# Euler scheme for SDEs driven
by fractional Brownian motions:
integrability and convergence in law

Jorge A. León  J. A. León: Departamento de Control Automático, Cinvestav-IPN,
Mexico<EMAIL_ADDRESS>,  Yanghui Liu∗ Y. Liu: Baruch College, New
York<EMAIL_ADDRESS>and  Samy Tindel  S. Tindel: Department of
Mathematics, Purdue University, West Lafayette<EMAIL_ADDRESS>
###### Abstract.

We prove that the modified Euler scheme for stochastic differential equations
driven by fractional Brownian motions (fBm) with Hurst parameter $H>1/3$,
together with its Malliavin derivatives, are integrable uniformly with respect
to the step size $n$. Then we use the integrability results to derive the
convergence rate in law $n^{1-4H+\varepsilon}$ for the Euler scheme. The proof
for integrability is based on a nontrivial generalization of the greedy
sequence argument in [8] to a quadratic functional of the fBm. The proof of
weak convergence applies Malliavin calculus and some upper-bound estimates for
weighted random sums.

###### Key words and phrases:

Rough paths, Discrete sewing lemma, Fractional Brownian motion, Stochastic
differential equations, Euler scheme, Asymptotic error distributions,
Malliavin calculus.
$\ast$ Corresponding author.

##  1\. Introduction

This note is concerned with the following stochastic differential equation
driven by a $d$-dimensional fractional Brownian motion (fBm in the sequel) $x$
with Hurst parameter $\frac{1}{3}<H<\frac{1}{2}$:

$\displaystyle dy_{t}=V_{0}(y_{t})dt+V(y_{t})dx_{t}\,,\qquad y_{0}=a,$ (1.1)

where we assume that $a\in\mathbb{R}^{m}$, the collection of vector field
$V_{0}=(V_{0}^{i})_{1\leq i\leq m}$ belongs to
$C^{2}_{b}(\mathbb{R}^{m},\mathbb{R}^{m})$ and ${V}=(V^{i}_{j})_{1\leq i\leq
m,1\leq j\leq d}$ sits in
$C^{3}_{b}(\mathbb{R}^{m},{\mathcal{L}}(\mathbb{R}^{d},\mathbb{R}^{m}))$.
Under this setting the theory of rough paths gives a framework allowing to get
existence and uniqueness results for equation (1.1), and the unique solution
$y$ in the rough paths sense has ${\gamma}$-Hölder continuity for all
${\gamma}<H$; see e.g. [13, 16].

One of the basic questions about systems like (1.1) concerns the existence of
a proper numerical scheme approximating the solution $y$. In case of a Hurst
parameter $H\in(\frac{1}{3},\frac{1}{2})$, the simplest possible solution to
this problem is to use a Milstein type scheme. However Milstein type schemes
involve second order expansions and iterated integrals of the fBm $x$, which
should be morally thought of as objects of the form

$x^{2}_{st}=\int_{s}^{t}\int_{s}^{u}dx_{v}\otimes dx_{u},$

and are notoriously uneasy to simulate. Therefore several contributions aimed
in the recent past at avoiding iterated integrals while still producing
convergent numerical schemes for rough differential equations. The first
article tackling this issue is [12], where the iterated integrals in $x^{2}$
were replaced by products of increments of $x$. The rate of convergence
obtained in [12] was then pushed to its optimal limit in [14]. Let us also
mention the article [32], which thoroughly explores Runge-Kutta methods based
on the same idea of replacing iterated integrals by products of increments.

In this paper we will focus our attention on another numerical approximation,
called first-order scheme in the sequel. The main idea behind this method is
to simply replace the second order terms $x^{2}$ by their expected values.
This yields simpler schemes than the aforementioned methods based on product
of increments, and at the same time produces optimal convergence rates.
Specifically, if $x$ is a fBm with Hurst parameter $H$ and one uses an
approximating grid with mesh of order $1/n$, then the rate of convergence is
of order $1/n^{2H-1/2}$. This method has first been introduced in [22] for a
Hurst parameter $H>1/2$, and has been extended to the rough path case in [28].
We also refer to [23, 24] for further extensions.

In order to describe our first-order numerical scheme, let us introduce some
basic settings. For simplicity, we are considering a finite time interval
$[0,T]$ and we take the uniform partition $\pi:0=t_{0}<t_{1}<\cdots<t_{n}=T$
on $[0,T]$. Specifically, for $k=0,\ldots,n$ we have $t_{k}=k\Delta$, where we
denote $\Delta=\frac{T}{n}$. In the sequel, the quantity $\delta x_{st}$ will
stand for the vector $x_{t}-x_{s}$. Our generic approximation is called
$y^{n}$, and it starts from the initial condition $y^{n}_{0}=y_{0}=a$. With
this notation in hand, we can now define our scheme recursively as follows
(here and below we set $\delta x_{t_{k}t_{k+1}}=x_{t_{k+1}}-x_{t_{k}}$):

