# Error distribution of the Euler approximation scheme for stochastic Volterra
equations

David Nualart, Bhargobjyoti Saikia  David Nualart: Department of Mathematics,
University of Kansas, 1450 Jayhawk Blvd., Lawrence, KS 66045. USA.
<EMAIL_ADDRESS>Bhargobjyoti Saikia: Department of Mathematics, University of
Kansas, 1450 Jayhawk Blvd., Lawrence, KS 66045. USA<EMAIL_ADDRESS>
###### Abstract.

The purpose of this paper is to establish the convergence in distribution of
the normalized error in the Euler approximation scheme for stochastic Volterra
equations driven by a standard Brownian motion, with a kernel of the form
$(t-s)^{\alpha}$, where $\alpha\in(-\frac{1}{2},\frac{1}{2})$.

###### Key words and phrases:

Stochastic Volterra equations, Euler approximation

###### 2020 Mathematics Subject Classification:

60H20,60F05

The work by D. Nualart has been supported by the NSF grant DMS-2054735

##  1\. Introduction

There have been several studies on stochastic Volterra equations starting from
the pioneering work by Berger and Mizel [1, 2]. We could mention, for
instance, the papers by Protter [11], Pardoux and Protter [10], Cochran, Lee
and Potthoff [4], Coutin and Decreusefond [6], Decreusefond [7], Wang [14],
Zhang [17], and the references therein.

In this paper we are interested in the following stochastic Volterra equation

(1.1) $\displaystyle
X_{t}=X_{0}+\int^{t}_{0}(t-s)^{\alpha}\sigma(X_{s})\,dW_{s},\qquad t\in[0,T],$

where $X_{0}\in\mathbb{R}$, $\alpha\in(-\frac{1}{2},\frac{1}{2})$, $\sigma$ is
a Lipschitz function and $W=\\{W_{t},t\geq[0,T]\\}$ is a standard Brownian
motion defined in a probability space $(\Omega,\mathcal{F},P)$. Our aim is to
study the fluctuations of the error in the Euler approximation scheme for this
equation. It is known that the rate of convergence of the Euler approximation
scheme is of the order $n^{-\alpha-\frac{1}{2}}$. We refer for this to the
work by Li, Huang and Hu [9], where they consider the more general
$\theta$-Euler and Milstein schemes in the singular case
$-\frac{1}{2}<\alpha<0$, and the papers by Zhang [16] and Richard, Tan and
Yang [13], where the Euler approximation scheme for more general stochastic
Volterra equations is studied. However, to the best of our knowledge, there
have been no results on the convergence in distribution of the normalized
error, analogous to the case of ordinary stochastic differential equations
(see, for instance, the work by Jacod and Protter [8]).

The Euler scheme approximation for the solution to equation (1.1) is defined
as

(1.2) $\displaystyle
X^{n}_{t}=X_{0}+\int^{t}_{0}(t-\eta_{n}(s))^{\alpha}\sigma(X^{n}_{\eta_{n}(s)})\,dW_{s},$

where $\eta_{n}(s)=\frac{\lfloor ns\rfloor}{n}$. That is,
$\eta_{n}(s)=\frac{j}{n}$ if $\frac{j}{n}\leq s<\frac{j+1}{n}$ for some
integer $j\geq 0$. Consider the normalized error defined by

(1.3) $Y^{n}_{t}=n^{\alpha+\frac{1}{2}}(X^{n}_{t}-X_{t}).$

It is known that $Y^{n}_{t}$ is bounded in $L^{p}$, for any $p\geq 2$,
uniformly in $t\in[0,T]$ and $n\geq 1$ (see Lemma 3.1) for a short proof. This
means that the rate of convergence to zero as $n\to\infty$ in $L^{p}$ of
$X^{n}_{t}-X_{t}$ is of the order $n^{-\alpha-\frac{1}{2}}$. The goal of this
paper is to study the asymptotic behavior of the normalized error (1.3).

We will describe here the approach and the challenging difficulties
encountered in handling the limit distribution of the normalized error
$Y^{n}_{t}$. This error can be decomposed as follows:

$\displaystyle Y^{n}_{t}$
$\displaystyle=n^{\alpha+\frac{1}{2}}\int^{t}_{0}[(t-\eta_{n}(s))^{\alpha}-(t-s)^{\alpha}]\sigma(X^{n}_{\eta_{n}(s)})\,dW_{s}$
$\displaystyle\qquad+n^{\alpha+\frac{1}{2}}\int^{t}_{0}(t-s)^{\alpha}[\sigma(X^{n}_{\eta_{n}(s)})-\sigma(X_{s})]\,dW_{s}$
(1.4) $\displaystyle:=A^{n}_{t}+Y^{n,1}_{t}.$

The term $A^{n}_{t}$ has an unexpected asymptotic behavior. Indeed, it only
converges at points of the form $t=\eta_{n}(t)$ (see Remark 1 below) and its
limit in law is $\kappa_{1}\sigma(X_{t})Z_{t}$, where $\kappa_{1}$ is a
constant defined in (2.1) and $Z_{t}$ is a $N(0,1)$ random variable
independent of the Brownian motion $W$. Moreover, (see Proposition 4.3) the
finite-dimensional distributions of the process
$\\{A^{n}_{\eta_{n}(t)},t\in[0,T]\\}$ converge to those of
$\\{\kappa_{1}\sigma(X_{t})Z_{t},t\in[0,T]\\}$, where $\\{Z_{t},t\in[0,T]\\}$
is a centered Gaussian process independent of $W$ with covariance

(1.5) $E[Z_{t_{1}}Z_{t_{2}}]=\mathbf{1}_{\\{t_{1}=t_{2}\\}}.$

Notice that the limit in distribution of $A^{n}_{\eta_{n}(t)}$ does not have
any Hölder continuity properties, which creates problems in the global
analysis of the error. For this reason, we have split the study of the error
in two parts, corresponding to both terms in the above decomposition. For the
term $Y^{n,1}_{t}$, the limit in distribution can be expressed as the solution
to a stochastic Volterra equation, with an input given by a stochastic
integral term with respect to an independent Brownian motion $B$. A precise
statement is given in Theorem 2.1. The proof of the convergence of the term
$Y^{n,1}_{t}$ relies on the application of the asymptotic version of Knight’s
theorem. Finally, in Theorem 2.2 we establish the convergence in distribution
of the couple of random variables $(A^{n}_{\eta_{n}(t)},Y^{n,1}_{t})$, which
implies, as a consequence, the limit in law of the normalized error
$Y^{n}_{\eta_{n}(t)}$ given in Corollary 2.1.

The paper is organized as follows. Section 2 contains the main result. In
Section 3 we introduce some preliminaries and in Section 4 we present the
proof of the main result. Some technical auxiliary results are included in the
Apppendix.

##  2\. Main result

Let $X=\\{X_{t},t\in[0,T]\\}$ be the solution to the stochastic Volterra
equation (1.1). Consider the Euler approximation scheme
$X^{n}=\\{X^{n}_{t},t\in[0,T]\\}$ defined in (1.2). The following theorem,
which deals with the convergence of the processes $A^{n}_{t}$ and
$Y^{n,1}_{t}$ appearing in the decomposition of the normalized error, is the
main result of this paper.

######  Theorem 2.1.

Let $Y^{n}=\\{Y^{n}_{t},t\in[0,T]\\}$ be the normalized error in the Euler
approximation scheme of equation (1.1) defined in (1.3). Suppose that $\sigma$
is continuously differentiable and $\sigma^{\prime}$ is Hölder continuous of
order $\beta$ for some $\beta\in(0,1]$. Consider the decomposition introduced
in (1.4). Then the following results hold true:

  * (i)

As $n$ tends to infinity, the finite-dimensional distributions of the process
$\\{A^{n}_{\eta_{n}(t)},t\in[0,T]\\}$ converge in distribution to those of
$\kappa_{1}\sigma(X_{t})Z_{t}$, where

(2.1)
$\kappa_{1}=\sqrt{\sum_{k=1}^{\infty}\int_{0}^{1}[k^{\alpha}-(k-x)^{\alpha}]^{2}dx}$

and $\\{Z_{t},t\in[0,T]\\}$ is a zero mean Gaussian process independent of $W$
with covariance (1.5).

  * (ii)

As $n$ tends to infinity, the finite-dimensional distributions of the process
$\\{Y^{n,1}_{t},t\in[0,T]\\}$ converge in distribution to those of
$\\{Y^{\infty,1}_{t},t\in[0,T]\\}$, where $Y^{\infty,1}_{t}$ is the solution
to the following stochastic Volterra equation

$Y^{\infty,1}_{t}=\kappa_{2}\int_{0}^{t}(t-s)^{\alpha}(\sigma^{\prime}\sigma)(X_{s})dB_{s}+\int_{0}^{t}(t-s)^{\alpha}\sigma^{\prime}(X_{s})Y^{\infty,1}_{s}dW_{s},$

where $B=\\{B_{t},t\in[0,T]\\}$ is a standard Brownian motion independent of
$W$ and

(2.2)
$\kappa_{2}=\sqrt{\kappa_{3}+\frac{2\alpha+1}{2(\alpha+1)^{2}}+\kappa_{4}+\kappa_{5}},$

with

(2.3)
$\kappa_{3}=\sum_{k=1}^{\infty}\int_{0}^{1}\left[(x+k)^{\alpha}-k^{\alpha}\right]^{2}dx,$
(2.4)
$\kappa_{4}=\sum_{k=0}^{\infty}\int_{0}^{1}\int_{0}^{1\wedge(y+k)}[(y+k)^{\alpha}-(y+k-z)^{\alpha}]^{2}dzdy.$

and

(2.5)
$\kappa_{5}=\sum_{k=1}^{\infty}\int_{0}^{1}\int_{0}^{1}[(y+k)^{\alpha}-(y+k-z)^{\alpha}][(y+k)^{\alpha}-k^{\alpha}]dzdy.$

In the next result we analyze the joint convergence of $A^{n}_{t}$ and
$Y^{n,1}_{t}$ for a fixed $t\in[0,T]$.

######  Theorem 2.2.

Under the assumptions of Theorem 2.1, for any $t\in[0,T]$, the two-dimensional
random variable $(A^{n}_{\eta_{n}(t)},Y^{n,1}_{t})$ converges in distribution
to $(\kappa_{1}\sigma(X_{t})Z_{t},Y^{\infty,1}_{t})$, where $Z_{t}$ and
$Y^{\infty,1}_{t}$ are the random variables described in points (i) and (ii)
of Theorem 2.1 and $Z_{t}$ and $Y^{\infty,1}_{t}$ are independent.

As a consequence of theorems 2.1 and 2.2 we obtain the following result.

######  Corollary 2.1.

Under the assumptions of Theorem 2.1, for any $t\in[0,T]$, the normalized
error converges in distribution to
$\kappa_{1}\sigma(X_{t})Z_{t}+Y^{\infty,1}_{t}$, where $Z_{t}$ and
$Y^{\infty,1}_{t}$ are the random variables described in points (i) and (ii)
of Theorem 2.1 and $Z_{t}$ and $Y^{\infty,1}_{t}$ are independent.

##  3\. Preliminaries

In this section we will present some basic properties of the solution to the
stochastic Volterra equation (1.1) and its associated Euler approximation
scheme (1.2). Throughout this paper $C$ will denote a generic constant, which
may depend on $\alpha$, the coefficient $\sigma$ and the terminal time $T$.
Also, $C_{p}$ will denote a generic constant, which may depend on $p$,
$\alpha$, the coefficient $\sigma$ and the terminal time $T$. Moreover, we
will denote by $\\{\mathcal{F}_{t},t\in[0,T]\\}$ the increasing family of
$\sigma$-algebras generated by the Brownian motion $W$.

Assuming that the coefficient $\sigma$ is Lipschitz, there exist a unique
solution to equation (1.1), such that for any $p\geq 2$,

(3.1) $\sup_{0\leq t\leq T}E[|X_{t}|^{p}]<C_{p}.$

Furthermore, for any $p\geq 2$ and for any $s,t\in[0,T]$,

(3.2) $E[|X_{t}-X_{s}|^{p}]\leq C_{p}|t-s|^{p(\alpha+\frac{1}{2})}.$

We refer, for instance, to Theorem 2.1 and Theorem 2.2 of [9] for the proof of
these estimates in the case $\alpha<0$. The case $\alpha\geq 0$ is easier and
can be obtained by similar arguments.

The Euler approximation $X^{n}_{t}$ satisfies similar estimates. That means,
for any $p\geq 2$, we have

(3.3) $\sup_{0\leq t\leq T}E[|X^{n}_{t}|^{p}]<C_{p}$

and for any $s,t\in[0,T]$,

(3.4) $E[|X^{n}_{t}-X^{n}_{s}|^{p}]\leq C_{p}|t-s|^{p(\alpha+\frac{1}{2})}.$

Again, these estimates can be obtained as in the proof of Theorem 2.1 and
Theorem 2.2 of [9], when $\alpha<0$, the case $\alpha\geq 0$ being simpler.

The following result provides the rate of convergence in $L^{p}$, for any
$p\geq 2$, for the Euler approximation scheme. The case $\alpha<0$ and
$t=\eta_{n}(t)$ was established in Theorem 2.3 of [9]. For the sake of
completeness we include the main ideas of the proof.

######  Lemma 3.1.

Let $Y^{n}_{t}$ be the normalized error defined in (1.3). Then, there exists a
constant $C_{p}>0$ such that for any $t\in[0,T]$, and for any $p\geq 2$, we
have

(3.5) $E[|Y^{n}_{t}|^{p}]\leq C_{p}.$

###### Proof.

We can write, using Burkholder and Minkowski’s inequalities, the Lipschitz
property of $\sigma$ and the estimate (3.1),

$\displaystyle\|Y^{n}_{t}\|_{p}^{2}$ $\displaystyle\leq
C_{p}n^{2\alpha+1}\int_{0}^{t}\|(t-s)^{\alpha}\sigma(X_{s})-(t-\eta_{n}(s))^{\alpha}\sigma(X_{\eta_{n}(s)})\|_{p}^{2}ds$
$\displaystyle\leq
C_{p}\int_{0}^{t}(t-s)^{\alpha}\|Y^{n}_{s}\|_{p}^{2}ds+C_{p}n^{2\alpha+1}\int_{0}^{t}[(t-s)^{\alpha}-(t-\eta_{n}(s))^{\alpha}]^{2}ds.$

Lemma 6.3 allows us to write

$\|Y^{n}_{t}\|_{p}^{2}\leq
C_{p}\int_{0}^{t}(t-s)^{\alpha}\|Y^{n}_{s}\|_{p}^{2}ds+C,$

which implies the desired estimate by an application of an extended version of
Gronwall’s inequality (see, for instance, [3, 15]). $\square$

As a consequence of (3.2) and (3.5), we obtain, for any $s,t\in[0,T]$,

(3.6) $E[|X^{n}_{t}-X_{s}|^{p}]\leq
C_{p}\left(|t-s|^{p(\alpha+\frac{1}{2})}+n^{-p(\alpha+\frac{1}{2})}\right).$

##  4\. Proof of Theorem 2.1

Recall that $X=\\{X_{t},t\in[0,T]\\}$ is the solution to the stochastic
Volterra equation (1.1) and the normalized Euler scheme approximation for $X$
is defined in (1.3).

In order to prove Theorem 2.1 we make the following decomposition of the
normalized error

$Y^{n}_{t}=A^{n}_{t}+Y^{n,1}_{t}=:A^{n}_{t}+C^{n}_{t}+Z^{n}_{t},$

where

(4.1)
$A^{n}_{t}=n^{\alpha+\frac{1}{2}}\int^{t}_{0}[(t-\eta_{n}(s))^{\alpha}-(t-s)^{\alpha}]\sigma(X^{n}_{\eta_{n}(s)})\,dW_{s},$
(4.2)
$C^{n}_{t}=n^{\alpha+\frac{1}{2}}\int^{t}_{0}(t-s)^{\alpha}[\sigma(X^{n}_{\eta_{n}(s)})-\sigma(X^{n}_{s})]\,dW_{s}$

and

(4.3)
$Z^{n}_{t}=n^{\alpha+\frac{1}{2}}\int^{t}_{0}(t-s)^{\alpha}[\sigma(X^{n}_{s})-\sigma(X_{s})]\,dW_{s}.$

Then, the proof of Theorem 2.1 will be done in several steps included in the
next subsections.

###  4.1. Modified error

The next proposition will allow us to replace the three terms $A^{n}_{t}$,
$C^{n}_{t}$ and $Z^{n}_{t}$ defined in (4.1), (4.2) and (4.3), respectively,
by simpler terms that have the same asymptotic behavior.

######  Proposition 4.1.

Let $A^{n}_{t}$, $C^{n}_{t}$ and $Z^{n}_{t}$ be defined in (4.1), (4.2) and
(4.3), respectively. Then we claim that the following convergences are true

(4.4)
$\lim_{n\rightarrow\infty}\sup_{t\in[0,T]}E[|A^{n}_{t}-\widetilde{A}^{n}_{t}|^{2}]=0,$
(4.5)
$\lim_{n\rightarrow\infty}\sup_{t\in[0,T]}E[|C^{n}_{t}-\widetilde{C}^{n}_{t}|^{2}]=0$

and

(4.6)
$\lim_{n\rightarrow\infty}\sup_{t\in[0,T]}E[|Z^{n}_{t}-\widetilde{Z}^{n}_{t}|^{2}]=0,$

where

(4.7)
$\widetilde{A}^{n}_{t}=n^{\alpha+\frac{1}{2}}\sigma(X_{t})\int^{t}_{0}[(t-\eta_{n}(s))^{\alpha}-(t-s)^{\alpha}]dW_{s},$
(4.8)
$\widetilde{C}^{n}_{t}=n^{\alpha+\frac{1}{2}}\int^{t}_{0}(t-s)^{\alpha}\sigma^{\prime}(X_{s})\left(X^{n}_{\eta_{n}(s)}-X^{n}_{s}\right)\,dW_{s}$

and

(4.9)
$\widetilde{Z}^{n}_{t}=n^{\alpha+\frac{1}{2}}\int^{t}_{0}(t-s)^{\alpha}\sigma^{\prime}(X_{s})\left(X^{n}_{s}-X_{s}\right)\,dW_{s}.$

###### Proof.

The proof will be done in three steps.

