#  Sharp $L^{1}$-Approximation of the log-Heston SDE by Euler-type methods

Annalena Mickel 111Mathematical Institute and DFG Research Training Group
1953, University of Mannheim, B6, 26, D-68131 Mannheim, Germany,
<EMAIL_ADDRESS>Andreas Neuenkirch 222Mathematical Institute,
University of Mannheim, B6, 26, D-68131 Mannheim, Germany<EMAIL_ADDRESS>mannheim.de

###### Abstract

We study the $L^{1}$-approximation of the log-Heston SDE at equidistant time
points by Euler-type methods. We establish the convergence order
$1/2-\epsilon$ for $\epsilon>0$ arbitrarily small, if the Feller index $\nu$
of the underlying CIR process satisfies $\nu>1$. Thus, we recover the standard
convergence order of the Euler scheme for SDEs with globally Lipschitz
coefficients. Moreover, we discuss the case $\nu\leq 1$ and illustrate our
findings by several numerical examples.

Key words:  CIR process, Heston model, Euler-type methods, $L^{1}$-error
2010 Mathematics Subject Classification: 65C30; 60H35; 91G60

##  1 Introduction and Main Results

The CIR process originates from the works of Feller in the 1950s, see e.g.
[16], and was used by Cox, Ingersoll and Ross [10] to model short term
interest rates. It is the solution to the following stochastic differential
equation (SDE)

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

where $V_{0}=v_{0}>0$ and $W=(W_{t})_{t\in[0,T]}$ is a Brownian motion. The
parameters can be interpreted as follows: $\theta>0$ is the long run mean of
the process, $\kappa>0$ is its speed of mean reversion and $\sigma>0$ is its
volatility. We assume the initial value $v_{0}$ to be deterministic. We denote
the Feller index of the CIR process by

$\nu:=\frac{2\kappa\theta}{\sigma^{2}}.$

Note that the CIR process takes positive values only. Additionally, if
$\nu\geq 1$ almost all sample paths are strictly positive. The CIR process is
used in particular to model the volatility of the asset price in the Heston
model [22]. Here, the SDE for the price process and its volatility are given
by

$\begin{aligned} dS_{t}&=\mu S_{t}dt+\sqrt{V_{t}}S_{t}\left(\rho
dW_{t}+\sqrt{1-\rho^{2}}dB_{t}\right),\\\ dV_{t}&=\kappa(\theta-
V_{t})dt+\sigma\sqrt{V_{t}}dW_{t},\end{aligned}\qquad t\in[0,T],$ (2)

where $S_{0}=s_{0}>0$ is deterministic, $\mu\in\mathbb{R}$ is the risk-free
interest rate, $\rho\in[-1,1]$ determines the correlation between the two
processes and $W=(W_{t})_{t\in[0,T]}$, $B=(B_{t})_{t\in[0,T]}$ are independent
Brownian motions. Usually, the log-Heston model instead of the Heston model is
considered in numerical practice. This yields the SDE

$\begin{aligned}
dX_{t}&=\left(\mu-\frac{1}{2}V_{t}\right)dt+\sqrt{V_{t}}\left(\rho
dW_{t}+\sqrt{1-\rho^{2}}dB_{t}\right),\\\ dV_{t}&=\kappa(\theta-
V_{t})dt+\sigma\sqrt{V_{t}}dW_{t},\end{aligned}\qquad t\in[0,T],$ (3)

where $X_{t}=\log(S_{t})$ and $x_{0}=\log(s_{0})\in\mathbb{R}$. Since the
right hand side of the SDE for the log-asset price does not depend on $X_{t}$,
its approximation reduces to the approximation of a Riemann integral of $V$
and an Itō integral of $\sqrt{V}$. So, the main difficulty is the
approximation of the CIR process $V=(V_{t})_{t\in[0,T]}$. Since the CIR
process takes positive values only and the diffusion coefficient is a square
root and thus not globally Lipschitz continuous, much effort has been devoted
to this problem in the last 25 years, see Subsection 1.1.

In this article, we are looking at Euler discretization schemes for the CIR
process and the log-Heston model. We will work with an equidistant
discretization

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

with $\Delta t=T/N$. A naive Euler discretization will give negative values
and is not well defined due to the square root coefficient. Therefore a ”fix”
is required. A summary of the existing Euler schemes for the CIR process and a
numerical comparison can be found in [28], where a general framework for Euler
schemes for the CIR process is proposed as

$\displaystyle\bar{v}_{t_{k+1}}$
$\displaystyle=f_{1}(\bar{v}_{t_{k}})+\kappa\left(\theta-
f_{2}(\bar{v}_{t_{k}})\right)(t_{k+1}-t_{k})+\sigma\sqrt{f_{3}(\bar{v}_{t_{k}})}\left(W_{t_{k+1}}-W_{t_{k}}\right)$
(4) $\displaystyle\hat{v}_{t_{k+1}}$ $\displaystyle=f_{3}(\bar{v}_{t_{k+1}})$

for $k=0,\ldots,N-1$ with $\hat{v}_{0}=\bar{v}_{0}=v_{0}$ and suitable
functions $f_{i}$ that are chosen from

$\displaystyle{\tt id}:\mathbb{R}\rightarrow\mathbb{R},$
$\displaystyle\quad{\tt id}(x)=x,$ $\displaystyle{\tt
abs}:\mathbb{R}\rightarrow[0,\infty),$ $\displaystyle\quad{\tt abs}(x)=x^{+},$
$\displaystyle{\tt sym}:\mathbb{R}\rightarrow[0,\infty),$
$\displaystyle\quad{\tt sym}(x)=|x|.$

Here we will study the Euler schemes with $f_{i}$ given by

$\displaystyle f_{1}={\tt id},\quad f_{2}\in\\{{\tt id,abs,sym}\\},\quad
f_{3}\in\\{{\tt abs,sym}\\}$ (5)

or

$\displaystyle f_{1}=f_{2}=f_{3}\in\\{\tt abs,sym\\}.$ (6)

The first set of conditions modifies the coefficients of the CIR process to
deal with negative values, which may arise in the computation. For example,
$\sqrt{v}$ is replaced by $\sqrt{v^{+}}$ or $\sqrt{|v|}$. After the
approximation $\bar{v}$ has been computed, $f_{3}$ is again applied to obtain
$\hat{v}$, since $\bar{v}$ may be still negative. The second set of conditions
is different. Here after each Euler step ${\tt abs}$ or ${\tt sym}$,
respectively, is applied to avoid negative values. See also Subsection 2.1 and
Subsection 2.2.

Table 1 shows all Euler schemes that are presented in [28] in detail. The Full
Truncation Euler was introduced in the same paper. The origin of the Euler
with Absorption fix is unknown, the Symmetrized Euler was analyzed in [7] for
example. The scheme from Higham and Mao was first analyzed in [23] and the
Partial Truncation Euler was first introduced in [14].

Table 1: Euler schemes from [28] Scheme | $f_{1}(x)$ | $f_{2}(x)$ | $f_{3}(x)$
---|---|---|---
Absorption (AE) | $(x)^{+}$ | $(x)^{+}$ | $(x)^{+}$
Symmetrized (SE) | $|x|$ | $|x|$ | $|x|$
Higham and Mao (HM) | $x$ | $x$ | $|x|$
Partial Truncation (PTE) | $x$ | $x$ | $(x)^{+}$
Full Truncation (FTE) | $x$ | $(x)^{+}$ | $(x)^{+}$

Results involving a (polynomial) convergence rate for these Euler schemes are
rare and usually come along with a strong restriction on the Feller index, see
Subsections 1.1 and 1.2. In this article, we will prove the
$L^{1}$-convergence rate of $\frac{1}{2}-\epsilon$ for all these schemes if
$\nu>1$ (with $\epsilon>0$ arbitrarily small). Furthermore, we will show that
this result carries over to the log-Heston model if the price process is
discretized with the standard Euler scheme, i.e. with

$\displaystyle\hat{x}_{t_{k+1}}=\hat{x}_{t_{k}}$
$\displaystyle+\left(\mu-\frac{1}{2}\hat{v}_{t_{k}}\right)(t_{k+1}-t_{k})$ (7)
$\displaystyle+\sqrt{\hat{v}_{t_{k}}}\left(\rho\left(W_{t_{k+1}}-W_{t_{k}}\right)+\sqrt{1-\rho^{2}}\left(B_{t_{k+1}}-B_{t_{k}}\right)\right),$

where $\hat{x}_{0}=x_{0}$ and $k=0,\ldots,N-1$.

######  Theorem 1.1.

Let $\nu>1$, $\epsilon>0$ and
$(\hat{v}_{t_{k}},\hat{x}_{t_{k}})_{k\in\\{0,\ldots,N\\}}$ given by Equations
(4), (5), (7) or by Equations (4), (6), (7). Then we have

$\displaystyle\lim_{N\rightarrow\infty}\,\,N^{1/2-\epsilon}\left(\max_{k\in\\{0,\ldots,N\\}}\mathbb{E}\left[|X_{t_{k}}-\hat{x}_{t_{k}}|\right]+\max_{k\in\\{0,\ldots,N\\}}\mathbb{E}\left[|V_{t_{k}}-\hat{v}_{t_{k}}|\right]\right)=0.$

Thus, we recover (up to an arbitrarily small $\epsilon>0$) the standard
convergence order of the Euler scheme for SDEs with globally Lipschitz
continuous coefficients.

For the case $\nu\leq 1$ we can obtain e.g. for the Euler schemes given by
Equations (4), (5), (7) convergence order $\nu/2-\epsilon$. However, this
estimate does not seem to be sharp, see our simulation study in Section 5.

######  Proposition 1.2.

Let $\nu\leq 1$, $\epsilon>0$ and
$(\hat{v}_{t_{k}},\hat{x}_{t_{k}})_{k\in\\{0,\ldots,N\\}}$ given by Equations
(4), (5), (7). Then we have

$\displaystyle\lim_{N\rightarrow\infty}\,\,N^{\nu/2-\epsilon}\left(\max_{k\in\\{0,\ldots,N\\}}\mathbb{E}\left[|X_{t_{k}}-\hat{x}_{t_{k}}|\right]+\max_{k\in\\{0,\ldots,N\\}}\mathbb{E}\left[|V_{t_{k}}-\hat{v}_{t_{k}}|\right]\right)=0.$

We conclude this section with a summary of previous results from the
literature, further new results and an outline of the remainder of this
article.

###  1.1 Previous results for the CIR process

The strong approximation of the CIR process has been intensively studied in
the last years. The first works on this topic are [14, 1, 23], which prove
strong convergence (without a polynomial rate) of various explicit and
implicit schemes using the Yamada-Watanabe approach.

One of the schemes of [1] is the drift-implicit square root Euler scheme which
is well defined and positivity preserving for $\nu\geq\frac{1}{2}$. This
scheme turned out to be accessible to a more detailed error analysis, see [15,
2, 32, 25]. In particular, for the $L^{1}$-approximation at the final time
point [2] establishes convergence order 1 for $\nu>2$, while [15] gives
convergence order $\frac{1}{2}$ for $\nu>1$, [25] yields order
$\nu-\frac{1}{2}-\epsilon$ for $\nu>\frac{1}{2}$ and [19] establishes order
$\frac{1}{2}$ for the particular case $\nu=\frac{1}{2}$.

A breakthrough for the (very challenging) case $\nu\leq 1$ was provided by
[19] and [18]. In particular, the truncated Milstein scheme of [18] attains
$L^{1}$-convergence order $\min\\{\frac{1}{2},\nu\\}-\epsilon$ in this regime.

