#  Strong convergence of the exponential Euler scheme for SDEs with
superlinear growth coefficients and one-sided Lipschitz drift

Mireille<EMAIL_ADDRESS>Université Côte d’Azur, Inria, France
Kerlyns<EMAIL_ADDRESS>Departamento de Ingeniería Matemática,
Universidad de Concepción, Chile

###### Abstract

We consider the problem of the discrete-time approximation of the solution of
a one-dimensional SDE with piecewise locally Lipschitz drift and continuous
diffusion coefficients with polynomial growth. In this paper, we study the
strong convergence of a (semi-explicit) exponential-Euler scheme previously
introduced in Bossy et al. (2021). We show the usual 1/2 rate of convergence
for the exponential-Euler scheme when the drift is continuous. When the drift
is discontinuous, the convergence rate is penalised by a factor $\varepsilon$
decreasing with the time-step. We examine the case of the diffusion
coefficient vanishing at zero, which adds a positivity preservation condition
and a convergence analysis that exploits the negative moments and exponential
moments of the scheme with the help of change of time technique introduced in
Berkaoui et al. (2008). Asymptotic behaviour and theoretical stability of the
exponential scheme, as well as numerical experiments, are also presented.

###### Contents

  1. 1 Introduction
    1. 1.1 Summary of the contributions and plan of the paper
    2. 1.2 Strong wellposedness for the solution to SDE (1.1)
  2. 2 The exponential scheme, moment bounds and local error estimations
    1. 2.1 Local error and negative moments for the Exp-EM scheme
    2. 2.2 Exponential moment for the exponential scheme
  3. 3 The strong rate of convergence for the exponential scheme
    1. 3.1 The Itô formula for the error process
    2. 3.2 Preliminary estimations for the treatment of the discontinuity of $b$
    3. 3.3 Proof of Theorem 3.1
    4. 3.4 Strong convergence rate of the unstopped scheme
    5. 3.5 Asymptotic behaviour and stability of the Exp-EM scheme on a prototypical case
  4. 4 Numerical experiments
  5. A Appendix
    1. A.1 Proof of Theorem 3.5
    2. A.2 Technical lemmas and proofs
    3. A.3 Proof of Proposition 1.3

##  1 Introduction

In this paper, we analyse the strong convergence of the exponential Euler
scheme for the simulation of the solution of SDE with coefficients having
superlinear growth. More precisely, we consider the one-dimensional SDE

$dX_{t}=b(X_{t})dt+\sigma(X_{t})dW_{t},~{}X_{0}=x_{0}>0,$ (1.1)

where $(W_{t};0\leq t\leq T)$ is a standard Brownian motion on the probability
space $(\Omega,\mathcal{F},\mathbb{P})$ equipped with its natural filtration
$(\mathcal{F}_{t};\,0\leq t\leq T)$. The diffusion coefficient
$\sigma:[0,+\infty)\rightarrow\mathbb{R}$ is assumed to be continuous with
polynomial growth:

$\sigma^{2}(x)\leq{\text{\scriptsize{$\Upsigma$}}}^{2}x^{2\alpha},\quad\text{for
}x\in\mathbb{R}^{+},\quad\text{ for some
}{\text{\scriptsize{$\Upsigma$}}}>0\text{ and }\alpha>1.$ (1.2)

In particular, we consider the situation where $\sigma(0)=0$, and
wellposedness hypotheses leading to a positive solution to (1.1). We show that
the existence and control of moments of the positive solution to (1.1) is
granted when the drift $b:[0,+\infty)\rightarrow\mathbb{R}$ is assumed to be
piecewise locally Lipschitz function, with a polynomial growth bound of the
form (see Proposition 1.3):

$b(x)\leq b(0)+B_{1}x-B_{2}x^{{\beta}},~{}\forall
x\in\mathbb{R}^{+},\quad\text{with }{\beta}\geq 2\alpha-1>1,$ (1.3)

for some positive constants $b(0),B_{1}$ and $B_{2}$.

Under assumptions ensuring the wellposedness and strict positivity of $X$, the
exponential scheme strategy draws on the dynamics artificially transformed
into a semi-linear form (1.4). A semi-exact scheme in exponential form is then
introduced, maintaining the strict positivity of the approximate solution.
Moreover, it possesses identical finite moments as the exact solution under
the same set of sufficient conditions.

We refer to Bossy et al. (2021) for some specific examples of applications
motivating the study of the exponential scheme for (1.1), as well as for
elements of review of the literature on numerical schemes adapted to SDEs with
both non-Lipschitz drift and diffusion coefficients. We complete this review
with works address the strong error for similar SDE situations. First, we
emphasise that Hutzenthaler et al. (2010) established the $L^{p}$-strong
divergence, for $p\in[1,+\infty)$, related to the Euler-Maruyama scheme for
SDEs with both drift and diffusion satisfying some superlinear growth
condition. In Hutzenthaler et al. (2012), the authors proposed a time-explicit
tamed-Euler scheme to overcome this _divergence_ problem of the Euler
approximation, based on renormalised-increments to the scheme. Recently
Hutzenthaler and Jentzen (2020) proved the ${1}/{2}$ rate of the
$L^{p}$-strong convergence for the tamed-Euler scheme for a family of SDE that
includes some locally Lipschitz cases for both continuous drift and diffusion
coefficients. A truncated version of the Euler-Maruyama scheme has also been
introduced and adapted for positivity preserving approximation (see Mao et al.
(2021) and references therein).

While Bossy et al. (2021) introduces the exponential Euler scheme and analyses
its convergence rate in the weak sense, this paper aims to prove a convergence
rate of 1/2 for the $L^{p}(\Omega)$-$\sup_{t\in[0,T]}$ error for the
exponential scheme, as described below. The main objectives are as follows:
(i) To expand the $C^{4}$ regularity requirement for the drift $b$ in weak
error analysis and start to encompass discontinuous case. (ii) To enhance the
control condition concerning the growth and domination of the drift and
diffusion to achieve convergence rates. (iii) To explore and expand the
proposition of schemes applicable to contexts of superlinear coefficients.
Rather than a priori control of the schema by a tamed/truncated strategy, we
aim to identify a robust a posteriori threshold for the approximated process,
defining a value range for the scheme that ensures its convergence. By
obtaining a threshold that expands as we refine the time-step, we pave the way
for exploring natural and explicit strategies for adaptive time-step schemes
handling increasingly explosive cases of SDEs.

#### Exponential scheme

The exponential-Euler Maruyama scheme (Exp-EM for short), originates from
rewriting the SDE (1.1) into a quasi-linear SDE, taking part of the strict
positivity of the solution

$dX_{t}=X_{t}\Big{(}\frac{b(X_{t})-b(0)}{X_{t}}dt+\frac{\sigma(X_{t})}{X_{t}}dW_{t}\Big{)}+b(0)dt,\quad
X_{0}=x_{0}>0.$ (1.4)

The semi-linear integration produces, for an homogeneous $N$-partition of the
time interval $[0,T]$ with time-step $\Delta t=t_{n+1}-t_{n}$, the
approximation scheme:

$\overline{X}_{t_{n+1}}=\overline{X}_{t_{n}}\exp\left\\{\frac{\sigma(\overline{X}_{t_{n}})}{\overline{X}_{t_{n}}}\;(W_{t_{n+1}}-W_{t_{n}})+\left(\frac{b(\overline{X}_{t_{n}})-b(0)}{\overline{X}_{t_{n}}}-\frac{1}{2}\frac{\sigma^{2}(\overline{X}_{t_{n}})}{\overline{X}_{t_{n}}^{2}}\right)\Delta
t\right\\}+b(0)\Delta t,\quad\overline{X}_{0}=x_{0},$ (1.5)

that preserves the positiveness of the solution. We refer the reader to Bossy
et al. (2021) for a detailed construction of (1.5) in the particular case
$\sigma(x)={\text{\scriptsize{$\Upsigma$}}}x^{\alpha}$. As pointed out, a
significant advantage of the Exp-EM scheme is its ability to preserve the
conditions under which the boundedness of moments holds compared to the
continuous time process (see Proposition 1.3 and Lemma 2.1) in contrast with
the classical Euler-Maruyama time-discretisation for some locally Lipschitz
coefficients (see, for instance, Hutzenthaler et al. (2010)).

The preservation properties (positivity, moments) and the ease of
computational handling (both numerical and analytical) make the Exp-EM version
a robust method that can be employed in diverse simulation problems, including
SPDE as proposed in Bréhier et al. (2023) and Erdogan and Lord (2023).
Specifically, it can prove highly beneficial in a splitting simulation
strategy aimed at accurately decomposing the equation to be solved into its
exact or semi-exact form.

While this paper primarily examines the Exp-EM scheme applied to 1D positive-
valued SDEs (involving possibly non-integer powers $\alpha$ and ${\beta}$), it
is worth noting that the Exp-EM scheme can readily be extended to handle SDEs
in $\mathbb{R}^{d}$, $d\geq 1$, featuring more generalized diffusion
coefficient class $\sigma(x)$. In such case, the Exp-EM scheme can be roughly
expressed as:

$dX_{t}=X_{t}\Big{(}((b(X_{t})-b(0))\div
X_{t})dt+((\sigma(X_{t})-\sigma(0))\div
X_{t})dW_{t}\Big{)}+b(0)dt+\sigma(0)dW_{t},$

where $b$ is $\mathbb{R}^{d}$\- valued and $\sigma$ is a $d\times r$ matrix,
consistent with the $r$-dimensional Brownian motion $W$. The notation $\div$
is used here for component-wise division. The Exp-EM scheme is then driven by
the following SDE

$\displaystyle d\overline{X}^{i}_{t}=$
$\displaystyle\left(\overline{X}_{t}-b(0){\delta(t)}-\sigma(0)\delta(W_{t})\right)^{i}\Big{(}\frac{(b(\overline{X}_{\eta(t)})-b(0))^{i}}{\overline{X}^{i}_{\eta(t)}}dt+\frac{(\sigma(\overline{X}_{\eta(t)})-\sigma(0))^{i,k}}{\overline{X}_{\eta(t)}^{i}}dW^{k}_{t}\Big{)}$
(1.6) $\displaystyle+(b(0)dt+\sigma(0)dW_{t})^{i},$

with $\delta(t)=t-\eta(t)$, and $\delta(W_{t})=W_{t}-W_{\eta(t)}$. The
simplicity of the 1D case is widely exploited in our proofs. Nevertheless, the
extension of the analysis to the multidimensional case is straightforward in
view of the main argument of stochastic time change to circumvent the problem
of stochastic Gronwall estimation. The extension to multidimensional drift
with discontinuity strongly depends on the topology of discontinuity sets.
Here also the 1D proof handles discontinuity points based on the ideas of
sojourn time estimation of $\mathbb{R}^{d}$-valued discrete time diffusion in
small ball (Bernardin et al. (2009)).

#### Preserving positivity

Preserving positivity in approximation schemes for multidimensional problems
with polynomial coefficients poses a significant challenge, particularly in
the context of epidemiological modelling. In this regard, we refer to the
works of Cai et al. (2023), Greenhalgh et al. (2016), and the references cited
therein.

The preserving positivity issue addressed here presents an additional
challenge for the scheme. Indeed, although chosen in the set of parameters
that ensure sufficient conditions for the Feller test on process $X$ (defining
the Feller zone of non-explosion and uniqueness), some parameters strongly
impact the behaviour of the exact and approximated processes. Large values of
the processes $X$ and $\overline{X}$ can abruptly return to zero. However, the
behaviour of $X$ in the vicinity of zero is well controlled in the Feller
zone, in particular when $b(0)=0$, allowing to upper bound all negative
moments of the exact process $X$ without additional restriction (see the first
item of Proposition 1.3). Unlike the control of positive moments, the control
of negative moments on the Exp-EM scheme requires constraining certain
parameters or limiting the maximum value of $\overline{X}$ below a threshold
(see Proposition 2.4).

Note that the $L^{p}$-hypothesis (or Hasmisnskii type condition), which is
commonly used to establish the existence of a superlinear diffusion in $L^{p}$
(see also Remark 1.4 and the discussion in Kelly and Lord (2022)), and that
typically provides also $L^{p}$-control for a reasonable approximation scheme,
is insufficient to get control on the scheme negative moments, at least for
the Exp-EM scheme.

###  1.1 Summary of the contributions and plan of the paper

##### Convergence results.

Beyond the usual motivations for the study of strong convergence of numerical
schemes – a crucial role for the convergence of parameter estimators for the
SDE model, as well as for the convergence of multilevel methods – a main
objective here is to broaden the prediction of the $L^{p}$-convergence of the
scheme concerning the SDE parameters, further than the theoretical conditions
established in Bossy et al. (2021) which seem far from being necessary. In
section 4 with the help of numerical tests, we explore the new theoretical
sufficient conditions for convergence obtained on the parameters of $X$.

The polynomial domination bounds (1.2)-(1.3) are deeply exploited in our proof
– together with polynomial growth 1 and piecewise locally Lipschitz behaviour
2 on $b$ – to identify explicit conditions on $p$ and on the set of parameters
$b(0)$, $B_{2}$, $\Upsigma$, $\beta$ and $\alpha$, for which the
$L^{p}(\Omega)$-theoretical rate is applicable.

In Theorem 3.1, we state that the convergence is of order $1/2$ for the
$L^{p}(\Omega)$-$[\sup_{t\in[0,T\wedge\overline{S}^{\Delta t}]}\cdot]$ norm,
where $\overline{S}^{\Delta t}$ is the first exit-time of the Exp-EM scheme
$\overline{X}$ from the set $(0,\Delta t^{-\frac{1}{{\beta}-1}}]$. This result
is obtained under a large set of parameters for the coefficients, including
the case $b(0)=0$. In such a case, the Exp-EM scheme (1.5) is still positive
but the control of negative moments of $\overline{X}$ escapes our analysis
(see Proposition 2.4). The threshold $\Delta t^{-\frac{1}{{\beta}-1}}$ is
going to infinity with $\Delta t$. It can be used in numerical experiments as
an indicator to locally decrease and adapt the time-step until the updated
value of $\overline{X}$ falls below the threshold. This approach will be the
subject of future research.

Convergence of the unstopped process is stated in Theorem 3.5, allowing an
additional domination bound on the derivative $b^{\prime}$ outside a compact.
This additional condition improves overpass the stochastic Gronwall estimation
step for which our technique requires the control of some exponential moments
for the Exp-EM process.

Theorems 3.1 and 3.5 are stated in Section 3. Their proofs are decomposed in
several steps that we summarise below.

The first step

lies in Section 2, where we analyse some properties of $\overline{X}$ such as
positive, negative and exponential moments, as well as rate of convergence of
the local error
$\|\overline{X}^{\gamma}_{t}-\overline{X}^{\gamma}_{\eta(t)}\|_{L^{p}(\Omega)}$.
Due to the polynomial dominance of the coefficients, the local error is
exploited for different values of $\gamma$ (see Proposition 2.2).

It is essential to highlight the importance of controlling exponential moments
of $\overline{X}$ in establishing the convergence conditions of the scheme,
particularly when the drift $b$ lacks adequate regularity. However, it is
important to note that we attain only partial control, as indicated in
Proposition 2.5, which specifically pertains to the stopped process
$\overline{X}_{\cdot\wedge\overline{S}^{\Delta t}}$.

Discontinuity.

Both Theorem 3.1 and 3.5 allow the drift $b$ to be discontinuous. The
convergence analysis consists of isolating the neighbourhood of this
discontinuity and showing that the time that the approximation process spends
in this small neighbourhood is itself small. This technique draws inspiration
from the methods employed in Bernardin et al. (2009) and relies primarily on
the occupation time formula and Bernstein’s inequality. This particular
treatment induces a $\varepsilon$ penalty in the convergence rate. Note that
the result of Bernardin et al. (2009) applies originally to the analysis of
weak convergence. We detail this aspect in the proof of Theorem 3.1 in Lemma
3.4.

Circumvent the stochastic Gronwall estimation step.

For the $L^{p}$-error analysis, a prerequisite is the boundedness of the
$p$-moments for the solution and for the scheme, that requires
$\mathds{1}_{\\{{\beta}=2\alpha-1\\}}\ 2p\leq
1+\tfrac{2B_{2}}{{\text{\scriptsize{$\Upsigma$}}}^{2}}$ (see Proposition 1.3
for $X$ and Lemma 2.1 for $\overline{X}$). However, the nonlinearity in the
error dynamics in the proof of Theorem 3.1 necessitates the application of a
stochastic Gronwall lemma, which requires the use of exponential moments. This
issue was already successfully addressed in Berkaoui et al. (2008) to obtain
the 1/2-strong rate of convergence associated with a Symmetrized Euler Scheme
for SDEs with diffusion of type $x^{\alpha}$, and $\alpha\in[\tfrac{1}{2},1)$.
Berkaoui et al. (2008) employs a time change step that necessitates the
control of exponential moments for $X^{2\alpha-1}$. Adapted to our case, this
requirement is now on the processes $(X\vee\overline{X})^{\beta-1}$ and
$\frac{1}{X}$. While the exponential moments are controlled for $X$ with
additional conditions identified on the parameter set, as highlighted in the
first step, the control over the scheme, such as it is, is only applicable to
the stopped process $\overline{X}_{\cdot\wedge\overline{S}^{\Delta t}}$. We
also emphasise that the technique of the change of time easily extends to
multidimensional cases.

Adding polynomial growth assumption on $b^{\prime}$ far from zero

allows extending the convergence proof to the unstopped $\overline{X}$ in
Theorem 3.5 improving the convergence rate and simplifying its proof (see
Appendix A.1).

##### Asymptotic stability.

We also explore the respective asymptotic behaviours of the trajectories of
the process and its approximation by the Exp-EM scheme, through a stability
analysis. In Section 3.5, for the polynomial coefficients case, we show that
the trajectories of $\overline{X}$ admit an interval of stability points, that
converges to the unique stability point of $X$.

##### Numerical experiments.

In Section 4, we assess the performance of the Exp-EM scheme in the case of
polynomial coefficients. The evaluation of the prototypical case confirms the
theoretical rate obtained in Theorem 3.5 and shows that the parameter
conditions for convergence are largely sufficient. Additionally, we will
examine discontinuous scenarios or cases with less regular behaviour covered
by Theorem 3.1 through numerical experiments. Lastly, we will illustrate the
stability of the numerical scheme established in Proposition 3.10.

### Notation

Throughout this paper, $T>0$ will refer to an arbitrary finite time horizon,
$C$ will denote a positive constant, possibly depending on the parameters of
the dynamic, which may change from line to line. Any process
$(Z_{t},t\in[0,T])$ will be simply denoted by $Z$.

For any $a,b\in\mathbb{R}$, $a\vee b$ and $a\wedge b$ denote respectively the
maximum and minimum between $a$ and $b$. Given the fixed non-negative discrete
time-step parameter $\Delta t$, we set $t_{k}=k\Delta t$ for $k\in\mathbb{N}$,
$\eta(t)=\Delta t\lfloor\frac{t}{\Delta t}\rfloor$, $\delta(t)=t-\eta(t)$, and
$\delta(W_{t})=W_{t}-W_{\eta(t)}$.

###  1.2 Strong wellposedness for the solution to SDE (1.1)

This study is restricted to the case where $\alpha>1$. The deterministic
initial position $x_{0}$ is strictly positive.

Going back to the proof proposed in Bossy et al. (2021), in Proposition 1.3
below we extend the sufficient conditions ensuring the strong wellposedness of
(1.1) as well as conditions ensuring moments. We first introduce the global
growth condition we consider on the drift $b$, contributing to the solution’s
positiveness and non-explosion: the growth rate around the origin is allowed
to go above the derivative of the squared diffusion, so $2\alpha-1$.

######  Hypothesis 1.

Polynomial Growths. (1).

  * $(b)$

The drift $b$ satisfies that $0\leq b(0)<+\infty$, and there exists some non-
negative constant $L_{\textit{G}}\,$ and ${\beta}\geq 2\alpha-1$ such that

$|b(x)-b(0)|\leq L_{\textit{G}}\,\ (x^{{\beta}}\vee x),~{}\forall
x\in\mathbb{R}^{+}.$ (1.7)

  * $(\sigma)$

The diffusion function $\sigma$ is locally Lipschitz continuous in
$\mathbb{R}^{+}$. Furthermore, there exists some non-negative constant
${\text{\scriptsize{$\Upsigma$}}}>0$ such that

$|\sigma(x)|^{2}\leq{\text{\scriptsize{$\Upsigma$}}}^{2}x^{2\alpha},~{}\forall
x\in\mathbb{R}^{+},$ (1.8)

and the map $x\mapsto\sigma^{2}(x)\,\sigma^{2}(x^{-1})$ is positively bounded
on the interval $(0,1)$.

Next, we introduce some conditions on the regularity of the drift $b$. First,
we introduce the notion of a piecewise Lipschitz function for the drift $b$,
allowing a finite number of discontinuities points. This kind of hypothesis
was recently considered in Müller-Gronbach et al. (2022) to extend the locally
Lipschitz condition to piecewise locally Lipschitz condition on $b$ in the
context of convergence rate analysis for the tamed Euler scheme.

######  Definition 1.1.

We say a function $f:\mathbb{R}^{+}\rightarrow\mathbb{R}$ is piecewise
Lipschitz continuous, if there are finitely many points
$0<\upchi_{1}<\ldots<\upchi_{m}<+\infty$ such that $f$ is Locally Lipschitz on
each of the intervals $(\upchi_{k},\upchi_{k+1})$, $k=0,\ldots m$, with
$\upchi_{0}=0$ and $\upchi_{m+1}=+\infty$.

The piecewise locally Lipschitz continuity, applied to the drift $b$, is
restricted to $\mathbb{R}^{+}$ since the Exp-EM scheme here preserves the
positiveness of the solution. Moreover, the already stated hypothesis 1
constrains the form of the local Lipschitz constant as follows:

######  Hypothesis 2.

Piecewise Locally Lipschitz condition. (2).

  * (i).

The drift $b$ is continuous at point $0$ (where the diffusion term vanishes).
Moreover $b$ is piecewise locally Lipschitz continuous, in the sense of
Definition 1.1: for any $k=0,\ldots,m$,

$|b(x)-b(y)|\leq C_{\upchi_{k}}(1+x^{{\beta}-1}\vee
y^{{\beta}-1})|x-y|,\quad\forall x,y\in(\upchi_{k},\upchi_{k+1}).$

We denote $\mathcal{C}_{\textit{max}}=\displaystyle{\max_{1\leq i\leq
m}}C_{\upchi_{i}}$.

  * (ii).

The discontinuity jumps are decreasing: for any $k=1,\ldots,m$,
$b(\upchi_{k}^{+})-b(\upchi_{k}^{-})<0.$

Note that that 2-(i) and 1 are compatible by taking the growth constant equal
to $\mathcal{C}_{\textit{max}}$.

Under 2–(i) existence and uniqueness up to an explosion time holds for the
solution of (1.1). We introduce a third condition that moves the explosion
time to infinity.