Proof of (4.4): Fix $\delta>0$. With the notation

(4.10) $\psi_{n,1}(s,t):=(t-\eta_{n}(s))^{\alpha}-(t-s)^{\alpha},\quad 0\leq
s<t\leq T,$

we can write

$\displaystyle A^{n}_{t}-\widetilde{A}^{n}_{t}$
$\displaystyle=n^{\alpha+\frac{1}{2}}\left(\int_{0}^{t}\psi_{n,1}(s,t)\sigma(X^{n}_{\eta_{n}(s)})\,dW_{s}-\sigma(X_{t})\int_{0}^{t}\psi_{n,1}(s,t)dW_{s}\right)$
$\displaystyle=n^{\alpha+\frac{1}{2}}\Bigg{(}\int_{0}^{(t-\delta)_{+}}\psi_{n,1}(s,t)\sigma(X^{n}_{\eta_{n}(s)})dW_{s}-\sigma(X_{t})\int_{0}^{(t-\delta)_{+}}\psi_{n,1}(s,t)dW_{s}$
$\displaystyle\qquad+\int_{(t-\delta)_{+}}^{t}\psi_{n,1}(s,t)\left[\sigma(X^{n}_{\eta_{n}(s)}-\sigma(X_{(t-\delta)_{+}})\right]dW_{s}$
$\displaystyle\qquad+\left[\sigma(X_{(t-\delta)_{+}})-\sigma(X_{t})\right]\int_{(t-\delta)_{+}}^{t}\psi_{n,1}(s,t)dW_{s}\Bigg{)}.$

Now, using triangle inequality and Cauchy-Schwartz inequality, we obtain

$\displaystyle E[|A^{n}_{t}-\widetilde{A}^{n}_{t}|^{2}]$ $\displaystyle\leq
Cn^{2\alpha+1}\int_{0}^{(t-\delta)_{+}}\psi^{2}_{n,1}(s,t)E(\sigma^{2}(X^{n}_{\eta_{n}(s)}))ds$
$\displaystyle\qquad+Cn^{2\alpha+1}\left[E\left(\sigma^{4}(X_{t})\right)E\left[\left(\int_{0}^{(t-\delta)_{+}}\psi_{n,1}(s,t)dW_{s}\right)^{4}\right]\right]^{\frac{1}{2}}$
$\displaystyle\qquad+Cn^{2\alpha+1}\int_{(t-\delta)_{+}}^{t}\psi^{2}_{n,1}(s)E\left[\left|\sigma(X^{n}_{\eta_{n}(s)})-\sigma(X_{(t-\delta)_{+}})\right|^{2}\right]ds$
$\displaystyle\qquad+Cn^{2\alpha+1}\left[E\left(\left|\sigma(X_{(t-\delta)_{+}})-\sigma(X_{t})\right|^{4}\right)E\left(\left|\int_{(t-\delta)_{+}}^{t}\psi_{n,1}(s,t)dW_{s}\right|^{4}\right)\right]^{\frac{1}{2}}$
$\displaystyle=:Cn^{2\alpha+1}(A_{1}+A_{2}+A_{3}+A_{4}).$

By the estimate (3.3), taking into account that $\sigma$ is Lipschitz, we have

$L_{1}:=\sup_{n\geq
1}\sup_{s\in[0,T]}E(\sigma^{2}(X^{n}_{\eta_{n}(s)}))<\infty.$

Therefore, applying the estimate (6.4), we can write

(4.11) $A_{1}\leq L_{1}\int_{0}^{(t-\delta)_{+}}\psi_{n,1}^{2}(s,t)ds\leq
L_{1}C_{2}n^{-2}\delta^{2\alpha-1}.$

For the term $A_{2}$, using that the random variable
$\int_{0}^{(t-\delta)_{+}}\psi_{n,1}(s,t)dW_{s}$ is Gaussian and applying
again the estimate (6.4), we obtain

$\displaystyle A_{2}$
$\displaystyle\leq\sqrt{3}\sup_{t\in[0,T]}\left[E\left(\sigma^{4}(X_{t})\right)\right]^{\frac{1}{2}}\int_{0}^{(t-\delta)_{+}}\psi_{n,1}^{2}(s,t)ds$
(4.12)
$\displaystyle\leq\sqrt{3}\sup_{t\in[0,T]}\left[E\left(\sigma^{4}(X_{t})\right)\right]^{\frac{1}{2}}C_{2}n^{-2}\delta^{2\alpha-1}.$

For the term $A_{3}$, using the Lipschitz property of $\sigma$ and inequality
(3.6), we obtain, assuming $\delta>\frac{1}{n}$,

$\displaystyle
E\left[\left|\sigma(X^{n}_{\eta_{n}(s)})-\sigma(X_{(t-\delta)_{+}})\right|^{2}\right]$
$\displaystyle\leq
C\left[\sup_{s\in[(t-\delta)_{+},t]}|\eta_{n}(s)-(t-\delta)_{+}|\right]^{2\alpha+1}+\frac{C}{n^{2\alpha+1}}$
$\displaystyle\leq C\delta^{2\alpha+1}+\frac{C}{n^{2\alpha+1}}$
$\displaystyle\leq C\delta^{2\alpha+1}.$

Therefore, using (6.3),

(4.13) $A_{3}\leq C\delta^{2\alpha+1}\int_{0}^{t}\psi_{n,1}^{2}(s,t)ds\leq
C\delta^{2\alpha+1}n^{-2\alpha-1}.$

Finally, for the term $A_{4}$, using the Lipschitz property of $\sigma$ and
the estimates (3.1) and (6.3), yields

(4.14) $A_{4}\leq C\delta^{2\alpha+1}\int_{0}^{t}\psi_{n,1}^{2}(s,t)ds\leq
C\delta^{2\alpha+1}n^{-2\alpha-1}.$

The estimates (4.11), (4.12), (4.13) and (4.14) allow us to write

$\limsup_{n\rightarrow\infty}\sup_{t\in[0,T]}E\left[|A^{n}_{t}-\widetilde{A}^{n}_{t}|^{2}\right]\leq
C\delta^{2\alpha+1},$

where the constant $C$ does not depend on $\delta$. Taking into account that
$\delta>0$ is arbitrary, we conclude the proof of (4.4).

Proof of (4.5): We can write

$C^{n}_{t}=n^{\alpha+\frac{1}{2}}\int^{t}_{0}(t-s)^{\alpha}\Psi^{n}_{s}\,\left(X^{n}_{\eta_{n}(s)}-X^{n}_{s}\right)\,dW_{s},$

where $\Psi^{n}_{s}=\int^{1}_{0}\sigma^{\prime}(\theta
X^{n}_{\eta_{n}(s)}+(1-\theta)X^{n}_{s})\,d\theta$. Since $\sigma^{\prime}$ is
$\beta$-Hölder continuous, using the definition of $\Psi^{n}_{s}$ we have

(4.15) $|\Psi^{n}_{s}-\sigma^{\prime}(X^{n}_{s})|\leq
C|X^{n}_{\eta_{n}(s)}-X^{n}_{s}|^{\beta}.$

Now using (3.4) and the fact that $|\eta_{n}(s)-s|\leq\frac{1}{n}$ we have

(4.16) $E[|\Psi^{n}_{s}-\sigma^{\prime}(X_{s})|^{p}]\leq
Cn^{-p\beta(\alpha+\frac{1}{2})}.$

Next coming back to proof of (4.5), we can write, using Cauchy-Schwartz
inequality,

$\displaystyle E\left[\left|C^{n}_{t}-\widetilde{C}^{n}_{t}\right|^{2}\right]$
$\displaystyle=n^{2\alpha+1}E\left[\left|\int^{t}_{0}(t-s)^{\alpha}\left(\sigma^{\prime}(X_{s})-\Psi^{n}_{s}\right)\left(X^{n}_{\eta_{n}(s)}-X^{n}_{s}\right)\,dW_{s}\right|^{2}\right]$
$\displaystyle=n^{2\alpha+1}\int^{t}_{0}(t-s)^{2\alpha}E\left[\left(\sigma^{\prime}(X_{s})-\Psi^{n}_{s}\right)^{2}\left(X^{n}_{\eta_{n}(s)}-X^{n}_{s}\right)^{2}\right]\,ds$
$\displaystyle\leq
n^{2\alpha+1}\int^{t}_{0}(t-s)^{2\alpha}\left(E[(\sigma^{\prime}(X_{s})-\Psi^{n}_{s})^{4}]\right)^{\frac{1}{2}}\left(E[(X^{n}_{\eta_{n}(s)}-X^{n}_{s})^{4}]\right)^{\frac{1}{2}}ds.$

Finally, from the estimates (3.4) and (4.16), we obtain

$E\left[\left|C^{n}_{t}-\widetilde{C}^{n}_{t}\right|^{2}\right]\leq
Cn^{-\beta(2\alpha+1)},$

which completes the proof of (4.5).

Proof of (4.6): Rewrite $Z^{n}_{t}$ as
$Z^{n}_{t}=n^{\alpha+\frac{1}{2}}\int^{t}_{0}(t-s)^{\alpha}\Psi^{n}_{s}(X^{n}_{s}-X_{s})\,dW_{s}$.
Then, we can write, using the estimates (3.5) and (4.16),

$\displaystyle E\left[Z^{n}_{t}-\widetilde{Z}^{n}_{t}\right]$
$\displaystyle=n^{2\alpha+1}E\left[\left|\int^{t}_{0}(t-s)^{\alpha}\left(\sigma^{\prime}(X_{s})-\Psi^{n}_{s}\right)\left(X^{n}_{s}-X_{s}\right)\,dW_{s}\right|^{2}\right]$
$\displaystyle\leq
n^{2\alpha+1}\int^{t}_{0}(t-s)^{\alpha}\left(E[(\sigma^{\prime}(X_{s})-\Psi^{n}_{s})^{4}]\right)^{\frac{1}{2}}\left(E[(X^{n}_{s}-X_{s})^{4}]\right)^{\frac{1}{2}}\,ds$
$\displaystyle\leq Cn^{-\beta(2\alpha+1)}.$

This completes the proof of (4.6). $\square$

Let us denote by $\widetilde{Y}^{n}_{t}$ the solution to the following
stochastic integral equation

(4.17)
$\displaystyle\widetilde{Y}^{n}_{t}=\widetilde{A}^{n}_{t}+\widetilde{C}^{n}_{t}+\int_{0}^{t}(t-s)^{\alpha}\sigma^{\prime}(X_{s})\widetilde{Y}^{n}_{s}\,dW_{s}.$

The next proposition tell us that the limit behavior of the normalized error
$Y^{n}_{t}$ is analogous to the limit of $\widetilde{Y}^{n}_{t}$.

######  Proposition 4.2.

Let $\widetilde{Y}^{n}$ be defined as in (4.17). Then, we have

$\lim_{n\rightarrow\infty}\sup_{t\in[0,T]}E[|\widetilde{Y}^{n}_{t}-Y^{n}_{t}|^{2}]=0.$

###### Proof.

By Proposition 4.1, we can write

$n^{\alpha+\frac{1}{2}}Y^{n}_{t}=R^{n}_{t}+\widetilde{A}^{n}_{t}+\widetilde{C}^{n}_{t}+\widetilde{Z}^{n}_{t},$

where $\lim_{n\rightarrow\infty}\sup_{t\in[0,T]}E[|R^{n}_{t}|^{2}]=0$. As a
consequence,

$\widetilde{Y}^{n}_{t}-Y^{n}_{t}=R^{n}_{t}+\int_{0}^{t}(t-s)^{\alpha}\sigma^{\prime}(X_{s})(\widetilde{Y}^{n}_{s}-Y^{n}_{s})\,dW_{s},$

which allow us to conclude easily the proof of the proposition. $\square$

###  4.2. Asymptotic behavior of $\widetilde{A}^{n}_{t}$

The term $\widetilde{A}^{n}_{t}$ is singular in the sense that for any fixed
$t\geq 0$ we cannot guarantee that $\widetilde{A}^{n}_{t}$ converges in
$L^{2}$. However, $\widetilde{A}^{n}_{\eta_{n}(t)}$ does converges in $L^{2}$,
as it is explained in the remark after the proof of the next proposition. The
main result of this section is the following proposition.

######  Proposition 4.3.

The finite-dimensional distributions of the process
$\\{\widetilde{A}^{n}_{\eta_{n}(t)},t\in[0,T]\\}$ converge in distribution, as
$n\to\infty$, to those of $\\{\kappa_{1}Z_{t}\sigma(X_{t}),t\in[0,T]\\}$,
where $\kappa_{1}$ has been defined in (2.1) and $\\{Z_{t},t\in[0,T]\\}$ is a
zero mean Gaussian process independent of $W$ with covariance given by (1.5).

###### Proof.

Let $\psi_{n,1}(s,t)$ be defined as in (4.10) and set

(4.18) $N_{t}^{n,1}=n^{\alpha+\frac{1}{2}}\int_{0}^{t}\psi_{n,1}(s,t)dW_{s}.$

It suffices to show that the finite-dimensional distributions of the process
$\\{N^{n,1}_{\eta_{n}(t)},t\in[0,T]\\}$ converge in distribution to those of
$\\{Z_{t},t\in[0,T]\\}$. Notice that $\\{N^{n,1}_{\eta_{n}(t)},t\in[0,T]\\}$
is a centered Gaussian process. So, we just need to show that the covariance
of $\\{N^{n,1}_{\eta_{n}(t)},t\in[0,T]\\}$ converges to the covariance of
$\\{Z_{t},t\in[0,T]\\}$ and also that the processes
$\\{N^{n,1}_{\eta_{n}(t)},t\in[0,T]\\}$ and $W$ are not correlated.

Let us first compute the variance of $N^{n,1}_{\eta_{n}(t)}$. We have, with
the change of variables $ns=x$ and $x-j=y$,

(4.19) $\displaystyle E[(N^{n,1}_{\eta_{n}(t)})^{2}]$
$\displaystyle=n^{2\alpha+1}\int_{0}^{\eta_{n}(t)}\psi_{n,1}^{2}(s,\eta_{n}(t))ds$
$\displaystyle=n^{2\alpha+1}\sum^{\lfloor
nt\rfloor-1}_{j=0}\int^{\frac{j+1}{n}}_{\frac{j}{n}}\left[\left(\frac{\lfloor
nt\rfloor}{n}-\frac{j}{n}\right)^{\alpha}-\left(\frac{\lfloor
nt\rfloor}{n}-s\right)^{\alpha}\right]^{2}\,ds$ $\displaystyle=\sum^{\lfloor
nt\rfloor-1}_{j=0}\int^{j+1}_{j}\left[(\lfloor nt\rfloor-j)^{\alpha}-(\lfloor
nt\rfloor-x)^{\alpha}\right]^{2}\,dx$ $\displaystyle=\sum^{\lfloor
nt\rfloor-1}_{j=0}\int^{1}_{0}\left[(\lfloor nt\rfloor-j)^{\alpha}-(\lfloor
nt\rfloor-j-y)^{\alpha}\right]^{2}\,dy$ $\displaystyle=\sum^{\lfloor
nt\rfloor}_{k=1}\int^{1}_{0}(k^{\alpha}-(k-y)^{\alpha})^{2}\,dy,$

which converges to $\kappa_{1}^{2}$ as $n\to\infty$, where $\kappa_{1}$ is
defined in (2.1). On the other hand, for any interval $[a,b]$, with $0\leq
a<b\leq T$, we have

$\lim_{n\rightarrow\infty}E[N^{n}_{\eta_{n}(t)}(W_{b}-W_{a})]=0.$

Indeed, with the same computations as above, we can write

$\displaystyle|E[N^{n}_{\eta_{n}(t)}(W_{b}-W_{a})]|$
$\displaystyle=n^{\alpha+\frac{1}{2}}\left|\sum^{\lfloor
nt\rfloor-1}_{j=0}\int_{[\frac{j}{n},\frac{j+1}{n}]\cap[a,b]}\left[\left(\frac{\lfloor
nt\rfloor}{n}-\frac{j}{n}\right)^{\alpha}-\left(\frac{\lfloor
nt\rfloor}{n}-s\right)^{\alpha}\right]\,ds\right|$
$\displaystyle=n^{-\frac{1}{2}}\sum^{\lfloor
nt\rfloor}_{k=1}\int^{1}_{0}|k^{\alpha}-(k-y)^{\alpha}|\,dy$
$\displaystyle\leq Cn^{\max\\{\alpha,0\\}-\frac{1}{2}},$

which converges to zero as $n\to\infty$. Finally, for any $0\leq
t_{1}<t_{2}\leq T$ we have

$\displaystyle|E[N^{n}_{\eta_{n}(t_{2})}N^{n}_{\eta_{n}(t_{1})}]|$
$\displaystyle=n^{2\alpha+1}\left|\int_{0}^{\eta_{n}(t_{1})}\psi_{n,1}(s,\eta_{n}(t_{1}))\psi_{n,1}(s,\eta_{n}(t_{2}))ds\right|$
$\displaystyle\leq n\sum^{\lfloor
nt_{1}\rfloor-1}_{j=0}\int_{\frac{j}{n}}^{\frac{j+1}{n}}|(\lfloor
nt_{2}\rfloor-j)^{\alpha}-(\lfloor nt_{2}\rfloor-ns)^{\alpha}|$
$\displaystyle\qquad\times|(\lfloor nt_{1}\rfloor-j)^{\alpha}-(\lfloor
nt_{1}\rfloor-ns)^{\alpha}|ds.$

The change of variables $ns=x$ and $x-j=y$ allow us to write

$\displaystyle|E[N^{n}_{\eta_{n}(t_{2})}N^{n}_{\eta_{n}(t_{1})}]|$
$\displaystyle\leq\sum^{\lfloor nt_{1}\rfloor-1}_{j=0}\int_{0}^{1}|(\lfloor
nt_{2}\rfloor-j)^{\alpha}-(\lfloor nt_{2}\rfloor-j-y)^{\alpha}|$
$\displaystyle\qquad\times|(\lfloor nt_{1}\rfloor-j)^{\alpha}-(\lfloor
nt_{1}\rfloor-j-y)^{\alpha}|dy$ $\displaystyle=\sum^{\lfloor
nt_{1}\rfloor}_{k=1}\int_{0}^{1}|(\lfloor nt_{2}\rfloor-\lfloor
nt_{1}\rfloor+k)^{\alpha}-(\lfloor nt_{2}\rfloor-\lfloor
nt_{1}\rfloor+k-y)^{\alpha}|$
$\displaystyle\qquad\times|k^{\alpha}-(k-y)^{\alpha}|dy,$

which converges to zero as $n\to\infty$. This completes the proof of the
proposition. $\square$

Remark. Notice that if we replace $\eta_{n}(t)$ by $t$ in the proof of
Proposition 4.3, then instead of (4.19), we obtain

$E[(N^{n,1}_{t})^{2}]=\sum^{\lfloor
nt\rfloor}_{k=0}\int^{1\wedge(nt-k)}_{0}\left[(nt-k)^{\alpha}-(nt-
k-y)^{\alpha}\right]^{2}\,dy,$

and this term does not seem to converge as $n$ tends to infinity due to the
fluctuation of $nt-\lfloor nt\rfloor$ between $0$ and $1$ as $n$ tends to
infinity.

###  4.3. Decomposition of the modified normalized error
$\widetilde{Y}^{n}_{t}$

Because of the particular behavior of $\widetilde{A}^{n}_{t}$, we will
consider the decomposition

$\widetilde{Y}^{n}_{t}=\widetilde{A}^{n}_{t}+\widetilde{Y}^{n,1}_{t}.$

From Proposition 4.2 we deduce that

$\lim_{n\rightarrow\infty}\sup_{t\in[0,T]}E[|\widetilde{Y}^{n,1}_{t}-Y^{n,1}_{t}|^{2}]=0,$

where $Y^{n,1}_{t}$ has been introduced in (1.4).

The process
$\widetilde{Y}^{n,1}_{t}=\widetilde{Y}^{n}_{t}-\widetilde{A}^{n}_{t}$
satisfies the equation

(4.20)
$\widetilde{Y}^{n,1}_{t}=\widetilde{C}^{n}_{t}+\int_{0}^{t}(t-s)^{\alpha}\sigma^{\prime}(X_{s})(\widetilde{Y}^{n,1}_{s}+\widetilde{A}^{n}_{s})\,dW_{s}.$

With the notation

$\displaystyle\widetilde{A}^{n,1}_{t}$
$\displaystyle=\int_{0}^{t}(t-s)^{\alpha}\sigma^{\prime}(X_{s})\widetilde{A}^{n}_{s}\,dW_{s}$
(4.21)
$\displaystyle=n^{\alpha+\frac{1}{2}}\int_{0}^{t}(t-s)^{\alpha}\sigma^{\prime}(X_{s})\sigma(X_{s})\left(\int_{0}^{s}\psi_{n,1}(u,s)dW_{u}\right)dW_{s},$

we can write equation (4.20) as

(4.22)
$\widetilde{Y}^{n,1}_{t}=\widetilde{A}^{n,1}_{t}+\widetilde{C}^{n}_{t}+\int_{0}^{t}(t-s)^{\alpha}\sigma^{\prime}(X_{s})\widetilde{Y}^{n,1}_{s}\,dW_{s}.$

###  4.4. Asymptotic behavior of
$\widetilde{A}^{n,1}_{t}+\widetilde{C}^{n}_{t}$

In this section we will study the limit as $n$ tends to infinity of the random
process $\\{\widetilde{A}^{n,1}_{t}+\widetilde{C}^{n}_{t},t\in[0,T]\\}$ where
$\widetilde{A}^{n,1}_{t}$ has been introduced in (4.21) and
$\widetilde{C}^{n}_{t}$ is defined in (4.8). The main result is the following
proposition.

######  Proposition 4.4.

The finite-dimensional distributions of the process
$\\{\widetilde{A}^{n,1}_{t}+\widetilde{C}^{n}_{t},t\in[0,T]\\}$ converge in
law to those of

$\left\\{\kappa_{2}\int^{t}_{0}(t-s)^{\alpha}(\sigma^{\prime}\sigma)(X_{s})dB_{s},t\in[0,T]\right\\},$

where $B=\\{B_{t},t\in[0,T]\\}$ is a standard Brownian motion independent of
$W$ and $\kappa_{2}$ is the constant defined in (2.2).