So, which rates are best possible for the (non-adaptive) $L^{1}$-approximation
of the CIR process at the final time point? This question has been answered by
the works [20] and [21], which yield

$\liminf_{N\rightarrow\infty}\,N^{\min\\{\nu,1\\}}\,\inf_{u\in\mathcal{U}}\,\mathbb{E}\left[\big{|}u(W_{t_{1}},W_{t_{2}},\ldots,W_{t_{N}})-V_{T}\big{|}\right]>0,$

where $\mathcal{U}$ is the set of measurable functions
$u:\mathbb{R}^{N}\rightarrow\mathbb{R}$. Thus, the convergence rate of the
truncated Milstein scheme for $\nu\leq\frac{1}{2}$ and the rate of the drift-
implicit square root Euler for $\nu>2$ are optimal.

In contrast to this, convergence rate results for the explicit Euler schemes
in Table 1 have been rare. In [6], the authors prove $L^{p}$-convergence order
$\frac{1}{2}$ for the Symmetrized Euler but with a strong restriction on the
Feller index. For FTE the $L^{p}$-convergence order $\frac{1}{2}$ for $2\leq
p<\nu-1$ and $\nu>3$ is shown in [12]. As mentioned, [28] provides a survey
and numerical comparison of Euler-type schemes. Further contributions on the
strong approximation of the CIR process can be found in [17, 9, 31, 8], where
[31] provides in particular a method for the uniform approximation of the CIR
process.

Clearly, the truncated Milstein scheme and the drift-implicit square root
Euler scheme have similar or even better convergence properties than the Euler
schemes studied in this article. However, Euler schemes for the approximation
of the CIR process are nevertheless popular and our analysis addresses a
problem that has been open for a long time, namely to establish (sharp)
convergence rates for these schemes under mild assumptions on the Feller
index. Moreover, for the log-Heston SDE the obtained convergence rate is
optimal for $\nu>1$, see the next subsection.

###  1.2 Previous results for the log-Heston SDE

We are not aware of any results concerning the strong approximation of the
log-Heston model except [27, 3]. In [3] the drift-implicit square root Euler
for the CIR process is combined with an Euler discretization of the log-Heston
process and $L^{2}$-convergence order $\frac{1}{2}$ is obtained for $\nu>2$,
while [27] uses a drift-implicit Milstein discretization of the CIR process
instead and obtains $L^{2}$-convergence for $\nu>1$.

In view of these results, Theorem 1.1 and Proposition 1.2 are a major
improvement, in particular with respect to the conditions on the Feller index.
Moreover, for $\nu>1$ the obtained convergence rate is best possible. In [30]
we study the optimal $L^{1}$-approximation of the log-Heston model by
equidistant discretizations of $W$ and $B$ and show in particular that

$\liminf_{N\rightarrow\infty}\,N^{\frac{1}{2}}\,\inf_{u\in\mathcal{U}}\,\mathbb{E}\left[\big{|}u(W_{t_{1}},W_{t_{2}},\ldots,W_{t_{N}},B_{t_{1}},B_{t_{2}},\ldots,B_{t_{N}})-X_{T}\big{|}\right]>0,$

for $\nu>1$, where $\mathcal{U}$ is the set of measurable functions from
$\mathbb{R}^{2N}$ to $\mathbb{R}$.

The strong approximation of the full Heston model, i.e. of $S_{T}$ instead of
$X_{T}$ carries an additional burden, since the SDE for the asset price has
superlinear coefficients and admits moment explosions, i.e.
${\mathbb{E}}(S_{T}^{p})=\infty$ for certain parameter constellations and
$p>1$, see e.g. [4]. The article [11], where exponential integrability results
for several Euler-type methods for the CIR process have been established, is
dedicated to this problem.

###  1.3 Further results

Our analysis is taylor-made for the $L^{1}$-approximation and based on the
Tanaka-Meyer formula combined with a clever control of the arising local time
of the error process. We found this approach in [13], where the approximation
of SDEs with irregular drift and additive noise has been studied. For the
$L^{p}$-approximation with $p\geq 1$ we could deduce the upper bound
$\min\\{1,\nu\\}/(2p)$ for the convergence order by a standard application of
the Hölder inequality. However, this bound is unlikely to be sharp, compare
e.g. [6] and [12], so we do not spell out this result in detail.

More importantly, our results can in particular be helpful for the Monte-Carlo
pricing of (path-dependent) European options, since they allow to control the
bias:

######  Proposition 1.3.

Let $\nu>1$, $\epsilon>0$ and $(\hat{x}_{t_{k}})_{k\in\\{0,\ldots,N\\}}$ as in
Theorem 1.1. Moreover, let $G:\mathbb{R}^{[0,T]}\rightarrow[0,\infty)$ be a
measurable mapping for which there exists $L_{G}>0$ such that

$|G(y)-G(z)|\leq L_{G}\sup_{t\in[0,T]}|y_{t}-z_{t}|$

for all measurable $y,z\in\mathbb{R}^{[0,T]}$. Finally, set
$\hat{x}^{pc,N}_{t}=\hat{x}_{\eta(t)}$ with
$\eta(t)=\max\\{k\in\\{0,...,N\\}:t_{k}\leq t\\}\Delta t$. Then we have

$\displaystyle\lim_{N\rightarrow\infty}\,\,N^{1/2-\epsilon}\,\big{|}\mathbb{E}\left[G(X)\right]-\mathbb{E}\left[G\left(\hat{x}^{pc,N}\right)\right]\big{|}=0.$

Examples for $G$ include lookback-put options as

$G(X)=\left[K-\sup_{t\in[0,T]}\exp(X_{t})\right]^{+}$

or arithmetic-asian-put options

$G(X)=\left[K-\frac{1}{T}\int_{0}^{T}\exp(X_{t})dt\right]^{+}$

with $K>0$. Note that for $X=W$ and $G(x)=\sup_{t\in[0,T]}x_{t}$, we have the
well known result

$\lim_{N\rightarrow\infty}{N}^{1/2}\left(\mathbb{E}\left[\max_{t\in[0,T]}W_{t}\right]-\mathbb{E}\left[\max_{k=0,\ldots,N}W_{t_{k}}\right]\right)=\sqrt{\frac{T}{2\pi}}\,\left|\zeta(\tfrac{1}{2})\right|,$

see e.g. pages 884 – 886 in [5], where $\zeta(\cdot)$ denotes the Riemann zeta
function. So, in Proposition 1.3 we can not expect to obtain a better decay of
the bias.

###  1.4 Notation and Outline

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
$T,x_{0},v_{0},\kappa,\theta,\sigma,\mu,\rho$ and the choice of
$f_{1},f_{2},f_{3}$ will be denoted in the following by $C$, regardless of
their value. Other dependencies will be denoted by subscripts, that is
$C_{h,p}$ means e.g. that this constant depends additionally on a function $h$
and a parameter $p$. Moreover, the value of all these constants can change
from line to line. Finally, 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.

The remainder of this article is organized as follows. We first show and
collect some preliminary results in Section 2. The proofs of Theorem 1.1 and
Proposition 1.2 are carried out in Section 3 and Section 4, while the proof of
Proposition 1.3 is also given in Section 4. Finally, our simulation study is
presented in Section 5.

##  2 Time-continuous extensions of the schemes and other preliminary results

In this section, we will present the discretization schemes in detail that we
are analyzing and a couple of preliminary results that are needed to prove our
main theorems. The first one is a well-known result for the CIR process (see
e.g. Section 3 in [15] and Theorem 3.1 in [24]).

######  Lemma 2.1.

Let $p>-\nu$. Then we have

$\sup\limits_{t\in[0,T]}\mathbb{E}\left[V^{p}_{t}\right]<\infty.$

The next auxiliary result on the smoothness of the CIR process and the log-
Heston-SDE is also well known:

######  Lemma 2.2.

Let $p\geq 1$. Then we have

$\sup_{0\leq s<t\leq
T}\mathbb{E}\left[\frac{|V_{t}-V_{s}|^{p}}{|t-s|^{p/2}}\right]+\sup_{0\leq
s<t\leq
T}\mathbb{E}\left[\frac{|X_{t}-X_{s}|^{p}}{|t-s|^{p/2}}\right]<\infty.$

###### Proof.

Let $p\geq 2$. The case $p\in[1,2)$ then follows by an application of the
Lyapunov inequality.

By Jensen’s inequality, it holds that $\left|x+y\right|^{p}\leq
2^{p-1}\left(|x|^{p}+|y|^{p}\right)$ and $\left|x+y+z\right|^{p}\leq
3^{p-1}\left(|x|^{p}+|y|^{p}+|z|^{p}\right)$. Together with Hölder’s
inequality we have that

$\displaystyle\sup_{0\leq s<t\leq
T}\mathbb{E}\left[\frac{|V_{t}-V_{s}|^{p}}{|t-s|^{p/2}}\right]$
$\displaystyle=\sup_{0\leq s<t\leq
T}\frac{1}{|t-s|^{p/2}}\mathbb{E}\left[\left|\int_{s}^{t}\kappa(\theta-
V_{u})du+\sigma\int_{s}^{t}\sqrt{V_{u}}dW_{u}\right|^{p}\right]$
$\displaystyle\leq C_{p}\left(\sup_{0\leq s<t\leq
T}\frac{|t-s|^{p-1}}{|t-s|^{p/2}}\mathbb{E}\left[\int_{s}^{t}\left|\kappa(\theta-
V_{u})\right|^{p}du\right]\right.$
$\displaystyle\qquad\qquad\left.+\sup_{0\leq s<t\leq
T}\frac{1}{|t-s|^{p/2}}\mathbb{E}\left[\left|\sigma\int_{s}^{t}\sqrt{V_{u}}dW_{u}\right|^{p}\right]\right).$

The first term is bounded by Lemma 2.1. For the second term, an application of
the Burkholder-Davis-Gundy inequality, Hölder’s inequality and Lemma 2.1 gives
the desired result. The log-price process can be treated analogously, since
its integral equation does not depend on $X$. ∎

The following lemma gives us a bound for the expected local time in zero of a
semimartingale. It can be found in Lemma 5.1 in [13].

######  Lemma 2.3.

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

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

where $L^{0}(Y)=(L_{t}^{0}(Y))_{t\in[0,T]}$ denotes the local time of $Y$ in
zero.

The following statement can be verified by a simple computation.

######  Lemma 2.4.

For $\lambda\in[0,1]$ and $x,y\geq 0$, we have

$\displaystyle\left|\sqrt{x}-\sqrt{y}\right|\leq
x^{-\frac{1}{2}(1-\lambda)}\left|x-y\right|^{1-\frac{\lambda}{2}}.$

We also require the well known Burkholder-Davis-Gundy inequalities, see e.g.
Theorem 3.28 in Chapter III of [26].

######  Proposition 2.5.

Let $M=(M_{t})_{t\in[0,T]}$ be a continuous martingale and $\alpha>0$. Then,
there exist constants $k_{\alpha},K_{\alpha}>0$ such that

$k_{\alpha}\,\mathbb{E}\left[\langle
M\rangle_{t}^{\alpha}\right]\leq\mathbb{E}\left[\sup_{u\in[0,t]}\left|M_{u}\right|^{2\alpha}\right]\leq
K_{\alpha}\,\mathbb{E}\left[\langle M\rangle_{t}^{{\alpha}}\right],\qquad
t\in[0,T].$

Here $\langle M\rangle_{t}$, $t\in[0,T]$, denotes the quadratic variation of
$M$.

Finally, we will also need Doob’s maximal inequality, see e.g. Theorem 3.8 in
Chapter I of [26].