######  Hypothesis 3.

Controls on $b$ (3). There exist some non-negative constants $b(0)$, $B_{1}$,
$B_{2}$, and ${\beta}\geq 2\alpha-1$, such that,

$b(x)\leq b(0)+B_{1}\ x-B_{2}\ x^{{\beta}},~{}\forall x\in\mathbb{R}^{+}.$
(1.9)

Moreover, $b$ is one-sided locally Lipschitz: for all $x\neq y$ positive,
there exists a positive constant $L_{b_{\text{loc}}}$ such that

$\frac{b(x)-b(y)}{x-y}\leq L_{b_{\text{loc}}}(1+x^{{\beta}-1}\vee
y^{{\beta}-1}).$ (1.10)

######  Remark 1.2.

We would like to emphasise the consequences of 3 on the parameter set under
consideration.

Note that 3, in particular $B_{2}\geq 0$ is a sufficient condition for the
non-explosion of $X$, but far to be a necessary one. Typically, when
$b(x)=b(0)+B_{1}\ x-B_{2}\ x^{\alpha}$, the Feller test extends the non-
explosion condition to
${B_{2}}>-\tfrac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2}$.

The one-sided locally Lipschitz property of $b$ stated in (1.10) was already a
consequence of 2 with $L_{b_{\text{loc}}}=\mathcal{C}_{\textit{max}}$. The
additional condition (1.9) now imposes $\mathcal{C}_{\textit{max}}\geq
B_{1}\vee B_{2}$. At the same time, we expect in many cases a much smaller
constant $L_{b_{\text{loc}}}$ in condition (1.10), as again when
$b(x)=b(0)+B_{1}\ x-B_{2}\ x^{2\alpha-1}$, we have even a sort of negative
$L_{b_{\text{loc}}}$ with (applying Lemma A.2).

$\frac{b(x)-b(y)}{x-y}\leq B_{1}-B_{2}(x^{2\alpha-2}\vee y^{2\alpha-2}).$

The following proposition regroups the properties of $X$ solution of (1.1). It
is derived from a similar proposition presented in Bossy et al. (2021), but
extends both the regularity assumptions on $b$ and the theoretical control of
the exponential moments of the solution to (1.1). The proof is postponed to
Appendix A.3.

######  Proposition 1.3.

Assume 2–(i) , 1 and 3. Then there exists a unique (strictly) positive strong
solution $X$ to the SDE (1.1) with the following moment bounds:
${\footnotesize\bullet}$ Negative moments of any order: for all $q>0$, there
exists $C_{q}>0$, depending on $q$, but not on $x_{0}$, such that

$\sup_{t\in[0,T]}\mathbb{E}\big{[}X_{t}^{-q}\big{]}\leq~{}C_{q}(1+x_{0}^{-q}).$
(1.11)

${\footnotesize\bullet}$ Some positive moments: for all exponent $p>0$ such
that $\mathds{1}_{\\{2\alpha-1\\}}({\beta})\
p\leq\frac{1}{2}+\tfrac{B_{2}}{{\text{\scriptsize{$\Upsigma$}}}^{2}}$, there
exists $C_{p}>0$, depending on $p$, but not on $x_{0}$, such that

$\sup_{t\in[0,T]}\mathbb{E}\big{[}X_{t}^{2p}\big{]}\leq~{}C_{p}(1+x_{0}^{2p}).$
(1.12)

${\footnotesize\bullet}$ Some exponential moments: assume
$(b(0),\alpha)\notin(0,\frac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2})\times[1,\frac{3}{2}]$.

$\displaystyle\mbox{ When $b(0)\neq 0$, ~{}for all $\nu\in\mathbb{R}$,
}\quad\sup_{t\in[0,T]}\mathbb{E}\Big{[}\exp\left\\{\nu\int_{0}^{t}\dfrac{ds}{X_{s}}\right\\}\Big{]}<+\infty.$
(1.13) For all $\mu<B_{2}$, for all $v$ such that
$3\upsilon\left(3\upsilon-1\right)\frac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2}<B_{2}$,
$\displaystyle\quad\sup_{t\in[0,T]}\mathbb{E}\left[\exp\left\\{\mu\int_{0}^{t}X_{s}^{{\beta}-1}ds\right\\}\right]+\sup_{t\in[0,T]}\mathbb{E}\left[\exp\left\\{-\upsilon\int_{0}^{t}\frac{b(X_{s})}{X_{s}}ds\right\\}\right]<+\infty.$

######  Remark 1.4.

It is worth notice that the combination of 3 with the parameters condition
$\mathds{1}_{\\{2\alpha-1\\}}({\beta})\
p\leq\frac{1}{2}+\tfrac{B_{2}}{{\text{\scriptsize{$\Upsigma$}}}^{2}}$, to get
$p$-moment bounds could be written differently. For example, recently Müller-
Gronbach et al. (2022) or Kelly and Lord (2022)) consider the following
combination of $xb(x)$ and quadratic variation term:

$2xb(x)+(p-1)\sigma^{2}(x)\leq c(1+x^{2}).$

However, specifying the coefficients in 3 helps keep track of the assumptions
to control the sufficient moments for any $L^{p}$-norm and delineates the role
of each parameter in this control. More precisely, the non-negative constants
$B_{1}$ and $b(0)$ in the SDE (1.1) serve to propel the solution away from
zero. Conversely, the constant $B_{2}$ counteracts the solution’s growth
induced by the diffusion term, manifesting a mean-reverting effect. This
reasoning aligns with condition (3.14) of (Mao, 2011, Ch. 4), and the
conditions imposed on the parameters to control the moments are consistent
with those of Theorem 4.1 of (Mao, 2011, Ch. 4).

Using the Lenglart Inequality (see Lemma A.1), it is possible to directly
derive the moment estimation (1.12) with the $\sup$ inside the expectation.
Then the condition leading to (1.12) is strengthened with
$\mathds{1}_{\\{2\alpha-1\\}}({\beta})\
\frac{p}{\epsilon}\leq\frac{1}{2}+\frac{B_{2}}{{\text{\scriptsize{$\Upsigma$}}}^{2}}$,
for some $\epsilon\in(0,1)$. For example, considering
$\epsilon=\tfrac{p}{p+1}$, the condition is replaced by
$\mathds{1}_{\\{2\alpha-1\\}}({\beta})\
(p+1)\leq\frac{1}{2}+\frac{B_{2}}{{\text{\scriptsize{$\Upsigma$}}}^{2}}$, and

$\mathbb{E}\big{[}\sup_{t\in[0,T]}X_{t}^{2p}\big{]}\leq~{}(p+1)^{\frac{2p+1}{(p+1)^{2}}}p^{-\frac{p}{(p+1)^{2}}}C_{p+1}(1+x_{0}^{2(p+1)}).$

##  2 The exponential scheme, moment bounds and local error estimations

The scheme $(\overline{X}_{t_{n}};n\geq 1)$, defined in (1.5) admits the
following continuous version

$\overline{X}_{t}=b(0){\delta(t)}+\overline{X}_{\eta(t)}\exp\Big{\\{}\tfrac{\sigma(\overline{X}_{\eta(t)})}{\overline{X}_{\eta(t)}}(W_{t}-W_{\eta(t)})+\Big{(}\tfrac{b\left(\overline{X}_{\eta(t)}\right)-b(0)}{\overline{X}_{\eta(t)}}-\tfrac{1}{2}\tfrac{\sigma^{2}(\overline{X}_{\eta(t)})}{\overline{X}_{\eta(t)}^{2}}\Big{)}{\delta(t)}\Big{\\}},$
(2.1)

driven by the SDE

$d\overline{X}_{t}=\left(\overline{X}_{t}-b(0){\delta(t)}\right)\Big{(}\tfrac{b\left(\overline{X}_{\eta(t)}\right)-b(0)}{\overline{X}_{\eta(t)}}dt+\tfrac{\sigma(\overline{X}_{\eta(t)})}{\overline{X}_{\eta(t)}}dW_{t}\Big{)}+b(0)dt.$
(2.2)

From (2.1), almost surely,

$0\leq\left(1-\frac{{b(0)}\delta(t)}{\overline{X}_{t}}\right)\leq 1\ \text{ or
equivalently, }\ ~{}0\leq{b(0)\delta(t)}\leq{\overline{X}_{t}}.$ (2.3)

For the Exp-EM process $\overline{X}$, we bound the same order of $2p$ th-
moments than for $X$, with the same sufficient condition
$0\leq\mathds{1}_{\\{2\alpha-1\\}}({\beta})\,p\leq\frac{1}{2}+\frac{B_{2}}{{\text{\scriptsize{$\Upsigma$}}}^{2}}$:

######  Lemma 2.1.

Assume 3 and 1-($\sigma$). For all exponents $p>0$ such that
$\mathds{1}_{\\{2\alpha-1\\}}({\beta})\,p\leq\frac{1}{2}+\frac{B_{2}}{{\text{\scriptsize{$\Upsigma$}}}^{2}}$,
there exists a non-negative constant $C_{p}$, depending on $p$ but not on
$x_{0}>0$, such that

$\sup_{t\in[0,T]}\mathbb{E}\big{[}{\overline{X}_{t}^{2p}}\big{]}\leq
C_{p}(1+x_{0}^{2p}).$

###### Proof.

Considering $p>0$, we apply the Itô formula to $\overline{X}_{t}^{2p}$ (with a
localization argument in a compact set of $\mathbb{R}^{+*}$ omitted here or
simplicity):

$\displaystyle\mathbb{E}[\overline{X}_{t}^{2p}]$
$\displaystyle=x_{0}^{2p}+2pb(0)\mathbb{E}\big{[}\int_{0}^{t}\overline{X}_{s}^{2p-1}ds\big{]}$
$\displaystyle+2p\mathbb{E}\Big{[}\int_{0}^{t}\overline{X}_{s}^{2p-2}(\overline{X}_{s}-b(0){\delta(s)})\Big{\\{}\overline{X}_{s}\tfrac{b(\overline{X}_{\eta(s)})-b(0)}{\overline{X}_{\eta(s)}}+\frac{(2p-1)}{2}\left(\overline{X}_{s}-b(0){\delta(s)}\right)\tfrac{\sigma^{2}(\overline{X}_{\eta(s)})}{\overline{X}_{\eta(s)}^{2}}\Big{\\}}ds\Big{]},$

and thus, from 3, 1-($\sigma$) and inequality (2.3), we get

$\displaystyle\mathbb{E}[\overline{X}_{t}^{2p}]\leq$ $\displaystyle
x_{0}^{2p}+2pb(0)\mathbb{E}\big{[}\int_{0}^{t}\overline{X}_{s}^{2p-1}ds\big{]}$
$\displaystyle+2p\mathbb{E}\Big{[}\int_{0}^{t}\overline{X}_{s}^{2p-2}(\overline{X}_{s}-b(0){\delta(s)})\Big{\\{}B_{1}\overline{X}_{s}-B_{2}\overline{X}_{s}\overline{X}_{\eta(s)}^{{\beta}-1}+(2p-1)\frac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2}\left(\overline{X}_{s}-b(0){\delta(s)}\right)\overline{X}_{\eta(s)}^{2(\alpha-1)}\Big{\\}}ds\Big{]}$
$\displaystyle\leq$ $\displaystyle
x_{0}^{2p}+2pb(0)\mathbb{E}\big{[}\int_{0}^{t}\overline{X}_{s}^{2p-1}ds\big{]}+2B_{1}p\mathbb{E}\Big{[}\int_{0}^{t}\overline{X}_{s}^{2p}ds\Big{]}$
$\displaystyle+2p\mathbb{E}\Big{[}\int_{0}^{t}\overline{X}_{s}^{2p-1}(\overline{X}_{s}-b(0){\delta(s)})\Big{\\{}-B_{2}\overline{X}_{\eta(s)}^{{\beta}-1}+(2p-1)\tfrac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2}\overline{X}_{\eta(s)}^{2(\alpha-1)}\Big{\\}}ds\Big{]}.$

When ${\beta}=2\alpha-1$, the last term above is negative provided that
$(2p-1)\leq 2B_{2}/{\text{\scriptsize{$\Upsigma$}}}^{2}$. Otherwise, if
${\beta}>2\alpha-1$, the map
$z\mapsto(2p-1)\tfrac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2}z^{2\alpha-2}-B_{2}z^{{\beta}-1}$
is bounded from above. In the two cases, from Young inequality, there exists a
constant $C$, independent of $t$, such that

$\mathbb{E}[\overline{X}_{t}^{2p}]\leq
C(1+x_{0}^{2p})+C\mbox{$\int_{0}^{t}$}\mathbb{E}[\overline{X}_{s}^{2p}]ds.$

The proof ends by applying Gronwall’s inequality. ∎

###  2.1 Local error and negative moments for the Exp-EM scheme

We analyse the local error of the Exp-EM scheme
$\|\overline{X}_{t}^{\gamma}-\overline{X}_{\eta(t)}^{\gamma}\|_{L^{2p}(\Omega)}$
for some exponent $\gamma>0$. Interesting values of $\gamma$ are indeed
$\gamma=1$, and the exponents appearing in the Itô formula applied to
$\overline{X}$, typically $\gamma={\beta}-1$ and $\gamma=\alpha-1$. The
convergence rate of the local error for $\gamma=1$ is stated in Proposition
2.2 below. It is expected that the local error bound requires sufficient
control on positive moments of $\overline{X}$.

######  Proposition 2.2.

Assume 1 and 3. For all $p>0$, integer exponent such that

$\mathds{1}_{\\{2\alpha-1\\}}({\beta})\times\ p\
(2\alpha-1)\leq\frac{1}{2}+\frac{B_{2}}{{\text{\scriptsize{$\Upsigma$}}}^{2}}.$
(2.4)

Then, there exists a non-negative constant $C_{p}$, independent of $\Delta t$,
such that

$\sup_{t\in[0,T]}\|\overline{X}_{t}-\overline{X}_{\eta(t)}\|_{L^{2p}(\Omega)}\leq
C_{p}\ \Delta t^{\frac{1}{2}}.$ (2.5)

In order to prove the strong convergence of the exponential scheme we use the
local error approximation stated in Proposition 2.2. However, we show below a
more general lemma that is useful for proving the finiteness of the negative
moments of the scheme:

######  Lemma 2.3.

Assume 1 and 3. For $\gamma>0$, for all $p>0$ such that
$\mathds{1}_{\\{2\alpha-1\\}}({\beta})\,p\,(\gamma+2(\alpha-1))\leq\frac{1}{2}+\frac{B_{2}}{{\text{\scriptsize{$\Upsigma$}}}^{2}}$,
there exists a non-negative constant $C_{p}$, independent of $\Delta t$, such
that

$\sup_{t\in[0,T]}\|\overline{X}_{t}^{\gamma}-\overline{X}_{\eta(t)}^{\gamma}\|_{L^{2p}(\Omega)}\leq
C_{p}\left(\delta(t)^{\frac{1}{2}}\ \mathds{1}_{\\{0\\}}(b(0))\ +\
\delta(t)^{\frac{1}{2}\wedge\gamma}\ \mathds{1}_{(0,+\infty)}(b(0))\right).$
(2.6)

Let $\Delta t<1$. Then, for any $0\leq\epsilon<\frac{1}{2}\wedge\gamma$ and
all $p>0$ such that $\mathds{1}_{\\{2\alpha-1\\}}({\beta})\,(p+\Delta
t^{2p\epsilon})\,(\gamma+2(\alpha-1))\leq\frac{1}{2}+\frac{B_{2}}{{\text{\scriptsize{$\Upsigma$}}}^{2}}$,
we have

$\big{\|}\sup_{t\in[0,T]}|\overline{X}_{t}^{\gamma}-\overline{X}_{\eta(t)}^{\gamma}|\big{\|}_{L^{2p}(\Omega)}\leq
C_{p+\Delta t^{2p\epsilon}}\left(\delta(t)^{\frac{1}{2}-\epsilon}\
\mathds{1}_{\\{0\\}}(b(0))\ +\ \delta(t)^{(\frac{1}{2}\wedge\gamma)-\epsilon}\
\mathds{1}_{(0,+\infty)}(b(0))\right).$ (2.7)

###### Proof.

Applying the Itô formula to $\overline{X}^{\gamma}_{t}$, we get

$\displaystyle d\overline{X}^{\gamma}_{t}$
$\displaystyle=\gamma\overline{X}^{\gamma}_{t}\left(1-\tfrac{b(0)\delta(t)}{\overline{X}_{t}}\right)\left[\tfrac{b(\overline{X}_{\eta(t)})-b(0)}{\overline{X}_{\eta(t)}}+(\gamma-1)\frac{1}{2}\left(1-\tfrac{b(0)\delta(t)}{\overline{X}_{t}}\right)\tfrac{\sigma^{2}(\overline{X}_{\eta(t)})}{\overline{X}_{\eta(t)}^{2}}\right]dt+\gamma
b(0)\overline{X}_{t}^{\gamma-1}dt$
$\displaystyle\qquad+\gamma\overline{X}^{\gamma}_{t}\left(1-\tfrac{b(0)\delta(t)}{\overline{X}_{t}}\right)\tfrac{\sigma(\overline{X}_{\eta(t)})}{\overline{X}_{\eta(t)}}dW_{t}.$

So for $t$ such that $\delta(t)>0$,

$\displaystyle\|\overline{X}_{t}^{\gamma}-\overline{X}_{\eta(t)}^{\gamma}\|^{2p}_{L^{2p}(\Omega)}$
(2.8)
$\displaystyle=\mathbb{E}\left[\left\\{\gamma\int_{\eta(t)}^{t}\overline{X}^{\gamma}_{s}\left(1-\tfrac{b(0)\delta(s)}{\overline{X}_{s}}\right)\left[\tfrac{b(\overline{X}_{\eta(s)})-b(0)}{\overline{X}_{\eta(s)}}+(\gamma-1)\tfrac{1}{2}\left(1-\tfrac{b(0)\delta(s)}{\overline{X}_{s}}\right)\tfrac{\sigma^{2}(\overline{X}_{\eta(s)})}{\overline{X}_{\eta(s)}^{2}}\right]ds\right.\right.$
$\displaystyle\qquad\left.\left.+\gamma
b(0)\int_{\eta(t)}^{t}\overline{X}_{s}^{\gamma-1}ds+\gamma\int_{\eta(t)}^{t}\overline{X}^{\gamma}_{s}\left(1-\tfrac{b(0)\delta(s)}{\overline{X}_{s}}\right)\tfrac{\sigma(\overline{X}_{\eta(s)})}{\overline{X}_{\eta(s)}}dW_{s}\right\\}^{2p}\right]$
$\displaystyle\quad\leq C(p)\gamma^{2p}\
\mathbb{E}\left[\left\\{\int_{\eta(t)}^{t}\overline{X}^{\gamma}_{s}\left(1-\tfrac{b(0)\delta(s)}{\overline{X}_{s}}\right)\left[\tfrac{b(\overline{X}_{\eta(s)})-b(0)}{\overline{X}_{\eta(s)}}+(\gamma-1)\tfrac{1}{2}\left(1-\tfrac{b(0)\delta(s)}{\overline{X}_{s}}\right)\tfrac{\sigma^{2}(\overline{X}_{\eta(s)})}{\overline{X}_{\eta(s)}^{2}}\right]ds+b(0)\overline{X}_{s}^{\gamma-1}ds\right\\}^{2p}\right]$
$\displaystyle\qquad+C(p)\gamma^{2p}\
\mathbb{E}\left[\left(\int_{\eta(t)}^{t}\overline{X}^{\gamma}_{s}\left(1-\tfrac{b(0)\delta(s)}{\overline{X}_{s}}\right)\tfrac{\sigma(\overline{X}_{\eta(s)})}{\overline{X}_{\eta(s)}}dW_{s}\right)^{2p}\right]$
$\displaystyle\qquad:=E_{1}(t)+E_{2}(t),$

with $E_{1}$ and $E_{2}$ separating the Lebesgue from the Itô integrals. For
$E_{1}$, using (2.3) and 1, we get

$E_{1}(t)\leq
C(p)\gamma^{2p}~{}\mathbb{E}\left[\left(\int_{\eta(t)}^{t}\overline{X}^{\gamma}_{s}\left[L_{\textit{G}}\,(\overline{X}_{\eta(s)}^{{\beta}-1}\vee
1)+|\gamma-1|\tfrac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2}\,\overline{X}_{\eta(s)}^{2(\alpha-1)}\right]ds+b(0)\int_{\eta(t)}^{t}\overline{X}_{s}^{\gamma-1}ds\right)^{2p}\right].$

$\bullet$ When $\gamma\geq 1$, all the terms involved above are integrable for
$B_{2}$ large enough, and by Jensen inequality,

$\displaystyle E_{1}(t)$ $\displaystyle\leq
C(p)\delta(t)^{2p-1}\mathbb{E}\left[\int_{\eta(t)}^{t}\overline{X}^{2p\gamma}_{s}\left[L_{\textit{G}}\,(\overline{X}_{\eta(s)}^{{\beta}-1}\vee
1)+(\gamma-1)\tfrac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2}\,\overline{X}_{\eta(s)}^{2(\alpha-1)}\right]^{2p}ds\right]+C(p)~{}\delta(t)^{2p}\sup_{0\leq
t\leq T}\mathbb{E}\left[\overline{X}_{t}^{2p(\gamma-1)}\right]$
$\displaystyle\leq C(p)~{}\delta(t)^{2p}\left\\{1+\sup_{0\leq t\leq
T}\mathbb{E}\left[\overline{X}_{t}^{2p(\gamma+{\beta}-1)}\right]+b(0)\sup_{0\leq
t\leq T}\mathbb{E}\left[\overline{X}_{t}^{2p(\gamma-1)}\right]\right\\},$

since ${\beta}-1\geq 2(\alpha-1)>0$. Then (2.6) holds when $(B_{2},{\beta})$
are such that $\sup_{0\leq t\leq
T}\mathbb{E}[\overline{X}^{2p(\gamma+{\beta}-1)}_{t}]$ is bounded. Obviously,
when $b(0)=0$, the last upper-bound for $E_{1}$ holds for all $\gamma>0$.

$\bullet$ When $0<\gamma<1$ and $b(0)>0$, using (2.3)

$b(0)\int_{\eta(t)}^{t}\overline{X}_{s}^{\gamma-1}ds\leq
b^{\gamma}(0)\int_{\eta(t)}^{t}(s-\eta(t))^{\gamma-1}ds=b^{\gamma}(0)\int_{0}^{\delta(t)}s^{\gamma-1}ds=\frac{b^{\gamma}(0)}{\gamma}\delta(t)^{\gamma}.$

Then

$\displaystyle E_{1}(t)$ $\displaystyle\leq
C(p)\delta(t)^{2p-1}\mathbb{E}\left[\int_{\eta(t)}^{t}\overline{X}^{2p\gamma}_{s}\left[L_{\textit{G}}\,(\overline{X}_{\eta(s)}^{{\beta}-1}\vee
1)+|\gamma-1|\tfrac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2}\,\overline{X}_{\eta(s)}^{2(\alpha-1)}\right]^{2p}ds\right]+C(p)~{}\delta(t)^{2p\gamma}$
$\displaystyle\leq C(p)~{}\delta(t)^{2p}\left\\{1+\sup_{0\leq t\leq
T}\mathbb{E}\left[\overline{X}_{t}^{2p(\gamma+{\beta}-1)}\right]\right\\}+C(p)~{}\delta(t)^{2p\gamma}.$

For the bound of the term $E_{2}(t)$ in (2.8), we use usual arguments from Itô
calculus, with some updated constant $C(p)$, and using again (2.3) with 1,
obtaining

$E_{2}(t)\leq C(p)\ \gamma^{2p}\ {\text{\scriptsize{$\Upsigma$}}}^{2p}\
\delta(t)^{p}\left(1+\sup_{0\leq t\leq
T}\mathbb{E}[\overline{X}^{2p(\gamma+\alpha-1)}_{t}]\right).$

Finally, from Lemma 2.1, a sufficient condition to control the highest moment
$\sup_{0\leq t\leq T}\mathbb{E}[\overline{X}^{2p(\gamma+{\beta}-1)}_{t}]$ when
${\beta}=2\alpha-1$ is that $2p(\gamma+2(\alpha-1))\leq
1+\frac{2B_{2}}{{\text{\scriptsize{$\Upsigma$}}}^{2}}$, whereas no conditions
are needed in the case ${\beta}>2\alpha-1$.

