#  On the convergence order of the Euler scheme for scalar SDEs with Hölder-
type diffusion coefficients

Annalena Mickel 111former member of Mathematical Institute and DFG RTG 1953,
University of Mannheim, B6, 26, D-68131 Mannheim, Germany Andreas Neuenkirch
222Mathematical Institute, University of Mannheim, B6, 26, D-68131 Mannheim,
Germany<EMAIL_ADDRESS>
###### Abstract

We study the Euler scheme for scalar non-autonomous stochastic differential
equations, whose diffusion coefficient is not globally Lipschitz but a
fractional power of a globally Lipschitz function. We analyse the strong error
and establish a criterion, which relates the convergence order of the Euler
scheme to an inverse moment condition for the diffusion coefficient. Our
result in particular applies to Cox-Ingersoll-Ross-, Chan-Karolyi-Longstaff-
Sanders- or Wright-Fisher-type stochastic differential equations and thus
provides a unifying framework.

Key words:  scalar SDEs, Euler scheme, non-standard assumptions, fractional
power, $L^{1}$-error, inverse moment condition
2010 Mathematics Subject Classification: 65C30; 60H35

##  1 Introduction and Main Result

Consider the scalar non-autonomous stochastic differential equation (SDE)

$dX_{t}=a(t,X_{t})dt+c(t,X_{t})dW_{t},\quad t\in[0,T],\qquad
X_{0}=x_{0}\in\mathbb{R},$ (1)

where $W=(W_{t})_{t\in[0,T]}$ is a Brownian motion and the drift and diffusion
coefficients $a,c:[0,T]\times\mathbb{R}\rightarrow\mathbb{R}$ satisfy suitable
assumptions. Explicit solutions of such SDEs are rarely known and a simple and
popular numerical method for this equation is the Euler scheme. For an
equidistant discretization with $N\in\mathbb{N}$ time steps, i.e.,

$t_{k}=k\frac{T}{N},\quad k=0,\ldots,N,$

the equidistant Euler scheme for (1) is given by $x^{(N)}_{k}=x_{k}$ with

$x_{k+1}=x_{k}+a(t_{k},x_{k})(t_{k+1}-t_{k})+c(t_{k},x_{k})(W_{t_{k+1}}-W_{t_{k}}),\qquad
k=0,\ldots,N-1.$ (2)

It is well known that the Euler scheme (2) for SDE (1) has strong convergence
order $1/2$, i.e., one has

$\limsup_{N\rightarrow\infty}\,\,N^{1/2}\,\sup_{k=0,\ldots,N}\mathbb{E}\left|X_{t_{k}}-x_{k}^{(N)}\right|<\infty,$

if the drift and diffusion coefficient satisfy the following standard
assumptions (HLG) and (L).

######  Assumption 1.1.

We will use the following assumptions on functions
$f:[0,T]\times\mathbb{R}\rightarrow\mathbb{R}$, which are measurable.

  * (HLG)

There exists a constant $K>0$ such that

$|f(t,x)-f(s,x)|\leq K(1+|x|)|t-s|^{1/2}$

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

  * (L)

There exists a constant $K>0$ such that

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

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

  * (P)

We have

$f(t,x)\geq 0$

for all $t\in[0,T]$ and all $x\in\mathbb{R}$.

In recent years, the analysis of numerical methods for SDEs under non-standard
assumptions has become a very active research field, see Subsection 1.1 for an
overview. In this article, we consider SDEs where the drift coefficient
satisfies the standard assumptions, but the diffusion coefficient is a
fractional power of a standard diffusion coefficient. More precisely, we will
study the SDE

$dX_{t}=a(t,X_{t})dt+\sigma(t,X_{t})^{\mathfrak{g}}dW_{t},\quad
t\in[0,T],\qquad X_{0}=x_{0}\in\mathbb{R},$ (3)

where $a,\sigma:[0,T]\times\mathbb{R}\rightarrow\mathbb{R}$ satisfy (HLG) and
(L), $\sigma$ satisfies additionally (P) and we have $\mathfrak{g}\in[1/2,1)$.
Since $a$ and $\sigma^{\mathfrak{g}}$ are continuous and of linear growth, the
existence of a weak solution to Equation (3) follows from the works of
Skorokhod (see [40, 41] and also [23]). Since moreover

$|\sigma(t,x)^{\mathfrak{g}}-\sigma(t,y)^{\mathfrak{g}}|\leq
K^{\mathfrak{g}}|x-y|^{\mathfrak{g}}$

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

$\int_{0}^{\varepsilon}x^{-2\mathfrak{g}}dx=\infty$

for all $\varepsilon>0$, we have strong uniqueness and consequently existence
of a unique strong solution to Equation (3) by the results of Yamada and
Watanabe (see, e.g., Proposition 2.13 and Corollary 3.23 in Chapter V of
[21]).

Typical examples for SDE (3) are the non-autonomous Cox-Ingersoll-Ross (CIR)
process (see, e.g., [4])

$dX_{t}=\kappa(t)\left(\lambda(t)-X_{t}\right)dt+\theta(t)\sqrt{X_{t}^{+}}dW_{t},$

the non-autonomous Chan-Karolyi-Longstaff-Sanders (CKLS) process (see, e.g.,
[3])

$dX_{t}=\kappa(t)\left(\lambda(t)-X_{t}\right)dt+\theta(t)\left(X_{t}^{+}\right)^{\mathfrak{g}}dW_{t},$

with $\mathfrak{g}\in(1/2,1)$ and the non-autonomous Wright-Fisher (WF)
process (see, e.g., [10])

$dX_{t}=\kappa(t)\left(\lambda(t)-X_{t}\right)dt+\theta(t)\sqrt{(X_{t}(1-X_{t}))^{+}}dW_{t},$

with $\kappa,\lambda,\theta:[0,T]\rightarrow[0,\infty)$. These SDEs have
applications in various fields, e.g., in biology and finance, to mention a
few.

To formulate our main result, we will need the following assumption:

######  Assumption 1.2.

Assume that $a,\sigma$ satisfy (HLG) and (L) and that $\sigma$ additionally
satisfies (P). Moreover let $\mathfrak{g}\in[1/2,1)$ and assume that there
exists $\mathfrak{s}\in[0,1-\mathfrak{g}]$ such that

$\int_{0}^{T}\mathbb{E}\left[\sigma(s,X_{s})^{2(\mathfrak{g}+\mathfrak{s}-1)}\right]ds<\infty.$
(4)

Our main result reads then as follows:

######  Theorem 1.3.

Let $\mathfrak{g}\in[1/2,1)$ be given and assume that Assumption 1.2 holds for
a fixed $\mathfrak{s}\in[0,1-\mathfrak{g}]$. Moreover, let
$X=(X_{t})_{t\in[0,T]}$ be the solution of (3) and let
$(x_{k})_{k=0,\ldots,N}$ be given by the Euler scheme (2) with diffusion
coefficient $c=\sigma^{\mathfrak{g}}$. Then we have

$\limsup_{N\rightarrow\infty}\,\,N^{\lambda}\,\sup_{k=0,\ldots,N}\mathbb{E}\left|X_{t_{k}}-x_{k}^{(N)}\right|=0$

for all $\lambda<1/2-\mathfrak{s}$.

How to interpret condition (4)? For this note first that the irregularity of
the diffusion coefficient $\sigma^{\mathfrak{g}}$ arises from the set

$\mathcal{I}_{\sigma}=\\{(t,x)\in[0,T]\times\mathbb{R}:\,\sigma(t,x)=0\\}.$

For $(t_{*},x_{*})\in\mathcal{I}_{\sigma}$ and a neighbourhood $U_{*}$ of
$(t_{*},x_{*})$ the map
$U_{*}\ni(t,x)\mapsto\sigma(t,x)^{\mathfrak{g}}\in[0,\infty)$ is only Hölder
continuous of order $\mathfrak{g}$ and not Lipschitz continuous. Moreover,
since $\mathfrak{g}+\mathfrak{s}\leq 1$ the exponent in condition (4) is zero
or negative. Thus, this condition is a weighted measure, how much the solution
$X$ takes values in or close to the set $\mathcal{I}_{\sigma}$. In the case
$\mathfrak{s}=0$, that is

$\int_{0}^{T}\mathbb{E}\left[\sigma(s,X_{s})^{2\mathfrak{g}-2}\right]ds<\infty,$

the process $X$ is sufficiently apart from the set $\mathcal{I}_{\sigma}$ and
we have directly convergence order $1/2-\varepsilon$ for all $\varepsilon>0$.
If on the other hand

$\int_{0}^{T}\mathbb{E}\left[\sigma(s,X_{s})^{2\mathfrak{g}-2}\right]ds=\infty,$

the process $X$ is too close to the set $\mathcal{I}_{\sigma}$ and we need a
compensation factor $\mathfrak{s}$ and obtain convergence order
$1/2-\mathfrak{s}-\varepsilon$.

Finally, let us comment on three limiting cases:

  * (i)

For $\mathfrak{g}=1$ condition (4) would be trivially satisfied for
$\mathfrak{s}=0$ and we would (formally) obtain convergence order
$1/2-\varepsilon$ in accordance with the classic result for the Euler scheme.

  * (ii)

If $\sigma$ is uniformly elliptic, i.e.,

$\inf_{(t,x)\in[0,T]\times\mathbb{R}}\sigma(t,x)>0,$

then condition (4) is trivially satisfied for $\mathfrak{s}=0$ and we obtain
convergence order $1/2-\varepsilon$. This is again in accordance with the
classic result for the Euler scheme, since $\sigma^{\mathfrak{g}}$ satisfies
(HLG) and (L), if $\sigma$ is uniformly elliptic.

