# On the randomized Euler algorithm under inexact information

Marcin Baranek  AGH University of Krakow, Faculty of Applied Mathematics, Al.
A. Mickiewicza 30, 30-059 Kraków, Poland<EMAIL_ADDRESS>,  Andrzej
Kałuża  AGH University of Krakow, Faculty of Applied Mathematics, Al. A.
Mickiewicza 30, 30-059 Kraków, Poland<EMAIL_ADDRESS>,  Paweł M. Morkisz
NVIDIA Corp. and AGH University of Krakow, Faculty of Applied Mathematics, Al.
A. Mickiewicza 30, 30-059 Kraków, Poland<EMAIL_ADDRESS>,  Paweł
Przybyłowicz  AGH University of Krakow, Faculty of Applied Mathematics, Al. A.
Mickiewicza 30, 30-059 Kraków, Poland<EMAIL_ADDRESS>corresponding
author and  Michał Sobieraj  AGH University of Krakow, Faculty of Applied
Mathematics, Al. A. Mickiewicza 30, 30-059 Kraków, Poland<EMAIL_ADDRESS>
###### Abstract.

This paper focuses on analyzing the error of the randomized Euler algorithm
when only noisy information about the coefficients of the underlying
stochastic differential equation (SDE) and the driving Wiener process is
available. Two classes of disturbed Wiener process are considered, and the
dependence of the algorithm’s error on the regularity of the disturbing
functions is investigated. The paper also presents results from numerical
experiments to support the theoretical findings.

Key words: stochastic differential equations, randomized Euler algorithm,
inexact information, Wiener process, lower bounds, optimality

MSC 2010: 65C30, 68Q25

##  1\. Introduction

We investigate the strong approximation of solutions of the following SDEs

$\left\\{\begin{array}[]{ll}\displaystyle{\,{\mathrm{d}}X(t)=a(t,X(t))\,{\mathrm{d}}t+b(t,X(t))\,{\mathrm{d}}W(t),\
t\in[0,T]},\\\ X(0)=\eta,\end{array}\right.$ (1)

where $T>0$, $W$ is an $m$-dimensional Wiener process, and
$\eta\in\mathbb{R}^{d}$. Our analysis is performed under the assumption that
only standard noisy information about $(a,b,W)$ is available. This means that
we have access to $a,b,W$ only through its inexact values at finite number of
discretization points.

Our interest lies in approximating the values of $X(T)$ using the inexact
information about the coefficients $(a,b)$ and the driving Wiener process $W$.
We consider algorithms that are based on values of $a,b$, and $W$ corrupted by
noise. This noise can arise from measurement errors, rounding procedures, etc.
The inspiration of considering such inexact information comes from various
sources, such as numerically solving SDEs on GPUs and understanding of impact
of low precision in computations (when switching from double to float and
half, see [6]), as well as modeling real-world phenomena that are described by
SDEs such as energy demand/production forecasting (where exact information is
rarely available).

The study of inexact information has been explored in the literature for
various problems, including function approximation and integration ([4], [11],
[12], [18]), approximate solving of ODEs ([5]) and PDEs ([21], [22]), see also
the related monograph [17]. In the context of stochastic integration and
approximation of solution of stochastic differential equations inexact
information about the integrands or coefficients of the underlying SDEs has
been considered in [7], [14], [15]. However, it is important to note that in
[14] and [15] the information considered about the process $W$ was exact. We
also refer to the article [2] where noisy information induced by the
approximation of normally distributed random variables is considered. However,
the computational setting (devoted for weak approximation of SDEs) is
different that the one considered in this paper (established in the context of
strong approximation of the solution $X$).

In this paper, we mainly extend the proof technique known from [14] and [15].
Namely, we cover the case when also the information about the Wiener process
$W$ is inexact. This assumption leads to a significant change in the proof
technique. It allows us to investigate the error behavior for the randomized
Euler scheme under inexact information about the tuple $(a,b,W)$ with
precision parameters $\delta_{1},\delta_{2},\delta_{3}\in[0,1]$ for $a,b,W$,
respectively. (See also, for example, [1], [13], [8], [9] where other
randomized algorithms for approximation of solutions of ODEs and SDEs have
been defined and investigated under exact information.) Roughly speaking, we
show that the $L^{r}(\Omega)$-error of the randomized Euler scheme, that use
$O(n)$ noisy evaluations of $(a,b,W)$, is
$O(n^{-\min\\{\varrho,1/2\\}}+\delta_{1}+\delta_{2}+\delta_{3})$, provided
that the corrupting functions for $W$ are sufficiently regular (see Theorem 1
(i). In the case of less regular corrupting functions for $W$ (assuming only
Hölder continuity) the error might increase due to the presence of
informational noise (see Theorem 1 (ii)).

The main contributions of this paper are as follows:

  * •

Upper error bounds on the randomized Euler algorithm in two classes of
corrupting functions for the Wiener process $W$ (Theorem 1),

  * •

Lower error bounds and optimality of the randomized Euler algorithm (Theorem
2),

  * •

Results of numerical experiments that confirm our theoretical findings
(Section 5).

The structure of the paper is as follows. Section 2 provides basic notions and
definitions, along with a description of the computation model used when
dealing with inexact information for drift and diffusion coefficients, as well
as for the driving Wiener process. In Section 3, we analyze the upper bounds
for the error of the randomized Euler algorithm. Lower bounds and some
optimality results are stated in Section 4. Section 5 contains the results of
numerical experiments conducted to validate our theoretical findings. Finally,
the Appendix provides auxiliary results used in the paper.

##  2\. Preliminaries

We denote by $\mathbb{N}=\\{1,2,\ldots\\}$. Let $W=\\{W(t)\\}_{t\geq 0}$ be a
standard $m$-dimensional Wiener process defined on a complete probability
space $(\Omega,\Sigma,\mathbb{P})$. By $\\{\Sigma_{t}\\}_{t\geq 0}$ we denote
a filtration, satisfying the usual conditions, such that $W$ is a Wiener
process with respect to $\\{\Sigma_{t}\\}_{t\geq 0}$. We set
$\Sigma_{\infty}=\sigma\Bigl{(}\bigcup_{t\geq 0}\Sigma_{t}\Bigr{)}$. We denote
by $\|\cdot\|$ the Frobenius norm in $\mathbb{R}^{m}$ or $\mathbb{R}^{d\times
m}$ respectively, where we treat a column vector in $\mathbb{R}^{m}$ as a
matrix of size $m\times 1$. For $x\in\mathbb{R}^{m}$ and
$\alpha\in\mathbb{R}$, by $x\cdot\alpha$ or $\alpha\cdot x$ we mean a
componentwise scalar-by-vector multiplication and for a matrix
$y\in\mathbb{R}^{d\times m}$ by $y\cdot x$ we mean a standard matrix-by-vector
multiplication. For a sufficiently smooth function
$f:[0,T]\times\mathbb{R}^{m}\to\mathbb{R}$ we denote by $\partial
f(t,y)/\partial y$ its gradient, while by $\partial^{2}f(t,y)/\partial y^{2}$
its Hessian matrix of size $m\times m$. Moreover, for a smooth function
$f:[0,T]\times\mathbb{R}^{m}\to\mathbb{R}^{m}$ we denote by $\partial
f(t,y)/\partial y$ its Jacobi matrix of size $m\times m$, computed also with
respect to the space variable $y$. For $r\in[2,+\infty)$ by the
$L^{r}(\Omega)$-norm, either for a random vector or a random matrix, we mean

$\|Y\|_{r}:=\left({\mathbb{E}}\|Y\|^{r}\right)^{1/r}\qquad\mbox{for}\qquad
Y:\Omega\to\mathbb{R}^{m}\mbox{ or }Y:\Omega\to\mathbb{R}^{d\times m}.$ (2)

We also make us of the following second order differential operator

$\mathcal{L}=\frac{\partial}{\partial
t}+\frac{1}{2}\sum\limits_{k=1}^{m}\frac{\partial^{2}}{\partial y_{k}^{2}}.$
(3)

We now define classes of drift and diffusion coefficients. Let $T>0$, $K>0$,
$\varrho\in(0,1]$. A function $a:[0,T]\times\mathbb{R}^{d}\to\mathbb{R}^{d}$
belongs to $\mathcal{A}_{K}$ if

  * •

the mapping $a:[0,T]\times\mathbb{R}^{d}\to\mathbb{R}^{d}$ is Borel
measurable,

  * •

for all $t\in[0,T]$

$\|a(t,0)\|\leq K,$ (4)

  * •

for all $t\in[0,T]$, $x,y\in\mathbb{R}^{d}$

$\|a(t,x)-a(t,y)\|\leq K\|x-y\|.$ (5)

Note that if $a\in\mathcal{A}_{K}$ then for all
$(t,y)\in[0,T]\times\mathbb{R}^{d}$ we have

$\|a(t,y)\|\leq K(1+\|y\|).$ (6)

A mapping $b:[0,T]\times\mathbb{R}^{d}\to\mathbb{R}^{d\times m}$ belongs to
$\mathcal{B}^{\varrho}_{K}$ if

  * •

$b$ is bounded in the origin $(0,0)$,

$\|b(0,0)\|\leq K,$ (7)

  * •

for all $t\in[0,T]$, $x,y\in\mathbb{R}^{d}$

$\|b(t,x)-b(t,y)\|\leq K\|x-y\|,$ (8)

  * •

for all $t,s\in[0,T]$, $x\in\mathbb{R}^{d}$

$\|b(t,x)-b(s,x)\|\leq K(1+\|x\|)\cdot|t-s|^{\varrho}.$ (9)

The above conditions imply that for all $(t,y)\in[0,T]\times\mathbb{R}^{d}$

$\|b(t,y)\|\leq\bar{K}(1+\|y\|),$ (10)

where $\bar{K}=K\cdot\max\\{1,T^{\varrho}\\}$. We also consider the following
class of initial values

$\mathcal{J}_{K}=\\{\eta\in\mathbb{R}^{d}\ |\ \|\eta\|\leq K\\}.$ (11)

The class of all admissible tuples $(a,b,\eta)$ is defined as

$\mathcal{F}(\varrho,K)=\mathcal{A}_{K}\times\mathcal{B}^{\varrho}_{K}\times\mathcal{J}_{K}.$
(12)

Let

$\delta_{1},\delta_{2},\delta_{3},\delta_{4}\in[0,1]$ (13)

We refer to $\delta_{1}$, $\delta_{2}$, $\delta_{3}$, and $\delta_{4}$ as to
precision parameters. We now describe what we mean by corrupted values and
information about $a,b,W$.

Let us set

$\displaystyle\mathcal{K}^{s}$ $\displaystyle=$
$\displaystyle\\{p:[0,T]\times\mathbb{R}^{d}\to\mathbb{R}^{d\times s}\ |\
p(\cdot,\cdot)-\hbox{Borel measurable},$ $\displaystyle\|p(t,y)\|\leq 1+\|y\|\
\hbox{for all}\ t\in[0,T],y\in\mathbb{R}^{d}\\},$

for $s\in\\{1,m\\}$. The classes $\mathcal{K}^{1}$, $\mathcal{K}^{m}$ are
nonempty and contain constant functions. Let

$V_{c}(\gamma)=\\{\tilde{c}\ |\
\exists_{p_{c}\in\mathcal{K}^{s}}:\tilde{c}=c+\gamma\cdot p_{c}\\},$ (14)

where $c\in\\{a,b\\}$, $(\gamma,s)=(\delta_{1},1)$ if $c=a$ and
$(\gamma,s)=(\delta_{2},m)$ if $c=b$. By $\tilde{a}$ and $\tilde{b}$ we mean
any functions $\tilde{a}\in V_{a}(\delta_{1})$ and $\tilde{b}\in
V_{b}(\delta_{2})$, respectively. We have that $\\{a\\}=V_{a}(0)\subset
V_{a}(\delta_{1})\subset V_{a}(\delta^{\prime}_{1})$ for
$0\leq\delta_{1}\leq\delta^{\prime}_{1}\leq 1$ and $\\{b\\}=V_{b}(0)\subset
V_{b}(\delta_{2})\subset V_{b}(\delta^{\prime}_{2})$ for
$0\leq\delta_{2}\leq\delta^{\prime}_{2}\leq 1$.

