# Drift-implicit Euler scheme for sandwiched processes
driven by Hölder noises

Giulia Di Nunno1,2
<EMAIL_ADDRESS>Yuliya Mishura3
<EMAIL_ADDRESS>Anton Yurchenko-Tytarenko1
<EMAIL_ADDRESS>
(1Department of Mathematics, University of Oslo
2Department of Business and Management Science, NHH Norwegian School of
Economics, Bergen
3Department of Probability Theory, Statistics and Actuarial Mathematics, Taras
Shevchenko National University of Kyiv
)

###### Abstract

In this paper, we analyze the drift-implicit (or backward) Euler numerical
scheme for a class of stochastic differential equations with unbounded drift
driven by an arbitrary $\lambda$-Hölder continuous process, $\lambda\in(0,1)$.
We prove that, under some mild moment assumptions on the Hölder constant of
the noise, the $L^{r}(\Omega;L^{\infty}([0,T]))$-rate of convergence is equal
to $\lambda$. To exemplify, we consider numerical schemes for the generalized
Cox–Ingersoll-Ross and Tsallis–Stariolo–Borland models. The results are
illustrated by simulations.

Keywords: sandwiched process, unbounded drift, Hölder continuous noise,
numerical scheme
MSC 2020: 60H10; 60H35; 60G22; 91G30

## Introduction

We analyze the drift-implicit (also known as _backward_) Euler numerical
scheme for stochastic differential equations (SDEs) of the form

$Y(t)=Y(0)+\int_{0}^{t}b(s,Y(s))ds+Z(t),\quad t\in[0,T],$ (0.1)

where $Z$ is a general $\lambda$-Hölder continuous noise, $\lambda\in(0,1)$,
and the drift $b$ is unbounded and has one of the following two properties:

  * (A)

$b(t,y)$ has an explosive growth of the type $(y-\varphi(t))^{-\gamma}$ as
$y\downarrow\varphi(t)$, where $\varphi$ is a given Hölder continuous function
of the same order $\lambda$ as $Z$ and $\gamma>\frac{1}{\lambda}-1$;

  * (B)

$b(t,y)$ has an explosive growth of the type $(y-\varphi(t))^{-\gamma}$ as
$y\downarrow\varphi(t)$ _and_ an explosive decrease of the type
$-(\psi(t)-y)^{-\gamma}$ as $y\uparrow\psi(t)$, where $\varphi$ and $\psi$ are
given Hölder continuous functions of the same order $\lambda$ as $Z$ such that
$\varphi(t)<\psi(t)$, $t\in[0,T]$, and $\gamma>\frac{1}{\lambda}-1$.

The SDEs of this type were extensively studied in [14]. It was shown that the
properties (A) or (B), along with some relatively weak additional assumptions,
ensure that the solution to (0.1) is bounded from below (one-sided sandwich
case) by the function $\varphi$ in the setting (A), i.e.

$Y(t)>\varphi(t),\quad t\in[0,T],$ (0.2)

or stays between $\varphi$ and $\psi$ (two-sided sandwich case) in the setting
(B), i.e.

$\varphi(t)<Y(t)<\psi(t),\quad t\in[0,T].$ (0.3)

We emphasize that the SDE type (0.1) includes and generalizes several
widespread stochastic models. For example, the process given by

$Y(t)=Y(0)-\int_{0}^{t}\frac{\kappa Y(s)}{1-Y^{2}(s)}ds+Z(t),\quad t\in[0,T],$

where $Z$ is $\lambda$-Hölder continuous with $\lambda>\frac{1}{2}$, fits into
the setting (B) and can be regarded as a natural extension of the
Tsallis–Stariolo–Borland (TSB) model employed in biophysics (for more details
on the standard Brownian TSB model see e.g. [15, Subsection 2.3] or [16,
Chapter 3 and Chapter 8]). Another important example is

$Y(t)=Y(0)+\int_{0}^{t}\left(\frac{\kappa_{1}}{Y^{\gamma}(s)}-\kappa_{2}Y(s)\right)ds+Z(t),\quad
t\in[0,T],$ (0.4)

where $Z$ is $\lambda$-Hölder continuous, $\lambda\in(0,1)$, and
$\gamma>\frac{1}{\lambda}-1$. It can be shown (see [14, Subsection 4.2]) that,
if $\lambda>\frac{1}{2}$, stochastic process $X(t):=Y^{1+\gamma}(t)$ satisfies
the SDE

$X(t)=X(0)+(1+\gamma)\int_{0}^{t}\left(\kappa_{1}-\kappa_{2}X(s)\right)ds+\int_{0}^{t}X^{\alpha}(s)dZ(s),\quad
t\in[0,T],$ (0.5)

where $\alpha:=\frac{\gamma}{1+\gamma}\in(0,1)$ and the integral w.r.t. $Z$
exists as a pathwise limit of Riemann-Stieltjes integral sums. Equations of
the type (0.5) are used in finance in the standard Brownian setting and are
called Chan–-Karolyi–-Longstaff–-Sanders (CKLS) or constant elasticity of
variance (CEV) model (see, e.g., [4, 8, 9]). If $\alpha=\frac{1}{2}$, the
equation (0.5) is also known as the Cox–Ingersoll–Ross (CIR) equation, see ,
e.g., [10, 11, 12].

In this work, we develop a numerical approximation (both pathwise and in
$L^{r}(\Omega;L^{\infty}([0,T]))$) for sandwiched processes (0.1) which is
similar to the drift-implicit (also known as backward) Euler scheme
constructed for the classical Cox-Ingersoll-Ross process in [2, 3, 13] and
extended to the case of the fractional Brownian motion with $H>\frac{1}{2}$ in
[18, 21, 22]. In this drift-implicit scheme, in order to generate
$\widehat{Y}(t_{k+1})$, one has to solve the equation of the type

$\widehat{Y}(t_{k+1})=\widehat{Y}(t_{k})+b(t_{k+1},\widehat{Y}(t_{k+1}))\Delta_{N}+(Z(t_{k+1})-Z(t_{k}))$
(0.6)

with respect to $\widehat{Y}(t_{k+1})$ which is in general a more
computationally heavy problem in comparison to the standard Euler-type
techniques (see e.g. [14, Section 5]). However, this drift-implicit numerical
method also has a substantial advantage: the approximation $\widehat{Y}$
maintains the property of being sandwiched, i.e., for all points $t_{k}$ of
the partition

$\widehat{Y}(t_{k})>\varphi(t_{k})$

in the setting (A) and

$\varphi(t_{k})<\widehat{Y}(t_{k})<\psi(t_{k})$

in the case (B). Having this in mind, we shall say that the drift-implicit
scheme is sandwich preserving.

We note that a similar approximation scheme was studied in [21] and [18, 22]
for processes of the type (0.4) driven by a fractional Brownian motion with
$H>1/2$. Our work can be seen as an extension of those. However, we emphasize
that our results have several elements of novelty. In particular, the paper
[21] discusses only pathwise convergence and not convergence in
$L^{r}(\Omega;L^{\infty}([0,T]))$. The approach of [18] and [22] is very noise
specific as both use Malliavin calculus techniques in the spirit of [19,
Proposition 3.4] to estimate inverse moments of the considered process (which
turns out to be crucial to control explosive growth of the drift). As a
result, two limitations appear: a restrictive condition involving the time
horizon $T$ (see e.g. [18, Eq. (8) and Remark 3.1]) and sensitivity to the
choice of the noise, i.e. their method cannot be applied directly for drivers
other then fBm with $H>1/2$. This lack of flexibility in terms of the choice
of the noise is a crucial disadvantage in e.g. finance where modern empirical
studies justify the use of fBm with extremely low Hurst index ($H<0.1$) [7] or
even drivers with time-varying roughness [1]. Our approach makes use of [14,
Theorem 3.2] based on the pathwise calculus and allows us to obtain strong
convergence with no limitations on $T$ for a substantially larger class of
noises. In fact, we require only Hölder continuity of the noise and some
moment condition on the corresponding Hölder coefficient which is often
satisfied and shared by e.g. _all_ Hölder continuous Gaussian processes.

The paper is organized as follows. Section 1 describes the setting in detail
and contains some necessary statements on the properties of the sandwiched
processes. In Section 2, we give the convergence results in the setting (B)
which turns out to be a bit simpler then (A) due to boundedness of the
process. Section 3 extends the scheme to the setting (A). In Section 4, we
give some examples and simulations; in particular we show that in some cases
(e.g. for the generalized TSB and CIR models) equations (0.6) can be solved
explicitly which drastically improves the computational efficiency of the
algorithm.

##  1 Preliminaries and assumptions

Fix $T>0$ and define

$\displaystyle\mathcal{D}_{a_{1}}$
$\displaystyle:=\\{(t,y)\in[0,T]\times\mathbb{R}_{+},~{}y\in(\varphi(t)+a_{1},\infty)\\},\quad$
$\displaystyle a_{1}$ $\displaystyle>0,$ (1.1)
$\displaystyle\mathcal{D}_{a_{1},a_{2}}$
$\displaystyle:=\\{(t,y)\in[0,T]\times\mathbb{R}_{+},~{}y\in(\varphi(t)+a_{1},\psi(t)-a_{2})\\},\quad$
$\displaystyle a_{1},a_{2}$
$\displaystyle\in\left[0,\frac{1}{2}\lVert\psi-\varphi\rVert_{\infty}\right).$

where $\varphi$, $\psi\in C([0,T])$ are such that $\varphi(t)<\psi(t)$,
$t\in[0,T]$.

Throughout the paper, we will be dealing with a stochastic differential
equation of the form

$Y(t)=Y(0)+\int_{0}^{t}b(s,Y(s))ds+Z(t),\quad t\in[0,T].$ (1.2)

The noise $Z=\\{Z(t),~{}t\in[0,T]\\}$ is always assumed to satisfy the
following conditions:

  * (Z1)

$Z(0)=0$ a.s.;

  * (Z2)

$Z$ has a.s. $\lambda$-Hölder continuous paths, $\lambda\in(0,1)$, i.e. there
exists a positive random variable $\Lambda$ such that

$|Z(t)-Z(s)|\leq\Lambda|t-s|^{\lambda},\quad s,t\in[0,T].$

Given the noise $Z$ satisfying (Z1)–(Z2), the initial value $Y(0)$ and the
drift $b$ satisfy one of the two lists of assumptions given below.

######  Assumptions A.

(One-sided sandwich case) There exists a $\lambda$-Hölder continuous function
$\varphi$: $[0,T]\to\mathbb{R}$ with $\lambda$ being the same as in (Z2) such
that

  * (A1)

$Y(0)$ is deterministic and $Y(0)>\varphi(0)$,

  * (A2)

$b$: $\mathcal{D}_{0}\to\mathbb{R}$ is continuous and for any
$\varepsilon\in\left(0,1\right)$

$|b(t_{1},y_{1})-b(t_{2},y_{2})|\leq\frac{c_{1}}{\varepsilon^{p}}\left(|y_{1}-y_{2}|+|t_{1}-t_{2}|^{\lambda}\right),\quad(t_{1},y_{1}),(t_{2},y_{2})\in\mathcal{D}_{\varepsilon},$

where $c_{1}>0$ and $p>1$ are some given constants and $\lambda$ is from (Z2),

  * (A3)

$b(t,y)\geq\frac{c_{2}}{(y-\varphi(t))^{\gamma}},\quad(t,y)\in\mathcal{D}_{0}\setminus\mathcal{D}_{y_{*}},$

where $y_{*}$, $c_{2}>0$ are some given constants and
$\gamma>\frac{1}{\lambda}-1$ with $\lambda$ being from (Z2),

  * (A4)

the partial derivative $\frac{\partial b}{\partial y}$ with respect to the
spacial variable exists, is continuous and bounded from above, i.e.

$\frac{\partial b}{\partial y}(t,y)<c_{3},\quad(t,y)\in\mathcal{D}_{0},$

for some $c_{3}>0$.