  * (iii)

If $\mathfrak{g}>1/2$ we can choose $\mathfrak{s}=1-\mathfrak{g}$. Then
condition (4) is satisfied, since $\mathfrak{g}+\mathfrak{s}-1=0$, and we
obtain convergence order $\mathfrak{g}-1/2-\varepsilon$. This recovers in our
setup the results of [42] and [14] for autonomous SDEs.

How to verify condition (4)? For this we present two approaches in Subsection
2.1 and Subsection 2.2.

The remainder of this article is structured as follows: In the next two
subsections, we give an overview of previous results in the literature and
outline our notation. Section 2 applies our main result to several SDEs, in
particular to the above mentioned CIR, CKLS and WF processes. Section 3 is
devoted to the proof of our main result, while in the Appendix we collect some
general auxiliary results.

###  1.1 Overview of previous results

Since we have $\gamma\in\left[\frac{1}{2},1\right)$, the diffusion coefficient
of (3) is not globally Lipschitz continuous and therefore standard convergence
results for the Euler scheme do not apply. Following the pioneering articles
[15, 13, 19, 42, 22], the strong approximation of the Euler scheme for SDEs
with non-standard coefficients has been extensively studied333To improve the
readability, we will slightly abuse notation and write in this overview
convergence order $\lambda$ also for results which established convergence
order $\lambda-\varepsilon$ for all $\varepsilon>0$ for a given $\lambda>0$..

In the case of an irregular drift coefficient, there has been a series of
articles including [36, 37, 38, 27, 1, 33, 12, 6, 32, 7, 24] that has
culminated so far in the works [29] and [5], which established under mild
assumptions on the drift coefficient the classical convergence order $1/2$ for
the Euler scheme. More precisely, for scalar autonomous SDEs with piecewise
Lipschitz drift and a globally Lipschitz diffusion coefficient, which does not
vanish in the discontinuity points of the drift, an $L^{p}$-error of order
$1/2$ is obtained in [29], while in the multi-dimensional case for a bounded
drift and an elliptic, bounded $\mathcal{C}^{2}_{b}$-diffusion coefficient the
same classical convergence order is obtained in [5].

Furthermore, the strong approximation by other schemes than the Euler scheme
has been analyzed, e.g., in [25, 26, 43, 30, 31, 11], and corresponding lower
error bounds for the strong approximation of SDEs with (possibly)
discontinuous drift coefficient have been studied in [17], [31] and [9].

The case of a general SDE with a non-Lipschitz diffusion coefficient has
received less attention. While the works [36, 37, 38] also allow weaker
assumptions on the diffusion coefficient, the strongest results so far are due
to [42] and [14].

In [42] the following Hölder- and Lipschitz-continuity assumptions on the
coefficients are used: There exist
$K>0,\beta_{1},\beta_{2}\in(0,1],\mathfrak{g}\in(1/2,1]$ such that

$\displaystyle|a(t,x)-a(s,y)|\leq
K\left(|t-s|^{\beta_{1}}+|x-y|\right),\qquad|c(t,x)-c(s,y)|\leq
K\left(|t-s|^{\beta_{2}}+|x-y|^{\mathfrak{g}}\right)$

for all $s,t\in[0,T],x,y\in\mathbb{R}$. Then, Theorem 4.1 in [42] gives for
the Euler scheme (2) the $L^{1}$-convergence order
$\min\\{\beta_{1},\mathfrak{g}-1/2,(2-1/\mathfrak{g})\beta_{2}\\}$ in the
discretization points.

The article [14] analyses the Euler-type polygonal approximation

$dY_{t}=a(t,Y_{\eta(t)})dt+c(t,Y_{\eta(t)})dW_{t},\quad t\in[0,T],\qquad
Y_{0}=x_{0}\in\mathbb{R},$ (5)

where $\eta(t)=\max\\{t_{k}\in\\{t_{0},t_{1},...,t_{N}\\}:t_{k}\leq t\\}$. For
autonomous SDEs this approximation coincides in the discretization points with
the Euler scheme (2), while for general non-autonomous coefficients this
scheme is typically not implementable. The assumptions in [14] are as follows:
We have $a=b+f$, where for all $t\in[0,T]$ the function $f(t,\cdot)$ is
monotonically decreasing,

$\sup_{t\in[0,T]}\left(|b(t,0)|+|c(t,0)|\right)<\infty$

and there exists $K>0,\beta_{3}\in(0,1],\mathfrak{g}\in(1/2,1]$ such that

$\displaystyle|b(t,x)-b(s,y)|\leq K|x-y|,\quad|c(t,x)-c(s,y)|\leq
K|x-y|^{\mathfrak{g}},\quad|f(t,x)-f(s,y)|\leq K|x-y|^{\beta_{3}}$

for all $t\in[0,T],x,y\in\mathbb{R}$. Under these assumptions, Theorem 2.1 in
[14] gives that the $L^{1}$-error of (5) is bounded by $C/\log(N)$ for a
constant $C>0$ if $\mathfrak{g}=1/2$ and is of order
$\min\\{\mathfrak{g}-1/2,\beta_{3}/2\\}$ for $\mathfrak{g}\in(1/2,1]$.

On the other hand, the classical autonomous CIR process has a
Hölder-1/2-diffusion coefficient and has received a lot of attention in recent
years, see, e.g., [16, 18, 28] and the references therein. In particular, it
turned out that Euler schemes attain for this SDE $L^{1}$-convergence order
$1/2$, if the Feller index $\nu:=\frac{2\kappa\lambda}{\theta^{2}}$ satisfies
$\nu>1$, see [28]. In this case our Assumption 1.2 is satisfied with
$\mathfrak{s}=0$. Remarkably, this is a much better decay of the error than
the logarithmic one provided in [14]. Similarly, for CKLS-type SDEs with
$\mathfrak{g}\in(1/2,1)$ and a symmetrized Euler scheme the work [2] also
establishes convergence order $1/2$.

Thus, Theorem 1.3 provides a unifying framework and also a criterion, i.e.,
condition (1.2), which explains and unites these different error bounds for
the Euler scheme.

###  1.2 Notation

As already mentioned, we will work with an equidistant discretization

$t_{k}=k\Delta t,\quad k=0,\ldots,N,$

with $\Delta t=T/N$ and $N\in\mathbb{N}$. Furthermore, we define
$n(t):=\max\\{k\in\\{0,...,N\\}:t_{k}\leq t\\}$ and $\eta(t):=t_{n(t)}$.
Constants whose values depend only on $x_{0},T$, $a,\sigma$, $\mathfrak{g}$
will be denoted in the following by $C$, regardless of their value. Other
dependencies will be denoted by subscripts, i.e., $C_{p}$ means that this
constant depends additionally on the parameter $p$. Moreover, the value of all
these constants can change from line to line. We will work on a filtered
probability space $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\in[0,T]},{P})$
where the filtration satisfies the usual conditions, and (in-)equalities
between random variables or random processes are understood ${P}$-a.s. unless
mentioned otherwise.

Finally, for $p\in\mathbb{N}$ we denote by $\mathcal{C}^{p}(M;\mathbb{R})$ the
set of $p$-times continuously differentiable functions from the set $M$ to
$\mathbb{R}$ and by $\mathcal{C}^{p}_{b}(M;\mathbb{R})$ the set of
continuously differentiable functions from $M$ to $\mathbb{R}$, whose
derivatives are bounded. Moreover, for $\gamma\in(0,1)$, we denote by
$\mathcal{C}^{\gamma}(M;\mathbb{R})$ the set of $\gamma$-Hölder continuous
functions from $M$ to $\mathbb{R}$.

##  2 Examples

Condition (4) from Assumption 1.2 gives structural insight in the convergence
behaviour of the Euler scheme, but how to verify it? Here we present two
approaches. The first one is based on the Feller test and Itō’s lemma. The
second one is based on a time change and the comparison lemma for SDEs.

###  2.1 Autonomous SDEs on domains

Here we consider the autonomous equation

$dX_{t}=a(X_{t})dt+\sigma(X_{t})^{\mathfrak{g}}dW_{t},\quad t\in[0,T],\qquad
X_{0}=x_{0},$ (6)

and we assume that $a\in C^{1}_{b}(\mathbb{R};\mathbb{R})$ and $\sigma\in
C^{2}_{b}(\mathbb{R};[0,\infty))$. To apply Itō’s lemma to derive an explicit
criterion for condition (4) we need that $\sigma^{\mathfrak{g}}$ is
differentiable on the support of $X$. To this end, we assume that there exists
$I=(l,r)$ with $-\infty\leq l\leq r\leq\infty$ such that

$\displaystyle\forall x\in I:\quad\sigma(x)>0,$

as well as

$P(X_{t}\in I,\,t\in[0,T])=1.$

The latter condition can be checked, e.g., by the Feller test, see Theorem 4.1
in the Appendix.