In order to introduce perturbed information about the Wiener process $W$, we
introduce the following classes of corrupting functions for $W$

$\displaystyle\mathcal{K}_{0}=\Bigl{\\{}p:[0,T]\times\mathbb{R}^{m}\to\mathbb{R}^{m}\
|\ p^{j}\in C^{1,2}([0,T]\times\mathbb{R}^{m};\mathbb{R}),\ |p^{j}(0,0)|\leq
1,$ $\displaystyle\quad\quad\quad\quad\max\Bigl{\\{}\Bigl{|}\frac{\partial
p^{j}}{\partial t}(t,y)\Bigl{|},\Bigl{\|}\frac{\partial p^{j}}{\partial
y}(t,y)\Bigl{\|},\Bigl{\|}\frac{\partial^{2}p^{j}}{\partial
y^{2}}(t,y)\Bigl{\|}\Bigr{\\}}\leq 1$
$\displaystyle\quad\quad\quad\quad\hbox{for all}\
t\in[0,T],y\in\mathbb{R}^{m},j=1,2,\ldots,m\Bigr{\\}},$ (15)

and

$\displaystyle\mathcal{K}_{\alpha,\beta}=\Bigl{\\{}p:[0,T]\times\mathbb{R}^{m}\to\mathbb{R}^{m}\
|\ \|p(t,x)-p(s,y)\|\leq|t-s|^{\alpha}+\|x-y\|^{\beta},$
$\displaystyle\quad\quad\quad\quad\hbox{for all}\
t,s\in[0,T],x,y\in\mathbb{R}^{m}\Bigr{\\}}.$ (16)

We consider the following classes of disturbed Wiener processes

$\mathcal{W}_{0}(\delta_{3})=\\{\tilde{W}\ |\
\exists_{p\in\mathcal{K}_{0}}:\forall_{(t,\omega)\in[0,T]\times\Omega}\
\tilde{W}(t,\omega)=W(t,\omega)+\delta_{3}\cdot p(t,W(t,\omega))\\},$ (17)

and

$\mathcal{W}_{\alpha,\beta}(\delta_{3})=\\{\tilde{W}\ |\
\exists_{p\in\mathcal{K}_{\alpha,\beta}}:\forall_{(t,\omega)\in[0,T]\times\Omega}\
\tilde{W}(t,\omega)=W(t,\omega)+\delta_{3}\cdot p(t,W(t,\omega))\\}.$ (18)

We have that
$\\{W\\}=\mathcal{W}_{0}(0)\subset\mathcal{W}_{0}(\delta_{3})\subset\mathcal{W}_{0}(\delta^{\prime}_{3})$
for $0\leq\delta_{3}\leq\delta^{\prime}_{3}\leq 1$, and similarly for
$\mathcal{W}_{\alpha,\beta}$. As in [7] the classes defined above allow us to
model the impact of regularity of noise on the error bound.

We assume that the algorithm is based on discrete noisy information about
$(a,b,W)$ and exact information about $\eta$. Hence, a vector of noisy
information has the following form

$\displaystyle\mathcal{N}(\tilde{a},\tilde{b},\tilde{W},\eta)$
$\displaystyle=$
$\displaystyle\Bigl{[}\tilde{a}(\xi_{0},y_{0}),\tilde{a}(\xi_{1},y_{1}),\ldots,\tilde{a}(\xi_{i_{1}-1},y_{i_{1}-1}),$
(19)
$\displaystyle\tilde{b}(t_{0},z_{0}),\tilde{b}(t_{1},z_{1}),\ldots,\tilde{b}(t_{i_{1}-1},z_{i_{1}-1}),$
$\displaystyle\tilde{W}(u_{0}),\tilde{W}(u_{1}),\ldots,\tilde{W}(u_{i_{2}-1}),\eta],$

where $i_{1},i_{2}\in\mathbb{N}$ and $(\xi_{0},\xi_{1},\ldots,\xi_{i_{1}-1})$
is a random vector on $(\Omega,\Sigma,\mathbb{P})$ which takes values in
$[0,T]^{i_{1}}$. We assume that the $\sigma$-fields
$\sigma(\xi_{0},\xi_{1},\ldots,\xi_{i_{1}-1})$ and $\Sigma_{\infty}$ are
independent. Moreover, $t_{0},t_{1},\ldots,t_{i_{1}-1}\in[0,T]$ and
$u_{0},u_{1},\ldots,u_{i_{2}-1}\in[0,T]$ are fixed time points. The evaluation
points $y_{j}$, $z_{j}$ for the spatial variables $y,z$ of $a(\cdot,y)$ and
$b(\cdot,z)$ can be computed in adaptive way with respect to $(a,b,\eta)$ and
$W$. Formally, it means that there exist Borel measurable mappings
$\psi_{0}:\mathbb{R}^{i_{2}\times m}\times\mathbb{R}^{d}\to\mathbb{R}^{2d}$,
$\psi_{j}:\mathbb{R}^{d\times j}\times\mathbb{R}^{d\times m\times
j}\times\mathbb{R}^{i_{2}\times m}\times\mathbb{R}^{d}\to\mathbb{R}^{2d}$,
$j=1,2,\ldots,i_{1}-1$, such that the successive points $y_{j},z_{j}$ are
given as follows

$(y_{0},z_{0})=\psi_{0}\Bigl{(}\tilde{W}(u_{0}),\tilde{W}(u_{1}),\ldots,\tilde{W}(u_{i_{2}-1}),\eta\Bigr{)},$
(20)

and for $j=1,2,\ldots,i_{1}-1$

$\displaystyle(y_{j},z_{j})$ $\displaystyle=$
$\displaystyle\psi_{j}\Bigl{(}\tilde{a}(\xi_{0},y_{0}),\tilde{a}(\xi_{1},y_{1}),\ldots,\tilde{a}(\xi_{j-1},y_{j-1}),$
(21)
$\displaystyle\quad\quad\tilde{b}(t_{0},z_{0}),\tilde{b}(t_{1},z_{1}),\ldots,\tilde{b}(t_{j-1},z_{j-1}),$
$\displaystyle\quad\quad\tilde{W}(u_{0}),\tilde{W}(u_{1}),\ldots,\tilde{W}(u_{i_{2}-1}),\eta\Bigr{)}.$

The total number of noisy evaluations of $(a,b,W)$ is $l=2i_{1}+i_{2}$.

The algorithm $\mathcal{A}$ that uses the noisy information
$\mathcal{N}(\tilde{a},\tilde{b},\tilde{W},\eta)$ and computes approximation
of $X(T)$ is defined as

$\mathcal{A}(\tilde{a},\tilde{b},\tilde{W},\eta)=\varphi\Bigl{(}\mathcal{N}(\tilde{a},\tilde{b},\tilde{W},\eta)\Bigr{)},$
(22)

for some Borel measurable function $\varphi:\mathbb{R}^{i_{1}\times
d}\times\mathbb{R}^{i_{1}\times d\times m}\times\mathbb{R}^{i_{2}\times
m}\times\mathbb{R}^{d}\to\mathbb{R}^{d}$. For a fixed $n\in\mathbb{N}$ by
$\Phi_{n}$ we denote a class of all algorithms (22) for which the total number
of evaluations $l$ is at most $n$.

Let $r\in[2,+\infty)$. The $L^{r}(\Omega)$-error of $\mathcal{A}\in\Phi_{n}$
for the fixed tuple $(a,b,\eta)\in\mathcal{G}$ is given by

$\displaystyle
e^{(r)}(\mathcal{A},a,b,\eta,\mathcal{W},\delta_{1},\delta_{2},\delta_{3})$
$\displaystyle\quad\quad=\sup\limits_{(\tilde{a},\tilde{b},\tilde{W})\in
V_{a}(\delta_{1})\times
V_{b}(\delta_{2})\times\mathcal{W}(\delta_{3})}\|X(a,b,W,\eta)(T)-\mathcal{A}(\tilde{a},\tilde{b},\tilde{W},\eta)\|_{r},$
(23)

where $\mathcal{W}\in\\{\mathcal{W}_{0},\mathcal{W}_{\alpha,\beta}\\}$ and
$\mathcal{G}$ is a subclass of $\mathcal{F}(\varrho,K)$. The worst case error
of $\mathcal{A}$ in $\mathcal{G}$ is

$e^{(r)}(\mathcal{A},\mathcal{G},\mathcal{W},\delta_{1},\delta_{2},\delta_{3})=\sup\limits_{(a,b,\eta)\in\mathcal{G}}e^{(r)}(\mathcal{A},a,b,\eta,\mathcal{W},\delta_{1},\delta_{2},\delta_{3}).$
(24)

Finally, we look for (essentially) sharp bounds for the $n$th minimal error,
defined as

$e^{(r)}_{n}(\mathcal{G},\mathcal{W},\delta_{1},\delta_{2},\delta_{3})=\inf\limits_{\mathcal{A}\in\Phi_{n}}e^{(r)}(\mathcal{A},\mathcal{G},\mathcal{W},\delta_{1},\delta_{2},\delta_{3}).$
(25)

In (25) we define the minimal possible error among all algorithms of the form
(22) that use at most $n$ noisy evaluations of $a,b$ and $W$ . Our aim is to
find possibly sharp bounds on the $n$th minimal error, i.e., lower and upper
bounds which match up to constants. We are also interested in defining an
algorithm for which the infimum in
$e^{(r)}_{n}(\mathcal{G},\mathcal{W},\delta_{1},\delta_{2},\delta_{3})$ is
asymptotically attained. We call such an algorithm the optimal one.

Unless otherwise stated, all constants appearing in this paper (including
those in the ’O’, ’$\Omega$’, and ’$\Theta$’ notation) will only depend on the
parameters of the class $\mathcal{F}(\varrho,K)$, $\alpha,\beta$ and $r$.
Furthermore, the same symbol may be used in order to denote different
constants.

##  3\. Error of the Euler scheme under inexact information

We investigate the error of the randomized Euler scheme in the case of inexact
information about $a$, $b$, and the driving Wiener process $W$.