######  Proposition 2.6.

Let $M=(M_{t})_{t\in[0,T]}$ be a continuous martingale and $p>1$. Then, it
holds that

$\mathbb{E}\left[\sup_{u\in[0,t]}\left|M_{u}\right|^{p}\right]\leq\left(\frac{p}{p-1}\right)^{p}\mathbb{E}|M_{t}|^{p},\qquad
t\in[0,T].$

###  2.1 Euler schemes – Case I

For the choice (4), (5), (7) the time-continuous extensions
$\bar{v}=(\bar{v}_{t})_{t\in[0,T]}$ and $\hat{v}=(\hat{v}_{t})_{t\in[0,T]}$
read as

$\begin{aligned}
\bar{v}_{t}&=\bar{v}_{\eta(t)}+\int_{\eta(t)}^{t}\kappa(\theta-
f_{2}(\bar{v}_{\eta(s)}))ds+\sigma\int_{\eta(t)}^{t}\sqrt{f_{3}(\bar{v}_{\eta(s)})}dW_{s},\\\
\hat{v}_{t}&=f_{3}\left(\bar{v}_{t}\right),\end{aligned}\qquad t\in[0,T],$ (8)

with $f_{2}\in\\{{\tt id},{\tt abs},{\tt sym}\\}$, $f_{3}\in\\{{\tt abs},{\tt
sym}\\}$ and $\bar{v}_{0}=v_{0}$. Note that $f_{2}$ and $f_{3}$ are globally
Lipschitz continuous with Lipschitz constant $L=1$ and satisfy

$|x-f_{i}(y)|\leq|x-y|,\qquad x\geq 0,y\in\mathbb{R},\quad i=2,3.$ (9)

Moreover note that

$\sqrt{|f_{i}(x)|}\leq 1+|x|,\qquad x\in\mathbb{R},\quad i=1,2,3.$ (10)

The next lemma can be shown by some tedious but straightforward computations,
since the coefficients of the Euler scheme are of linear growth.

######  Lemma 2.7.

Let $p\geq 1$. There exists $C_{p}>0$ such that

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

###### Proof.

Let $p\geq 2$. The case $p\in[1,2)$ then follows by an application of the
Lyapunov inequality.

The boundedness of the second term follows from the boundedness of the first
term by applying the same technique as in the proof of Lemma 2.2, where we use
that Equation (10) holds for $f_{2}$ and $f_{3}$.

For the first term, we will start with establishing

$\sup_{t\in[0,T]}\mathbb{E}\left[|\bar{v}_{t}|^{p}\right]<\infty.$ (11)

Let $\tau_{n}$ be the stopping time defined by
$\tau_{n}:=\inf\\{0<t<T;\bar{v}_{t}\geq n\\}$ with $\inf\\{\emptyset\\}=T$. We
have $|f_{i}(\bar{v}_{\eta(t)})|\leq|\bar{v}_{\eta(t)}|$ for $i\in\\{2,3\\}$.
Thus, it follows

$\displaystyle\mathbb{E}\left[|\bar{v}_{t\wedge\tau_{n}}|^{p}\right]$
$\displaystyle\leq\mathbb{E}\left[\left|v_{0}+\int_{0}^{t\wedge\tau_{n}}\kappa(\theta-
f_{2}(\bar{v}_{\eta(s)}))ds+\sigma\int_{0}^{t\wedge\tau_{n}}\sqrt{f_{3}(\bar{v}_{\eta(s)})}dW_{s}\right|^{p}\right]$
$\displaystyle\leq
C_{p}\left(v_{0}^{p}+\mathbb{E}\left[\left|\int_{0}^{t\wedge\tau_{n}}\kappa(\theta-
f_{2}(\bar{v}_{\eta(s)}))ds\right|^{p}\right]+\mathbb{E}\left[\left|\sigma\int_{0}^{t\wedge\tau_{n}}\sqrt{f_{3}(\bar{v}_{\eta(s)})}dW_{s}\right|^{p}\right]\right)$
$\displaystyle\leq
C_{p}\left(1+\mathbb{E}\left[\left|\int_{0}^{t\wedge\tau_{n}}\kappa(\theta-
f_{2}(\bar{v}_{\eta(s)}))ds\right|^{p}\right]+\mathbb{E}\left[\left|\sigma^{2}\int_{0}^{t\wedge\tau_{n}}(1+|\bar{v}_{\eta(s)}|)^{2}ds\right|^{p/2}\right]\right)$
$\displaystyle\leq
C_{p}\left(1+\mathbb{E}\left[\int_{0}^{t\wedge\tau_{n}}|\bar{v}_{\eta(s)}|^{p}ds\right]+\mathbb{E}\left[\left|\int_{0}^{t\wedge\tau_{n}}|\bar{v}_{\eta(s)}|^{2}ds\right|^{p/2}\right]\right)$
$\displaystyle\leq
C_{p}\left(1+\mathbb{E}\left[\int_{0}^{t\wedge\tau_{n}}|\bar{v}_{\eta(s)}|^{p}ds\right]\right)$

by applications of the Hölder and the Burkholder-Davis-Gundy inequality.
Therefore, we have shown that

$\displaystyle\mathbb{E}\left[|\bar{v}_{t\wedge\tau_{n}}|^{p}\right]\leq
C_{p}\left(1+\int_{0}^{t}\mathbb{E}\left[|\bar{v}_{\eta(s)\wedge\tau_{n}}|^{p}\right]ds\right)$

and we consequently obtain

$\displaystyle\sup_{s\in[0,t]}\mathbb{E}\left[|\bar{v}_{s\wedge\tau_{n}}|^{p}\right]\leq
C_{p}\left(1+\int_{0}^{t}\sup_{u\in[0,s]}\mathbb{E}\left[|\bar{v}_{u\wedge\tau_{n}}|^{p}\right]ds\right).$

The Gronwall inequality now yields

$\displaystyle\sup_{t\in[0,T]}\mathbb{E}\left[|\bar{v}_{t\wedge\tau_{n}}|^{p}\right]\leq
C_{p},$

where the constant $C_{p}>0$ does in particular not depend on $n$. Taking the
limit $n\rightarrow\infty$, we obtain (11). Due to

$\displaystyle\sup_{t\in[0,T]}|\bar{v}_{t}|^{p}\leq
C_{p}\left(1+\int_{0}^{T}\left|\bar{v}_{\eta(s)}\right|^{p}ds+\sup_{t\in[0,T]}\left|\int_{0}^{t}\sigma\sqrt{f_{3}(\bar{v}_{\eta(s)})}dW_{s}\right|^{p}\right),$

the assertion now follows from the properties of $f_{3}$, an application of
the Burkholder-Davis-Gundy inequality and Equation (11). ∎

###  2.2 Euler schemes – Case II

For (4), (6), (7) we obtain the Symmetrized Euler (SE) and the Euler with
Absorption (AE). We can write the time-continuous extension
$\hat{v}^{sym}=(\hat{v}^{sym}_{t})_{t\in[0,T]}$ of (SE) as

$\hat{v}^{sym}_{t}=\left|\hat{v}^{sym}_{\eta(t)}+\kappa\left(\theta-\hat{v}^{sym}_{\eta(t)}\right)(t-\eta(t))+\sigma\sqrt{\hat{v}^{sym}_{\eta(t)}}\left(W_{t}-W_{\eta(t)}\right)\right|$

on each interval $[t_{k},t_{k+1})$ and the time-continuous extension
$\hat{v}^{abs}=(\hat{v}^{abs}_{t})_{t\in[0,T]}$ of (AE) as

$\hat{v}^{abs}_{t}=\left(\hat{v}^{abs}_{\eta(t)}+\kappa\left(\theta-\hat{v}^{abs}_{\eta(t)}\right)(t-\eta(t))+\sigma\sqrt{\hat{v}^{abs}_{\eta(t)}}\left(W_{t}-W_{\eta(t)}\right)\right)^{+},$

respectively. Now, let $\star\in\\{sym,abs\\}$. We define

$z_{t}^{\star}:=\hat{v}_{\eta(t)}^{\star}+\kappa\big{(}\theta-\hat{v}_{\eta(t)}^{\star}\big{)}(t-\eta(t))+\sigma\sqrt{\hat{v}_{\eta(t)}^{\star}}\left(W_{t}-W_{\eta(t)}\right)$

and use the Tanaka-Meyer formula for
$\hat{v}^{sym}_{t}=\bigl{|}z_{t}^{sym}\bigr{|}$ and for
$\hat{v}^{abs}_{t}=\left(z_{t}^{abs}\right)^{+}$ to obtain

$\displaystyle\hat{v}^{sym}_{t}=\hat{v}_{\eta(t)}^{sym}$
$\displaystyle+\int_{\eta(t)}^{t}\operatorname{sign}\left(z_{s}^{sym}\right)\kappa\left(\theta-\hat{v}_{\eta(s)}^{sym}\right)ds+\sigma\int_{\eta(t)}^{t}\operatorname{sign}\left(z_{s}^{sym}\right)\sqrt{\hat{v}_{\eta(s)}^{sym}}dW_{s}$
$\displaystyle+L_{t}^{0}(z^{sym})-L_{\eta(t)}^{0}(z^{sym}),\qquad t\in[0,T],$

and

$\displaystyle\hat{v}^{abs}_{t}=\text{ }\hat{v}_{\eta(t)}^{abs}$
$\displaystyle+\int_{\eta(t)}^{t}\mathbbm{1}_{\\{z_{s}^{abs}>0\\}}\kappa\left(\theta-\hat{v}_{\eta(s)}^{abs}\right)ds+\sigma\int_{\eta(t)}^{t}\mathbbm{1}_{\\{z_{s}^{abs}>0\\}}\sqrt{\hat{v}_{\eta(s)}^{abs}}dW_{s}$
$\displaystyle+\frac{1}{2}\left(L_{t}^{0}(z^{abs})-L_{\eta(t)}^{0}(z^{abs})\right),\qquad
t\in[0,T].$

Here $L^{0}(z^{\star})=(L_{t}^{0}(z^{\star}))_{t\in[0,T]}$ is the local time
of $z^{\star}$ in $x=0$. For almost all $\omega\in\Omega$ the map $[0,T]\ni
t\mapsto[L_{t}^{0}(z^{\star})](\omega)\in\mathbb{R}$ is continuous and non-
decreasing with $L_{0}^{0}(z^{\star})=0$. See e.g. Theorem 7.1 in Chapter III
of [26]. We can rewrite both schemes as

$\displaystyle\hat{v}^{\star}_{t}=$
$\displaystyle\hat{v}_{\eta(t)}^{\star}+\int_{\eta(t)}^{t}\kappa\left(\theta-\hat{v}_{\eta(s)}^{\star}\right)ds+\sigma\int_{\eta(t)}^{t}\sqrt{\hat{v}_{\eta(s)}^{\star}}dW_{s}$
(12)
$\displaystyle-2c^{\star}\sigma\int_{\eta(t)}^{t}\mathbbm{1}_{\\{z_{s}^{\star}\leq
0\\}}\sqrt{\hat{v}_{\eta(s)}^{\star}}dW_{s}-2c^{\star}\int_{\eta(t)}^{t}\mathbbm{1}_{\\{z_{s}^{\star}\leq
0\\}}\kappa\left(\theta-\hat{v}_{\eta(s)}^{\star}\right)ds$
$\displaystyle+c^{\star}\left(L_{t}^{0}(z^{\star})-L_{\eta(t)}^{0}(z^{\star})\right),\qquad
t\in[0,T],$

with $c^{sym}=1$ and $c^{abs}=\frac{1}{2}$.