###### Proof.

Fix points $t_{1},\dots,t_{Q}\in[0,T]$ and real numbers $\rho_{i}$,$1\leq
i\leq Q$. Define the martingale

$\displaystyle M^{(n)}_{\tau}$
$\displaystyle=\sum_{i=1}^{Q}\rho_{i}\int_{0}^{\tau\wedge
t_{i}}(t_{i}-s)^{\alpha}\sigma^{\prime}(X_{s})\widetilde{A}^{n}_{s}dW_{s}$
$\displaystyle\qquad+n^{\alpha+\frac{1}{2}}\sum_{i=1}^{Q}\rho_{i}\int_{0}^{\tau\wedge
t_{i}}(t_{i}-s)^{\alpha}\sigma^{\prime}(X_{s})(X^{n}_{\eta_{n}(s)}-X^{n}_{s})dW_{s}$
$\displaystyle=n^{\alpha+\frac{1}{2}}\sum_{i=1}^{Q}\rho_{i}\int_{0}^{\tau\wedge
t_{i}}(t_{i}-s)^{\alpha}\sigma^{\prime}(X_{s})\Theta^{n}_{s}dW_{s},$

where

(4.23)
$\Theta^{n}_{s}:=\sigma(X_{s})\left(\int_{0}^{s}\psi_{n,1}(u,s)dW_{u}\right)+X^{n}_{\eta_{n}(s)}-X^{n}_{s}.$

Then, in order to prove the proposition, it suffices to prove that, as $n$
tends to infinity, $M^{(n)}_{\tau}$ converges in distribution to

$\kappa_{2}^{2}\sum_{i=1}^{Q}\rho_{i}\int^{\tau\wedge
t_{i}}_{0}(t_{i}-s)^{\alpha}(\sigma^{\prime}\sigma)(X_{s})dB_{s},$

where $B=\\{B_{t},t\in[0,T]\\}$ is a standard Brownian motion independent of
$W$. By the asymptotic version of Knight’s theorem (as presented in [12,
Chapter XIII, Theorem 2.3]), it suffices to show that the following two
conditions hold:

(C1) As $n$ tends to infinity, the quadratic variation $\langle
M^{(n)}\rangle_{\tau}$ given by

(4.24) $\langle
M^{(n)}\rangle_{\tau}=n^{2\alpha+1}\int_{0}^{\tau}\left|\sum_{i=1}^{Q}\rho_{i}\mathbf{1}_{[0,t_{i}]}(\tau)(t_{i}-s)^{\alpha}\sigma^{\prime}(X_{s})\Theta^{n}_{s}\right|^{2}ds,$

converges in $L^{2}$, uniformly in $\tau\in[0,T]$, to

$\kappa_{2}^{2}\int_{0}^{\tau}\left|\sum_{i=1}^{Q}\rho_{i}\mathbf{1}_{[0,t_{i}]}(\tau)(t_{i}-s)^{\alpha}(\sigma^{\prime}\sigma)(X_{s})\right|^{2}ds.$

(C2) As $n$ tends to infinity, the joint variation

$\langle
M^{(n)},W\rangle_{\tau}=n^{\alpha+\frac{1}{2}}\int_{0}^{\tau}\sum_{i=1}^{Q}\rho_{i}\mathbf{1}_{[0,t_{i}]}(\tau)(t_{i}-s)^{\alpha}\sigma^{\prime}(X_{s})\Theta^{n}_{s}ds$

converges to zero in $L^{2}$, uniformly in $\tau\in[0,T]$.

By linearity, to show properties (C1) and (C2), it suffices to prove that the
following two properties hold true:

(C3) For any $t_{1},t_{2}\in[0,T]$

$\displaystyle\lim_{n\rightarrow\infty}n^{2\alpha+1}\int_{0}^{\tau\wedge
t_{1}\wedge
t_{2}}(t_{1}-s)^{\alpha}(t_{2}-s)^{\alpha}(\sigma^{\prime}(X_{s})\Theta^{n}_{s})^{2}ds$
$\displaystyle\qquad=\kappa_{2}^{2}\int_{0}^{\tau\wedge t_{1}\wedge
t_{2}}(t_{1}-s)^{\alpha}(t_{2}-s)^{\alpha}(\sigma^{\prime}\sigma)^{2}(X_{s})ds,$

where the convergence holds in $L^{2}$, uniformly in $\tau\in[0,T]$.

(C4) For any $t\in[0,T]$

$\lim_{n\rightarrow\infty}n^{\alpha+\frac{1}{2}}\int_{0}^{\tau\wedge
t}(t-s)^{\alpha}\sigma^{\prime}(X_{s})\Theta^{n}_{s}ds=0,$

where the convergence holds in $L^{2}$, uniformly in $\tau\in[0,T]$.

Proof of (C3): We will make use of the notation

(4.25)
$\psi_{n,2}(u,s)=(s-\eta_{n}(u))^{\alpha}-(\eta_{n}(s)-\eta_{n}(u))^{\alpha}.$

Recall that $\Theta^{n}_{s}$ has been defined in (4.23). Then, we can write

$\displaystyle\Theta^{n}_{s}$
$\displaystyle=\sigma(X_{s})\int_{0}^{s}\psi_{n,1}(u,s)dW_{u}+X^{n}_{\eta_{n}(s)}-X^{n}_{s}$
$\displaystyle=\sigma(X_{s})\int_{0}^{s}\psi_{n,1}(u,s)dW_{u}+\int^{s}_{\eta_{n}(s)}\left(s-\eta_{n}(u)\right)^{\alpha}\sigma(X^{n}_{\eta_{n}(u)})\,dW_{u}$
$\displaystyle\qquad+\int^{\eta_{n}(s)}_{0}\psi_{n,2}(u,s)\,\sigma(X^{n}_{\eta_{n}(u)})\,dW_{u}.$

The first step in the proof of (C3) consists on replacing
$\sigma(X^{n}_{\eta_{n}(u)})$ by $\sigma(X_{\eta_{n}(u)})$ in the expression
of $\Theta^{n}_{s}$. That is, if we define

$\displaystyle\widehat{\Theta}^{n}_{s}$
$\displaystyle=\sigma(X_{s})\int_{0}^{s}\psi_{n,1}(u,s)dW_{u}+X^{n}_{\eta_{n}(s)}-X^{n}_{s}$
$\displaystyle=\sigma(X_{s})\int_{0}^{s}\psi_{n,1}(u,s)dW_{u}+\int^{s}_{\eta_{n}(s)}\left(s-\eta_{n}(u)\right)^{\alpha}\sigma(X_{\eta_{n}(u)})\,dW_{u}$
(4.26)
$\displaystyle\qquad+\int^{\eta_{n}(s)}_{0}\psi_{n,2}(u,s)\,\sigma(X_{\eta_{n}(u)})\,dW_{u},$

then

(4.27)
$\lim_{n\rightarrow\infty}\sup_{s\in[0,T]}n^{2\alpha+1}E[|\widehat{\Theta}^{n}_{s}-\Theta^{n}_{s}|^{2}]=0.$

It is easy to check that (4.27) will guarantee that if (C3) holds for
$\widehat{\Theta}^{n}_{s}$, it also holds for $\Theta^{n}_{s}$. The
convergence (4.27) is a consequence of the limits

(4.28)
$\lim_{n\rightarrow\infty}\sup_{s\in[0,T]}n^{2\alpha+1}\int_{\eta_{n}(s)}^{s}(s-\eta_{n}(u))^{2\alpha}E[|\sigma(X^{n}_{\eta_{n}(u)})-\sigma(X_{\eta_{n}(u)})|^{2}]du=0$

and

(4.29)
$\lim_{n\rightarrow\infty}\sup_{s\in[0,T]}n^{2\alpha+1}\int_{0}^{\eta_{n}(s)}\psi_{n,2}^{2}(u,s)E[|\sigma(X^{n}_{\eta_{n}(u)})-\sigma(X_{\eta_{n}(u)})|^{2}]du=0.$

The convergences (4.28) and (4.29) can be proved using the Lipschitz property
of $\sigma$, the estimate (3.5) and Lemma 6.3. Next, we make the following
decomposition of $(\widehat{\Theta}^{n}_{s})^{2}$:

$(\widehat{\Theta}^{n}_{s})^{2}=\sum_{j=1}^{6}\Psi^{n,j}_{s},$

where

$\Psi^{n,1}_{s}=\sigma^{2}(X_{s})\left(\int_{0}^{s}\psi_{n,1}(u,s)dW_{u}\right)^{2},$
$\Psi^{n,2}_{s}=\left(\int^{s}_{\eta_{n}(s)}\left(s-\eta_{n}(u)\right)^{\alpha}\sigma(X_{\eta_{n}(u)})\,dW_{u}\right)^{2},$
$\Psi^{n,3}_{s}=\left(\int^{\eta_{n}(s)}_{0}\psi_{n,2}(u,s)\,\sigma(X_{\eta_{n}(u)})\,dW_{u}\right)^{2},$
$\Psi^{n,4}_{s}=2\sigma(X_{s})\left(\int_{0}^{s}\psi_{n,1}(u,s)dW_{u}\right)\left(\int^{s}_{\eta_{n}(s)}\left(s-\eta_{n}(u)\right)^{\alpha}\sigma(X_{\eta_{n}(u)})\,dW_{u}\right),$
$\Psi^{n,5}_{s}=2\sigma(X_{s})\left(\int_{0}^{s}\psi_{n,1}(u,s)dW_{u}\right)\left(\int^{\eta_{n}(s)}_{0}\psi_{n,2}(u,s)\,\sigma(X_{\eta_{n}(u)})\,dW_{u}\right)$

and

$\Psi^{n,6}_{s}=2\left(\int^{s}_{\eta_{n}(s)}\left(s-\eta_{n}(u)\right)^{\alpha}\sigma(X_{\eta_{n}(u)})\,dW_{u}\right)\left(\int^{\eta_{n}(s)}_{0}\psi_{n,2}(u,s)\,\sigma(X_{\eta_{n}(u)})\,dW_{u}\right).$

Then, condition (C3) will be a consequence of the following limits in $L^{2}$,
uniformly in $\tau\in[0,T]$,

$\displaystyle\lim_{n\rightarrow\infty}n^{2\alpha+1}\int_{0}^{\tau\wedge
t_{1}\wedge
t_{2}}(t_{1}-s)^{\alpha}(t_{2}-s)^{\alpha}(\sigma^{\prime}(X_{s}))^{2}\Psi^{n,1}_{s}ds$
(4.30) $\displaystyle\qquad=\kappa_{3}\int_{0}^{\tau\wedge t_{1}\wedge
t_{2}}(t_{1}-s)^{\alpha}(t_{2}-s)^{\alpha}(\sigma^{\prime}\sigma)^{2}(X_{s})ds,$
$\displaystyle\lim_{n\rightarrow\infty}n^{2\alpha+1}\int_{0}^{\tau\wedge
t_{1}\wedge
t_{2}}(t_{1}-s)^{\alpha}(t_{2}-s)^{\alpha}(\sigma^{\prime}(X_{s}))^{2}\Psi^{n,2}_{s}ds$
(4.31) $\displaystyle\qquad=\frac{1}{2\alpha+2}\int_{0}^{\tau\wedge
t_{1}\wedge
t_{2}}(t_{1}-s)^{\alpha}(t_{2}-s)^{\alpha}(\sigma^{\prime}\sigma)^{2}(X_{s})ds,$
$\displaystyle\lim_{n\rightarrow\infty}n^{2\alpha+1}\int_{0}^{\tau\wedge
t_{1}\wedge
t_{2}}(t_{1}-s)^{\alpha}(t_{2}-s)^{\alpha}(\sigma^{\prime}(X_{s}))^{2}\Psi^{n,3}_{s}ds$
(4.32) $\displaystyle\qquad=\kappa_{4}\int_{0}^{\tau\wedge t_{1}\wedge
t_{2}}(t_{1}-s)^{\alpha}(t_{2}-s)^{\alpha}(\sigma^{\prime}\sigma)^{2}(X_{s})ds,$
$\displaystyle\lim_{n\rightarrow\infty}n^{2\alpha+1}\int_{0}^{\tau\wedge
t_{1}\wedge
t_{2}}(t_{1}-s)^{\alpha}(t_{2}-s)^{\alpha}(\sigma^{\prime}(X_{s}))^{2}\Psi^{n,4}_{s}ds$
(4.33) $\displaystyle\qquad=\frac{\alpha}{2(\alpha+1)^{2}}\int_{0}^{\tau\wedge
t_{1}\wedge
t_{2}}(t_{1}-s)^{\alpha}(t_{2}-s)^{\alpha}(\sigma^{\prime}\sigma)^{2}(X_{s})ds,$
$\displaystyle\lim_{n\rightarrow\infty}n^{2\alpha+1}\int_{0}^{\tau\wedge
t_{1}\wedge
t_{2}}(t_{1}-s)^{\alpha}(t_{2}-s)^{\alpha}(\sigma^{\prime}(X_{s}))^{2}\Psi^{n,5}_{s}ds$
(4.34) $\displaystyle\qquad=\kappa_{5}\int_{0}^{\tau\wedge t_{1}\wedge
t_{2}}(t_{1}-s)^{\alpha}(t_{2}-s)^{\alpha}(\sigma^{\prime}\sigma)^{2}(X_{s})ds,$

and

(4.35) $\lim_{n\rightarrow\infty}n^{2\alpha+1}\int_{0}^{\tau\wedge t_{1}\wedge
t_{2}}(t_{1}-s)^{\alpha}(t_{2}-s)^{\alpha}(\sigma^{\prime}(X_{s}))^{2}\Psi^{n,6}_{s}ds=0,$

where $\kappa_{3}$, $\kappa_{4}$ and $\kappa_{5}$ have been introduced in
(2.3), (2.4) and (2.5), respectively.

Proof of (4.30): To simplify the presentation we write

(4.36) $\gamma_{s}:=\mathbf{1}_{[0,t_{1}\wedge
t_{2}]}(s)(t_{1}-s)^{\alpha}(t_{2}-s)^{\alpha}.$

Then,

$\displaystyle R^{n,1}_{\tau}$
$\displaystyle:=n^{2\alpha+1}\int_{0}^{\tau\wedge t_{1}\wedge
t_{2}}(t_{1}-s)^{\alpha}(t_{2}-s)^{\alpha}(\sigma^{\prime}(X_{s}))^{2}\Psi^{n,1}_{s}ds$
$\displaystyle=n^{2\alpha+1}\int_{0}^{\tau}\gamma_{s}(\sigma^{\prime}\sigma)^{2}(X_{s})\left(\int_{0}^{s}\psi_{n,1}(u,s)dW_{u}\right)^{2}ds$
$\displaystyle=:R^{n,2}_{\tau}+R^{n,3}_{\tau},$

where

$R^{n,2}_{\tau}=n^{2\alpha+1}\int_{0}^{\tau}\gamma_{s}(\sigma^{\prime}\sigma)^{2}(X_{s})\left[\left(\int_{0}^{s}\psi_{n,1}(u,s)dW_{u}\right)^{2}-\int_{0}^{s}\psi^{2}_{n,1}(u,s)du\right]ds,$

and

$R^{n,3}_{\tau}=n^{2\alpha+1}\int_{0}^{\tau}\gamma_{s}(\sigma^{\prime}\sigma)^{2}(X_{s})\int_{0}^{s}\psi^{2}_{n,1}(u,s)duds.$

We claim that

(4.37)
$\lim_{n\rightarrow\infty}\sup_{\tau\in[0,T]}E[|R^{n,2}_{\tau}|^{2}]=0.$

Proof of (4.37): The proof of (4.37) is rather involved and will be done in
two steps. First we will show that we have convergence to zero in $L^{2}$ if
we remove the random factor $(\sigma^{\prime}\sigma)^{2}(X_{s})$ and we
integrate on any fixed interval $[\tau_{1},\tau_{2}]\subset[0,\tau]$.
Secondly, we will use the small blocks / big blocks argument (see, for
instance, [5]) to show that we have convergence to zero when we include the
random weight $(\sigma^{\prime}\sigma)^{2}(X_{s})$.

Step (i): We claim that for any interval $[\tau_{1},\tau_{2}]\subset[0,\tau]$

(4.38)
$R^{n,4}:=n^{2\alpha+1}\int_{\tau_{1}}^{\tau_{2}}\gamma_{s}\left[\left(\int^{s}_{0}\psi_{n,1}(u,s)dW_{u}\right)^{2}-\int^{s}_{0}\psi_{n,1}^{2}(u,s)du\,\right]ds\rightarrow
0,$

where the convergence holds in $L^{2}$, uniformly in $\tau_{1},\tau_{2}$ and
$\tau$. Notice that

$\left(\int^{s}_{0}\psi_{n,1}(u,s)dW_{u}\right)^{2}-\int^{s}_{0}\psi_{n,1}^{2}(u,s)du=I_{2}\left(\psi_{n,1}^{\otimes
2}(\cdot,s)\mathbf{1}_{[0,s]^{2}}\right),$

where $I_{2}$ denotes the double stochastic integral with respect to the
Brownian motion $W$. Therefore,

$\displaystyle E[|R^{n,4}|^{2}]$
$\displaystyle=2n^{4\alpha+2}\int_{\tau_{1}}^{\tau_{2}}\int_{\tau_{1}}^{\tau_{2}}\gamma_{s_{1}}\gamma_{s_{2}}\langle\psi^{\otimes
2}_{n,1}(\cdot,s_{1})\mathbf{1}^{\otimes 2}_{[0,s_{1}]},\psi^{\otimes
2}_{n,1}(\cdot,s_{2})\mathbf{1}^{\otimes
2}_{[0,s_{2}]}\rangle_{L^{2}(\mathbb{R}^{2})}ds_{1}ds_{2}$
$\displaystyle=4n^{4\alpha+2}\int_{\tau_{1}}^{\tau_{2}}\int_{\tau_{1}}^{s_{2}}\gamma_{s_{1}}\gamma_{s_{2}}\left(\int_{0}^{s_{1}}\psi_{n,1}(u,s_{1})\psi_{n,1}(u,s_{2})du\right)^{2}ds_{1}ds_{2}.$

Fix $\delta>0$ such that $\delta>\frac{1}{n}$. On the set
$\\{(s_{1},s_{2}):s_{2}-s_{1}>\delta\\}$ we have, by the Mean Value Theorem,

$|\psi_{n,1}(u,s_{2})|=|(s_{2}-\eta_{n}(u))^{\alpha}-(s_{2}-u)^{\alpha}|\leq\frac{\alpha}{n}\delta^{\alpha-1}.$

As a consequence, using the inequality (6.3), we obtain

$\displaystyle E[|R^{n,4}|^{2}]$ $\displaystyle\leq
Cn^{2\alpha-1}\delta^{2\alpha-2}\int_{\tau_{1}}^{\tau_{2}}\int_{\tau_{1}}^{s_{2}}\gamma_{s_{1}}\gamma_{s_{2}}\mathbf{1}_{\\{s_{2}-s_{1}>\delta\\}}ds_{1}ds_{2}$
$\displaystyle\qquad+C\int_{\tau_{1}}^{\tau_{2}}\int_{\tau_{1}}^{s_{2}}\gamma_{s_{1}}\gamma_{s_{2}}\mathbf{1}_{\\{s_{2}-s_{1}\leq\delta\\}}ds_{1}ds_{2}$
$\displaystyle\leq
Cn^{2\alpha-1}\delta^{2\alpha-2}\left(\int_{0}^{T}\gamma_{s}ds\right)^{2}+C\int_{0}^{T}\int_{0}^{s_{2}}\gamma_{s_{1}}\gamma_{s_{2}}\mathbf{1}_{\\{s_{2}-s_{1}\leq\delta\\}}ds_{1}ds_{2}.$

This implies

$\limsup_{n\rightarrow\infty}\sup_{\tau_{1},\tau_{2}\leq\tau\in[0,T]}E[|R^{n,4}|^{2}]\leq
C\int_{0}^{T}\int_{0}^{s_{2}}\gamma_{s_{1}}\gamma_{s_{2}}\mathbf{1}_{\\{s_{2}-s_{1}\leq\delta\\}}ds_{1}ds_{2}.$

Taking into account that $\delta>0$ is arbitrary, this completes the proof of
(4.38).

Step (ii): Fix an integer $M\geq 1$ and consider a partition
$0=\tau_{0}<\tau_{1}<\cdots<\tau_{M}=\tau$, where $\tau_{i}=\frac{i}{M}\tau$
for $i=0,1,\dots,M$. With the notation

$\Xi^{n,1}_{s}=\left[\left(\int^{s}_{0}\psi_{n,1}(u,s)dW_{u}\right)^{2}-\int^{s}_{0}\psi_{n,1}^{2}(u,s)du\,\right],$

we can write

$\displaystyle R^{n,3}_{\tau}$
$\displaystyle=n^{2\alpha+1}\int^{\tau}_{0}\gamma_{s}(\sigma^{\prime}\sigma)^{2}(X_{s})\Xi^{n,1}_{s}ds$
$\displaystyle=n^{2\alpha+1}\sum_{i=0}^{M-1}\int_{\tau_{i}}^{\tau_{i+1}}\gamma_{s}(\sigma^{\prime}\sigma)^{2}(X_{s})\Xi^{n,1}_{s}ds$
$\displaystyle=:R^{n,M,1}_{\tau}+R^{n,M,2}_{\tau},$

where

$R^{n,M,1}_{\tau}=n^{2\alpha+1}\sum_{i=0}^{M-1}(\sigma^{\prime}\sigma)^{2}(X_{\tau_{i}})\int_{\tau_{i}}^{\tau_{i+1}}\gamma_{s}\Xi^{n,1}_{s}ds$

and

$R^{n,M,2}_{\tau}=n^{2\alpha+1}\sum_{i=0}^{M-1}\int_{\tau_{i}}^{\tau_{i+1}}\gamma_{s}\left[(\sigma^{\prime}\sigma)^{2}(X_{s})-(\sigma^{\prime}\sigma)^{2}(X_{\tau_{i}})\right]\Xi^{n,1}_{s}ds$

From Step 1, we know that for any fixed $M$

(4.39)
$\lim_{n\rightarrow\infty}\sup_{\tau\in[0,T]}E[|R^{n,M,1}_{\tau}|^{2}]=0.$

Indeed, for each $0\leq i\leq M-1$, the term
$n^{2\alpha+1}\int_{\tau_{i}}^{\tau_{i+1}}\gamma_{s}\Xi^{n,1}_{s}ds$ converges
to zero in $L^{2}$ as $n\to\infty$, uniformly in $i=0,\dots,M-1$ and
$\tau\in[0,T]$, and this convergence also holds in $L^{p}$ for any $p\geq 2$,
because $\Xi^{n,1}_{s}$ belongs to the second Wiener chaos.