Fix $n\in\mathbb{N}$, $t_{i}=iT/n$ for $i=0,1,\ldots,n$. Let
$\\{\xi_{i}\\}_{i=0}^{n-1}$ be independent random variables on
$(\Omega,\Sigma,\mathbb{P})$, such that the $\sigma$-fields
$\sigma(\xi_{0},\xi_{1},\ldots,\xi_{n-1})$ and $\Sigma_{\infty}$ are
independent, with $\xi_{i}$ being uniformly distributed on $[t_{i},t_{i+1}]$.
Let us fix $(a,b,W)\in\mathcal{F}(\varrho,K)$ and take any
$(\tilde{a},\tilde{b},\tilde{W})\in V_{a}(\delta_{1})\times
V_{b}(\delta_{2})\times\mathcal{W}(\delta_{3})$, where
$\mathcal{W}\in\\{\mathcal{W}_{0},\mathcal{W}_{\alpha,\beta}\\}$. The
randomized Euler scheme under inexact information is defined by taking

$\bar{X}^{RE}_{n}(0)=\eta,$ (26)

and

$\bar{X}^{RE}_{n}(t_{i+1})=\bar{X}^{RE}_{n}(t_{i})+\tilde{a}(\xi_{i},\bar{X}^{RE}_{n}(t_{i}))\cdot\frac{T}{n}+\tilde{b}(t_{i},\bar{X}^{RE}_{n}(t_{i}))\cdot\Delta\tilde{W}_{i},$
(27)

for $i=0,1,\ldots,n-1$, where
$\Delta\tilde{W}_{i}=\tilde{W}(t_{i+1})-\tilde{W}(t_{i})$. The randomized
Euler algorithm $\mathcal{\bar{A}}^{RE}_{n}$ is defined as

$\mathcal{\bar{A}}^{RE}_{n}(\tilde{a},\tilde{b},\tilde{W},\eta):=\bar{X}^{RE}_{n}(T).$
(28)

The informational cost of the randomized Euler algorithm is $O(n)$ noisy
evaluations of $a,b,W$. By $X^{RE}_{n}$ we denote the randomized Euler
algorithm $\bar{X}^{RE}_{n}$ under the case when information is exact, i.e.,
when $\delta_{1}=\delta_{2}=\delta_{3}=0$.

Let $\mathcal{G}^{n}=\sigma(\xi_{0},\xi_{1},\ldots,\xi_{n-1})$ and
$\tilde{\Sigma}_{t}^{n}=\sigma\Bigl{(}\Sigma_{t}\cup\mathcal{G}^{n}\Bigr{)}$,
$t\geq 0$. Since the $\sigma$-fields $\Sigma_{\infty}$ and $\mathcal{G}^{n}$
are independent, the process $W$ is still the $m$-dimensional Wiener process
on $(\Omega,\Sigma,\mathbb{P})$ with respect to
$\\{\tilde{\Sigma}_{t}^{n}\\}_{t\geq 0}$.

######  Proposition 1.

Let $r\in[2,+\infty)$, $\varrho\in(0,1]$.

  * (i)

There exists $C\in(0,+\infty)$, depending only on the parameters of the class
$\mathcal{F}(\varrho,K)$ and $r$, such that for all $n\in\mathbb{N}$,
$\delta_{1},\delta_{2},\delta_{3}\in[0,1]$,
$(a,b,\eta)\in\mathcal{F}(\varrho,K)$, $(\tilde{a},\tilde{b},\tilde{W})\in
V_{a}(\delta_{1})\times V_{b}(\delta_{2})\times\mathcal{W}_{0}(\delta_{3})$ it
holds

$\Bigl{\|}\max\limits_{0\leq i\leq
n}\|X^{RE}_{n}(t_{i})-\bar{X}^{RE}_{n}(t_{i})\|\Bigl{\|}_{r}\leq
C(\delta_{1}+\delta_{2}+\delta_{3}).$ (29)

  * (ii)

Let $\alpha,\beta\in(0,1]$. There exists $C\in(0,+\infty)$, depending only on
the parameters of the class $\mathcal{F}(\varrho,K)$, $\alpha,\beta$, and $r$,
such that for all $n\in\mathbb{N}$,
$\delta_{1},\delta_{2},\delta_{3}\in[0,1]$,
$(a,b,\eta)\in\mathcal{F}(\varrho,K)$, $(\tilde{a},\tilde{b},\tilde{W})\in
V_{a}(\delta_{1})\times
V_{b}(\delta_{2})\times\mathcal{W}_{\alpha,\beta}(\delta_{3})$ it holds

$\displaystyle\Bigl{\|}\max\limits_{0\leq i\leq
n}\|X^{RE}_{n}(t_{i})-\bar{X}^{RE}_{n}(t_{i})\|\Bigl{\|}_{r}$
$\displaystyle\leq C(\delta_{1}+\delta_{2}+\delta_{3}\cdot
n^{1-\gamma})\cdot(1+\delta_{3}n^{1-\gamma})\cdot
e^{C(1+(\delta_{3}n^{1-\gamma})^{r})},$ (30)

where $\gamma=\min\\{\alpha,\beta/2\\}$.

###### Proof.

Firstly, we prove (i). For $\tilde{W}\in\mathcal{W}_{0}(\delta_{3})$ we have
that

$\tilde{W}(t)=W(t)+\delta_{3}\cdot Z(t),$ (31)

with $Z(t)=p_{W}(t,W(t))$ and $p_{W}\in\mathcal{K}_{0}$. Then, by the Itô
formula we get that

$Z(t)=Z(0)+M(t)+V(t),\quad t\in[0,T],$ (32)

where $M(t)=[M^{1}(t),M^{2}(t),\ldots,M^{m}(t)]$,
$V(t)=[V^{1}(t),V^{2}(t),\ldots,V^{m}(t)]$ and

$\displaystyle V^{j}(t)$ $\displaystyle=$
$\displaystyle\int\limits_{0}^{t}\mathcal{L}p^{j}_{W}(z,W(z))\,{\mathrm{d}}z,$
(33) $\displaystyle M^{j}(t)$ $\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{m}\int\limits_{0}^{t}\frac{\partial
p^{j}_{W}}{\partial y_{i}}(z,W(z))\,{\mathrm{d}}W^{i}(z),$ (34)

for $j=1,2,\ldots,m$. We stress that $\\{V(t)\\}_{t\in[0,T]}$ is a continuous
process of bounded variation that is adapted to
$\\{\tilde{\Sigma}_{t}^{n}\\}_{t\geq 0}$. Moreover, since
$(M(t),\tilde{\Sigma}_{t}^{n})_{t\in[0,T]}$ is a martingale, $Z$ is still a
continuous semimartingale with respect to the extended filtration
$\\{\tilde{\Sigma}_{t}^{n}\\}_{t\geq 0}$. In the sequel we will consider
stochastic integrals, with respect to the semimartingales $W$ and $Z$, of
processes that are adapted to the filtration
$\\{\tilde{\Sigma}_{t}^{n}\\}_{t\geq 0}$.

From (26), (27) for $i=0,1,\ldots,n$ we can write that

$\displaystyle\bar{X}^{RE}_{n}(t_{i})=\eta+\frac{T}{n}\sum\limits_{j=0}^{i-1}\tilde{a}(\xi_{j},\bar{X}^{RE}_{n}(t_{j}))+\
\sum\limits_{j=0}^{i-1}\tilde{b}(t_{j},\bar{X}^{RE}_{n}(t_{j}))\cdot\Delta
W_{j}$
$\displaystyle\quad\quad\quad\quad\quad\quad+\delta_{3}\sum\limits_{j=0}^{i-1}\tilde{b}(t_{j},\bar{X}^{RE}_{n}(t_{j}))\cdot\Delta
Z_{j},$ (35)

and

$\displaystyle
X^{RE}_{n}(t_{i})=\eta+\frac{T}{n}\sum\limits_{j=0}^{i-1}a(\xi_{j},X^{RE}_{n}(t_{j}))+\
\sum\limits_{j=0}^{i-1}b(t_{j},X^{RE}_{n}(t_{j}))\cdot\Delta W_{j}.$ (36)

Therefore

$\displaystyle\bar{e}_{i}:=X^{RE}_{n}(t_{i})-\bar{X}^{RE}_{n}(t_{i})=\sum\limits_{j=0}^{i-1}\int\limits_{t_{j}}^{t_{j+1}}\Bigl{(}a(\xi_{j},X^{RE}_{n}(t_{j}))-\tilde{a}(\xi_{j},\bar{X}^{RE}_{n}(t_{j}))\Bigr{)}\,{\mathrm{d}}s$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\
\sum\limits_{j=0}^{i-1}\int\limits_{t_{j}}^{t_{j+1}}\Bigl{(}b(t_{j},X^{RE}_{n}(t_{j}))-\tilde{b}(t_{j},\bar{X}^{RE}_{n}(t_{j}))\Bigr{)}\,{\mathrm{d}}W(s)$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\
(-\delta_{3})\sum\limits_{j=0}^{i-1}\int\limits_{t_{j}}^{t_{j+1}}\tilde{b}(t_{j},\bar{X}^{RE}_{n}(t_{j}))\,{\mathrm{d}}Z(s)$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad=\
\sum\limits_{j=0}^{i-1}\Bigl{(}A_{j}+B_{j}+C_{1,j}+C_{2,j}+C_{3,j}\Bigr{)},$
(37)

where

$\displaystyle
A_{j}=\int\limits_{t_{j}}^{t_{j+1}}\Bigl{(}a(\xi_{j},X^{RE}_{n}(t_{j}))-a(\xi_{j},\bar{X}^{RE}_{n}(t_{j}))\Bigr{)}\,{\mathrm{d}}s$
$\displaystyle
B_{j}=\int\limits_{t_{j}}^{t_{j+1}}\Bigl{(}b(t_{j},X^{RE}_{n}(t_{j}))-b(t_{j},\bar{X}^{RE}_{n}(t_{j}))\Bigr{)}\,{\mathrm{d}}W(s)$
$\displaystyle
C_{1,j}=(-\delta_{1})\int\limits_{t_{j}}^{t_{j+1}}p_{a}(\xi_{j},\bar{X}^{RE}_{n}(t_{j}))\,{\mathrm{d}}s$
$\displaystyle
C_{2,j}=(-\delta_{2})\int\limits_{t_{j}}^{t_{j+1}}p_{b}(t_{j},\bar{X}^{RE}_{n}(t_{j}))\,{\mathrm{d}}W(s),$
(38) $\displaystyle
C_{3,j}=(-\delta_{3})\int\limits_{t_{j}}^{t_{j+1}}\tilde{b}(t_{j},\bar{X}^{RE}_{n}(t_{j}))\,{\mathrm{d}}Z(s)$
$\displaystyle=(-\delta_{3})\int\limits_{t_{j}}^{t_{j+1}}\tilde{b}(t_{j},\bar{X}^{RE}_{n}(t_{j}))\,{\mathrm{d}}M(s)+(-\delta_{3})\int\limits_{t_{j}}^{t_{j+1}}\tilde{b}(t_{j},\bar{X}^{RE}_{n}(t_{j}))\,{\mathrm{d}}V(s)$
$\displaystyle=(-\delta_{3})\int\limits_{t_{j}}^{t_{j+1}}\tilde{b}(t_{j},\bar{X}^{RE}_{n}(t_{j}))\frac{\partial
p_{W}}{\partial y}(s,W(s))\,{\mathrm{d}}W(s)$
$\displaystyle+(-\delta_{3})\int\limits_{t_{j}}^{t_{j+1}}\tilde{b}(t_{j},\bar{X}^{RE}_{n}(t_{j}))\mathcal{L}p_{W}(s,W(s))\,{\mathrm{d}}s.$
(39)