######  Lemma 2.8.

Let $\star\in\\{sym,abs\\}$ and $p\geq 1$. Then, there exists $C_{p}>0$ such
that

$\mathbb{E}\left[\sup_{t\in[0,T]}|\hat{v}_{t}^{\star}|^{p}\right]+\sup_{t\in[0,T]}\mathbb{E}\left[\frac{|\hat{v}^{\star}_{t}-\hat{v}^{\star}_{\eta(t)}|^{p}}{\,|t-\eta(t)|^{p/2}}\right]\leq
C_{p}.$

###### Proof.

The finiteness of the first summand can be found in Lemma 2.1 in [7] for the
symmetrized Euler scheme and can obtained analogously for the absorbed Euler
scheme. The finiteness of the second summand is established in Lemma 3.7 in
[29]. ∎

The next two lemmas are Propositions 3.6 and 3.9 from [29].

######  Lemma 2.9.

For $\Delta t<\frac{1}{\kappa}$ and $\varepsilon\in(0,1/2]$ we have that

$P(z_{t}^{\star}\leq 0)\leq c\left(\frac{\Delta
t}{\varepsilon}\right)^{\nu(1-\varepsilon)},\qquad
t\in[0,T]\setminus\\{t_{0},t_{1},\ldots,t_{N}\\},$ (13)

for $\star\in\\{sym,abs\\}$, where $c$ is given by

$\displaystyle c=\exp\left(\kappa\left(\nu
T+\frac{2v_{0}}{\sigma^{2}}\right)\right)\left(\max\left\\{1,\frac{\sigma^{2}\nu}{v_{0}e}\right\\}\right)^{\nu}.$

######  Lemma 2.10.

Let $\beta>0$, $\varepsilon\in(0,1/2]$, $\Delta t\leq\frac{1}{2\kappa}$ and
$\star\in\\{sym,abs\\}$. Then, there exists a constant $C_{\beta}>0$ such that

$\mathbb{E}\left[L_{t}^{0}\left(z^{\star}\right)-L_{\eta(t)}^{0}\left(z^{\star}\right)\right]\leq
C_{\beta}\Delta t\left(\frac{\Delta
t}{\varepsilon}\right)^{\nu\frac{1-\varepsilon}{1+\beta}},\qquad t\in[0,T].$

The following Lemma gives a control of the non-martingale terms, which arise
additionally in the expansion of SE and AE, i.e. in (12).

######  Lemma 2.11.

Let $\Delta t\leq\frac{1}{2\kappa}$, $\varepsilon\in(0,1/2]$, $\beta>0$ and
$\star\in\\{sym,abs\\}$. Moreover, let $g:\mathbb{R}^{2}\rightarrow\mathbb{R}$
be measurable and bounded, and let $h:\mathbb{R}\rightarrow\mathbb{R}$ be
measurable and of linear growth. Then there exist constants $C_{g,h,\beta}>0$
and $C_{g,\beta}>0$ such that

$\sup_{t\in[0,T]}\mathbb{E}\left[\left|\int_{0}^{t}g(V_{u},\hat{v}^{\star}_{u})h(\hat{v}^{\star}_{\eta(u)})\mathbbm{1}_{\\{z_{u}^{\star}\leq
0\\}}du\right|\right]\leq C_{g,h,\beta}\left(\frac{\Delta
t}{\varepsilon}\right)^{\nu\frac{1-\varepsilon}{1+\beta}}$

and

$\sup_{t\in[0,T]}\mathbb{E}\left[\left|\int_{0}^{t}g(V_{u},\hat{v}^{\star}_{u})dL^{0}_{u}(z^{\star})\right|\right]\leq
C_{g,\beta}\left(\frac{\Delta
t}{\varepsilon}\right)^{\nu\frac{1-\varepsilon}{1+\beta}}.$

###### Proof.

(a) We start with the second assertion. Note that the integral under
consideration is a pathwise Riemann-Stieltjes integral, since
$L^{0}(z^{\star})$ is positive and non-decreasing with
$L_{0}^{0}(z^{\star})=0$. With
$\|g\|_{\infty}=\sup_{x,y\in\mathbb{R}}|g(x,y)|$ we then have

$-\|g\|_{\infty}L^{0}_{T}(z^{\star})\leq\int_{0}^{t}g(V_{u},\hat{v}^{\star}_{u})dL^{0}_{u}(z^{\star})\leq\|g\|_{\infty}L^{0}_{T}(z^{\star}),\qquad
t\in[0,T].$

It follows

$\sup_{t\in[0,T]}\mathbb{E}\left[\left|\int_{0}^{t}g(V_{u},\hat{v}^{\star}_{u})dL^{0}_{u}(z^{\star})\right|\right]\leq\|g\|_{\infty}\sum_{k=0}^{N-1}\mathbb{E}\left[L_{t_{k+1}}^{0}\left(z^{\star}\right)-L_{t_{k}}^{0}\left(z^{\star}\right)\right]$

and Lemma 2.10 gives

$\sup_{t\in[0,T]}\mathbb{E}\left[\left|\int_{0}^{t}g(V_{u},\hat{v}^{\star}_{u})dL^{0}_{u}(z^{\star})\right|\right]\leq
C_{g,\beta}\left(\frac{\Delta
t}{\varepsilon}\right)^{\nu\frac{1-\varepsilon}{1+\beta}}.$

(b) For the first assertion note that

$\displaystyle\sup_{t\in[0,T]}\mathbb{E}\left[\left|\int_{0}^{t}g(V_{u},\hat{v}^{\star}_{u})h(\hat{v}^{\star}_{\eta(u)})\mathbbm{1}_{\\{z_{u}^{\star}\leq
0\\}}du\right|\right]$ $\displaystyle\qquad\leq
C_{h}\|g\|_{\infty}\int_{0}^{T}\mathbb{E}\left[\left(1+\sup_{t\in[0,T]}|\hat{v}^{\star}_{t}|\right)\mathbbm{1}_{\\{z_{u}^{\star}\leq
0\\}}\right]du.$

An application of Hölder’s inequality together with Lemma 2.8 then yields

$\sup_{t\in[0,T]}\mathbb{E}\left[\left|\int_{0}^{t}g(V_{u},\hat{v}^{\star}_{u})h(\hat{v}^{\star}_{\eta(u)})\mathbbm{1}_{\\{z_{u}^{\star}\leq
0\\}}du\right|\right]\leq C_{g,h,\beta}\int_{0}^{T}(P(z_{u}^{\star}\leq
0))^{\frac{1}{1+\beta}}du$

for all $\beta>0$. Lemma 2.9 implies now that

$\displaystyle\sup_{t\in[0,T]}\mathbb{E}\left[\left|\int_{0}^{t}g(V_{u},\hat{v}^{\star}_{u})h(\hat{v}^{\star}_{\eta(u)})\mathbbm{1}_{\\{z_{u}^{\star}\leq
0\\}}du\right|\right]\quad\leq C_{g,h,{\beta}}\left(\frac{\Delta
t}{\varepsilon}\right)^{\nu\frac{1-\varepsilon}{1+\beta}}.$

which finishes the proof. ∎

###  2.3 The Euler scheme for the log-price process

The time-continuous extension $\hat{x}=(\hat{x}_{t})_{t\in[0,T]}$ of the Euler
scheme for the log-price process in the Heston model is given by

$\displaystyle\hat{x}_{t}=\hat{x}_{\eta(t)}$
$\displaystyle+\left(\mu-\frac{1}{2}\hat{v}_{\eta(t)}\right)(t-\eta(t))+\rho\sqrt{\hat{v}_{\eta(t)}}\left(W_{t}-W_{\eta(t)}\right)$
(14)
$\displaystyle+\sqrt{1-\rho^{2}}\sqrt{\hat{v}_{\eta(t)}}\left(B_{t}-B_{\eta(t)}\right).$

For $\hat{v}$, we can choose one of the previously introduced schemes for the
CIR process. We have the same results concerning the moment stability and the
local smoothness as before.

######  Lemma 2.12.

Let $p\geq 1$. For the Euler scheme (14) together with the scheme (8) or (12),
there exists $C_{p}>0$ such that

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

###### Proof.

Again, let $p\geq 2$. The case $p\in[1,2)$ then follows by an application of
the Lyapunov inequality.

Using the results from Lemma 2.7 and Lemma 2.8, both statements follow by
standard computations. For the first term, we use the Hölder and the
Burkholder-Davis-Gundy inequality, which leads to

$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}|\hat{x}_{t}|^{p}\right]$
$\displaystyle=\mathbb{E}\left[\sup_{t\in[0,T]}\left|x_{0}+\mu
t-\frac{1}{2}\int_{0}^{t}\hat{v}_{\eta(s)}ds+\rho\int_{0}^{t}\sqrt{\hat{v}_{\eta(s)}}dW_{s}+\sqrt{1-\rho^{2}}\int_{0}^{t}\sqrt{\hat{v}_{\eta(s)}}dB_{s}\right|^{p}\right]$
$\displaystyle\leq
C_{p}\left(1+\mathbb{E}\left[\sup_{t\in[0,T]}\left|\int_{0}^{t}\hat{v}_{\eta(s)}ds\right|^{p}\right]\right.$
$\displaystyle\left.\qquad\qquad+\mathbb{E}\left[\sup_{t\in[0,T]}\left|\int_{0}^{t}\sqrt{\hat{v}_{\eta(s)}}dW_{s}\right|^{p}\right]+\mathbb{E}\left[\sup_{t\in[0,T]}\left|\int_{0}^{t}\sqrt{\hat{v}_{\eta(s)}}dB_{s}\right|^{p}\right]\right)$
$\displaystyle\leq C_{p}.$

The statement for the second term follows by the first statement and by
analogous computations as in the proof of Lemma 2.2.

∎

##  3 CIR process: convergence rates

In this section, we will prove the announced $L^{1}$-convergence rates for the
Euler schemes presented in the introduction.

###  3.1 Error analysis - Case I and $\mathbf{\nu>1}$

We will first look at the discretization from (8) under the condition $\nu>1$.

######  Theorem 3.1.

Let $\bar{v}$ be given by (8) and $\nu>1$. Then, for all $\varepsilon>0$ there
exists a constant $C_{\varepsilon}>0$ such that

$\sup\limits_{t\in[0,T]}\mathbb{E}\left[\left|V_{t}-\bar{v}_{t}\right|\right]\leq
C_{\varepsilon}\left(\Delta t\right)^{\frac{1}{2}-\varepsilon}.$

###### Proof.

Define $e=(e_{t})_{t\in[0,T]}$ by $e_{t}=V_{t}-\bar{v}_{t}$.