######  Assumptions B.

(Two-sided sandwich case) There exist $\lambda$-Hölder continuous functions
$\varphi$, $\psi$: $[0,T]\to\mathbb{R}$, $\varphi(t)<\psi(t)$, $t\in[0,T]$,
with $\lambda$ being the same as in (Z2) such that

  * (B1)

$Y(0)$ is deterministic and $\varphi(0)<Y(0)<\psi(0)$,

  * (B2)

$b$: $\mathcal{D}_{0,0}\to\mathbb{R}$ is continuous and for any
$\varepsilon\in\left(0,\min\left\\{1,\frac{1}{2}\lVert\psi-\varphi\rVert_{\infty}\right\\}\right)$

$|b(t_{1},y_{1})-b(t_{2},y_{2})|\leq\frac{c_{1}}{\varepsilon^{p}}\left(|y_{1}-y_{2}|+|t_{1}-t_{2}|^{\lambda}\right),\quad(t_{1},y_{1}),(t_{2},y_{2})\in\mathcal{D}_{\varepsilon,\varepsilon},$

where $c_{1}>0$ and $p>1$ are some given constants and $\lambda$ is from (Z2),

  * (B3)

$b(t,y)\geq\frac{c_{2}}{(y-\varphi(t))^{\gamma}},\quad(t,y)\in\mathcal{D}_{0,0}\setminus\mathcal{D}_{y_{*},0},$
$b(t,y)\leq-\frac{c_{2}}{(\psi(t)-y)^{\gamma}},\quad(t,y)\in\mathcal{D}_{0,0}\setminus\mathcal{D}_{0,y_{*}},$

where $y_{*}$, $c_{2}>0$ are some given constants and
$\gamma>\frac{1}{\lambda}-1$ with $\lambda$ being from (Z2),

  * (B4)

the partial derivative $\frac{\partial b}{\partial y}$ with respect to the
spacial variable exists, is continuous and bounded from above, i.e.

$\frac{\partial b}{\partial y}(t,y)<c_{3},\quad(t,y)\in\mathcal{D}_{0,0},$

for some $c_{3}>0$.

Both Assumptions A and B along with (Z1)–(Z2) ensure that the SDE (1.2) has a
unique solution. In the theorem below, we provide some relevant results
related to sandwiched processes (see [14, Theorems 2.3, 2.5, 2.6, 3.1 and
3.2]).

######  Theorem 1.1.

Let $Z=\\{Z(t),~{}t\in[0,T]\\}$ be a stochastic process satisfying (Z1)–(Z2).

  * 1)

If the initial value $Y(0)$ and the drift $b$ satisfy assumptions (A1)–(A3),
then the SDE has a unique strong pathwise solution such that for all
$t\in[0,T]$

$Y(t)>\varphi(t)\quad a.s.$ (1.3)

Moreover, there exist deterministic constants $L_{1}$, $L_{2}$, $L_{3}$ and
$L_{4}>0$ depending only on $Y(0)$, the shape of $b$ and $\lambda$, such that
for all $t\in[0,T]$ the estimate (1.3) can be refined as follows:

$\varphi(t)+\frac{L_{1}}{(L_{2}+\Lambda)^{\frac{1}{\gamma\lambda+\lambda-1}}}\leq
Y(t)\leq L_{3}+L_{4}\Lambda\quad a.s.,$ (1.4)

where $\Lambda$ is from (Z2) and $\gamma$ is from (A3). In particular, if
$\Lambda$ is such that

$\mathbb{E}\left[\Lambda^{\frac{r}{\gamma\lambda+\lambda-1}}\right]<\infty$
(1.5)

for some $r>0$, then

$\mathbb{E}\left[\sup_{t\in[0,T]}\frac{1}{(Y(t)-\varphi(t))^{r}}\right]<\infty,$

and, if

$\mathbb{E}\Lambda^{r}<\infty$ (1.6)

for some $r>0$, then

$\mathbb{E}\left[\sup_{t\in[0,T]}|Y(t)|^{r}\right]<\infty.$

  * 2)

If the initial value $Y(0)$ and the drift $b$ satisfy assumptions (B1)–(B3),
then the SDE has a unique strong pathwise solution such that for all
$t\in[0,T]$

$\varphi(t)<Y(t)<\psi(t)\quad a.s.$ (1.7)

Moreover, there exist deterministic constants $L_{1}$ and $L_{2}>0$ depending
only on $Y(0)$, the shape of $b$ and $\lambda$, such that for all $t\in[0,T]$
the estimate (1.7) can be refined as follows:

$\varphi(t)+\frac{L_{1}}{(L_{2}+\Lambda)^{\frac{1}{\gamma\lambda+\lambda-1}}}\leq
Y(t)\leq\psi(t)-\frac{L_{1}}{(L_{2}+\Lambda)^{\frac{1}{\gamma\lambda+\lambda-1}}}\quad
a.s.,$ (1.8)

where $\Lambda$ is from (Z2) and $\gamma$ is from (B3). In particular, if
$\Lambda$ can be chosen in such a way that

$\mathbb{E}\left[\Lambda^{\frac{r}{\gamma\lambda+\lambda-1}}\right]<\infty$
(1.9)

for some $r>0$, then

$\mathbb{E}\left[\sup_{t\in[0,T]}\frac{1}{(Y(t)-\varphi(t))^{r}}\right]<\infty,\quad\mathbb{E}\left[\sup_{t\in[0,T]}\frac{1}{(\psi(t)-Y(t))^{r}}\right]<\infty.$

######  Remark 1.2.

Properties (1.3)–(1.4) and (1.7)–(1.8) hold on each $\omega\in\Omega$ such
that $Z(\omega;t)$, $t\in[0,T]$, is Hölder continuous and we always consider
only such $\omega\in\Omega$ in all proofs with pathwise arguments. For
notational simplicity, we will also omit $\omega$ in brackets.

######  Remark 1.3.

Due to the property (1.7), the setting described in Assumptions B will be
referred to as the two-sided sandwich case since the solution is “sandwiched”
between $\varphi$ and $\psi$ a.s. Similarly, the property (1.3) justifies the
name one-sided sandwich case for the setting corresponding to Assumptions A.
In both cases A and B, the solution to (1.2) will be referred to as a
sandwiched process.

######  Remark 1.4.

Note that assumptions (A4) and (B4) are not required for Theorem 1.1 to hold
and will be used later on.

In what follows, conditions (1.5), (1.6) and (1.9) will play an important role
since the $L^{r}(\Omega;L^{\infty}([0,T]))$-convergence of the approximation
scheme will directly follow from the integrability of $\Lambda$. However it
should be noted that these conditions are not very restricting as indicated in
the following example.

######  Example 1.5.

(Hölder Gaussian noises) Let $Z=\\{Z(t),~{}t\in[0,T]\\}$ be an arbitrary
Hölder continuous Gaussian process satisfying (Z1)–(Z2). In this case, by [6],
the random variable $\Lambda$ from (Z2) can be chosen to have moments of all
orders.

We now complete the Section with some examples of the sandwiched processes.

######  Example 1.6.

(_Generalized CIR and CKLS/CEV models_) Let $\varphi\equiv 0$, $Z$ satisfy
(Z1)–(Z2) with $\lambda\in(0,1)$ and $Y(0)$, $\kappa_{1}$, $\kappa_{2}>0$,
$\gamma>\frac{1}{\lambda}-1$ be given. Then, by Theorem 1.1, 1), the SDE of
the form

$Y(t)=Y(0)+\int_{0}^{t}\left(\frac{\kappa_{1}}{Y^{\gamma}(s)}-\kappa_{2}Y(s)\right)ds+Z(t),\quad
t\in[0,T],$ (1.10)

has a unique positive solution. Moreover, it can be shown (see [14, Subsection
4.2]) that, if $\lambda>\frac{1}{2}$, stochastic process
$X(t):=Y^{1+\gamma}(t)$, $t\in[0,T]$, a.s. satisfies the SDE of the form

$X(t)=X(0)+(1+\gamma)\int_{0}^{t}\left(\kappa_{1}-\kappa_{2}X(s)\right)ds+\int_{0}^{t}X^{\alpha}(s)dZ(s),\quad
t\in[0,T],$ (1.11)

where $\alpha:=\frac{\gamma}{1+\gamma}\in(0,1)$ and the integral w.r.t. $Z$
exists a.s. as a pathwise limit of Riemann-Stieltjes integral sums. As
mentioned already, the (1.11) appears in finance in the standard Brownian
setting and is called Chan–Karolyi–Longstaff–Sanders (CKLS) or constant
elasticity of variance (CEV) model (see e.g. [4, 8, 9]). If
$\alpha=\frac{1}{2}$ (i.e. when $\gamma=1$), the equation (1.11) is also known
as the Cox-Ingersoll-Ross (CIR) equation [10, 11, 12].

######  Example 1.7.

(Generalized TSB model) Let $\varphi\equiv-1$, $\psi\equiv 1$,
$Y(0)\in(-1,1)$, $Z$ satisfy (Z1)–(Z2) with $\lambda>\frac{1}{2}$ and
$\kappa>0$. Then, by Theorem 1.1, 2), the SDE of the form

$Y(t)=Y(0)-\int_{0}^{t}\frac{\kappa Y(s)}{1-Y^{2}(s)}ds+Z(t),\quad t\in[0,T],$
(1.12)

has a unique solution such that $-1<Y(t)<1$ for all $t\in[0,T]$ a.s. In the
standard Brownian setting, the SDE of the type (1.12) is known as the
Tsallis–Stariolo–Borland (TSB) model and is used in biophysics (for more
details, see e.g. [15, Subsection 2.3] or [16, Chapter 3 and Chapter 8]).

######  Example 1.8.

For the given $Z$ satisfying (Z1)–(Z2) with $\lambda\in(0,1)$,
$\lambda$-Hölder continuous functions $\varphi$, $\psi$, $\varphi(t)<\psi(t)$,
$t\in[0,T]$, and $Y(0)\in(\varphi(0),\psi(0))$ consider the SDE of the form

$Y(t)=Y(0)+\int_{0}^{t}\left(\frac{\kappa_{1}}{(Y(s)-\varphi(s))^{\gamma}}-\frac{\kappa_{2}}{(\psi(s)-Y(s))^{\gamma}}-\kappa_{3}Y(s)\right)ds+Z(t),\quad
t\in[0,T],$

where $\kappa_{1}$, $\kappa_{2}>0$, $\kappa_{3}\in\mathbb{R}$ and
$\gamma>\frac{1}{\lambda}-1$. By Theorem 1.1, 2), this SDE has a unique
solution such that $\varphi(t)<Y(t)<\psi(t)$ a.s. Note that the TSB drift from
(1.12) also has this shape with $\varphi\equiv-1$, $\psi\equiv 1$, $\gamma=1$,
$\kappa_{1}=\kappa_{2}=\frac{\kappa}{2}$ and $\kappa_{3}=0$ since

$-\frac{\kappa
y}{1-y^{2}}=\frac{\kappa}{2}\left(\frac{1}{y+1}-\frac{1}{1-y}\right).$

######  Notation 1.9.

In what follows, $C$ denotes any positive deterministic constant that does not
depend on the partition and the exact value of which is not relevant. Note
that $C$ may change from line to line (or even within one line).