Then for all $i=0,1,\ldots,n$

$\displaystyle\|\bar{e}_{i}\|\leq\sum\limits_{j=0}^{i-1}\|A_{j}\|+\Bigl{\|}\sum\limits_{j=0}^{i-1}B_{j}\Bigl{\|}+\sum\limits_{j=0}^{i-1}\|C_{1,j}\|+\Bigl{\|}\sum\limits_{j=0}^{i-1}C_{2,j}\Bigl{\|}+\Bigl{\|}\sum\limits_{j=0}^{i-1}C_{3,j}\Bigl{\|},$
(40)

where

$\Bigl{\|}\sum\limits_{j=0}^{i-1}C_{3,j}\Bigl{\|}\leq\delta_{3}\cdot\Bigl{(}C^{1}_{3,i}+C^{2}_{3,i}\Bigr{)},$
(41)

with

$\displaystyle
C^{1}_{3,i}=\Bigl{\|}\sum\limits_{j=0}^{i-1}\int\limits_{t_{j}}^{t_{j+1}}\tilde{b}(t_{j},\bar{X}^{RE}_{n}(t_{j}))\frac{\partial
p_{W}}{\partial y}(s,W(s))\,{\mathrm{d}}W(s)\Bigl{\|},$ (42) $\displaystyle
C^{2}_{3,i}=\sum\limits_{j=0}^{i-1}\Bigl{\|}\int\limits_{t_{j}}^{t_{j+1}}\tilde{b}(t_{j},\bar{X}^{RE}_{n}(t_{j}))\mathcal{L}p_{W}(s,W(s))\,{\mathrm{d}}s\Bigl{\|}.$
(43)

Hence, for $i=0,1,\ldots,n$

$\displaystyle\|\bar{e}_{i}\|\leq\sum\limits_{j=0}^{i-1}\|A_{j}\|+\Bigl{\|}\sum\limits_{j=0}^{i-1}B_{j}\Bigl{\|}+\sum\limits_{j=0}^{n-1}\|C_{1,j}\|$
$\displaystyle\quad\quad\quad+\max\limits_{1\leq i\leq
n}\Bigl{\|}\sum\limits_{j=0}^{i-1}C_{2,j}\Bigl{\|}+\delta_{3}\cdot\max\limits_{1\leq
i\leq n}C^{1}_{3,i}+\delta_{3}\cdot C^{2}_{3,n},$ (44)

and for all $k=0,1,\ldots,n$

$\displaystyle\mathbb{E}\Bigl{(}\max\limits_{0\leq i\leq
k}\|\bar{e}_{i}\|^{r}\Bigr{)}\leq
c_{r}\mathbb{E}\Bigl{(}\sum\limits_{j=0}^{k-1}\|A_{j}\|\Bigr{)}^{r}+c_{r}\mathbb{E}\Bigl{(}\max\limits_{1\leq
i\leq
k}\Bigl{\|}\sum\limits_{j=0}^{i-1}B_{j}\Bigl{\|}^{r}\Bigr{)}+c_{r}\mathbb{E}\Bigl{(}\sum\limits_{j=0}^{n-1}\|C_{1,j}\|\Bigr{)}^{r}$
$\displaystyle\quad\quad\quad+c_{r}\mathbb{E}\Bigl{(}\max\limits_{1\leq i\leq
n}\Bigl{\|}\sum\limits_{j=0}^{i-1}C_{2,j}\Bigl{\|}^{r}\Bigr{)}+c_{r}\delta_{3}^{r}\cdot\mathbb{E}\Bigl{(}\max\limits_{1\leq
i\leq
n}(C^{1}_{3,i})^{r}\Bigr{)}+c_{r}\delta_{3}^{r}\cdot\mathbb{E}(C^{2}_{3,n})^{r}.$
(45)

By the Jensen inequality we have for $k=0,1,\ldots,n$

$\Bigl{(}\sum\limits_{j=0}^{k-1}\|\bar{e}_{j}\|\Bigr{)}^{r}\leq
k^{r-1}\sum\limits_{j=0}^{k-1}\|e_{j}\|^{r}\leq
n^{r-1}\sum\limits_{j=0}^{k-1}\|e_{j}\|^{r},$ (46)

which implies that

$\Bigl{(}\frac{1}{n}\sum\limits_{j=0}^{k-1}\|\bar{e}_{j}\|\Bigr{)}^{r}\leq\frac{1}{n}\sum\limits_{j=0}^{k-1}\|e_{j}\|^{r}.$
(47)

Moreover

$\|A_{j}\|\leq\frac{KT}{n}\|\bar{e}_{j}\|,$ (48)

and hence

$\Bigl{(}\sum\limits_{j=0}^{k-1}\|A_{j}\|\Bigr{)}^{r}\leq
K^{r}T^{r}\Bigl{(}\frac{1}{n}\sum\limits_{j=0}^{k-1}\|\bar{e}_{j}\|\Bigr{)}^{r}\leq\frac{K^{r}T^{r}}{n}\sum\limits_{j=0}^{k-1}\|\bar{e}_{j}\|^{r}.$
(49)

It holds that
$\displaystyle{\Bigl{(}\sum\limits_{j=0}^{k}B_{j},\tilde{\Sigma}^{n}_{t_{k+1}}\Bigr{)}_{k=0,1,\ldots,n-1}}$
is a discrete-time martingale. To see that let us denote by
$\displaystyle{M_{k}\coloneqq\sum\limits_{j=0}^{k}B_{j}}$. By the basic
properties of the Itô integral we get for $k=0,1,\ldots,n-1$ that
$\sigma(M_{k})\subset\tilde{\Sigma}^{n}_{t_{k+1}}$,

$\displaystyle\mathbb{E}(M_{k+1}-M_{k}|\tilde{\Sigma}^{n}_{t_{k+1}})=\mathbb{E}(B_{k+1}|\tilde{\Sigma}^{n}_{t_{k+1}})=0,\quad
k=0,1,\ldots,n-2,$

and for $j=0,1,\ldots,n-1$

$\mathbb{E}\|B_{j}\|^{r}\leq
C(T/n)^{r/2}\mathbb{E}\|\bar{e}_{j}\|^{r}<+\infty.$ (50)

Hence, by the Burkholder and Jensen inequalities we get for $k=0,1,\ldots,n$

$\displaystyle\mathbb{E}\Bigl{(}\max\limits_{1\leq i\leq
k}\Bigl{\|}\sum\limits_{j=0}^{i-1}B_{j}\Bigl{\|}^{r}\Bigr{)}=\mathbb{E}\Bigl{(}\max\limits_{0\leq
i\leq k-1}\Bigl{\|}\sum\limits_{j=0}^{i}B_{j}\Bigl{\|}^{r}\Bigr{)}\leq
C_{r}^{r}\mathbb{E}\Bigl{(}\sum\limits_{j=0}^{k-1}\|B_{j}\|^{2}\Bigr{)}^{r/2}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\leq
C_{r}^{r}k^{\frac{r}{2}-1}\sum\limits_{j=0}^{k-1}\mathbb{E}\|B_{j}\|^{r}\leq\frac{C}{n}\sum\limits_{j=0}^{k-1}\mathbb{E}\|\bar{e}_{j}\|^{r}.$
(51)

From (3), (49), (3), and the fact that $\bar{e}_{0}=0$ we get for
$k=0,1,\ldots,n$ that

$\displaystyle\mathbb{E}\Bigl{(}\max\limits_{0\leq i\leq
k}\|\bar{e}_{i}\|^{r}\Bigr{)}\leq\frac{C}{n}\sum\limits_{j=0}^{k-1}\mathbb{E}\|\bar{e}_{j}\|^{r}+c_{r}R_{n}\leq\frac{C}{n}\sum\limits_{j=0}^{k-1}\mathbb{E}\Bigl{(}\max\limits_{0\leq
i\leq j}\|\bar{e}_{i}\|^{r}\Bigr{)}+c_{r}R_{n}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad=\frac{C}{n}\sum\limits_{j=1}^{k-1}\mathbb{E}\Bigl{(}\max\limits_{0\leq
i\leq j}\|\bar{e}_{i}\|^{r}\Bigr{)}+c_{r}R_{n},$ (52)

where

$\displaystyle
R_{n}=\mathbb{E}\Bigl{(}\sum\limits_{j=0}^{n-1}\|C_{1,j}\|\Bigr{)}^{r}+\mathbb{E}\Bigl{(}\max\limits_{1\leq
i\leq n}\Bigl{\|}\sum\limits_{j=0}^{i-1}C_{2,j}\Bigl{\|}^{r}\Bigr{)}$
$\displaystyle\quad\quad+\delta_{3}^{r}\cdot\mathbb{E}\Bigl{(}\max\limits_{1\leq
i\leq
n}(C^{1}_{3,i})^{r}\Bigr{)}+\delta_{3}^{r}\cdot\mathbb{E}(C^{2}_{3,n})^{r}.$
(53)

By the discrete version of the Gronwall’s lemma (see, for example, Lemma 2.1
in [8]) we get

$\mathbb{E}\Bigl{(}\max\limits_{0\leq i\leq n}\|\bar{e}_{i}\|^{r}\Bigr{)}\leq
KR_{n}.$ (54)

By the Jensen inequality and Lemma 1 we get

$\displaystyle\mathbb{E}\Bigl{(}\sum\limits_{j=0}^{n-1}\|C_{1,j}\|\Bigr{)}^{r}\leq\delta_{1}^{r}T^{r}n^{-1}\sum\limits_{j=0}^{n-1}\mathbb{E}\|p_{a}(\xi_{j},\bar{X}^{RE}_{n}(t_{j}))\|^{r}$
$\displaystyle\leq C\delta_{1}^{r}(1+\max\limits_{0\leq j\leq
n}\mathbb{E}\|\bar{X}^{RE}_{n}(t_{j})\|^{r})\leq K_{1}\delta_{1}^{r}.$ (55)

The process
$\displaystyle{\Bigl{(}\sum\limits_{j=0}^{k}C_{2,j},\tilde{\Sigma}^{n}_{t_{k+1}}\Bigr{)}_{k=0,1,\ldots,n-1}}$
is a discrete-time martingale - this can be justified in analogous way as for
$\displaystyle{\Bigl{(}\sum\limits_{j=0}^{k}B_{j},\tilde{\Sigma}^{n}_{t_{k+1}}\Bigr{)}_{k=0,1,\ldots,n-1}}$.
Hence, again by the Burkholder and Jensen inequalities, we obtain

$\displaystyle\mathbb{E}\Bigl{(}\max\limits_{1\leq i\leq
n}\Bigl{\|}\sum\limits_{j=0}^{i-1}C_{2,j}\Bigl{\|}^{r}\Bigr{)}\leq
C_{r}^{r}n^{\frac{r}{2}-1}\sum\limits_{j=0}^{n-1}\mathbb{E}\|C_{2,j}\|^{r}$
$\displaystyle\quad\quad\leq K_{1}(1+\max\limits_{0\leq j\leq
n}\mathbb{E}\|\bar{X}^{RE}_{n}(t_{j})\|^{r})\delta_{2}^{r}\leq
K_{2}\delta_{2}^{r}.$ (56)

Let us denote by

$D_{j}=\int\limits_{t_{j}}^{t_{j+1}}\tilde{b}(t_{j},\bar{X}^{RE}_{n}(t_{j}))\frac{\partial
p_{W}}{\partial y}(s,W(s))\,{\mathrm{d}}W(s),$ (57)

then
$\displaystyle{\Bigl{(}\sum\limits_{j=0}^{k}D_{j},\tilde{\Sigma}^{n}_{t_{k+1}}\Bigr{)}_{k=0,1,\ldots,n-1}}$
is also a discrete-time martingale. Therefore, by the Burkholder and Jensen
inequalities, we obtain