(i) The Tanaka-Meyer formula, see e.g. Equation 7.9 in Chapter III in [26],
yields

$\displaystyle\mathbb{E}\left[\left|e_{t}\right|\right]=$
$\displaystyle\mathbb{E}\left[\int_{0}^{t}\operatorname{sign}(e_{u})de_{u}\right]+\mathbb{E}\left[L^{0}_{t}(e)\right]$
(15) $\displaystyle=$
$\displaystyle\mathbb{E}\left[\int_{0}^{t}\operatorname{sign}(e_{u})\left(-\kappa\left(V_{u}-f_{2}(\bar{v}_{\eta(u)})\right)\right)du\right]$
$\displaystyle\quad+\mathbb{E}\left[\int_{0}^{t}\operatorname{sign}(e_{u})\sigma\left(\sqrt{V_{u}}-\sqrt{f_{3}(\bar{v}_{\eta(u)})}\right)dW_{u}\right]+\mathbb{E}\left[L^{0}_{t}(e)\right].$

We have

$\mathbb{E}\left[\int_{0}^{t}\operatorname{sign}(e_{u})\sigma\left(\sqrt{V_{u}}-\sqrt{f_{3}(\bar{v}_{\eta(u)})}\right)dW_{u}\right]=0$

due to Lemma 2.1, Lemma 2.7, the properties of $f_{3}$ and the martingale
property of the Itō integral. Looking at the first term, we have

$\displaystyle\mathbb{E}\left[\int_{0}^{t}\operatorname{sign}(e_{u})\left(-\kappa\left(V_{u}-f_{2}(\bar{v}_{\eta(u)})\right)\right)du\right]$
$\displaystyle=-\kappa\,\mathbb{E}\left[\int_{0}^{t}\operatorname{sign}(e_{u})\left(V_{u}-f_{2}(\bar{v}_{u})\right)du\right]-\kappa\,\mathbb{E}\left[\int_{0}^{t}\operatorname{sign}(e_{u})\left(f_{2}(\bar{v}_{u})-f_{2}(\bar{v}_{\eta(u)})\right)du\right]$
$\displaystyle\leq\kappa\int_{0}^{t}\mathbb{E}\left[|e_{u}|\right]du+\kappa\int_{0}^{t}\mathbb{E}\left[|\bar{v}_{u}-\bar{v}_{\eta(u)}|\right]du$
$\displaystyle\leq C(\Delta
t)^{\frac{1}{2}}+\kappa\int_{0}^{t}\mathbb{E}\left[|e_{u}|\right]du$

due to Lemma 2.7 and $|x-f_{2}(y)|\leq|x-y|$ for $x\geq 0,y\in\mathbb{R}$ as
well as $|f_{2}(x)-f_{2}(y)|\leq|x-y|$ for $x,y\in\mathbb{R}$. Therefore, we
obtain

$\sup_{u\in[0,t]}\mathbb{E}\left[\left|e_{u}\right|\right]\leq C(\Delta
t)^{\frac{1}{2}}+\kappa\int_{0}^{t}\sup_{v\in[0,u]}\mathbb{E}\left[|e_{v}|\right]du+\mathbb{E}\left[L^{0}_{t}(e)\right].$
(16)

(ii) With Lemma 2.3 we can derive a bound for the expected local time in 0 of
$e$. Let $\delta\in(0,1)$, then

$\displaystyle\mathbb{E}\left[L^{0}_{t}(e)\right]\leq 4\delta$
$\displaystyle-2\mathbb{E}\left[\int_{0}^{t}\left(\mathbbm{1}_{\\{e_{s}\in(0,\delta)\\}}+\mathbbm{1}_{\\{e_{s}\geq\delta\\}}e^{1-\frac{e_{s}}{\delta}}\right)de_{s}\right]$
(17)
$\displaystyle+\frac{1}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{e_{s}>\delta\\}}e^{1-\frac{e_{s}}{\delta}}d\langle
e\rangle_{s}\right].$

We define
$Y_{s}:=\mathbbm{1}_{\\{e_{s}\in(0,\delta)\\}}+\mathbbm{1}_{\\{e_{s}\geq\delta\\}}e^{1-\frac{e_{s}}{\delta}}$
and look at the second term of (17), i.e. at

$\mathbb{E}\left[\int_{0}^{t}Y_{s}de_{s}\right]=-\kappa\,\mathbb{E}\left[\int_{0}^{t}Y_{s}\left(V_{s}-f_{2}(\bar{v}_{\eta(s)})\right)ds\right],$

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

$\displaystyle\left|\mathbb{E}\left[\int_{0}^{t}Y_{s}de_{s}\right]\right|$
$\displaystyle\leq\kappa\int_{0}^{t}\mathbb{E}\left[|V_{s}-\bar{v}_{\eta(s)}|\right]ds$
(18) $\displaystyle\leq C(\Delta
t)^{\frac{1}{2}}+\kappa\int_{0}^{t}\mathbb{E}\left[|e_{u}|\right]du.$

The third term of (17) can be bounded as follows with Lemma 2.4 (using
$\lambda=0$) and the properties of $f_{3}$:

$\displaystyle\frac{1}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{e_{s}>\delta\\}}e^{1-\frac{e_{s}}{\delta}}d\langle
e\rangle_{s}\right]$
$\displaystyle=\frac{1}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{e_{s}>\delta\\}}e^{1-\frac{e_{s}}{\delta}}\sigma^{2}\left(\sqrt{V_{s}}-\sqrt{f_{3}(\bar{v}_{\eta(s)}})\right)^{2}ds\right]$
$\displaystyle\leq\frac{1}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{e_{s}>\delta\\}}e^{1-\frac{e_{s}}{\delta}}\sigma^{2}\frac{\left|V_{s}-f_{3}(\bar{v}_{\eta(s)})\right|^{2}}{V_{s}}ds\right]$
$\displaystyle\leq\frac{1}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{e_{s}>\delta\\}}e^{1-\frac{e_{s}}{\delta}}\sigma^{2}\frac{\left|V_{s}-\bar{v}_{\eta(s)}\right|^{2}}{V_{s}}ds\right].$

With Lemma 2.7, Lemma 2.1, the Minkowski inequality and Hölder’s inequality it
follows

$\displaystyle\frac{1}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{e_{s}>\delta\\}}e^{1-\frac{e_{s}}{\delta}}d\langle
e\rangle_{s}\right]\leq$
$\displaystyle\frac{C}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{e_{s}>\delta\\}}e^{1-\frac{e_{s}}{\delta}}\
\frac{\left|V_{s}-\bar{v}_{s}\right|^{2}}{{V_{s}}}ds\right]$ (19)
$\displaystyle+\frac{C}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{e_{s}>\delta\\}}e^{1-\frac{e_{s}}{\delta}}\frac{\left|\bar{v}_{s}-\bar{v}_{\eta(s)}\right|^{2}}{V_{s}}ds\right]$
$\displaystyle\leq$
$\displaystyle\frac{C}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{e_{s}>\delta\\}}e^{1-\frac{e_{s}}{\delta}}\
\frac{\left|e_{s}\right|^{2}}{V_{s}}ds\right]$ $\displaystyle+C\frac{\Delta
t}{\delta}\int_{0}^{t}\left(\mathbb{E}\left[\frac{1}{V_{s}^{(1+\nu)/2}}\right]\right)^{\frac{2}{1+\nu}}ds$
$\displaystyle\leq$
$\displaystyle\frac{C}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{e_{s}>\delta\\}}e^{1-\frac{e_{s}}{\delta}}\frac{\left|e_{s}\right|^{2}}{V_{s}}ds\right]$
$\displaystyle+C\left(\frac{\Delta t}{\delta}\right).$

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

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

for all $p\geq 1$ due to Lemma 2.1 and Lemma 2.7, we have that

$\displaystyle\frac{1}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{e_{s}>\delta\\}}e^{1-\frac{e_{s}}{\delta}}\frac{\left|e_{s}\right|^{2}}{V_{s}}ds\right]$
(20)
$\displaystyle\quad=\frac{1}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{e_{s}\in(\delta,\delta^{\alpha})\\}}e^{1-\frac{e_{s}}{\delta}}\frac{\left|e_{s}\right|^{2}}{V_{s}}ds\right]$
$\displaystyle\qquad\quad+\frac{1}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{e_{s}\geq\delta^{\alpha}\\}}e^{1-\frac{e_{s}}{\delta}}\frac{\left|e_{s}\right|^{2}}{V_{s}}ds\right]$
$\displaystyle\quad\leq\delta^{2\alpha-1}\int_{0}^{t}\mathbb{E}\left[\frac{1}{V_{s}}\right]ds+C\frac{e^{-\delta^{\alpha-1}}}{\delta}\int_{0}^{t}\left(\mathbb{E}\left[\frac{1}{V_{s}^{(1+\nu)/2}}\right]\right)^{\frac{2}{1+\nu}}ds$
$\displaystyle\quad\leq C_{\alpha}\delta^{2\alpha-1}$

by another application of Hölder’s inequality and $\limsup_{\delta\rightarrow
0}\frac{e^{-\delta^{\alpha-1}}}{\delta^{2\alpha}}=0$.

Summarizing (17) – (20) we have shown that

$\mathbb{E}\left[L^{0}_{t}(e)\right]\leq 4\delta+C(\Delta
t)^{\frac{1}{2}}+\kappa\int_{0}^{t}\mathbb{E}\left[|e_{u}|\right]du+C_{\alpha}\delta^{2\alpha-1}+C\frac{\Delta
t}{\delta}.$ (21)

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

$\displaystyle\mathbb{E}\left[L^{0}_{t}(e)\right]$
$\displaystyle\leq\kappa\int_{0}^{t}\mathbb{E}\left[|e_{u}|\right]du+C_{\alpha}(\Delta
t)^{\alpha-1/2}$ (22)
$\displaystyle\leq\kappa\int_{0}^{t}\sup_{v\in[0,u]}\mathbb{E}\left[|e_{v}|\right]du+C_{\alpha}(\Delta
t)^{\alpha-1/2}.$

Combining this with (16) yields

$\sup_{u\in[0,t]}\mathbb{E}\left[\left|e_{u}\right|\right]\leq
C_{\alpha}(\Delta
t)^{\alpha-1/2}+2\kappa\int_{0}^{t}\sup_{v\in[0,u]}\mathbb{E}\left[|e_{v}|\right]du$

and the assertion follows by choosing $\alpha=1-\varepsilon$ and an
application of Gronwall’s lemma.

∎

###  3.2 Error analysis - Case II and $\mathbf{\nu>1}$

In this section, we will analyze both schemes from (12) under the condition
$\nu>1$.

######  Theorem 3.2.

Let $\hat{v}$ be given by (12) and $\nu>1$. Then, for all $\varepsilon>0$
there exists a constant $C_{\varepsilon}>0$ such that

$\sup\limits_{t\in[0,T]}\mathbb{E}\left[\left|V_{t}-\hat{v}_{t}\right|\right]\leq
C_{\varepsilon}\left(\Delta t\right)^{\frac{1}{2}-\varepsilon}.$

###### Proof.

The proof is very similar to the proof of Theorem 3.1. Differences are only
due to the additional terms in the expansion of the schemes and we will give
the required additional steps in the following. For $e_{t}=V_{t}-\hat{v}_{t}$
we have

$\displaystyle e_{t}=$
$\displaystyle-\int_{0}^{t}\kappa\left(V_{s}-\hat{v}_{\eta(s)}^{\star}\right)ds+\sigma\int_{0}^{t}\left(\sqrt{V_{s}}-\sqrt{\hat{v}_{\eta(s)}^{\star}}\right)dW_{s}$
(23)
$\displaystyle+2c^{\star}\sigma\int_{0}^{t}\mathbbm{1}_{\\{z_{s}^{\star}\leq
0\\}}\sqrt{\hat{v}_{\eta(s)}^{\star}}dW_{s}+2c^{\star}\int_{0}^{t}\mathbbm{1}_{\\{z_{s}^{\star}\leq
0\\}}\kappa\left(\theta-\hat{v}_{\eta(s)}^{\star}\right)ds$
$\displaystyle-c^{\star}L_{t}^{0}(z^{\star}).$