Under these assumptions it is well known that the Euler scheme has convergence
order $1/2$ in law, that is

$\lim_{N\rightarrow\infty}\,\,N^{1/2}\,\sup_{k=0,\ldots,N}\big{|}X_{t_{k}}-x_{k}^{(N)}\big{|}\stackrel{{\scriptstyle\mathcal{L}}}{{=}}\max_{t\in[0,T]}|U_{t}|,$

where the limiting process $U$ is the solution of the SDE

$\displaystyle dU_{t}$
$\displaystyle=a^{\prime}(X_{t})U_{t}dt+\mathfrak{g}\sigma^{\mathfrak{g}-1}(X_{t})\sigma^{\prime}(X_{t})U_{t}dW_{t}+\frac{\mathfrak{g}}{\sqrt{2}}\sigma(X_{t})^{2\mathfrak{g}-1}\sigma^{\prime}(X_{t})dB_{t},\qquad
U_{0}=0,$ (7)

where $B=(B_{t})_{t\in[0,T]}$ is another Brownian motion which is independent
of $W$. See [35] and [39]. Thus it is reasonable to expect that in this case
the Euler scheme has also strong convergence order $1/2$. The limiting SDE (7)
also illustrates that the bottleneck is the error propagation and not the
local error, since we have the exponent $\mathfrak{g}-1$ in the error
propagation term and the less restrictive exponent $2\mathfrak{g}-1$ in the
local error term.

######  Proposition 2.1.

Let $\mathfrak{g}\in[1,2,1)$, $a\in C^{1}_{b}(\mathbb{R};\mathbb{R})$ and
$\sigma\in C^{2}_{b}(\mathbb{R};[0,\infty))$. Moreover let
$I=(l,r),-\infty\leq l\leq r\leq\infty$, and assume that the unique solution
of the SDE

$dX_{t}=a(X_{t})dt+\sigma(X_{t})^{\mathfrak{g}}dW_{t},\quad t\in[0,T],\qquad
X_{0}=x_{0}\in I,$

satisfies

$P(X_{t}\in I,\,t\in[0,T])=1$

as well as

$\displaystyle\forall x\in I:\quad\sigma(x)>0.$

Finally, assume that

$\displaystyle\inf_{x\in
I}\left[\frac{\sigma^{\prime}(x)a(x)}{\sigma(x)}+\frac{1}{2}\frac{\sigma^{\prime\prime}(x)\sigma(x)^{2\mathfrak{g}}}{\sigma(x)}+\left(\mathfrak{g}-\frac{3}{2}\right)\frac{(\sigma^{\prime}(x))^{2}\sigma(x)^{2\mathfrak{g}}}{\sigma(x)^{2}}\right]>-\infty.$
(8)

Then we have

$\limsup_{N\rightarrow\infty}\,\,N^{\lambda}\,\sup_{k=0,\ldots,N}\mathbb{E}\left|X_{t_{k}}-x_{k}^{(N)}\right|=0$

for all $\lambda<1/2$.

###### Proof.

Let $k\in\mathbb{N}$ such that $k>\sigma(x_{0})^{2(\mathfrak{g}-1)}$ and set

$\tau_{k}=\inf\\{t\in[0,T]:\,\sigma(X_{t})^{2(\mathfrak{g}-1)}\geq k\\}$

with the convention $\inf\emptyset=T$. An application of Itō’s lemma and the
local martingale property of the Itō integral give

$\displaystyle\mathbb{E}\left(\sigma(X_{t\wedge\tau_{k}})^{2(\mathfrak{g}-1)}\right)$
$\displaystyle=\sigma(x_{0})^{2(\mathfrak{g}-1)}+\mathbb{E}\left(\int_{0}^{t\wedge\tau_{k}}h_{a,\sigma,\mathfrak{g}}(X_{s})ds\right)$

with

$\displaystyle h_{a,\sigma,\mathfrak{g}}(x)$
$\displaystyle=(2\mathfrak{g}-2)\sigma(x)^{2\mathfrak{g}-2}$
$\displaystyle\qquad\times\left(\sigma^{\prime}(x)\sigma^{-1}(x)a(x)+\left(\mathfrak{g}-\frac{3}{2}\right)\sigma(x)^{2\mathfrak{g}-2}(\sigma^{\prime}(x))^{2}+\frac{1}{2}\sigma(x)^{2\mathfrak{g}-1}\sigma^{\prime\prime}(x)\right).$

Assumption (8) now yields the existence of constant $C>0$, which is in
particular independent of $k$, such that

$h_{a,\sigma,\mathfrak{g}}(x)\leq C\sigma(x)^{2(\mathfrak{g}-1)}$

for all $x\in I$. Gronwall’s lemma gives then the existence of another
constant $C>0$, which is in particular independent of $k$, such that

$\sup_{t\in[0,T]}\mathbb{E}\left(\sigma(X_{t\wedge\tau_{k}})^{2(\mathfrak{g}-1)}\right)\leq
C.$

Assumption (4) with $\mathfrak{s}=0$ now follows by taking
$k\rightarrow\infty$. ∎

###  2.2 Prototype SDEs

In this subsection we consider the SDEs

$dX_{t}=a(t,X_{t})dt+c_{i}(t,X_{t})dW_{t},\qquad t\in[0,T],$ (9)

for $i=1,2,3$, where

$c_{i}(t,x)=\theta(t)h_{i}(x)$

with $\theta\in C^{1/2}([0,T];(0,\infty))$ and

$h_{1}(x)=\sqrt{x^{+}},\qquad h_{2}(x)=\sqrt{(x(1-x))^{+}},\qquad
h_{3}(x)=(x^{+})^{\mathfrak{g}},$

where $\mathfrak{g}\in(1/2,1)$.

Theses SDEs are extensions of the classical CIR, CKLS and WF processes. For
verifying Assumption 1.2, the following time change procedure will be helpful.
So, let

$\Theta:[0,T]\rightarrow\mathbb{R},\qquad\Theta(t)=\int_{0}^{t}\theta^{2}(s)ds.$

Then $\Theta$ is strictly increasing and has an inverse $A:=\Theta^{-1}$. Now
consider the time-changed process

$\widetilde{X}_{t}=X_{A(t)},\qquad t\in[0,A^{-1}(T)].$

Since $X$ satisfies

$X_{t}=x_{0}+\int_{0}^{t}a(s,X_{s})ds+\int_{0}^{t}\theta(s)h_{i}(X_{s})dW_{s},\qquad
t\in[0,T],$

we have

$\displaystyle\widetilde{X}_{t}=x_{0}+\int_{0}^{A(t)}a(s,X_{s})ds+\int_{0}^{A(t)}\theta(s)h_{i}(X_{s})dW_{s},\qquad
t\in[0,T].$

Now set

$\widetilde{W}_{t}=\int_{0}^{A(t)}\theta(s)dW_{s},\qquad t\in[0,A^{-1}(T)].$

Then $\widetilde{W}$ is a Brownian motion on $[0,A^{-1}(T)]$ and the rules for
the time change for Riemann and Itō integrals, see Proposition 4.2 in the
Appendix, yield that

$\displaystyle\widetilde{X}_{t}$
$\displaystyle=x_{0}+\int_{0}^{t}\frac{a(A(s),\widetilde{X}_{s})}{\theta(A(s))^{2}}ds+\int_{0}^{t}h_{i}(\widetilde{X}_{s})d\widetilde{W}_{s},\qquad
t\in[0,A^{-1}(T)].$

This enables us to exploit the classical bounds for the inverse moments of the
CIR, CKLS and WF processes. See Subsection 4.3 in the Appendix.

For $c_{1}$ and $c_{2}$ we obtain the following result:

######  Proposition 2.2.

(i) Let $x_{0}>0$ and $\theta\in\mathcal{C}^{1/2}([0,T];(0,\infty))$.
Moreover, assume that the drift coefficient $a$ satisfies (HLG), (L) and
assume furthermore that

$\mu_{0}:=\min_{t\in[0,T]}\frac{a(t,0)}{\theta(t)^{2}}>0.$

Then the unique strong solution of

$dX_{t}=a(t,X_{t})dt+\theta(t)\sqrt{X_{t}^{+}}dW_{t},\quad t\in[0,T],\qquad
X_{0}=x_{0},$

satisfies $P(X_{t}\geq 0,t\in[0,T])=1$ and we have

$\limsup_{N\rightarrow\infty}\,\,N^{\lambda}\,\sup_{k=0,\ldots,N}\mathbb{E}\left|X_{t_{k}}-x_{k}^{(N)}\right|=0$

for all

$\lambda<\min\\{1/2,\mu_{0}\\}.$

(ii) Let $x_{0}\in(0,1)$ and $\theta\in\mathcal{C}^{1/2}([0,T];(0,\infty))$.
Moreover, assume that the drift coefficient $a$ satisfies (HLG), (L) and
assume furthermore that

$\mu_{0}:=\min_{t\in[0,T]}\frac{a(t,0)}{\theta(t)^{2}}>0\qquad\textrm{and}\qquad\mu_{1}:=-\max_{t\in[0,T]}\frac{a(t,1)}{\theta(t)^{2}}>0.$

Then the unique strong solution of

$dX_{t}=a(t,X_{t})dt+\theta(t)\sqrt{(X_{t}(1-X_{t}))^{+}}dW_{t},\quad
t\in[0,T],\qquad X_{0}=x_{0},$

satisfies $P(X_{t}\in[0,1],t\in[0,T])=1$ and we have

$\limsup_{N\rightarrow\infty}\,\,N^{\lambda}\,\sup_{k=0,\ldots,N}\mathbb{E}\left|X_{t_{k}}-x_{k}^{(N)}\right|=0$

for all

$\lambda<\min\\{1/2,\mu_{0},\mu_{1}\\}.$

###### Proof.

(1) Let us first check the assumptions (HLG) and (L) on $a$ and $\sigma$ for
both equations. For the drift coefficient there is nothing to verify.