Assuming that $n\geq M$, the term $R^{n,M,2}_{\tau}$ can be estimated as
follows

$\|R^{n,M,2}_{\tau}\|_{2}\leq
C\sup_{u,v\in[0,1],|u-v|\leq\frac{\tau}{M}}\|(\sigma^{\prime}\sigma)^{2}(X_{u})-(\sigma^{\prime}\sigma)^{2}(X_{v})\|_{4}n^{2\alpha+1}\int_{0}^{\tau}\gamma_{s}\|\Xi^{n,1}_{s}\|_{4}ds.$

From Lemma 6.4 we get

$\sup_{n\geq
1}n^{2\alpha+1}\int_{0}^{T}\gamma_{s}\|\Xi^{n,1}_{s}\|_{4}ds<\infty.$

Therefore,

(4.40) $\lim_{M\rightarrow\infty}\sup_{n\geq
M}\sup_{\tau\in[0,T]}\|R^{n,M,2}_{\tau}\|_{2}=0.$

In conclusion, (4.39) and (4.40) imply (4.37).

It remains to compute the limit as $n$ tends to infinity of the term
$R^{n,3}_{\tau}$. First notice that the term $R^{n,3}_{\tau}$ can be expressed
as follows

$\displaystyle R^{n,3}_{\tau}$
$\displaystyle=n^{2\alpha+1}\int_{0}^{\tau}\gamma_{s}(\sigma^{\prime}\sigma)^{2}(X_{s})\int_{0}^{s}[(s-\eta_{n}(u))^{\alpha}-(s-u)^{\alpha}]^{2}duds$
$\displaystyle=n^{2\alpha+1}\sum_{j=0}^{\lfloor
n\tau\rfloor}\int_{\frac{j}{n}}^{\frac{j+1}{n}\wedge\tau}\gamma_{s}(\sigma^{\prime}\sigma)^{2}(X_{s})\sum_{i=0}^{\lfloor
ns\rfloor}\int_{\frac{i}{n}}^{\frac{i+1}{n}\wedge
s}[(s-\frac{i}{n})^{\alpha}-(s-u)^{\alpha}]^{2}duds.$

The change of variables $nu=v$ and $ns=x$ yields

$R^{n,3}_{\tau}=\frac{1}{n}\sum_{j=0}^{\lfloor
n\tau\rfloor}\int_{j}^{(j+1)\wedge
n\tau}\gamma_{\frac{x}{n}}(\sigma^{\prime}\sigma)^{2}(X_{\frac{x}{n}})\sum_{i=0}^{\lfloor
x\rfloor}\int_{i}^{(i+1)\wedge x}[(x-i)^{\alpha}-(x-v)^{\alpha}]^{2}dvdx.$

With the additional change of variables $x-j=y$ and $v-i=z$, we can write

$\displaystyle R^{n,3}_{\tau}$ $\displaystyle=\frac{1}{n}\sum_{j=0}^{\lfloor
n\tau\rfloor}\int_{0}^{1\wedge(n\tau-j)}\gamma_{\frac{y+j}{n}}(\sigma^{\prime}\sigma)^{2}(X_{\frac{y+j}{n}})$
$\displaystyle\qquad\times\sum_{i=0}^{\lfloor
y+j\rfloor}\int_{0}^{1\wedge(y+j-i)}[(y+j-i)^{\alpha}-(y+j-i-z)^{\alpha}]^{2}dzdy$
$\displaystyle=\frac{1}{n}\sum_{j=0}^{\lfloor
n\tau\rfloor}\int_{0}^{1\wedge(n\tau-j)}\gamma_{\frac{y+j}{n}}(\sigma^{\prime}\sigma)^{2}(X_{\frac{y+j}{n}})\sum_{k=j-\lfloor
y+j\rfloor}^{j}\int_{0}^{1\wedge(y+k)}[(y+k)^{\alpha}-(y+k-z)^{\alpha}]^{2}dzdy.$

Because $y\leq 1$, we have that $\lfloor y+j\rfloor=j$. Therefore, we obtain

$\displaystyle\lim_{n\rightarrow\infty}R^{n,3}_{\tau}$
$\displaystyle=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{j=0}^{\lfloor
n\tau\rfloor}\gamma_{\frac{j}{n}}(\sigma^{\prime}\sigma)^{2}(X_{\frac{j}{n}})\int_{0}^{1}\sum_{k=0}^{j}\int_{0}^{1\wedge(y+k)}[(y+k)^{\alpha}-(y+k-z)^{\alpha}]^{2}dzdy$
$\displaystyle=\left(\sum_{k=0}^{\infty}\int_{0}^{1}\int_{0}^{1\wedge(y+k)}[(y+k)^{\alpha}-(y+k-z)^{\alpha}]^{2}dzdy\right)$
$\displaystyle\qquad\times\int_{0}^{\tau\wedge t_{1}\wedge
t_{2}}(t_{1}-s)^{\alpha}(t_{2}-s)^{\alpha}(\sigma^{\prime}\sigma)^{2}(X_{s})ds,$

where the convergence is in $L^{2}$, uniformly in $\tau\in[0,T]$. This
completes the proof of (4.30).

Proof of (4.31): With the notation (4.36), we have

$\displaystyle Q^{n}_{\tau}$ $\displaystyle:=n^{2\alpha+1}\int_{0}^{\tau\wedge
t_{1}\wedge
t_{2}}(t_{1}-s)^{\alpha}(t_{2}-s)^{\alpha}(\sigma^{\prime}(X_{s}))^{2}\Psi^{n,2}_{s}ds$
$\displaystyle=n^{2\alpha+1}\int_{0}^{\tau}\gamma_{s}(\sigma^{\prime}(X_{s}))^{2}\left(\int^{s}_{\eta_{n}(s)}\left(s-\eta_{n}(u)\right)^{\alpha}\sigma(X_{\eta_{n}(u)})\,dW_{u}\right)^{2}ds$
$\displaystyle=n^{2\alpha+1}\int^{\tau}_{0}\gamma_{s}(s-\eta_{n}(s))^{2\alpha}\left(\sigma^{\prime}(X_{s})\right)^{2}\sigma^{2}(X_{\eta_{n}(s)})\,\left(W_{s}-W_{\eta_{n}(s)}\right)^{2}\,ds$
$\displaystyle=n^{2\alpha+1}\int^{\tau}_{0}\gamma_{s}(s-\eta_{n}(s))^{2\alpha}\left(\sigma^{\prime}(X_{s})\right)^{2}\sigma^{2}(X_{\eta_{n}(s)})\,\left[\left(W_{s}-W_{\eta_{n}(s)}\right)^{2}-(s-\eta_{n}(s))\right]\,ds$
$\displaystyle\qquad+n^{2\alpha+1}\int^{\tau}_{0}\gamma_{s}(s-\eta_{n}(s))^{2\alpha+1}\left(\sigma^{\prime}(X_{s})\right)^{2}\,\sigma^{2}(X_{\eta_{n}(s)})ds$
$\displaystyle=:Q^{n,1}_{\tau}+Q^{n,2}_{\tau}.$

We claim that

(4.41)
$\lim_{n\rightarrow\infty}\sup_{\tau\in[0,T]}E\left[|Q^{n,1}_{\tau}|^{2}\right]=0.$

In order to show the convergence (4.41), we first replace
$\sigma^{\prime}(X_{s})$ by $\sigma^{\prime}(X_{\eta_{n}(s)})$ in
$Q^{n,1}_{\tau}$ and then use the fact that we are dealing with a discrete
martingale sequence. That is, we make the decomposition

$Q^{n,1}_{\tau}=:Q^{n,3}_{\tau}+Q^{n,4}_{\tau},$

where

$\displaystyle Q^{n,3}_{\tau}$
$\displaystyle=n^{2\alpha+1}\int^{\tau}_{0}\gamma_{s}(s-\eta_{n}(s))^{2\alpha}\left((\sigma^{\prime}(X_{s}))^{2}-(\sigma^{\prime}(X_{\eta_{n}(s)}))^{2}\right)\sigma^{2}(X_{\eta_{n}(s)})\,$
$\displaystyle\qquad\times\left[\left(W_{s}-W_{\eta_{n}(s)}\right)^{2}-(s-\eta_{n}(s))\right]\,ds$

and

$Q^{n.4}_{\tau}=n^{2\alpha+1}\int^{\tau}_{0}\gamma_{s}(s-\eta_{n}(s))^{2\alpha}(\sigma^{\prime}\sigma)^{2}(X_{\eta_{n}(s)})\,\left[\left(W_{s}-W_{\eta_{n}(s)}\right)^{2}-(s-\eta_{n}(s))\right]\,ds.$

Then, using Minkowski’s inequality, we obtain

$\displaystyle\|Q^{n,3}_{\tau}\|_{2}$ $\displaystyle\leq
n^{2\alpha+1}\int^{\tau}_{0}\gamma_{s}(s-\eta_{n}(s))^{2\alpha}\|\left((\sigma^{\prime}(X_{s}))^{2}-(\sigma^{\prime}(X_{\eta_{n}(s)}))^{2}\right)\sigma^{2}(X_{\eta_{n}(s)})\|_{4}$
$\displaystyle\qquad\times\|[(W_{s}-W_{\eta_{n}(s)})^{2}-(s-\eta_{n}(s))]\|_{4}ds.$

Because $\sigma^{\prime}$ is $\beta$-Hölder continuous and $\sigma$ has linear
growth, we can write using the estimates (3.1) and (3.2)

$\|Q^{n,3}_{\tau}\|_{2}\leq
Cn^{2\alpha+1-\beta(\alpha+\frac{1}{2})}\int^{\tau}_{0}\gamma_{s}(s-\eta_{n}(s))^{2\alpha+1}ds\leq
Cn^{-\beta(\alpha+\frac{1}{2})},$

which shows that

(4.42)
$\lim_{n\rightarrow\infty}\sup_{\tau\in[0,T]}E\left[|Q^{n,3}_{\tau}|^{2}\right]=0.$

To handle the term $Q^{n,4}_{\tau}$ we write

$Q^{n,4}_{\tau}=n^{2\alpha+1}\sum_{j=0}^{\lfloor n\tau\rfloor}\xi_{j,n},$

where

$\xi_{j,n}=\int_{\frac{j}{n}}^{\frac{j+1}{n}\wedge\tau}\gamma_{s}(s-\frac{j}{n})^{2\alpha}(\sigma^{\prime}\sigma)^{2}(X_{\frac{j}{n}})\left[(W_{s}-W_{\frac{j}{n}})^{2}-(s-\frac{j}{n})\right]ds.$

The random variables $\xi_{j,n}$ for $j=0,\dots,\lfloor n\tau\rfloor$, are
$\mathcal{F}_{\frac{j+1}{n}}$-measurable and satisfy
$E[\xi_{j,n}|\mathcal{F}_{\frac{j}{n}}]=0$. That means, they form a discrete
martingale sequence. As a consequence, for $j\not=k$ we have
$E[\xi_{j,n}\xi_{k,n}]=0$ and this implies

(4.43) $E[|Q^{n,4}_{\tau}|^{2}]=n^{4\alpha+2}\sum_{j=0}^{\lfloor
n\tau\rfloor}E[\xi_{j,n}^{2}].$

By Minkowski’s inequality

(4.44) $\|\xi_{j,n}\|_{2}\leq
C\int_{\frac{j}{n}}^{\frac{j+1}{n}\wedge\tau}\gamma_{s}(s-\frac{j}{n})^{2\alpha+1}ds\leq
Cn^{-2\alpha-1}\int_{\frac{j}{n}}^{\frac{j+1}{n}\wedge\tau}\gamma_{s}ds.$

Substituting (4.44) into (4.43), we obtain

$E[|Q^{n,4}_{\tau}|^{2}]\leq C\sum_{j=0}^{\lfloor
n\tau\rfloor}\left(\int_{\frac{j}{n}}^{\frac{j+1}{n}\wedge\tau}\gamma_{s}ds\right)^{2}\leq
C\sup_{0\leq j\leq\lfloor
n\tau\rfloor}\left(\int_{\frac{j}{n}}^{\frac{j+1}{n}\wedge\tau}\gamma_{s}ds\right)\int_{0}^{\tau}\gamma_{s}ds,$

which implies

(4.45)
$\lim_{n\rightarrow\infty}\sup_{\tau\in[0,T]}E\left[|Q^{n,4}_{\tau}|^{2}\right]=0.$

Therefore, (4.42) and (4.45) imply (4.41).

In order to find the limit of the term $Q^{n,2}_{\tau}$, we first make the
following decomposition

$Q^{n,2}_{\tau}=Q^{n,5}_{\tau}+Q^{n,6}_{\tau},$

where

$Q^{n,5}_{\tau}=n^{2\alpha+1}\int^{\tau}_{0}\gamma_{s}(s-\eta_{n}(s))^{2\alpha+1}[(\sigma^{\prime}(X_{s}))^{2}-(\sigma^{\prime}(X_{\eta_{n}(s)}))^{2}]\,\sigma^{2}(X_{\eta_{n}(s)})ds,$

and

$Q^{n,6}_{\tau}=n^{2\alpha+1}\int^{\tau}_{0}\gamma_{s}(s-\eta_{n}(s))^{2\alpha+1}(\sigma^{\prime}\sigma)^{2}(X_{\eta_{n}(s)})ds.$

The $\beta$-Hölder continuity of $\sigma^{\prime}$ together with the estimates
(3.1) and (3.2) imply

(4.46)
$\lim_{n\rightarrow\infty}\sup_{\tau\in[0,T]}E\left[|Q^{n,5}_{\tau}|^{2}\right]=0.$

Then, we make a further decomposition of the term $Q^{6,n}_{\tau}$ as follows

$Q^{n,6}_{\tau}=Q^{n,7}_{\tau}+Q^{n,8}_{\tau},$

where

$Q^{n,7}_{\tau}=n^{2\alpha+1}\int^{\tau}_{0}[\gamma_{s}-\gamma_{\eta_{n}(s)}](s-\eta_{n}(s))^{2\alpha+1}(\sigma^{\prime}\sigma)^{2}(X_{\eta_{n}(s)})ds,$

and

$Q^{n,8}_{\tau}=n^{2\alpha+1}\int^{\tau}_{0}\gamma_{\eta_{n}(s)}(s-\eta_{n}(s))^{2\alpha+1}(\sigma^{\prime}\sigma)^{2}(X_{\eta_{n}(s)})ds.$

We have, using (6.3)

$\|Q^{n,7}_{\tau}\|_{2}\leq C\int^{t_{1}\wedge
t_{2}}_{0}|(t_{1}-s)^{\alpha}(t_{2}-s)^{\alpha}-(t_{1}-\eta_{n}(s))^{\alpha}(t_{2}-\eta_{n}(s))^{\alpha}|ds,$

which implies

(4.47) $\lim_{n\rightarrow\infty}\sup_{\tau\in[0,T]}\|Q^{n,7}_{\tau}\|_{2}=0.$

Moreover, the term $Q^{n,8}_{\tau}$ can be expressed as follows

$\displaystyle Q^{n,8}_{\tau}$ $\displaystyle=n^{2\alpha+1}\sum_{j=0}^{\lfloor
n\tau\rfloor}\gamma(\frac{j}{n})(\sigma^{\prime}\sigma)^{2}(X_{\frac{j}{n}})\int_{\frac{j}{n}}^{\frac{j+1}{n}\wedge\tau}(s-\frac{j}{n})^{2\alpha+1}ds$
$\displaystyle=\frac{n^{2\alpha+1}}{2\alpha+2}\sum_{j=0}^{\lfloor
n\tau\rfloor}\gamma(\frac{j}{n})(\sigma^{\prime}\sigma)^{2}(X_{\frac{j}{n}})\left(\frac{j+1}{n}\wedge\tau-\frac{j}{n}\right)^{2\alpha+2}$
$\displaystyle=\frac{1}{(2\alpha+2)n}\sum_{j=0}^{\lfloor
n\tau\rfloor-1}\gamma(\frac{j}{n})(\sigma^{\prime}\sigma)^{2}(X_{\frac{j}{n}})$
$\displaystyle\qquad+\frac{n^{2\alpha+1}}{2\alpha+2}(\sigma^{\prime}\sigma)^{2}(X_{\frac{\lfloor
n\tau\rfloor}{n}})\gamma(\frac{\lfloor
n\tau\rfloor}{n})\left(\tau-\frac{\lfloor
n\tau\rfloor}{n}\right)^{2\alpha+2}.$

Notice that

$\gamma(\frac{\lfloor n\tau\rfloor}{n})\left(\tau-\frac{\lfloor
n\tau\rfloor}{n}\right)^{2\alpha+2}=\mathbf{1}_{\\{\frac{\lfloor
n\tau\rfloor}{n}\leq t_{1}\wedge t_{2}\\}}\left(t_{1}-\frac{\lfloor
n\tau\rfloor}{n}\right)^{\alpha}\left(t_{2}-\frac{\lfloor
n\tau\rfloor}{n}\right)^{\alpha}\left(\tau-\frac{\lfloor
n\tau\rfloor}{n}\right)^{2\alpha+2}.$

If $n$ is large enough, this quantity vanishes if $\tau>t_{1}\wedge t_{2}$, is
bounded by $Cn^{-2\alpha-2}$ if $\tau<t_{1}\wedge t_{2}$, is bounded by
$n^{-\alpha-2}$ if $\tau=t_{1}<t_{2}$ and, finally it is bounded by $n^{-2}$
if $\tau=t_{1}=t_{2}$. As a consequence,

$\displaystyle\lim_{n\rightarrow\infty}Q^{n,8}_{\tau}$
$\displaystyle=\frac{1}{2\alpha+2}\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{j=0}^{\lfloor
n\tau\rfloor-1}\gamma(\frac{j}{n})(\sigma^{\prime}\sigma)^{2}(X_{\frac{j}{n}})$
$\displaystyle=\frac{1}{2\alpha+2}\lim_{n\rightarrow\infty}\int_{0}^{\frac{\lfloor
n\tau\rfloor}{n}}\gamma(\eta_{n}(s))(\sigma^{\prime}\sigma)^{2}(X_{\eta_{n}(s)})ds$
(4.48) $\displaystyle=\frac{1}{2\alpha+2}\int_{0}^{\tau\wedge t_{1}\wedge
t_{2}}(t_{1}-s)^{\alpha}(t_{2}-s)^{\alpha}(\sigma^{\prime}\sigma)^{2}(X_{s})ds,$

where the convergence is in $L^{2}$, uniformly in $\tau\in[0,T]$. The limit in
the last equality of the above equation can be proved by arguments similar to
those used before based on the estimate (6.3) and the Hölder continuity of
$\sigma^{\prime}$. From (4.46), (4.47) and (4.48) we deduce that

$\lim_{n\rightarrow\infty}Q^{n,2}_{\tau}=\frac{1}{2\alpha+2}\int_{0}^{\tau\wedge
t_{1}\wedge
t_{2}}(t_{1}-s)^{\alpha}(t_{2}-s)^{\alpha}(\sigma^{\prime}\sigma)^{2}(X_{s})ds,$

where the convergence is in $L^{2}$, uniformly in $\tau\in[0,T]$. Together
with (4.41), this completes the proof of (4.31).

Proof of (4.32): We have

$N^{n}_{\tau}:=n^{2\alpha+1}\int^{\tau}_{0}\gamma_{s}\left(\sigma^{\prime}(X_{s})\right)^{2}\left(\int^{\eta_{n}(s)}_{0}\psi_{n,2}(u,s)\sigma(X_{\eta_{n}(u)})\,dW_{u}\right)^{2}\,ds.$

First we are going to show that

$N^{n}_{\tau}=N^{n,1}_{\tau}+N^{n,2}_{\tau},$

where

$N^{n,1}_{\tau}=n^{2\alpha+1}\int^{\tau}_{0}\gamma_{s}(\sigma^{\prime}\sigma)^{2}(X_{\eta_{n}(s)})\left(\int^{\eta_{n}(s)}_{0}\psi_{n,2}(u,s)dW_{u}\right)^{2}\,ds$

and

(4.49) $\lim_{n\rightarrow\infty}\sup_{\tau\in[0,T}E[|N^{n,2}_{\tau}|^{2}]=0.$

Because $\sigma(X_{\eta_{n}(u)})$ lies inside the stochastic integral with
respect to $dW_{u}$ and $\sigma(X_{\eta_{n}(s)})$ is not adapted to
$\mathcal{F}_{u}$, to replace the term $\sigma(X_{\eta_{n}(u)})$ by
$\sigma(X_{\eta_{n}(s)})$ requires a delicate argument. The idea is to show
that if we consider the stochastic integral in an interval of the form
$[0,\eta_{n}(s)-\delta]$, where $\delta>0$, then the corresponding term tends
to zero in $L^{1}$ and the integral over $[\eta_{n}(s)-\delta,\eta_{n}(s)]$
can be made arbitrarily small in $L^{2}$-norm if $\delta$ is small.