$\displaystyle\mathbb{E}\Bigl{(}\max\limits_{1\leq i\leq
n}(C^{1}_{3,i})^{r}\Bigr{)}=\mathbb{E}\Bigl{(}\max\limits_{1\leq i\leq
n}\Bigl{\|}\sum\limits_{j=0}^{i-1}D_{j}\Bigl{\|}^{r}\Bigr{)}\leq
C_{r}^{r}n^{\frac{r}{2}-1}\sum\limits_{j=0}^{n-1}\mathbb{E}\|D_{j}\|^{r},$
(58)

where, by (83) and submultiplicativity of the Frobenius norm, we get

$\displaystyle\mathbb{E}\|D_{j}\|^{r}=\mathbb{E}\Bigl{\|}\int\limits_{t_{j}}^{t_{j+1}}\tilde{b}(t_{j},\bar{X}^{RE}_{n}(t_{j}))\frac{\partial
p_{W}}{\partial y}(s,W(s))\,{\mathrm{d}}W(s)\Bigl{\|}^{r}$ $\displaystyle\leq
C(T/n)^{\frac{r}{2}-1}\mathbb{E}\int\limits_{t_{j}}^{t_{j+1}}\Bigl{\|}\tilde{b}(t_{j},\bar{X}^{RE}_{n}(t_{j}))\frac{\partial
p_{W}}{\partial y}(s,W(s))\Bigl{\|}^{r}\,{\mathrm{d}}s$ $\displaystyle\leq
C(T/n)^{\frac{r}{2}-1}\mathbb{E}\int\limits_{t_{j}}^{t_{j+1}}\Bigl{\|}\tilde{b}(t_{j},\bar{X}^{RE}_{n}(t_{j}))\Bigl{\|}^{r}\cdot\Bigl{\|}\frac{\partial
p_{W}}{\partial y}(s,W(s))\Bigl{\|}^{r}\,{\mathrm{d}}s\leq K_{3}n^{-r/2},$
(59)

for $j=0,1,\ldots,n-1$. This implies that

$\mathbb{E}\Bigl{(}\max\limits_{1\leq i\leq n}(C^{1}_{3,i})^{r}\Bigr{)}\leq
K_{4}.$ (60)

Finally, from (84) and Lemma 1

$\displaystyle\mathbb{E}(C^{2}_{3,n})^{r}=\mathbb{E}\Bigl{(}\sum\limits_{j=0}^{n-1}\Bigl{\|}\int\limits_{t_{j}}^{t_{j+1}}\tilde{b}(t_{j},\bar{X}^{RE}_{n}(t_{j}))\mathcal{L}p_{W}(s,W(s))\,{\mathrm{d}}s\Bigl{\|}\Bigr{)}^{r}$
$\displaystyle\leq
n^{r-1}\sum\limits_{j=0}^{n-1}\mathbb{E}\Bigl{\|}\int\limits_{t_{j}}^{t_{j+1}}\tilde{b}(t_{j},\bar{X}^{RE}_{n}(t_{j}))\mathcal{L}p_{W}(s,W(s))\,{\mathrm{d}}s\Bigl{\|}^{r}$
$\displaystyle\leq
n^{r-1}\sum\limits_{j=0}^{n-1}\mathbb{E}\Bigl{(}\int\limits_{t_{j}}^{t_{j+1}}\|\tilde{b}(t_{j},\bar{X}^{RE}_{n}(t_{j}))\|\cdot\|\mathcal{L}p_{W}(s,W(s))\|\,{\mathrm{d}}s\Bigr{)}^{r}$
$\displaystyle\leq\frac{C}{n}\sum\limits_{j=0}^{n-1}\mathbb{E}\|\tilde{b}(t_{j},\bar{X}^{RE}_{n}(t_{j}))\|^{r}\leq
K_{5}.$ (61)

Combining (68), (3), (3), (60), (3) we obtain

$\mathbb{E}\Bigl{(}\max\limits_{0\leq i\leq n}\|\bar{e}_{i}\|^{r}\Bigr{)}\leq
K_{6}(\delta_{1}^{r}+\delta_{2}^{r}+\delta_{3}^{r}),$ (62)

which proves (i).

We now show (ii). Let $\tilde{W}\in\mathcal{W}_{\alpha,\beta}(\delta_{3})$.
Note that in this case $Z$ might not be semimartingale nor even a process of
bounded variation. Hence, we have that

$\bar{e}_{i}=X^{RE}_{n}(t_{i})-\bar{X}^{RE}_{n}(t_{i})=\sum\limits_{j=0}^{i-1}\Bigl{(}A_{j}+B_{j}+C_{1,j}+C_{2,j}+C_{3,j}\Bigr{)},$
(63)

where

$\displaystyle
A_{j}=\int\limits_{t_{j}}^{t_{j+1}}\Bigl{(}a(\xi_{j},X^{RE}_{n}(t_{j}))-a(\xi_{j},\bar{X}^{RE}_{n}(t_{j}))\Bigr{)}\,{\mathrm{d}}s$
$\displaystyle
B_{j}=\int\limits_{t_{j}}^{t_{j+1}}\Bigl{(}b(t_{j},X^{RE}_{n}(t_{j}))-b(t_{j},\bar{X}^{RE}_{n}(t_{j}))\Bigr{)}\,{\mathrm{d}}W(s)$
$\displaystyle
C_{1,j}=(-\delta_{1})\int\limits_{t_{j}}^{t_{j+1}}p_{a}(\xi_{j},\bar{X}^{RE}_{n}(t_{j}))\,{\mathrm{d}}s$
$\displaystyle
C_{2,j}=(-\delta_{2})\int\limits_{t_{j}}^{t_{j+1}}p_{b}(t_{j},\bar{X}^{RE}_{n}(t_{j}))\,{\mathrm{d}}W(s),$
(64)

and

$C_{3,j}=(-\delta_{3})\cdot\tilde{b}(t_{j},\bar{X}^{RE}_{n}(t_{j}))\cdot\Delta
Z_{j}.$ (65)

Using similar arguments as for the proof of (3) we have

$\mathbb{E}\Bigl{(}\max\limits_{0\leq i\leq
k}\|\bar{e}_{i}\|^{r}\Bigr{)}\leq\frac{C}{n}\sum\limits_{j=1}^{k-1}\mathbb{E}\Bigl{(}\max\limits_{0\leq
i\leq j}\|\bar{e}_{i}\|^{r}\Bigr{)}+c_{r}\bar{R}_{n},$ (66)

where this time we get from (3), (3), and Lemma 2 that

$\displaystyle\bar{R}_{n}=\mathbb{E}\Bigl{(}\sum\limits_{j=0}^{n-1}\|C_{1,j}\|\Bigr{)}^{r}+\mathbb{E}\Bigl{(}\max\limits_{1\leq
i\leq
n}\Bigl{\|}\sum\limits_{j=0}^{i-1}C_{2,j}\Bigl{\|}^{r}\Bigr{)}+\mathbb{E}\Bigl{(}\max\limits_{1\leq
i\leq n}\Bigl{\|}\sum\limits_{j=0}^{i-1}C_{3,j}\Bigl{\|}^{r}\Bigr{)}$
$\displaystyle\leq
C(\delta_{1}^{r}+\delta_{2}^{r})(1+\delta_{3}^{r}n^{r(1-\gamma)})\cdot
e^{C(1+\delta_{3}^{r}n^{r(1-\gamma)})}+\mathbb{E}\Bigl{(}\max\limits_{1\leq
i\leq n}\Bigl{\|}\sum\limits_{j=0}^{i-1}C_{3,j}\Bigl{\|}^{r}\Bigr{)}.$ (67)

Agian by the discrete version of the Gronwall’s lemma we get

$\mathbb{E}\Bigl{(}\max\limits_{0\leq i\leq n}\|\bar{e}_{i}\|^{r}\Bigr{)}\leq
K\bar{R}_{n}.$ (68)

Moreover,

$\Biggl{\|}\max\limits_{1\leq i\leq
n}\Bigl{\|}\sum\limits_{j=0}^{i-1}C_{3,j}\Bigl{\|}\Biggl{\|}_{r}\leq\Bigl{\|}\sum\limits_{j=0}^{n-1}\|C_{3,j}\|\Bigl{\|}_{r}\leq\sum\limits_{j=0}^{n-1}\Bigl{\|}\|C_{3,j}\|\Bigl{\|}_{r},$
(69)

and, since $\bar{X}^{RE}_{n}(t_{j})$ and $\Delta W_{j}$ are independent, we
have by Lemma 2

$\displaystyle\Bigl{\|}\|C_{3,j}\|\Bigl{\|}_{r}\leq
C\delta_{3}\Bigl{\|}1+\|\bar{X}^{RE}_{n}(t_{j})\|\Bigl{\|}_{r}\cdot\Bigl{\|}(T/n)^{\alpha}+\|\Delta
W_{j}\|^{\beta}\Bigl{\|}_{r}$ $\displaystyle\leq C\delta_{3}\cdot
n^{-\gamma}\cdot\Bigl{(}1+\max\limits_{0\leq i\leq
n}\|\bar{X}^{RE}_{n}(t_{i})\|_{r}\Bigr{)}$ $\displaystyle\leq
C\delta_{3}n^{-\gamma}\cdot(1+\delta_{3}n^{1-\gamma})\cdot
e^{C(1+(\delta_{3}n^{1-\gamma})^{r})},$ (70)

and

$\Biggl{\|}\max\limits_{1\leq i\leq
n}\Bigl{\|}\sum\limits_{j=0}^{i-1}C_{3,j}\Bigl{\|}\Biggl{\|}_{r}\leq
C\delta_{3}n^{1-\gamma}\cdot(1+\delta_{3}n^{1-\gamma})\cdot
e^{C(1+(\delta_{3}n^{1-\gamma})^{r})}.$ (71)

From (3), (68), and (71) we get the thesis of (i). ∎

######  Theorem 1.

Let $r\in[2,+\infty)$, $\varrho\in(0,1]$.

  * (i)

There exists $C\in(0,+\infty)$, depending only on the parameters of the class
$\mathcal{F}(\varrho,K)$ and $r$, such that for all $n\in\mathbb{N}$,
$\delta_{1},\delta_{2},\delta_{3}\in[0,1]$,
$(a,b,\eta)\in\mathcal{F}(\varrho,K)$, $(\tilde{a},\tilde{b},\tilde{W})\in
V_{a}(\delta_{1})\times V_{b}(\delta_{2})\times\mathcal{W}_{0}(\delta_{3})$ it
holds

$\|X(a,b,W,\eta)(T)-\mathcal{\bar{A}}^{RE}_{n}(\tilde{a},\tilde{b},\tilde{W},\eta)\|_{r}\leq
C(n^{-\min\\{\varrho,1/2\\}}+\delta_{1}+\delta_{2}+\delta_{3}).$ (72)

  * (ii)

Let $\alpha,\beta\in(0,1]$. There exists $C\in(0,+\infty)$, depending only on
the parameters of the class $\mathcal{F}(\varrho,K)$ and $r$, such that for
all $n\in\mathbb{N}$, $\delta_{1},\delta_{2},\delta_{3}\in[0,1]$,
$(a,b,\eta)\in\mathcal{F}(\varrho,K)$, $(\tilde{a},\tilde{b},\tilde{W})\in
V_{a}(\delta_{1})\times
V_{b}(\delta_{2})\times\mathcal{W}_{\alpha,\beta}(\delta_{3})$ it holds

$\displaystyle\|X(a,b,W,\eta)(T)-\mathcal{\bar{A}}^{RE}_{n}(\tilde{a},\tilde{b},\tilde{W},\eta)\|_{r}\leq
Cn^{-\min\\{\varrho,1/2\\}}$
$\displaystyle\quad\quad+C(\delta_{1}+\delta_{2}+\delta_{3}\cdot
n^{1-\gamma})\cdot(1+\delta_{3}n^{1-\gamma})\cdot
e^{C(1+(\delta_{3}n^{1-\gamma})^{r})},$ (73)

where $\gamma=\min\\{\alpha,\beta/2\\}$.