To prove the estimation (2.7), we consider $\epsilon>0$ and the estimation
(2.6) with $p^{\prime}=p+\Delta t^{2p\epsilon}$. Then, assuming the parameters
satisfy $2(p+\Delta t^{2p\epsilon})(\gamma+2(\alpha-1))\leq
1+\frac{2B_{2}}{{\text{\scriptsize{$\Upsigma$}}}^{2}}$, we apply Lenglart’s
inequality in Lemma A.1, with $r=\frac{p}{p+\Delta t^{2p\epsilon}}$, obtaining

$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}|\overline{X}_{t}^{\gamma}-\overline{X}_{\eta(t)}^{\gamma}|^{2p}\
\right]$
$\displaystyle=\mathbb{E}\left[\left(\sup_{t\in[0,T]}|\overline{X}_{t}^{\gamma}-\overline{X}_{\eta(t)}^{\gamma}|^{2(p+\Delta
t^{2p\epsilon})}\right)^{r}\ \right]$ $\displaystyle\leq C_{p+\Delta
t^{2p\epsilon}}C_{r}\left(\Delta t^{p}\ \mathds{1}_{\\{0\\}}(b(0))\ +\ \Delta
t^{p\wedge 2p\gamma}\ \mathds{1}_{(0,+\infty)}(b(0))\right),$

with $C_{r}=o(\Delta t^{-2p\epsilon})$. Therefore,

$\big{\|}\sup_{t\in[0,T]}|\overline{X}_{t}^{\gamma}-\overline{X}_{\eta(t)}^{\gamma}|\big{\|}_{L^{2p}(\Omega)}\leq
C_{p+\Delta t^{2p\epsilon}}\left(\Delta t^{\frac{1}{2}-\epsilon}\
\mathds{1}_{\\{0\\}}(b(0))\ +\ \Delta t^{(\frac{1}{2}\wedge\gamma)-\epsilon}\
\mathds{1}_{(0,+\infty)}(b(0))\right),$

under assumption $2(p+\Delta t^{2p\epsilon})(\gamma+2(\alpha-1))\leq
1+\frac{2B_{2}}{{\text{\scriptsize{$\Upsigma$}}}^{2}}$, obtaining (2.7). ∎

With this lemma, we deduce some bounds on the negative moments of a stopped
version of the scheme. More precisely, we introduce the stopping time

$\overline{S}^{\Delta t}=\inf\\{s\geq 0;\overline{X}_{\eta(s)}>\Delta
t^{-\frac{1}{{\beta}-1}}\\},$ (2.9)

that goes to infinity, when $\Delta t$ is going to zero. Also, we observe that
$\eta(\overline{S}^{\Delta t})=\overline{S}^{\Delta t}$.

######  Proposition 2.4.

Assume 1 and 3. Then, for $\overline{S}^{\Delta t}$ defined in (2.9), for any
$\kappa>0$

$\sup_{t\leq T}\mathbb{E}[(\overline{X}_{t\wedge\overline{S}^{\Delta
t}}^{-\kappa})]\leq\sup_{t\leq
T}\mathbb{E}[(\overline{X}_{\cdot}-b(0)\delta(\cdot))_{t\wedge\overline{S}^{\Delta
t}}^{-\kappa}]\leq C(\kappa,\alpha,T).$ (2.10)

In addition, assume $b(0)>0$, then for any $\kappa>0$, such that,

$\mathds{1}_{\\{2\alpha-1\\}}(\beta)\times\Big{(}\mathds{1}_{(1,\frac{5}{4}]}(\alpha)+4(\alpha-1)\mathds{1}_{(\frac{5}{4},+\infty)}(\alpha)\Big{)}\times\left(\kappa\vee(2\alpha-2)\right)\leq\frac{1}{2}+\frac{B_{2}}{{\text{\scriptsize{$\Upsigma$}}}^{2}},$
$\sup_{t\leq T}\mathbb{E}[\overline{X}^{-\kappa}_{t}]\ \leq
C(\kappa,\alpha,T).$ (2.11)

###### Proof.

Applying the Itô formula to $\overline{X}_{t}^{-\kappa}$, and from hypothesis
1 and (2.3):

$\displaystyle d\overline{X}^{-\kappa}_{t}$
$\displaystyle=-\kappa\overline{X}^{-\kappa}_{t}\left(1-\tfrac{{b(0)}\delta(t)}{\overline{X}_{t}}\right)\left[\tfrac{b(\overline{X}_{\eta(t)})-b(0)}{\overline{X}_{\eta(t)}}-(\kappa+1)\frac{1}{2}\left(1-\tfrac{{b(0)}\delta(t)}{\overline{X}_{t}}\right)\tfrac{\sigma^{2}(\overline{X}_{\eta(t)})}{\overline{X}_{\eta(t)}^{2}}\right]dt-\kappa{b(0)}\overline{X}_{t}^{-(\kappa+1)}dt$
$\displaystyle\qquad-\kappa\overline{X}^{-\kappa}_{t}\left(1-\tfrac{{b(0)}\delta(t)}{\overline{X}_{t}}\right)\tfrac{\sigma(\overline{X}_{\eta(t)})}{\overline{X}_{\eta(t)}}dW_{t}$
$\displaystyle\leq-\kappa\overline{X}^{-\kappa}_{t}\left[-L_{\textit{G}}\,(\overline{X}_{\eta(t)}^{{\beta}-1}\vee
1)-(\kappa+1)\tfrac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2}\overline{X}_{\eta(t)}^{2(\alpha-1)}\right]dt-\kappa{b(0)}\overline{X}_{t}^{-(\kappa+1)}dt-\kappa\overline{X}^{-\kappa}_{t}\left(1-\tfrac{{b(0)}\delta(t)}{\overline{X}_{t}}\right)\tfrac{\sigma(\overline{X}_{\eta(t)})}{\overline{X}_{\eta(t)}}dW_{t}.$

Taking the expectation, (with a stopping time if needed to handle the local
martingale), and keeping only the positive terms, we introduce the local error
as follows

$\displaystyle d\ \mathbb{E}[\overline{X}^{-\kappa}_{t}]$
$\displaystyle\leq\kappa\Big{(}(\kappa+1)\frac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2}\mathbb{E}[\overline{X}^{-\kappa}_{t}\overline{X}_{\eta(t)}^{2(\alpha-1)}]+L_{\textit{G}}\,\,\mathbb{E}[\overline{X}^{-\kappa}_{t}(\overline{X}_{\eta(t)}^{{\beta}-1}\vee
1)]\Big{)}\ dt$
$\displaystyle\leq\kappa\Big{(}(\kappa+1)\frac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2}+(L_{\textit{G}}\,+(\kappa+1)\tfrac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2})\,\mathbb{E}[\overline{X}^{-\kappa}_{t}(\overline{X}_{\eta(t)}^{{\beta}-1}\vee
1)]\Big{)}\ dt$ $\displaystyle\leq
C(\kappa)\mathbb{E}[\overline{X}^{-\kappa+{\beta}-1}_{t}]\
dt+C(\kappa)\mathbb{E}[\overline{X}^{-\kappa}_{t}|\overline{X}_{\eta(t)}^{{\beta}-1}-\overline{X}_{t}^{{\beta}-1}|]\
dt+C(\kappa)(1+\mathbb{E}[\overline{X}_{t}^{-\kappa}])dt,$

with
$C(\kappa):=\kappa\big{(}(\kappa+1)\frac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2}+L_{\textit{G}}\,\big{)}$.
Assuming first $\kappa\geq{\beta}-1$, Jensen inequality in the first term
gives us:

$\mathbb{E}[\overline{X}^{-\kappa+{\beta}-1}_{t}]\leq\left(\mathbb{E}[{\overline{X}_{t}}^{-\kappa}]\right)^{\frac{\kappa-({\beta}-1)}{\kappa}}.$

For the second term we use Hölder inequality and (2.3): assuming $b(0)>0$, let
$\ell<\kappa$, then for

$\mathbb{E}[\overline{X}^{-\kappa}_{t}(\overline{X}_{\eta(t)}^{{\beta}-1}-\overline{X}_{t}^{{\beta}-1})]\leq\tfrac{1}{(b(0)\delta(t))^{\ell}}\mathbb{E}[\overline{X}^{-\kappa+\ell}_{t}|\overline{X}_{\eta(t)}^{{\beta}-1}-\overline{X}_{t}^{{\beta}-1}|]\leq\tfrac{1}{(b(0)\delta(t))^{\ell}}\mathbb{E}[\overline{X}^{-\kappa}_{t}]^{\frac{\kappa-\ell}{\kappa}}\
\mathbb{E}[|\overline{X}_{\eta(t)}^{{\beta}-1}-\overline{X}_{t}^{{\beta}-1}|^{\frac{\kappa}{\ell}}]^{\frac{\ell}{\kappa}}.$

So we can use the previous local error bound, provided that we control the
$\tfrac{\kappa}{\ell}4(\alpha-1)$th-moment of $\overline{X}$ (requiring
additional parameter control when ${\beta}=2\alpha-1$, see Lemma 2.3),
obtaining

$\mathbb{E}[|\overline{X}_{\eta(t)}^{{\beta}-1}-\overline{X}_{t}^{{\beta}-1}|^{\frac{\kappa}{\ell}}]^{\frac{\ell}{\kappa}}\leq
C(\frac{\kappa}{\ell})\delta(t)^{\frac{1}{2}\wedge({\beta}-1)}.$

Choosing $\ell=\tfrac{1}{2}\wedge({\beta}-1)$, then for any $\Delta t\leq 1$,

$\mathbb{E}[\overline{X}^{-\kappa}_{t}|\overline{X}_{\eta(t)}^{{\beta}-1}-\overline{X}_{t}^{{\beta}-1}|]\leq
C(\kappa,\alpha)\mathbb{E}[\overline{X}^{-\kappa}_{t}]^{\frac{\kappa-\ell}{\kappa}}.$

Since $\frac{\kappa-({\beta}-1)}{\kappa}\vee\frac{\kappa-\ell}{\kappa}\leq 1$,
we obtain the following bound

$d\ \mathbb{E}[\overline{X}^{-\kappa}_{t}]\leq
C(\kappa,\alpha)\left[1+1\vee\mathbb{E}[\overline{X}^{-\kappa}_{t}]\right]dt,$

from which, applying Gronwall inequality we get the desired inequality

$\sup_{t\leq T}\mathbb{E}[\overline{X}^{-\kappa}_{t}]\leq
C(x_{0},\kappa,\alpha,T).$

To guarantee the estimation, when ${\beta}=2\alpha-1$, and according to the
choice $\ell=\tfrac{1}{2}\wedge({\beta}-1)$, we need to impose that
$\frac{\kappa}{\frac{1}{2}\wedge(2\alpha-2)}(2\alpha-2)\leq\frac{B_{2}}{{\text{\scriptsize{$\Upsigma$}}}^{2}}+\frac{1}{2}$.

Finally, when $\kappa<{\beta}-1$, the first term is a positive moment, and
using the bound on the second term, under the sufficient condition that
$\frac{1}{\frac{1}{2}\wedge(2\alpha-2)}(2\alpha-2)^{2}\leq\frac{B_{2}}{{\text{\scriptsize{$\Upsigma$}}}^{2}}+\frac{1}{2}$
we have the same conclusion. Putting together the required conditions leads to
the following

$\kappa\vee(2\alpha-2)\left(\mathds{1}_{(1,\frac{5}{4}]}(\alpha)+4(\alpha-1)\mathds{1}_{(\frac{5}{4},+\infty)}(\alpha)\right)\leq\frac{1}{2}+\frac{B_{2}}{{\text{\scriptsize{$\Upsigma$}}}^{2}}.$

We prove now the estimation (2.10) for any negative moment of the stopped Exp-
EM scheme. Assume now, $b(0)\geq 0$. For $\kappa>0$, we apply Itô formula to
the process

$d(\overline{X}_{t}-b(0)\delta(t))=\left(\overline{X}_{t}-b(0){\delta(t)}\right)\Big{(}\tfrac{b\left(\overline{X}_{\eta(t)}\right)-b(0)}{\overline{X}_{\eta(t)}}dt+\tfrac{\sigma(\overline{X}_{\eta(t)})}{\overline{X}_{\eta(t)}}dW_{t}\Big{)}.$
$d(\overline{X}_{t}-b(0)\delta(t))^{-\kappa}=-\kappa\left(\overline{X}_{t}-b(0){\delta(t)}\right)^{-\kappa}\Big{(}\tfrac{b\left(\overline{X}_{\eta(t)}\right)-b(0)}{\overline{X}_{\eta(t)}}\
dt+\tfrac{\sigma(\overline{X}_{\eta(t)})}{\overline{X}_{\eta(t)}}\
dW_{t}-\tfrac{1}{2}(\kappa+1)\tfrac{\sigma^{2}(\overline{X}_{\eta(t)})}{\overline{X}_{\eta(t)}^{2}}\
dt\Big{)}.$

We rewrite $\overline{X}_{t}-b(0)\delta(t)=\overline{X}_{\eta(t)}\
Z(\delta(t))$, denoting

$Z(\delta(t)):=\exp\Big{\\{}\tfrac{\sigma(\overline{X}_{\eta(t)})}{\overline{X}_{\eta(t)}}(W_{t}-W_{\eta(t)})+\left(\tfrac{b(\overline{X}_{\eta(t)})-b(0)}{\overline{X}_{\eta(t)}}-\tfrac{1}{2}\tfrac{\sigma^{2}(\overline{X}_{\eta(t)})}{\overline{X}_{\eta(t)}^{2}}\right){\delta(t)}\Big{\\}}.$

From 1, observe that, for any $\Delta t\leq 1$:

$\displaystyle\mathbb{E}\left[\mathds{1}_{\\{\overline{X}_{\eta(t)}\leq\Delta
t^{-\frac{1}{{\beta}-1}}\\}}Z^{-\kappa}(\delta(t))\Big{|}\mathcal{F}_{\eta(t)}\right]$
$\displaystyle\leq\mathds{1}_{\\{\overline{X}_{\eta(t)}\leq\Delta
t^{-\frac{1}{{\beta}-1}}\\}}\exp\\{L_{G}(\Delta t^{-1}\vee 1)\Delta
t\kappa\\}~{}\exp\Big{\\{}\tfrac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2}\kappa(\kappa+1)\overline{X}_{\eta(t)}^{2(\alpha-1)}{\delta(t)}\Big{\\}}$
$\displaystyle\leq\mathds{1}_{\\{\overline{X}_{\eta(t)}\leq\Delta
t^{-\frac{1}{{\beta}-1}}\\}}~{}\exp\Big{\\{}L_{G}\kappa+\tfrac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2}\kappa(\kappa+1)\Big{\\}}.$

For $\overline{S}^{\Delta t}=\inf\\{s\geq 0;\overline{X}_{\eta{(s)}}>\Delta
t^{-\frac{1}{{\beta}-1}}\\}$, we consider the stopped process
$(\overline{X}^{-\kappa}_{\cdot\wedge\overline{S}^{\Delta
t}}-b(0)\delta(\cdot\wedge\overline{S}^{\Delta t}))$. From 1 and the previous
estimation we obtain

$\displaystyle d\
\mathbb{E}[(\overline{X}_{\cdot}-b(0)\delta(\cdot))^{-\kappa}_{t\wedge\overline{S}^{\Delta
t}}]$
$\displaystyle\leq\kappa\mathbb{E}\left[\mathds{1}_{\\{\overline{X}_{\eta(t)}\leq\Delta
t^{-\frac{1}{{\beta}-1}}\\}}Z^{-\kappa}(\delta(t))\overline{X}_{\eta(t)}^{-\kappa}\Big{(}L_{G}\left(\overline{X}_{\eta(t)}^{{\beta}-1}\vee
1\right)+\tfrac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2}(\kappa+1)\overline{X}_{\eta(t)}^{2(\alpha-1)}\Big{)}\right]\
dt$
$\displaystyle\quad=\kappa\mathbb{E}\left[\mathbb{E}\left[\mathds{1}_{\\{\overline{X}_{\eta(t)}\leq\Delta
t^{-\frac{1}{{\beta}-1}}\\}}Z^{-\kappa}(\delta(t))\Big{|}\mathcal{F}_{\eta(t)}\right]\overline{X}_{\eta(t)}^{-\kappa}\Big{(}L_{G}\left(\overline{X}_{\eta(t)}^{{\beta}-1}\vee
1\right)+\tfrac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2}(\kappa+1)\overline{X}_{\eta(t)}^{2(\alpha-1)}\Big{)}\right]\
dt$ $\displaystyle\qquad\leq
C\mathbb{E}\left[\mathds{1}_{\\{\overline{X}_{\eta(t)}\leq\Delta
t^{-\frac{1}{{\beta}-1}}\\}}\big{(}\overline{X}_{\eta(t)}^{-\kappa}+\overline{X}_{\eta(t)}^{-\kappa+{\beta}-1}+\overline{X}_{\eta(t)}^{-\kappa+2(\alpha-1)}\big{)}\right]\
dt.$

Thus, for any $\kappa>{\beta}-1\geq 2(\alpha-1)$, (otherwise we can use Hölder
inequality):

$\displaystyle\sup_{0\leq s\leq
t}\mathbb{E}[(\overline{X}_{\cdot}-b(0)\delta(\cdot))^{-\kappa}_{s\wedge\overline{S}^{\Delta
t}}]$ $\displaystyle\leq C\ \left(x_{0}^{-\kappa}+\int_{0}^{t}\sup_{0\leq
r\leq s}\mathbb{E}\left[\overline{X}^{-\kappa}_{r\wedge\overline{S}^{\Delta
t}}\right]ds\right)$ $\displaystyle\leq C\
\left(x_{0}^{-\kappa}+\int_{0}^{t}\sup_{0\leq r\leq
s}\mathbb{E}\left[(\overline{X}_{\cdot}-b(0)\delta(\cdot))^{-\kappa}_{r\wedge\overline{S}^{\Delta
t}}\right]ds\right).$

From Gronwall’s inequality, we conclude that there exists a positive constant
$C$ independent on $\Delta t$, such that

$\sup_{t\in[0,T]}\mathbb{E}[(\overline{X}_{\cdot}-b(0)\delta(\cdot))^{-\kappa}_{t\wedge\overline{S}^{\Delta
t}}]\leq C\ (x_{0}^{-\kappa}+1).$

∎

###  2.2 Exponential moment for the exponential scheme

######  Proposition 2.5.

Assume 1 and 3. Then, for $\overline{S}^{\Delta t}$ defined in (2.9), for any
$\mu<B_{2}$,

$\mathbb{E}\exp\left\\{\mu\int_{0}^{t\wedge\overline{S}^{\Delta
t}}\overline{X}_{\eta(s)}^{{\beta}-1}ds\right\\}\ \leq C(\mu,{\beta},T).$
(2.12)

###### Proof.

Applying the Itô formula with $\log$ function to

$d(\overline{X}_{t}-b(0)\delta(t))=\left(\overline{X}_{t}-b(0){\delta(t)}\right)\Big{(}\tfrac{b\left(\overline{X}_{\eta(t)}\right)-b(0)}{\overline{X}_{\eta(t)}}dt+\tfrac{\sigma(\overline{X}_{\eta(t)})}{\overline{X}_{\eta(t)}}dW_{t}\Big{)},$
(2.13)

and using 3, we get

$\log(\overline{X}_{t}-b(0){\delta(t)})\leq\log(x_{0})+B_{1}T-B_{2}\int_{0}^{t}\overline{X}_{\eta(s)}^{{\beta}-1}ds+\int_{0}^{t}\tfrac{\sigma(\overline{X}_{\eta(s)})}{\overline{X}_{\eta(s)}}dW_{s}-\tfrac{1}{2}\int_{0}^{t}\tfrac{\sigma^{2}(\overline{X}_{\eta(s)})}{\overline{X}_{\eta(s)}^{2}}ds.$

Then, for any $\mu>0$,

$\exp\left\\{\mu\int_{0}^{t}\overline{X}_{\eta(s)}^{{\beta}-1}ds\right\\}\leq
C(\overline{X}_{t}-b(0)\delta(t))^{-\tfrac{\mu}{B_{2}}}\exp\left\\{\tfrac{\mu}{B_{2}}\int_{0}^{t}\tfrac{\sigma(\overline{X}_{\eta(s)})}{\overline{X}_{\eta(s)}}dW_{s}-\tfrac{\mu}{B_{2}}\tfrac{1}{2}\int_{0}^{t}\tfrac{\sigma^{2}(\overline{X}_{\eta(s)})}{\overline{X}_{\eta(s)}^{2}}ds\right\\},$

where $C=x_{0}^{\tfrac{\mu}{B_{2}}}\exp\\{\tfrac{\mu}{B_{2}}B_{1}T\\}$.
Stopping the time at $\overline{S}^{\Delta t}$ defined in (2.9), taking
expectations and applying Hölder inequality, we get, for all $0<\mu<B_{2}$,

$\displaystyle\mathbb{E}\left[\exp\left\\{\mu\int_{0}^{t\wedge\overline{S}^{\Delta
t}}\overline{X}_{\eta(s)}^{{\beta}-1}ds\right\\}\right]$ $\displaystyle\leq
C~{}\mathbb{E}\left[(\overline{X}_{t\wedge\overline{S}^{\Delta
t}}-b(0)\delta({t\wedge\overline{S}^{\Delta
t}}))^{-\tfrac{\mu}{B_{2}}}\exp\left\\{\tfrac{\mu}{B_{2}}\int_{0}^{t\wedge\overline{S}^{\Delta
t}}\tfrac{\sigma(\overline{X}_{\eta(s)})}{\overline{X}_{\eta(s)}}dW_{s}-\tfrac{\mu}{B_{2}}\tfrac{1}{2}\int_{0}^{t\wedge\overline{S}^{\Delta
t}}\tfrac{\sigma^{2}(\overline{X}_{\eta(s)})}{\overline{X}_{\eta(s)}^{2}}ds\right\\}\right]$
$\displaystyle\leq
C~{}\mathbb{E}^{\tfrac{(B_{2}-\mu)}{B_{2}}}\left[(\overline{X}_{t\wedge\overline{S}^{\Delta
t}}-b(0)\delta({t\wedge\overline{S}^{\Delta
t}}))^{-\tfrac{\mu}{B_{2}-\mu}}\right].$

We end the proof by applying the Proposition 2.4. ∎

##  3 The strong rate of convergence for the exponential scheme

We introduce the error process $Y_{t}=X_{t}-\overline{X}_{t}$, with dynamics

$\displaystyle dY_{t}$
$\displaystyle=Y_{t}\left(\frac{b(X_{t})-b(0)}{X_{t}}dt+\frac{\sigma(X_{t})}{X_{t}}dW_{t}\right)+b(0)\delta(t)\frac{\sigma(\overline{X}_{\eta(t)})}{\overline{X}_{\eta(t)}}dW_{t}+b(0)\delta(t)\left[\frac{b(\overline{X}_{\eta(t)})-b(0)}{\overline{X}_{\eta(t)}}\right]dt$
(3.1)
$\displaystyle\quad+\overline{X}_{t}\left(\Big{[}\frac{b(X_{t})-b(0)}{X_{t}}-\frac{b(\overline{X}_{\eta(t)})-b(0)}{\overline{X}_{\eta(t)}}\Big{]}dt+\Big{[}\frac{\sigma(X_{t})}{X_{t}}-\frac{\sigma(\overline{X}_{\eta(t)})}{\overline{X}_{\eta(t)}}\Big{]}dW_{t}\right).$

This error process involves the diffusion coefficient
$\overline{X}_{t}[\tfrac{\sigma(X_{t})}{X_{t}}-\tfrac{\sigma(\overline{X}_{\eta(t)})}{\overline{X}_{\eta(t)}}]$
which requires further regularity from $\sigma$. For simplicity, we assume the
differentiability of $\sigma$ with a compatible polynomial growth that
reinforces 1-$(\sigma)$:

######  Hypothesis 4.

Control on $\sigma^{\prime}$. (4). In addition to Hypothesis 1-$(\sigma)$, the
diffusion function $\sigma$ is differentiable in $\mathbb{R}^{+}$ and there
exists some non-negative constant $\Upsigma^{\prime}$ such that, for all
$x\in\mathbb{R}^{+}$,

$|\sigma^{\prime}(x)|\leq\alpha\ {\text{\scriptsize{$\Upsigma^{\prime}$}}}\
x^{\alpha-1}.$ (3.2)

We are now in a position to state our first convergence results.