Fix $\delta>0$. By Lemma 6.3 and Burkholder and Minkowski’s inequalities, we
obtain

$\displaystyle
n^{2\alpha+1}\left\|\int_{0}^{(\delta+\frac{1}{n})\wedge\tau}\gamma_{s}\left(\sigma^{\prime}(X_{s})\right)^{2}\left(\int^{\eta_{n}(s)}_{0}\psi_{n,2}(u,s)\sigma(X_{\eta_{n}(u)})\,dW_{u}\right)^{2}\,ds\right\|_{2}$
(4.50) $\displaystyle\leq
Cn^{2\alpha+1}\int_{0}^{(\delta+\frac{1}{n})\wedge\tau}\gamma_{s}\int^{\eta_{n}(s)}_{0}\psi^{2}_{n,2}(u,s)duds\leq
C(\delta+\frac{1}{n})^{2\alpha+1}.$

Define

$\displaystyle K^{n,1}_{\tau}$
$\displaystyle=n^{2\alpha+1}\int^{\tau}_{(\delta+\frac{1}{n})\wedge\tau}\gamma_{s}\left(\sigma^{\prime}(X_{s})\right)^{2}$
$\displaystyle\times\left[\left(\int_{0}^{\eta_{n}(s)}\psi_{n,2}(u,s)\sigma(X_{\eta_{n}(u)})\,dW_{u}\right)^{2}-\left(\int_{\eta_{n}(s)-\delta}^{\eta_{n}(s)}\psi_{n,2}(u,s)\sigma(X_{\eta_{n}(u)})\,dW_{u}\right)^{2}\right]\,ds.$

Then, using Burkholder’s inequality we can estimate the $L^{2}$-norm of
$K^{n,1}_{\tau}$ as follows

$\displaystyle\|K^{n,1}_{\tau}\|_{2}$ $\displaystyle\leq
Cn^{2\alpha+1}\int^{\tau}_{(\delta+\frac{1}{n})\wedge\tau}\gamma_{s}\left\|\int_{0}^{\eta_{n}(s)-\delta}\psi_{n,2}(u,s)\sigma(X_{\eta_{n}(u)})\,dW_{u}\right\|_{4}$
$\displaystyle\qquad\times\left\|\int_{0}^{\eta_{n}(s)}\psi_{n,2}(u,s)\sigma(X_{\eta_{n}(u)})dW_{u}+\int_{\eta_{n}(s)-\delta}^{\eta_{n}(s)}\psi_{n,2}(u,s)\sigma(X_{\eta_{n}(u)})dW_{u}\right\|_{4}ds$
$\displaystyle\leq
Cn^{2\alpha+1}\int^{\tau}_{(\delta+\frac{1}{n})\wedge\tau}\gamma_{s}\left(\int_{0}^{\eta_{n}(s)-\delta}\psi_{n,2}^{2}(u,s)du\right)^{\frac{1}{2}}\left(\int_{0}^{\eta_{n}(s)}\psi_{n,2}^{2}(u,s)du\right)^{\frac{1}{2}}ds$
$\displaystyle\leq
Cn^{\alpha+\frac{1}{2}}\int^{\tau}_{(\delta+\frac{1}{n})\wedge\tau}\gamma_{s}\left(\int_{0}^{\eta_{n}(s)-\delta}\psi_{n,2}^{2}(u,s)du\right)^{\frac{1}{2}}ds,$

where in the last inequality we have used Lemma 6.3. Finally, applying the
Mean Value Theorem, we obtain

$\displaystyle\|K^{n,1}_{\tau}\|_{2}$ $\displaystyle\leq
Cn^{\alpha+\frac{1}{2}}\int^{\tau}_{(\delta+\frac{1}{n})\wedge\tau}\gamma_{s}\left(\int^{\eta_{n}(s)-\delta}_{0}(s-\eta_{n}(s))^{2}((\eta_{n}(s)-\eta_{n}(u))^{2\alpha-2}du\right)^{\frac{1}{2}}ds$
(4.51) $\displaystyle\leq C\delta^{\alpha-1}n^{\alpha-\frac{1}{2}}.$

Consider now the term

$\displaystyle K^{n,2}_{\tau}$
$\displaystyle=n^{2\alpha+1}\int^{\tau}_{(\delta+\frac{1}{n})\wedge\tau}\gamma_{s}\left(\sigma^{\prime}(X_{s})\right)^{2}$
$\displaystyle\times\left[\left(\int_{\eta_{n}(s)-\delta}^{\eta_{n}(s)}\psi_{n,2}(u,s)\sigma(X_{\eta_{n}(u)})\,dW_{u}\right)^{2}-\left(\int_{\eta_{n}(s)-\delta}^{\eta_{n}(s)}\psi_{n,2}(u,s)\sigma(X_{\eta_{n}(s)-\delta})\,dW_{u}\right)^{2}\right]\,ds.$

We can estimate the $L^{2}$-norm of $K^{n,2}_{\tau}$ as follows

$\displaystyle\|K^{n,2}_{\tau}\|_{2}$ $\displaystyle\leq
n^{2\alpha+1}\int^{\tau}_{(\delta+\frac{1}{n})\wedge\tau}\gamma_{s}\left\|\int_{\eta_{n}(s)-\delta}^{\eta_{n}(s)}\psi_{n,2}(u,s)[\sigma(X_{\eta_{n}(u)})-\sigma(X_{\eta_{n}(s)-\delta})]\,dW_{u}\right\|_{4}$
$\displaystyle\times\left\|\int_{\eta_{n}(s)-\delta}^{\eta_{n}(s)}\psi_{n,2}(u,s)[\sigma(X_{\eta_{n}(u)})+\sigma(X_{\eta_{n}(s)-\delta})]\,dW_{u}\right\|_{4}ds$
(4.52) $\displaystyle\leq
Cn^{2\alpha+1}\delta^{\alpha+\frac{1}{2}}\int^{T}_{0}\gamma_{s}\int_{0}^{\eta_{n}(s)}\psi_{n,2}^{2}(u,s)duds\leq
C\delta^{\alpha+\frac{1}{2}},$

where in the first inequality we have used the Lipschitz property of $\sigma$,
inequality (3.2), together with Burkholder’s inequality for the stochastic
integral and in the second inequality we have used Lemma 6.3. Define now

$\displaystyle K^{n,3}_{\tau}$
$\displaystyle=n^{2\alpha+1}\int^{\tau}_{(\delta+\frac{1}{n})\wedge\tau}\gamma_{s}[(\sigma^{\prime}(X_{s})\sigma(X_{\eta_{n}(s)-\delta}))^{2}-(\sigma^{\prime}\sigma)^{2}(X_{\eta_{n}(s)})]\left(\int_{\eta_{n}(s)-\delta}^{\eta_{n}(s)}\psi_{n,2}(u,s)dW_{u}\right)^{2}ds$

By the same arguments as before, we can write

(4.53) $\|K^{n,3}_{\tau}\|_{2}\leq
C\left(n^{-\beta(\alpha+\frac{1}{2})}+\delta^{\alpha+\frac{1}{2}}\right).$

The term

$\displaystyle K^{n,4}_{\tau}$
$\displaystyle:=n^{2\alpha+1}\int^{\tau}_{(\delta+\frac{1}{n})\wedge\tau}\gamma_{s}(\sigma^{\prime}\sigma)^{2}(X_{\eta_{n}(s)})$
$\displaystyle\qquad\times\left[\left(\int_{\eta_{n}(s)-\delta}^{\eta_{n}(s)}\psi_{n,2}(u,s)dW_{u}\right)^{2}-\left(\int_{0}^{\eta_{n}(s)}\psi_{n,2}(u,s)dW_{u}\right)^{2}\right]ds$

can be handled as the term $K^{n,1}_{\tau}$ and we get

(4.54) $\|K^{n,4}_{\tau}\|_{2}\leq C\delta^{\alpha-1}n^{\alpha-\frac{1}{2}}.$

Finally, as in the proof of (4.50), we can write

(4.55)
$n^{2\alpha+1}\left\|\int_{0}^{(\delta+\frac{1}{n})\wedge\tau}\gamma_{s}(\sigma^{\prime}\sigma)^{2}(X_{\eta_{n}(s)})\left(\int_{0}^{\eta_{n}(s)}\psi_{n,2}(u,s)dW_{u}\right)^{2}ds\right\|_{2}\leq
C(\delta+\frac{1}{n})^{2\alpha+1}.$

In conclusion, form estimates (4.50), (4.51), (4.52), (4.53), (4.54) and
(4.55), by taking first the limit as $n$ tends to infinity and later taking
$\delta\downarrow 0$, we obtain (4.49).

Next we will make a decomposition of the term $N^{n,1}_{\tau}$ into two parts:

$N^{n,1}_{\tau}=N^{n,3}_{\tau}+N^{n,4}_{\tau},$

where

$N^{n,3}_{\tau}=n^{2\alpha+1}\int^{\tau}_{0}\gamma_{s}(\sigma^{\prime}\sigma)^{2}(X_{\eta_{n}(s)}))\left[\left(\int^{\eta_{n}(s)}_{0}\psi_{n,2}(u,s)dW_{u}\right)^{2}-\int^{\eta_{n}(s)}_{0}\psi_{n,2}^{2}(u,s)du\,\right]ds$

and

$N^{n,4}_{\tau}=n^{2\alpha+1}\int^{\tau}_{0}\gamma_{s}(\sigma^{\prime}\sigma)^{2}(X_{\eta_{n}(s)})\int^{\eta_{n}(s)}_{0}\psi_{n,2}^{2}(u,s)du\,ds.$

Using the same arguments as in the proof of (4.37) we can show that

(4.56)
$\lim_{n\rightarrow\infty}\sup_{\tau\in[0,T]}E[|N^{n,3}_{\tau}|^{2}]=0.$

It only remains to show that $N^{n,4}_{\tau}$ converges in $L^{2}$, uniformly
in $\tau\in[0,T]$, as $n\to\infty$ to

$\kappa_{3}\int_{0}^{\tau\wedge t_{1}\wedge
t_{2}}(t_{1}-s)^{\alpha}(t_{2}-s)^{\alpha}(\sigma^{\prime}\sigma)^{2}(X_{s})ds,$

where

$\kappa_{3}=\sum_{k=1}^{\infty}\int_{0}^{1}[(x+k)^{\alpha}-k^{\alpha}]^{2}dx.$

We can write

$\displaystyle N^{n,4}_{\tau}$ $\displaystyle=n^{2\alpha+1}\sum_{j=0}^{\lfloor
n\tau\rfloor}\int_{\frac{j}{n}}^{\frac{j+1}{n}\wedge\tau}\gamma_{s}(\sigma^{\prime}\sigma)^{2}(X_{\frac{j}{n}})\int_{0}^{\frac{j}{n}}[(s-\eta_{n}(u))^{\alpha}-(\frac{j}{n}-\eta_{n}(u))^{\alpha}]^{2}duds$
$\displaystyle=n^{2\alpha}\sum_{j=0}^{\lfloor
n\tau\rfloor}(\sigma^{\prime}\sigma)^{2}(X_{\frac{j}{n}})\int_{\frac{j}{n}}^{\frac{j+1}{n}\wedge\tau}\gamma_{s}\sum_{i=0}^{j-1}[(s-\frac{i}{n})^{\alpha}-(\frac{j-i}{n})^{\alpha}]^{2}ds.$

Making the change of indices $j-i=k$ and the change of variables
$s=\frac{y}{n}$ and $y-j=x$, we obtain

$\displaystyle N^{n,4}_{\tau}$ $\displaystyle=n^{2\alpha}\sum_{j=0}^{\lfloor
n\tau\rfloor}(\sigma^{\prime}\sigma)^{2}(X_{\frac{j}{n}})\int_{\frac{j}{n}}^{\frac{j+1}{n}\wedge\tau}\gamma_{s}\sum_{k=1}^{j}[(s-\frac{j}{n}+\frac{k}{n})^{\alpha}-(\frac{k}{n})^{\alpha}]^{2}ds$
$\displaystyle=n^{-1}\sum_{j=0}^{\lfloor
n\tau\rfloor}(\sigma^{\prime}\sigma)^{2}(X_{\frac{j}{n}})\int_{j}^{(j+1)\wedge
n\tau}\gamma_{\frac{y}{n}}\sum_{k=1}^{j}[(y-j+k)^{\alpha}-k^{\alpha}]^{2}dy$
$\displaystyle=n^{-1}\sum_{j=0}^{\lfloor
n\tau\rfloor}(\sigma^{\prime}\sigma)^{2}(X_{\frac{j}{n}})\int_{0}^{1\wedge(n\tau-j)}\gamma_{\frac{x+j}{n}}\sum_{k=1}^{j}[(x+k)^{\alpha}-k^{\alpha}]^{2}dx.$

We have
$\lim_{n\rightarrow\infty}\sup_{\tau\in[0,T]}E[|N^{n,5}_{\tau}|^{2}]=0$, where

$N^{n,5}_{\tau}:=\frac{1}{n}\sum_{j=0}^{\lfloor
n\tau\rfloor}(\sigma^{\prime}\sigma)^{2}(X_{\frac{j}{n}})\int_{0}^{1\wedge(n\tau-j)}\gamma_{\frac{x+j}{n}}\sum_{k=j+1}^{\infty}[(x+k)^{\alpha}-k^{\alpha}]^{2}dx=0.$

Indeed, using the Mean Value Theorem, we can write

$\|N^{n,5}_{\tau}\|_{2}\leq\alpha^{2}\sup_{0\leq
s\leq\tau}\|(\sigma^{\prime}\sigma)^{2}(X_{s})\|_{2}\frac{1}{n}\sum_{j=0}^{\lfloor
n\tau\rfloor}\int_{0}^{1\wedge(n\tau-j)}\gamma_{\frac{x+j}{n}}dx\sum_{k=j+1}^{\infty}k^{2\alpha-2}.$

Taking into account that $\int_{0}^{1\wedge(n\tau-j)}\gamma_{\frac{x+j}{n}}dx$
is bounded by $\frac{2}{2\alpha+1}(t_{1}\vee t_{2})^{2\alpha+1}$, we deduce
that

$\|N^{n,5}_{\tau}\|_{2}\leq C\frac{1}{n}\sum_{j=0}^{\lfloor
nT\rfloor}\sum_{k=j+1}^{\infty}k^{2\alpha-2}\to 0,$

which converges to zero a $n\to\infty$. As a consequence,

$\displaystyle\lim_{n\rightarrow\infty}N^{n,4}_{\tau}$
$\displaystyle=\lim_{n\rightarrow\infty}n^{-1}\sum_{j=0}^{\lfloor
n\tau\rfloor}(\sigma^{\prime}\sigma)^{2}(X_{\frac{j}{n}})\int_{0}^{1\wedge(n\tau-j)}\gamma_{\frac{x+j}{n}}\sum_{k=1}^{\infty}[(x+k)^{\alpha}-k^{\alpha}]^{2}dx$
$\displaystyle=\left(\sum_{k=1}^{\infty}\int_{0}^{1}[(x+k)^{\alpha}-k^{\alpha}]^{2}dx\right)\int_{0}^{\tau\wedge
t_{1}\wedge
t_{2}}(t_{1}-s)^{\alpha}(t_{2}-s)^{\alpha}(\sigma^{\prime}\sigma)^{2}(X_{s})ds,$

where the convergence holds in $L^{2}$, uniformly in $\tau\in[0,T]$. This
implies the desired result.

Proof of (4.33): Set

$\displaystyle L^{n}_{\tau}$
$\displaystyle:=n^{2\alpha+1}\int^{\tau}_{0}\gamma_{s}\left(\sigma^{\prime}(X_{s})\right)^{2}\sigma(X_{s})\left(\int_{0}^{s}\psi_{n,1}(u,s)dW_{u}\right)$
$\displaystyle\qquad\times\left(\int^{s}_{\eta_{n}(s)}(s-\eta_{n}(u))^{\alpha}\sigma(X_{\eta_{n}(u)})\,dW_{u}\right)ds$
$\displaystyle=n^{2\alpha+1}\int^{\tau}_{0}\gamma_{s}\left(\sigma^{\prime}(X_{s})\right)^{2}\sigma(X_{s})\sigma(X_{\eta_{n}(s)})(s-\eta_{n}(s))^{\alpha}(W_{s}-W_{\eta_{n}(s)})$
$\displaystyle\qquad\times\left(\int_{0}^{s}\psi_{n,1}(u,s)dW_{u}\right)ds.$

Using the Lipschitz property of $\sigma$ and the Hölder continuity of
$\sigma^{\prime}$ and taking into account Lemma 6.4, we have that
$L^{n}_{\tau}$ has the same asymptotic behavior in $L^{2}$, uniformly in
$\tau\in[0,T]$, as

$L^{n,1}_{\tau}=n^{2\alpha+1}\int^{\tau}_{0}\gamma_{s}(\sigma^{\prime}\sigma)^{2}(X_{\eta_{n}(s)})(s-\eta_{n}(s))^{\alpha}(W_{s}-W_{\eta_{n}(s)})\left(\int_{0}^{s}\psi_{n,1}(u,s)dW_{u}\right)ds.$

Consider the decomposition
$L^{n,1}_{\tau}=:L^{n,2}_{\tau}+L^{n,3}_{\tau}+L^{n,4}_{\tau}$, where

$L^{n,2}_{\tau}=n^{2\alpha+1}\int^{\tau}_{0}\gamma_{s}(\sigma^{\prime}\sigma)^{2}(X_{\eta_{n}(s)})(s-\eta_{n}(s))^{\alpha}(W_{s}-W_{\eta_{n}(s)})\left(\int_{0}^{\eta_{n}(s)}\psi_{n,1}(u,s)dW_{u}\right)ds,$
$\displaystyle L^{n,3}_{\tau}$
$\displaystyle=n^{2\alpha+1}\int^{\tau}_{0}\gamma_{s}(\sigma^{\prime}\sigma)^{2}(X_{\eta_{n}(s)})(s-\eta_{n}(s))^{\alpha}$
$\displaystyle\qquad\times\left[(W_{s}-W_{\eta_{n}(s)})\left(\int_{\eta_{n}(s)}^{s}\psi_{n,1}(u,s)dW_{u}\right)-\int_{\eta_{n}(s)}^{s}\psi_{n,1}(u,s)du\right]ds.$

and

$L^{n,4}_{\tau}=n^{2\alpha+1}\int^{\tau}_{0}\gamma_{s}(\sigma^{\prime}\sigma)^{2}(X_{\eta_{n}(s)})(s-\eta_{n}(s))^{\alpha}\int_{\eta_{n}(s)}^{s}\psi_{n,1}(u,s)duds.$

We claim that

(4.57)
$\lim_{n\rightarrow\infty}\sup_{\tau\in[0,T]}\|L^{n,2}_{\tau}+L^{n,3}_{\tau}\|_{2}=0.$

Indeed, we can express the sum of these two terms as follows

$L^{n,2}_{\tau}+L^{n,3}_{\tau}=n^{2\alpha+1}\int^{\tau}_{0}\gamma_{s}(\sigma^{\prime}\sigma)^{2}(X_{\eta_{n}(s)})(s-\eta_{n}(s))^{\alpha}\Lambda^{n}_{s}ds,$

where

$\displaystyle\Lambda^{n}_{s}$
$\displaystyle=(W_{s}-W_{\eta_{n}(s)})\left(\int_{0}^{s}\psi_{n,1}(u,s)dW_{u}\right)$
$\displaystyle\qquad+(W_{s}-W_{\eta_{n}(s)})\left(\int_{\eta_{n}(s)}^{s}\psi_{n,1}(u,s)dW_{u}\right)-\int_{\eta_{n}(s)}^{s}\psi_{n,1}(u,s)du.$

Notice that $\Lambda^{n}_{s}$ is a random variable in the second Wiener chaos
that satisfies $E[\Lambda^{n}_{s}|\mathcal{F}_{\eta_{n}(s)}]=0$. As a
consequence, if $s_{1}+\frac{1}{n}<s_{2}$, then $s_{1}<\eta_{n}(s_{2})$ we
have

$\displaystyle
E[(\sigma^{\prime}\sigma)^{2}(X_{\eta_{n}(s_{1})})\Lambda^{n}_{s_{1}}(\sigma^{\prime}\sigma)^{2}(X_{\eta_{n}(s_{2})})\Lambda^{n}_{s_{2}}]$
$\displaystyle=E[(\sigma^{\prime}\sigma)^{2}(X_{\eta_{n}(s_{1})})\Lambda^{n}_{s_{1}}(\sigma^{\prime}\sigma)^{2}(X_{\eta_{n}(s_{2})})E[\Lambda^{n}_{s_{2}}|\mathcal{F}_{\eta_{n}(s_{2})}]]=0.$

This implies that

$\displaystyle E[|L^{n,2}_{\tau}+L^{n,3}_{\tau}|^{2}]$
$\displaystyle=n^{4\alpha+2}\int_{[0,\tau]^{2}}\gamma_{s_{1}}\gamma_{s_{2}}\mathbf{1}_{|s_{1}-s_{2}|\leq\frac{1}{n}}(s_{1}-\eta_{n}(s_{1}))^{\alpha}(s_{2}-\eta_{n}(s_{2}))^{\alpha}$
$\displaystyle\qquad\times
E[(\sigma^{\prime}\sigma)^{2}(X_{\eta_{n}(s_{1})})\Lambda^{n}_{s_{1}}(\sigma^{\prime}\sigma)^{2}(X_{\eta_{n}(s_{2})})\Lambda^{n}_{s_{2}}]ds_{1}ds_{2}$

We can write

$\Lambda^{n}_{s}=(W_{s}-W_{\eta_{n}(s)})\left(\int_{0}^{s}\psi_{n,1}(u,s)dW_{u}\right)+I_{2}\left(\mathbf{1}_{[\eta_{n}(s),s]}\otimes\psi_{n,1}(\cdot,s)\right),$

where $I_{2}$ denotes the double Wiener-Itô stochastic integral with respect
to $W$. Therefore, $\|\Lambda^{n}_{s}\|_{2}\leq Cn^{-\alpha-1}$, and we obtain
the following estimate

$E[|L^{n,2}_{\tau}+L^{n,3}_{\tau}|^{2}]\leq\int_{[0,T]^{2}}\gamma_{s_{1}}\gamma_{s_{2}}\mathbf{1}_{\\{|s_{1}-s_{2}|\leq\frac{1}{n}\\}}ds_{1}ds_{2},$

which converges to zero as $n$ tends to infinity. This proves (4.57). It
remains to study the limit of the term $L^{n,4}_{\tau}$. We have

$\displaystyle L^{n,4}_{\tau}$ $\displaystyle=n^{2\alpha+1}\sum_{j=0}^{\lfloor
n\tau\rfloor}\int_{\frac{j}{n}}^{\frac{j+1}{n}\wedge\tau}\gamma_{s}(\sigma^{\prime}\sigma)^{2}(X_{\frac{j}{n}})(s-\frac{j}{n})^{\alpha}\int_{\frac{j}{n}}^{s}[(s-\frac{j}{n})^{\alpha}-(s-u)^{\alpha}]duds$
$\displaystyle=\frac{\alpha}{\alpha+1}n^{2\alpha+1}\sum_{j=0}^{\lfloor
n\tau\rfloor}\int_{\frac{j}{n}}^{\frac{j+1}{n}\wedge\tau}\gamma_{s}(\sigma^{\prime}\sigma)^{2}(X_{\frac{j}{n}})(s-\frac{j}{n})^{2\alpha+1}ds.$

We can replace $\gamma_{s}$ by $\gamma_{\frac{j}{n}}$, and we conclude that

$\lim_{n\rightarrow\infty}L^{n,4}_{\tau}=\frac{\alpha}{2(\alpha+1)^{2}}\int_{0}^{\tau}\gamma_{s}(\sigma^{\prime}\sigma)^{2}(X_{s})ds,$

where the convergence holds in $L^{2}$, uniformly in $\tau\in[0,T]$.