(i) The Tanaka-Meyer formula yields

$\displaystyle\mathbb{E}\left[\left|e_{t}\right|\right]=$
$\displaystyle\mathbb{E}\left[\int_{0}^{t}\operatorname{sign}(e_{u})de_{u}\right]+\mathbb{E}\left[L^{0}_{t}(e)\right]$
$\displaystyle=$
$\displaystyle-\kappa\mathbb{E}\left[\int_{0}^{t}\operatorname{sign}(e_{u})(V_{u}-\hat{v}^{\star}_{\eta(u)})du\right]$
$\displaystyle+\mathbb{E}\left[\sigma\int_{0}^{t}\operatorname{sign}(e_{u})\left(\sqrt{V_{u}}-\sqrt{\hat{v}^{\star}_{\eta(u))}}\right)dW_{u}\right]+\mathbb{E}\left[L^{0}_{t}(e)\right]$
$\displaystyle+\mathbb{E}\left[2c^{\star}\sigma\int_{0}^{t}\operatorname{sign}(e_{u})\mathbbm{1}_{\\{z_{u}^{\star}\leq
0\\}}\sqrt{\hat{v}_{\eta(u)}^{\star}}dW_{u}\right]$
$\displaystyle+\mathbb{E}\left[2c^{\star}\int_{0}^{t}\operatorname{sign}(e_{u})\mathbbm{1}_{\\{z_{u}^{\star}\leq
0\\}}\kappa\left(\theta-\hat{v}_{\eta(u)}^{\star}\right)du-c^{\star}\int_{0}^{t}\operatorname{sign}(e_{u})dL^{0}_{u}(z^{\star})\right].$

However, Lemma 2.8 and the martingale property of Itō integrals imply

$\mathbb{E}\left[2c^{\star}\sigma\int_{0}^{t}\operatorname{sign}(e_{u})\mathbbm{1}_{\\{z_{u}^{\star}\leq
0\\}}\sqrt{\hat{v}_{\eta(u)}^{\star}}dW_{u}\right]=0$

and Lemma 2.11 gives

$\left|\mathbb{E}\left[\int_{0}^{t}\operatorname{sign}(e_{u})\left(\mathbbm{1}_{\\{z_{u}^{\star}\leq
0\\}}2\kappa(\theta-\hat{v}_{\eta(u)}^{\star})du-
dL^{0}_{u}(z^{\star})\right)\right]\right|\leq C_{\varepsilon}\left(\Delta
t\right)^{\nu\frac{1-\varepsilon}{1+\varepsilon}}$

by choosing $\beta=\varepsilon$. Thus, we have

$\sup_{u\in[0,t]}\mathbb{E}\left[\left|e_{u}\right|\right]\leq C(\Delta
t)^{\frac{1}{2}}+\kappa\int_{0}^{t}\sup_{v\in[0,u]}\mathbb{E}\left[|e_{v}|\right]du+\mathbb{E}\left[L^{0}_{t}(e)\right]$
(24)

as in the first step of the previous proof by choosing $\varepsilon$
appropriately and exploiting that $\nu>1$.

(ii) In the same way Lemma 2.11 and Lemma 2.8 also yield

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

Finally, we have

$\displaystyle\langle
e\rangle_{t}=\sigma^{2}\int_{0}^{t}\left(\sqrt{V_{s}}-\sqrt{\hat{v}^{\star}_{\eta(s)}}+2c^{\star}\mathbbm{1}_{\\{z_{s}^{\star}\leq
0\\}}\sqrt{\hat{v}_{\eta(s)}^{\star}}\right)^{2}ds.$

But again Lemma 2.11 with $\beta=\varepsilon$ gives that

$\displaystyle\frac{1}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{e_{s}>\delta\\}}e^{1-\frac{e_{s}}{\delta}}d\langle
e\rangle_{s}\right]\leq$
$\displaystyle\frac{2}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{e_{s}>\delta\\}}e^{1-\frac{e_{s}}{\delta}}\left(\sqrt{V}_{s}-\sqrt{\hat{v}_{\eta(s)}}\right)^{2}\right]ds+C_{\varepsilon}\frac{(\Delta
t)^{\nu\frac{1-\varepsilon}{1+\varepsilon}}}{\delta}$

and proceeding as in the previous proof we obtain that

$\displaystyle\mathbb{E}\left[L^{0}_{t}(e)\right]\leq 4\delta+C(\Delta
t)^{\frac{1}{2}}$
$\displaystyle+\kappa\int_{0}^{t}\mathbb{E}\left[|e_{u}|\right]du+C_{\alpha}\delta^{2\alpha-1}+C\frac{\Delta
t}{\delta}+C_{\varepsilon}\frac{(\Delta
t)^{\nu\frac{1-\varepsilon}{1+\varepsilon}}}{\delta}.$

(iii) Setting $\delta=(\Delta t)^{1/2}$ and using $\nu>1$ now yield

$\mathbb{E}\left[L^{0}_{t}(e)\right]\leq\kappa\int_{0}^{t}\mathbb{E}\left[|e_{u}|\right]du+C_{\alpha}(\Delta
t)^{\alpha-1/2}+C_{\varepsilon}(\Delta
t)^{\frac{1-\varepsilon}{1+\varepsilon}-1/2}.$

Combining this with (24) gives

$\sup_{u\in[0,t]}\mathbb{E}\left[\left|e_{u}\right|\right]\leq
C_{\alpha}(\Delta t)^{\alpha-1/2}+C_{\varepsilon}(\Delta
t)^{\frac{1-\varepsilon}{1+\varepsilon}-1/2}+2\kappa\int_{0}^{t}\sup_{v\in[0,u]}\mathbb{E}\left[|e_{v}|\right]du.$

Choosing $\alpha=1-\varepsilon$ and observing that
$\frac{1}{2}-\varepsilon\geq\frac{1-\varepsilon}{1+\varepsilon}-\frac{1}{2}$
for all $\varepsilon\in(0,1)$ an application of Gronwall’s lemma yields then

$\sup_{u\in[0,T]}\mathbb{E}\left[\left|e_{u}\right|\right]\leq
C_{\varepsilon}(\Delta t)^{\frac{1-\varepsilon}{1+\varepsilon}-1/2}$

and the assertion follows by choosing $\varepsilon$ appropriately.

∎

###  3.3 Error analysis - Case I and $\mathbf{\nu\leq 1}$

Now we will study again the discretization from (8) but under the condition
$\nu\leq 1$.

######  Proposition 3.3.

Let $\bar{v}$ be given by (8) and $\nu\leq 1$. Then, for all $\varepsilon>0$
there exists a constant $C_{\varepsilon}>0$ such that

$\sup\limits_{t\in[0,T]}\mathbb{E}\left[\left|V_{t}-\bar{v}_{t}\right|\right]\leq
C_{\varepsilon}\left(\Delta t\right)^{\frac{\nu}{2}-\varepsilon}.$

###### Proof.

Define again $e=(e_{t})_{t\in[0,T]}$ by $e_{t}=V_{t}-\bar{v}_{t}$.

(i) Proceeding as in the proof of Theorem 3.1 we obtain

$\sup_{u\in[0,t]}\mathbb{E}\left[\left|e_{u}\right|\right]\leq C(\Delta
t)^{\frac{1}{2}}+\kappa\int_{0}^{t}\sup_{v\in[0,u]}\mathbb{E}\left[|e_{v}|\right]du+\mathbb{E}\left[L^{0}_{t}(e)\right]$
(25)

and

$\displaystyle\mathbb{E}\left[L^{0}_{t}(e)\right]\leq 4\delta$
$\displaystyle+C(\Delta
t)^{\frac{1}{2}}+\kappa\int_{0}^{t}\mathbb{E}\left[|e_{u}|\right]du+\frac{1}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{e_{s}>\delta\\}}e^{1-\frac{e_{s}}{\delta}}d\langle
e\rangle_{s}\right]$ (26)

with

$\langle
e\rangle_{t}=\sigma^{2}\int_{0}^{t}\left(\sqrt{V_{s}}-\sqrt{f_{3}(\bar{v}_{\eta(s)})}\right)^{2}ds.$

(ii) For the remaining term in (26) we apply Lemma 2.4 with
$\lambda=1-\nu(1-\zeta)$ for $\zeta\in(0,1)$ and Lemma 2.1, Lemma 2.7,
Hölder’s and Minkowski’s inequality as well as the properties of $f_{3}$ to
obtain

$\displaystyle\frac{1}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{e_{s}>\delta\\}}e^{1-\frac{e_{s}}{\delta}}d\langle
e\rangle_{s}\right]$ (27)
$\displaystyle\quad\leq\frac{C_{\zeta}}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{e_{s}>\delta\\}}e^{1-\frac{e_{s}}{\delta}}\
\frac{\left|V_{s}-\bar{v}_{s}\right|^{1+\nu(1-\zeta)}}{{V_{s}}^{\nu(1-\zeta)}}ds\right]$
$\displaystyle\qquad+\frac{C_{\zeta}}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{e_{s}>\delta\\}}e^{1-\frac{e_{s}}{\delta}}\frac{\left|\bar{v}_{s}-\bar{v}_{\eta(s)}\right|^{1+\nu(1-\zeta)}}{V_{s}^{\nu(1-\zeta)}}ds\right]$
$\displaystyle\quad\leq\frac{C_{\zeta}}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{e_{s}>\delta\\}}e^{1-\frac{e_{s}}{\delta}}\
\frac{\left|e_{s}\right|^{1+\nu(1-\zeta)}}{V_{s}^{\nu(1-\zeta)}}ds\right]$
$\displaystyle\qquad+C_{\zeta}\frac{\left(\Delta
t\right)^{(1+\nu(1-\zeta))/2}}{\delta}\int_{0}^{t}\left(\mathbb{E}\left[\frac{1}{V_{s}^{\nu(1-\zeta^{2})}}\right]\right)^{\frac{1}{1+\zeta}}ds$
$\displaystyle\quad\leq\frac{C_{\zeta}}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{e_{s}>\delta\\}}e^{1-\frac{e_{s}}{\delta}}\frac{\left|e_{s}\right|^{1+\nu(1-\zeta)}}{V_{s}^{\nu(1-\zeta)}}ds\right]$
$\displaystyle\qquad+C_{\zeta}\left(\frac{\left(\Delta
t\right)^{(1+\nu(1-\zeta))/2}}{\delta}\right).$

Now let again $\alpha\in(0,1)$. Since

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

for all $p\geq 1$ due to Lemma 2.1 and Lemma 2.7, we have that

$\displaystyle\frac{1}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{e_{s}>\delta\\}}e^{1-\frac{e_{s}}{\delta}}\frac{\left|e_{s}\right|^{1+\nu(1-\zeta)}}{V_{s}^{\nu(1-\zeta)}}ds\right]$
$\displaystyle\quad=\frac{1}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{e_{s}\in(\delta,\delta^{\alpha})\\}}e^{1-\frac{e_{s}}{\delta}}\frac{\left|e_{s}\right|^{1+\nu(1-\zeta)}}{V_{s}^{\nu(1-\zeta)}}ds\right]$
$\displaystyle\qquad\quad+\frac{1}{\delta}\mathbb{E}\left[\int_{0}^{t}\mathbbm{1}_{\\{e_{s}\geq\delta^{\alpha}\\}}e^{1-\frac{e_{s}}{\delta}}\frac{\left|e_{s}\right|^{{1+\nu(1-\zeta)}}}{V_{s}^{\nu(1-\zeta)}}ds\right]$
$\displaystyle\quad\leq\delta^{({1+\nu(1-\zeta)})\alpha-1}\int_{0}^{t}\mathbb{E}\left[\frac{1}{V_{s}^{\nu(1-\zeta)}}\right]ds$
$\displaystyle\qquad\quad+C\frac{e^{-\delta^{\alpha-1}}}{\delta}\int_{0}^{t}\left(\mathbb{E}\left[\frac{1}{V_{s}^{\nu(1-\zeta^{2})}}\right]\right)^{\frac{1}{1+\zeta}}ds$
$\displaystyle\quad\leq C_{\zeta,\alpha}\delta^{(1+\nu(1-\zeta))\alpha-1}$

by another application of Hölder’s inequality, Lemma 2.1 and
$\limsup_{\delta\rightarrow
0}\frac{e^{-\delta^{\alpha-1}}}{\delta^{2\alpha}}=0$.