######  Theorem 3.1.

Assume 2, 1, 3 and 4. Then, for all $0<\varepsilon<\frac{1}{2}$, there exists
$0<\Delta(\varepsilon)\leq 1$, such that for all $\Delta
t\leq\Delta(\varepsilon)$, for $\overline{S}^{\Delta t}$ defined in (2.9), for
any integer $p\geq 1$ satisfying

$\displaystyle\mathds{1}_{\\{2\alpha-1\\}}({\beta})\ 2(p+1+\Delta
t^{2\varepsilon(p+1)})\
(2\alpha-1)\leq\tfrac{1}{2}+\tfrac{B_{2}}{{\text{\scriptsize{$\Upsigma$}}}^{2}},$
(3.3) $\displaystyle 2\
(p+1)^{2}\left(L_{b_{\text{loc}}}+2(2p-1)(6{\text{\scriptsize{$\Upsigma$}}}^{2}+4(\alpha{\text{\scriptsize{$\Upsigma^{\prime}$}}})^{2}\right)<B_{2},$
$\displaystyle\quad\mbox{when
}(b(0),\alpha)\notin(0,\tfrac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2})\times[1,\tfrac{3}{2}],$
(3.4)

there exists some positive constant $C(p)$, independent of $\Delta t$ and
$\varepsilon$, such that

$\sup_{0\leq t\leq T}\big{\|}X_{t\wedge\overline{S}^{\Delta
t}}-\overline{X}_{t\wedge\overline{S}^{\Delta
t}}\big{\|}_{L^{2p}(\Omega)}+\big{\|}\sup_{0\leq t\leq
T}|X_{t\wedge\overline{S}^{\Delta
t}}-\overline{X}_{t\wedge\overline{S}^{\Delta t}}|\big{\|}_{L^{p}(\Omega)}\leq
C(p)\,\Delta t^{\frac{1}{2}-\varepsilon}.$

Furthermore, when $b$ is continuous on $\mathbb{R}^{+}$, the convergence above
takes place for $\Delta t\leq 1$, ($\varepsilon=0$), and

$\sup_{0\leq t\leq T}\big{\|}X_{t\wedge\overline{S}^{\Delta
t}}-\overline{X}_{t\wedge\overline{S}^{\Delta
t}}\big{\|}_{L^{2p}(\Omega)}+\big{\|}\sup_{0\leq t\leq
T}|X_{t\wedge\overline{S}^{\Delta
t}}-\overline{X}_{t\wedge\overline{S}^{\Delta t}}|\big{\|}_{L^{p}(\Omega)}\leq
C(p)\,\Delta t^{\frac{1}{2}}.$

with conditions (3.3) and (3.4) enlarged to

$\displaystyle\mathds{1}_{\\{2\alpha-1\\}}({\beta})\ 2(p+1)\
(2\alpha-1)\leq\frac{1}{2}+\frac{B_{2}}{{\text{\scriptsize{$\Upsigma$}}}^{2}},$
(3.5) $\displaystyle 2\ p\
(p+1)\left(L_{b_{\text{loc}}}+2(2p-1)(6{\text{\scriptsize{$\Upsigma$}}}^{2}+4(\alpha{\text{\scriptsize{$\Upsigma^{\prime}$}}})^{2}\right)<B_{2},$
$\displaystyle\quad\mbox{when
}(b(0),\alpha)\notin(0,\tfrac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2})\times[1,\tfrac{3}{2}].$
(3.6)

The proof of Theorem 3.1 is detailed in Section 3.3, after some preliminaries
estimates. In the proof, we draw the following estimation for
$\varepsilon\mapsto\Delta(\varepsilon)$, behaving mainly with an exponential
decay:

$\Delta(\varepsilon):=\exp(-\frac{1}{2\varepsilon}\log(\log(\tfrac{1}{2\varepsilon})))\
\wedge\
\left\\{\frac{1}{16{\text{\scriptsize{$\Upsigma$}}}}(\tfrac{2}{3})^{\alpha}\underline{\Delta\upchi}^{1-\alpha}\right\\}^{\tfrac{2}{1-2\varepsilon}}\wedge\
\left\\{\tfrac{8}{3}\frac{{\text{\scriptsize{$\Upsigma$}}}}{L_{G}}(\tfrac{3}{2}\underline{\Delta\upchi})^{\alpha-{\beta}}\right\\}^{\tfrac{2}{1+2\varepsilon}}\wedge\
\frac{\underline{\Delta\upchi}}{4b(0)},$ (3.7)

where
$\underline{\Delta\upchi}=\min\\{(\upchi_{k+1}-\upchi_{k});{k=0,\ldots,m-1}\\}$.

######  Remark 3.2.

The extension of Theorem 3.1 to $p\geq 1/2$ with value in $\mathbb{R}^{+}$ can
be achieved without too much effort using the Itô Tanaka’s formula, setting
$Z_{t}=|Y_{t}|=|X_{t}-\overline{X}_{t}|$, with dynamics of the form

$dZ_{t}=\int_{0}^{t}\displaystyle{\rm
sgn}(Y_{s})b_{Y}(X_{t},\overline{X}_{\eta(s)})ds+\sigma\int_{0}^{t}\displaystyle{\rm
sgn}(Y_{s})\sigma_{Y}(X_{t},\overline{X}_{\eta(s)})dW_{s}+L^{0}_{t}(Y),$

where $\displaystyle{\rm sgn}(x):=1-2\,\mathds{1}_{(x\leq 0)}$, and
$L^{0}(Y)=L^{0}(Z)$. Since $\int_{0}^{t}Z_{s}^{2p-1}dL^{0}_{s}(Y)=0$, we have

$d\mathbb{E}[Z_{t}^{2p}]=\int_{0}^{t}2p\mathbb{E}[\displaystyle{\rm
sgn}(Y_{s})Z_{t}^{2p-1}b_{Y}(X_{t},\overline{X}_{\eta(s)})]ds+\int_{0}^{t}p(2p-1)\sigma_{Y}^{2}(X_{t},\overline{X}_{\eta(s)})Y^{2p-2}_{s}ds$

There is no major difficulty in proceeding in this way, provided we carefully
add the conditions for controlling negative moments. However, the proof,
already rather long, would almost double in size due to the separation of the
different cases to be discussed, which motivates us to work with
$p\in\mathbb{N}$.

###  3.1 The Itô formula for the error process

For simplicity, the integer $p\geq 1$ allows us to consider equivalently,
$|Y_{t}|^{2p}=Y_{t}^{2p}$, for the error process
$(Y_{t}=X_{t}-\overline{X}_{t},t\in[0,T])$, satisfying (3.1). Then, applying
Itô formula to $Y_{t}^{2p}$, we get

$\displaystyle dY_{t}^{2p}$
$\displaystyle={2p}Y_{t}^{2p-1}\left[Y_{t}\left(\tfrac{b(X_{t})-b(0)}{X_{t}}\right)+\overline{X}_{t}\left(\tfrac{b(X_{t})-b(0)}{X_{t}}-\tfrac{b(\overline{X}_{\eta(t)})-b(0)}{\overline{X}_{\eta(t)}}\right)+b(0)\delta(t)\left(\tfrac{b(\overline{X}_{\eta(t)})-b(0)}{\overline{X}_{\eta(t)}}\right)\right]dt$
$\displaystyle\quad+{2p}Y_{t}^{2p-1}\left(Y_{t}\tfrac{\sigma(X_{t})}{X_{t}}+\overline{X}_{t}\Big{[}\tfrac{\sigma(X_{t})}{X_{t}}-\tfrac{\sigma(\overline{X}_{\eta(t)})}{\overline{X}_{\eta(t)}}\Big{]}+b(0)\delta(t)\tfrac{\sigma(\overline{X}_{\eta(t)})}{\overline{X}_{\eta(t)}}\right)dW_{t}$
$\displaystyle\quad+p(2p-1)Y_{t}^{2p-2}\left(Y_{t}\tfrac{\sigma(X_{t})}{X_{t}}+\overline{X}_{t}\Big{[}\tfrac{\sigma(X_{t})}{X_{t}}-\tfrac{\sigma(\overline{X}_{\eta(t)})}{\overline{X}_{\eta(t)}}\Big{]}+b(0)\delta(t)\tfrac{\sigma(\overline{X}_{\eta(t)})}{\overline{X}_{\eta(t)}}\right)^{2}dt.$

On the last line, containing the squared sum, keeping the same terms order and
putting the smallest weigh of the second term, we apply the inequality
$(a+b+c)^{2}\leq 2b^{2}+4a^{2}+4c^{2}$ and obtain the upper-bound of the cross
variation terms. Isolating the Brownian integrals, we enumerate the part of
the drift term as follows:

$\displaystyle dY_{t}^{2p}$
$\displaystyle\leq\sum_{i=1}^{5}T_{i}(t)\;dt+{2p}Y_{t}^{2p-1}\left[Y_{t}\frac{\sigma(X_{t})}{X_{t}}+\overline{X}_{t}\Big{[}\frac{\sigma(X_{t})}{X_{t}}-\frac{\sigma(\overline{X}_{\eta(t)})}{\overline{X}_{\eta(t)}}\Big{]}+b(0)\delta(t)\frac{\sigma(\overline{X}_{\eta(t)})}{\overline{X}_{\eta(t)}}\right]dW_{t},$
(3.8)

with

$\begin{array}[]{lr}T_{1}(t)={2p}Y_{t}^{2p}\left(\tfrac{b(X_{t})-b(0)}{X_{t}}+2(2p-1)\tfrac{\sigma^{2}(X_{t})}{X_{t}^{2}}\right),&\quad
T_{4}(t)=2pb(0)\delta(t)\
Y_{t}^{2p-1}\tfrac{b(\overline{X}_{\eta(t)})-b(0)}{\overline{X}_{\eta(t)}},\\\
T_{2}(t)=2pY_{t}^{2p-1}\overline{X}_{t}\left(\tfrac{b(X_{t})-b(0)}{X_{t}}-\tfrac{b(\overline{X}_{\eta(t)})-b(0)}{\overline{X}_{\eta(t)}}\right),&\quad
T_{5}(t)=4p(2p-1)b(0)^{2}\
\delta(t)^{2}Y_{t}^{2p-2}\tfrac{\sigma^{2}(\overline{X}_{\eta(t)})}{\overline{X}_{\eta(t)}^{2}},\\\
T_{3}(t)=2p(2p-1)Y_{t}^{2p-2}\overline{X}_{t}^{2}\Big{[}\tfrac{\sigma(X_{t})}{X_{t}}-\tfrac{\sigma(\overline{X}_{\eta(t)})}{\overline{X}_{\eta(t)}}\Big{]}^{2}.&\end{array}$

Notice that the stochastic integral in (3.8) is a square-integrable martingale
under the sufficient condition that the $2(2p+\alpha-1)$th-moments for $X$ and
$\overline{X}$ are bounded. With Proposition 1.3 and Lemma 2.1, this leads to
the assumption that
$2p+\alpha-1\leq\frac{1}{2}+\frac{B_{2}}{{\text{\scriptsize{$\Upsigma$}}}^{2}}$,
when ${\beta}=2\alpha-1$, largely covered by (3.3) or (3.5).

###  3.2 Preliminary estimations for the treatment of the discontinuity of $b$

We use the short notation $\\{\overline{X}_{[s,t]}\in B(\upchi,\delta)\\}$ for
the event of the entire trajectory $(\overline{X}_{u},u\in[s,t])$ in the ball
$B(\upchi,\delta)$.

With the help of the occupation time formula for diffusions, we estimate some
statistics of the amount of time the approximation process spends in a ball of
radius $h$, centered at a given discontinuity point $\upchi$:

######  Lemma 3.3.

[Sojourn time of the scheme in a given ball]. Assume 1, 3, 4. Let $t\in[0,T]$
and $p>0$. Let $\upchi>0$ and $\Delta t\leq\frac{\upchi}{4b(0)}$. Then, for
any $0<h\leq\frac{1}{4}\upchi$, for any $q\geq 1$, there exists a constant $C$
(independent of $\Delta t$ and $h$) such that

$\displaystyle\left\|\int_{0}^{t}\mathds{1}_{\\{\overline{X}_{[\eta(s),\eta(s)+\Delta
t]}\in B(\upchi,2h)\\}}ds\right\|_{L^{q}(\Omega)}$
$\displaystyle\leq\frac{C}{\displaystyle\inf_{y\in
B(\upchi,\frac{\upchi}{2})}\sigma^{2}(y)}\ h,$ (3.9) $\displaystyle\text{and
for all $\mu>0$,
}\quad\mathbb{E}\left[\exp\left\\{\mu\int_{0}^{t}\mathds{1}_{\\{\overline{X}_{[\eta(s),\eta(s)+\Delta
t]}\in B(\upchi,2h)\\}}ds\right\\}\right]$ $\displaystyle\leq\exp\Big{\\{}\mu\
\frac{C}{\displaystyle\inf_{y\in B(\upchi,\frac{\upchi}{2})}\sigma^{2}(y)}\
h\Big{\\}}.$

###### Proof.

We focus the proof of the first statement in (3.9), the second one being
implied by this first. We introduce the following interpolated dynamics of
$\overline{X}$, coinciding with (2.2):

$\begin{array}[]{c}d\overline{X}_{t}:=b^{\text{exp}}(\overline{X}_{t},\overline{X}_{\eta(t)})dt+{\sigma^{\text{exp}}}(\overline{X}_{t},\overline{X}_{\eta(t)})dW_{t}\\\\[5.69046pt]
b^{\text{exp}}(x,y)=(x-b(0)\delta(t))\Big{(}\dfrac{b(y)-b(0)}{y}\Big{)}+b(0),\qquad\qquad\quad{\sigma^{\text{exp}}}(x,y)=\dfrac{\sigma(y)}{y}(x-b(0)\delta(t)).\end{array}$
(3.10)

We observe that

$\int_{0}^{t}\mathds{1}_{\\{\overline{X}_{[\eta(s),\eta(s)+\Delta t]}\in
B(\upchi,2h)\\}}ds=\sum_{k=0}^{\frac{\eta(t)}{\Delta
t}}\mathds{1}_{\\{\overline{X}_{[t_{k},t\wedge t_{k+1}]}\in
B(\upchi,2h)\\}}\int_{t_{k}}^{t\wedge
t_{k+1}}\mathds{1}_{\\{\overline{X}_{s}\in B(\upchi,2h)\\}}ds.$

Since $\sigma$ is continuous, then it is bounded on $B(\upchi,2h)$ and the
diffusion ${\sigma^{\text{exp}}}^{2}$ is bounded below on the ball
$B(\upchi,2h)^{2}$ by $\frac{(\upchi-2h-b(0)\Delta t)^{2}}{(\upchi+2h)^{2}}\
\inf_{y\in B(\upchi,2h)}{\sigma^{2}(y)}$. With the imposed conditions that
$h\leq\upchi/4$ and $\Delta t\leq\frac{\upchi}{4b(0)}$, we have

$\int_{t_{k}}^{t\wedge t_{k+1}}\mathds{1}_{\\{\overline{X}_{s}\in
B(\upchi,2h)\\}}ds\leq\frac{36}{\displaystyle\inf_{y\in
B(\upchi,\upchi/2)}\sigma^{2}(y)}\int_{t_{k}}^{t\wedge
t_{k+1}}{\sigma^{\text{exp}}}(\overline{X}_{s},\overline{X}_{\eta(s)})^{2}\mathds{1}_{\\{\overline{X}_{s}\in
B(\upchi,2h)\\}}ds.$

The occupation time formula (see, e.g. (Revuz and Yor, 1999, Ex.1.15,Chap.VI))
leads to representation

$\int_{t_{k}}^{t\wedge
t_{k+1}}{\sigma^{\text{exp}}}(\overline{X}_{s},\overline{X}_{\eta(s)})^{2}\mathds{1}_{\\{\overline{X}_{s}\in
B(\upchi,2h)\\}}ds=\int_{\mathbb{R}}\mathds{1}_{[\upchi-2h,\upchi+2h]}(y)(L^{y}_{t\wedge
t_{k+1}}(\overline{X})-L^{y}_{t_{k}}(\overline{X}))\ dy$

where $L^{y}(\overline{X})$ is the (increasing) local time in $y$ of the Itô
process $\overline{X}$. Then

$\int_{t_{k}}^{t\wedge t_{k+1}}\mathds{1}_{\\{\overline{X}_{s}\in
B(\upchi,2h)\\}}ds\leq{C}h\left(\sup_{y\in
B(\upchi,2h)}\tfrac{1}{2}(L^{y}_{t\wedge
t_{k+1}}(\overline{X})-L^{y}_{t_{k}}(\overline{X}))\right).$

By the Itô-Tanaka-Meyer formula,

$0\leq\tfrac{1}{2}(L^{y}_{t\wedge
t_{k+1}}(\overline{X})-L^{y}_{t_{k}}(\overline{X}))=(\overline{X}_{t\wedge
t_{k+1}}-y)_{+}-(\overline{X}_{t_{k}}-y)_{+}-\int_{t_{k}}^{t\wedge
t_{k+1}}\mathds{1}_{\\{\overline{X}_{s-}>y\\}}d\overline{X}_{s}.$

But for all $y$, from the Lipschitz property of the positive part function, it
is easy to check that

$(\overline{X}_{t_{k+1}}-y)_{+}-(\overline{X}_{t_{k}}-y)_{+}\leq(\overline{X}_{t_{k+1}}-\overline{X}_{t_{k}})\
\mathds{1}_{\\{\overline{X}_{t_{k}}\leq\overline{X}_{t_{k+1}}\\}}=\
\int_{t_{k}}^{t_{k+1}}\
\mathds{1}_{\\{\overline{X}_{\eta(s)}\leq\overline{X}_{\eta(s)+\Delta
t}\\}}d\overline{X}_{s}.$

By summing on the time intervals, we have obtained that

$\int_{0}^{t}\mathds{1}_{\\{\overline{X}_{[\eta(s),\eta(s)+\Delta t]}\in
B(\upchi,2h)\\}}ds\leq Ch\ H_{t}(\overline{X}),$

with

$\displaystyle H_{t}(\overline{X}):=$ $\displaystyle\sup_{y\in
B(\upchi,2h)}\sum_{k=1}^{\frac{\eta(t)}{\Delta
t}}\mathds{1}_{\\{\overline{X}_{[t_{k},t\wedge t_{k+1}]}\in
B(\upchi,2h)\\}}\frac{1}{2}(L^{y}_{t\wedge
t_{k+1}}(\overline{X})-L^{y}_{t_{k}}(\overline{X})),$

and $|H_{t}(\overline{X})|$ further bounded by

$\displaystyle|H_{t}(\overline{X})|$
$\displaystyle\leq\left|\int_{0}^{t}\mathds{1}_{\\{\overline{X}_{[\eta(s),\eta(s)+\Delta
t]}\in
B(\upchi,2h)\\}}\mathds{1}_{\\{\overline{X}_{\eta(s)}\leq\overline{X}_{\eta(s)+\Delta
t}\\}}d\overline{X}_{s}\right|$ $\displaystyle\qquad+\sup_{y\in
B(\upchi,2h)}\left|\int_{0}^{t}\mathds{1}_{\\{\overline{X}_{[\eta(s),t\wedge\eta(s)+\Delta
t]}\in
B(\upchi,2h)\\}}\mathds{1}_{\\{\overline{X}_{s-}>y\\}}d\overline{X}_{s}\right|$
$\displaystyle\leq
2\int_{0}^{t}\mathds{1}_{\\{\overline{X}_{[\eta(s),\eta(s)+\Delta t]}\in
B(\upchi,2h)\\}}d\overline{X}_{s}.$

To complete the proof, it remains to verify that for any $q\geq 1$,
$\mathbb{E}[H^{q}(\overline{X})]$ is bounded with a polynomial dependency in
$q$ for the right-hand side constant:

$\displaystyle\mathbb{E}[H^{q}(\overline{X})]$ $\displaystyle\leq
2^{q}\mathbb{E}\left[\left(\int_{0}^{t}\mathds{1}_{\\{\overline{X}_{[\eta(s),\eta(s)+\Delta
t]}\in B(\upchi,2h)\\}}d\overline{X}_{s}\right)^{q}\right]$ $\displaystyle\leq
2^{2q}\mathbb{E}\left[\left(\int_{0}^{t}\mathds{1}_{\\{\overline{X}_{[\eta(s),\eta(s)+\Delta
t]}\in
B(\upchi,2h)\\}}b^{\text{exp}}(\overline{X}_{s},\overline{X}_{\eta(s)})ds\right)^{q}\right]$
$\displaystyle\quad+2^{2q}\mathbb{E}^{\frac{1}{2}}\left[\int_{0}^{t}\mathds{1}_{\\{\overline{X}_{[\eta(s),\eta(s)+\Delta
t]}\in
B(\upchi,2h)\\}}{\sigma^{\text{exp}}}(\overline{X}_{t},\overline{X}_{\eta(t)})^{2q}ds\right]$
$\displaystyle\leq 2^{2q}t^{q}(C\ \upchi^{{\beta}}\vee\upchi+b_{0})^{q}\ +\
2^{2q}t^{\frac{q}{2}}{\text{\scriptsize{$\Upsigma$}}}^{q}(\tfrac{3}{2}\upchi)^{\alpha
q}.$