###### Proof.

By Proposition 1 in [14] (for the case $\delta_{1}=\delta_{2}=\delta_{3}=0$)
and from Proposition 1 (i) we get

$\displaystyle\|X(a,b,W,\eta)(T)-\mathcal{\bar{A}}^{RE}_{n}(\tilde{a},\tilde{b},\tilde{W},\eta)\|_{r}\leq\|X(a,b,W,\eta)(T)-X_{n}^{RE}(a,b,W,\eta)(T)\|_{r}$
$\displaystyle+\|X_{n}^{RE}(a,b,W,\eta)(T)-\bar{X}_{n}^{RE}(\tilde{a},\tilde{b},\tilde{W},\eta)(T)\|_{r}\leq
C_{1}n^{-\min\\{\varrho,1/2\\}}+C_{2}(\delta_{1}+\delta_{2}+\delta_{3}),$ (74)

which implies (i). Similarly, by using the above error decomposition together
with Proposition 1 (ii) we obtain the thesis of (ii). ∎

##  4\. Lower bounds and optimality of the randomized Euler algorithm

In this section, we investigate lower error bound for an arbitrary method (22)
from the class $\Phi_{n}$. We focus only on the class $\mathcal{W}_{0}$ of
disturbed Wiener processes $\tilde{W}$. Essentially, sharp lower bounds in the
class $\mathcal{W}_{\alpha,\beta}$ are left as an open problem. For some
special cases we also show optimality of the randomized Euler algorithm
$\mathcal{\bar{A}}^{RE}_{n}$.

The following result follows from Lemma 3 in [14] and Theorem 1.

######  Theorem 2.

Let $r\in[2,+\infty)$, $K\in(0,+\infty)$, $\varrho\in(0,1]$. Then there exist
$C_{1},C_{2}\in(0,+\infty)$, depending only on the parameter of the class
$\mathcal{F}(\varrho,K)$ and $r$, such that for all $n\in\mathbb{N}$,
$\delta_{1},\delta_{2},\delta_{3}\in[0,1]$ it holds

$C_{1}(n^{-\min\\{\varrho,1/2\\}}+\delta_{1}+\delta_{2})\leq
e^{(r)}_{n}(\mathcal{F}(\varrho,K),\mathcal{W}_{0},\delta_{1},\delta_{2},\delta_{3})\leq
C_{2}(n^{-\min\\{\varrho,1/2\\}}+\delta_{1}+\delta_{2}+\delta_{3}).$

In particular, the $n$th minimal error satisfies

$e^{(r)}_{n}(\mathcal{F}(\varrho,K),\mathcal{W}_{0},\delta_{1},\delta_{2},0)=\Theta(n^{-\min\\{\varrho,1/2\\}}+\delta_{1}+\delta_{2}),$
(75)

and

$e^{(r)}_{n}(\mathcal{F}(\varrho,K),\mathcal{W}_{0},\delta_{1},\delta_{2},\max\\{\delta_{1},\delta_{2}\\})=\Theta(n^{-\min\\{\varrho,1/2\\}}+\delta_{1}+\delta_{2}),$
(76)

as $n\to+\infty$, $\max\\{\delta_{1},\delta_{2}\\}\to 0^{+}$. In both cases
(75), (76) an optimal algorithm is the randomized Euler algorithm
$\mathcal{\bar{A}}^{RE}_{n}$.

In general we have a gap between upper and lower bounds, and sharp bounds
appear only in special cases (i.e.: when $\delta_{3}=0$ or
$\delta_{3}=\max\\{\delta_{1},\delta_{2}\\}$). However, in the particular case
for the randomized Euler algorithm we have the following bounds for its worst-
case error (the proof follows from Proposition 1 in [7]).

######  Proposition 2.

Let $r\in[2,+\infty)$, $K\in(0,+\infty)$, $\varrho\in(0,1]$. Then for the
randomized Euler algorithm $\mathcal{\bar{A}}^{RE}_{n}$ it holds

$e^{(r)}(\mathcal{\bar{A}}^{RE}_{n},\mathcal{F}(\varrho,K),\mathcal{W}_{0},\delta_{1},\delta_{2},\delta_{3})=\Theta(n^{-\min\\{\varrho,1/2\\}}+\delta_{1}+\delta_{2}+\delta_{3}),$

as $n\to+\infty$, $\max\\{\delta_{1},\delta_{2},\delta_{3}\\}\to 0+$.

##  5\. Numerical experiments

Let us consider the following linear SDE that describes the well-known
multidimensional Black-Scholes model

$\left\\{\begin{array}[]{ll}\mathrm{d}X(t)=\begin{pmatrix}\mu_{1}X_{1}(t)\\\
\mu_{2}X_{2}(t)\\\ \vdots\\\
\mu_{d}X_{d}(t)\end{pmatrix}\mathrm{d}t+\begin{pmatrix}\sigma^{1,1}X_{1}(t)&\sigma^{1,2}X_{1}(t)&\cdots&\sigma^{1,m}X_{1}(t)\\\
\sigma^{2,1}X_{2}(t)&\sigma^{2,2}X_{2}(t)&\cdots&\sigma^{2,m}X_{2}(t)\\\
\vdots&\vdots&\ddots&\vdots\\\
\sigma^{d,1}X_{d}(t)&\sigma^{d,2}X_{d}(t)&\cdots&\sigma^{d,m}X_{d}(t)\end{pmatrix}\mathrm{d}W(t),&\\\
X(0)=x_{0},\quad t\in[0,T],\end{array}\right.$ (77)

where $\sigma^{i,j}>0$ for $i\in\\{1,\ldots,d\\}$, $j\in\\{1,\ldots,m\\}$,
$\mu_{i}\in{\mathbb{R}}$ , $x_{0}\in{\mathbb{R}}_{+}^{d}$. Functions $a$ and
$b$ take the following forms
$a(t,x)=(\mu_{1}x_{1},\ldots,\mu_{d}x_{d})^{\text{T}},b(t,x)=\begin{pmatrix}\sigma^{1,1}x_{1}&\sigma^{1,2}x_{1}&\cdots&\sigma^{1,m}x_{1}\\\
\sigma^{2,1}x_{2}&\sigma^{2,2}x_{2}&\cdots&\sigma^{2,m}x_{2}\\\
\vdots&\vdots&\ddots&\vdots\\\
\sigma^{d,1}x_{d}&\sigma^{d,2}x_{d}&\cdots&\sigma^{d,m}x_{d}\end{pmatrix}.$
The exact solution of problem (77) has the following form

$X_{i}(t)=X_{i}(0)\cdot\exp\Biggl{(}\Biggl{(}\mu-\frac{1}{2}\sum_{j=1}^{m}\Big{(}\sigma^{i,j}\Big{)}^{2}\Biggr{)}t+\sum_{j=1}^{m}\sigma^{i,j}W_{j}(t)\Biggr{)}$

for $i=1,\ldots,d.$

To perform numerical experiments, we choose two examples.

Example 1

$a(t,x)=\begin{pmatrix}0.5x_{1},0.7x_{2}\\\ \end{pmatrix}^{T},$ $\
b(t,x)=\begin{pmatrix}0.5x_{1},0.7x_{1},0.2x_{1}\\\
-0.5x_{2},-0.7x_{2},-0.2x_{2}\\\ \end{pmatrix}$, $x_{0}=(1,2)^{T}$, $T=1$.

Example 2

$a(t,x)=\begin{pmatrix}0.5x_{1},0.7x_{2},0.4x_{3}\\\ \end{pmatrix}^{T},$ $\
b(t,x)=\begin{pmatrix}0.5x_{1},0.7x_{1},0.2x_{1}\\\
0.1x_{2},0.x_{2},0.013x_{2}\\\ 0.x_{3},0.75x_{3},0.013x_{3}\\\ \end{pmatrix}$,
$x_{0}=(1,0.1,0.4)^{T}$, $T=1$. We take an estimator of the error of
$\|X(T)-\bar{X}_{n}^{RE}(T)\|_{2}$

$\varepsilon_{K}(\bar{X}^{RE}_{n}(T))=\Biggl{(}\frac{1}{K}\sum_{j=1}^{K}\|X_{(j)}(T)-\\-\bar{X}^{RE}_{(j),n}(T)\|^{2}\Biggr{)}^{1/2}.$
(78)

We also conduct numerical experiments for an equation in which the exact
solution is unknown (Example 3). In this case, to estimate the error
$\|X(T)-\bar{X}_{n}^{RE}(T)\|_{2}$, the exact solution $X(T)$ is approximated
by $\bar{X}^{RE}_{n}$ computed under exact information with for
$n=1310720=10\cdot 2^{17}$, and then

$\varepsilon_{K}(\bar{X}^{RE}_{n}(T))=\Biggl{(}\frac{1}{K}\sum_{j=1}^{K}\|\bar{X}^{RE}_{(j),1310720}(T)-\\-\bar{X}^{RE}_{(j),n}(T)\|^{2}\Biggr{)}^{1/2}.$
(79)

Example 3 $a(t,x)=0.5\begin{pmatrix}t\sin(10x_{1})\\\
\cos(7x_{2})\end{pmatrix}^{T},$ $\
b(t,x)=\begin{pmatrix}tx_{1}&tx_{2}&\sin(x_{2})\\\
t\cos(x_{1})&x_{2}&-x_{1}\end{pmatrix}$, $x_{0}=(1,2)^{T}$, $T=1$.

For Examples 1 - 3 we used $K=20000$.

###  5.1. Linear disturbing function

To obtain results in the numerical simulations related to the Theorem 1 (ii)
we propose the following disruptive function

$p_{a}(t,x)=\textrm{U}_{1}\cdot a(t,x)$

and

$p_{b}(t,x)=\textrm{U}_{2}\cdot b(t,x),$

where $\textrm{U}_{1},\textrm{U}_{2}$ has uniform distribution over the
interval $[-1,1]$.

As a corrupting function for the Wiener process $W$ in Examples 1 - 3 we take

$p_{w}(t,x)=\textrm{U}_{3}\cdot x,$ (80)

where $\textrm{U}_{3}$ is a random variable with a uniform distribution over
the interval $[-1,1]$. We also assume that $\textrm{U}_{1}$, $\textrm{U}_{2}$,
$\textrm{U}_{3}$ are independent of $W$ and $\mathcal{G}^{n}$. We use those
uniform distributions to better approximate the worst case scenario setting of
error.

Figure 1. Log-log plot of the randomized error of the Euler algorithm for the
Example 1 with $p_{w}$ as disruption of the Wiener process Figure 2. Log-log
plot of the randomized error of the Euler algorithm for the Example 2 with
$p_{w}$ as disruption of the Wiener process Figure 3. Log-log plot of the
randomized error of the Euler algorithm for the Example 3 with $p_{w}$ as
disruption of the Wiener process

###  5.2. Nonlinear disturbing function

To conduct illustrative numerical simulations, as per Theorem 1 (ii), we
propose the following disruptive functions

$p_{a}(t,x)=\textrm{U}_{1}\cdot a(t,x)$

and

$p_{b}(t,x)=\textrm{U}_{2}\cdot b(t,x),$

where $\textrm{U}_{1},\textrm{U}_{2}$ is a random variable with a uniform
distribution over the interval $[-1,1]$.