For the diffusion coefficient in (i) note that $\sigma$ is given by
$\sigma(t,x)=\theta(t)^{2}x^{+}$. So clearly, we have

$|\sigma(t,x)-\sigma(t,y)|\leq\left(\max_{\tau\in[0,T]}|\theta(\tau)|^{2}\right)|x^{+}-y^{+}|\leq\|\theta\|_{\infty}^{2}|x-y|$

with

$\|\theta\|_{\infty}:=\max_{t\in[0,T]}|\theta(t)|<\infty$

and

$\displaystyle|\sigma(t,x)-\sigma(s,x)|$ $\displaystyle\leq
2\left(\max_{\tau\in[0,T]}\theta(\tau)\right)|\theta(t)-\theta(s)||x^{+}|$
$\displaystyle\leq 2\|\theta\|_{\infty}\|\theta\|_{1/2}|t-s|^{1/2}|x|$

with

$\|\theta\|_{1/2}:=\sup_{0\leq s<t\leq
T}\frac{|\theta(t)-\theta(s)|}{|t-s|^{1/2}}<\infty.$

For (ii) note that $\sigma(t,x)=\theta(t)^{2}(x(1-x))^{+}$. In this case we
have

$|\sigma(t,x)-\sigma(t,y)|\leq\max_{\tau\in[0,T]}|\theta(\tau)|^{2}|(x(1-x))^{+}-(y(1-y))^{+}|\leq\|\theta\|_{\infty}^{2}|x-y|$

and

$\displaystyle|\sigma(t,x)-\sigma(s,x)|$ $\displaystyle\leq
2\|\theta\|_{\infty}\|\theta\|_{1/2}|t-s|^{1/2}.$

(2) Thus it remains to relate the inverse moment condition to the quantities
$\mu_{0}$ and $\mu_{1}$. We start with assertion (i). First note that

$0<\min_{t\in[0,T]}\theta(t)\leq\max_{t\in[0,T]}\theta(t)<\infty.$

Using this and the time change described above we have that

$\sup_{t\in[0,T]}\mathbb{E}\left[\left(\theta(t)\sqrt{X_{t}^{+}}\right)^{p}\right]<\infty\quad\Longleftrightarrow\sup_{t\in[0,A^{-1}(T)]}\mathbb{E}\left[\widetilde{X}_{t}^{p/2}\right]<\infty$

for $p\in\mathbb{R}$, where

$\displaystyle\widetilde{X}_{t}$
$\displaystyle=x_{0}+\int_{0}^{t}\frac{a(A(s),\widetilde{X}_{s})}{\theta(A(s))^{2}}ds+\int_{0}^{t}\sqrt{\widetilde{X}_{s}^{+}}d\widetilde{W}_{s},\qquad
t\in[0,A^{-1}(T)].$

Recall that

$\mu_{0}:=\min_{t\in[0,T]}\frac{a(t,0)}{\theta(t)^{2}}>0$

and note that

$\min_{t\in[0,T]}\frac{a(t,0)}{\theta(t)^{2}}=\min_{t\in[0,A^{-1}(T)]}\frac{a(A(t),0)}{\theta(A(t))^{2}}.$

Define

$\displaystyle
a_{cp}(s,x)=\mu_{0}-K\left(\max_{\tau\in[0,A^{-1}(T)]}\frac{1}{\theta(A(\tau))^{2}}\right)x,$
(10)

where $K$ is the Lipschitz constant of $a$ with respect to $x$ from assumption
(L). Since

$a(A(s),x)=a(A(s),0)+a(A(s),x)-a(A(s),0)$

we now have that

$\frac{a(A(s),x)}{\theta^{2}(A(s))}\geq a_{cp}(s,x),\qquad
s\in[0,A^{-1}(T)],\,\ x\in\mathbb{R}.$

Therefore, the comparison result for SDEs, see Proposition 4.4 in the
Appendix, yields that

$P(\widetilde{X}_{t}\geq Z_{t},\,t\in[0,A^{-1}(T)])=1,$

where $Z$ is given by the SDE

$Z_{t}=x_{0}+\int_{0}^{t}a_{cp}(s,Z_{s})ds+\int_{0}^{t}\sqrt{Z_{s}^{+}}d\widetilde{W}_{s},\qquad
t\in[0,A^{-1}(T)].$

We have

$P(Z_{t}\geq 0,\,t\in[0,A^{-1}(T)])=1$

and

$\sup_{t\in[0,A^{-1}(T)]}\mathbb{E}\left[Z_{t}^{p}\right]<\infty\qquad\textrm{for}\qquad
p>-2\mu_{0},$

see Lemma 4.3(i). Since

$\frac{1}{T}\int_{0}^{T}\mathbb{E}\big{[}X_{t}^{2\mathfrak{s}-1}\big{]}dt\leq\sup_{t\in[0,A^{-1}(T)]}\mathbb{E}\big{[}\widetilde{X}_{t}^{2\mathfrak{s}-1}\big{]}\leq\sup_{t\in[0,A^{-1}(T)]}\mathbb{E}\big{[}Z_{t}^{2\mathfrak{s}-1}\big{]},$

condition (1.2) with $\mathfrak{g}=1/2$, that is

$\mathbb{E}\int_{0}^{T}\left(\theta(t)^{2}X_{t}\right)^{2\mathfrak{s}-1}dt<\infty,$

holds if

$\mathfrak{s}+\mu_{0}>1/2.$

Hence, Theorem 1.3 yields convergence order $\lambda$ for all $\lambda<1/2$,
if $\mu_{0}>1/2$. Otherwise, i.e., if $\mu_{0}\leq 1/2$, Theorem 1.3 yields
convergence order $\lambda$ for all $\lambda<\mu_{0}$.

(3) For assertion (ii) note that

$\sup_{t\in[0,T]}\mathbb{E}\left[\left(\theta(t)\sqrt{X_{t}(1-X_{t})^{+}}\right)^{p}\right]<\infty\quad\Longleftrightarrow\sup_{t\in[0,A^{-1}(T)]}\mathbb{E}\left[\left(\widetilde{X}_{t}(1-\widetilde{X}_{t})\right)^{p/2}\right]<\infty$

for $p\in\mathbb{R}$ where

$\displaystyle\widetilde{X}_{t}$
$\displaystyle=x_{0}+\int_{0}^{t}\frac{a(A(s),\widetilde{X}_{s})}{\theta(A(s))^{2}}ds+\int_{0}^{t}\sqrt{(\widetilde{X}_{s}(1-\widetilde{X}_{s}))^{+}}d\widetilde{W}_{s},\qquad
t\in[0,A^{-1}(T)].$

(a) Proceeding analogously as for case (i) we obtain that

$P(\widetilde{X}_{t}\geq Z_{t}\geq 0,\,t\in[0,A^{-1}(T)])=1$

where $Z$ is given by the SDE

$Z_{t}=x_{0}+\int_{0}^{t}a_{cp}(s,Z_{s})ds+\int_{0}^{t}\sqrt{(Z_{s}(1-Z_{s}))^{+}}d\widetilde{W}_{s},\qquad
t\in[0,A^{-1}(T)],$

with $a_{cp}$ as in (10) and

$\sup_{t\in[0,A^{-1}(T)]}\mathbb{E}\left[Z_{t}^{p}\right]<\infty\qquad\textrm{for
}\quad p>-2\mu_{0},$

see Lemma 4.3(ii), which yields that

$\mathbb{E}\int_{0}^{T}\left(\theta(t)^{2}X_{t}\right)^{2\mathfrak{s}-1}dt<\infty$

holds if

$\mathfrak{s}+\mu_{0}>1/2.$

Here we have used again

$\frac{1}{T}\int_{0}^{T}\mathbb{E}\big{[}X_{t}^{2\mathfrak{s}-1}\big{]}dt\leq\sup_{t\in[0,A^{-1}(T)]}\mathbb{E}\big{[}\widetilde{X}_{t}^{2\mathfrak{s}-1}\big{]}\leq\sup_{t\in[0,A^{-1}(T)]}\mathbb{E}\big{[}Z_{t}^{2\mathfrak{s}-1}\big{]}.$

(b) Now we need to establish that

$\mathbb{E}\int_{0}^{T}\left(\theta(t)^{2}(1-X_{t})\right)^{2\mathfrak{s}-1}dt<\infty$

holds if

$\mathfrak{s}+\mu_{1}>1/2$

where

$\mu_{1}:=-\max_{t\in[0,T]}\frac{a(t,1)}{\theta(t)^{2}}.$

This can be done, e.g., by using the reflected process
$\widetilde{X}_{t}^{r}=1-\widetilde{X}_{t}$, which satisfies the SDE

$\displaystyle\widetilde{X}_{t}^{r}$
$\displaystyle=1-x_{0}-\int_{0}^{t}\frac{a(A(s),1-\widetilde{X}^{r}_{s})}{\theta(A(s))^{2}}ds+\int_{0}^{t}\sqrt{(\widetilde{X}_{s}^{r}(1-\widetilde{X}^{r}_{s}))^{+}}d\widetilde{W}^{r}_{s},\qquad
t\in[0,A^{-1}(T)],$

where $\widetilde{W}^{r}$ is a Brownian motion given by

$\widetilde{W}^{r}_{t}=-\widetilde{W}_{t},\qquad t\in[0,A^{-1}(T)],$

and

$a_{cp}(s,x)=\mu_{1}-K\left(\max_{\tau\in[0,A^{-1}(T)]}\frac{1}{\theta(A(\tau))^{2}}\right)x.$

Here $K$ is again the Lipschitz constant of $a$ with respect to $x$ from
assumption (L). Since

$a(A(s),1-x)=a(A(s),1)+a(A(s),1-x)-a(A(s),1)$

we have that

$-\frac{a(A(s),1-x)}{\theta(A(s))^{2}}\geq a_{cp}(s,x),\qquad
s\in[0,A^{-1}(T)],\,\ x\in\mathbb{R}.$