Summarizing the previous steps we have shown that

$\displaystyle\mathbb{E}\left[L^{0}_{t}(e)\right]\leq 4\delta$
$\displaystyle+C_{\zeta}\frac{\left(\Delta
t\right)^{(1+\nu(1-\zeta))/2}}{\delta}+C_{\zeta,\alpha}\delta^{(1+\nu(1-\zeta))\alpha-1}$
$\displaystyle+\kappa\int_{0}^{t}\mathbb{E}\left[|e_{u}|\right]du+C(\Delta
t)^{1/2}.$

Setting $\delta=(\Delta t)^{1/2}$ and $\alpha=1-\zeta$ gives

$\mathbb{E}\left[L^{0}_{t}(e)\right]\leq\kappa\int_{0}^{t}\sup_{v\in[0,u]}\mathbb{E}\left[|e_{v}|\right]du+C_{\zeta}\left(\Delta
t\right)^{\nu(1-\zeta)/2}+C_{\zeta}(\Delta t)^{(\nu(1-\zeta)^{2}-\zeta)/2}.$

Combining this with (25) yields

$\sup_{u\in[0,t]}\mathbb{E}\left[\left|e_{u}\right|\right]\leq
C_{\zeta}(\Delta
t)^{(\nu(1-\zeta)^{2}-\zeta)/2}+2\kappa\int_{0}^{t}\sup_{v\in[0,u]}\mathbb{E}\left[|e_{v}|\right]du$

and the assertion follows by choosing $\zeta$ sufficiently small and an
application of Gronwall’s lemma.

∎

######  Remark 3.4.

We are not able to establish the analogous result to Proposition 3.3 for Case
II of the Euler schemes, since we have in that case

$\frac{1}{\delta}\langle
e\rangle_{t}=\frac{\sigma^{2}}{\delta}\int_{0}^{t}\left(\sqrt{V_{s}}-\sqrt{\hat{v}^{\star}_{\eta(s)}}+2c^{\star}\mathbbm{1}_{\\{z_{s}^{\star}\leq
0\\}}\sqrt{\hat{v}_{\eta(s)}^{\star}}\right)^{2}ds$

instead of

$\frac{1}{\delta}\langle
e\rangle_{t}=\frac{\sigma^{2}}{\delta}\int_{0}^{t}\left(\sqrt{V_{s}}-\sqrt{f_{3}(\bar{v}_{\eta(s)})}\right)^{2}ds.$

The additional term gives a contribution of order $\frac{1}{\delta}\Delta
t^{\nu\frac{1-\varepsilon}{1+\varepsilon}}$, which will lead to a worse error
bound than the one given in Proposition 3.3.

###  3.4 Summary

The error bounds in Theorem 1.1 and Proposition 1.2 for the CIR process follow
now from Theorem 3.1, Theorem 3.2, Proposition 3.3 and $|x-f_{3}(y)|\leq|x-y|$
for $x\geq 0$, $y\in\mathbb{R}$.

##  4 Log-Heston model: convergence rates

In this section, we will in particular show that the results from Theorem 3.1,
Theorem 3.2 and Proposition 3.3 carry over to a discretization of the log-
Heston model, where the log-price process is discretized with the Euler scheme
(14), that is

$\displaystyle\hat{x}_{t}=\hat{x}_{\eta(t)}$
$\displaystyle+\left(\mu-\frac{1}{2}\hat{v}_{\eta(t)}\right)(t-\eta(t))+\sqrt{\hat{v}_{\eta(t)}}\left(U_{t}-U_{\eta(t)}\right)$
(28)

with

$U_{t}=\rho W_{t}+\sqrt{1-\rho^{2}}B_{t},\qquad t\in[0,T].$

Recall that our first set of Euler schemes is given by

$\displaystyle\bar{v}_{t}$
$\displaystyle=\bar{v}_{\eta(t)}+\int_{\eta(t)}^{t}\kappa(\theta-
f_{2}(\bar{v}_{\eta(s)}))ds+\sigma\int_{\eta(t)}^{t}\sqrt{f_{3}(\bar{v}_{\eta(s)})}dW_{s},$
(29) $\displaystyle\hat{v}_{t}$ $\displaystyle=f_{3}\left(\bar{v}_{t}\right),$

where

$\displaystyle f_{2}\in\\{{\tt id,abs,sym}\\},\qquad f_{3}\in\\{{\tt
abs,sym}\\},$ (30)

while SE and AE can be expressed as

$\displaystyle\hat{v}^{\star}_{t}=$
$\displaystyle\hat{v}_{\eta(t)}^{\star}+\int_{\eta(t)}^{t}\kappa\left(\theta-\hat{v}_{\eta(s)}^{\star}\right)ds+\sigma\int_{\eta(t)}^{t}\sqrt{\hat{v}_{\eta(s)}^{\star}}dW_{s}$
(31)
$\displaystyle-2c^{\star}\sigma\int_{\eta(t)}^{t}\mathbbm{1}_{\\{z_{s}^{\star}\leq
0\\}}\sqrt{\hat{v}_{\eta(s)}^{\star}}dW_{s}-2c^{\star}\int_{\eta(t)}^{t}\mathbbm{1}_{\\{z_{s}^{\star}\leq
0\\}}\kappa\left(\theta-\hat{v}_{\eta(s)}^{\star}\right)ds$
$\displaystyle+c^{\star}\left(L_{t}^{0}(z^{\star})-L_{\eta(t)}^{0}(z^{\star})\right)$

where $c^{sym}=1$ and $c^{abs}=\frac{1}{2}$.

###  4.1 Error analysis - Case I

The key ingredient here and also for the second case is the observation that
two continuous martingales $M=(M_{t})_{t\in[0,T]}$ and
$\tilde{M}=(\tilde{M}_{t})_{t\in[0,T]}$, whose quadratic variation coincides,
have equivalent moments for their supremum norm. This directly follows from
Proposition 2.5.

######  Theorem 4.1.

Let $(\hat{x},\hat{v})$ be given by (28), (29) and (30). Then, for all
$\epsilon>0$ there exists a constant $C_{\epsilon}>0$ such that

$\mathbb{E}\left[\sup\limits_{t\in[0,T]}\left|X_{t}-\hat{x}_{t}\right|\right]\leq
C_{\epsilon}\left(\Delta t\right)^{\frac{\min\\{1,\nu\\}}{2}-\epsilon}.$

###### Proof.

(i) Without loss of generality, we can assume $\mu=0$. We have to analyze

$\mathbb{E}\left[\sup\limits_{t\in[0,T]}\left|X_{t}-\hat{x}_{t}\right|\right]=\mathbb{E}\left[\sup\limits_{t\in[0,T]}\left|\frac{1}{2}\int_{0}^{t}\left(\hat{v}_{\eta(s)}-V_{s}\right)ds+\int_{0}^{t}\left(\sqrt{V_{s}}-\sqrt{\hat{v}_{\eta(s)}}\right)dU_{s}\right|\right].$

Using Theorem 3.1, Proposition 3.3 and Lemma 2.7, we obtain

$\displaystyle\mathbb{E}\left[\sup\limits_{t\in[0,T]}\left|X_{t}-\hat{x}_{t}\right|\right]$
$\displaystyle\leq\frac{1}{2}\int_{0}^{T}\mathbb{E}\left[\left|V_{s}-\hat{v}_{s}\right|\right]ds+\frac{1}{2}\int_{0}^{T}\mathbb{E}\left[\left|\hat{v}_{s}-\hat{v}_{\eta(s)}\right|\right]ds+\mathbb{E}\left[\sup\limits_{t\in[0,T]}\left|M_{t}\right|\right]$
$\displaystyle\leq C_{\varepsilon}\left(\Delta
t\right)^{\frac{\min\\{1,\nu\\}}{2}-\varepsilon}+\mathbb{E}\left[\sup\limits_{t\in[0,T]}\left|M_{t}\right|\right],$
(32)

where

$M_{t}=\int_{0}^{t}\left(\sqrt{V_{s}}-\sqrt{\hat{v}_{\eta(s)}}\right)dU_{s}.$

(ii) Let

$\tilde{M}_{t}=\int_{0}^{t}\left(\sqrt{V_{s}}-\sqrt{\hat{v}_{\eta(s)}}\right)dW_{s},\qquad
t\in[0,T].$

Clearly, we have

$\langle M\rangle_{t}=\langle\tilde{M}\rangle_{t},\qquad t\in[0,T],$

and so Proposition 2.5 yields

$\mathbb{E}\left[\sup\limits_{t\in[0,T]}\left|M_{t}\right|\right]\leq
K_{1/2}\,\mathbb{E}\left[\langle
M\rangle_{T}^{\frac{1}{2}}\right]=K_{1/2}\,\mathbb{E}\left[\langle\tilde{M}\rangle_{T}^{\frac{1}{2}}\right]\leq\frac{K_{1/2}}{k_{1/2}}\,\mathbb{E}\left[\sup\limits_{t\in[0,T]}|\tilde{M}_{t}|\right].$
(33)

Now, the Lyapunov inequality and an application of Doob’s maximal inequality,
i.e. Proposition 2.6, give

$\mathbb{E}\left[\sup\limits_{t\in[0,T]}\left|M_{t}\right|\right]\leq\frac{K_{1/2}(1+\beta)}{k_{1/2}\beta}\,\left(\mathbb{E}\left[|\tilde{M}_{T}|^{1+\beta}\right]\right)^{1/(1+\beta)}$
(34)

for $\beta>0$.

Using (29) and the SDE for the CIR process we have

$\tilde{M}_{T}=\frac{1}{\sigma}\left(V_{T}-\overline{v}_{T}+\kappa\int_{0}^{T}\left(V_{s}-f_{2}(\overline{v}_{\eta(s)})\right)ds\right).$

Thus, we obtain

$\displaystyle\mathbb{E}\left[|\tilde{M}_{T}|\right]$
$\displaystyle\leq\frac{1}{\sigma}\left(\mathbb{E}\left[\left|V_{T}-\overline{v}_{T}\right|\right]+\kappa\int_{0}^{T}\mathbb{E}\left[\left|V_{s}-\overline{v}_{s}\right|\right]ds+\kappa\int_{0}^{T}\mathbb{E}\left[\left|\overline{v}_{s}-\overline{v}_{\eta(s)}\right|\right]ds\right)$
(35) $\displaystyle\leq C_{\varepsilon}\left(\Delta
t\right)^{\frac{\min\\{1,\nu\\}}{2}-\varepsilon},$

where we used Theorem 3.1, Proposition 3.3, Lemma 2.7 and the properties of
$f_{2}$. Moreover, for all $p\geq 1$ there exists a constant $C_{p}>0$ such
that

$\mathbb{E}\left[|\tilde{M}_{T}|^{p}\right]\leq C_{p}$

due to Lemma 2.7 and Lemma 2.1. Thus, a standard application of Hölder’s
inequality yields

$\mathbb{E}\left[|\tilde{M}_{T}|^{1+\beta}\right]\leq
C_{\beta}\,\left(\mathbb{E}\left[|\tilde{M}_{T}|\right]\right)^{\frac{1}{1+\beta}},$

which in turn together with (34) and (35) gives

$\mathbb{E}\left[\sup\limits_{t\in[0,T]}\left|M_{t}\right|\right]\leq
C_{\beta,\varepsilon}\left(\Delta
t\right)^{\left(\frac{\min\\{1,\nu\\}}{2}-\varepsilon\right)\frac{1}{(1+\beta)^{2}}}.$
(36)

(iii) The assertion follows now from (32) and (36) by choosing $\varepsilon$
and $\beta$ sufficiently small. ∎

###  4.2 Error analysis - Case II

The second case can be treated analogously for $\nu>1$, except at one point.
Here the martingale $\tilde{M}$ is given by