∎

In the convergence analysis, the only instance where the discontinuity of the
drift $b$ needs to be addressed is within the local error term singled out on
the right-hand side of (3.12) below, and examined in the following lemma:

######  Lemma 3.4.

Assume 2, 1, 3 and 4. For all $0<\varepsilon<\frac{1}{2}$, consider
$0<\Delta(\varepsilon)\leq 1$, given by (3.7). Then, for all $\Delta
t\leq\Delta(\varepsilon)$, for any integer $p\geq 1$ satisfying

$\displaystyle\mathds{1}_{\\{2\alpha-1\\}}({\beta})\ (\ 2p+\Delta
t^{2p\varepsilon})\ (2\alpha-1)$
$\displaystyle\leq\tfrac{1}{2}+\tfrac{B_{2}}{{\text{\scriptsize{$\Upsigma$}}}^{2}},$
(3.11)

there exits a positive constant $C$, independent of $\Delta t$ and $\epsilon$,
such that

$\displaystyle\mathbb{E}\left[\left|Y_{t}^{2p-1}\overline{X}_{t}\left(\frac{b(\overline{X}_{t})-b(0)}{\overline{X}_{t}}-\frac{b(\overline{X}_{\eta(t)})-b(0)}{\overline{X}_{\eta(t)}}\right)\right|\right]$
(3.12) $\displaystyle\leq C\mathbb{E}[Y_{t}^{2p}]+C\Delta
t^{p(1-2\varepsilon)}+\mathbb{E}\left[|Y_{t}|^{2p-1}\sum_{i=1}^{m}\mathds{1}_{\left\\{\overline{X}_{[\eta(t),\eta(t)+\Delta
t]}\in
B(\upchi_{i},8{\text{\scriptsize{$\Upsigma$}}}(\tfrac{3}{2}\underline{\Delta\upchi})^{\alpha}\
\Delta t^{1/2-\varepsilon})\right\\}}\right].$

When $b$ is a continuous function, for all $\Delta t\leq 1$, for any integer
$p\geq 1$ satisfying

$\displaystyle\mathds{1}_{\\{2\alpha-1\\}}({\beta})\ \ p\
(4\alpha-3)\leq\tfrac{1}{2}+\tfrac{B_{2}}{{\text{\scriptsize{$\Upsigma$}}}^{2}},$
(3.13)

there exits a positive constant $C$, independent of $\Delta t$, such that

$\displaystyle\mathbb{E}\left[\left|Y_{t}^{2p-1}\overline{X}_{t}\left(\frac{b(\overline{X}_{t})-b(0)}{\overline{X}_{t}}-\frac{b(\overline{X}_{\eta(t)})-b(0)}{\overline{X}_{\eta(t)}}\right)\right|\right]\leq
C\mathbb{E}[Y_{t}^{2p}]+C\Delta t^{p}.$ (3.14)

###### Proof.

Without loss of generality, we simplify the discussion in this proof, by
assuming that $b$ has only one discontinuity point $\upchi_{m}$. We consider a
ball $B(\upchi_{m},f(\Delta t))$, with $f(\Delta t)\leq\upchi_{m}/4$ to be
chosen later small enough. We decompose

$A:=Y_{t}^{2p-1}\overline{X}_{t}\left(\frac{b(\overline{X}_{t})-b(0)}{\overline{X}_{t}}-\frac{b(\overline{X}_{\eta(t)})-b(0)}{\overline{X}_{\eta(t)}}\right)$
(3.15)

discussing the relative position of $\overline{X}_{t}$ and
$\overline{X}_{\eta(t)}$ w.r.t $\upchi_{m}$. First we introduce the two events

$A\left(\mathds{1}_{\\{\overline{X}_{\eta(t)}\in B(\upchi_{m},f(\Delta
t))\\}}+\mathds{1}_{\\{\overline{X}_{\eta(t)}\notin B(\upchi_{m},f(\Delta
t))\\}}\right).$

These events are further decomposed. For the event
$\\{\overline{X}_{\eta(t)}\notin B(\upchi_{m},f(\Delta t))\\}$, we isolate the
discontinuity point $\upchi_{m}$, while discussing the distance
$|\overline{X}_{\cdot}-\overline{X}_{\eta(t)}|$, compared to the threshold
$\frac{\upchi_{m}}{4}$:

$\displaystyle\mathds{1}_{\\{\overline{X}_{\eta(t)}\notin
B(\upchi_{m},f(\Delta
t))\\}}\leq\mathds{1}_{\\{(\overline{X}_{\eta(t)}\wedge\overline{X}_{t})>\upchi_{m}\\}}+\mathds{1}_{\\{(\overline{X}_{\eta(t)}\vee\overline{X}_{t})<\upchi_{m}\\}}+\mathds{1}_{\\{|\overline{X}_{t}-\overline{X}_{\eta(t)}|\geq
f(\Delta t)\\}}$
$\displaystyle\leq\left(\mathds{1}_{\\{(\overline{X}_{\eta(t)}\wedge\overline{X}_{t})>\upchi_{m}\\}}+\mathds{1}_{\\{(\overline{X}_{\eta(t)}\vee\overline{X}_{t})<\upchi_{m}\\}}\right)$
$\displaystyle\quad+\mathds{1}_{\\{|\overline{X}_{t}-\overline{X}_{\eta(t)}|\geq
f(\Delta
t)\\}}\left(\mathds{1}_{\\{\sup_{s\in[\eta(t),t)}|\overline{X}_{s}-\overline{X}_{\eta(t)}|\geq\frac{\upchi_{m}}{4}\\}}+\mathds{1}_{\\{\sup_{s\in[\eta(t),t)}|\overline{X}_{s}-\overline{X}_{\eta(t)}|\leq\frac{\upchi_{m}}{4}\\}\cap\\{(\overline{X}_{\eta(t)}\wedge\overline{X}_{t})\leq\upchi_{m}\leq(\overline{X}_{\eta(t)}\vee\overline{X}_{t})\\}}\right).$

Notice that $\upchi_{m}-\frac{\upchi_{m}}{4}-f(\Delta t)\geq\upchi_{m}/2$ is
far from zero, and

$\displaystyle\mathds{1}_{\\{\overline{X}_{\eta(t)}\notin
B(\upchi_{m},f(\Delta t))\\}}$
$\displaystyle\leq\left(\mathds{1}_{\\{(\overline{X}_{\eta(t)}\wedge\overline{X}_{t})>\upchi_{m}\\}}+\mathds{1}_{\\{(\overline{X}_{\eta(t)}\vee\overline{X}_{t})<\upchi_{m}\\}}\right)$
$\displaystyle\quad+\left(\mathds{1}_{\\{\sup_{s\in[\eta(t),\eta(t)+\Delta
t)}|\overline{X}_{s}-\overline{X}_{\eta(t)}|\geq\frac{\upchi_{m}}{4}\\}}+\mathds{1}_{\\{|\overline{X}_{t}-\overline{X}_{\eta(t)}|\geqslant
f(\Delta t)\\}}\ \mathds{1}_{\\{\overline{X}_{[\eta(t),\eta(t)+\Delta t)}\in
B(\upchi_{m},\frac{\upchi_{m}}{4})|\\}}\right).$

We decompose the event $\\{\overline{X}_{\eta(t)}\in B(\upchi_{m},f(\Delta
t))\\}$, following the same idea:

$\displaystyle\mathds{1}_{\\{\overline{X}_{\eta(t)}\in B(\upchi_{m},f(\Delta
t))\\}}$ $\displaystyle=\mathds{1}_{\\{\overline{X}_{\eta(t)}\in
B(\upchi_{m},f(\Delta
t))\\}}\left(\mathds{1}_{\\{\sup_{s\in[\eta(t),\eta(t)+\Delta
t)}|\overline{X}_{s}-\overline{X}_{\eta(t)}|\geq\frac{\upchi_{m}}{4}\\}}+\mathds{1}_{\\{\sup_{s\in[\eta(t),\eta(t)+\Delta
t)}|\overline{X}_{s}-\overline{X}_{\eta(t)}|\leq\frac{\upchi_{m}}{4}\\}}\right)$
$\displaystyle\quad\leq\mathds{1}_{\\{\sup_{s\in[\eta(t),\eta(t)+\Delta
t)}|\overline{X}_{s}-\overline{X}_{\eta(t)}|\geq\frac{\upchi_{m}}{4}\\}}$
$\displaystyle\quad\quad+\mathds{1}_{\\{\overline{X}_{\eta(t)}\in
B(\upchi_{m},f(\Delta
t))\\}}\left(\mathds{1}_{\\{\overline{X}_{[\eta(t),\eta(t)+\Delta t]}\in
B(\upchi_{m},2f(\Delta
t))\\}}+\mathds{1}_{\\{\overline{X}_{[\eta(t),\eta(t)+\Delta t]}\in
B(\upchi_{m},\frac{\upchi_{m}}{4}+f(\Delta t))-B(\upchi_{m},2f(\Delta
t))\\}}\right)$
$\displaystyle\quad\leq\mathds{1}_{\\{\sup_{s\in[\eta(t),\eta(t)+\Delta
t)}|\overline{X}_{s}-\overline{X}_{\eta(t)}|\geq\frac{\upchi_{m}}{4}\\}}$
$\displaystyle\quad+\mathds{1}_{\\{\overline{X}_{\eta(t)}\in
B(\upchi_{m},f(\Delta
t))\\}}\mathds{1}_{\\{\overline{X}_{[\eta(t),\eta(t)+\Delta t]}\in
B(\upchi_{m},2f(\Delta t))\\}}$
$\displaystyle\quad+\mathds{1}_{\\{\overline{X}_{\eta(t)}\in
B(\upchi_{m},f(\Delta
t))\\}}\mathds{1}_{\\{\overline{X}_{[\eta(t),\eta(t)+\Delta t]}\in
B(\upchi_{m},\frac{\upchi_{m}}{4}+f(\Delta
t))\\}}\mathds{1}_{\\{\sup_{s\in[\eta(t),\eta(t)+\Delta
t)}|\overline{X}_{s}-\overline{X}_{\eta(t)}|\geq f(\Delta t)\\}}.$

After small bound in the indicators and joining common terms, we can gather
the two decompositions as follows, identifying four terms that we evaluate
with different techniques:

$\displaystyle A$
$\displaystyle\leq\left(\mathds{1}_{\\{(\overline{X}_{\eta(t)}\wedge\overline{X}_{t})>\upchi_{m}\\}}+\mathds{1}_{\\{(\overline{X}_{\eta(t)}\vee\overline{X}_{t})<\upchi_{m}\\}}\right)\
A$ $\displaystyle\quad+2\ \mathds{1}_{\\{\sup_{s\in[\eta(t),\eta(t)+\Delta
t)}|\overline{X}_{s}-\overline{X}_{\eta(t)}|\geq\frac{\upchi_{m}}{4}\\}}\ A$
$\displaystyle\quad+\mathds{1}_{\\{\overline{X}_{[\eta(t),\eta(t)+\Delta
t]}\in B(\upchi_{m},2f(\Delta t))\\}}\ A$ $\displaystyle\quad+2\
\mathds{1}_{\\{\overline{X}_{[\eta(t),\eta(t)+\Delta t]}\in
B(\upchi_{m},\frac{\upchi_{m}}{4}+f(\Delta
t))\\}}\mathds{1}_{\\{\sup_{s\in[\eta(t),\eta(t)+\Delta
t)}|\overline{X}_{s}-\overline{X}_{\eta(t)}|\geq f(\Delta t)\\}}\ A$
$\displaystyle\qquad:=A_{1}+A_{2}+A_{3}+A_{4}.$

In what follows, the changing generic constant $C$ remains uniformly bounded
in $\Delta t$, but may depend on $\upchi_{m}$.

#### Bound for $|A_{3}|$.

The path $\overline{X}_{[\eta(t),\eta(t)+\Delta t]}$ being confined in the
ball $B(\upchi_{m},2f(\Delta t))\\}$, this term is just bounded as

$\displaystyle|A_{3}|=$
$\displaystyle\left|Y_{t}^{2p-1}\overline{X}_{t}\left(\frac{b(\overline{X}_{t})-b(0)}{\overline{X}_{t}}-\frac{b(\overline{X}_{\eta(t)})-b(0)}{\overline{X}_{\eta(t)}}\right)\right|\mathds{1}_{\\{\overline{X}_{[\eta(t),\eta(t)+\Delta
t]}\in B(\upchi_{m},2f(\Delta t))\\}}$ $\displaystyle\leq
C|Y_{t}|^{2p-1}\mathds{1}_{\\{\overline{X}_{[\eta(t),\eta(t)+\Delta t]}\in
B(\upchi_{m},2f(\Delta t))\\}}.$

#### Bound for $\mathbb{E}|A_{1}|$, far from the discontinuity.

The indicators locate the arguments of $b$ in its intervals of continuity, as
$\upchi_{m}\notin[\overline{X}_{t}\wedge\overline{X}_{\eta(t)},\overline{X}_{t}\vee\overline{X}_{\eta(t)}]$,
where $b$ is locally Lipschitz. We observe that

$\overline{X}_{t}^{2p}\left(\frac{b(\overline{X}_{t})-b(0)}{\overline{X}_{t}}-\frac{b(\overline{X}_{\eta(t)})-b(0)}{\overline{X}_{\eta(t)}}\right)^{2p}=(\overline{X}_{t}-\overline{X}_{\eta(t)})^{2p}\left\\{\frac{\overline{X}_{t}}{\overline{X}_{t}-\overline{X}_{\eta(t)}}\left(\frac{b(\overline{X}_{t})-b(0)}{\overline{X}_{t}}-\frac{b(\overline{X}_{\eta(t)})-b(0)}{\overline{X}_{\eta(t)}}\right)\right\\}^{2p}.$

From 2 in the continuity interval
$[\overline{X}_{t}\wedge\overline{X}_{\eta(t)},\overline{X}_{t}\vee\overline{X}_{\eta(t)}]$,
and 1

$\displaystyle\left|\frac{\overline{X}_{t}}{\overline{X}_{t}-\overline{X}_{\eta(t)}}\left(\frac{b(\overline{X}_{t})-b(0)}{\overline{X}_{t}}-\frac{b(\overline{X}_{\eta(t)})-b(0)}{\overline{X}_{\eta(t)}}\right)\right|=\left|\frac{b(\overline{X}_{t})-b(\overline{X}_{\eta(t)})}{{\overline{X}_{t}-\overline{X}_{\eta(t)}}}-\frac{b(\overline{X}_{\eta(t)})-b(0)}{\overline{X}_{\eta(t)}}\right|$
$\displaystyle\quad\leq\mathcal{C}_{\textit{max}}(1+\overline{X}_{t}^{{\beta}-1}+\overline{X}_{\eta(t)}^{{\beta}-1})+L_{\textit{G}}\,(\overline{X}_{\eta(t)}^{{\beta}-1}\vee
1).$

Therefore, for any $r_{2}>1$ (to be chosen later), for another constant $C$

$\mathbb{E}\left[\overline{X}_{t}^{2p}\left(\frac{b(\overline{X}_{t})-b(0)}{\overline{X}_{t}}-\frac{b(\overline{X}_{\eta(t)})-b(0)}{\overline{X}_{\eta(t)}}\right)^{2p}\right]\leq
C\left(1+\sup_{t\in[0,T]}\mathbb{E}^{\frac{r_{2}-1}{r_{2}}}\Big{[}\overline{X}_{t}^{2p({\beta}-1)\frac{r_{2}}{r_{2}-1}}\Big{]}\right)\mathbb{E}^{\frac{1}{r_{2}}}\Big{[}|\overline{X}_{t}-\overline{X}_{\eta(t)}|^{2pr_{2}}\Big{]},$

and we get, applying a Young inequality in $A_{1}$ first, (and removing the
indicators)

$\mathbb{E}[|A_{1}|]\leq
C\mathbb{E}[Y_{t}^{2p}]+C\left(1+\sup_{t\in[0,T]}\mathbb{E}^{\frac{r_{2}-1}{r_{2}}}\Big{[}\overline{X}_{t}^{2p({\beta}-1)\frac{r_{2}}{r_{2}-1}}\Big{]}\right)\|\overline{X}_{t}-\overline{X}_{\eta(t)}\|_{L^{2pr_{2}}(\Omega)}^{2p}.$

We apply Lemma 2.1 and Lemma 2.3 to control
$\|\overline{X}_{t}-\overline{X}_{\eta(t)}\|_{L^{2pr_{2}}}\leq C\Delta
t^{\frac{1}{2}}$ and
$\sup_{s\in[0,T]}\|\overline{X}_{s}\|_{L^{2p({\beta}-1)\frac{r_{2}}{r_{2}-1}}}\leq
C$, implying to satisfy the combined constraint that

$\mathds{1}_{\\{2\alpha-1\\}}({\beta})\,p\
\Big{(}r_{2}{\beta}\vee[\tfrac{r_{2}}{r_{2}-1}({\beta}-1)]\Big{)}\leq\frac{1}{2}+\frac{B_{2}}{{\text{\scriptsize{$\Upsigma$}}}^{2}},$

leading to the well balanced choice of $r_{2}=1+\frac{{\beta}-1}{{\beta}}$,
and the constraint
$\mathds{1}_{\\{2\alpha-1\\}}({\beta})\,(4\alpha-3)p\leq\frac{1}{2}+\frac{B_{2}}{{\text{\scriptsize{$\Upsigma$}}}^{2}}$
in (3.13) covered by the condition (3.11). Then

$\mathbb{E}[|A_{1}|]\leq C\mathbb{E}[Y_{t}^{2p}]+C\Delta t^{p}.$

When $b$ is continuous, $A=A_{1}$ and the latter estimation gives (3.14) under
the condition (3.13).

#### Bernstein inequality for the bound of $\mathbb{E}|A_{4}|$.

From 1, we have the following bound

$\displaystyle\overline{X}_{t}\left(\frac{b(\overline{X}_{t})-b(0)}{\overline{X}_{t}}-\frac{b(\overline{X}_{\eta(t)})-b(0)}{\overline{X}_{\eta(t)}}\right)\mathds{1}_{\\{\overline{X}_{[\eta(t),\eta(t)+\Delta
t]}\in B(\upchi_{m},\frac{\upchi_{m}}{4}+f(\Delta
t))\\}}\mathds{1}_{\\{\sup_{s\in[\eta(t),\eta(t)+\Delta
t)}|\overline{X}_{s}-\overline{X}_{\eta(t)}|\geq f(\Delta t)\\}}$
$\displaystyle\qquad\leq
C\mathds{1}_{\left\\{\overline{X}_{[\eta(t),\eta(t)+\Delta t]}\in
B(\upchi_{m},\frac{\upchi_{m}}{4}+f(\Delta
t))\right\\}}\mathds{1}_{\left\\{\sup_{s\in[\eta(t),\eta(t)+\Delta
t)}|\overline{X}_{s}-\overline{X}_{\eta(t)}|\geq f(\Delta t)\right\\}},$

where $C$ above is a polynomial on $\upchi_{m}^{{\beta}}$. So

$|A_{4}|\leq
C|Y_{t}|^{2p-1}\mathds{1}_{\\{\overline{X}_{[\eta(t),\eta(t)+\Delta t]}\in
B(\upchi_{m},\frac{\upchi_{m}}{4}+f(\Delta
t))\\}}\mathds{1}_{\\{\sup_{s\in[\eta(t),\eta(t)+\Delta
t)}|\overline{X}_{s}-\overline{X}_{\eta(t)}|\geq f(\Delta t)\\}}.$ (3.16)

Using the definition of ${\sigma^{\text{exp}}}$ in (3.10), we denote by $M$
the martingale

$M_{t}=\int_{0}^{t}\mathds{1}_{\\{\overline{X}_{[\eta(s),s)}\in
B(\upchi_{m},\frac{\upchi_{m}}{4}+f(\Delta
t))\\}}{\sigma^{\text{exp}}}(\overline{X}_{s},\overline{X}_{\eta(s)})dW_{s},$

which from 4 satisfies, for all $t\leq T$

${{\displaystyle{\langle}{M,M}\displaystyle{\rangle}}}_{t}\leq{\text{\scriptsize{$\Upsigma$}}}^{2}\int_{0}^{t}\mathbb{E}[\mathds{1}_{\\{\overline{X}_{[\eta(s),s)}\in
B(\upchi_{m},\frac{\upchi_{m}}{4}+f(\Delta t))\\}}\
\overline{X}_{\eta(s)}^{2(\alpha-1)}\overline{X}_{s}^{2}]\ ds\leq c^{2}\ t,$

with $c={\text{\scriptsize{$\Upsigma$}}}(\tfrac{3}{2}\upchi_{m})^{\alpha}$.
Then, we have

$\displaystyle\left\\{\sup_{s\in[\eta(t),\eta(t)+\Delta
t)}|\overline{X}_{s}-\overline{X}_{\eta(t)}|\geq f(\Delta t)\right\\}\ \cap\
\left\\{\overline{X}_{[\eta(t),\eta(t)+\Delta t]}\in
B(\upchi_{m},\frac{\upchi_{m}}{4}+f(\Delta t))\right\\}$
$\displaystyle\subset\left\\{\sup_{s\in[\eta(t),\eta(t)+\Delta
t)}|\int_{\eta(t)}^{s}\mathds{1}_{\\{\overline{X}_{[\eta(t),\theta)}\in
B(\upchi_{m},\frac{\upchi_{m}}{4}+f(\Delta
t))\\}}b^{\text{exp}}(\overline{X}_{\theta},\overline{X}_{\eta(t)})\
d\theta+M_{s}-M_{\eta(s)}|\geq f(\Delta t)\right\\}$
$\displaystyle\quad\subset\left\\{\sup_{s\in[\eta(t),\eta(t)+\Delta
t)}|M_{s}-M_{\eta(t)}|\geq f(\Delta t)-\Delta
t\sup_{B(\upchi_{m},\frac{\upchi_{m}}{4}+f(\Delta
t))}|b^{\text{exp}}|\right\\}$
$\displaystyle\quad\subset\left\\{\sup_{s\in[\eta(t),\eta(t)+\Delta
t)}|M_{s}-M_{\eta(s)}|\geq\frac{1}{2}f(\Delta t)\right\\},$

provided that $\sup_{x,y\in B(\upchi_{m},\frac{\upchi_{m}}{4}+f(\Delta
t))}|b^{exp}(x,y)|\leq\frac{1}{2}f(\Delta t)/\Delta t$, in addition to the
previous condition $f(\Delta t)\leq\frac{\upchi_{m}}{4}$. We now use the
Bernstein inequality (see, e.g., Revuz and Yor (Revuz and Yor, 1999,
Ex.3.16,Chap.IV, p. 153)):

$\mathbb{P}\left(\sup_{t_{k}\leq s\leq
t_{k+1}}|M_{s}-M_{\eta(s)}|\geq\frac{f(\Delta t)}{2\Delta t}\Delta
t\right)\leq\exp\left(-\dfrac{1}{c^{2}}\dfrac{f^{2}(\Delta t)}{8\Delta
t}\right).$

Thus, taking expectation and applying Young inequality in (3.16), we can use
the estimation above as follows:

$\displaystyle\mathbb{E}[|A_{4}|]$ $\displaystyle\leq C\
\mathbb{E}[Y_{t}^{2p-1}\mathds{1}_{\\{\sup_{s\in[\eta(t),\eta(t)+\Delta
t)}|M_{s}-M_{\eta(t)}|\geq\frac{f(\Delta t)}{2}\\}}]$ $\displaystyle\leq C\
\mathbb{E}[Y_{t}^{2p}]+C\ \left(\mathbb{P}\left(\sup_{t_{k}\leq s\leq
t_{k+1}}|M_{s}-M_{\eta(s)}|\geq\frac{f(\Delta t)}{2\Delta t}\Delta
t\right)\right)^{\frac{1}{2p}}$ $\displaystyle\leq C\
\mathbb{E}[Y_{t}^{2p}]+C\ \exp\left(-\dfrac{1}{c^{2}}\dfrac{f^{2}(\Delta
t)}{16p\Delta t}\right).$

Considering $f(\Delta t):=4c\ \Delta t^{1/2-\epsilon_{1}}$, for some
$\epsilon_{1}\in(0,\frac{1}{2})$, we get:

$\mathbb{E}[|A_{4}|]\leq C\ \mathbb{E}[Y_{t}^{2p}]+C\
\exp\left(-\dfrac{1}{p}\Delta t^{-2\epsilon_{1}}\right).$

Notice that, the map
$y\mapsto\exp(\frac{1}{p^{2}}\frac{2\epsilon_{1}}{1-2\epsilon_{1}}y)-y$ hits
zero, its minimum value, at
$y^{*}=\log(p^{2}\frac{(1-2\epsilon_{1})}{2\epsilon_{1}})$. So, for a given
$p$ and $\epsilon_{1}$, if $\Delta t$ is chosen smaller than
$\exp(-\frac{1}{2\epsilon_{1}}\log(\log(p^{2}\frac{(1-2\epsilon_{1})}{2\epsilon_{1}})))$,
then $\Delta t^{-2\epsilon_{1}}>y^{*}$ and the map is increasing, leading to
the inequality

$\exp\left(-\frac{1}{p}\Delta t^{-2\epsilon_{1}}\right)\leq\Delta
t^{p-2\epsilon_{1}p}.$

Putting all the constraint on $\Delta t$ together, a sufficient condition on
$\Delta t$ is that $\Delta t\leq\Delta(\epsilon_{1})$, with

$\Delta(\epsilon_{1}):=\exp(-\frac{1}{2\epsilon_{1}}\log(\log(\tfrac{1}{2\epsilon_{1}})))\
\wedge\
\left\\{\frac{1}{16{\text{\scriptsize{$\Upsigma$}}}}(\tfrac{2}{3})^{\alpha}\upchi_{m}^{1-\alpha}\right\\}^{\tfrac{2}{1-2\epsilon_{1}}}\wedge\
\left\\{\tfrac{8}{3}\frac{{\text{\scriptsize{$\Upsigma$}}}}{L_{G}}(\tfrac{3}{2}\upchi_{m})^{\alpha-{\beta}}\right\\}^{\tfrac{2}{1+2\epsilon_{1}}}\wedge\
\frac{\upchi_{m}}{4b(0)},$ (3.17)

using $f(\Delta t)\leq\upchi_{m}/4$ to impose
$\sup_{B(\upchi_{m},\frac{\upchi_{m}}{4}+f(\Delta t))}|b^{\exp}|\leq
3L_{G}(\tfrac{3}{2}\upchi_{m}^{\beta}\vee 1)\leq\tfrac{1}{2}{f(\Delta
t)}/{\Delta t}$, as required. In this case we obtain the estimation:

$\mathbb{E}[|A_{4}|]\leq C\ \mathbb{E}[Y_{t}^{2p}]+C\ \Delta
t^{p-2\epsilon_{1}p}.$

#### Markov inequality argument for the bound of $\mathbb{E}|A_{2}|$.

Applying Young inequality with 1:

$\displaystyle A_{2}=$ $\displaystyle
2Y_{t}^{2p-1}\overline{X}_{t}\left(\frac{b(\overline{X}_{t})-b(0)}{\overline{X}_{t}}-\frac{b(\overline{X}_{\eta(t)})-b(0)}{\overline{X}_{\eta(t)}}\right)\mathds{1}_{\\{\sup_{s\in[\eta(t),\eta(t)+\Delta
t)}|\overline{X}_{s}-\overline{X}_{\eta(t)}|\geq\frac{\upchi_{m}}{4}\\}}$
$\displaystyle\leq
C|Y_{t}|^{2p-1}\left(1+\overline{X}_{t}^{{\beta}}+\overline{X}_{\eta(t)}^{{\beta}}\right)\mathds{1}_{\\{\sup_{s\in[\eta(t),\eta(t)+\Delta
t)}|\overline{X}_{s}-\overline{X}_{\eta(t)}|\geq\frac{\upchi_{m}}{4}\\}}$
$\displaystyle\leq
C|Y_{t}|^{2p}+C\left(1+\overline{X}_{t}^{2p{\beta}}+\overline{X}_{\eta(t)}^{2p{\beta}}\right)\mathds{1}_{\\{\sup_{s\in[\eta(t),\eta(t)+\Delta
t)}|\overline{X}_{s}-\overline{X}_{\eta(t)}|\geq\frac{\upchi_{m}}{4}\\}}.$

For some constant exponent $r_{3}>1$,

$\displaystyle\mathbb{E}\Big{[}\left(1+\overline{X}_{t}^{2p{\beta}}+\overline{X}_{\eta(t)}^{2p{\beta}}\right)\mathds{1}_{\\{\sup_{s\in[\eta(t),\eta(t)+\Delta
t)}|\overline{X}_{s}-\overline{X}_{\eta(t)}|\geq\frac{\upchi_{m}}{4}\\}}\Big{]}$
$\displaystyle\qquad\leq\Big{(}1+\sup_{s\leq
T}\mathbb{E}^{\frac{r_{3}-1}{r_{3}}}[\overline{X}_{s}^{2p{\beta}\frac{r_{3}}{r_{3}-1}}]\Big{)}\mathbb{P}^{\frac{1}{r_{3}}}\left(\sup_{s\in[\eta(t),\eta(t)+\Delta
t)}|\overline{X}_{s}-\overline{X}_{\eta(t)}|\geq\frac{\upchi_{m}}{4}\right).$

By the Markov inequality combined with the local error in Lemma 2.3, for
$\gamma=1$, $\epsilon_{2}\in(0,\tfrac{1}{2})$ and $p$ substitute by $2pr_{3}$:

$\mathbb{P}^{\frac{1}{r_{3}}}\left(\sup_{s\in[\eta(t),\eta(t)+\Delta
t)}|\overline{X}_{s}-\overline{X}_{\eta(t)}|\geq\frac{\upchi_{m}}{4}\right)\leq\mathbb{E}^{\frac{1}{r_{3}}}\left[\sup_{s\in[\eta(t),\eta(t)+\Delta
t)}|\overline{X}_{s}-\overline{X}_{\eta(t)}|^{2pr_{3}}\right]\leq C\ \Delta
t^{p-2\epsilon_{2}p}.$

Notice that, in order to get $\|\sup_{\eta(s)\leq\theta\leq
s}|\overline{X}_{\theta}-\overline{X}_{\eta(s)}|\|_{L^{2pr_{3}}(\Omega)}\leq
C\Delta t^{{\frac{1}{2}}-\epsilon_{2}}$ from Lemma 2.3 and the upperbound of
$\sup_{s\leq
T}\|\overline{X}_{s}\|_{L^{2p{\beta}\frac{r_{3}}{r_{3}-1}}(\Omega)}$ from
Lemma 2.1, we need to assume the combined constraint that

$\mathds{1}_{\\{2\alpha-1\\}}({\beta})\,p{\beta}\left(r_{3}\frac{p+\Delta
t^{2p\epsilon_{2}}}{p}\vee\frac{r_{3}}{r_{3}-1}\right)\leq\frac{1}{2}+\frac{B_{2}}{{\text{\scriptsize{$\Upsigma$}}}^{2}},$

that coincides with the condition (3.11) with $r_{3}=1+\frac{p}{p+\Delta
t^{2p\epsilon_{2}}}$ and $\varepsilon=\epsilon_{2}$. Then, the contribution of
this terms is of the form

$\mathbb{E}|A_{2}|\leq C\mathbb{E}[Y_{t}^{2p}]+C\Delta t^{p-2\epsilon_{2}p}.$

The proof is completed by joining the four bounds for $\mathbb{E}|A_{i}|$
together and choosing $\varepsilon=\epsilon_{1}=\epsilon_{2}$ and $f(\Delta
t)=4{\text{\scriptsize{$\Upsigma$}}}(\tfrac{3}{2}\upchi_{m})^{\alpha}\Delta
t^{1/2-\varepsilon}$. ∎

###  3.3 Proof of Theorem 3.1

In this proof and as before, the positive constant $C$ may change from line to
line. It depends on $T$, $p$ and all the parameters in the hypotheses, but not
on $\Delta t$ and $\epsilon$.

To simplify the presentation, we consider in 2 only one point of discontinuity
$\upchi_{m}$ for $b$, the other cases being a sum of similar contributions.

From the five drift contributions $T_{i}$ in (3.8), the two lasts $T_{4}$ and
$T_{5}$ are residual. They can be bounded in a similar manner, with the help
of the minoration (2.3), Young and Cauchy Schawrz inequalities. Note also that
they vanish with $b(0)$. More precisely using 1,

$\displaystyle T_{5}$
$\displaystyle\leq{\text{\scriptsize{$\Upsigma$}}}^{2}\frac{p-1}{p}\Delta
t\,Y_{t}^{2p}+C\ \Delta t^{p}\ \overline{X}_{\eta(t)}^{2p(\alpha-1)},$
$\displaystyle T_{4}$ $\displaystyle\leq 2pb(0)\ \delta(t)\ |Y_{t}|^{2p-1}\
L_{\textit{G}}\,(\overline{X}_{\eta(t)}^{{\beta}-1}\vee
1)\leq\frac{2p-1}{2p}\sqrt{\Delta t}\ Y_{t}^{2p}+C\ \Delta t^{p}\
(1+\overline{X}_{\eta(t)}^{2p({\beta}-1)}),$

for some constant $C$ depending on $b(0)$, $\sigma$ and $p$. In the above
inequalities,
$\sup_{t\in[0,T]}(\mathbb{E}[\overline{X}_{t}^{2p(\alpha-1)}]+\mathbb{E}[\overline{X}_{t}^{2p({\beta}-1)}])$
is finite under (3.3) or (3.5). Therefore,

$\mathbb{E}[|T_{4}|+\mathbb{E}[|T_{5}|]\leq\left\\{\left({\text{\scriptsize{$\Upsigma$}}}^{2}\tfrac{p-1}{p}+\tfrac{2p-1}{2p}\right)\sqrt{\Delta
t}\ \mathbb{E}[Y_{t}^{2p}]+C\,\Delta
t^{p}\right\\}\mathds{1}_{\\{{b(0)}>0\\}}.$ (3.18)

The main idea of this proof is to make use of the control of some exponential
moments of $\int_{0}^{t}X_{s}^{{\beta}-1}ds$ and
$\int_{0}^{t}\overline{X}_{\overline{S}^{\Delta
t}\wedge\eta(s)}^{{\beta}-1}ds$ given in Propositions 1.3 and 2.5 by the mean
of a change in time formula in the dynamics of the error process $Y$.

For the stopped process $\overline{X}_{\cdot\wedge\overline{S}^{\Delta t}}$,
this exponential control is unconstrained on $\beta$, whereas it induces an
additional constraint on $B_{2}$ for the exact process $X$. Thus, a part of
the game consists in tracking the terms containing the factor
$Y_{t}^{2p}X_{t}^{{\beta}-1}$, and upper-bound whenever is possible, in favour
of terms containing $Y_{t}^{2p}\overline{X}_{\eta(t)}^{{\beta}-1}$.

We first state the intermediate bound for the sum $T_{1}+T_{2}+T_{3}$ in
(3.8). We consider first the term
$T_{3}=2p(2p-1)Y_{t}^{2p-2}\overline{X}_{t}^{2}\left[\frac{\sigma(X_{t})}{X_{t}}-\frac{\sigma(\overline{X}_{\eta(t)})}{\overline{X}_{\eta(t)}}\right]^{2}$,
exchanging $\overline{X}_{t}$ with $X_{t}$,

$T_{3}\leq
4p(2p-1){\text{\scriptsize{$\Upsigma$}}}^{2}Y_{t}^{2p}[X_{t}^{\alpha-1}\vee\overline{X}_{\eta(t)}^{\alpha-1}]^{2}+4p(2p-1)Y_{t}^{2p-2}(X_{t}-\overline{X}_{\eta(t)})^{2}\left[\tfrac{X_{t}}{X_{t}-\overline{X}_{\eta(t)}}\left(\tfrac{\sigma(X_{t})}{X_{t}}-\tfrac{\sigma(\overline{X}_{\eta(t)})}{\overline{X}_{\eta(t)}}\right)\right]^{2}.$

From Lemma 3.7-($\sigma$),
$\left[\frac{X_{t}}{X_{t}-\overline{X}_{\eta(t)}}\left(\frac{\sigma(X_{t})}{X_{t}}-\frac{\sigma(\overline{X}_{\eta(t)})}{\overline{X}_{\eta(t)}}\right)\right]^{2}\leq
2((\alpha{\text{\scriptsize{$\Upsigma^{\prime}$}}})^{2}+{\text{\scriptsize{$\Upsigma$}}}^{2})(X_{t}\vee\overline{X}_{\eta(t)})^{2(\alpha-1)}$.
So, with Young and Cauchy-Schwarz inequalities,

$\displaystyle\mathbb{E}[T_{3}\mathds{1}_{\\{t\leq\overline{S}^{\Delta
t}\\}}]$ $\displaystyle\leq
4p(2p-1)(5{\text{\scriptsize{$\Upsigma$}}}^{2}+4(\alpha{\text{\scriptsize{$\Upsigma^{\prime}$}}})^{2})\mathbb{E}[Y_{t}^{2p}(X_{t}\vee\overline{X}_{\eta(t)})^{2(\alpha-1)}\mathds{1}_{\\{t\leq\overline{S}^{\Delta
t}\\}}]$
$\displaystyle\qquad+16p(2p-1)((\alpha{\text{\scriptsize{$\Upsigma^{\prime}$}}})^{2}+{\text{\scriptsize{$\Upsigma$}}}^{2})\mathbb{E}\left[Y_{t}^{2p-2}(\overline{X}_{t}-\overline{X}_{\eta(t)})^{2}(X_{t}\vee\overline{X}_{\eta(t)})^{2(\alpha-1)}\mathds{1}_{\\{t\leq\overline{S}^{\Delta
t}\\}}\right]$ $\displaystyle\leq C\ \mathbb{E}[Y_{t\wedge\overline{S}^{\Delta
t}}^{2p}]+4p(2p-1)(5{\text{\scriptsize{$\Upsigma$}}}^{2}+4(\alpha{\text{\scriptsize{$\Upsigma^{\prime}$}}})^{2})\mathbb{E}[Y_{t\wedge\overline{S}^{\Delta
t}}^{2p}(X_{t\wedge\overline{S}^{\Delta
t}}\vee\overline{X}_{\eta({t\wedge\overline{S}^{\Delta t}})})^{2(\alpha-1)}]$
$\displaystyle\qquad+C\
\mathbb{E}^{1/2}\left[(\overline{X}_{t\wedge\overline{S}^{\Delta
t}}-\overline{X}_{\eta({t\wedge\overline{S}^{\Delta
t}})})^{4p}\right]\mathbb{E}^{1/2}\left[(X_{t\wedge\overline{S}^{\Delta
t}}\vee\overline{X}_{\eta({t\wedge\overline{S}^{\Delta
t}})})^{4(\alpha-1)p}\right].$

With the assumption that
$\mathds{1}_{\\{2\alpha-1\\}}({\beta})\,2p(2\alpha-1)\leq\tfrac{1}{2}+\tfrac{B_{2}}{{\text{\scriptsize{$\Upsigma$}}}^{2}}$,
which is covered by (3.3), the control of the $L^{4p}$-local error is obtained
from Proposition 2.2. This hypothesis also allowed to apply Proposition 1.3
and Lemma 2.1 for the control of the $4(\alpha-1)p$-moments. We obtain the
estimation:

$\mathbb{E}[T_{3}\mathds{1}_{\\{t\leq\overline{S}^{\Delta t}\\}}]\leq
C\mathbb{E}[Y_{t\wedge\overline{S}^{\Delta
t}}^{2p}]+4p(2p-1)(5{\text{\scriptsize{$\Upsigma$}}}^{2}+4(\alpha{\text{\scriptsize{$\Upsigma^{\prime}$}}})^{2})\mathbb{E}[Y_{t\wedge\overline{S}^{\Delta
t}}^{2p}(X_{t\wedge\overline{S}^{\Delta
t}}\vee\overline{X}_{\eta({t\wedge\overline{S}^{\Delta
t}})})^{2(\alpha-1)}]+C\Delta t^{p}.$

Next, we consider
$T_{2}(t)=2pY_{t}^{2p-1}\overline{X}_{t}\left(\frac{b(X_{t})-b(0)}{X_{t}}-\frac{b(\overline{X}_{\eta(t)})-b(0)}{\overline{X}_{\eta(t)}}\right)$.
In view of Lemma 3.4, we decompose $T_{2}$ as follows, using the notation $A$
in (3.15),

$T_{2}(t)=-2pY_{t}^{2p}\frac{b(X_{t})-b(0)}{X_{t}}+\ A\
+2pY_{t}^{2p-1}\left(b(X_{t})-b(\overline{X}_{t})\right).$

So, with
$T_{1}=2pY_{t}^{2p}(\frac{b(X_{t})-b(0)}{X_{t}}+2(2p-1)\frac{\sigma^{2}(X_{t})}{X_{t}^{2}})$,
we have

$T_{1}(t)+T_{2}(t)=4p(2p-1)Y_{t}^{2p}\frac{\sigma^{2}(X_{t})}{X_{t}^{2}}+\ A\
+2pY_{t}^{2p}\left(\frac{b(X_{t})-b(\overline{X}_{t})}{X_{t}-\overline{X}_{t}}\right).$

Applying 1 and 3, we obtain

$T_{1}(t)+T_{2}(t)\leq
4p(2p-1)Y_{t}^{2p}{\text{\scriptsize{$\Upsigma$}}}^{2}X_{t}^{2(\alpha-1)}+\ A\
+2pY_{t}^{2p}\
L_{b_{\text{loc}}}(1+X_{t}^{{\beta}-1}\vee\overline{X}_{t}^{{\beta}-1}).$

It remains to apply Lemma 3.4 for the term $A$, accompanied with the
constraint that $\Delta t\leq\Delta(\varepsilon)$, for $\Delta(\varepsilon)$
in (3.7), and $\varepsilon\in(0,\tfrac{1}{2})$, while the condition (3.11) is
covered by (3.3) (and (3.13) by (3.5) when $b$ is continuous).

Adding the estimation of $T_{3}$ above and (3.18) for $T_{4}+T_{5}$, we can
summarize the bound we get as

$\displaystyle\mathbb{E}[\sum_{i=1}^{5}T_{i}(t)]$
$\displaystyle\leq\mathbb{E}[\ \mathcal{K}_{p}(X_{t},\overline{X}_{\eta(t)})\
Y^{2p}_{t}]+C\Delta t^{p-2\varepsilon
p}+C\mathbb{E}\left[|Y_{t}|^{2p-1}\mathds{1}_{\left\\{\overline{X}_{[\eta(t),\eta(t)+\Delta
t]}\in\mathcal{B}(\Delta t^{1/2-\varepsilon})\right\\}}\right],$ (3.19)

with, from the application of Lemma 3.4, $\mathcal{B}(\Delta
t^{1/2-\varepsilon})=B(\upchi_{m},8{\text{\scriptsize{$\Upsigma$}}}(\frac{3}{2}\upchi_{m})^{\alpha}\
\Delta t^{1/2-\varepsilon})$, and

$\displaystyle\mathcal{K}_{p}(x,y)$
$\displaystyle=C+2p\mathcal{K}_{\star}(y\vee x)^{{\beta}-1},\quad\text{with
~{}}\mathcal{K}_{\star}:=L_{b_{\text{loc}}}+2(2p-1)(6{\text{\scriptsize{$\Upsigma$}}}^{2}+4(\alpha{\text{\scriptsize{$\Upsigma^{\prime}$}}})^{2}).$

We bound
$|Y_{t}|^{2p-1}\mathds{1}_{\left\\{\overline{X}_{[\eta(t),\eta(t)+\Delta
t]}\in\mathcal{B}(\Delta t^{1/2-\varepsilon})\right\\}}$ as follows,
discussing whenever $|Y_{t}|>\Delta t^{\frac{1}{2}-\varepsilon}$ or not:

$\displaystyle|Y_{t}|^{2p-1}\mathds{1}_{\\{\overline{X}_{[\eta(t),\eta(t)+\Delta
t]}\in\mathcal{B}(\Delta t^{1/2-\varepsilon})\\}}$ $\displaystyle\leq\Delta
t^{(\frac{1}{2}-\varepsilon)(2p-1)}\
\mathds{1}_{\\{\overline{X}_{[\eta(t),\eta(t)+\Delta t]}\in\mathcal{B}(\Delta
t^{1/2-\varepsilon})\\}}+|Y_{t}|^{2p}\Delta
t^{-(\frac{1}{2}-\varepsilon)}\mathds{1}_{\\{\overline{X}_{[\eta(t),\eta(t)+\Delta
t]}\in\mathcal{B}(\Delta t^{1/2-\varepsilon})\\}}.$

From Lemma 3.3 applied with the norm $L^{1}(\Omega)$ to the first term of the
above inequality and the estimation of $\mathbb{E}[\sum_{i=1}^{5}T_{i}(t)]$,
we obtain

$\mathbb{E}[Y_{t\wedge\overline{S}^{\Delta
t}}^{2p}]\leq\mathbb{E}\Big{[}\int_{0}^{t\wedge\overline{S}^{\Delta
t}}\widetilde{\mathcal{K}}_{p}(X_{s},\overline{X}_{[\eta(s),\eta(s)+\Delta
t]})\ Y_{s}^{2p}ds\Big{]}+C\ \Delta t^{p-2\varepsilon p},$

with

$\widetilde{\mathcal{K}}_{p}(X_{t},\overline{X}_{[\eta(t),\eta(t)+\Delta
t]})=\mathcal{K}_{p}(X_{t},\overline{X}_{t})+C\ \Delta
t^{-(\frac{1}{2}-\varepsilon)}\mathds{1}_{\\{\overline{X}_{[\eta(t),\eta(t)+\Delta
t]}\in\mathcal{B}(\Delta t^{1/2-\varepsilon})\\}}.$

In order to apply the Gronwall Lemma in presence of the stochastic coefficient
$\widetilde{\mathcal{K}}_{p}(X_{s},\overline{X}_{[\eta(s),\eta(s)+\Delta
t]})$, we define now the stopping time

$\tau_{\lambda}=\inf\\{t\geq 0:~{}{\bf\textrm{H}}(t)\geq\lambda\\}\wedge
T\wedge\overline{S}^{\Delta t},$

such that ${\bf\textrm{H}}(\tau_{\lambda})=\lambda$, when
${\bf\textrm{H}}^{-1}(\lambda)\leq T\wedge\overline{S}^{\Delta t}$, with
$t\mapsto{\bf\textrm{H}}(t)$ defined as