Therefore, another application of the comparison result for SDEs, see
Proposition 4.4 in the Appendix, yields that

$P(\widetilde{X}_{t}^{r}\geq Z_{t}\geq 0,\,t\in[0,A^{-1}(T)])=1,$

where $Z$ is given by the SDE

$Z_{t}=x_{0}+\int_{0}^{t}a_{cp}(s,Z_{s})ds+\int_{0}^{t}\sqrt{(Z_{s}(1-Z_{s}))^{+}}d\widetilde{W}^{r}_{s},\qquad
t\in[0,A^{-1}(T)].$

Now we have

$\sup_{t\in[0,A^{-1}(T)]}\mathbb{E}\left[Z_{t}^{p}\right]<\infty\qquad\textrm{for
}\quad p>-2\mu_{1},$

see Lemma 4.3(ii), which yields that

$\displaystyle\mathbb{E}\int_{0}^{T}\left(\theta(t)^{2}(1-X_{t})\right)^{2\mathfrak{s}-1}dt<\infty,$

if

$\mathfrak{s}+\mu_{1}>1/2.$

(c) Since

$\displaystyle(x(1-x))^{+})^{2\mathfrak{s}-1}$ $\displaystyle\leq
2^{1-2\mathfrak{s}}\left((x^{+})^{2\mathfrak{s}-1}\mathbbm{1}_{\\{x\leq
1/2\\}}+((1-x)^{+})^{2\mathfrak{s}-1}\mathbbm{1}_{\\{x>1/2\\}}\right)$
$\displaystyle\leq
2^{1-2\mathfrak{s}}\left((x^{+})^{2\mathfrak{s}-1}+((1-x)^{+})^{2\mathfrak{s}-1}\right),$

we obtain from (a), (b) and the Minkowski inequality that

$\mathbb{E}\int_{0}^{T}\left(\theta(t)^{2}X_{t}(1-X_{t})\right)^{2\mathfrak{s}-1}dt<\infty$

if $\mathfrak{s}+\mu_{i}>1/2$ for $i=0,1$. Assertion (ii) now follows from
Theorem 1.3. ∎

Thus, the relative strength of the upward and downward drift, respectively,
with respect to $\theta$ at the boundaries of the domain of the SDE influences
the convergence rate for these prototype equations with a square root
diffusion coefficient.

However, for $c_{3}$ already $\mu_{0}>0$ is sufficient to obtain convergence
order $1/2-\varepsilon$.

######  Proposition 2.3.

Let $x_{0}>0$, $\mathfrak{g}\in(1/2,1)$ and
$\theta\in\mathcal{C}^{1/2}([0,T];(0,\infty))$. Moreover, assume that the
drift coefficient $a$ satisfies (HLG), (L) and assume furthermore that

$\mu_{0}:=\min_{t\in[0,T]}\frac{a(t,0)}{\theta(t)^{2}}>0.$

Then the unique strong solution of

$dX_{t}=a(t,X_{t})dt+\theta(t)\left(X_{t}^{+}\right)^{\mathfrak{g}}dW_{t},\quad
t\in[0,T],\qquad X_{0}=x_{0},$

satisfies $P(X_{t}\geq 0,t\in[0,T])=1$ and we have

$\limsup_{N\rightarrow\infty}\,\,N^{\lambda}\,\sup_{k=0,\ldots,N}\mathbb{E}\left|X_{t_{k}}-x_{k}^{(N)}\right|=0$

for all

$\lambda<1/2.$

###### Proof.

The proof can be done along the same lines as the proof of case (i) of the
previous proposition. The only and crucial difference is, that Lemma 4.3(iii)
gives us the existence off all inverse moments of the CKLS process. Thus we
can take always $\mathfrak{s}=0$ in Theorem 1.3. ∎

##  3 Proof of Theorem 1.3

###  3.1 Preliminaries

Before we start with the proof of our main result, we will present some
results that will help us later. First, we define a time-continuous version of
the Euler scheme by $\bar{x}^{(N)}_{t}=\bar{x}_{t}$ with

$\bar{x}_{t}=\bar{x}_{\eta(t)}+a(\eta(t),\bar{x}_{\eta(t)})(t-\eta(t))+\sigma(\eta(t),\bar{x}_{\eta(t)})^{\mathfrak{g}}(W_{t}-W_{\eta(t)}),\qquad
t\in[0,T].$ (11)

Here, we denote $\eta(t)=\max\\{t_{k}\in\\{t_{0},t_{1},...,t_{N}\\}:t_{k}\leq
t\\}$. Clearly, we have $x_{k}=\overline{x}_{t_{k}}$ for $k=0,\ldots,N$.

Note that (HLG) and (L) imply that $a$ and $\sigma$ satisfy a linear growth
condition, that is there exists $C>0$ such that

$|a(t,x)|+|\sigma(t,x)|\leq C(1+|x|)\qquad\textrm{for all}\quad
t\in[0,T],\,x\in\mathbb{R}.$

Since

$x^{\mathfrak{g}}\leq 1+x\qquad\textrm{for all}\quad x\in[0,\infty),$

also $\sigma^{\mathfrak{g}}$ satisfies a linear growth condition. Thus, we
obtain the following result by standard computations.

######  Lemma 3.1.

Let $p\geq 1$ and Assumption 1.1 hold. Then, there exist constants $C_{p}>0$
such that

$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}|X_{t}|^{p}\right]+\sup_{0\leq
s<t\leq T}\mathbb{E}\left[\frac{|X_{t}-X_{s}|^{p}}{|t-s|^{p/2}}\right]\leq
C_{p}$

and

$\sup_{N\in\mathbb{N}}\mathbb{E}\left[\sup_{t\in[0,T]}|\bar{x}_{t}^{(N)}|^{p}\right]+\sup_{N\in\mathbb{N}}\sup_{0\leq
s<t\leq
T}\mathbb{E}\left[\frac{|\bar{x}_{t}^{(N)}-\bar{x}_{s}^{(N)}|^{p}}{|t-s|^{p/2}}\right]\leq
C_{p}.$

The next lemma gives us a bound for the expected local time in zero of a
semimartingale. It is taken from [7].

######  Lemma 3.2.

For any $\delta\in(0,1)$ and any real-valued, continuous semimartingale
$Y=(Y_{t})_{t\in[0,T]}$, we have

$\displaystyle\mathbb{E}\left[L^{0}_{t}(Y)\right]\leq 4\delta$
$\displaystyle-2\mathbb{E}\left[\int_{0}^{t}\left(\mathbbm{1}_{\\{Y_{s}\in(0,\delta)\\}}+\mathbbm{1}_{\\{Y_{s}\geq\delta\\}}e^{1-\frac{Y_{s}}{\delta}}\right)dY_{s}\right]$
$\displaystyle+\frac{1}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{Y_{s}>\delta\\}}e^{1-\frac{Y_{s}}{\delta}}d\langle
Y\rangle_{s}\right],\qquad\qquad t\in[0,T].$

Here $(L_{t}^{0}(Y))_{t\in[0,T]}$ is the local time of $Y$ in $x=0$. For
almost all $\omega\in\Omega$ the map $[0,T]\ni
t\mapsto[L_{t}^{0}(Y)](\omega)\in\mathbb{R}$ is continuous and non-decreasing
with $L_{0}^{0}(Y)=0$. For more background on local times see, e.g., Chapter
III.7 in [21]. For our purposes, it suffices to mention the Tanaka-Meyer
formula for continuous semimartingales of the form

$Y_{t}=y_{0}+\int_{0}^{t}y^{A}_{s}ds+\int_{0}^{t}y_{s}^{M}dW_{s},\qquad
t\in[0,T],$

where $(y_{t}^{A})_{t\in[0,T]}$ and $(y_{t}^{M})_{t\in[0,T]}$ are continuous,
square-integrable and adapted processes and $y_{0}\in\mathbb{R}$. Then the
Tanaka-Meyer formula, see, e.g., Equation 7.9 in Chapter III in [21], states
that

$|Y_{t}|=|y_{0}|+\int_{0}^{t}\text{sign}(Y_{s})y_{s}^{A}ds+\int_{0}^{t}\text{sign}(Y_{s})y_{s}^{M}dW_{s}+2L_{t}^{0}(Y),\qquad
t\in[0,T].$ (12)

Finally, the following inequalities will be very useful in our computations.