##  2 The approximation scheme for the two-sided sandwich

We will start by considering the numerical scheme for the two-sided sandwich
case which turns out to be slightly simpler due to boundedness of $Y$. Let the
noise $Z$ satisfy (Z1)–(Z2), $Y(0)$ and $b$ satisfy Assumptions B and
$Y=\\{Y(t),~{}t\in[0,T]\\}$ be the unique solution of the SDE (1.2). Consider
a uniform partition $\\{0=t_{0}<t_{1}<...<t_{N}=T\\}$ of $[0,T]$,
$t_{k}:=\frac{Tk}{N}$, $k=0,1,...,N$, with the mesh $\Delta_{N}:=\frac{T}{N}$
such that

$c_{3}\Delta_{N}<1,$ (2.1)

where $c_{3}$ is an upper bound for $\frac{\partial b}{\partial y}$ from (B4).
Let us define $\widehat{Y}(t)$ as follows:

$\displaystyle\widehat{Y}(0)$ $\displaystyle=Y(0),$ (2.2)
$\displaystyle\widehat{Y}(t_{k+1})$
$\displaystyle=\widehat{Y}(t_{k})+b(t_{k+1},\widehat{Y}(t_{k+1}))\Delta_{N}+(Z(t_{k+1})-Z(t_{k})),$
$\displaystyle\widehat{Y}(t)$ $\displaystyle=\widehat{Y}(t_{k}),\quad
t\in[t_{k},t_{k+1}),$

where the second expression is considered as an equation with respect to
$\widehat{Y}(t_{k+1})$.

######  Remark 2.1.

Equation with respect to $\widehat{Y}(t_{k+1})$ from (2.2) has a unique
solution such that $\widehat{Y}(t_{k+1})\in(\varphi(t_{k+1}),\psi(t_{k+1}))$.
Indeed, for any fixed $t\in[0,T]$ and any $z\in\mathbb{R}$, consider the
equation

$y-b(t,y)\Delta_{N}=z$ (2.3)

w.r.t. $y$. Assumption (B4) together with condition (2.1) imply that
$(y-b(t,y)\Delta_{N})^{\prime}_{y}>0$ and, by (B3),

$\displaystyle y-b(t,y)\Delta_{N}\to-\infty,\quad y\to\varphi(t)+,$
$\displaystyle y-b(t,y)\Delta_{N}\to\infty,\quad y\to\psi(t)-.$

Thus there exists a unique $y\in(\varphi(t),\psi(t))$ satisfying (2.3).

######  Remark 2.2.

The value of $\widehat{Y}(t)$ for $t\in[0,T]\setminus\\{t_{0},...,t_{N}\\}$
can also be defined via linear interpolation as

$\widehat{Y}(t)=\frac{1}{\Delta_{N}}\left((t_{k+1}-t)\widehat{Y}(t_{k})+(t-t_{k})\widehat{Y}(t_{k+1})\right),\quad
t\in[t_{k},t_{k+1}),\quad k=0,...,N-1.$

In such case all results of this section hold with almost no changes in the
proofs.

######  Remark 2.3.

The algorithms of the type (2.2) are sometimes called the drift-implicit [2,
3, 13] or backward [18] Euler approximation schemes.

Before presenting the main results of this section, we require some auxiliary
lemmas. First of all, we note that the values $\widehat{Y}(t_{n})$,
$n=0,1,...,N$, of the discretized process are bounded away from both $\varphi$
and $\psi$ by random variables that do not depend on the partition. Namely, we
have the following result that can be regarded as a discrete modification of
arguments in [14, Theorem 3.2].

######  Lemma 2.4.

Let $Z$ satisfy (Z1)–(Z2), Assumptions B hold and the mesh of the partition
$\Delta_{N}$ satisfy (2.1). Then there exist deterministic constants $L_{1}$
and $L_{2}>0$ depending only on $Y(0)$, the shape of the drift $b$ and
$\lambda$, such that

$\varphi(t_{n})+\frac{L_{1}}{(L_{2}+\Lambda)^{\frac{1}{\gamma\lambda+\lambda-1}}}\leq\widehat{Y}(t_{n})\leq\psi(t_{n})-\frac{L_{1}}{(L_{2}+\Lambda)^{\frac{1}{\gamma\lambda+\lambda-1}}},\quad
n=0,1,...,N,\quad a.s.,$

where $\Lambda$ is from (Z2) and $\gamma$ is from (B3).

###### Proof.

We will prove that

$\varphi(t_{n})+\frac{L_{1}}{(L_{2}+\Lambda)^{\alpha}}\leq\widehat{Y}(t_{n}),\quad
n=0,1,...,N,\quad a.s.$ (2.4)

by using the pathwise argument (see Remark 1.2). The other inequality can be
derived in a similar manner. Recall that, by Assumptions B, $\varphi$ and
$\psi$ are $\lambda$-Hölder continuous, i.e. there exists $K>0$ such that

$|\varphi(t)-\varphi(s)|+|\psi(t)-\psi(s)|\leq K|t-s|^{\lambda},\quad
t,s\in[0,T].$

Denote also

$\beta:=\frac{\lambda^{\frac{\lambda}{1-\lambda}}-\lambda^{\frac{1}{1-\lambda}}}{c_{2}^{\frac{\lambda}{1-\lambda}}}>0,$

where $c_{2}$ is from (B3),

$L_{2}:=K+\left(2\beta\right)^{\lambda-1}\left(\frac{(Y(0)-\varphi(0))\wedge
y_{*}\wedge(\psi(0)-Y(0))}{2}\right)^{1-\lambda-\gamma\lambda}>0,$

with the constants $y_{*}$ and $\gamma$ also from (B3), and

$\varepsilon:=\frac{1}{(2\beta)^{\frac{1-\lambda}{\gamma\lambda+\lambda-1}}(L_{2}+\Lambda)^{\frac{1}{\gamma\lambda+\lambda-1}}}.$

Note that, with probability 1,

$|\varphi(t)-\varphi(s)|+|\psi(t)-\psi(s)|+|Z(t)-Z(s)|\leq(L_{2}+\Lambda)|t-s|^{\lambda},\quad
t,s\in[0,T],$

and, furthermore, it is easy to check that $\varepsilon<Y(0)-\varphi(0)$,
$\varepsilon<\psi(0)-Y(0)$ and $\varepsilon<y_{*}$.

If $\widehat{Y}(t_{n})\geq\varphi(t_{n})+\varepsilon$ for a particular
$n=0,1,...,N$, then, by definition of $\varepsilon$, the bound of the type
(2.4) holds automatically. Suppose that there exists $n=1,...,N$ such that
$\widehat{Y}({t_{n}})<\varphi(t_{n})+\varepsilon$. Denote by $\kappa(n)$ the
last point of the partition before $t_{n}$ on which $\widehat{Y}$ stays above
$\varepsilon$, i.e.

$\kappa(n):=\max\\{k=0,...,n-1~{}|~{}\widehat{Y}_{t_{k}}\geq\varphi(t_{k})+\varepsilon\\}$

(note that such point exists since
$\widehat{Y}(t_{0})-\varphi(0)=Y(0)-\varphi(0)>\varepsilon$). Then, for all
$k=\kappa(n)+1,...,n$ we have that $\widehat{Y}(t_{k})<\varepsilon<y_{*}$ and
therefore, using (B3), we obtain that, with probability 1,

$\displaystyle\widehat{Y}(t_{n})-\varphi(t_{n})$
$\displaystyle=\widehat{Y}(t_{\kappa(n)})-\varphi(t_{n})+\Delta_{N}\sum_{k=\kappa(n)+1}^{n}b(t_{k},\widehat{Y}(t_{k}))+Z(t_{n})-Z(t_{\kappa(n)})$
$\displaystyle\geq\varepsilon+\varphi(t_{\kappa(n)})-\varphi(t_{n})+\frac{c_{2}}{\varepsilon^{\gamma}}(t_{n}-t_{\kappa(n)})+Z(t_{n})-Z(t_{\kappa(n)})$
$\displaystyle\geq\varepsilon+\frac{c_{2}}{\varepsilon^{\gamma}}(t_{n}-t_{\kappa(n)})-(L_{2}+\Lambda)(t_{n}-t_{\kappa(n)})^{\lambda}.$

Consider a function $F_{\varepsilon}:\mathbb{R}_{+}\to\mathbb{R}$ such that

$F_{\varepsilon}(t)=\varepsilon+\frac{c_{2}}{\varepsilon^{\gamma}}t-(L_{2}+\Lambda)t^{\lambda}.$

It is straightforward to verify that $F_{\varepsilon}$ attains its minimum at

$t_{*}:=\left(\frac{\lambda}{c_{2}}\right)^{\frac{1}{1-\lambda}}\varepsilon^{\frac{\gamma}{1-\lambda}}(L_{2}+\Lambda)^{\frac{1}{1-\lambda}}$

and, taking into account the explicit form of $\varepsilon$,

$\displaystyle F_{\varepsilon}(t_{*})$
$\displaystyle=\varepsilon+\frac{\lambda^{\frac{1}{1-\lambda}}}{c_{2}^{\frac{\lambda}{1-\lambda}}}\varepsilon^{\frac{\gamma\lambda}{1-\lambda}}(L_{2}+\Lambda)^{\frac{1}{1-\lambda}}-\frac{\lambda^{\frac{\lambda}{1-\lambda}}}{c_{2}^{\frac{\lambda}{1-\lambda}}}\varepsilon^{\frac{\gamma\lambda}{1-\lambda}}(L_{2}+\Lambda)^{\frac{1}{1-\lambda}}$
$\displaystyle=\varepsilon-\beta\varepsilon^{\frac{\gamma\lambda}{1-\lambda}}(L_{2}+\Lambda)^{\frac{1}{1-\lambda}}=\frac{1}{2^{\frac{\gamma\lambda}{\gamma\lambda+\lambda-1}}\beta^{\frac{1-\lambda}{\gamma\lambda+\lambda-1}}(L_{2}+\Lambda)^{\frac{1}{\gamma\lambda+\lambda-1}}}$
$\displaystyle=\frac{\varepsilon}{2}.$

Namely, even if $\widehat{Y}(t_{n})<\varphi(t_{n})+\varepsilon$, we still have
that, with probability 1,

$\widehat{Y}(t_{n})-\varphi(t_{n})\geq
F_{\varepsilon}(t_{n}-t_{\kappa(n)})\geq
F_{\varepsilon}(t_{*})=\frac{\varepsilon}{2},$

and thus, with probability 1, for any $n=0,1,...,N$

$\displaystyle\widehat{Y}(t_{n})$
$\displaystyle\geq\varphi(t_{n})+\frac{\varepsilon}{2}=\varphi(t_{n})+\frac{1}{2^{\frac{\gamma\lambda}{\gamma\lambda+\lambda-1}}\beta^{\frac{1-\lambda}{\gamma\lambda+\lambda-1}}(L_{2}+\Lambda)^{\frac{1}{\gamma\lambda+\lambda-1}}}$
$\displaystyle=:\varphi(t_{n})+\frac{L_{1}}{(L_{2}+\Lambda)^{\frac{1}{\gamma\lambda+\lambda-1}}},$

where
$L_{1}:=\frac{1}{2^{\frac{\gamma\lambda}{\gamma\lambda+\lambda-1}}\beta^{\frac{1-\lambda}{\gamma\lambda+\lambda-1}}}$.
∎

######  Remark 2.5.

It is clear that constants $L_{1}$ and $L_{2}$ in Lemma 2.4 can be chosen
jointly for $Y$ and $\widehat{Y}$, so that the inequalities

$\varphi(t)+\frac{L_{1}}{(L_{2}+\Lambda)^{\alpha}}\leq
Y(t)\leq\psi(t)-\frac{L_{1}}{(L_{2}+\Lambda)^{\alpha}},\quad t\in[0,T],$

and

$\varphi(t_{n})+\frac{L_{1}}{(L_{2}+\Lambda)^{\alpha}}\leq\widehat{Y}(t_{n})\leq\psi(t_{n})-\frac{L_{1}}{(L_{2}+\Lambda)^{\alpha}},\quad
n=0,1,...,N,$

hold simultaneously with probability 1.

Next, we proceed with a simple property of the sandwiched process $Y$ in
(1.2).

######  Lemma 2.6.

Let $Z$ satisfy (Z1)–(Z2) and assumptions (B1)–(B3) hold.