${\bf\textrm{H}}(t)=\int_{0}^{t}\widetilde{\mathcal{K}}_{p}(X_{s},\overline{X}_{[\eta(s),\eta(s)+\Delta
t]})\,ds.$ (3.20)

For any $\lambda>0$, through the change of time $u={\bf\textrm{H}}(s)$,
$\mathbb{E}\left[\int_{0}^{\tau_{\lambda}}Y_{s}^{2p}d{\bf\textrm{H}}(s)\right]=\mathbb{E}\left[\int_{0}^{\lambda}Y_{\tau_{u}}^{2p}du\right]$,
and we obtain

$\mathbb{E}[Y_{\tau_{\lambda}}^{2p}]\leq\mathbb{E}\left[\int_{0}^{\lambda}Y_{\tau_{u}}^{2p}du\right]+C\Delta
t^{p-2\varepsilon p}.$

From Gronwall inequality we derive the first estimation

$\displaystyle\mathbb{E}[Y_{\tau_{\lambda}}^{2p}]$ $\displaystyle\leq C\
\Delta t^{p-2\varepsilon p}\ \exp\\{\lambda\\}.$ (3.21)

Now, choosing $\lambda={\bf\textrm{H}}({T\wedge\overline{S}^{\Delta t}})$,

$\displaystyle\mathbb{E}[Y_{T\wedge\overline{S}^{\Delta
t}}^{2p}]\leq\mathbb{E}\left[\int_{0}^{{\bf\textrm{H}}({T\wedge\overline{S}^{\Delta
t}})}Y_{\tau_{u}}^{2p}du\right]+C\Delta t^{p-2\varepsilon p}.$

It remains to bound the first term in the right-hand side above. Applying
Hölder inequality with exponent $\frac{p+1}{p}$,

$\mathbb{E}\left[\int_{0}^{{\bf\textrm{H}}({T\wedge\overline{S}^{\Delta
t}})}Y_{\tau_{u}}^{2p}du\right]\leq\int_{0}^{+\infty}\left(\mathbb{P}({\bf\textrm{H}}({T\wedge\overline{S}^{\Delta
t}})\geq
u)\right)^{\tfrac{1}{p+1}}~{}\mathbb{E}^{\tfrac{p}{p+1}}\left[Y_{\tau_{u}}^{2(p+1)}\right]du.$

By using the preliminary upper bound (3.21) (increasing the sufficient
condition in Lemma 3.4 to get this upper bound for
$\mathbb{E}[Y_{\tau_{u}}^{2(p+1)}]$, with
$\mathds{1}_{\\{2\alpha-1\\}}({\beta})\,2(p+1+\Delta t^{2(p+1)\varepsilon})\
(2\alpha-1)\leq\tfrac{1}{2}+\tfrac{B_{2}}{{\text{\scriptsize{$\Upsigma$}}}^{2}}$),
we get

$\mathbb{E}^{\tfrac{p}{p+1}}\left[Y_{\tau_{u}}^{2(p+1)}\right]\leq C\ \Delta
t^{p-2\varepsilon p}\ \exp\Big{\\{}u\frac{p}{p+1}\Big{\\}}.$

Next, applying Markov’s inequality, we observe that, for any $\mu>0$,

$\mathbb{P}({\bf\textrm{H}}({T\wedge\overline{S}^{\Delta t}})\geq
u)\leq\mathbb{E}[\exp\\{\mu\ {\bf\textrm{H}\,}({T\wedge\overline{S}^{\Delta
t}})\\}]\ \exp(-\mu u).$

Thus, considering $\mu>p$, we obtain

$\displaystyle\mathbb{E}\left[\int_{0}^{{\bf\textrm{H}}({T\wedge\overline{S}^{\Delta
t}})}Y_{\tau_{u}}^{2p}du\right]$ $\displaystyle\leq C\ \Delta
t^{p-2\varepsilon p}\ \mathbb{E}^{\frac{1}{p+1}}[\exp\\{\mu\
{\bf\textrm{H}\,}({T\wedge\overline{S}^{\Delta
t}})\\}]\int_{0}^{+\infty}\exp\left\\{\frac{u\
p}{p+1}-\mu\frac{u}{p+1}\right\\}du$ $\displaystyle\quad=C\ \Delta
t^{p-2\varepsilon p}\ \left(\frac{p+1}{\mu-p}\right)\
\mathbb{E}^{\frac{1}{p+1}}[\exp\\{\mu\
{\bf\textrm{H}\,}({T\wedge\overline{S}^{\Delta t}})\\}],$

and therefore, for an updated constant $C$, for all $t\in[0,T]$,

$\mathbb{E}[Y_{t\wedge\overline{S}^{\Delta t}}^{2p}]\leq C\ \Delta
t^{p-2\varepsilon p}\left(1+\left(\frac{p+1}{\mu-p}\right)\
\mathbb{E}^{\frac{1}{p+1}}\Big{[}\exp\Big{\\{}\mu\
{\bf\textrm{H}\,}({T\wedge\overline{S}^{\Delta t}})\Big{\\}}\Big{]}\right).$
(3.22)

Applying the Hölder inequality with exponent $\epsilon_{0}>1$,

$\displaystyle\mathbb{E}[\exp\\{\mu{\bf\textrm{H}\,}({T\wedge\overline{S}^{\Delta
t}})\\}]$ $\displaystyle\quad\leq C\
\mathbb{E}\left[\exp\left\\{\mu\,2p\mathcal{K}_{\star}\int_{0}^{T\wedge\overline{S}^{\Delta
t}}(\overline{X}_{\eta(s)}\vee
X_{s})^{{\beta}-1}ds+\mu\int_{0}^{T\wedge\overline{S}^{\Delta t}}\Delta
t^{-(\frac{1}{2}-\varepsilon)}\mathds{1}_{\\{\overline{X}_{[\eta(t),\eta(t)+\Delta
t]}\in\mathcal{B}(\Delta t^{1/2-\varepsilon})\\}}ds\right\\}\right]$
$\displaystyle\quad\leq
C\,\mathbb{E}^{\tfrac{1}{\epsilon_{0}}}\left[\exp\left\\{\mu\epsilon_{0}\
2p\mathcal{K}_{\star}\int_{0}^{T\wedge\overline{S}^{\Delta
t}}(\overline{X}_{\eta(s)}\vee X_{s})^{{\beta}-1}ds\right\\}\right]$
$\displaystyle\qquad\qquad\times\
\mathbb{E}^{\frac{\epsilon_{0}-1}{\epsilon_{0}}}\left[\exp\left\\{\mu\tfrac{\epsilon_{0}}{\epsilon_{0}-1}\int_{0}^{T\wedge\overline{S}^{\Delta
t}}\Delta
t^{-(\frac{1}{2}-\varepsilon)}\mathds{1}_{\\{\overline{X}_{[\eta(t),\eta(t)+\Delta
t]}\in\mathcal{B}(\Delta t^{1/2-\varepsilon})\\}}ds\right\\}\right].$

From Lemma 3.3, we derive the following bound for the second exponential term:

$\mathbb{E}\left[\exp\left\\{\mu\tfrac{\epsilon_{0}}{\epsilon_{0}-1}\Delta
t^{-(\frac{1}{2}-\varepsilon)}\int_{0}^{t}\mathds{1}_{\\{\overline{X}_{[\eta(s),\eta(s)+\Delta
t]}\in\mathcal{B}(\Delta
t^{1/2-\varepsilon})\\}}ds\right\\}\right]\leq\exp\left\\{C\,\mu\tfrac{\epsilon_{0}}{\epsilon_{0}-1}\right\\}.$

It remains to apply Proposition 2.5 for the first term, provided that $\mu>p$
and $\epsilon_{0}>1$. Taking $\epsilon_{0}=1+\tfrac{1}{p}$ and $\mu=p+1$,
imposing

$2(p+1)^{2}\mathcal{K}_{\star}<B_{2},\quad\mbox{when
}(b(0),\alpha)\notin(0,\tfrac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2})\times[1,\tfrac{3}{2}],$

we obtain

$\sup_{t\in[0,T]}\mathbb{E}[Y_{t\wedge\overline{S}^{\Delta t}}^{2p}]\leq
C~{}\Delta t^{p-2\varepsilon p},$

for any $\varepsilon\in(0,1/2)$ and $\Delta t\leq\Delta(\varepsilon)$.

Applying the same reasoning as in Remark 1.4-(ii), with the help of the
Lenglart’s inequality, we obtain

$\mathbb{E}\left[\sup_{0\leq t\leq T}Y_{t\wedge\overline{S}^{\Delta
t}}^{p}\right]\leq C\,\Delta t^{\frac{p}{2}(1-2\varepsilon)}.$

Here we choose $\frac{p}{\epsilon}=\frac{1}{2}$, in order to avoid further
reinforcing the conditions of convergence.

When $b$ is continuous, the proof is simpler. Repeating the steps, to apply
Proposition 2.2 and Lemma 3.4, we need only that
$\mathds{1}_{\\{2\alpha-1\\}}({\beta})2(p+1)(2\alpha-1)\leq\tfrac{1}{2}+\tfrac{B_{2}}{{\text{\scriptsize{$\Upsigma$}}}^{2}}$
to get

$\mathbb{E}[Y_{t\wedge\overline{S}^{\Delta
t}}^{2p}]\leq\mathbb{E}\Big{[}\int_{0}^{t\wedge\overline{S}^{\Delta
t}}\left(C+2p\
\mathcal{K}_{\star}(X_{s}\vee\overline{X}_{\eta(s)})^{{\beta}-1}\right)Y_{s}^{2p}ds\Big{]}+C\Delta
t^{p}.$

Proceeding as before with the change of time argument, we end with the
expression (3.22) when $\varepsilon=0$, and

$\displaystyle\mathbb{E}[\exp\\{\mu{\bf\textrm{H}\,}({T\wedge\overline{S}^{\Delta
t}})\\}]\leq C\ \mathbb{E}\left[\exp\left\\{\mu\,2p\
\mathcal{K}_{\star}\int_{0}^{T\wedge\overline{S}^{\Delta
t}}(\overline{X}_{\eta(s)}\vee X_{s})^{{\beta}-1}ds\right\\}\right].$

Taking $\mu=p+1$, imposing $2p\ (p+1)\ \mathcal{K}_{\star}<B_{2}$, when
$(b(0),\alpha)\notin(0,\tfrac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2})\times[1,\tfrac{3}{2}]$,
we obtain

$\sup_{t\in[0,T]}\mathbb{E}[Y_{t\wedge\overline{S}^{\Delta t}}^{2p}]\leq
C~{}\Delta t^{p}.$

As before, from Lenglart’s inequality with $r=\frac{1}{2}$ and under the same
conditions, we get

$\mathbb{E}\left[\sup_{0\leq t\leq T}Y_{t\wedge\overline{S}^{\Delta
t}}^{p}\right]\leq C\,\Delta t^{p/2}.$

###  3.4 Strong convergence rate of the unstopped scheme

In exchange for more control over the drift behaviour, we extend the
convergence result to the unstopped scheme. We also extend the parameter
domain of Theorem 3.1 for which we prove the convergence rate. Similarly to
the weak error analysis in Bossy et al. (2021), we introduce some smoothness
on $b$ such that, at least for large values of $x$, $b^{\prime}(x)$ exists and
inherits from the inward effect assumed in 3. In other words, $b^{\prime}(x)$
is bounded by a polynomial function of order ${\beta}-1$, with negative main
coefficient of large enough amplitude:

######  Hypothesis 5.

Control on $b^{\prime}$. (5). For the same $L_{\textit{G}}\,$ constant in 1,
there exist positive constants ${B_{1}^{\prime}}$ and a point
$\upzeta>\upchi_{m}$, such that, for any $x\geq\upzeta$ (so outside any
discontinuity interval of $b$),

$b^{\prime}(x)\leq{B_{1}^{\prime}}-{\beta}L_{\textit{G}}\,\ x^{{\beta}-1}.$
(3.23)

A typical case where 3 and 5 are together is when $b$ is a polynomial
function, and so 5 is just the derivative of 3. However, it is easy to get out
of this type of case, for example when the polynomial function has oscillating
coefficients (see Section 4). Under this additional hypothesis, we prove the
following convergence result, the proof of which is detailed in Appendix A.1.

######  Theorem 3.5.

Assume 2, 1, 3, 5 and 4. For all $0<\varepsilon<\frac{1}{2}$, there exists
$0<\Delta(\varepsilon)\leq 1$, estimated with (3.7), such that for all $\Delta
t\leq\Delta(\varepsilon)$, for $p\geq 1$ any integer satisfying

$\displaystyle\mathds{1}_{\\{2\alpha-1\\}}({\beta})\,\left(\\{(\ 2(p+1)+\Delta
t^{2(p+1)\varepsilon})\ (2\alpha-1)\\}\vee\\{2\ (2(p+1)-\tfrac{3}{4})\\}\
\vee\
\\{2\alpha(p+1)\\}\right)\leq\frac{1}{2}+\frac{B_{2}}{{\text{\scriptsize{$\Upsigma$}}}^{2}}$
(3.24)
$\displaystyle\quad\mbox{and}\quad\mathds{1}_{\\{2\alpha-1\\}}({\beta})\
4(2(p+1)-1)({\text{\scriptsize{$\Upsigma$}}}^{2}+(\alpha{\text{\scriptsize{$\Upsigma^{\prime}$}}})^{2})\leq
L_{G}\ (({\beta}-1)\wedge 1).$ (3.25)

there exists some positive constant $C(p)$, independent of $\Delta t$ and
$\varepsilon$, such that

$\sup_{t\in[0,T]}\left\|X_{t}-\overline{X}_{t}\right\|_{L^{2p}(\Omega)}+\Big{\|}\sup_{t\in[0,T]}|X_{t}-\overline{X}_{t}|\Big{\|}_{L^{p}(\Omega)}\leq
C(p)\,\Delta t^{\frac{1}{2}-\varepsilon}.$

Furthermore, when $b$ is continuous on $\mathbb{R}^{+}$, replacing (3.25) and
(3.24) by the condition

$\mathds{1}_{\\{2\alpha-1\\}}({\beta})\,\left(\\{p\
(4\alpha-3)\\}\vee\\{2\alpha p\\}\vee\\{2\
(2p-\tfrac{3}{4})\\}\right)\leq\frac{1}{2}+\frac{B_{2}}{{\text{\scriptsize{$\Upsigma$}}}^{2}},$
(3.26)

the following convergence takes place for $\Delta t\leq 1$:

$\sup_{t\in[0,T]}\left\|X_{t}-\overline{X}_{t}\right\|_{L^{2p}(\Omega)}+\Big{\|}\sup_{t\in[0,T]}|X_{t}-\overline{X}_{t}|\Big{\|}_{L^{p}(\Omega)}\leq
C(p)\,\Delta t^{\frac{1}{2}}.$

######  Remark 3.6.

The weak error analysis in Bossy et al. (2021) gives a convergence (with a
rate one) whenever $b$ in $C^{4}$,
$\frac{B_{2}}{{\text{\scriptsize{$\Upsigma$}}}^{2}}+\frac{1}{2}>5\alpha+C$,
where the positive constant $C$ depends on the polynomial decays of the
successive derivatives of $b$. Moreover, some of the parameters were
forbidden. Typically, when $b(0)>0$, the theoretical weak convergence was
proven under the condition that $\alpha>3/2$. With Theorem 3.5, the quadratic
error is going to zero (with rate 1/2) for a larger set of parameters,
allowing all the values of $b(0)\geq 0$ for all the values of $\alpha>1$.

The following lemma, the proof of which is postponed in Appendix A.2,
summarizes how we use the hypotheses 5 and 4 in order to derive the strong
error. Hypothesis 5 strengthens 3, putting even more weight on the inward
behaviour of the drift $b$. The factor ${\beta}$ in front of
$L_{\textit{G}}\,$ dramatically simplifies the proof of this technical Lemma,
by using the conservation of inequality by integration. On the other hand,
hypothesis 4 maintains the polynomial growth of the map
$x\mapsto\tfrac{\sigma(x)}{x}$ while keeping the driven power in terms of the
parameter $\alpha$.

######  Lemma 3.7.

  * $(b)$.

Assume 2, 1 and 5. For all $x,y\in(0,+\infty)$, we have

$\displaystyle\frac{y}{x-y}\left(\frac{b(x)-b(0)}{x}-\frac{b(y)-b(0)}{y}\right)\leq\mathcal{L}^{\scriptsize{\tiny{0}⃝}}-\
L_{\textit{G}}\,(({\beta}-1)\wedge 1)\ y\ (x\vee y)^{{\beta}-2},$ (3.27)

and, with $\upzeta$ given in 5, such that $\upzeta>\upchi_{m}$, we estimate
the non-negative constant $\mathcal{L}^{\scriptsize{\tiny{0}⃝}}$ as

$\displaystyle\mathcal{L}^{\scriptsize{\tiny{0}⃝}}$
$\displaystyle=\left({B_{1}^{\prime}}\vee[L_{b_{\text{loc}}}(\upzeta^{{\beta}-1}+1)]+L_{\textit{G}}\,({\beta}\upzeta^{{\beta}-1}+1)\right)\
\vee\
\big{(}L_{b_{\text{loc}}}+L_{\textit{G}}\,\big{)}((\upzeta^{{\beta}-1}\vee
1)+1).$ (3.28)

When (3.23)-5 is satisfied in the whole space $\mathbb{R}^{+}$, then $b$ is
locally Lipschitz on $\mathbb{R}^{+}$, and (3.27) is valid with the constant
$\mathcal{L}^{\scriptsize{\tiny{0}⃝}}=\left({{B_{1}^{\prime}}}+L_{\textit{G}}\,\right)$.

  * $(\sigma)$.

Assume 4. Then for all $x,y$ in $\mathbb{R}^{+}$

$\frac{y}{(x-y)^{2}}\left(\frac{\sigma(x)}{x}-\frac{\sigma(y)}{y}\right)^{2}\leq
2({\text{\scriptsize{$\Upsigma$}}}^{2}+\alpha^{2}{\text{\scriptsize{$\Upsigma^{\prime}$}}}^{2})\
(x\vee y)^{2\alpha-3}.$ (3.29)

###  3.5 Asymptotic behaviour and stability of the Exp-EM scheme on a
prototypical case

Understanding the asymptotic behaviour of a stochastic system is crucial in
various applications as it enables the identification of suitable actions to
be taken in different situations. For instance, it is relevant in modelling
asset returns or volatility Ait-Sahalia (1996); Higham et al. (2010),
estimating the proportion of individuals infected during an epidemic Gray et
al. (2011), and studying the equilibrium regime in turbulent kinetic energy
modeling Bossy et al. (2022). Equally important is the guarantee that an
approximation scheme will behave like the model.

We restrict our analysis to the prototypical case where the SDE (1.1) has
drift $b$ and diffusion $\sigma$ of the form

$b(x)=b(0)+B_{1}x-B_{2}\ x^{2\alpha-1}\quad\mbox{and
}\quad\sigma(x)={\text{\scriptsize{$\Upsigma$}}}x^{\alpha}.$ (3.30)

The stability results presented in this section can be extended
straightforwardly to the case ${\beta}>2\alpha-1$. The following proposition
states that the solution $X$ of (1.1), with (3.30), approaches a steady state
$\xi^{*}$ an infinite number of times in the asymptotic regime. The same
behaviour is expected to be reached by the approximation $\overline{X}$, as
the time-step $\Delta t$ tends to zero.

######  Proposition 3.8.

Let $B_{1},b(0)$ some non-negative constants and $B_{2}>0$, with
$2(\alpha-1)\leq 1+2\tfrac{B_{2}}{{\text{\scriptsize{$\Upsigma$}}}^{2}}$. Then
the solution of (1.1) with the coefficients $b$ and $\sigma$ defined in (3.30)
satisfies

$\liminf_{t\rightarrow+\infty}X_{t}\leq\xi^{*}\qquad\mbox{ and
}\qquad\limsup_{t\rightarrow+\infty}X_{t}\geq\xi^{*},\qquad\mathbb{P}\mbox{-a.s.}$

where $\xi^{*}$ is the unique positive solution of
$\\{x\in\mathbb{R}^{+};b(x)/x-\tfrac{1}{2}(\sigma(x)/x))^{2}=0\\}$.

###### Proof.

For the sake of completeness, we describe below the main steps of the proof;
the details are obtained by following the ideas proposed in Gray et al.
(2011).

The Itô formula applied to the map $\log(X_{t})$ in $[0,t]$ leads to

$\frac{\log(X_{t})}{t}=\frac{\log(x)}{t}+\frac{1}{t}\int_{0}^{t}\varphi(X_{s})ds+\frac{{\text{\scriptsize{$\Upsigma$}}}}{t}\int_{0}^{t}X_{s}^{\alpha-1}dW_{s},$
(3.31)

where
$\varphi(x)=\frac{b(0)}{x}+B_{1}-\left(B_{2}+\frac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2}\right)x^{2(\alpha-1)}$.
Notice that $\varphi^{\prime}(x)<0$ for all $x>0$, implying that

$\lim_{x\rightarrow+\infty}\varphi(x)<0<\lim_{x\rightarrow 0^{+}}\varphi(x).$

Therefore, there exists a unique positive $\xi^{*}$ solution to
$\\{x;\varphi(x)=0\\}$. Moreover $\varphi$ changes signs in the neighbourhood
of $\xi^{*}$.

Let us assume that $\limsup_{t\rightarrow+\infty}X_{t}<\xi^{*}$. Then we
consider $\epsilon>0$ small enough such that, for
$\omega_{0}\in\\{\omega:~{}\limsup_{t\rightarrow+\infty}X_{t}(\omega)\leq\xi^{*}-2\epsilon\\}$
there exists $T(\omega_{0})$ such that,

$X_{t}(\omega_{0})\leq\xi^{*}-\epsilon,\mbox{ for all }t\geq T(\omega_{0}).$

Since $x\mapsto\varphi(x)$ is non-increasing, we obtain

$\varphi(X_{t}(\omega_{0}))\geq\varphi(\xi^{*}-\epsilon),\mbox{ for all }t\geq
T(\omega_{0}).$ (3.32)

From the Law of Large Numbers (LLN) for continuous martingales, we also have

$\omega_{0}\in\\{\omega:~{}\limsup_{t\rightarrow+\infty}X_{t}(\omega)\leq\xi^{*}-2\epsilon,\quad\mbox{
and
}\quad\lim_{t\rightarrow+\infty}\tfrac{1}{t}\int_{0}^{t}X_{s}^{\alpha-1}(\omega)dW_{s}(\omega)=0\\}.$

Then, from Equation (3.31) and (3.32), we obtain

$\displaystyle\liminf_{t\rightarrow+\infty}\frac{\log(X_{t}(\omega_{0}))}{t}\geq\liminf_{t\rightarrow+\infty}\frac{1}{t}\int_{0}^{T(\omega_{0})}\varphi(X_{s}(\omega_{0}))ds+\liminf_{t\rightarrow+\infty}\frac{1}{t}\int_{T(\omega_{0})}^{t}\varphi(\xi^{*}-\epsilon)ds=\varphi(\xi^{*}-\epsilon)>0.$

The above implies that
$\limsup_{t\rightarrow+\infty}X_{t}(\omega_{0})\geq\liminf_{t\rightarrow+\infty}X_{t}(\omega_{0})=+\infty,$
which is a contradiction since we assumed
$\limsup_{t\rightarrow+\infty}X_{t}<\xi^{*}<+\infty$. Hence, it must be the
case that, $\mathbb{P}$-a.s., $\limsup_{t\rightarrow+\infty}X_{t}\geq\xi^{*}$.