######  Lemma 3.3.

Let $\mathfrak{g}\in\left[\frac{1}{2},1\right)$ and $\beta\in(0,1]$. Then, we
have

$\displaystyle\left|x^{\mathfrak{g}}-y^{\mathfrak{g}}\right|\leq
2x^{-(1-\mathfrak{g})}\left|x-y\right|\qquad\textrm{for all }\quad x>0,\,y\geq
0,$ (13)

and

$\displaystyle\left|x^{\mathfrak{g}}-y^{\mathfrak{g}}\right|\leq
2x^{-(1-\mathfrak{g})\beta}\left|x-y\right|^{\beta+\mathfrak{g}(1-\beta)}\qquad\textrm{for
all }\quad x>0,\,y\geq 0.$ (14)

###### Proof.

(i) We first prove the assertion

$|x^{\mathfrak{g}}-y^{\mathfrak{g}}|\left(x^{1-\mathfrak{g}}+y^{1-\mathfrak{g}}\right)\leq
2|x-y|$

from which inequality (13) directly follows. For $x=y$, the statement holds
trivially. Without loss of generality, we can assume that $x>y$ and prove the
assertion

$(x^{\mathfrak{g}}-y^{\mathfrak{g}})\left(x^{1-\mathfrak{g}}+y^{1-\mathfrak{g}}\right)\leq
2(x-y).$

We start with

$\displaystyle(x^{\mathfrak{g}}-y^{\mathfrak{g}})\left(x^{1-\mathfrak{g}}+y^{1-\mathfrak{g}}\right)$
$\displaystyle=y^{\mathfrak{g}}\left(\left(\frac{x}{y}\right)^{\mathfrak{g}}-1\right)y^{1-\mathfrak{g}}\left(\left(\frac{x}{y}\right)^{1-\mathfrak{g}}+1\right)$
$\displaystyle=y\left(\left(\frac{x}{y}\right)^{\mathfrak{g}}-1\right)\left(\left(\frac{x}{y}\right)^{1-\mathfrak{g}}+1\right).$

Now set $z:=\frac{x}{y}$. Notice that $z>1$. Therefore, it remains to show
that

$\displaystyle(z^{\mathfrak{g}}-1)\left(z^{1-\mathfrak{g}}+1\right)\leq
2(z-1)$ (15)

for $z>1$. Next, we define

$\displaystyle
f(z):=2(z-1)-(z^{\mathfrak{g}}-1)\left(z^{1-\mathfrak{g}}+1\right).$

We have $f(1)=0$ and if we are able to show that $f$ is increasing for all
$z\geq 1$, then Equation (15) follows. We have

$\displaystyle
f^{\prime}(z)=1-\mathfrak{g}z^{\mathfrak{g}-1}+(1-\mathfrak{g})z^{-\mathfrak{g}}$

and consequently

$f^{\prime}(z)\geq 1-\mathfrak{g}>0$

for all $z\geq 1$. This concludes the proof of (15) and the first assertion.

(ii) For the second assertion note that

$|x^{\mathfrak{g}}-y^{\mathfrak{g}}|\leq|x-y|^{\mathfrak{g}}$

by concavity. Equation (14) follows then from

$|x^{\mathfrak{g}}-y^{\mathfrak{g}}|=|x^{\mathfrak{g}}-y^{\mathfrak{g}}|^{\beta}|x^{\mathfrak{g}}-y^{\mathfrak{g}}|^{1-\beta}\leq|x^{\mathfrak{g}}-y^{\mathfrak{g}}|^{\beta}|x-y|^{\mathfrak{g}(1-\beta)}$

and inequality (13). ∎

###  3.2 Proof of Theorem 1.3

Theorem 1.3 is a direct consequence of the following Theorem 3.4 with

$\beta=1-\frac{1}{1-\mathfrak{g}}\left(\mathfrak{s}+\frac{\varepsilon}{2}\right)$

and $\varepsilon>0$ arbitrarily small. The strategy of the proof of Theorem
3.4 is based on the strategy of [28], which in turn was inspired by [7].

######  Theorem 3.4.

Let $\bar{x}$ be given as in (11) and let Assumption 1.1 be fulfilled.
Moreover, assume that there exists $\beta\in(0,1]$ and $\varepsilon>0$ such
that

$\mathbb{E}\left[\int_{0}^{T}\sigma(s,X_{s})^{2(\mathfrak{g}-1)\beta-\varepsilon}ds\right]<\infty.$
(16)

Then, we have

$\limsup_{N\rightarrow\infty}\,N^{\lambda}\sup\limits_{t\in[0,T]}\mathbb{E}\left|X_{t}-\bar{x}_{t}^{(N)}\right|=0$

for all $\lambda<\mathfrak{g}-1/2+\beta(1-\mathfrak{g})$.

###### Proof.

Define the error process $R=(R_{t})_{t\in[0,T]}$ by $R_{t}=X_{t}-\bar{x}_{t}$.

(i) The Tanaka-Meyer formula, see Equation (12), yields

$\displaystyle\mathbb{E}\left[\left|R_{t}\right|\right]=$
$\displaystyle\mathbb{E}\left[\int_{0}^{t}\text{sign}(R_{u})dR_{u}\right]+\mathbb{E}\left[L^{0}_{t}(R)\right]$
(17) $\displaystyle=$
$\displaystyle\mathbb{E}\left[\int_{0}^{t}\text{sign}(R_{u})\left(a(u,X_{u})-a(\eta(u),\bar{x}_{\eta(u)})\right)du\right]$
$\displaystyle+\mathbb{E}\left[\int_{0}^{t}\text{sign}(R_{u})\left(\sigma(u,X_{u})^{\mathfrak{g}}-\sigma(\eta(u),\bar{x}_{\eta(u)})^{\mathfrak{g}}\right)dW_{u}\right]+\mathbb{E}\left[L^{0}_{t}(R)\right].$

We have

$\mathbb{E}\left[\int_{0}^{t}\text{sign}(R_{u})\left(\sigma(u,X_{u})^{\mathfrak{g}}-\sigma(\eta(u),\bar{x}_{\eta(u)})^{\mathfrak{g}}\right)dW_{u}\right]=0$

due to Lemma 3.1, the linear growth of the diffusion coefficient and the
martingale property of the Itō integral. Looking at the first term, we have

$\displaystyle\mathbb{E}\left[\int_{0}^{t}\text{sign}(R_{u})\left(a(u,X_{u})-a(\eta(u),\bar{x}_{\eta(u)})\right)du\right]$
$\displaystyle=\mathbb{E}\left[\int_{0}^{t}\text{sign}(R_{u})\left(a(u,X_{u})-a(u,\bar{x}_{u})\right)du\right]$
$\displaystyle\qquad+\mathbb{E}\left[\int_{0}^{t}\text{sign}(R_{u})\left(a(u,\bar{x}_{u})-a(\eta(u),\bar{x}_{u})\right)du\right]$
$\displaystyle\qquad+\mathbb{E}\left[\int_{0}^{t}\text{sign}(R_{u})\left(a(\eta(u),\bar{x}_{u})-a(\eta(u),\bar{x}_{\eta(u)})\right)du\right]$
$\displaystyle\leq C\left(\int_{0}^{t}\mathbb{E}\left[|R_{u}|\right]du+(\Delta
t)^{1/2}\int_{0}^{t}\mathbb{E}\left[1+|\bar{x}_{u}|\right]du+\int_{0}^{t}\mathbb{E}\left[|\bar{x}_{u}-\bar{x}_{\eta(u)}|\right]du\right)$

using (HLG) and (L). Lemma 3.1 now implies

$\sup_{u\in[0,t]}\mathbb{E}\left[\left|R_{u}\right|\right]\leq C(\Delta
t)^{\frac{1}{2}}+C\int_{0}^{t}\sup_{v\in[0,u]}\mathbb{E}\left[|R_{v}|\right]du+\mathbb{E}\left[L^{0}_{t}(R)\right].$
(18)

(ii) With Lemma 3.2 we can derive a bound for the expected local time of the
error process $R$ in zero. Let $\delta\in(0,1)$, then

$\displaystyle\mathbb{E}\left[L^{0}_{t}(R)\right]\leq 4\delta$
$\displaystyle-2\mathbb{E}\left[\int_{0}^{t}\left(\mathbbm{1}_{\\{R_{s}\in(0,\delta)\\}}+\mathbbm{1}_{\\{R_{s}\geq\delta\\}}e^{1-\frac{R_{s}}{\delta}}\right)dR_{s}\right]$
(19)
$\displaystyle+\frac{1}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{R_{s}>\delta\\}}e^{1-\frac{R_{s}}{\delta}}d\langle
R\rangle_{s}\right].$

We define
$Y_{s}:=\mathbbm{1}_{\\{R_{s}\in(0,\delta)\\}}+\mathbbm{1}_{\\{R_{s}\geq\delta\\}}e^{1-\frac{R_{s}}{\delta}}$
and look at the second term of (19), i.e.,

$\mathbb{E}\left[\int_{0}^{t}Y_{s}dR_{s}\right]=\,\mathbb{E}\left[\int_{0}^{t}Y_{s}\left(a(s,X_{s})-a(\eta(s),\bar{x}_{\eta(s)})\right)ds\right],$

where we already used the martingale property of the Itō integral. Since
$0\leq Y_{s}\leq 1$, we obtain proceeding as above that

$\displaystyle\left|\mathbb{E}\left[\int_{0}^{t}Y_{s}dR_{s}\right]\right|\leq
C(\Delta t)^{\frac{1}{2}}+C\int_{0}^{t}\mathbb{E}\left[|R_{u}|\right]du.$ (20)

For the third term of Equation (19) note that

$\langle
R\rangle_{t}=\int_{0}^{t}\left(\sigma(s,X_{s})^{\mathfrak{g}}-\sigma(\eta(s),\bar{x}_{\eta(s)})^{\mathfrak{g}}\right)^{2}ds,\qquad
t\in[0,T].$

Condition (16) implies that

$P(\sigma(x,X_{s})=0)=0$

for all $s\in[0,T]$. Thus, we can apply Lemma 3.3, i.e.,

$\left|x^{\mathfrak{g}}-y^{\mathfrak{g}}\right|\leq
2x^{-(1-\mathfrak{g})\beta}\left|x-y\right|^{\beta+\mathfrak{g}(1-\beta)},\qquad
x>0,\,y\geq 0,$

and obtain

$\displaystyle\frac{1}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{R_{s}>\delta\\}}e^{1-\frac{R_{s}}{\delta}}d\langle
R\rangle_{s}\right]$
$\displaystyle=\frac{1}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{R_{s}>\delta\\}}e^{1-\frac{R_{s}}{\delta}}\left(\sigma(s,X_{s})^{\mathfrak{g}}-\sigma(\eta(s),\bar{x}_{\eta(s)})^{\mathfrak{g}}\right)^{2}ds\right]$
$\displaystyle\leq\frac{4}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{R_{s}>\delta\\}}e^{1-\frac{R_{s}}{\delta}}\frac{|\sigma(s,X_{s})-\sigma(\eta(s),\bar{x}_{\eta(s)})|^{2\beta+2\mathfrak{g}(1-\beta)}}{\sigma(s,X_{s})^{2(1-\mathfrak{g})\beta}}ds\right].$