  * 1)

There exists a positive random variable $\Upsilon$ such that, with probability
1,

$|Y(t)-Y(s)|\leq\Upsilon|t-s|^{\lambda},\quad t,s\in[0,T].$

  * 2)

If, for some $r\geq 1$,

$\mathbb{E}\left[\Lambda^{\frac{r\max\\{p,\gamma\lambda+\lambda-1\\}}{\gamma\lambda+\lambda-1}}\right]<\infty,$
(2.5)

where $\lambda$ and $\Lambda$ are from (Z2), $p$ is from (B2) and $\gamma$ is
from (B3), then one can choose $\Upsilon$ such that

$\mathbb{E}[\Upsilon^{r}]<\infty.$

###### Proof.

Denote $\phi(t):=\frac{1}{2}(\psi(t)+\varphi(t))$, $t\in[0,T]$. By (1.8),

$\varphi(t)+\frac{L_{1}}{(L_{2}+\Lambda)^{\frac{1}{\gamma\lambda+\lambda-1}}}\leq
Y(t)\leq\psi(t)-\frac{L_{1}}{(L_{2}+\Lambda)^{\frac{1}{\gamma\lambda+\lambda-1}}},\quad
t\in[0,T],\quad a.s.,$

i.e. with probability 1
$(t,Y(t))\in\mathcal{D}_{\frac{1}{\xi},\frac{1}{\xi}}$, $t\in[0,T]$, where

$\xi:=\frac{(L_{2}+\Lambda)^{\frac{1}{\gamma\lambda+\lambda-1}}}{L_{1}}$ (2.6)

and $\mathcal{D}_{\frac{1}{\xi},\frac{1}{\xi}}$ is defined by (1.1). It is
evident that $(t,\phi(t))\in\mathcal{D}_{\frac{1}{\xi},\frac{1}{\xi}}$,
$t\in[0,T]$, therefore, using (Z2), (B2) and (1.7), we can write that, with
probability 1, for all $0\leq s<t\leq T$:

$\displaystyle|Y(t)-Y(s)|$
$\displaystyle\leq\int_{s}^{t}|b(u,Y(u))|du+|Z(t)-Z(s)|$ (2.7)
$\displaystyle\leq\int_{s}^{t}|b(u,Y(u))-b(u,\phi(u))|du+\int_{s}^{t}|b(u,\phi(u))|du+\Lambda(t-s)^{\lambda}$
$\displaystyle\leq
c_{1}\xi^{p}\int_{s}^{t}|Y(u)-\phi(u)|du+\max_{u\in[0,T]}|b(u,\phi(u))|(t-s)+\Lambda(t-s)^{\lambda}$
$\displaystyle\leq\left(c_{1}\xi^{p}\lVert\psi-\varphi\rVert_{\infty}+\max_{u\in[0,T]}|b(u,\phi(u))|\right)(t-s)+\Lambda(t-s)^{\lambda}$
$\displaystyle\leq C(\xi^{p}+\Lambda+1)(t-s)^{\lambda},$

where $C$ is a positive constant. Now one can put

$\Upsilon:=C(\xi^{p}+\Lambda+1)$ (2.8)

and observe that the definition of $\Upsilon$, (2.5) and (2.6) imply that

$\mathbb{E}[\Upsilon^{r}]<\infty.$

∎

######  Corollary 2.7.

Under (Z1)–(Z2) and Assumptions B, using Lemma 2.4 and following the proof of
Lemma 2.6, it is easy to obtain that there exists a random variable $\Upsilon$
independent of the partition such that with probability 1

$|\widehat{Y}(t_{k})-\widehat{Y}(t_{n})|\leq\Upsilon|t_{k}-t_{n}|^{\lambda},\quad
k,n=0,...,N.$ (2.9)

Furthermore, just like in Lemma 2.6, if (2.5) for some for $r\geq 1$, then

$\mathbb{E}[\Upsilon^{r}]<\infty.$

Finally, just as in Remark 2.5, $\Upsilon$ can be chosen jointly for $Y$ and
$\widehat{Y}$, so that

$|Y(t)-Y(s)|\leq\Upsilon|t-s|^{\lambda},\quad t,s\in[0,T],$

holds simultaneously with (2.9) with probability 1.

######  Lemma 2.8.

Let $Z$ satisfy (Z1)–(Z2), Assumptions B hold and the mesh of the partition
$\Delta_{N}$ satisfy (2.1).

  * 1)

For any $r\geq 1$, there exists a positive random variable $\mathcal{C}_{1}$
that does not depend on the partition such that

$\sup_{k=0,1,...,N}|Y(t_{k})-\widehat{Y}(t_{k})|^{r}\leq\mathcal{C}_{1}\Delta_{N}^{\lambda
r}\quad a.s.$

  * 2)

If, additionally,

$\mathbb{E}\left[\Lambda^{\frac{r(p+\max\\{p,\gamma\lambda+\lambda-1\\})}{\gamma\lambda+\lambda-1}}\right]<\infty,$
(2.10)

where $\lambda$ and $\Lambda$ are from (Z2), $p$ is from (B2) and $\gamma$ is
from (B3), then one can choose $\mathcal{C}_{1}$ such that
$\mathbb{E}[\mathcal{C}_{1}]<\infty$, i.e. there exists a deterministic
constant $C$ that does not depend on the partition such that

$\mathbb{E}\left[\sup_{k=0,1,...,N}|Y(t_{k})-\widehat{Y}(t_{k})|^{r}\right]\leq
C\Delta_{N}^{\lambda r}.$

###### Proof.

Fix $\omega\in\Omega$ such that $Z(\omega,t)$, $t\in[0,T]$, is Hölder
continuous (for simplicity of notation, we will omit $\omega$ in the
brackets). Denote $e_{n}:=Y(t_{n})-\widehat{Y}(t_{n})$, $\Delta
Z_{n}:=Z(t_{n})-Z(t_{n-1})$. Then

$\displaystyle e_{n}$
$\displaystyle=Y(t_{n-1})+\int_{t_{n-1}}^{t_{n}}b(s,Y(s))ds+\Delta Z_{n}$
(2.11)
$\displaystyle\quad-\widehat{Y}(t_{n-1})-b(t_{n},\widehat{Y}(t_{n}))\Delta_{N}-\Delta
Z_{n}$
$\displaystyle=e_{n-1}+\left(b(t_{n},Y(t_{n}))-b(t_{n},\widehat{Y}(t_{n}))\right)\Delta_{N}$
$\displaystyle\quad+\int_{t_{n-1}}^{t_{n}}(b(s,Y(s))-b(t_{n},Y(t_{n})))ds.$

By the mean value theorem,

$\displaystyle\left(b(t_{n},Y(t_{n}))-b(t_{n},\widehat{Y}(t_{n}))\right)\Delta_{N}$
$\displaystyle=\frac{\partial b}{\partial
y}(t_{n},\Theta_{n})\Delta_{N}(Y(t_{n})-\widehat{Y}(t_{n}))$
$\displaystyle=\frac{\partial b}{\partial y}(t_{n},\Theta_{n})\Delta_{N}e_{n}$

with
$\Theta_{n}\in(Y(t_{n})\wedge\widehat{Y}(t_{n}),Y(t_{n})\vee\widehat{Y}(t_{n}))$.
Using this, we can rewrite (2.11) as follows:

$\displaystyle\left(1-\frac{\partial b}{\partial
y}(t_{n},\Theta_{n})\Delta_{N}\right)e_{n}$
$\displaystyle=e_{n-1}+\int_{t_{n-1}}^{t_{n}}(b(s,Y(s))-b(t_{n},Y(t_{n})))ds,$
(2.12)

where

$1-\frac{\partial b}{\partial
y}(t_{n},\Theta_{n})\Delta_{N}>1-c_{3}\Delta_{N}>0$

by (B4) and (2.1).

Next, denote

$\zeta_{0}:=1,\quad\zeta_{n}:=\prod_{i=1}^{n}\left(1-\frac{\partial
b}{\partial y}(t_{i},\Theta_{i})\Delta_{N}\right)$

and define $\tilde{e}_{n}:=\zeta_{n}e_{n}$. By multiplying both sides of
(2.12) by $\zeta_{n-1}$, we obtain that

$\displaystyle\tilde{e}_{n}=\tilde{e}_{n-1}$
$\displaystyle+\zeta_{n-1}\int_{t_{n-1}}^{t_{n}}(b(s,Y(s))-b(t_{n},Y(t_{n})))ds$
(2.13)

and, expanding the terms $\tilde{e}_{i-1}$ in (2.13) one by one,
$i=n,n-1,...,1$, and taking into account that $\tilde{e}_{0}=0$, we obtain
that

$\displaystyle\tilde{e}_{n}$
$\displaystyle=\sum_{i=1}^{n}\zeta_{i-1}\int_{t_{i-1}}^{t_{i}}(b(s,Y(s))-b(t_{i},Y(t_{i})))ds.$

Therefore

$\displaystyle e_{n}$
$\displaystyle=\sum_{i=1}^{n}\frac{\zeta_{i-1}}{\zeta_{n}}\int_{t_{i-1}}^{t_{i}}(b(s,Y(s))-b(t_{i},Y(t_{i})))ds.$

Observe that, by assumption (B4) and (2.1), for any $i,n\in\mathbb{N}$, $i<n$,

$\displaystyle\frac{\zeta_{k}}{\zeta_{n}}$
$\displaystyle=\prod_{i=k+1}^{n}\left(1-\frac{\partial b}{\partial
y}(t_{i},\Theta_{i})\Delta_{N}\right)^{-1}\leq\prod_{i=k+1}^{N}(1-c_{3}\Delta_{N})^{-1}$
$\displaystyle\leq(1-c_{3}\Delta_{N})^{-N}=\left(1-\frac{c_{3}T}{N}\right)^{-N}\to
e^{c_{3}T},\quad N\to\infty,$

whence there exists a constant $C$ that does not depend on $i$, $n$ or $N$
such that

$\frac{\zeta_{k}}{\zeta_{n}}\leq C.$

Using this, one can deduce that

$\displaystyle|e_{n}|^{r}$ $\displaystyle\leq
C\left|\sum_{i=1}^{n}\frac{\zeta_{i-1}}{\zeta_{n}}\int_{t_{i-1}}^{t_{i}}(b(s,Y(s))-b(t_{i},Y(t_{i})))ds\right|^{r}$
$\displaystyle\leq
C\left(\sum_{i=1}^{n}\int_{t_{i-1}}^{t_{i}}\left|b(s,Y(s))-b(t_{i},Y(t_{i}))\right|ds\right)^{r}.$

Note that $(t,Y(t))\in\mathcal{D}_{\frac{1}{\xi},\frac{1}{\xi}}$, where $\xi$
is defined by (2.6) and $\mathcal{D}_{\frac{1}{\xi},\frac{1}{\xi}}$ is defined
via (1.1), hence, by (B2) as well as Lemma 2.6, we can deduce that

$\displaystyle\left(\sum_{i=1}^{n}\int_{t_{i-1}}^{t_{i}}\left|b(s,Y(s))-b(t_{i},Y(t_{i}))\right|ds\right)^{r}$
$\displaystyle\leq
C\xi^{pr}\left(\sum_{i=1}^{n}\int_{t_{i-1}}^{t_{i}}\left|s-t_{i}\right|^{\lambda}ds\right)^{r}+C\xi^{pr}\left(\sum_{i=1}^{n}\int_{t_{i-1}}^{t_{i}}\left|Y(s)-Y(t_{i})\right|ds\right)^{r}$
$\displaystyle\leq
C\xi^{pr}\left(\sum_{i=1}^{n}\int_{t_{i-1}}^{t_{i}}\left|s-t_{i}\right|^{\lambda}ds\right)^{r}+C\xi^{pr}\Upsilon^{r}\left(\sum_{i=1}^{n}\int_{t_{i-1}}^{t_{i}}\left|s-t_{i}\right|^{\lambda}ds\right)^{r}$
$\displaystyle=C\xi^{pr}(1+\Upsilon^{r})\left(\sum_{i=1}^{n}\frac{1}{(1+\lambda)}\Delta_{N}^{1+\lambda}\right)^{r}$
$\displaystyle\leq C\xi^{pr}(1+\Upsilon^{r})\Delta_{N}^{\lambda r}.$

In other words, there exists a constant $C$ that does not depend on the
partition such that

$|e_{n}|^{r}=|Y(t_{n})-\widehat{Y}(t_{n})|^{r}\leq
C\xi^{pr}(1+\Upsilon^{r})\Delta_{N}^{\lambda r}$

and, since the right-hand side of the relation above does not depend on $n$ or
$N$, we have

$\sup_{n=0,...,N}|Y(t_{n})-\widehat{Y}(t_{n})|^{r}\leq
C\xi^{pr}(1+\Upsilon^{r})\Delta_{N}^{\lambda
r}=:\mathcal{C}_{1}\Delta_{N}^{\lambda r}.$ (2.14)

It remains to notice that, by (2.6) and (2.8),

$\mathbb{E}\left[\xi^{pr}(1+\Upsilon^{r})\right]<\infty$

whenever (2.10) holds, which finally implies

$\mathbb{E}\left[\sup_{n=0,...,N}|Y(t_{n})-\widehat{Y}(t_{n})|^{r}\right]\leq\mathbb{E}[\mathcal{C}_{1}]\Delta_{N}^{\lambda
r}=:C\Delta_{N}^{\lambda r}.$

∎

Now we are ready to proceed to the main results of this subsection.

