d}s+\int_{t}^{t+h}g(x)-\bar{g}_{h}(x)\,\text{d}W(s)\Big{|}^{2q}\Big{]}$
$\displaystyle\leq$ $\displaystyle
Ch^{2q-1}\int_{t}^{t+h}\big{\|}f(x)-\bar{f}_{h}(x)\big{\|}_{L^{2q}(\Omega,\mathbb{R}^{d})}^{2q}\,\text{d}s$
$\displaystyle\quad+C\big{\|}g(x)-\bar{g}_{h}(x)\big{\|}_{L^{4q}(\Omega,\mathbb{R}^{d\times
m})}^{2q}\cdot\big{\|}W(t+h)-W(t)\big{\|}_{L^{4q}(\Omega,\mathbb{R}^{m})}^{2q}$
$\displaystyle\leq$ $\displaystyle
C(1+|x|^{2qr_{1}})h^{2q(\mathsf{a}+1)}+C(1+|x|^{2q\rho_{1}})h^{2q(\mathsf{b}+\frac{1}{2})},$
(D.2)

which implies the first assertion. Similar arguments as above lead to

$\displaystyle\mathbb{E}\big{[}\big{|}\delta_{Y,x}\big{|}^{2q}\big{]}$
$\displaystyle\leq$ $\displaystyle
Ch^{2q}\big{\|}\bar{f}_{h}(x)\big{\|}_{L^{2q}(\Omega,\mathbb{R}^{d})}^{2q}+C\big{\|}\bar{g}_{h}(x)\big{\|}_{L^{4q}(\Omega,\mathbb{R}^{d\times
m})}^{2q}\cdot\big{\|}W(t+h)-W(t)\big{\|}_{L^{4q}(\Omega,\mathbb{R}^{m})}^{2q}$
(D.3) $\displaystyle\leq$ $\displaystyle
C(1+|x|^{2q(2r+r_{1}+\rho_{1}+1)})h^{q}.$

This gives the second assertion and finishes the proof. ∎

##  Appendix E Proof of Lemma 5.5

A triangle inequality yields

$\displaystyle\bigg{|}\mathbb{E}\Big{[}\prod_{j=1}^{s}(\delta_{X,x})^{i_{j}}-\prod_{j=1}^{s}(\delta_{Y,x})^{i_{j}}\Big{]}\bigg{|}$
(E.1) $\displaystyle\leq$
$\displaystyle\bigg{|}\mathbb{E}\Big{[}\prod_{j=1}^{s}(\delta_{X,x})^{i_{j}}-\prod_{j=1}^{s}(\delta_{Y_{E},x})^{i_{j}}\Big{]}\bigg{|}+\bigg{|}\mathbb{E}\Big{[}\prod_{j=1}^{s}(\delta_{Y_{E},x})^{i_{j}}-\prod_{j=1}^{s}(\delta_{Y,x})^{i_{j}}\Big{]}\bigg{|}$
$\displaystyle=:$ $\displaystyle A_{1}+A_{2},\qquad s=1,2,3.$

Thanks to (4.6), for any $s=1,2,3$ we have

$A_{1}\leq C(1+|x|^{8r+3})h^{2}.$ (E.2)

For $s=1,$ the second term $A_{2}$ can be estimated as follows:

$\displaystyle\big{|}\mathbb{E}\big{[}(\delta_{Y_{E},x})^{i_{1}}-(\delta_{Y,x})^{i_{1}}\big{]}\big{|}$
$\displaystyle=$
$\displaystyle\bigg{|}\mathbb{E}\bigg{[}\big{(}f^{i_{1}}(x)-\bar{f}^{i_{1}}_{h}(x)\big{)}h+\big{(}g_{i_{1}}(x)-\bar{g}^{i_{1}}_{h}(x)\big{)}\big{(}W(t+h)-W(t)\big{)}\bigg{]}\bigg{|}$
(E.3) $\displaystyle\leq$ $\displaystyle
C(1+|x|^{r_{1}})h^{\mathsf{a}+1}+C(1+|x|^{\rho_{1}})h^{\mathsf{b}+1}$
$\displaystyle\leq$ $\displaystyle
C(1+|x|^{r_{1}+\rho_{1}})h^{(\mathsf{a}+1)\land(\mathsf{b}+1)}.$

For $s=2,3$, with the Hölder inequality, (4.4) and Lemma 5.4 at hand, one can
readily get

$\displaystyle\bigg{|}\mathbb{E}\bigg{[}\prod_{j=1}^{2}(\delta_{Y_{E},x})^{i_{j}}-\prod_{j=1}^{2}(\delta_{Y,x})^{i_{j}}\bigg{]}\bigg{|}$
$\displaystyle\leq
C(1+|x|^{2r+2\rho_{1}+2r_{1}+1})h^{(\mathsf{a}+\frac{3}{2})\land(\mathsf{b}+1)},$
(E.4)
$\displaystyle\bigg{|}\mathbb{E}\bigg{[}\prod_{j=1}^{3}(\delta_{Y_{E},x})^{i_{j}}-\prod_{j=1}^{3}(\delta_{Y,x})^{i_{j}}\bigg{]}\bigg{|}$
$\displaystyle\leq
C(1+|x|^{4r+3\rho_{1}+3r_{1}+2})h^{(\mathsf{a}+2)\land(\mathsf{b}+\frac{3}{2})}.$
(E.5)

This together with (E.3), (E.4) gives

$A_{2}\leq
C(1+|x|^{4r+3\rho_{1}+3r_{1}+2})h^{(\mathsf{a}+1)\land(\mathsf{b}+1)},\,\,s=1,2,3.$
(E.6)

This combined with (E.2) promises (5.13). Using the Hölder inequality and
(4.5) yields

$\bigg{\|}\prod_{j=1}^{4}(\delta_{X,x})^{i_{j}}\bigg{\|}_{L^{2}(\Omega;\mathbb{R})}\leq\prod_{j=1}^{4}\Big{\|}(\delta_{X,x})^{i_{j}}\Big{\|}_{L^{8}(\Omega;\mathbb{R})}\leq
C(1+|x|^{8r+4})h^{2}.$ (E.7)

Armed with (5.12), we can derive (5.15) in a similar way. ∎