Following a similar argument, it can be shown that
$\liminf_{t\rightarrow+\infty}X_{t}\leq\xi^{*}$. ∎

######  Remark 3.9.

For a more general drift $b$ satisfying 1, it is still possible to show that
the function $\psi(x)=\frac{b(x)}{x}-\tfrac{1}{2}(\sigma(x)/x)^{2}$ satisfies
$\lim_{x\rightarrow+\infty}\psi(x)<0$ and $\lim_{x\rightarrow
0^{+}}\psi(x)>0$, i.e. $\psi(x)=0$ has at least one solution $\xi^{*}$.
However, in order to extend the results from Proposition 3.8, we need
$\xi^{*}$ to be unique. For this we could consider some additional hypotheses
on the growth of $\tfrac{d}{dx}\left(b(x)/x\right)$ or impose the uniqueness
for the root of $\psi$.

####  3.5.1 Stability of the Exp-EM scheme

For the prototypical case (3.30), given an homogeneous $N$-partition of the
time interval $[0,T]$ with time-step $\Delta t=t_{n+1}-t_{n}$, the Exp-EM
scheme simplifies as

$d\overline{X}_{t}=\left(\overline{X}_{t}-b(0){\delta(t)}\right)\Big{[}(B_{1}-B_{2}\overline{X}_{\eta(t)}^{2(\alpha-1)})\
dt+{\text{\scriptsize{$\Upsigma$}}}\
\overline{X}_{\eta(t)}^{\alpha-1}dW_{t}\Big{]}+b(0)dt.$ (3.33)

######  Proposition 3.10.

Let $B_{1},b(0)$ some non-negative constants, $B_{2}>0$ such that
$2(\alpha-1)\leq 1+2\tfrac{B_{2}}{{\text{\scriptsize{$\Upsigma$}}}^{2}}$ and
$\Delta t\leq\frac{1}{B_{1}}$. Then, the scheme $\overline{X}$ solution to
(3.33) satisfies

$\liminf_{t\rightarrow+\infty}\overline{X}_{t}\leq\xi_{\Delta t},\qquad\mbox{
and }\qquad\limsup_{t\rightarrow+\infty}\overline{X}_{t}\geq\xi^{\Delta
t},\qquad\mathbb{P}\mbox{-a.s.}$

where $\xi^{\Delta t}$ is the unique positive solution of
$\\{x\in\mathbb{R}^{+};\varphi^{\Delta
t}(x):=\frac{b(x)}{x}-\tfrac{1}{2}(\frac{\sigma(x)}{x}))^{2}-\frac{b(0)}{x}B_{1}\Delta
t=0\\}$, and $\xi_{\Delta t}$ is the unique positive solution of
$\\{x\in\mathbb{R}^{+};\varphi_{\Delta
t}(x):=\frac{b(x)}{x}-\tfrac{1}{2}(\frac{\sigma(x)}{x}))^{2}+b(0)({\text{\scriptsize{$\Upsigma$}}}^{2}+B_{2})x^{2\alpha-3}\Delta
t=0\\}$.

###### Proof.

Define the map
$\varphi(t,x)=B_{1}-\left(1-\frac{b(0)\delta(t)}{x}\right)\left(\frac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2}+B_{2}-\frac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2}\frac{b(0)\delta(t)}{x}\right)x^{2(\alpha-1)}+\tfrac{b(0)(1-B_{1}\Delta
t)}{x}$. Since, for all $t>0$,
$\overline{X}_{t}>\overline{X}_{t}-b(0)\delta(t)\geq 0$, we have

$\varphi(t,\overline{X}_{t})\geq\varphi(0,\overline{X}_{t})=\varphi^{\Delta
t}(\overline{X}_{t}).$ (3.34)

Applying Itô formula and reorganizing some terms, we get

$\displaystyle\log(\overline{X}_{t})$
$\displaystyle=\log(x)+\int_{0}^{t}\frac{\left(\overline{X}_{s}-b(0){\delta(s)}\right)}{\overline{X}_{s}}\left(B_{1}-B_{2}\overline{X}_{\eta(s)}^{2(\alpha-1)}\right)ds+\int_{0}^{t}\frac{b(0)}{\overline{X}_{s}}ds+\int_{0}^{t}\frac{{\text{\scriptsize{$\Upsigma$}}}\left(\overline{X}_{s}-b(0){\delta(s)}\right)}{\overline{X}_{s}}\overline{X}_{\eta(s)}^{\alpha-1}dW_{s}$
$\displaystyle\qquad\qquad-\frac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2}\int_{0}^{t}\frac{\left(\overline{X}_{t}-b(0){\delta(s)}\right)^{2}}{\overline{X}_{s}^{2}}\overline{X}_{\eta(s)}^{2(\alpha-1)}ds$
$\displaystyle=\log(x)+\int_{0}^{t}\left[B_{1}-\frac{\left(\overline{X}_{s}-b(0){\delta(s)}\right)}{\overline{X}_{s}}\left(\frac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2}\frac{\left(\overline{X}_{s}-b(0){\delta(s)}\right)}{\overline{X}_{s}}+B_{2}\right)\overline{X}_{\eta(s)}^{2(\alpha-1)}\right]ds+\int_{0}^{t}\frac{b(0)(1-B_{1}\delta(s))}{\overline{X}_{s}}ds$
$\displaystyle\qquad\qquad+\int_{0}^{t}\frac{{\text{\scriptsize{$\Upsigma$}}}\left(\overline{X}_{s}-b(0){\delta(t)}\right)}{\overline{X}_{s}}\overline{X}_{\eta(s)}^{\alpha-1}dW_{s}.$

If we assume $\Delta t$ is such that $\Delta t\leq\frac{1}{B_{1}}$, then, for
all $t\geq 0$, $1-B_{1}\delta(t)\geq 1-B_{1}\Delta t\geq 0$ and

$\displaystyle\log(\overline{X}_{t})$
$\displaystyle\geq\log(x)+\int_{0}^{t}\left[B_{1}-\frac{\left({\overline{X}_{s}\vee\overline{X}_{\eta(s)}}-b^{+}(0){\delta(s)}\right)}{{\overline{X}_{s}\vee\overline{X}_{\eta(s)}}}\left(\frac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2}\frac{\left({\overline{X}_{s}\vee\overline{X}_{\eta(s)}}-b^{+}(0){\delta(s)}\right)}{\overline{X}_{s}\vee\overline{X}_{\eta(s)}}+B_{2}\right)({\overline{X}_{s}\vee\overline{X}_{\eta(s)}})^{2(\alpha-1)}\right]ds$
$\displaystyle\qquad+\int_{0}^{t}\frac{b^{+}(0)(1-B_{1}\Delta
t)}{\overline{X}_{s}\vee\overline{X}_{\eta(s)}}ds+\int_{0}^{t}\frac{{\text{\scriptsize{$\Upsigma$}}}\left(\overline{X}_{s}-b^{+}(0){\delta(t)}\right)}{\overline{X}_{s}}\overline{X}_{\eta(s)}^{\alpha-1}dW_{s}$
$\displaystyle\quad=\log(x)+\int_{0}^{t}\varphi(s,\overline{X}_{s}\vee\overline{X}_{\eta(s)})ds+\int_{0}^{t}\frac{{\text{\scriptsize{$\Upsigma$}}}\left(\overline{X}_{s}-b^{+}(0){\delta(t)}\right)}{\overline{X}_{s}}\overline{X}_{\eta(s)}^{\alpha-1}dW_{s}$
$\displaystyle\qquad\geq\log(x)+\int_{0}^{t}\varphi^{\Delta
t}(\overline{X}_{s}\vee\overline{X}_{\eta(s)})ds+\int_{0}^{t}\frac{{\text{\scriptsize{$\Upsigma$}}}\left(\overline{X}_{s}-b^{+}(0){\delta(t)}\right)}{\overline{X}_{s}}\overline{X}_{\eta(s)}^{\alpha-1}dW_{s},$

where the last inequality is due to (3.34).

It is easy to verify that the map $x\mapsto\varphi^{\Delta t}(x)$ is strictly
decreasing in $(0,+\infty)$ and satisfies

$\lim_{x\rightarrow+\infty}\varphi^{\Delta t}(x)<0<\lim_{x\rightarrow
0^{+}}\varphi^{\Delta t}(x).$

Therefore, there exists a unique (positive) solution $\xi^{\Delta t}$ to the
equation $\varphi^{\Delta t}(\xi^{\Delta t})=0,$ and $\varphi^{\Delta
t}(\xi^{\Delta t}+\epsilon)<0<\varphi^{\Delta t}(\xi^{\Delta t}-\epsilon)$ for
any $\epsilon>0.$

We assume now that $\limsup_{t\rightarrow+\infty}\overline{X}_{t}<\xi^{\Delta
t}$, and consider $\epsilon>0$ such that

$\mathbb{P}(\\{\omega:~{}\limsup_{t\rightarrow+\infty}\overline{X}_{\eta(t)}(\omega)=\limsup_{t\rightarrow+\infty}\overline{X}_{t}(\omega)\leq\xi^{\Delta
t}-2\epsilon\\})>\epsilon.$

Take the pair $(\omega_{0},T(\omega_{0}))$ and $\epsilon$ small enough such
that, for all $t\geq T(\omega_{0}),$
$\overline{X}_{t}(\omega_{0})\vee\overline{X}_{\eta(t)}(\omega_{0})\leq\xi^{\Delta
t}-\epsilon$. From the monotonicity of $\varphi^{\Delta t}(\cdot)$, we obtain

$\varphi^{\Delta
t}(\overline{X}_{t}(\omega_{0})\vee\overline{X}_{\eta(t)}(\omega_{0}))\geq\varphi^{\Delta
t}(\xi^{\Delta t}-\epsilon)>0,\qquad\mbox{for all }t\geq T(\omega_{0}).$
(3.35)

Thus, from LLN for continuous martingales, we can consider $\omega_{0}$ such
that,

$\liminf_{t\rightarrow+\infty}\frac{\log(\overline{X}_{t}(\omega_{0}))}{t}\geq\liminf_{t\rightarrow+\infty}\frac{1}{t}\int_{T(\omega_{0})}^{t}\varphi^{\Delta
t}(\xi^{\Delta t}-\epsilon)ds=\varphi(0,\xi^{\Delta t}-\epsilon)>0,$

implying $\limsup_{t\rightarrow+\infty}\overline{X}_{t}(\omega_{0})=+\infty,$
and contradicting the inequality
$\limsup_{t\rightarrow+\infty}\overline{X}_{t}<\xi^{\Delta t}$.

In order to prove
$\liminf_{t\rightarrow+\infty}\overline{X}_{t}\leq\xi_{\Delta t}$ we first
bound $\log(\overline{X}_{t})$ from above:

$\displaystyle\log(\overline{X}_{t})$
$\displaystyle=\log(x)+\int_{0}^{t}\left[B_{1}-\left(1-\tfrac{b(0){\delta(s)}}{\overline{X}_{s}}\right)\left(\frac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2}+B_{2}-\frac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2}\frac{b(0){\delta(s)}}{\overline{X}_{s}}\right)\overline{X}_{\eta(s)}^{2(\alpha-1)}\right]ds+\int_{0}^{t}\frac{b(0)(1-B_{1}\delta(s))}{\overline{X}_{s}}ds$
$\displaystyle\qquad\qquad+\int_{0}^{t}\frac{{\text{\scriptsize{$\Upsigma$}}}\left(\overline{X}_{s}-b(0){\delta(t)}\right)}{\overline{X}_{s}}\overline{X}_{\eta(s)}^{\alpha-1}dW_{s}$
$\displaystyle\leq\log(x)+\int_{0}^{t}\left[B_{1}-\left(1-\tfrac{b(0){\delta(s)}}{\overline{X}_{s}}\right)\left(\frac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2}+B_{2}-\frac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2}\frac{b(0){\delta(s)}}{\overline{X}_{s}}\right)(\overline{X}_{s}\wedge\overline{X}_{\eta(s)})^{2(\alpha-1)}\right]ds+\int_{0}^{t}\frac{b(0)}{\overline{X}_{s}\wedge\overline{X}_{\eta(s)}}ds$
$\displaystyle\qquad\qquad+\int_{0}^{t}\frac{{\text{\scriptsize{$\Upsigma$}}}\left(\overline{X}_{s}-b(0){\delta(t)}\right)}{\overline{X}_{s}}\overline{X}_{\eta(s)}^{\alpha-1}dW_{s}.$

Then, using that $\overline{X}_{t}\geq b(0)\delta(t)$, we get

$\displaystyle\log(\overline{X}_{t})$
$\displaystyle\leq\log(x)+\int_{0}^{t}\left[B_{1}-\left(\frac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2}+B_{2}\right)(\overline{X}_{\eta(s)}\wedge\overline{X}_{s})^{2(\alpha-1)}+\frac{b(0){\delta(s)}}{\overline{X}_{s}}\left({\text{\scriptsize{$\Upsigma$}}}^{2}+B_{2}\right)(\overline{X}_{\eta(s)}\wedge\overline{X}_{s})^{2(\alpha-1)}\right]ds$
$\displaystyle\qquad\qquad+\int_{0}^{t}\frac{b(0)}{\overline{X}_{s}\wedge\overline{X}_{\eta(s)}}ds+\int_{0}^{t}\frac{{\text{\scriptsize{$\Upsigma$}}}\left(\overline{X}_{s}-b(0){\delta(t)}\right)}{\overline{X}_{s}}\overline{X}_{\eta(s)}^{\alpha-1}dW_{s}$
$\displaystyle=\log(x)+\int_{0}^{t}\varphi_{\Delta
t}(\overline{X}_{s}\wedge\overline{X}_{\eta(s)})ds+\int_{0}^{t}\frac{\sigma\left(\overline{X}_{s}-b(0){\delta(t)}\right)}{\overline{X}_{s}}\overline{X}_{\eta(s)}^{\alpha-1}dW_{s}.$

Notice that $\lim_{x\rightarrow 0^{+}}\varphi_{\Delta t}(x)=+\infty$ and
$\lim_{x\rightarrow+\infty}\varphi_{\Delta t}(x)=-\infty$, form which we
deduce that there exist at least one positive solution to the equation
$\varphi_{\Delta t}(\xi_{\Delta t})=0$, although $\varphi_{\Delta t}$ is no
more non-increasing (as show in Figure 1). Let us consider the case
$\varphi_{\Delta t}$ not monotone and take $x_{0}$ a critical point of the
map. Then, using $\varphi_{\Delta t}^{\prime}(x_{0})=0$, we deduce that

$\varphi_{\Delta
t}(x_{0})=B_{1}+\frac{(2\alpha-1)b(0)}{2(\alpha-1)x_{0}}+\frac{b(0){\Delta
t}\left({\text{\scriptsize{$\Upsigma$}}}^{2}+B_{2}\right)}{2(\alpha-1)}x_{0}^{2\alpha-3}+\frac{x_{0}}{2(\alpha-1)}\varphi_{\Delta
t}^{\prime}(x_{0})>0,$

and so either $\varphi_{\Delta t}$ is non-increasing or its minimum is
positive (see Figure 1). As a consequence, there exists a unique (positive)
solution $\xi_{\Delta t}$ to the equation $\varphi_{\Delta t}(\xi_{\Delta
t})=0,$ and

$\varphi_{\Delta t}(\xi_{\Delta t}+\epsilon)<0<\varphi_{\Delta t}(\xi_{\Delta
t}-\epsilon),\quad\mbox{ for any }\epsilon>0.$ (3.36)

Assuming that $\liminf_{t\rightarrow+\infty}\overline{X}_{t}>\xi_{\Delta t}$,
we can consider $\epsilon>0$ such that

$\mathbb{P}(\\{\omega:~{}\liminf_{t\rightarrow+\infty}\overline{X}_{\eta(t)}(\omega)=\liminf_{t\rightarrow+\infty}\overline{X}_{t}(\omega)\geq\xi_{\Delta
t}+2\epsilon\\})>\epsilon.$

Take the pair $(\omega_{1},T(\omega_{1}))$ and $\epsilon$ small enough such
that,

$\overline{X}_{t}(\omega_{1})\wedge\overline{X}_{\eta(t)}(\omega_{1})\geq\xi_{\Delta
t}+\epsilon,\qquad\mbox{ for all }t\geq T(\omega_{1}).$

$-1$$1$$2$$3$$4$$-4$$-2$$2$$4$$6$$8$$10$$\xi_{\Delta t}$$\xi_{\Delta
t}+\epsilon$$x$$\psi(x)$ Figure 1: Behaviour of function
$\varphi_{\Delta}t(x)=B_{1}-(\frac{{\text{\scriptsize{$\Upsigma$}}}^{2}}{2}+B_{2})x^{2(\alpha-1)}+\frac{b(0)}{x}+b(0)\left({\text{\scriptsize{$\Upsigma$}}}^{2}+B_{2}\right)x^{2\alpha-3}\Delta
t$.

From the behaviour of $x\mapsto\varphi_{\Delta t}(x)$ in the interval
$(\xi_{\Delta t}+\epsilon,+\infty)$ (see Figure 1), we obtain

$\varphi_{\Delta
t}(\overline{X}_{t}(\omega_{1})\wedge\overline{X}_{\eta(t)}(\omega_{1}))\leq\varphi_{\Delta
t}(\xi_{\Delta t}+\epsilon)<0,\quad\mbox{for all }t\geq T(\omega_{1}).$ (3.37)

Thus, from LLN for martingales and the inequality (3.37), we can consider
$\omega_{1}$ such that,

$\limsup_{t\rightarrow+\infty}\frac{\log(\overline{X}_{t}(\omega_{1}))}{t}\leq\varphi_{\Delta
t}(\xi_{\Delta t}+\epsilon)<0,$

implying $\liminf_{t\rightarrow+\infty}\overline{X}_{t}(\omega_{0})=0,$ and
contradicting the inequality
$\liminf_{t\rightarrow+\infty}\overline{X}_{t}>\xi_{\Delta t}$. ∎

From Propositions 3.8 and 3.10 we have $\xi^{\Delta
t}\leq\xi^{*}\leq\xi_{\Delta t}$. Remarkably, as $\Delta t$ tends to 0, the
Exp-EM process attains the same stable state than the solution $(X_{t};t\geq
0)$ of the SDE (1.1) since

$|\varphi^{\Delta t}(\xi^{*})|=\frac{B_{1}b(0)}{\xi^{*}}\Delta
t,\quad\mbox{and}\quad|\varphi_{\Delta
t}(\xi^{*})|=b(0)({\text{\scriptsize{$\Upsigma$}}}^{2}+B_{2})(\xi^{*})^{2\alpha-3}\Delta
t.$

Moreover, for $b(0)=0$ we have $\xi^{\Delta t}=\xi^{*}=\xi_{\Delta t}$. When
$b(0)>0$, is it possible to control the asymptotic bias between the scheme and
the true stationary state $\xi^{*}$ having

$|\varphi^{\Delta t}(\xi^{*})|\vee|\varphi_{\Delta
t}(\xi^{*})|\leq\epsilon_{0},$

with a certain threshold $\epsilon_{0}>0$, we can choose $\Delta t$ such that
$\Delta t\leq\epsilon_{0}\ {\xi^{*}}[b(0)\
\\{({\text{\scriptsize{$\Upsigma$}}}^{2}+B_{2})(\xi^{*})^{2(\alpha-1)}\vee
B_{1}\\}]^{-1}$.

##  4 Numerical experiments

In this section, we investigate the numerical rate of convergence of the Exp-
EM scheme and the optimality of the parameter conditions, as expounded in
Theorem 3.5 and Theorem 3.1. Most of the cases under consideration delve into
the prototypical scenario of decreasing polynomial coefficients, encompassing
both continuous and discontinuous cases satisfying assumptions of Theorem 3.5.
A distinctive case has been incorporated, where the driving term of the drift
is a function bounded above by a decreasing polynomial outside a compact set
(according to 3) but the derivative of this function is unbounded
(contradicting 5) , thereby illustrating the applicability of Theorem 3.1.

The following class of model allows to restrict the set of parameters involved
in the condition of convergence to the only parameter $B_{2}>0$:

$dX_{t}=(1+X_{t}-B_{2}\;X_{t}^{2\alpha-1})dt+X_{t}^{2}dW_{t},\;X_{0}=x>0.$
(4.1)

We vary the value of parameter $B_{2}$ in the range $[1.0,6.5]$, such that at
least, according to Lemma 2.1, the second moment of the solution is finite
($B_{2}\geq 1/2$). Note that expression (A.12) in the Feller test applied in
the Appendix, allows to extract the condition
$2B_{2}\geq-(2\alpha-1)\Sigma^{2}$ for $X$ to not explode at infinity in
finite time, here $B_{2}\geq-1/2$.

The model (4.1) serves as a first test bench for the numerical performance of
the Exp-EM scheme and its behaviour in terms of the theoretical condition on
the parameters, allowing us to obtain a more nuanced understanding of its
convergence characteristics. In addition, this model class enables us to
compare the range of cases encompassed in the strong error convergence with
the convergence obtained in Bossy et al. (2021) for the weak error of the Exp-
EM scheme. The theoretical condition (3.26) for the strong convergence with
order 1/2 in Theorem 3.5 is expressed as $\kappa_{\texttt{Strong}({2p})}\geq
0$, while the conditions in Corollary 4.1 of Bossy et al. (2021) are presented
as $\kappa_{\texttt{Weak}}\geq 0$ with, in term of $(\alpha,B_{2})$,

$\displaystyle\kappa_{\texttt{Strong}({2p})}$
$\displaystyle:=2B_{2}+1-2p[(4\alpha-3)\vee(2\alpha)]\vee\\{4\
(2p-\tfrac{3}{4})\\},$ (4.2) $\displaystyle\kappa_{\texttt{Weak}}$
$\displaystyle:=2B_{2}-6\alpha-\alpha^{2}\vee[(12\alpha-19)\vee(8\alpha-10)\vee\tfrac{5\alpha^{2}}{2\alpha-1}].$

To simplify our comparison, $\alpha$ is fixed to 2 in the model (4.1). We also
limit the experiments to the $L^{2}(\Omega)$-error, choosing $p=1$ and we
expect to observe numerically that

$\sqrt{\mathbb{E}\left[\sup_{0\leq t\leq
T}|X_{t}-\overline{X}_{t}|^{2}\right]}\leq C\;\sqrt{\Delta t},$ (4.3)

where now the conditions in (4.2) simplify as
$\kappa_{\texttt{Strong}({2})}=2B_{2}-9\geq 0$ and the
$\kappa_{\texttt{Weak}}=2B_{2}-{56}/{3}\geq 0$. Note then that
$\kappa_{\texttt{Strong}({2})}$ reduces to the condition (3.13) that allows to
bound the local error term in Lemma 3.4.

##### Numerical parameters.