######  Theorem 2.9.

Let $Z$ satisfy (Z1)–(Z2), Assumptions B hold and the mesh of the partition
$\Delta_{N}$ satisfy (2.1).

  * 1)

For any $r\geq 1$, there exists a random variable $\mathcal{C}_{2}$ that does
not depend on the partition such that

$\sup_{t\in[0,T]}|Y(t)-\widehat{Y}(t)|^{r}\leq\mathcal{C}_{2}\Delta_{N}^{\lambda
r}\quad a.s.$

  * 2)

If, additionally,

$\mathbb{E}\left[\Lambda^{\frac{r(p+\max\\{p,\gamma\lambda+\lambda-1\\})}{\gamma\lambda+\lambda-1}}\right]<\infty,$

where $\lambda$ and $\Lambda$ are from (Z2), $p$ is from (B2) and $\gamma$ is
from (B3), then one can choose $\mathcal{C}_{2}$ such that
$\mathbb{E}[\mathcal{C}_{2}]<\infty$, i.e. there exists a deterministic
constant $C$ that does not depend on the partition such that

$\mathbb{E}\left[\sup_{t\in[0,T]}|Y(t)-\widehat{Y}(t)|^{r}\right]\leq
C\Delta_{N}^{\lambda r}.$

###### Proof.

Fix $\omega\in\Omega$ such that $Z(\omega,t)$, $t\in[0,T]$, is Hölder
continuous (for simplicity of notation, we again omit $\omega$ in the
brackets) and consider an arbitrary $t\in[0,T]$. Denote

$n(t):=\max\\{n=0,1,...,N~{}|~{}t\geq t_{n}\\},$

i.e. $t\in[t_{n(t)},t_{n(t)+1})$. Then

$\displaystyle|Y(t)-\widehat{Y}(t)|^{r}$ $\displaystyle\leq
C\left(|Y(t)-Y(t_{n(t)})|^{r}+|Y(t_{n(t)})-\widehat{Y}(t_{n(t)})|^{r}\right)$
$\displaystyle\leq C\Upsilon^{r}(t-t_{n(t)})^{\lambda
r}+C(L_{2}+\Lambda)^{\frac{pr}{\gamma\lambda+\lambda-1}}(1+\Upsilon^{r})\Delta_{N}^{\lambda
r}$ $\displaystyle\leq
C\left(\Upsilon^{r}+(1+\Upsilon^{r})(L_{2}+\Lambda)^{\frac{pr}{\gamma\lambda+\lambda-1}}\right)\Delta_{N}^{\lambda
r},$

where we used Lemma 2.6 to estimate $|Y(t)-Y(t_{n(t)})|^{r}$ and bound (2.14)
to estimate $|Y(t_{n(t)})-\widehat{Y}(t_{n(t)})|^{r}$. Therefore

$\sup_{t\in[0,T]}|Y(t)-\widehat{Y}(t)|^{r}\leq
C\left(\Upsilon^{r}+(1+\Upsilon^{r})(L_{2}+\Lambda)^{\frac{pr}{\gamma\lambda+\lambda-1}}\right)\Delta_{N}^{\lambda
r}=:\mathcal{C}_{2}\Delta_{N}^{\lambda r}.$

Finally, using the same arguments as in Lemma 2.6 and Lemma 2.8, it is easy to
see that the condition

$\mathbb{E}\left[\Lambda^{\frac{r(p+\max\\{p,\gamma\lambda+\lambda-1\\})}{\gamma\lambda+\lambda-1}}\right]<\infty$

implies that

$\mathbb{E}\left[\Upsilon^{r}+(1+\Upsilon^{r})(L_{2}+\Lambda)^{\frac{pr}{\gamma\lambda+\lambda-1}}\right]<\infty,$

therefore

$\mathbb{E}\left[\sup_{t\in[0,T]}|Y(t)-\widehat{Y}(t)|^{r}\right]\leq
C\left(\Upsilon^{r}+(1+\Upsilon^{r})(L_{2}+\Lambda)^{\frac{pr}{\gamma\lambda+\lambda-1}}\right)\Delta_{N}^{\lambda
r}$

for some constant $C>0$ that does not depend on the partition. ∎

######  Theorem 2.10.

  * 1)

Let $Z$ satisfy (Z1)–(Z2), Assumptions B hold and the mesh of the partition
$\Delta_{N}$ satisfy (2.1). Then, for any $r\geq 1$, there exists a random
variable $\mathcal{C}_{3}$ that does not depend on the partition such that

$\sup_{n=0,1,...,N}\left|\frac{1}{Y(t_{n})-\varphi(t_{n})}-\frac{1}{\widehat{Y}(t_{n})-\varphi(t_{n})}\right|^{r}\leq\mathcal{C}_{3}\Delta_{N}^{\lambda
r}\quad a.s.$

and

$\sup_{n=0,1,...,N}\left|\frac{1}{\psi(t_{n})-Y(t_{n})}-\frac{1}{\psi(t_{n})-\widehat{Y}(t_{n})}\right|^{r}\leq\mathcal{C}_{3}\Delta_{N}^{\lambda
r}\quad a.s.$

  * 2)

If, additionally,

$\mathbb{E}\left[\Lambda^{\frac{r(2+p+\max\\{p,\gamma\lambda+\lambda-1\\})}{\gamma\lambda+\lambda-1}}\right]<\infty,$
(2.15)

where $\lambda$ and $\Lambda$ are from (Z2), $p$ is from (B2) and $\gamma$ is
from (B3), then one can choose $\mathcal{C}_{3}$ such that
$\mathbb{E}[\mathcal{C}_{3}]<\infty$, i.e. there exists a deterministic
constant $C$ that does not depend on the partition such that

$\mathbb{E}\left[\sup_{n=0,1,...,N}\left|\frac{1}{Y(t_{n})-\varphi(t_{n})}-\frac{1}{\widehat{Y}(t_{n})-\varphi(t_{n})}\right|^{r}\right]\leq
C\Delta_{N}^{\lambda r}$

and

$\mathbb{E}\left[\sup_{n=0,1,...,N}\left|\frac{1}{\psi(t_{n})-Y(t_{n})}-\frac{1}{\psi(t_{n})-\widehat{Y}(t_{n})}\right|^{r}\right]\leq
C\Delta_{N}^{\lambda r}.$

###### Proof.

By Remark 2.5 and estimate (2.14), with probability 1 for any $n=0,...,N$:

$\displaystyle\left|\frac{1}{Y(t_{n})-\varphi(t_{n})}-\frac{1}{\widehat{Y}(t_{n})-\varphi(t_{n})}\right|^{r}$
$\displaystyle=\frac{|Y(t_{n})-\widehat{Y}(t_{n})|^{r}}{(Y_{t_{n}}-\varphi(t_{n}))^{r}(\widehat{Y}_{t_{n}}-\varphi(t_{n}))^{r}}$
$\displaystyle\leq\frac{(L_{2}+\Lambda)^{\frac{2r}{\gamma\lambda+\lambda-1}}}{L_{1}^{2r}}\sup_{n=0,1,...,N}|Y(t_{n})-\widehat{Y}(t_{n})|^{r}$
$\displaystyle\leq
C(L_{2}+\Lambda)^{\frac{2r}{\gamma\lambda+\lambda-1}}\xi^{pr}(1+\Upsilon^{r})\Delta_{N}^{\lambda
r}$ $\displaystyle=:\mathcal{C}_{3}\Delta_{N}^{\lambda r}.$

It remains to notice that, by (2.6) and (2.8), the condition (2.15) implies
that $\mathbb{E}[\mathcal{C}_{3}]<\infty$. The second estimate can be obtained
in a similar manner. ∎

##  3 One-sided sandwich case

The drift-implicit Euler approximation scheme described in Section 2 for the
two-sided sandwich can also be adapted for the one-sided setting that
corresponds to Assumptions A on the SDE (0.1). However, in the two-sided
sandwich case the process $Y$ was bounded (which was utilized, e.g., in Lemma
2.6) and, moreover, the behaviour of $Y$ was similar near both $\varphi$ and
$\psi$ so that it was sufficient to analyze only one of the bounds. In the
one-sided case, each $Y(t)$, for $t\in[0,T]$, is not a bounded random
variable, therefore the approach from Section 2 has to be adjusted. For this,
we will be using the inequalities (1.4).

Let the noise $Z$ satisfy (Z1)–(Z2), $Y(0)$ and $b$ satisfy Assumptions A and
$Y=\\{Y(t),~{}t\in[0,T]\\}$ be the unique solution of the SDE (1.2). In line
with Section 2, we consider a uniform partition
$\\{0=t_{0}<t_{1}<...<t_{N}=T\\}$ of $[0,T]$, $t_{k}:=\frac{Tk}{N}$,
$k=0,1,...,N$, with the mesh $\Delta_{N}:=\frac{T}{N}$ such that

$c_{3}\Delta_{N}<1,$ (3.1)

where $c_{3}$ is an upper bound for $\frac{\partial b}{\partial y}$ from
assumption (A4). The backward Euler approximation $\widehat{Y}(t)$ is defined
in a manner similar to (2.2), i.e.

$\displaystyle\widehat{Y}(0)$ $\displaystyle=Y(0),$ (3.2)
$\displaystyle\widehat{Y}(t_{k+1})$
$\displaystyle=\widehat{Y}(t_{k})+b(t_{k+1},\widehat{Y}(t_{k+1}))\Delta_{N}+(Z(t_{k+1})-Z(t_{k})),$
$\displaystyle\widehat{Y}(t)$ $\displaystyle=\widehat{Y}(t_{k}),\quad
t\in[t_{k},t_{k+1}),$

where the second expression is considered as an equation with respect to
$\widehat{Y}(t_{k+1})$.