Since

$|x|+|y|+|z|\leq
C_{\alpha}\left(|x|^{\alpha}+|y|^{\alpha}+|z|^{\alpha}\right)^{1/\alpha}\qquad\textrm{for
all}\quad x,y,z\in\mathbb{R},$

for $\alpha\geq 1$ and

$2\beta+2\mathfrak{g}(1-\beta)=2(1-\mathfrak{g})\beta+2\mathfrak{g},$

it follows that

$\displaystyle\frac{1}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{R_{s}>\delta\\}}e^{1-\frac{R_{s}}{\delta}}d\langle
R\rangle_{s}\right]$
$\displaystyle\leq\frac{C_{\beta,\mathfrak{g}}}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{R_{s}>\delta\\}}e^{1-\frac{R_{s}}{\delta}}\frac{|\sigma(s,X_{s})-\sigma(s,\bar{x}_{s})|^{2(1-\mathfrak{g})\beta+2\mathfrak{g}}}{\sigma(s,X_{s})^{2(1-\mathfrak{g})\beta}}ds\right]$
$\displaystyle\qquad+\frac{C_{\beta,\mathfrak{g}}}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{R_{s}>\delta\\}}e^{1-\frac{R_{s}}{\delta}}\frac{|\sigma(s,\bar{x}_{s})-\sigma(\eta(s),\bar{x}_{s})|^{2(1-\mathfrak{g})\beta+2\mathfrak{g}}}{\sigma(s,X_{s})^{2(1-\mathfrak{g})\beta}}ds\right]$
$\displaystyle\qquad+\frac{C_{\beta,\mathfrak{g}}}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{R_{s}>\delta\\}}e^{1-\frac{R_{s}}{\delta}}\frac{|\sigma(\eta(s),\bar{x}_{s})-\sigma(\eta(s),\bar{x}_{\eta(s)})|^{2(1-\mathfrak{g})\beta+2\mathfrak{g}}}{\sigma(s,X_{s})^{2(1-\mathfrak{g})\beta}}ds\right].$

Moreover, exploiting (HLG) and (L) yields

$\displaystyle\frac{1}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{R_{s}>\delta\\}}e^{1-\frac{R_{s}}{\delta}}d\langle
R\rangle_{s}\right]$
$\displaystyle\leq\frac{C_{\beta,\mathfrak{g}}}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{R_{s}>\delta\\}}e^{1-\frac{R_{s}}{\delta}}\frac{|R_{s}|^{2(1-\mathfrak{g})\beta+2\mathfrak{g}}}{\sigma(s,X_{s})^{2(1-\mathfrak{g})\beta}}ds\right]$
$\displaystyle\qquad+\frac{C_{\beta,\mathfrak{g}}}{\delta}(\Delta
t)^{\beta(1-\mathfrak{g})+\mathfrak{g}}\,\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{R_{s}>\delta\\}}e^{1-\frac{R_{s}}{\delta}}\frac{(1+|\bar{x}_{s}|)^{2(1-\mathfrak{g})\beta+2\mathfrak{g}}}{\sigma(s,X_{s})^{2(1-\mathfrak{g})\beta}}ds\right]$
$\displaystyle\qquad+\frac{C_{\beta,\mathfrak{g}}}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{R_{s}>\delta\\}}e^{1-\frac{R_{s}}{\delta}}\frac{|\bar{x}_{s}-\bar{x}_{\eta(s)}|^{2(1-\mathfrak{g})\beta+2\mathfrak{g}}}{\sigma(s,X_{s})^{2(1-\mathfrak{g})\beta}}ds\right].$

Applying Lemma 3.1, condition (16) and Hölder’s inequality with
$p=\frac{2(1-\mathfrak{g})\beta+\varepsilon}{2(1-\mathfrak{g})\beta}$ and
$q=\frac{p}{p-1}$ to the second and third term on the right side now gives

$\displaystyle\frac{1}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{R_{s}>\delta\\}}e^{1-\frac{R_{s}}{\delta}}d\langle
R\rangle_{s}\right]$
$\displaystyle\leq\frac{C_{\beta,\mathfrak{g}}}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{R_{s}>\delta\\}}e^{1-\frac{R_{s}}{\delta}}\frac{|R_{s}|^{2(1-\mathfrak{g})\beta+2\mathfrak{g}}}{\sigma(s,X_{s})^{2(1-\mathfrak{g})\beta}}ds\right]$
(21)
$\displaystyle\qquad+\frac{C_{\beta,\mathfrak{g},\varepsilon}}{\delta}(\Delta
t)^{\beta(1-\mathfrak{g})+\mathfrak{g}}.$

(iii) It remains to study

$\displaystyle\frac{C_{\beta,\mathfrak{g}}}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{R_{s}>\delta\\}}e^{1-\frac{R_{s}}{\delta}}\frac{|R_{s}|^{2(1-\mathfrak{g})\beta+2\mathfrak{g}}}{\sigma(s,X_{s})^{2(1-\mathfrak{g})\beta}}ds\right].$

For this let $\alpha\in(0,1)$. Since

$\sup_{s\in[0,T]}\mathbb{E}\left[|R_{s}|^{p}\right]\leq C_{p}$

for all $p\geq 1$ due to our assumptions and Lemma 3.1, we have that

$\displaystyle\frac{1}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{R_{s}>\delta\\}}e^{1-\frac{R_{s}}{\delta}}\frac{|R_{s}|^{2(1-\mathfrak{g})\beta+2\mathfrak{g}}}{\sigma(s,X_{s})^{2(1-\mathfrak{g})\beta}}ds\right]$
$\displaystyle\quad=\frac{1}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{R_{s}\in(\delta,\delta^{\alpha})\\}}e^{1-\frac{R_{s}}{\delta}}\frac{|R_{s}|^{2(1-\mathfrak{g})\beta+2\mathfrak{g}}}{\sigma(s,X_{s})^{2(1-\mathfrak{g})\beta}}ds\right]$
$\displaystyle\qquad\quad+\frac{1}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{R_{s}\geq\delta^{\alpha}\\}}e^{1-\frac{R_{s}}{\delta}}\frac{|R_{s}|^{2(1-\mathfrak{g})\beta+2\mathfrak{g}}}{\sigma(s,X_{s})^{2(1-\mathfrak{g})\beta}}ds\right]$
$\displaystyle\quad\leq\delta^{\alpha(2(1-\mathfrak{g})\beta+2\mathfrak{g})-1}\int_{0}^{t}\mathbb{E}\left[\sigma(s,X_{s})^{-2(1-\mathfrak{g})\beta}\right]ds$
$\displaystyle\qquad\quad+C_{\beta,\mathfrak{g},\varepsilon}\frac{e^{-\delta^{\alpha-1}}}{\delta}\int_{0}^{t}\left(\mathbb{E}\left[\sigma(s,X_{s})^{-2(1-\mathfrak{g})\beta-\varepsilon}\right]\right)^{\frac{2(1-\mathfrak{g})\beta}{2(1-\mathfrak{g})\beta+\varepsilon}}ds$

by another application of the Hölder inequality. Since

$\limsup_{\delta\rightarrow 0}\frac{e^{-\delta^{\alpha-1}}}{\delta^{p}}=0$

for all $p\geq 0$, condition (16) now yields

$\displaystyle\frac{1}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{R_{s}>\delta\\}}e^{1-\frac{R_{s}}{\delta}}\frac{|R_{s}|^{2(1-\mathfrak{g})\beta+2\mathfrak{g}}}{\sigma(s,X_{s})^{2(1-\mathfrak{g})\beta}}ds\right]\leq
C_{\beta,\mathfrak{g},\alpha,\varepsilon}\delta^{\alpha(2(1-\mathfrak{g})\beta+2\mathfrak{g})-1}.$
(22)

Summarizing the Equations (19) – (22) we have shown that

$\displaystyle\mathbb{E}\left[L^{0}_{t}(R)\right]\leq 4\delta$
$\displaystyle+C(\Delta
t)^{\frac{1}{2}}+C\int_{0}^{t}\mathbb{E}\left[|R_{u}|\right]du$ (23)
$\displaystyle+C_{\beta,\mathfrak{g},\alpha,\varepsilon}\delta^{\alpha(2(1-\mathfrak{g})\beta+2\mathfrak{g})-1}+\frac{C_{\beta,\mathfrak{g},\varepsilon}}{\delta}(\Delta
t)^{\beta(1-\mathfrak{g})+\mathfrak{g}}.$

(iv) Setting $\delta=(\Delta t)^{1/2}$ gives

$\mathbb{E}\left[L^{0}_{t}(R)\right]\leq C(\Delta
t)^{1/2}+C\int_{0}^{t}\mathbb{E}\left[|R_{u}|\right]du+C_{\beta,\mathfrak{g},\alpha,\varepsilon}(\Delta
t)^{\alpha((1-\mathfrak{g})\beta+\mathfrak{g})-\frac{1}{2}}+C_{\beta,\mathfrak{g},\varepsilon}(\Delta
t)^{\beta(1-\mathfrak{g})+\mathfrak{g}-\frac{1}{2}}.$

Note that

$\alpha(\beta(1-\mathfrak{g})+\mathfrak{g})-\frac{1}{2}<\beta(1-\mathfrak{g})+\mathfrak{g}-\frac{1}{2}\leq\frac{1}{2},$

since we have $\alpha\in(0,1)$ and $\beta\in(0,1]$. Consequently, we have

$\mathbb{E}\left[L^{0}_{t}(R)\right]\leq
C\int_{0}^{t}\mathbb{E}\left[|R_{u}|\right]du+C_{\beta,\mathfrak{g},\alpha,\varepsilon}(\Delta
t)^{\alpha(\beta(1-\mathfrak{g})+\mathfrak{g})-\frac{1}{2}}.$

Combining this with Equation (18) yields

$\sup_{u\in[0,t]}\mathbb{E}\left[\left|R_{u}\right|\right]\leq
C_{\beta,\mathfrak{g},\alpha,\varepsilon}(\Delta
t)^{\alpha(\beta(1-\mathfrak{g})+\mathfrak{g})-\frac{1}{2}}+C\int_{0}^{t}\sup_{v\in[0,u]}\mathbb{E}\left[|R_{v}|\right]du$

and the assertion follows by choosing $\alpha\in(0,1)$ sufficiently large and
an application of Gronwall’s lemma.