Proof of (4.34): Set

$\displaystyle T^{n}_{\tau}$
$\displaystyle:=n^{2\alpha+1}\int^{\tau}_{0}\gamma_{s}\left(\sigma^{\prime}(X_{s})\right)^{2}\sigma(X_{s})\left(\int_{0}^{s}\psi_{n,1}(u,s)dW_{u}\right)$
$\displaystyle\qquad\times\left(\int_{0}^{\eta_{n}(s)}\psi_{n,2}(u,s)\sigma(X_{\eta_{n}(u)})\,dW_{u}\right)ds.$

By the same arguments as before, it suffices to analyze the asymptotic
behavior of

$\displaystyle T^{n,1}_{\tau}$
$\displaystyle:=n^{2\alpha+1}\int^{\tau}_{0}\gamma_{s}(\sigma^{\prime}\sigma)^{2}(X_{\eta_{n}(s)})\int_{0}^{\eta_{n}(s)}\psi_{n,1}(u,s)\psi_{n,2}(u,s)duds$
$\displaystyle=n^{2\alpha+1}\sum_{j=0}^{\lfloor
n\tau\rfloor}\int_{\frac{j}{n}}^{\frac{j+1}{n}\wedge\tau}\gamma_{s}(\sigma^{\prime}\sigma)^{2}(X_{\frac{j}{n}})$
$\displaystyle\qquad\times\sum_{i=0}^{j-1}\int_{\frac{i}{n}}^{\frac{i+1}{n}}[(s-\frac{i}{n})^{\alpha}-(s-u)^{\alpha}][(s-\frac{i}{n})^{\alpha}-(\frac{j}{n}-\frac{i}{n})^{\alpha}]duds.$

Making the change of variables $ns=x$, $nu=v$ and later $x-j=y$ and $v-i=z$
allows us to write

$\displaystyle T^{n,1}_{\tau}$ $\displaystyle=\frac{1}{n}\sum_{j=0}^{\lfloor
n\tau\rfloor}\int_{j}^{(j+1)\wedge
n\tau}\gamma_{\frac{x}{n}}(\sigma^{\prime}\sigma)^{2}(X_{\frac{j}{n}})\sum_{i=0}^{j-1}\int_{i}^{i+1}[(x-i)^{\alpha}-(x-v)^{\alpha}][(x-i)^{\alpha}-(j-i)^{\alpha}]dvdx$
$\displaystyle=\frac{1}{n}\sum_{j=0}^{\lfloor
n\tau\rfloor}\int_{0}^{1\wedge(n\tau-j)}\gamma_{\frac{y+j}{n}}(\sigma^{\prime}\sigma)^{2}(X_{\frac{j}{n}})$
$\displaystyle\qquad\times\sum_{i=0}^{j-1}\int_{0}^{1}[(y+j-i)^{\alpha}-(y+j-i-z)^{\alpha}][(j-i+y)^{\alpha}-(j-i)^{\alpha}]dzdy$
$\displaystyle=\frac{1}{n}\sum_{j=0}^{\lfloor
n\tau\rfloor}\int_{0}^{1\wedge(n\tau-j)}\gamma_{\frac{y+j}{n}}(\sigma^{\prime}\sigma)^{2}(X_{\frac{j}{n}})$
$\displaystyle\qquad\times\sum_{k=1}^{j}\int_{0}^{1}[(y+k)^{\alpha}-(y+k-z)^{\alpha}][(k+y)^{\alpha}-k^{\alpha}]dzdy,$

which converges in $L^{2}$, uniformly in $\tau\in[0,T]$, as $n$ tends to
infinity, to

$\left(\sum_{k=1}^{\infty}\int_{0}^{1}\int_{0}^{1}[(y+k)^{\alpha}-(y+k-z)^{\alpha}][(k+y)^{\alpha}-k^{\alpha}]dzdy\right)\int_{0}^{\tau}\gamma_{s}(\sigma^{\prime}\sigma)^{2}(X_{s})ds.$

Proof of (4.35): Set

$\displaystyle G^{n}_{\tau}$
$\displaystyle:=n^{2\alpha+1}\int^{\tau}_{0}\gamma_{s}\left(\sigma^{\prime}(X_{s})\right)^{2}\,\left(\int^{s}_{\eta_{n}(s)}(s-\eta_{n}(u))^{\alpha}\sigma(X_{\eta_{n}(u)})\,dW_{u}\right)\,$
$\displaystyle\qquad\times\left(\int^{\eta_{n}(s)}_{0}\psi_{n,2}(u,s)\sigma(X_{\eta_{n}(u)})\,dW_{u}\right)ds,$

where $\psi_{n,2}(u,s)$ is defined in (4.25). With the notation

$M^{n,1}_{s}=\int^{s}_{\eta_{n}(s)}(s-\eta_{n}(u))^{\alpha}\sigma(X_{\eta_{n}(u)})\,dW_{u}$

and

$M^{n,2}_{s}=\int^{\eta_{n}(s)}_{0}\psi_{n,2}(u,s)\sigma(X_{\eta_{n}(u)})\,dW_{u},$

we can write

$G^{n}_{\tau}=n^{2\alpha+1}\int^{\tau}_{0}\gamma_{s}\left(\sigma^{\prime}(X_{s})\right)^{2}M^{n,1}_{s}M^{n,2}_{s}ds.$

In order to show that this quantity tends to zero in $L^{2}$, uniformly in
$\tau\in[0,T]$, we will show that for any interval
$[\tau_{1},\tau_{2}]\subset[0,\tau]$, we have

(4.58)
$\lim_{n\rightarrow\infty}n^{2\alpha+1}\int^{\tau_{2}}_{\tau_{1}}\gamma_{s}M^{n,1}_{s}M^{n,2}_{s}ds=0,$

where the convergence holds in $L^{2}$ uniformly in $\tau_{1},\tau_{2}$ and
$\tau$, the case of the random factor can be handled as before by the usual
small blocks / big blocks argument. For the sake of simplicity we omit the
details. In order to show (4.58), we notice that
$E[M^{n,1}_{s_{1}}M^{n,2}_{s_{1}}M^{n,1}_{s_{2}}M^{n,2}_{s_{2}}]=0$ if
$|s_{1}-s_{2}|>\frac{1}{n}$. In fact, if, for instance,
$s_{1}+\frac{1}{n}<s_{2}$, then $\eta_{n}(s_{1})<\eta_{n}(s_{2})$ and

$E[M^{n,1}_{s_{1}}M^{n,2}_{s_{1}}M^{n,1}_{s_{2}}M^{n,2}_{s_{2}}]=E[M^{n,1}_{s_{1}}M^{n,2}_{s_{1}}M^{n,2}_{s_{2}}E[M^{n,1}_{s_{2}}|\mathcal{F}_{\eta_{n}(s_{2})}]]=0.$

As a consequence,

$\displaystyle
n^{4\alpha+2}E\left[\left|\int^{\tau_{2}}_{\tau_{1}}\gamma_{s}M^{n,1}_{s}M^{n,2}_{s}ds\right|^{2}\right]$
$\displaystyle\qquad=n^{4\alpha+2}\int_{[\tau_{1},\tau_{2}]^{2}}\gamma_{s_{1}}\gamma_{s_{2}}\mathbf{1}_{\\{|s_{1}-s_{2}|\leq\frac{1}{n}\\}}E[M^{n,1}_{s_{1}}M^{n,2}_{s_{1}}M^{n,1}_{s_{2}}M^{n,2}_{s_{2}}]ds_{1}ds_{2}$
$\displaystyle\qquad\leq
n^{4\alpha+2}\int_{[\tau_{1},\tau_{2}]^{2}}\gamma_{s_{1}}\gamma_{s_{2}}\mathbf{1}_{\\{|s_{1}-s_{2}|\leq\frac{1}{n}\\}}\|M^{n,1}_{s_{1}}\|_{4}\|M^{n,2}_{s_{1}}\|_{4}\|M^{n,1}_{s_{2}}\|_{4}\|M^{n,2}_{s_{2}}\|_{4}ds_{1}ds_{2}.$

Taking into account that for all $s\in[0,\tau]$, $\|M^{n,1}_{s}\|_{4}\leq
Cn^{-\alpha-\frac{1}{2}}$ and $\|M^{n,2}_{s}\|_{4}\leq
C(\left(\int_{0}^{\eta_{n}(s)}\psi_{n,2}^{2}(u,s)du\right)^{\frac{1}{2}}$, we
obtain, using Lemma 6.3

$\displaystyle
n^{4\alpha+2}E\left[\left|\int^{\tau_{2}}_{\tau_{1}}\gamma_{s}M^{n,1}_{s}M^{n,2}_{s}ds\right|^{2}\right]$
$\displaystyle\qquad\leq
Cn^{2\alpha+1}\int_{[\tau_{1},\tau_{2}]^{2}}\gamma_{s}\gamma_{s_{2}}\mathbf{1}_{\\{|s_{1}-s_{2}|\leq\frac{1}{n}\\}}\int_{0}^{\eta_{n}(s)}\psi_{n,2}^{2}(u,s)du$
$\displaystyle\qquad\leq
C\int_{[0,T]^{2}}\gamma_{s_{1}}\gamma_{s_{2}}\mathbf{1}_{\\{|s_{1}-s_{2}|\leq\frac{1}{n}\\}}ds_{1}ds_{2},$

which clearly converges to zero as $n\to\infty$. This completes the proof of
(4.35).

Proof of (C4): As in the proof of (C3) we can replace $\Theta^{n}_{s}$ defined
in (4.23) by $\widehat{\Theta}_{s}^{n}$ defined in (4.26). Then, condition
(C4) will be a consequence of the following convergences in $L^{2}$, which are
uniform in $\tau\in[0,t]$:

(4.59)
$\lim_{n\rightarrow\infty}n^{\alpha+\frac{1}{2}}\int_{0}^{\tau}(t-s)^{\alpha}\sigma^{\prime}(X_{s})\left(\int_{0}^{s}\psi_{n,1}(u,s)dW_{u}\right)ds=0,$
(4.60)
$\lim_{n\rightarrow\infty}n^{\alpha+\frac{1}{2}}\int_{0}^{\tau}(t-s)^{\alpha}\sigma^{\prime}(X_{s})\left(\int^{s}_{\eta_{n}(s)}\left(s-\eta_{n}(u)\right)^{\alpha}\sigma(X_{\eta_{n}(u)})\,dW_{u}\right)ds=0$

and

(4.61)
$\lim_{n\rightarrow\infty}n^{\alpha+\frac{1}{2}}\int_{0}^{\tau}(t-s)^{\alpha}\sigma^{\prime}(X_{s})\left(\int^{\eta_{n}(s)}_{0}\psi_{n,2}(u,s)\,\sigma(X_{\eta_{n}(u)})\,dW_{u}\right)ds=0.$

Proof of (4.59): We will show first that for any interval
$[\tau_{1},\tau_{2}]\subset[0,\tau]$, the term

$S^{n}=n^{\alpha+\frac{1}{2}}\int_{[\tau_{1},\tau_{2}]}(t-s)^{\alpha}\left(\int_{0}^{s}\psi_{n,1}(u,s)dW_{u}\right)ds$

converges in $L^{2}$ to zero, uniformly in $\tau_{1},\tau_{2}$ and $\tau$, as
$n$ tends to infinity. We have

$\displaystyle E[|S^{n}|^{2}]$
$\displaystyle=2n^{2\alpha+1}\int_{\tau_{1}}^{\tau_{2}}\int_{\tau_{1}}^{s_{2}}(t-s_{1})^{\alpha}(t-s_{2})^{\alpha}\int_{0}^{s_{1}}\psi_{n,1}(u,s_{1})\psi_{n,1}(u,s_{2})duds_{1}ds_{2}$
$\displaystyle\leq
2n^{2\alpha+1}\int_{\tau_{1}}^{\tau_{2}}\int_{0}^{s_{2}}(t-s_{1})^{\alpha}(t-s_{2})^{\alpha}$
$\displaystyle\qquad\times\sum_{i=0}^{\lfloor
ns_{1}\rfloor}\int_{\frac{i}{n}}^{\frac{i+1}{n}\wedge
s_{1}}[(s_{1}-\frac{i}{n})^{\alpha}-(s_{1}-u)^{\alpha}][(s_{2}-\frac{i}{n})^{\alpha}-(s_{2}-u)^{\alpha}]duds_{1}ds_{2}.$

With the change of variables $nu=x+i$, we obtain

$\displaystyle E[|S^{n}|^{2}]$ $\displaystyle\leq
2\int_{\tau_{1}}^{\tau_{2}}\int_{0}^{s_{2}}(t-s_{1})^{\alpha}(t-s_{2})^{\alpha}\sum_{i=0}^{\lfloor
ns_{1}\rfloor}\int_{0}^{1\wedge(ns_{1}-i)}[(ns_{1}-i)^{\alpha}-(ns_{1}-i-x)^{\alpha}]$
$\displaystyle\qquad\times[(ns_{2}-i)^{\alpha}-(ns_{2}-i-x)^{\alpha}]dxds_{1}ds_{2}$
$\displaystyle\leq
2\int_{0}^{t}\int_{0}^{s_{2}}(t-s_{1})^{\alpha}(t-s_{2})^{\alpha}$
$\displaystyle\qquad\times\sum_{k=0}^{\infty}\int_{0}^{1\wedge(ns_{1}-\lfloor
ns_{1}\rfloor+k)}[(ns_{1}-\lfloor ns_{1}\rfloor+k)^{\alpha}-(ns_{1}-\lfloor
ns_{1}\rfloor+k-x)^{\alpha}]$ $\displaystyle\qquad\times[(ns_{2}-\lfloor
ns_{1}\rfloor+k)^{\alpha}-(ns_{2}-\lfloor
ns_{1}\rfloor+k-x)^{\alpha}]dxds_{1}ds_{2},$

which converges to zero as $n$ tends to infinity.

Set

$S^{n,1}_{\tau}:=n^{\alpha+\frac{1}{2}}\int_{0}^{\tau}(t-s)^{\alpha}\sigma^{\prime}(X_{s})\left(\int_{0}^{s}\psi_{n,1}(u,s)dW_{u}\right)ds.$

Fix an integer $M\geq 1$ and consider a partition
$0=\tau_{0}<\tau_{1}<\cdots<\tau_{M}=\tau$, where $\tau_{i}=\frac{i}{M}\tau$
for $i=0,1,\dots,M$. We can write

$S^{n,1}_{\tau}=S^{n,M,1}_{\tau}+S^{n,M,2}_{\tau},$

where

$S^{n,M,1}_{\tau}=n^{\alpha+\frac{1}{2}}\sum_{i=0}^{M-1}\sigma^{\prime}(X_{\tau_{i}})\int_{\tau_{i}}^{\tau_{i+1}}(t-s)^{\alpha}\left(\int_{0}^{s}\psi_{n,1}(u,s)dW_{u}\right)ds$

and

$S^{n,M,2}_{\tau}=n^{\alpha+\frac{1}{2}}\sum_{i=0}^{M-1}\int_{\tau_{i}}^{\tau_{i+1}}(t-s)^{\alpha}[\sigma^{\prime}(X_{s})-\sigma^{\prime}(X_{\tau_{i}})]\left(\int_{0}^{s}\psi_{n,1}(u,s)dW_{u}\right)ds$

From Step 1, we know that for any fixed $M$

(4.62)
$\lim_{n\rightarrow\infty}\sup_{\tau\in[0,t]}E[|S^{n,M,1}_{\tau}|^{2}]=0.$

Assuming that $n\geq M$, the term $S^{n,M,2}_{\tau}$ can be estimated as
follows

$\|S^{n,M,2}_{\tau}\|_{2}\leq
C\sup_{u,v\in[0,t],|u-v|\leq\frac{2t}{M}}\|(\sigma^{\prime})^{2}(X_{u})-(\sigma^{\prime})^{2}(X_{v})\|_{4}n^{\alpha+\frac{1}{2}}\int_{0}^{\tau}(t-s)^{\alpha}\left\|\int_{0}^{s}\psi_{n,1}(u,s)dW_{u}\right\|_{4}ds$

From Lemma 6.4 we get

$\sup_{n\geq
1}n^{\alpha+\frac{1}{2}}\int_{0}^{t}(t-s)^{\alpha}\|\int_{0}^{s}\psi_{n,1}(u,s)dW_{u}\|_{4}ds<\infty.$

Therefore,

(4.63) $\lim_{M\rightarrow\infty}\sup_{n\geq
M}\sup_{\tau\in[0,t]}\|S^{n,M,2}_{\tau}\|_{2}=0.$

In conclusion, (4.62) and (4.63) imply (4.59).