######  Remark 3.1.

Just as in the two-sided sandwich case, each $\widehat{Y}(t_{k})$,
$k=1,...,N$, is well defined since the equation

$y-b(t,y)\Delta_{N}=z$

has a unique solution w.r.t. $y$ such that $y>\varphi(t)$ for any fixed
$t\in[0,T]$ and any $z\in\mathbb{R}$. To understand this, note that assumption
(A4) together with (3.1) imply that

$(y-b(t,y)\Delta_{N})^{\prime}_{y}>0.$ (3.3)

Second, by (A3),

$\displaystyle y-b(t,y)\Delta_{N}\to-\infty,\quad y\to\varphi(t)+.$ (3.4)

Next, by (A2), for any $(s,y_{1})$,
$(s,y_{2})\in\overline{\mathcal{D}_{1}}:=\\{(u,y)\in[0,T]\times\mathbb{R}_{+},y\in[\varphi(u)+1,\infty)\\}$
we have that

$|b(s,y_{1})-b(s,y_{2})|\leq c_{1}|y_{1}-y_{2}|,$

i.e.

$\sup_{(s,y)\in\overline{\mathcal{D}_{1}}}\left|\frac{\partial b}{\partial
y}(s,y)\right|<\infty.$

Using this, (A4) and the mean value theorem, for any positive
$y\geq\varphi(t)+1$

$\displaystyle b(t,y)$ $\displaystyle=b(t,\varphi(t)+1)+\frac{\partial
b}{\partial y}(t,\theta_{y})(y-1-\varphi(t))$
$\displaystyle\leq\max_{s\in[0,T]}b(t,\varphi(t)+1)+\max_{s\in[0,T]}|1+\varphi(s)|\sup_{(s,y)\in\overline{\mathcal{D}_{1}}}\left|\frac{\partial
b}{\partial y}(s,y)\right|+c_{3}y$ $\displaystyle=:C+c_{3}y,$

whence

$y-b(t,y)\Delta_{N}\geq-C\Delta_{N}+(1-c_{3}\Delta_{N})y\to\infty,\quad
y\to\infty.$ (3.5)

Existence and uniqueness of the solution then follows from (3.3)–(3.5).

######  Remark 3.2.

Similarly to the two-sided sandwich case, the value of $\widehat{Y}(t)$ for
$t\in[0,T]\setminus\\{t_{0},...,t_{N}\\}$ can also be defined via linear
interpolation with no changes in formulations of the results and almost no
variations in the proofs.

Our strategy for proving the convergence of $\widehat{Y}$ to $Y$ will be
similar to what we have done in section 2. Therefore we will be omitting the
details highlighting only the points which are different from the two-sided
sandwich case. We start with some useful properties of $\widehat{Y}$ and $Y$.

######  Lemma 3.3.

Let $Z$ satisfy (Z1)–(Z2), Assumptions A hold and the mesh of the partition
$\Delta_{N}$ satisfy (3.1). Then there exist deterministic constants $L_{1}$,
$L_{2}>0$ depending only on $Y(0)$, the shape of the drift $b$ and $\lambda$,
such that

$\widehat{Y}(t_{n})\geq\varphi(t_{n})+\frac{L_{1}}{(L_{2}+\Lambda)^{\frac{1}{\gamma\lambda+\lambda-1}}}\quad
a.s.,$

where $\Lambda$ is from assumption (Z2) and $\gamma$ is from assumption (A3).
Moreover, there exist constants $L_{3}$, $L_{4}>0$ that also depend only on
$Y(0)$, the shape of the drift $b$ and $\lambda$ such that

$\widehat{Y}(t_{n})\leq L_{3}+L_{4}\Lambda,\quad n=0,1,...,N,\quad a.s.$

for all partitions with the mesh satisfying
$\frac{c_{1}}{(Y(0)-\varphi(0))^{p}}\Delta_{N}<1$ with $c_{1}$ and $p$ being
from (A2).

###### Proof.

The proof of

$\widehat{Y}(t_{n})\geq\varphi(t_{n})+\frac{L_{1}}{(L_{2}+\Lambda)^{\frac{1}{\gamma\lambda+\lambda-1}}}$

is identical to the corresponding one in Lemma 2.4 and will be omitted. Let us
prove that

$\widehat{Y}(t_{n})\leq L_{3}+L_{4}\Lambda\quad a.s.$

Fix $\omega\in\Omega$ for which $Z(\omega,t)$ is Hölder continuous, consider a
partition with the mesh satisfying
$\frac{c_{1}}{(Y(0)-\varphi(0))^{p}}\Delta_{N}<1$ and fix an arbitrary
$n=0,1,...,N-1$. Assume that
$\widehat{Y}(t_{n+1})>\varphi(t_{n+1})+(Y(0)-\varphi(0))$ (otherwise the claim
of the lemma holds automatically). Put

$\kappa(n):=\max\\{k=0,1,...,n~{}|~{}\widehat{Y}(t_{k})\leq\varphi(t_{k})+(Y(0)-\varphi(0))\\}$

and observe that $(t_{k},\widehat{Y}(t_{k}))\in\mathcal{D}_{Y(0)-\varphi(0)}$
for any $k=\kappa(n)+1,...,n+1$, where $\mathcal{D}_{Y(0)-\varphi(0)}$ is
defined via (1.1). Next, by (A2), for any $y\in\mathcal{D}_{Y(0)-\varphi(0)}$

$\displaystyle|b(t,y)|-\left|b\Big{(}t,\varphi(t)+(Y(0)-\varphi(0))\Big{)}\right|$
$\displaystyle\leq\left|b(t,y)-b\Big{(}t,\varphi(t)+(Y(0)-\varphi(0))\Big{)}\right|$
$\displaystyle\leq\frac{c_{1}}{(Y(0)-\varphi(0))^{p}}|y-\varphi(t)-(\varphi(0)-Y(0))|$
$\displaystyle\leq\frac{c_{1}}{(Y(0)-\varphi(0))^{p}}|y|+\frac{c_{1}}{(Y(0)-\varphi(0))^{p}}|\varphi(t)+(\varphi(0)-Y(0))|,$

i.e. there exists a constant $C>0$ that does not depend on the partition such
that

$|b(t,y)|\leq C+\frac{c_{1}}{(Y(0)-\varphi(0))^{p}}|y|.$ (3.6)

Next, observe that, for any $k=\kappa(n)+1,...,n+1$, we have

$\displaystyle\widehat{Y}(t_{k})$
$\displaystyle=\widehat{Y}(t_{\kappa(n)})+\sum_{i=\kappa(n)+1}^{k}b(t_{i},\widehat{Y}(t_{i}))\Delta_{N}+Z(t_{k})-Z(t_{\kappa(n)})$
$\displaystyle\leq\varphi(t_{\kappa(n)})+(Y(0)-\varphi(0))+\sum_{i=\kappa(n)+1}^{k}b(t_{i},\widehat{Y}(t_{i}))\Delta_{N}+\Lambda(t_{k}-t_{\kappa(n)})^{\lambda}$
$\displaystyle\leq\left|\max_{s\in[0,T]}\varphi(s)+(Y(0)-\varphi(0))\right|+T^{\lambda}\Lambda+\sum_{i=\kappa(n)+1}^{k}b(t_{i},\widehat{Y}(t_{i}))\Delta_{N}.$

Therefore, using (3.6) and

$\widehat{Y}(t_{k})>\varphi(t_{k})\geq\min_{s\in[0,T]}\varphi(s),$

one can write

$\displaystyle|\widehat{Y}(t_{k})|$
$\displaystyle\leq\left|\min_{s\in[0,T]}\varphi(s)\right|+\left|\max_{s\in[0,T]}\varphi(s)+(Y(0)-\varphi(0))\right|+T^{\lambda}\Lambda+\sum_{i=\kappa(n)+1}^{k}|b(t_{i},\widehat{Y}(t_{i}))|\Delta_{N}$
$\displaystyle\leq\left|\min_{s\in[0,T]}\varphi(s)\right|+\left|\max_{s\in[0,T]}\varphi(s)+(Y(0)-\varphi(0))\right|+T^{\lambda}\Lambda+C\sum_{i=\kappa(n)+1}^{k}\Delta_{N}$
$\displaystyle\qquad+\frac{c_{1}}{(Y(0)-\varphi(0))^{p}}\sum_{i=\kappa(n)+1}^{k}|\widehat{Y}(t_{i})|\Delta_{N},$

where $C>0$ is some positive constant that does not depend on the partition.

Now we want to apply the discrete version of the Gronwall inequality from [20,
Lemma A.3]. In order to do that, we observe that

$\displaystyle|\widehat{Y}(t_{\kappa(n)+1})|$ $\displaystyle\leq
C+T^{\lambda}\Lambda+\frac{c_{1}}{(Y(0)-\varphi(0))^{p}}\Delta_{N}|\widehat{Y}(t_{k})|,$

and, for any $k=\kappa(n)+2,...,n+1$,

$\displaystyle|\widehat{Y}(t_{k})|$ $\displaystyle\leq
C+T^{\lambda}\Lambda+\frac{c_{1}}{(Y(0)-\varphi(0))^{p}}\sum_{i=\kappa(n)+1}^{k-1}|\widehat{Y}(t_{i})|\Delta_{N}+\frac{c_{1}}{(Y(0)-\varphi(0))^{p}}\Delta_{N}|\widehat{Y}(t_{k})|.$

Now, since $\frac{c_{1}}{(Y(0)-\varphi(0))^{p}}\Delta_{N}<1$, we can write
that

$\left(1-\frac{c_{1}}{(Y(0)-\varphi(0))^{p}}\Delta_{N}\right)|\widehat{Y}(t_{\kappa(n)+1})|\leq
C+T^{\lambda}\Lambda$

and, for all $k=\kappa(n)+2,...,n+1$,

$\left(1-\frac{c_{1}}{(Y(0)-\varphi(0))^{p}}\Delta_{N}\right)|\widehat{Y}(t_{k})|\leq
C+T^{\lambda}\Lambda+\frac{c_{1}}{(Y(0)-\varphi(0))^{p}}\sum_{i=\kappa(n)+1}^{k-1}|\widehat{Y}(t_{i})|\Delta_{N}.$

Put

$N_{0}:=\min\left\\{N\geq
1:~{}\frac{c_{1}}{(Y(0)-\varphi(0))^{p}}\Delta_{N}<1\right\\}=\left[\frac{Tc_{1}}{(Y(0)-\varphi(0))^{p}}\right]+1$

with $[x]$ being the greatest integer less than or equal to $x$ and observe
that, for all $N\geq N_{0}$,

$1-\frac{c_{1}}{(Y(0)-\varphi(0))^{p}}\Delta_{N}\geq
1-\frac{c_{1}}{(Y(0)-\varphi(0))^{p}}\Delta_{N_{0}}.$

Therefore,

$\displaystyle|\widehat{Y}(t_{\kappa(n)+1})|$
$\displaystyle\leq\frac{C}{1-\frac{c_{1}}{(Y(0)-\varphi(0))^{p}}\Delta_{N}}+\frac{T^{\lambda}}{1-\frac{c_{1}}{(Y(0)-\varphi(0))^{p}}\Delta_{N}}\Lambda$
$\displaystyle\leq\frac{C}{1-\frac{c_{1}}{(Y(0)-\varphi(0))^{p}}\Delta_{N_{0}}}+\frac{T^{\lambda}}{1-\frac{c_{1}}{(Y(0)-\varphi(0))^{p}}\Delta_{N_{0}}}\Lambda$
$\displaystyle=:C_{1}+C_{2}\Lambda$

and, for all $k=\kappa(n)+2,...,n+1$,

$\displaystyle|\widehat{Y}(t_{k})|$
$\displaystyle\leq\frac{C}{1-\frac{c_{1}}{(Y(0)-\varphi(0))^{p}}\Delta_{N}}+\frac{T^{\lambda}}{1-\frac{c_{1}}{(Y(0)-\varphi(0))^{p}}\Delta_{N}}\Lambda$
$\displaystyle\quad+\frac{c_{1}}{(Y(0)-\varphi(0))^{p}}\sum_{i=\kappa(n)+1}^{k-1}|\widehat{Y}(t_{i})|\frac{\Delta_{N}}{1-\frac{c_{1}}{(Y(0)-\varphi(0))^{p}}\Delta_{N}}$
$\displaystyle\leq\frac{C}{1-\frac{c_{1}}{(Y(0)-\varphi(0))^{p}}\Delta_{N_{0}}}+\frac{T^{\lambda}}{1-\frac{c_{1}}{(Y(0)-\varphi(0))^{p}}\Delta_{N_{0}}}\Lambda$
$\displaystyle\quad+\frac{c_{1}}{(Y(0)-\varphi(0))^{p}}\sum_{i=\kappa(n)+1}^{k-1}|\widehat{Y}(t_{i})|\frac{\Delta_{N}}{1-\frac{c_{1}}{(Y(0)-\varphi(0))^{p}}\Delta_{N_{0}}}$
$\displaystyle=:C_{1}+C_{2}\Lambda+C_{3}\sum_{i=\kappa(n)+1}^{k-1}|\widehat{Y}(t_{i})|\Delta_{N}.$