As a corrupting function for the Wiener process, we consider

$p_{w,\beta}(t,x)=\textrm{U}_{3}\cdot\text{sgn}(\sin({100\|x\|}))\cdot|\sin({100\|x\|})|^{\beta},$
(81)

where $\textrm{U}_{3}$ is a random variable with a uniform distribution over
the interval $[-1,1]$. We also assume that $\textrm{U}_{1}$, $\textrm{U}_{2}$,
$\textrm{U}_{3}$ are independent of $W$ and $\mathcal{G}^{n}$.

Figure 4. Example 1 with $\delta:=\delta_{1}=\delta_{2}=\delta_{3}\in\\{0,\
0.1,\ 0.01,\ 0.0001\\}$ and $p_{w,\beta}$ disruption. Figure 5. Example 2 with
$\delta:=\delta_{1}=\delta_{2}=\delta_{3}\in\\{0,\ 0.1,\ 0.01,\ 0.0001\\}$ and
$p_{w,\beta}$ disruption. Figure 6. Example 3 with
$\delta:=\delta_{1}=\delta_{2}=\delta_{3}\in\\{0,\ 0.1,\ 0.01,\ 0.0001\\}$ and
$p_{w,\beta}$ disruption. Figure 7. Log-log plot with a doubly logarithmic
y-axis for Example 3 with (81) as the disruptive function for the Wiener
process, with $\delta=0.8,\ 0.9$.

In Figures 6 and 7, we present results for (81) as the disruptive function for
Wiener process in Example 3.

In this case the, logarithmic error exhibits exponential growth, necessitating
the use of doubly logarithmic y-axis. Notably, such error behavior was not
observed for disruptive functions from the class $\mathcal{K}_{0}$ for the
Wiener process.

##  6\. Conclusions

We have investigated the error and optimality of the randomized Euler scheme
in the case when we have access only to noisy standard information about the
coefficients $a$, $b$, and driving Wiener process $W$. We considered two
classes of disturbed Wiener processes for which we derived upper error bounds
for the randomized Euler algorithm. These bounds indicate that the error
significantly depends on the regularity of the disturbing functions. The
numerical experiments demonstrate that beyond a certain value of $n$, which
depends on the size of the disturbance, the error of the randomized Euler
algorithm stabilizes, and increasing number of discretization points $n$ does
not lead to reduction in error.

One particularly interesting observation is depicted in Figure 7. When using
function (81) as a perturbation for the Wiener process with sufficiently high
$\delta$, we observe an exponential increase in error as $n$ increases.

In future research, we plan to investigate the error of (multilevel) Monte
Carlo method under inexact information for the weak approximation of solutions
of SDEs.

##  7\. Appendix

The proof of the following fact is straightforward and, therefore, omitted.

######  Fact 1.

If $p\in\mathcal{K}_{0}$ then for all $t\in[0,T],x,y\in\mathbb{R}^{m}$ it
holds

$\|p(t,x)\|\leq m^{1/2}(1+\|x\|),\quad\|p(t,x)-p(t,y)\|\leq m^{1/2}\|x-y\|,$
(82) $\Bigl{\|}\frac{\partial p}{\partial y}(t,y)\Bigl{\|}\leq m^{1/2},$ (83)
$\|\mathcal{L}p(t,y)\|\leq\Bigl{(}2m+\frac{m^{2}}{2}\Bigr{)}^{1/2}.$ (84)

In order to estimate absolute moments of $\bar{X}^{RE}_{n}(t_{i})$, in the
case when the disturbed Wiener process $\tilde{W}$ belongs to the class
$\mathcal{W}_{0}(\delta_{3})$, we use the following time-continuous randomized
Euler process

$\left\\{\begin{array}[]{ll}\tilde{\bar{X}}^{RE}_{n}(0)&=\eta,\\\
\tilde{\bar{X}}^{RE}_{n}(t)&=\tilde{\bar{X}}^{RE}_{n}(t_{i})+\tilde{a}(\xi_{i},\tilde{\bar{X}}^{RE}_{n}(t_{i}))\cdot(t-t_{i})+\tilde{b}(t_{i},\tilde{\bar{X}}^{RE}_{n}(t_{i}))\cdot(\tilde{W}(t)-\tilde{W}(t_{i})),\end{array}\right.$
(85)

for $t\in[t_{i},t_{i+1}]$, $i=0,1,\ldots,n-1$, where
$\tilde{W}(t)=W(t)+\delta_{3}\cdot Z(t)$, $Z(t)=p_{W}(t,W(t))$,
$p_{W}\in\mathcal{K}_{0}$. It is easy to see that
$\tilde{\bar{X}}^{RE}_{n}(t_{i})=\bar{X}^{RE}_{n}(t_{i})$ for
$i=0,1,\ldots,n$. Moreover, the process
$(\tilde{\bar{X}}^{RE}_{n}(t))_{t\in[0,T]}$ is adapted to
$(\tilde{\Sigma}^{n}_{t})_{t\in[0,T]}$, which can be shown by induction.

######  Lemma 1.

Let $r\in[2,+\infty)$. There exists $C\in(0,+\infty)$, depending only on the
parameters of the class $\mathcal{F}(\varrho,K)$ and $r$, such that for all
$n\in\mathbb{N}$, $\delta_{1},\delta_{2},\delta_{3}\in[0,1]$,
$(a,b,\eta)\in\mathcal{F}(\varrho,K)$, $(\tilde{a},\tilde{b},\tilde{W})\in
V_{a}(\delta_{1})\times V_{b}(\delta_{2})\times\mathcal{W}_{0}(\delta_{3})$ it
holds

$\sup\limits_{0\leq t\leq T}\mathbb{E}\|\tilde{\bar{X}}^{RE}_{n}(t)\|^{r}\leq
C(1+\delta_{1}^{r}+\delta_{2}^{r}+\delta_{3}^{r})e^{CT(1+\delta_{1}^{r}+\delta_{2}^{r}+\delta_{3}^{r})}.$
(86)

###### Proof.

We denote by

$\bar{V}_{i}=(\xi_{i},\tilde{\bar{X}}^{RE}_{n}(t_{i})),\quad\bar{U}_{i}=(t_{i},\tilde{\bar{X}}^{RE}_{n}(t_{i})).$
(87)

Firstly, we show by induction that

$\max\limits_{0\leq i\leq
n}\mathbb{E}\|\tilde{\bar{X}}^{RE}_{n}(t_{i})\|^{r}<+\infty.$ (88)

Let us assume that there exists $l\in\\{0,1,\ldots,n-1\\}$ such that
$\max\limits_{0\leq i\leq
l}\mathbb{E}\|\tilde{\bar{X}}^{RE}_{n}(t_{i})\|^{r}<+\infty$. (This obviously
holds for $l=0$.) Due to the fact that
$\sigma(\bar{U}_{l})\subset\tilde{\Sigma}^{n}_{t_{l}}$ and, by (84),

$\|V(t_{l+1})-V(t_{l})\|\leq\int\limits_{t_{l}}^{t_{l+1}}\|\mathcal{L}p_{W}(s,W(s))\|\,{\mathrm{d}}s\leq
c(m)\cdot(t_{l+1}-t_{l}),$ (89)

we get

$\displaystyle\mathbb{E}\|\tilde{\bar{X}}^{RE}_{n}(t_{l+1})\|^{r}\leq
C\mathbb{E}\|\tilde{\bar{X}}^{RE}_{n}(t_{l})\|^{r}+C(T/n)^{r}\cdot\mathbb{E}\|\tilde{a}(\bar{U}_{l})\|^{r}$
$\displaystyle\quad\quad\quad\quad\quad\quad+\
C\mathbb{E}\|\tilde{b}(\bar{U}_{l})\|^{r}\cdot\mathbb{E}\|W(t_{l+1})-W(t_{l})\|^{r}$
(90) $\displaystyle\quad\quad\quad\quad\quad+\
C\delta_{3}\mathbb{E}\Bigl{\|}\int\limits_{t_{l}}^{t}\tilde{b}(\bar{U}_{l})\frac{\partial
p_{W}}{\partial y}(s,W(s))\,{\mathrm{d}}W(s)\Bigl{\|}^{r}$
$\displaystyle\quad\quad\quad\quad\quad\
+\delta_{3}\mathbb{E}\Bigl{(}\|\tilde{b}(\bar{U}_{l})\|^{r}\cdot\|V(t_{l+1})-V(t_{l})\|^{r}\Bigr{)}\leq
K(1+\mathbb{E}\|\tilde{\bar{X}}^{RE}_{n}(t_{l})\|^{r})<\infty.$ (91)

Hence, $\displaystyle{\max\limits_{0\leq i\leq
l+1}\mathbb{E}\|\tilde{\bar{X}}^{RE}_{n}(t_{i})\|^{r}=\max\\{\max\limits_{0\leq
i\leq
l}\mathbb{E}\|\tilde{\bar{X}}^{RE}_{n}(t_{i})\|^{r},\mathbb{E}\|\tilde{\bar{X}}^{RE}_{n}(t_{l+1})\|^{r}\\}<+\infty}$
and the inductive step is completed. Hence, we have shown (88). Moreover, by
(88) we get

$\sup\limits_{0\leq t\leq T}\mathbb{E}\|\tilde{\bar{X}}^{RE}_{n}(t)\|^{r}\leq
C\Bigl{(}1+\max\limits_{0\leq i\leq
n-1}\mathbb{E}\|\tilde{\bar{X}}^{RE}_{n}(t_{i})\|^{r}\Bigr{)}<+\infty$ (92)

The constant in (92) depends on $n$. In the second part of the proof we will
show that we can obtain the bound (86) with $C$ that does not depend on $n$.