∎

##  4 Appendix

For the sake of convenience, we collect here some general auxiliary results.

###  4.1 Feller test

Consider the autonomous SDE

$dX_{t}=a(X_{t})dt+c(X_{t})dW_{t},\quad t\in[0,T],\qquad X_{0}=x_{0}.$ (24)

We define

$\displaystyle S:=\inf\\{t\geq 0:X_{t}\notin(l,r)\\}$

as the exit time from $I=(l,r)$ with $-\infty\leq l<r\leq\infty$. Now,
Feller’s test for explosions (see Theorem 5.29 in Chapter V in [21]) gives
necessary and sufficient conditions for the finiteness of $S$.

######  Theorem 4.1.

Assume that the coefficients $a:I\rightarrow\mathbb{R}$ and
$c:I\rightarrow\mathbb{R}$ satisfy

$\displaystyle\forall x\in I:\quad c^{2}(x)>0,$

and

$\displaystyle\forall x\in
I\,\,\,\,\exists\varepsilon>0:\int_{x-\varepsilon}^{x+\varepsilon}\frac{1+|a(y)|}{c^{2}(y)}dy<\infty.$

Define the scale function $p$ by

$\displaystyle
p(x):=\int_{o}^{x}\exp\left(-2\int_{o}^{y}\frac{a(z)}{c^{2}(z)}dz\right)dy,\quad
x\in I,$

for a fixed $o\in I$ and the function $v$ by

$\displaystyle
v(x):=\int_{o}^{x}p^{\prime}(y)\int_{o}^{y}\frac{2}{p^{\prime}(z)c^{2}(z)}dzdy,\quad
x\in I.$

Let $(X,W)$ be a weak solution of (24) in $I$ with deterministic
$X_{0}=x_{0}\in I$. Then, $P\left(S=\infty\right)=1$ or
${P}\left(S=\infty\right)<1$ according to whether

$\displaystyle v(l+)=\lim_{x\downarrow l}v(x)=\lim_{x\uparrow
r}v(x)=v(r-)=\infty$

or not.

###  4.2 A deterministic time change

The following auxiliary result deals with deterministic time changes for
Riemann and Itō integrals. For this, let
$\theta\in\mathcal{C}([0,T];(0,\infty))$ and define

$\Theta:[0,T]\rightarrow\mathbb{R},\qquad\Theta(t)=\int_{0}^{t}\theta^{2}(s)ds.$

Since $\theta^{2}(t)>0$ for all $t\in[0,T]$, the function $\Theta$ is strictly
increasing. Therefore its inverse function $A=\Theta^{-1}:D\rightarrow[0,T]$
with ${D}=\Theta([0,T])$ exists. Then,

$\widetilde{W}_{t}=\int_{0}^{A(t)}\theta(s)dW_{s},\qquad t\in[0,A^{-1}(T)],$

is a Brownian motion adapted to the filtration
$(\widetilde{\mathcal{F}}_{t})_{t\in[0,A^{-1}(T))]}$ with
$\widetilde{\mathcal{F}}_{t}=\mathcal{F}_{A(t)}$. See, e.g., Proposition 4.6
in Chapter III of [21]. Moreover, we have the following:

######  Proposition 4.2.

Consider an continuous and adapted process $Y=(Y_{t})_{t\in[0,T]}$ such that

$\int_{0}^{T}\mathbb{E}|Y_{s}|^{2}ds<\infty$

and let $\theta\in\mathcal{C}([0,T];(0,\infty))$. Then, we have

$\int_{0}^{A(t)}Y_{s}ds=\int_{0}^{t}Y_{A(\tau)}\frac{1}{\theta^{2}(A(\tau))}d\tau,\qquad
t\in[0,A^{-1}(T)],\qquad P-a.s.,$

and

$\int_{0}^{A(t)}\theta(s)Y_{s}dW_{s}=\int_{0}^{t}Y_{A(\tau)}d\widetilde{W}_{\tau},\qquad
t\in[0,A^{-1}(T)],\qquad P-a.s..$

###### Proof.

The first statement is the classical substitution rule for Riemann integrals,
while the second statement follows from Proposition 4.8 in Chapter III of
[21]. ∎

###  4.3 Moment bounds for CIR, WF and CKLS

Here we collect moment bounds for the CIR, WF and CKLS process. Note that
under the assumptions below the Feller test implies that with probability one
the CIR and CKLS process are positive, i.e., take values in $[0,\infty)$,
while the WF process takes values in $[0,1]$.

######  Lemma 4.3.

  * (i)

Consider the CIR process given by

$dV_{t}=\kappa\left(\lambda-V_{t}\right)dt+\theta\sqrt{V_{t}}dW_{t}\quad
t\in[0,T],\qquad V_{0}=v_{0}>0,$

with $\kappa,\lambda,\theta>0$. Then we have

$\sup_{s\in[0,T]}\mathbb{E}|V_{s}|^{p}<\infty\quad\textrm{for all}\quad
p>-\frac{2\kappa\lambda}{\theta^{2}}.$

  * (ii)

Consider the WF process given by

$dV_{t}=\kappa\left(\lambda-
V_{t}\right)dt+\theta\sqrt{V_{t}(1-V_{t})}dW_{t}\quad t\in[0,T],\qquad
V_{0}=v_{0}\in(0,1),$

with $\kappa,\lambda,\theta>0$. Then we have

$\sup_{s\in[0,T]}\mathbb{E}|V_{s}|^{p}<\infty\quad\textrm{for all}\quad
p>-\frac{2\kappa\lambda}{\theta^{2}}\ $

and

$\sup_{s\in[0,T]}\mathbb{E}|1-V_{s}|^{p}<\infty\quad\textrm{for all}\quad
p>-\frac{2\kappa}{\theta^{2}}(1-\lambda).$

  * (iii)

Let $\mathfrak{g}\in(1/2,1)$. Consider the CKLS process given by

$dV_{t}=\kappa\left(\lambda-V_{t}\right)dt+\theta
V_{t}^{\mathfrak{g}}dW_{t}\quad t\in[0,T],\qquad V_{0}=v_{0}>0,$

with $\kappa,\lambda,\theta>0$. Then we have

$\sup_{s\in[0,T]}\mathbb{E}|V_{s}|^{p}<\infty\quad\textrm{for all}\quad
p\in\mathbb{R}.$

###### Proof.

(i) This statement can be found, e.g., in Section 3 in [8] and Theorem 3.1 in
[20], respectively.
(ii) This statement can be found, e.g., in Section 3.5 in [34]. (iii) This
statement can be found, e.g., in Section 3.4 in [34] and [2], respectively.

∎

###  4.4 Comparison result

Proposition 2.18 in Chapter V of [21] gives a comparison result for one-
dimensional SDEs, which we use in Section 2.

######  Proposition 4.4.

Consider two continuous, adapted process $X^{(j)},j=1,2$, such that

$\displaystyle
X^{(j)}_{t}=X^{(j)}_{0}+\int_{0}^{t}a_{j}(s,X^{(j)}_{s})ds+\int_{0}^{t}c(s,X^{(j)}_{s})dW_{s},\qquad
t\in[0,T],$

holds $P$-a.s.. Assume that

  * (i)

the coefficients $c(t,x),a_{1}(t,x),a_{2}(t,x)$ are continuous and real-valued
functions on $[0,T]\times\mathbb{R}$,

  * (ii)

the diffusion coefficient $c(t,x)$ satisfies

$\displaystyle\left|c(t,x)-c(t,y)\right|\leq h\left(|x-y|\right)$

for every $t\in[0,T]$, $x,y\in\mathbb{R}$, where
$h:[0,\infty)\rightarrow[0,\infty)$ is strictly increasing with $h(0)=0$ and

$\displaystyle\int_{0}^{\varepsilon}h^{-2}(u)du=\infty\qquad\text{for
all}\quad\varepsilon>0,$

  * (iii)

$X^{(1)}_{0}\leq X^{(2)}_{0}$ $P$-a.s.,

  * (iv)

$a_{1}(t,x)\leq a_{2}(t,x)$ for all $t\in[0,T]$ and all $x\in\mathbb{R}$,

  * (v)

either $a_{1}(t,x)$ or $a_{2}(t,x)$ satisfies condition (L), i.e., is globally
Lipschitz in $x$, uniformly in $t$.

Then,

$\displaystyle{P}\left(X^{(1)}_{t}\leq X^{(2)}_{t}\,\,\text{for all
}\,\,t\in[0,T]\right)=1.$

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