Using a discrete version of the Gronwall inequality, we now obtain that for
all $k=\kappa(n)+1,...,n+1$

$\displaystyle|\widehat{Y}(t_{k})|$
$\displaystyle\leq\left(C_{1}+C_{2}\Lambda\right)\exp\left\\{C_{3}\sum_{i=\kappa(n)+1}^{k-1}\Delta_{N}\right\\}\leq\left(C_{1}+C_{2}\Lambda\right)\exp\left\\{TC_{3}\right\\}$
$\displaystyle=:L_{3}+L_{4}\Lambda.$

which ends the proof. ∎

######  Remark 3.4.

It is clear that constants $L_{1}$, $L_{2}$, $L_{3}$ and $L_{4}$ can be chosen
jointly for $Y$ and $\widehat{Y}$, so that the inequalities

$\varphi(t)+\frac{L_{1}}{(L_{2}+\Lambda)^{\alpha}}\leq Y(t)\leq
L_{3}+L_{4}\Lambda,\quad t\in[0,T],$

and

$\varphi(t_{n})+\frac{L_{1}}{(L_{2}+\Lambda)^{\alpha}}\leq\widehat{Y}(t_{n})\leq
L_{3}+L_{4}\Lambda,\quad n=0,1,...,N,$

hold simultaneously with probability 1.

Next, corresponding to Lemma 2.6 in the two-sided case, $Y$ enjoys Hölder
continuity with the Hölder constant being integrable provided that $\Lambda$
has moments of sufficiently high order. This is summarized in the lemma below.

######  Lemma 3.5.

Let $Z$ satisfy (Z1)–(Z2) and assumptions (A1)–(A3) hold.

  * 1)

There exists a positive random variable $\Upsilon$ such that with probability
1

$|Y(t)-Y(s)|\leq\Upsilon|t-s|^{\lambda},\quad t,s\in[0,T].$

  * 2)

If, for some $r\geq 1$,

$\mathbb{E}\left[\Lambda^{\frac{r(p+\gamma\lambda+\lambda-1)}{\gamma\lambda+\lambda-1}}\right]<\infty,$
(3.7)

where $\lambda$ and $\Lambda$ are from (Z2), $p$ is from (A2) and $\gamma$ is
from (A3), then one can choose $\Upsilon$ such that

$\mathbb{E}[\Upsilon^{r}]<\infty.$

###### Proof.

By (1.4),

$Y(t)\geq\varphi(t)+\frac{L_{1}}{(L_{2}+\Lambda)^{\frac{1}{\gamma\lambda+\lambda-1}}}\quad
a.s.,$

i.e. with probability 1 $(t,Y(t))\in\mathcal{D}_{\frac{1}{\xi}}$, $t\in[0,T]$,
where

$\xi:=\frac{(L_{2}+\Lambda)^{\frac{1}{\gamma\lambda+\lambda-1}}}{L_{1}}$ (3.8)

and $\mathcal{D}_{\frac{1}{\xi}}$ is defined in (1.1). Denote
$\phi(t):=\varphi(t)+1$ and notice that
$(t,\phi(t))\in\mathcal{D}_{\frac{1}{\xi}}$, $t\in[0,T]$, since
$\frac{1}{\xi}\leq Y(0)-\varphi(0)$. Thus, using the same arguments as applied
in (2.7), we can write that, with probability 1, for any $0\leq s<t\leq T$:

$|Y(t)-Y(s)|\leq
c_{1}\xi^{p}\int_{s}^{t}|Y(u)-\phi(u)|du+\max_{u\in[0,T]}|b(u,\phi(u))|(t-s)+\Lambda(t-s)^{\lambda},$

where $c_{1}$ is from (A2). Now, again by (1.4),

$Y(t)\leq L_{3}+L_{4}\Lambda\quad a.s.,$

hence with probability 1

$\displaystyle|Y(t)-Y(s)|$ $\displaystyle\leq
c_{1}\xi^{p}\int_{s}^{t}|Y(u)-\phi(u)|du+\max_{u\in[0,T]}|b(u,\phi(u))|(t-s)+\Lambda(t-s)^{\lambda}$
$\displaystyle\leq
c_{1}\xi^{p}(L_{3}+L_{4}\Lambda)(t-s)+c_{1}\xi^{p}\max_{u\in[0,T]}|\phi(u)|(t-s)$
$\displaystyle\quad+\max_{u\in[0,T]}|b(u,\phi(u))|(t-s)+\Lambda(t-s)^{\lambda}$
$\displaystyle\leq C(1+\xi^{p}\Lambda+\xi^{p}+\Lambda)(t-s)^{\lambda},$

where $C$ is a positive constant. Now one can put

$\Upsilon:=C(1+\xi^{p}\Lambda+\xi^{p}+\Lambda)$ (3.9)

and observe that

$\mathbb{E}[\Upsilon^{r}]<\infty$

whenever (3.7) holds. ∎

######  Corollary 3.6.

Using Lemma 3.3 and following the proof of Lemma 3.5, it is easy to obtain
that, for any partition with the mesh satisfying

$\max\left\\{c_{3},\frac{c_{1}}{(Y(0)-\varphi(0))^{p}}\right\\}\Delta_{N}<1$
(3.10)

there is a random variable $\Upsilon$ independent of the partition such that
with probability 1

$|\widehat{Y}(t_{k})-\widehat{Y}(t_{n})|\leq\Upsilon|t_{k}-t_{n}|^{\lambda},\quad
k,n=0,...,N.$ (3.11)

Furthermore, just like in Lemma 2.6, for $r>0$

$\mathbb{E}[\Upsilon^{r}]<\infty$

provided that

$\mathbb{E}\left[\Lambda^{\frac{r(p+\gamma\lambda+\lambda-1)}{\gamma\lambda+\lambda-1}}\right]<\infty.$

Finally, such $\Upsilon$ can be chosen jointly for $Y$ and $\widehat{Y}$, so
that

$|Y(t)-Y(s)|\leq\Upsilon|t-s|^{\lambda},\quad t,s\in[0,T],$

holds simultaneously with (3.11) with probability 1.

######  Lemma 3.7.

Let $Z$ satisfy (Z1)–(Z2), Assumptions A hold and the mesh of the partition
$\Delta_{N}$ satisfy (3.1).

  * 1)

For any $r\geq 1$, there exists a positive random variable $\mathcal{C}_{4}$
that does not depend on the partition such that

$\sup_{k=0,1,...,N}|Y(t_{k})-\widehat{Y}(t_{k})|^{r}\leq\mathcal{C}_{4}\Delta_{N}^{\lambda
r}\quad a.s.$

  * 2)

If, additionally,

$\mathbb{E}\left[\Lambda^{\frac{r(2p+\gamma\lambda+\lambda-1)}{\gamma\lambda+\lambda-1}}\right]<\infty,$
(3.12)

where $\lambda$ and $\Lambda$ are from (Z2), $p$ is from (A2) and $\gamma$ is
from (A3), then one can choose $\mathcal{C}_{4}$ such that
$\mathbb{E}[\mathcal{C}_{4}]<\infty$, i.e. there exists a deterministic
constant $C$ that does not depend on the partition such that

$\mathbb{E}\left[\sup_{k=0,1,...,N}|Y(t_{k})-\widehat{Y}(t_{k})|^{r}\right]\leq
C\Delta_{N}^{\lambda r}.$

###### Proof.

Following the proof of Lemma 2.8, one can easily obtain that for any
$n=0,1,...,N$

$|Y(t_{n})-\widehat{Y}(t_{n})|\leq
C\left(\sum_{i=1}^{n}\int_{t_{i-1}}^{t_{i}}\left|b(s,Y(s))-b(t_{i},Y(t_{i}))\right|ds\right)^{r}.$

Next, note that $(t,Y(t))\in\mathcal{D}_{\frac{1}{\xi}}$, where $\xi$ is
defined by (3.8), so, by (A2) and Lemma 3.5,

$\displaystyle\left(\sum_{i=1}^{n}\int_{t_{i-1}}^{t_{i}}\left|b(s,Y(s))-b(t_{i},Y(t_{i}))\right|ds\right)^{r}$
$\displaystyle\leq
C\xi^{pr}\left(\sum_{i=1}^{n}\int_{t_{i-1}}^{t_{i}}\left|s-t_{i}\right|^{\lambda}ds\right)^{r}+C\xi^{pr}\left(\sum_{i=1}^{n}\int_{t_{i-1}}^{t_{i}}\left|Y(s)-Y(t_{i})\right|ds\right)^{r}$
$\displaystyle\leq
C\xi^{pr}\left(\sum_{i=1}^{n}\int_{t_{i-1}}^{t_{i}}\left|s-t_{i}\right|^{\lambda}ds\right)^{r}+C\xi^{pr}\Upsilon^{r}\left(\sum_{i=1}^{n}\int_{t_{i-1}}^{t_{i}}\left|s-t_{i}\right|^{\lambda}ds\right)^{r}$
$\displaystyle=C\xi^{pr}(1+\Upsilon^{r})\left(\sum_{i=1}^{n}\frac{1}{(1+\lambda)}\Delta_{N}^{1+\lambda}\right)^{r}$
$\displaystyle\leq C\xi^{pr}(1+\Upsilon^{r})\Delta_{N}^{\lambda r},$

i.e.

$\sup_{n=0,...,N}|Y(t_{n})-\widehat{Y}(t_{n})|^{r}\leq
C\xi^{pr}(1+\Upsilon^{r})\Delta_{N}^{\lambda r}.$ (3.13)

In order to conclude the proof, it remains to notice that (3.8), (3.9) and
(3.12) imply that

$\mathbb{E}\left[\xi^{pr}(1+\Upsilon^{r})\right]<\infty.$

∎

Now we are ready to formulate the two main results of this section.

######  Theorem 3.8.

Let $Z$ satisfy (Z1)–(Z2), Assumptions A hold and the mesh of the partition
$\Delta_{N}$ satisfy (3.10).

  * 1)

For any $r\geq 1$, there exists a random variable $\mathcal{C}_{5}$ that does
not depend on the partition such that

$\sup_{t\in[0,T]}|Y(t)-\widehat{Y}(t)|^{r}\leq\mathcal{C}_{5}\Delta_{N}^{\lambda
r}\quad a.s.$

  * 2)

If, additionally,

$\mathbb{E}\left[\Lambda^{\frac{r(2p+\gamma\lambda+\lambda-1)}{\gamma\lambda+\lambda-1}}\right]<\infty,$

where $\lambda$ and $\Lambda$ are from (Z2), $p$ is from (A2) and $\gamma$ is
from (A3), then one can choose $\mathcal{C}_{5}$ such that
$\mathbb{E}[\mathcal{C}_{5}]<\infty$, i.e. there exists a deterministic
constant $C$ that does not depend on the partition such that

$\mathbb{E}\left[\sup_{t\in[0,T]}|Y(t)-\widehat{Y}(t)|^{r}\right]\leq
C\Delta_{N}^{\lambda r}.$

###### Proof.

The proof is similar to the one of Theorem 2.9 but instead of Lemmas 2.6, 2.8
and bound (2.14) one should apply Lemmas 3.5, 3.7 and bound (3.13). ∎

######  Theorem 3.9.

Let $Z$ satisfy (Z1)–(Z2), Assumptions A hold and the mesh of the partition
$\Delta_{N}$ satisfy (3.10).