Let for $t\in[0,T]$

$\phi_{n}(t)=\sum\limits_{i=0}^{n-1}\tilde{a}(\bar{V}_{i})\cdot\mathbf{1}_{(t_{i},t_{i+1}]}(t),$
(93)
$\psi_{n}(t)=\sum\limits_{i=0}^{n-1}\tilde{b}(\bar{U}_{i})\cdot\mathbf{1}_{(t_{i},t_{i+1}]}(t).$
(94)

Note that $\\{\psi_{n}(t)\\}_{t\in[0,T]}$ is
$\\{\tilde{\Sigma}^{n}_{t}\\}_{t\geq 0}$-progressively measurable simple
process. Hence, we have for all $t\in[0,T]$ that

$\tilde{\bar{X}}^{RE}_{n}(t)=\eta+\tilde{\bar{A}}^{RE}_{n}(t)+\tilde{\bar{B}}^{RE}_{n}(t)+\tilde{\bar{C}}^{RE}_{n}(t),$
(95)

where

$\displaystyle\tilde{\bar{A}}^{RE}_{n}(t)=\int\limits_{0}^{t}\phi_{n}(s)\,{\mathrm{d}}s,$
$\displaystyle\tilde{\bar{B}}^{RE}_{n}(t)=\int\limits_{0}^{t}\psi_{n}(s)\,{\mathrm{d}}W(s),$

and

$\displaystyle\tilde{\bar{C}}^{RE}_{n}(t)=\delta_{3}\cdot\int\limits_{0}^{t}\psi_{n}(s)\,{\mathrm{d}}Z(s)$
$\displaystyle\quad\quad\quad\quad=\delta_{3}\cdot\int\limits_{0}^{t}\psi_{n}(s)\frac{\partial
p_{W}}{\partial
y}(s,W(s))\,{\mathrm{d}}W(s)+\delta_{3}\cdot\int\limits_{0}^{t}\psi_{n}(s)\mathcal{L}p_{W}(s,W(s))\,{\mathrm{d}}s.$
(96)

and all above stochastic integrals are well-defined. Hence, we have for
$t\in[0,T]$

$\displaystyle\tilde{\bar{X}}^{RE}_{n}(t)=\eta+\int\limits_{0}^{t}\Bigl{[}\phi_{n}(s)+\delta_{3}\cdot\psi_{n}(s)\cdot\mathcal{L}p_{W}(s,W(s))\Bigr{]}\,{\mathrm{d}}s$
$\displaystyle+\int\limits_{0}^{t}\psi_{n}(s)\cdot\Bigl{[}I+\delta_{3}\cdot\frac{\partial
p_{W}}{\partial y}(s,W(s))\Bigr{]}\,{\mathrm{d}}W(s),$ (97)

where $I$ is an identity matrix of size $m\times m$. Hence, by (83), (84)

$\displaystyle\mathbb{E}\|\tilde{\bar{X}}^{RE}_{n}(t)\|^{r}\leq
C_{1}\|\eta\|^{r}+C_{2}\mathbb{E}\int\limits_{0}^{t}\|\phi_{n}(s)\|^{r}\,{\mathrm{d}}s+C_{3}(1+\delta_{3}^{r})\mathbb{E}\int\limits_{0}^{t}\|\psi_{n}(s)\|^{r}\,{\mathrm{d}}s$
$\displaystyle\leq
K_{1}\cdot(1+\|\eta\|^{r})\cdot(1+\delta_{1}^{r}+\delta_{2}^{r}+\delta_{3}^{r})$
$\displaystyle\quad\quad\quad\quad+K_{2}\cdot(1+\delta_{1}^{r}+\delta_{2}^{r}+\delta_{3}^{r})\cdot\int\limits_{0}^{t}\sum\limits_{i=0}^{n-1}\mathbb{E}\|\tilde{\bar{X}}^{RE}_{n}(t_{i})\|^{r}\cdot\mathbf{1}_{(t_{i},t_{i+1}]}(s)\,{\mathrm{d}}s$
(98)

and therefore for all $t\in[0,T]$

$\displaystyle\sup\limits_{0\leq u\leq
t}\mathbb{E}\|\tilde{\bar{X}}^{RE}_{n}(u)\|^{r}\leq
K_{1}\cdot(1+\|\eta\|^{r})\cdot(1+\delta_{1}^{r}+\delta_{2}^{r}+\delta_{3}^{r})$
$\displaystyle+K_{2}\cdot(1+\delta_{1}^{r}+\delta_{2}^{r}+\delta_{3}^{r})\int\limits_{0}^{t}\sup\limits_{0\leq
u\leq s}\mathbb{E}\|\tilde{\bar{X}}^{RE}_{n}(u)\|^{r}\,{\mathrm{d}}s$ (99)

where $K_{1}$ and $K_{2}$ depends only on the parameters of the class
$\mathcal{F}(\varrho,K)$ and $r$. Since the function $[0,T]\ni
t\to\sup\limits_{0\leq u\leq t}\mathbb{E}\|\tilde{\bar{X}}^{RE}_{n}(u)\|^{r}$
is bounded (by (92)) and Borel measurable (as a nondecreasing function), by
using the Gronwall’s lemma we get (86). ∎

In the case of the class $\mathcal{W}_{\alpha,\beta}(\delta_{3})$ we have the
following absolute moments estimate for $\bar{X}_{n}^{RE}(t_{i})$. The proof
technique is different from the one used in the proof of Lemma 1, since for
$p_{W}\in\mathcal{K}_{\alpha,\beta}$ the process $Z(t)=p_{W}(t,W(t))$ might
not be a semimartingale nor a process of bounded variation.

######  Lemma 2.

Let $r\in[2,+\infty)$. There exists $C\in(0,+\infty)$, depending only on the
paramters of the class $\mathcal{F}(\varrho,K)$ and $r$, such that for all
$n\in\mathbb{N}$, $\delta_{1},\delta_{2},\delta_{3}\in[0,1]$,
$(a,b,\eta)\in\mathcal{F}(\varrho,K)$, $(\tilde{a},\tilde{b},\tilde{W})\in
V_{a}(\delta_{1})\times
V_{b}(\delta_{2})\times\mathcal{W}_{\alpha,\beta}(\delta_{3})$ it holds

$\mathbb{E}\Bigl{[}\max\limits_{0\leq i\leq
n}\|\bar{X}_{n}^{RE}(t_{i})\|^{r}\Bigr{]}\leq
C(1+\delta_{3}^{r}n^{r(1-\gamma)})\cdot
e^{C(1+\delta_{3}^{r}n^{r(1-\gamma)})},$ (100)

where $\gamma=\min\\{\alpha,\beta/2\\}$.

###### Proof.

We have that for $i=0,1,\ldots,n$ we can write that

$\displaystyle\bar{X}^{RE}_{n}(t_{i})=\eta+\frac{T}{n}\sum\limits_{j=0}^{i-1}\tilde{a}(\xi_{j},\bar{X}^{RE}_{n}(t_{j}))+\
\sum\limits_{j=0}^{i-1}\tilde{b}(t_{j},\bar{X}^{RE}_{n}(t_{j}))\cdot\Delta
W_{j}$
$\displaystyle\quad\quad\quad\quad\quad\quad+\delta_{3}\sum\limits_{j=0}^{i-1}\tilde{b}(t_{j},\bar{X}^{RE}_{n}(t_{j}))\cdot\Delta
Z_{j},$ (101)

and we have that

$\|\bar{X}_{n}^{RE}(t_{i})\|\leq
K+\sum\limits_{j=0}^{i-1}\|\tilde{A}_{j}\|+\Bigl{\|}\sum\limits_{j=0}^{i-1}\tilde{B}_{j}\Bigl{\|}+\sum\limits_{j=0}^{i-1}\|\tilde{C}_{j}\|,$
(102)

where

$\displaystyle\tilde{A}_{j}=\frac{T}{n}\tilde{a}(\xi_{j},\bar{X}^{RE}_{n}(t_{j})),$
(103)
$\displaystyle\tilde{B}_{j}=\tilde{b}(t_{j},\bar{X}^{RE}_{n}(t_{j}))\cdot\Delta
W_{j},$ (104)
$\displaystyle\tilde{C}_{j}=\delta_{3}\cdot\tilde{b}(t_{j},\bar{X}^{RE}_{n}(t_{j}))\cdot\Delta
Z_{j}.$ (105)

Hence, for all $k=0,1,\ldots,n$

$\displaystyle\mathbb{E}\Bigl{(}\max\limits_{0\leq i\leq
k}\|\bar{X}_{n}^{RE}(t_{i})\|^{r}\Bigr{)}\leq
c_{r}K^{r}+c_{r}\mathbb{E}\Bigl{(}\sum\limits_{j=0}^{k-1}\|\tilde{A}_{j}\|\Bigr{)}^{r}$
$\displaystyle+c_{r}\mathbb{E}\Bigl{[}\max\limits_{1\leq i\leq
k}\Bigl{\|}\sum\limits_{j=0}^{i-1}\tilde{B}_{j}\Bigl{\|}^{r}\Bigr{]}+c_{r}\mathbb{E}\Bigl{(}\sum\limits_{j=0}^{k-1}\|\tilde{C}_{j}\|\Bigr{)}^{r}.$
(106)

By Jensen inequality we have that

$\mathbb{E}\Bigl{(}\sum\limits_{j=0}^{k-1}\|\tilde{A}_{j}\|\Bigr{)}^{r}\leq
C_{1}+\frac{C_{1}}{n}\sum\limits_{j=0}^{k-1}\mathbb{E}\|\bar{X}^{RE}_{n}(t_{j})\|^{r}.$
(107)

From Burkholder and Jensen inequality we obtain that

$\mathbb{E}\Bigl{[}\max\limits_{1\leq i\leq
k}\Bigl{\|}\sum\limits_{j=0}^{i-1}\tilde{B}_{j}\Bigl{\|}^{r}\Bigr{]}\leq
C_{2}+\frac{C_{2}}{n}\sum\limits_{j=0}^{k-1}\mathbb{E}\|\bar{X}^{RE}_{n}(t_{j})\|^{r}.$
(108)

Finally, since $\bar{X}^{RE}_{n}(t_{j})$ and $\Delta W_{j}$ are independent,
and

$\|\|\Delta W_{j}\|^{\beta}\|_{r}\leq c(T/n)^{\beta/2},$ (109)

we get that

$\displaystyle\mathbb{E}\|\tilde{C}_{j}\|^{r}\leq\bar{K}_{1}\delta_{3}^{r}\mathbb{E}\Bigl{[}(1+\|\bar{X}^{RE}_{n}(t_{j})\|)^{r}\cdot\Bigl{(}(T/n)^{\alpha}+\|\Delta
W_{j}\|^{\beta}\Bigr{)}^{r}\Bigr{]}$
$\displaystyle\leq\bar{K}_{2}\delta_{3}^{r}\Bigl{(}1+\mathbb{E}\|\bar{X}^{RE}_{n}(t_{j})\|^{r}\Bigr{)}\cdot\Bigl{\|}(T/n)^{\alpha}+\|\Delta
W_{j}\|^{\beta}\Bigl{\|}_{r}^{r}$
$\displaystyle\leq\bar{K}_{3}\delta_{3}^{r}n^{-r\gamma}+\bar{K}_{4}\delta^{r}n^{-r\gamma}\mathbb{E}\|\bar{X}^{RE}_{n}(t_{j})\|^{r},$
(110)

and hence

$\displaystyle\mathbb{E}\Bigl{(}\sum\limits_{j=0}^{k-1}\|\tilde{C}_{j}\|\Bigr{)}^{r}\leq
n^{r-1}\sum\limits_{j=0}^{k-1}\mathbb{E}\|\tilde{C}_{j}\|^{r}\leq
C_{3}\delta_{3}^{r}n^{r(1-\gamma)}$
$\displaystyle+C_{4}\delta_{3}^{r}n^{r(1-\gamma)-1}\sum\limits_{j=0}^{k-1}\mathbb{E}\|\bar{X}^{RE}_{n}(t_{j})\|^{r}.$
(111)

Combining (7), (107), (108), (7) we arrive at

$\displaystyle\mathbb{E}\Bigl{(}\max\limits_{0\leq i\leq
k}\|\bar{X}_{n}^{RE}(t_{i})\|^{r}\Bigr{)}\leq
C_{5}(1+\delta_{3}^{r}n^{r(1-\gamma)})$
$\displaystyle+C_{6}(\delta_{3}^{r}n^{r(1-\gamma)-1}+n^{-1})\sum\limits_{j=1}^{k-1}\mathbb{E}\Bigl{(}\max\limits_{0\leq
i\leq j}\|\bar{X}_{n}^{RE}(t_{i})\|^{r}\Bigr{)}.$ (112)

By the discrete version of the Gronwall’s lemma we get the thesis. ∎

Acknowledgements
This research was realized as a part of joint research project between AGH UST
and NVIDIA.

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