Proof of (4.60): We have

$\displaystyle P^{n}_{\tau}$
$\displaystyle:=n^{\alpha+\frac{1}{2}}\int^{\tau}_{0}(t-s)^{\alpha}\sigma^{\prime}(X_{s})\,\left(\int^{s}_{\eta_{n}(s)}(s-\eta_{n}(u))^{\alpha}\sigma(X_{\eta_{n}(u)})\,dW_{u}\right)ds$
$\displaystyle=n^{\alpha+\frac{1}{2}}\int^{\tau}_{0}(t-s)^{\alpha}\sigma^{\prime}(X_{s})\sigma(X_{\eta_{n}(s)})\Xi^{n,2}_{s}ds,$

where $\Xi^{n,2}_{s}=\int^{s}_{\eta_{n}(s)}(s-\eta_{n}(u))^{\alpha}dW_{u}$.
Consider the decomposition

$P^{n}_{\tau}=P^{n,1}_{\tau}+P^{n,2}_{\tau},$

where

$P^{n,1}_{\tau}=n^{\alpha+\frac{1}{2}}\int^{\tau}_{0}(t-s)^{\alpha}(\sigma^{\prime}\sigma)(X_{\eta_{n}(s)})\Xi^{n,2}_{s}ds$

and

$P^{n,2}_{\tau}=n^{\alpha+\frac{1}{2}}\int^{\tau}_{0}(t-s)^{\alpha}[(\sigma^{\prime}(X_{s})-\sigma^{\prime}(X_{\eta_{n}(s)})]\sigma(X_{\eta_{n}(s)})\Xi^{n,2}_{s}ds.$

The $\beta$-Hölder continuity of $\sigma^{\prime}$, Lemma 3.4 and the fact
that $\|\Xi^{n,2}_{s}\|_{2}\leq Cn^{-\alpha-\frac{1}{2}}$ imply that

$\lim_{n\rightarrow\infty}\sup_{\tau\in[0,t]}E[|P^{n,2}_{\tau}|^{2}]=0.$

To handle the term $P^{n,1}_{\tau}$ we write

$E[|P^{n,1}_{\tau}|^{2}]=n^{2\alpha+1}\int_{[0,\tau]^{2}}(t-s_{1})^{\alpha}(t-s_{2})^{\alpha}E\left[(\sigma^{\prime}\sigma)(X_{\eta_{n}(s_{1})})(\sigma^{\prime}\sigma)(X_{\eta_{n}(s_{2})})\Xi^{n,2}_{s_{1}}\Xi^{n,2}_{s_{2}}\right]ds_{1}ds_{2}$

We claim that the expectation inside the integral vanishes if
$|s_{1}-s_{2}|>\frac{1}{n}$. Indeed, if, for instance
$s_{1}+\frac{1}{n}<s_{2}$, then $\eta_{n}(s_{1})<\eta_{n}(s_{2})$ and

$\displaystyle
E\left[(\sigma^{\prime}\sigma)(X_{\eta_{n}(s_{1})})(\sigma^{\prime}\sigma)(X_{\eta_{n}(s_{2})})\Xi^{n,2}_{s_{1}}\Xi^{n,2}_{s_{2}}\right]$
$\displaystyle\qquad=E\left[(\sigma^{\prime}\sigma)(X_{\eta_{n}(s_{1})})(\sigma^{\prime}\sigma)(X_{\eta_{n}(s_{2})})\Xi^{n,2}_{s_{1}}E[\Xi^{n,2}_{s_{2}}|\mathcal{F}_{\eta_{n}(s_{2})}\right]]=0.$

Therefore,

$\displaystyle E[|P^{n,1}_{\tau}|^{2}]$
$\displaystyle=n^{2\alpha+1}\int_{[0,\tau]^{2}}(t-s_{1})^{\alpha}(t-s_{2})^{\alpha}\mathbf{1}_{\\{|s_{1}-s_{2}|\leq\frac{1}{n}\\}}$
$\displaystyle\qquad\times
E\left[(\sigma^{\prime}\sigma)(X_{\eta_{n}(s_{1})})(\sigma^{\prime}\sigma)(X_{\eta_{n}(s_{2})})\Xi^{n,2}_{s_{1}}\Xi^{n,2}_{s_{2}}\right]ds_{1}ds_{2}$
$\displaystyle\leq
Cn^{2\alpha+1}\int_{[0,\tau]^{2}}(t-s_{1})^{\alpha}(t-s_{2})^{\alpha}\mathbf{1}_{\\{|s_{1}-s_{2}|\leq\frac{1}{n}\\}}\|\Xi^{n,2}_{s_{1}}\|_{4}\|\Xi^{n,2}_{s_{2}}\|_{4}ds_{1}ds_{2}$
$\displaystyle\leq
C\int_{[0,t]}(t-s_{1})^{\alpha}(t-s_{2})^{\alpha}\mathbf{1}_{\\{|s_{1}-s_{2}|\leq\frac{1}{n}\\}}ds_{1}ds_{2},$

which tends to zero as $n\to\infty$. This completes the proof of (4.60).

Proof of (4.61): Set

$F^{n}_{\tau}=n^{\alpha+\frac{1}{2}}\int^{\tau}_{0}(t-s)^{\alpha}\sigma^{\prime}(X_{s})\,\left(\int^{\eta_{n}(s)}_{0}\psi_{n,2}(u,s)\sigma(X_{\eta_{n}(u)})\,dW_{u}\right)ds,$

where $\psi_{n,2}(u,s)$ is defined in (4.25). Fix $\delta>0$ such that
$\delta<t$ and suppose that $n$ satisfies $\delta+\frac{1}{n}<t$ and
$\frac{1}{n}<\delta$. We decompose the term $F^{n}_{\tau}$ as follows

$\displaystyle F^{n}_{\tau}$
$\displaystyle=n^{\alpha+\frac{1}{2}}\int^{(\delta+\frac{1}{n})\wedge\tau}_{0}(t-s)^{\alpha}\sigma^{\prime}(X_{s})\left(\int^{\eta_{n}(s)}_{0}\psi_{n,2}(u,s)\sigma(X_{\eta_{n}(u)})\,dW_{u}\right)ds$
$\displaystyle\qquad+n^{\alpha+\frac{1}{2}}\int^{\tau}_{(\delta+\frac{1}{n})\wedge\tau}(t-s)^{\alpha}[\sigma^{\prime}(X_{s})-\sigma^{\prime}(X_{\eta_{n}(s)-\delta})]\,\left(\int^{\eta_{n}(s)}_{0}\psi_{n,2}(u,s)\sigma(X_{\eta_{n}(u)})\,dW_{u}\right)ds$
$\displaystyle\qquad+n^{\alpha+\frac{1}{2}}\int_{(\delta+\frac{1}{n})\wedge\tau}^{\tau}(t-s)^{\alpha}\sigma^{\prime}(X_{\eta_{n}(s)-\delta})\left(\int^{\eta_{n}(s)-\delta}_{0}\psi_{n,2}(u,s)\sigma(X_{\eta_{n}(u)})\,dW_{u}\right)ds$
$\displaystyle\qquad+n^{\alpha+\frac{1}{2}}\int_{(\delta+\frac{1}{n})\wedge\tau}^{\tau}(t-s)^{\alpha}\sigma^{\prime}(X_{\eta_{n}(s)-\delta})\left(\int_{\eta_{n}(s)-\delta}^{\eta_{n}(s)}\psi_{n,2}(u,s)\sigma(X_{\eta_{n}(u)})\,dW_{u}\right)ds$
$\displaystyle=:B^{n,1}_{\tau}+B^{n,2}_{\tau}+B^{n,3}_{\tau}+B^{n,4}_{\tau}.$

We have the following estimates. Using Minkowkski’s, Hölder and Burkholder’s
inequalities together with Lemma 6.3, we obtain

$\displaystyle\|B^{n,1}_{\tau}\|_{2}$ $\displaystyle\leq
Cn^{\alpha+\frac{1}{2}}\int^{\delta+\frac{1}{n}}_{0}(t-s)^{\alpha}\left(\int^{\eta_{n}(s)}_{0}\psi_{n,2}^{2}(u,s)du\right)^{\frac{1}{2}}ds$
$\displaystyle\leq
C\left(t^{\alpha+1}-(t-\delta-\frac{1}{n})^{\alpha+1}\right)$ (4.64)
$\displaystyle\leq C\left(\delta+\frac{1}{n}\right)^{(\alpha+1)\wedge 1}.$

For the term $B^{n,2}_{\tau}$ the $\beta$-Hölder continuity of
$\sigma^{\prime}$ allows us to write

$\displaystyle\|B^{n,2}_{\tau}\|_{2}$ $\displaystyle\leq
C\sup_{|u-v|\leq\delta+\frac{1}{n}}\|X_{u}-X_{v}\|_{4}^{\beta}n^{\alpha+\frac{1}{2}}\int^{t}_{0}(t-s)^{\alpha}\left(\int^{\eta_{n}(s)}_{0}\psi_{n,2}^{2}(u,s)du\right)^{\frac{1}{2}}ds$
(4.65) $\displaystyle\leq
C\left(\delta+\frac{1}{n}\right)^{\beta(\alpha+\frac{1}{2})}.$

For the term $B^{n,3}_{\tau}$, we have

(4.66) $\|B^{n,3}_{\tau}\|_{2}\leq
Cn^{\alpha+\frac{1}{2}}\int_{(\delta+\frac{1}{n})\wedge
t}^{t}(t-s)^{\alpha}\left(\int^{\eta_{n}(s)-\delta}_{0}\psi_{n,2}^{2}(u,s)du\right)^{\frac{1}{2}}ds.$

Applying the Mean Value Theorem and taking into account that
$\delta>\frac{1}{n}$, we obtain

$\displaystyle\int^{\eta_{n}(s)-\delta}_{0}\psi_{n,2}^{2}(u,s)du$
$\displaystyle=\int^{\eta_{n}(s)-\delta}_{0}[(s-\eta_{n}(u))^{\alpha}-(\eta_{n}(s)-\eta_{n}(u))^{\alpha}]^{2}du$
(4.67) $\displaystyle\leq Cn^{-2}\delta^{2\alpha-2}.$

Substituting (4.67) into (4.66), yields

(4.68) $\|B^{n,3}_{\tau}\|_{2}\leq n^{\alpha-\frac{1}{2}}\delta^{\alpha-1}.$

The estimation of the term $B^{n,4}_{\tau}$ is more involved. First, we write

$\displaystyle
E[|B^{n,4}_{\tau}|^{2}]=n^{2\alpha+1}\int_{[(\delta+\frac{1}{n})\wedge\tau,\tau]^{2}}(t-s_{1})^{\alpha}(t-s_{2})^{\alpha}E\left[\sigma^{\prime}(X_{\eta_{n}(s_{1})-\delta})M_{s_{1}}^{n,3}\sigma^{\prime}(X_{\eta_{n}(s_{2})-\delta})M_{s_{2}}^{n,3}\right]ds_{1}ds_{2},$

where, for any $s\in[0,\tau]$, we use the notation

$M^{n,3}_{s}=\int_{\eta_{n}(s)-\delta}^{\eta_{n}(s)}\psi_{n,2}(u,s)\sigma(X_{\eta_{n}(u)})\,dW_{u}.$

We claim that for $|s_{1}-s_{2}|>\delta+\frac{1}{n}$, the expectation
$E\left[\sigma^{\prime}(X_{\eta_{n}(s_{1})-\delta})M_{s_{1}}^{n,3}\sigma^{\prime}(X_{\eta_{n}(s_{2})-\delta})M_{s_{2}}^{n,3}\right]$
vanishes. Indeed, if, for instance $s_{1}<s_{2}-\delta-\frac{1}{n}$, we have

$\displaystyle
E\left[\sigma^{\prime}(X_{\eta_{n}(s_{1})-\delta})M_{s_{1}}^{n,3}\sigma^{\prime}(X_{\eta_{n}(s_{2})-\delta})M_{s_{2}}^{n,3}\right]$
$\displaystyle\qquad=E\left[\sigma^{\prime}(X_{\eta_{n}(s_{1})-\delta})M_{s_{1}}^{n,3}\sigma^{\prime}(X_{\eta_{n}(s_{2})-\delta})E[M_{s_{2}}^{n,3}|\mathcal{F}_{\eta_{n}(s_{2})-\delta}]\right]=0.$

As a consequence,

$\displaystyle E[|B^{n,4}_{\tau}|^{2}]$
$\displaystyle=n^{2\alpha+1}\int_{[(\delta+\frac{1}{n})\wedge\tau,\tau]^{2}}(t-s_{1})^{\alpha}(t-s_{2})^{\alpha}\mathbf{1}_{\\{|s_{1}-s_{2}|\leq\delta+\frac{1}{n}\\}}$
$\displaystyle\qquad\times
E\left[\sigma^{\prime}(X_{\eta_{n}(s_{1})-\delta})M_{s_{1}}^{n,3}\sigma^{\prime}(X_{\eta_{n}(s_{2})-\delta})M_{s_{2}}^{n,3}\right]ds_{1}ds_{2}$
$\displaystyle\leq
Cn^{2\alpha+1}\int_{[(\delta+\frac{1}{n})\wedge\tau,\tau]^{2}}(t-s_{1})^{\alpha}(t-s_{2})^{\alpha}\mathbf{1}_{\\{|s_{1}-s_{2}|\leq\delta+\frac{1}{n}\\}}\|M_{s_{1}}^{n,3}\|_{4}\|M_{s_{2}}^{n,3}\|_{4}ds_{1}ds_{2}$
(4.69) $\displaystyle\leq
C\int_{[0,t]^{2}}(t-s_{1})^{\alpha}(t-s_{2})^{\alpha}\mathbf{1}_{\\{|s_{1}-s_{2}|\leq\delta+\frac{1}{n}\\}}ds_{1}ds_{2},$

where in the last inequality we used Burkholder’s inequality and Lemma 6.3.
From the estimates (4.64), (4.65), (4.68) and (4.69), letting first $n$ tend
to infinity and later $\delta$ tend to zero, we obtain that (4.61) holds true.
$\square$

###  4.5. Asymptotic behavior of $\widetilde{Y}^{n,1}_{t}$

We recall that $\widetilde{Y}^{n,1}_{t}$ satisfies the equation

$\widetilde{Y}^{n,1}_{t}=\widetilde{A}^{n,1}_{t}+\widetilde{C}^{n}_{t}+\int_{0}^{t}(t-s)^{\alpha}\sigma^{\prime}(X_{s})\widetilde{Y}^{n,1}_{s}\,dW_{s}.$

In this section we will prove the following result.

######  Proposition 4.5.

The finite-dimensional distributions of the process
$\\{\widetilde{Y}^{n,1}_{t},t\in[0,T]\\}$ converges in distribution to those
of the process
$\widetilde{Y}^{\infty,1}=\\{\widetilde{Y}^{\infty,1}_{t},t\in[0,T]\\}$, where
$\widetilde{Y}^{\infty,1}$ is the solution to the stochastic Volterra equation

$\widetilde{Y}^{\infty,1}_{t}=\kappa_{2}\int^{t}_{0}(t-s)^{\alpha}(\sigma^{\prime}\sigma)(X_{s})dB_{s}+\int_{0}^{t}(t-s)^{\alpha}\sigma^{\prime}(X_{s})\widetilde{Y}^{\infty,1}_{s}\,dW_{s},$

where $B$ is a standard Brownian motion independent of $W$.

###### Proof.

Fix an integer $m\geq 1$. Set

$\widetilde{Y}^{n,m,1}_{t}=\widetilde{A}^{n,1}_{t}+\widetilde{C}^{n}_{t}+\int_{0}^{t}(t-s)^{\alpha}\sigma^{\prime}(X_{s})\widetilde{Y}^{n,m,1}_{\beta_{m}(s)}\,dW_{s},$

where $\beta_{m}(s)=\frac{i}{m}$ if $\frac{i}{m}\leq s<\frac{i+1}{m}$. From
Section 4.4 we know that, for any fixed $m\geq 1$, the finite-dimensional
distributions of the process $\widetilde{Y}^{n,m,1}$ converges in distribution
to those of the process
$\widetilde{Y}^{\infty,m,1}=\\{\widetilde{Y}^{\infty,m,1}_{t},t\in[0,T]\\}$
that satisfies

$\widetilde{Y}^{\infty,m,1}_{t}=\kappa_{2}\int^{t}_{0}(t-s)^{\alpha}(\sigma^{\prime}\sigma)(X_{s})dB_{s}+\int_{0}^{t}(t-s)^{\alpha}\sigma^{\prime}(X_{s})\widetilde{Y}^{\infty,m,1}_{\beta_{m}(s)}\,dW_{s},$

where $B$ is a standard Brownian motion independent of $W$. Then, the
proposition follows immediately form the following convergences:

(4.70) $\lim_{m\rightarrow\infty}\sup_{n\geq
1}E[|\widetilde{Y}^{n,1}_{t}-\widetilde{Y}^{n,m,1}_{t}|^{2}]=0$

and

(4.71)
$\lim_{m\rightarrow\infty}E[|\widetilde{Y}^{\infty,1}_{t}-\widetilde{Y}^{\infty,m,1}_{t}|^{2}]=0.$

Let us first show (4.70). We have

$\displaystyle E[|\widetilde{Y}^{n,1}_{t}-\widetilde{Y}^{n,m,1}_{t}|^{2}]$
$\displaystyle=\int_{0}^{t}(t-s)^{2\alpha}E\left[(\sigma^{\prime}(X_{s}))^{2}[\widetilde{Y}^{n,1}_{s}-\widetilde{Y}^{n,m,1}_{\beta_{m}(s)}]^{2}\,\right]ds$
$\displaystyle\leq\|\sigma^{\prime}\|_{\infty}^{2}\int_{0}^{t}(t-s)^{2\alpha}E[|\widetilde{Y}^{n,1}_{s}-\widetilde{Y}^{n,m,1}_{\beta_{m}(s)}|]^{2}\,ds$
$\displaystyle\leq
2\|\sigma^{\prime}\|_{\infty}^{2}\int_{0}^{t}(t-s)^{2\alpha}E[|\widetilde{Y}^{n,1}_{s}-\widetilde{Y}^{n,m,1}_{s}|]^{2}\,ds$
(4.72)
$\displaystyle\qquad+2\|\sigma^{\prime}\|_{\infty}^{2}\int_{0}^{t}(t-s)^{2\alpha}E[|\widetilde{Y}^{n,m,1}_{s}-\widetilde{Y}^{n,m,1}_{\beta_{m}(s)}|]^{2}\,ds.$

Applying Lemma 6.1 and the fact that $\sup_{t\in[0,T]}\sup_{n,m\geq
1}E[|\widetilde{Y}^{n,m,1}_{t}|^{2}]<\infty$, we can show that the process
$\widetilde{Y}^{n,m,1}_{t}$ is Hölder continuous, uniformly in $n$ and $m$.
Indeed, for any $0\leq t_{1}<t_{2}\leq T$,

$\displaystyle
E[|\widetilde{Y}^{n,m,1}_{t_{1}}-\widetilde{Y}^{n,m,1}_{t_{2}}|^{2}]$
$\displaystyle\leq
CE[|\widetilde{A}^{n,1}_{t_{1}}+\widetilde{C}^{n}_{t_{1}}-\widetilde{A}^{n,1}_{t_{2}}-\widetilde{C}^{n}_{t_{2}}|^{2}]$
$\displaystyle\qquad+C\int_{0}^{t_{1}}|(t_{2}-s)^{\alpha}-(t_{1}-s)^{\alpha}|^{2}E[|\widetilde{Y}^{n,m,2}_{\beta_{m}(s)}|^{2}]ds$
$\displaystyle\qquad+C\int_{t_{1}}^{t_{2}}(t_{2}-s)^{2\alpha}E[|\widetilde{Y}^{n,m,1}_{\beta_{m}(s)}|^{2}]ds$
(4.73) $\displaystyle\leq C|t_{1}-t_{2}|^{2\alpha+1}.$

Substituting (4.73) into (4.72), we obtain

$E[|\widetilde{Y}^{n,1}_{t}-\widetilde{Y}^{n,m,1}_{t}|^{2}]\leq
2\|\sigma^{\prime}\|_{\infty}^{2}\int_{0}^{t}(t-s)^{2\alpha}E[|\widetilde{Y}^{n,1}_{s}-\widetilde{Y}^{n,m,1}_{s}|]^{2}\,ds+Cm^{-2\alpha-1}.$

which implies (4.70). The proof of the convergence (4.71) is analogous.
$\square$

###  4.6. Conclusion

Part (i) of Theorem 2.1 follows from the convergence (4.4) and Proposition
4.3. Then, the convergence (4.4) and Proposition 4.2 imply that

$\lim_{n\rightarrow\infty}\sup_{t\in[0,T]}E[|\widetilde{Y}^{n,1}_{t}Y^{n,1}_{t}|^{2}]=0.$

Therefore, Part (ii) of Theorem 2.1 is a consequence of Proposition 4.5. This
concludes the proof of Theorem 2.1.

##  5\. Proof of Theorem 2.2

Theorem 2.2 follows from the next proposition which ensures the asymptotic
independence of the random variables $N^{n,1}_{\eta_{n}(t)}$ and
$Y^{n,1}_{t}$.

######  Proposition 5.1.

For any $t\in(0,T]$ and real numbers $\lambda,\mu\in{\mathbb{R}}$, the
following convergence holds true

$\lim_{n\rightarrow\infty}E\left(\exp\left(i\lambda N^{n,1}_{\eta_{n}(t)}+i\mu
Y^{n,1}_{t}\right)\right)=E\left(\exp\left(i\lambda
Z_{t}\right)\right)E\left(\exp\left(i\mu Y^{\infty,1}_{t}\right)\right),$

where $N^{n,1}_{t}$ has been defined in (4.18) and $Y^{n,1}_{t}$ is defined in
(1.4).