  * 1)

For any $r\geq 1$, there exists a random variable $\mathcal{C}_{6}$ that does
not depend on the partition such that

$\sup_{n=0,1,...,N}\left|\frac{1}{Y(t_{n})-\varphi(t_{n})}-\frac{1}{\widehat{Y}(t_{n})-\varphi(t_{n})}\right|^{r}\leq\mathcal{C}_{6}\Delta_{N}^{\lambda
r}\quad a.s.$

  * 2)

If, additionally,

$\mathbb{E}\left[\Lambda^{\frac{r(2+2p+\gamma\lambda+\lambda-1)}{\gamma\lambda+\lambda-1}}\right]<\infty,$
(3.14)

where $\lambda$ and $\Lambda$ are from (Z2), $p$ is from (A2) and $\gamma$ is
from (A3), then one can choose $\mathcal{C}_{6}$ such that
$\mathbb{E}[\mathcal{C}_{6}]<\infty$, i.e. there exists a deterministic
constant $C$ that does not depend on the partition such that

$\mathbb{E}\left[\sup_{n=0,1,...,N}\left|\frac{1}{Y(t_{n})-\varphi(t_{n})}-\frac{1}{\widehat{Y}(t_{n})-\varphi(t_{n})}\right|^{r}\right]\leq
C\Delta_{N}^{\lambda r}.$

###### Proof.

The proof is similar to Theorem 2.10 and is omitted. ∎

##  4 Examples and simulations

The algorithms presented in (2.2) and (3.2) imply that, in order to generate
$\widehat{Y}(t_{n+1})$, one has to solve an equation that potentially can be
challenging from the computational point of view. However, in some cases that
are relevant for applications this equation has a simple explicit solution.

Regarding the numerical examples that follow, we remark that:

  * 1)

all the simulations are performed in the R programming language on the system
with Intel Core i9-9900K CPU and 64 Gb RAM;

  * 2)

in order to simulate paths of fractional Brownian motion, R package somebm is
used;

  * 3)

in Example 4.3, discrete samples of the multifractional Brownian motion (mBm)
values are simulated using the Cholesky decomposition of the corresponding
covariance matrix (for covariance structure of the mBm, see e.g. [5,
Proposition 4]) and the R package nleqslv is used for solving (2.2)
numerically.

######  Example 4.1.

(_Generalized CIR processes_) Let $\varphi\equiv 0$, $Z$ satisfy (Z1)–(Z2)
with $\lambda\in\left(\frac{1}{2},1\right)$, $Y(0)$, $\kappa_{1}$,
$\kappa_{2}>0$, $\gamma>\frac{1}{\lambda}-1$ be given and
$\\{Y(t),~{}t\in[0,T]\\}$ satisfy the SDE of the form

$Y(t)=Y(0)+\int_{0}^{t}\left(\frac{\kappa_{1}}{Y(s)}-\kappa_{2}Y(s)\right)ds+Z(t),\quad
t\in[0,T].$ (4.1)

This process fits into the framework of Section 3 and the equation for
$\hat{Y}(t_{k+1})$ from (3.2) reads as follows:

$\widehat{Y}(t_{k+1})=\widehat{Y}(t_{k})+\left(\frac{\kappa_{1}}{\widehat{Y}(t_{k+1})}-\kappa_{2}\widehat{Y}(t_{k+1})\right)\Delta_{N}+Z(t_{k+1})-Z(t_{k}).$

It is easy to see that it has a unique positive solution

$\widehat{Y}(t_{k+1})=\frac{\widehat{Y}(t_{k})+(Z(t_{k+1})-Z(t_{k}))+\sqrt{\Big{(}\widehat{Y}(t_{k})+(Z(t_{k+1})-Z(t_{k}))\Big{)}^{2}+4\kappa_{1}\Delta_{N}(1+\kappa_{2}\Delta_{N})}}{2(1+\kappa_{2}\Delta_{N})}.$

Fig. 1 contains 10 sample paths of the process (4.1) driven by a fractional
Brownian motion with $H=0.7$. In all simulation we take $N=10000$, $T=1$ and
$Y(0)=1=\kappa_{1}=\kappa_{2}=1$. Based on 10000 simulations, the average time
for simulating one path is 0.005388308 seconds.

Figure 1: Ten sample paths of (4.1) generated using the backward Euler
approximation scheme; $N=10000$, $T=1$, $Y(0)=1=\kappa_{1}=\kappa_{2}=1$, $Z$
is a fractional Brownian motion with $H=0.7$.

Note that the drift-implicit Euler scheme for (4.1) driven by the fractional
Brownian motion was the main subject of [18] and [22] but in both these works
the convergence of $\hat{Y}$ to $Y$ is established only on $[0,T]$ with $T$
being small (see e.g. [18, Eq. (8) and Remark 3.1]). Our results fill this gap
and convergence holds on arbitrary $[0,T]$ for any model parameters.

######  Example 4.2.

(Sandwiched process of the TSB type) Consider a sandwiched SDE of the form

$Y(t)=Y(0)+\int_{0}^{t}\left(\frac{\kappa_{1}}{Y(s)-\varphi(s)}-\frac{\kappa_{2}}{\psi(s)-Y(s)}-\kappa_{3}Y(s)\right)ds+Z(t),\quad
t\in[0,T],$ (4.2)

where $Z$ satisfies (Z1)–(Z2) with $\lambda\in\left(\frac{1}{2},1\right)$.
This equation fits into the framework of Section 3 and the scheme (2.2) leads
to $N$ cubic equations of the form

$\widehat{Y}^{3}(t_{n+1})+B_{2,n}\widehat{Y}^{2}(t_{n+1})+B_{1,n}\widehat{Y}(t_{n+1})+B_{0,n}=0,\quad
n=0,...,N-1,$ (4.3)

where

$\displaystyle B_{0,n}$
$\displaystyle:=\frac{-\varphi(t_{n+1})\psi(t_{n+1})\left(\widehat{Y}(t_{n})+\Delta
Z_{n}\right)+\Delta_{N}\left(\kappa_{1}\psi(t_{n+1})+\kappa_{2}\varphi(t_{n+1})\right)}{1+\Delta_{N}\kappa_{3}},$
$\displaystyle B_{1,n}$
$\displaystyle:=\varphi(t_{n+1})\psi(t_{n+1})+\frac{(\varphi(t_{n+1})+\psi(t_{n+1}))(\widehat{Y}(t_{n})+\Delta
Z_{n})-\Delta_{N}(\kappa_{1}+\kappa_{2})}{1+\Delta_{N}\kappa_{3}},$
$\displaystyle B_{2,n}$
$\displaystyle:=-\varphi(t_{n+1})-\psi(t_{n+1})-\frac{\widehat{Y}(t_{n})+\Delta
Z_{n}}{1+\Delta_{N}\kappa_{3}},$

Note this equation can be solved explicitly using, e.g., the celebrated
Cardano method. Namely, define

$Q_{1,n}:=B_{1,n}-\frac{B_{2,n}^{2}}{3},\quad
Q_{2,n}:=\frac{2B_{2,n}}{27}-\frac{B_{2,n}B_{1,n}}{3}+B_{0,n}$

and put

$Q_{n}=\left(\frac{Q_{1,n}}{3}\right)^{3}+\left(\frac{Q_{2,n}}{2}\right)^{2},$
$\alpha_{n}:=\sqrt[3]{-\frac{Q_{2,n}}{2}+\sqrt{Q_{n}}},\quad\beta_{n}:=\sqrt[3]{-\frac{Q_{2,n}}{2}-\sqrt{Q_{n}}},$

where among possible complex values of $\alpha_{n}$ and $\beta_{n}$ one should
take those for which $\alpha_{n}\beta_{n}=-\frac{Q_{1,n}}{3}$. Then the three
roots of the cubic equation (4.3) are

$\displaystyle y_{1,n}$ $\displaystyle=\alpha_{n}+\beta_{n},$ $\displaystyle
y_{2,n}$
$\displaystyle=-\frac{\alpha_{n}+\beta_{n}}{2}+i\frac{\alpha_{n}-\beta_{n}}{2}\sqrt{3},$
$\displaystyle y_{3,n}$
$\displaystyle=-\frac{\alpha_{n}+\beta_{n}}{2}-i\frac{\alpha_{n}-\beta_{n}}{2}\sqrt{3},$

and $\widehat{Y}(t_{n+1})$ is equal to the root which belongs to
$(\varphi(t_{n+1}),\psi(t_{n+1}))$ (note that there is exactly one root in
that interval).

Fig. 2 contains 10 sample paths of the process (4.2) driven by a fractional
Brownian motion with $H=0.7$. In all simulation we take $N=10000$, $T=1$ and
$Y(0)=0$, $\kappa_{1}=\kappa_{2}=\frac{1}{2}$, $\kappa_{3}=0$ (this case
corresponds to the TSB equation described in Example 1.7). Simulation is
performed by direct implementation of the Cardano’s method in R; based on
10000 simulations, the average time for simulating one path is 0.03700142
seconds.

Figure 2: Ten sample paths of (4.2) generated using the backward Euler
approximation scheme; $N=10000$, $T=1$, $Y(0)=0$,
$\kappa_{1}=\kappa_{2}=\frac{1}{2}$, $\kappa_{3}=0$, $Z$ is a fractional
Brownian motion with $H=0.7$.

In both Examples 4.1 and 4.2, equations for computing $\widehat{Y}$ could be
explicitly solved but the Hölder continuity of the noise could not be less
then $1/2$. The next example shows that the drift-implicit Euler scheme can be
applied in the rough case as well.

######  Example 4.3.

(Sandwiched process driven by multifractional Brownian motion) Consider the
sandwiched SDE of the form

$Y(t)=Y(0)+\int_{0}^{t}\left(\frac{\kappa_{1}}{(Y(s)-\varphi(s))^{4}}-\frac{\kappa_{2}}{(\psi(s)-Y(s))^{4}}\right)ds+Z(t),\quad
t\in[0,T].$ (4.4)

In this case, Theorem 1.1 guarantees existence and uniqueness of the solution
for $\lambda$-Hölder $Z$ with $\lambda>\frac{1}{5}$ (note that this equation
fits the framework of Example 1.8 from Section 1). On Fig. 3, one can see
paths of the process (4.4) with $\kappa_{1}=\kappa_{2}=1$,
$\varphi(t)=\sin(10t)$, $\psi(t)=\sin(10t)+2$ driven by multifractional
Brownian motion (mBm) with functional Hurst parameter
$H(t)=\frac{1}{5}\sin(2\pi t)+\frac{1}{2}$ (note that the lowest value of the
functional Hurst parameter is $H\left(\frac{3}{4}\right)=0.3$). For more
details on mBm, see [5] as well as [17, Lemma 3.1] for results on Hölder
continuity of its paths. Based on 10000 simulations, the average time for
simulating one path is 0.4968714 seconds.

Figure 3: A sample path of (4.4) generated using the backward Euler
approximation scheme; $N=10000$, $T=1$, $Y(0)=1$, $\kappa_{1}=\kappa_{2}=1$,
$\varphi(t)=\sin(10t)$, $\psi(t)=\sin(10t)+2$, $Z$ is a multifractional
Brownian motion with functional Hurst parameter $H(t)=\frac{1}{5}\sin(2\pi
t)+\frac{1}{2}$.

## Acknowledgements

The present research is carried out within the frame and support of the
ToppForsk project nr. 274410 of the Research Council of Norway with title
STORM: Stochastics for Time-Space Risk Models. The second author was supported
by Japan Science and Technology Agency CREST JPMJCR21.

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