$\displaystyle\tilde{M}_{t}$
$\displaystyle=\int_{0}^{t}\left(\sqrt{V_{s}}-\sqrt{\hat{v}^{\star}_{\eta(s)}}\right)dW_{s}$
$\displaystyle=\frac{1}{\sigma}\left(V_{t}-\hat{v}_{t}^{\star}+\kappa\int_{0}^{t}\left(V_{s}-\hat{v}^{\star}_{\eta(s)}\right)ds\right)+2c^{\star}\int_{0}^{t}\mathbbm{1}_{\\{z_{s}^{\star}\leq
0\\}}\sqrt{v_{\eta(s)}^{\star}}dW_{s}$
$\displaystyle\quad+\frac{2c^{\star}}{\sigma}\int_{0}^{t}\mathbbm{1}_{\\{z_{s}^{\star}\leq
0\\}}\kappa\left(\theta-v^{\star}_{\eta(s)}\right)ds-\frac{c^{\star}}{\sigma}L^{0}_{t}(z^{\star}),\qquad
t\in[0,T].$

However, the additional terms can be treated with Lemma 2.10 and Lemma 2.11
and are (at least) of order $(\Delta t)^{\frac{1}{2}-\varepsilon}$. Using
Theorem 3.2 and Lemma 2.8 instead of Theorem 3.1 and Lemma 2.7 and proceeding
as in Case I we obtain

$\mathbb{E}\left[|\tilde{M}_{T}|\right]\leq
C_{\varepsilon}(\Delta)^{\frac{1}{2}-\varepsilon}.$

Therefore we also have the following result:

######  Theorem 4.2.

Let $\nu>1$ and $(\hat{x},\hat{v})$ be given by (28) and (31). Then, for all
$\epsilon>0$ there exists a constant $C_{\epsilon}>0$ such that

$\mathbb{E}\left[\sup\limits_{t\in[0,T]}\left|X_{t}-\hat{x}_{t}\right|\right]\leq
C_{\epsilon}\left(\Delta t\right)^{\frac{1}{2}-\epsilon}.$

###  4.3 Summary

The error bounds in Theorem 1.1 and Proposition 1.2 for the log-Heston process
follow now from Theorem 4.1 and Theorem 4.2.

###  4.4 Proof of Proposition 1.3

Recall that $\Delta t=\frac{T}{N}$. The Lipschitz continuity of $G$ implies
that

$\left|\mathbb{E}\left[G\left(X^{pc,N}\right)\right]-\mathbb{E}\left[G\left(\hat{x}^{pc,N}\right)\right]\right|\leq
L_{G}\,\mathbb{E}\left[\sup_{t\in[0,T]}|X_{t}-\hat{x}_{t}|\right],$

where we have set $X^{pc,N}_{t}=X_{\eta(t)}$ with
$\eta(t)=\max\\{k\in\\{0,...,N\\}:t_{k}\leq t\\}\Delta t$. Now Theorem 4.1 and
Theorem 4.2, respectively, yield that

$\lim_{N\rightarrow\infty}\,\,N^{1/2-\epsilon}\,\big{|}\mathbb{E}\left[G\left(X^{pc,N}\right)\right]-\mathbb{E}\left[G\left(\hat{x}^{pc,N}\right)\right]\big{|}=0$

for $\nu>1$. Moreover, the Lipschitz continuity of $G$ also implies that

$\big{|}\mathbb{E}\left[G(X)\right]-\mathbb{E}\left[G\left(X^{pc,N}\right)\right]\big{|}\leq
L_{G}\,\mathbb{E}\left[\sup_{t\in[0,T]}|X_{t}-X_{\eta(t)}|\right].$

Finally, recall that due to Lemma 2.2 we have that

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

for all $p\geq 1$. Thus, the classical Garsia-Rodemich-Rumsey Lemma, see e.g.
Equation A.10 with $m=\gamma(\frac{1}{2}-\varepsilon)$ and
$\gamma>\frac{1}{\varepsilon}$ in Appendix A.3 in [33], implies that

$\mathbb{E}\left[\sup_{0\leq s<t\leq
T}\frac{|X_{t}-X_{s}|}{\,|t-s|^{\frac{1}{2}-\varepsilon}}\right]<\infty$

for all $\varepsilon>0$. Therefore, we obtain

$\lim_{N\rightarrow\infty}\,\,N^{1/2-\epsilon}\,\big{|}\mathbb{E}\left[G(X)\right]-\mathbb{E}\left[G\left(X^{pc,N}\right)\right]\big{|}=0,$

which concludes the proof of Proposition 1.3.

##  5 Simulation study

In this section, we will test numerically our results from Theorem 1.1 and
Proposition 1.2 for some sets of exemplary parameters. We will perform the
tests for all schemes from Table 1. First, we will describe the design of the
numerical experiments. We would like to estimate the rate of the decay of the
errors

$\displaystyle e_{v}(N)$
$\displaystyle=\mathbb{E}\left[\left|V_{T}-\hat{v}_{t_{N}}^{(N)}\right|\right],\qquad
e_{x}(N)=\mathbb{E}\left[\left|X_{T}-\hat{x}_{t_{N}}^{(N)}\right|\right],$

for the numerical scheme $\hat{v}^{(N)},\hat{x}^{(N)}$ with step size $\Delta
t=\frac{T}{N}$. Since we cannot compute these quantities exactly, we
approximate their decay, see e.g. [1], by calculating

$\displaystyle{\tt err}_{v}(N)$
$\displaystyle=\frac{1}{M}\sum_{i=1}^{M}\left|\left(\hat{v}_{t_{N}}^{(N)}-\hat{v}_{t_{2N}}^{(2N)}\right)^{(i)}\right|,\qquad{\tt
err}_{x}(N)=\frac{1}{M}\sum_{i=1}^{M}\left|\left(\hat{x}_{t_{N}}^{(N)}-\hat{x}_{t_{2N}}^{(2N)}\right)^{(i)}\right|,$

where $M$ is the number of Monte Carlo repetitions and
$(\hat{v}_{t_{N}}^{(N)}-\hat{v}_{t_{2N}}^{(2N)})^{(i)},i=1,...,M$, are iid
copies of $\hat{v}_{t_{N}}^{(N)}-\hat{v}_{t_{2N}}^{(2N)}$. The same holds for
$(\hat{x}_{t_{N}}^{(N)}-\hat{x}_{t_{2N}}^{(2N)})^{(i)},i=1,...,M$. In our
simulations, we chose $M=10^{5}$ and $N\in\\{2^{1},...,2^{15}\\}$. To cover a
wide range of different Feller indices, we will perform numerical simulations
with five different parameter sets. We always choose $T=1$ and $s_{0}=100$.
The other parameters can be found in Table 2.

Table 2: Parameters Model | $v_{0}$ | $\kappa$ | $\theta$ | $\sigma$ | $\rho$ | $\mu$ | $\nu$
---|---|---|---|---|---|---|---
1 | 0.04 | 5 | 0.04 | 0.61 | -0.7 | 0.0319 | 1.075
2 | 0.0457 | 5.07 | 0.0457 | 0.48 | -0.767 | 0 | 2.0113
3 | 0.04 | 2.6 | 0.04 | 0.2 | -0.6 | 0 | 5.2
4 | 0.010201 | 6.21 | 0.019 | 0.61 | -0.7 | 0.0319 | 0.63
5 | 0.09 | 2 | 0.09 | 1 | -0.3 | 0.05 | 0.36

The estimates ${\tt err}_{v}(N)$ and ${\tt err}_{x}(N)$ for the five Euler
schemes are plotted in Figures 1-5 against the corresponding (inverse) step
sizes of $2N$. For each model, we show first the convergence behavior of the
error for the CIR process and then for the Heston model. Additionally, we
plotted a reference line with a suitable slope together with the error
estimates. We also estimated the rate of convergence by the slope of a least
squares fit, see Tables 3 – 7. Here, we only take errors with (inverse) step
sizes $N\in\\{2^{6},...,2^{15}\\}$ into account to get a stable result. For
all models, our simulation study shows that the numerical convergence rates do
not change significantly, if we extend the simulation from the CIR process to
the Heston model. This indicates that the parameters of the CIR process and
especially the Feller index solely determine the convergence behavior.

(a) CIR, $\nu=1.075$

(b) Heston, $\nu=1.075$

Figure 1: Error estimates for Model 1 Table 3: Estimated convergence rates model 1 Scheme | Rate CIR | Rate Heston
---|---|---
SE | 0.5139 | 0.5175
AE | 0.5089 | 0.5171
FTE | 0.5236 | 0.5249
PTE | 0.5204 | 0.5226
HM | 0.5117 | 0.5318

For the first model which has a Feller index around $1$, our main result
provides a strong convergence rate of $0.5$. This can be numerically confirmed
in Figure 1.

(a) CIR, $\nu=2.0113$

(b) Heston, $\nu=2.0113$

Figure 2: Error estimates for Model 2 Table 4: Estimated convergence rates model 2 Scheme | Rate CIR | Rate Heston
---|---|---
SE | 0.5213 | 0.5310
AE | 0.5216 | 0.5342
FTE | 0.5251 | 0.5354
PTE | 0.5243 | 0.5342
HM | 0.5222 | 0.5357

(a) CIR, $\nu=5.2$

(b) Heston, $\nu=5.2$

Figure 3: Error estimates for Model 3 Table 5: Estimated convergence rates model 3 Scheme | Rate CIR | Rate Heston
---|---|---
SE | 0.5184 | 0.5038
AE | 0.5184 | 0.5038
FTE | 0.5184 | 0.5038
PTE | 0.5184 | 0.5038
HM | 0.5184 | 0.5038

For models 2 and 3, which have higher Feller indices, Figure 2 and Figure 3
confirm again the expected strong convergence rate of $0.5$. Note that the
differences between the different Euler schemes vanish for small step sizes
and for high Feller indices. The Euler schemes only differ, if the
approximation of the CIR process becomes negative. For small step sizes and
for high Feller indices this is unlikely to happen in a Monte Carlo
simulation.

(a) CIR, $\nu=0.63$

(b) Heston, $\nu=0.63$

Figure 4: Error estimates for Model 4 Table 6: Estimated convergence rates model 4 Scheme | Rate CIR | Rate Heston
---|---|---
SE | 0.4878 | 0.4988
AE | 0.4675 | 0.4819
FTE | 0.4868 | 0.5023
PTE | 0.4790 | 0.4929
HM | 0.4726 | 0.5053

Model 4 has a Feller index of $0.63$ and we have shown that we can expect a
strong convergence rate of at least $0.315$ for FTE, PTE and HM. However,
looking at Figure 4 we can see that the rate is still around $0.5$, even for
SE and AE, for which we did not derive a convergence rate in this case.

(a) CIR, $\nu=0.36$

(b) Heston, $\nu=0.36$

Figure 5: Error estimates for Model 5 Table 7: Estimated convergence rates model 5 Scheme | Rate CIR | Rate Heston
---|---|---
SE | 0.4082 | 0.4128
AE | 0.3811 | 0.4042
FTE | 0.3701 | 0.3863
PTE | 0.3661 | 0.3828
HM | 0.3765 | 0.3770

The last model has the lowest Feller index. Again, we can see an estimated
convergence rate of the error that is better than expected. Here, we chose
$\nu$ as the slope of the reference line.

The last two examples indicate that it might be possible to obtain a
convergence rate of $\min\\{\nu,\frac{1}{2}\\}$ for all the considered Euler
schemes.

Acknowledgments. The authors are grateful to the referees for their insightful
comments and remarks, which helped to improve this article. Annalena Mickel
has been supported by the DFG Research Training Group 1953 ”Statistical
Modeling of Complex Systems”.

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