###### Proof.

Set

$\Lambda_{n}:=\exp\left(i\lambda N^{n,1}_{\eta_{n}(t)}+i\mu
Y^{n,1}_{t}\right).$

Recall that, for any $t\in[0,T]$,

$N^{n,1}_{t}=n^{\alpha+\frac{1}{2}}\int_{0}^{\eta_{n}(t)}\psi_{n,1}(s,\eta_{n}(t))dW_{s}$

and

$Y^{n,1}_{t}=n^{\alpha+\frac{1}{2}}\int_{0}^{t}(t-s)^{\alpha}\xi^{n}_{s}dW_{s},$

where $\xi^{n}_{s}=\sigma(X^{n}_{\eta_{n}(s)})-\sigma(X_{s})$. Let $\delta>0$
be such that $\delta<t$. Assume that $\frac{1}{n}<\frac{\delta}{2}$. We can
decompose the random variable $\Lambda_{n}$ as follows

$\Lambda_{n}=\Lambda^{(1)}_{n,\delta}+\Lambda^{(2)}_{n,\delta},$

where

$\displaystyle\Lambda^{(1)}_{n,\delta}$ $\displaystyle=\exp\left(i\mu
Y^{n,1}_{t}+i\lambda
n^{\alpha+\frac{1}{2}}\int_{t-\delta}^{\eta_{n}(t)}\psi_{n,1}(s,\eta_{n}(t))dW_{s}\right)$
$\displaystyle\qquad\times\left[\exp\left(i\lambda
n^{\alpha+\frac{1}{2}}\int^{t-\delta}_{0}\psi_{n,1}(s,\eta_{n}(t))dW_{s}\right)-1\right]$

and

$\Lambda^{(2)}_{n,\delta}=\exp\left(i\mu Y^{n,1}_{t}+i\lambda
n^{\alpha+\frac{1}{2}}\int_{t-\delta}^{\eta_{n}(t)}\psi_{n,1}(s,\eta_{n}(t))dW_{s}\right).$

We claim that for any fixed $\delta>0$,

(5.1) $\lim_{n\rightarrow\infty}E[\Lambda^{(1)}_{n,\delta}]=0.$

Indeed, we can write

$\displaystyle|E[\Lambda^{(1)}_{n,\delta}]|$
$\displaystyle\leq|\lambda|n^{\alpha+\frac{1}{2}}E\left[\left|\int^{t-\delta}_{0}\psi_{n,1}(s,\eta_{n}(t))dW_{s}\right|\right]$
$\displaystyle\leq|\lambda|n^{\alpha+\frac{1}{2}}\left(\int^{t-\delta}_{0}\psi^{2}_{n,1}(s,\eta_{n}(t))ds\right)^{\frac{1}{2}}.$

By the Mean Value Theorem, taking into account that $s\leq t-\delta$ and
$\frac{1}{n}<\frac{\delta}{2}$, we obtain

$\displaystyle\psi^{2}_{n,1}(s,\eta_{n}(t))$
$\displaystyle=[(\eta_{n}(t)-\eta_{n}(s))^{\alpha}-(\eta_{n}(t)-s)^{\alpha}]^{2}$
$\displaystyle\leq(\eta_{n}(t)-s)^{2\alpha-2}(s-\eta_{n}(s))^{2}\leq(\delta/2)^{2\alpha-2}n^{-2},$

and the convergence (5.1) follows easily. We now make a further decomposition

$\Lambda^{(2)}_{n}=\Lambda^{(3)}_{n,\delta}+\Lambda^{(4)}_{n,\delta},$

where

$\displaystyle\Lambda^{(3)}_{n,\delta}$ $\displaystyle=\exp\left(i\mu
n^{\alpha+\frac{1}{2}}\int_{0}^{t-\delta}(t-s)^{\alpha}\xi^{n}_{s}dW_{s}+i\lambda
n^{\alpha+\frac{1}{2}}\int_{t-\delta}^{\eta_{n}(t)}\psi_{n,1}(s,\eta_{n}(t))dW_{s}\right)$
$\displaystyle\qquad\times\left[\exp\left(i\mu
n^{\alpha+\frac{1}{2}}\int_{t-\delta}^{t}(t-s)^{\alpha}\xi^{n}_{s}dW_{s}\right)-1\right]$

and

$\Lambda^{(4)}_{n,\delta}=\exp\left(i\mu
n^{\alpha+\frac{1}{2}}\int_{0}^{t-\delta}(t-s)^{\alpha}\xi^{n}_{s}dW_{s}+i\lambda
n^{\alpha+\frac{1}{2}}\int_{t-\delta}^{\eta_{n}(t)}\psi_{n,1}(s,\eta_{n}(t))dW_{s}\right).$

We claim that

(5.2) $\lim_{\delta\rightarrow
0}\limsup_{n\rightarrow\infty}E[\Lambda^{(3)}_{n,\delta}]=0.$

Indeed, we can write

$\displaystyle|E[\Lambda^{(3)}_{n,\delta}]|$
$\displaystyle\leq|\mu|n^{\alpha+\frac{1}{2}}E\left[\left|\int_{t-\delta}^{t}(t-s)^{\alpha}\xi^{n}_{s}dW_{s}\right|\right]$
$\displaystyle\leq|\mu|n^{\alpha+\frac{1}{2}}\left(\int_{t-\delta}^{t}(t-s)^{2\alpha}E[|\xi^{n}_{s}|^{2}]ds\right)^{\frac{1}{2}}.$

Taking into account that $E[|\xi^{n}_{s}|^{2}]ds\leq Cn^{-2\alpha-1}$, the
limit (5.2) follows immediately. Finally, we can write

$\displaystyle
E[\Lambda^{(4)}_{n,\delta}]=E[E[\Lambda^{(4)}_{n,\delta}|\mathcal{F}_{t-\delta}]]$
$\displaystyle=E\left[\exp\left(i\mu
n^{\alpha+\frac{1}{2}}\int_{0}^{t-\delta}(t-s)^{\alpha}\xi^{n}_{s}dW_{s}\right)E\left[\exp\left(i\lambda
n^{\alpha+\frac{1}{2}}\int_{t-\delta}^{\eta_{n}(t)}\psi_{n,1}(s,\eta_{n}(t))dW_{s}\right)|\mathcal{F}_{t-\delta}\right]\right]$
$\displaystyle=E\left[\exp\left(i\mu
n^{\alpha+\frac{1}{2}}\int_{0}^{t-\delta}(t-s)^{\alpha}\xi^{n}_{s}dW_{s}\right)\right]\exp\left(-\frac{\lambda^{2}}{2}n^{2\alpha+1}\int_{t-\delta}^{\eta_{n}(t)}\psi_{n,1}^{2}(s,\eta_{n}(t))ds\right).$

We claim that

(5.3)
$\lim_{n\rightarrow\infty}\exp\left(-\frac{\lambda^{2}}{2}n^{2\alpha+1}\int_{t-\delta}^{\eta_{n}(t)}\psi_{n,1}^{2}(s,\eta_{n}(t))ds\right)=\exp\left(-\frac{\lambda^{2}}{2}\kappa_{1}^{2}\right)$

and

(5.4) $\lim_{\delta\rightarrow
0}\limsup_{n\rightarrow\infty}\left|E\left[\exp\left(i\mu
n^{\alpha+\frac{1}{2}}\int_{0}^{t-\delta}(t-s)^{\alpha}\xi^{n}_{s}dW_{s}\right)\right]-E\left[\exp\left(i\mu
Y^{n,1}_{t}\right)\right]\right|=0.$

The convergence (5.3) follows from Proposition 4.3 and the previous
computations. The proof of (5.4) is analogous to that of (4.38). In
conclusion, from (5.1), (4.38), (5.3) and (5.4), we can write

$\displaystyle\lim_{n\rightarrow\infty}E[\Lambda_{n}]$
$\displaystyle=\exp\left(-\frac{\lambda^{2}}{2}\kappa_{1}\right)\lim_{n\rightarrow\infty}E\left[\exp\left(i\mu
Y^{n,1}_{t}\right)\right]$ $\displaystyle=E\left[\exp\left(i\lambda
Z_{t}\right)\right]E\left[\exp\left(i\mu Y^{\infty,1}_{t}\right)\right],$

which completes the proof of the proposition. $\square$

##  6\. Appendix

In the next lemma we establish the Hölder continuity in $L^{2}$ of the
processes $\widetilde{A}^{n,1}_{t}$ and $\widetilde{C}_{t}$ defined in (4.66)
and (4.8), respectively.

######  Lemma 6.1.

Consider the processes $\widetilde{A}^{n,1}_{t}$ and $\widetilde{C}_{t}$
defined in (4.66) and (4.8), respectively. There exists a constant $C>0$, such
that for all $t_{1},t_{2}\in[0,T]$,

(6.1) $E[|\widetilde{A}^{n,1}_{t_{1}}-\widetilde{A}^{n,1}_{t_{2}}|^{2}]\leq
C|t_{1}-t_{2}|^{2\alpha+1},$

and

(6.2) $E[|\widetilde{C}_{t_{1}}-\widetilde{C}_{t_{2}}|^{2}]\leq
C|t_{1}-t_{2}|^{2\alpha+1}.$

###### Proof.

Let us first show (6.1). For any $0\leq t_{1}<t_{2}\leq T$ we have

$\displaystyle
E[|\widetilde{A}^{n,1}_{t_{1}}-\widetilde{A}^{n,1}_{t_{2}}|^{2}]$
$\displaystyle=n^{2\alpha+1}\int_{0}^{t_{1}}|(t_{2}-s)^{\alpha}-(t_{1}-s)^{\alpha}|^{2}E\left[(\sigma^{\prime}\sigma)^{2}(X_{s})\left(\int_{0}^{s}\psi_{n,1}(u,s)dW_{u}\right)^{2}\right]ds$
$\displaystyle\qquad+n^{2\alpha+1}\int_{t_{1}}^{t_{2}}(t_{2}-s)^{2\alpha}E\left[(\sigma^{\prime}\sigma)^{2}(X_{s})\left(\int_{0}^{s}\psi_{n,1}(u,s)dW_{u}\right)^{2}\right]ds$
$\displaystyle\leq
Cn^{2\alpha+1}\int_{0}^{t_{1}}|(t_{2}-s)^{\alpha}-(t_{1}-s)^{\alpha}|^{2}\int_{0}^{s}\psi^{2}_{n,1}(u,s)duds$
$\displaystyle\qquad+Cn^{2\alpha+1}\int_{t_{1}}^{t_{2}}(t_{2}-s)^{2\alpha}\int_{0}^{s}\psi^{2}_{n,1}(u,s)duds.$

Applying Lemma 6.4, yields,

$\displaystyle
E[|\widetilde{A}^{n,1}_{t_{1}}-\widetilde{A}^{n,1}_{t_{2}}|^{2}]$
$\displaystyle\leq
C\int_{0}^{t_{1}}|(t_{2}-s)^{\alpha}-(t_{1}-s)^{\alpha}|^{2}ds+C\int_{t_{1}}^{t_{2}}(t_{2}-s)^{2\alpha}ds$
$\displaystyle\leq C|t_{1}-t_{2}|^{2\alpha+1}.$

The second inequality in the previous display can be deduced from the $L^{2}$
norm of the increment on the interval $[t_{1},t_{2}]$ of a fractional Brownian
motion with Hurst parameter $H=\alpha+\frac{1}{2}$. The proof of the
inequality (6.2) is analogous, but instead of Lemma 6.4 we make use of
inequality (3.2). $\square$

The next results are technical lemmas that have been used several time in the
proofs.

######  Lemma 6.2.

For any $\alpha\in(-\frac{1}{2},\frac{1}{2})$ and for any $\delta>0$,there are
positive constants $C_{1}$, $C_{2}$ and $C_{3}$, such that for all
$t\in[0,1]$,

(6.3)
$\int^{t}_{0}\left[(t-\eta_{n}(s))^{\alpha}-(t-s)^{\alpha}\right]^{2}\,ds\leq
C_{1}\,n^{-2\alpha-1}$

and

(6.4)
$\int^{(t-\delta)_{+}}_{0}\left[(t-\eta_{n}(s))^{\alpha}-(t-s)^{\alpha}\right]^{2}\,ds\leq
C_{2}n^{-2}\delta^{2\alpha-1}.$

###### Proof.

By the Mean Value Theorem, for any $0\leq s<t\leq 1$, we have

(6.5)
$\left[(t-\eta_{n}(s))^{\alpha}-(t-s)^{\alpha}\right]^{2}\leq(t-s)^{2\alpha-2}(s-\eta_{n}(s))^{2}.$

Let us first show the inequality (6.3). We can assume that $t>\frac{1}{n}$,
because in the case $t\leq\frac{1}{n}$, the inequality can be easily proved.
We decompose the interval $[0,t]$ into the union of the intervals
$\left[0,\frac{\lfloor nt\rfloor-1}{n}\right]$ and $\left[\frac{\lfloor
nt\rfloor-1}{n},t\right]$. For the first interval we can write

$\displaystyle\int^{\frac{\lfloor
nt\rfloor-1}{n}}_{0}\left[(t-\eta_{n}(s))^{\alpha}-(t-s)^{\alpha}\right]^{2}\,ds$
$\displaystyle\leq\sum_{i=0}^{\lfloor
nt\rfloor-2}\int_{\frac{i}{n}}^{\frac{i+1}{n}}(t-s)^{2\alpha-2}(s-\eta_{n}(s))^{2}ds$
$\displaystyle\leq\sum_{i=0}^{\lfloor
nt\rfloor-2}\int_{\frac{i}{n}}^{\frac{i+1}{n}}(t-\frac{i+1}{n})^{2\alpha-2}n^{-2}ds$
$\displaystyle\leq\sum_{i=0}^{\lfloor
nt\rfloor-2}\frac{(nt-i-1)^{2\alpha-2}}{n^{2\alpha-2}}n^{-3}$
$\displaystyle=\frac{1}{n^{2\alpha+1}}\sum_{i=0}^{\lfloor
nt\rfloor-2}(nt-i-1)^{2\alpha-2}$ (6.6)
$\displaystyle\leq\frac{1}{n^{2\alpha+1}}\sum_{j=1}^{\infty}j^{2\alpha-2}.$

Next for any $s\in\left[\frac{\lfloor nt\rfloor-1}{n},t\right]$, we have

$\displaystyle\left[(t-\eta_{n}(s))^{\alpha}-(t-s)^{\alpha}\right]^{2}$
$\displaystyle\leq C\left[(t-\eta_{n}(s))^{2\alpha}+(t-s)^{2\alpha}\right]$
$\displaystyle\leq\begin{cases}Cn^{-2\alpha}&\alpha\geq 0\\\
C(t-s)^{2\alpha}&\alpha<0\end{cases}.$

Therefore,

(6.7) $\int_{\frac{\lfloor
nt\rfloor-1}{n}}^{t}\left[(t-\eta_{n}(s))^{\alpha}-(t-s)^{\alpha}\right]^{2}ds\leq
Cn^{-2\alpha-1}.$

Thus combining (6.6) and (6.7) we obtain (6.3).

The inequality (6.4) follows easily from (6.5). Indeed, we can assume that
$t>\delta$, and in this case

$\displaystyle\int_{0}^{t-\delta}\left[(t-\eta_{n}(s))^{\alpha}-(t-s)^{\alpha}\right]^{2}\,ds$
$\displaystyle\leq\int_{0}^{t-\delta}(t-s)^{2\alpha-2}(s-\eta_{n}(s))^{2}ds$
$\displaystyle\leq\frac{1}{n^{2}}\int_{0}^{t-\delta}(t-s)^{2\alpha-2}ds$
$\displaystyle=\frac{1}{n^{2}}\int_{\delta}^{t}s^{2\alpha-2}ds$
$\displaystyle\leq\frac{2\delta^{2\alpha-1}}{n^{2}(1-2\alpha)}=C_{2}{n^{-2}}\delta^{2\alpha-1}.$

$\square$

######  Lemma 6.3.

Let $\psi_{n,2}(u,s)$ defined as in (4.25). There exists a constant $C>0$ such
that for any $s\geq 0$, we have

$n^{2\alpha+1}\int_{0}^{\eta_{n}(s)}\psi^{2}_{n,2}(u,s)du\leq C.$

###### Proof.

We have, using the Mean Value Theorem,

$\displaystyle\int_{0}^{\eta_{n}(s)}\psi^{2}_{n,2}(u,s)du$
$\displaystyle=\int_{0}^{\eta_{n}(s)}[(s-\eta_{n}(u))^{\alpha}-(\eta_{n}(s)-\eta_{n}(u))^{\alpha}]^{2}du$
$\displaystyle\leq\alpha^{2}(s-\eta_{n}(s))^{2}\int_{0}^{\eta_{n}(s)}(\eta_{n}(s)-\eta_{n}(u))^{2\alpha-2}du$
$\displaystyle\leq Cn^{-3}\sum_{i=0}^{\lfloor
ns\rfloor-1}\left(\eta_{n}(s)-\frac{i}{n}\right)^{2\alpha-2}$
$\displaystyle=Cn^{-2\alpha-1}\sum_{i=0}^{\lfloor
ns\rfloor-1}(\eta_{n}-i)^{2\alpha-2}.$

Since $2\alpha-2<-1$ the series $\sum_{j=1}^{\infty}j^{2\alpha-2}$ is
convergent, and this completes the proof of the lemma. $\square$

######  Lemma 6.4.

Let $\psi_{n,1}(u,s)$ defined as in (4.10). There exists a constant $C>0$ such
that for any $s\geq 0$, we have

$n^{2\alpha+1}\int_{0}^{s}\psi^{2}_{n,1}(u,s)du\leq C.$

###### Proof.

We have, using the Mean Value Theorem,

$\displaystyle\int_{0}^{s}\psi^{2}_{n,1}(u,s)du$
$\displaystyle=\int_{0}^{s}[(s-\eta_{n}(u))^{\alpha}-(s-u)^{\alpha}]^{2}du$
$\displaystyle=\sum_{i=0}^{\lfloor
ns\rfloor}\int_{\frac{i}{n}}^{\frac{i+1}{n}\wedge
s}[(s-\frac{i}{n})^{\alpha}-(s-u)^{\alpha}]^{2}du$
$\displaystyle=n^{-2\alpha-1}\sum_{i=0}^{\lfloor
ns\rfloor}\int_{i}^{(i+1)\wedge ns}[(ns-i)^{\alpha}-(ns-x)^{\alpha}]^{2}dx$
$\displaystyle\leq
n^{-2\alpha-1}\sum_{k=0}^{\infty}\int_{0}^{1\wedge(ns-\lfloor
ns\rfloor+k)}[(ns-\lfloor ns\rfloor+k)^{\alpha}-(ns-\lfloor
ns\rfloor+k-x)^{\alpha}]^{2}dx,$

which is uniformly bounded by a constant $C$ that does not depend on $n$ and
$s$. $\square$

## References

  * [1] Berger, M. and Mizel, V.: Volterra equations with Itô integrals I. J. Integral Equations 2 (1980)187-245.
  * [2] Berger, M. and Mizel, V.: Volterra equations with Itô integrals II. J. Integral Equations 4 (1980) 319-337.
  * [3] Brunner, H.: Collocation methods for Volterra integral and related functional differential equations, Vol. 15 of Cambridge Monographs an Applied and Computational Mathematics. Cambridge University Press, Cambridge. 2004.
  * [4] Cochran, W. G., Lee, J. S. and Potthoff, J.: Stochastic Volterra equations with singular kernels. Stochastic Processes Appl. 56 (1995) 337-349.
  * [5] Corcuera, J.M., Nualart, D. and Podolskij, M.: Asymptotics of weighted random sums. Commun. Appl. Ind. Math.  6 (2014) e-486, 11pp.
  * [6] Coutin, L. and Decreusefond, L.: Stochastic Volterra equations with singular kernels. In: Stochastic analysis and mathematical physics, pp. 39-50, Birkhäuser, Boston, 2001.
  * [7] Decreusefond, L.: Regularity properties of some stochastic Volterra equations with singular kernels. Potential Anal. 16 (2002) 139-149.
  * [8] Jacod, J. and Protter, P.: Asymptotic errors distributions for the Euler method for stochastic differential equations. Ann Probab. 26 (1998) 267-307.
  * [9] Li, M., Huang, C. and Hu, Y.: Numerical methods for stochastic Volterra integral equations with doubly weakly singular kernels. IMA Journal of Numerical Analysis (2021), drab407.
  * [10] Pardoux, E. and Protter, P.: Stochastic Volterra equations with anticipating coefficients. Ann. Probab. 18 (1990) 1635-1655.
  * [11] Protter, P.: Volterra equations driven by semimartingales. Ann. Probab. 13 (1985) 519-503.
  * [12] Revuz, D. and Yor, M.: Continuous martingales and Brownian motion, Vol. 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, third edition, 1999.
  * [13] Richard, A., Tan X. and Yang, F.: Discrete-time simulation of stochastic Volterra equations. Stochastic Process. Appl. 141 (2021) 109-138.
  * [14] Wang, Z.: Existence and uniqueness of solutions to stochastic Volterra equations with singular kernels and non-Lipschtiz coefficients. Stat. Probab. Lett. 78 (2008) 1062-1071.
  * [15] Ye, H.P, Gao, J.M. and Ding, Y.S.: A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 327 (2007)1075-1081.
  * [16] Zhang, X. C.: Euler schemes and large deviations for stochastic Volterra equations with singular kernels. J. Differ. Equations 244 (2008) 2226-2250.
  * [17] Zhang, X. C.: Stochastic Volterra equations in Banach spaces and stochastic partial differential equation. J. Funct. Anal. 258 (2010) 1361-1425.