$y^{n}_{t_{k+1}}=y^{n}_{t_{k}}+V_{0}(y^{n}_{t_{k}})\Delta+V(y^{n}_{t_{k}})\delta
x_{t_{k}t_{k+1}}+\frac{1}{2}\sum_{j=1}^{d}\partial
V_{j}V_{j}(y^{n}_{t_{k}})\,\Delta^{2H},$ (1.2)

where the notation $\partial V_{i}V_{j}$ stands for a vector field of the form

$\displaystyle\partial
V_{i}V_{j}=\left(\sum_{l=1}^{m}\partial_{l}V_{i}^{k}V_{j}^{l};\,k=1,\dots,m\,,\right)$
(1.3)

and $\partial_{l}$ stands for the partial derivative in the $y_{l}$ direction:
$\partial_{l}=\frac{\partial}{\partial{y_{i}}}$. As mentioned above, the rate
of convergence of $y^{n}$ to $y$ is of order $1/n^{2H-1/2}$. One of the key
results in [28] is a functional central limit theorem of the form

$\lim_{n\to\infty}n^{2H-1/2}\left(y^{n}-y\right)\stackrel{{\scriptstyle(d)}}{{=}}U,$

where $U$ is solution to a rough differential equation driven by $x$ plus an
additional Brownian term.

In the current contribution, we are mostly interested in the convergence in
distribution of the approximation $y^{n}$ defined by (1.2). This endeavor is
motivated by three main reasons which can be summarized as follows:

  1. (i)

The weak convergence of a numerical scheme is directly related to the
performance of simulation for stochastic models, which is a center issue in
mathematical finance and engineering.

  2. (ii)

For diffusions processes, that is stochastic differential equations driven by
a Brownian motion, the convergence in distribution for numerical schemes is a
classical problem. This is assessed e.g by the remarkable publications [2, 3].
As mentioned in those two references, a good knowledge about the weak
convergence is useful in order to evaluate probabilities that $y$ reaches a
certain level, or to get some information about the moments of $y$.

  3. (iii)

In [4, 7, 18] we have started a long term program aiming at understanding the
law of Gaussian rough differential systems. The current result might play an
important role in this approach.

Let us now describe the main result contained in this paper.

######  Theorem 1.1.

Suppose that $V\in C^{4}_{b}$ and $x$ is a fBm with Hurst parameter $H>1/3$.
Let $y$ be the solution of the rough differential equation (1.1) and let
$y^{n}$ be the corresponding Euler scheme (1.2). Then for any $\varepsilon>0$,
$f\in C^{4}_{b}(\mathbb{R}^{m})$ and $t\in[0,T]$ there is a constant $C>0$
independent of $n$ such that

$\left|\mathbb{E}f(y^{n}_{t})-\mathbb{E}f(y_{t})\right|\leq\frac{C}{n^{4H-1-\varepsilon}}.$
(1.4)

To the best of our knowledge, Theorem 1.1 is the first weak convergence result
for numerical schemes of differential equations driven by a fBm with $H<1/2$.
In order to get a broader perspective on weak convergence for stochastic
differential systems, let us recall some of the rates obtained in previous
contributions:

  1. (a)

It is well-known that the weak convergence rate for an equation like (1.1)
driven by a Brownian motion is $n^{-1}$, versus a rate $n^{-1/2}$ for the
strong rate; This has been established in the classical references [2, 3].

  2. (b)

In [23] the authors consider differential equations driven by a fBm for the
range of Hurst parameter $H\in(1/2,1)$. It is shown that the weak rate is
$n^{-1}$ like in the Brownian case, regardless of the value of $H$. This rate
is optimal in the sense that the normalized error
$n[\mathbb{E}f(y^{n}_{t})-\mathbb{E}f(y_{t})]$ converges to a nonzero limit
for any test function $f\in C^{4}$.

  3. (c)

The recent articles [15, 17] consider the weak convergence of Euler schemes
for a mixed stochastic integral model
${\mathcal{I}}=\int_{0}^{T}\varphi\left(B_{t}^{H}\right)\,dW_{s}$, where $W$
is a Wiener process and $B^{H}$ is a Liouville type fractional Brownian motion
driven by $W$ (with Hurst parameter $H\in(0,1)$). It is proved that for a
general choice of functions $f,\varphi$, the weak rate is $1/n^{(3H+1/2)\wedge
1}$ (to be contrasted with the strong rate of the Euler scheme for
${\mathcal{I}}$ above, which is $1/n^{H}$). Our result shows that this
surprising behavior is probably due to some specific cancellations for mixed
quantities like ${\mathcal{I}}$ (see further remarks about this fact in [15]).

Compared to this body of literature, our Theorem 1.1 shows that when
$1/3<H<1/2$ the weak rate for equation (1.1) is $n^{1-4H}$ (note that we
believe that our rate is optimal for a generic test function). This
generalizes in a very natural way the convergence rate $n^{-1}$ obtained for
$H\geq 1/2$, except for the slightly non optimal $\varepsilon$ in relation
(1.4). Notice that this small $\varepsilon$ is due to the fact that our
analysis of the scheme is mostly pathwise, in spite of dealing with a
convergence in distribution. It is interesting to mention that Theorem 1.4
agrees with the rule of thumb in the martingale framework (see e.g. Heston’s
model [1], Schrödinger’s equation [10], reflected diffusions [5] or the
stochastic heat equation [11]), namely that the weak rate $n^{1-4H}$ is twice
the strong rate $n^{1/2-2H}$ (see [28]).

At the core of our methodology for the proof of Theorem 1.1 lies a combination
of rough paths and Malliavin techniques, plus some specific tools for discrete
rough paths that have been developed by two of the authors in [28, 29]. Those
elements are summarized in Section 2 and Sections 4.1-4.2. Our main additional
ingredient in the current contribution is to show the integrability of the
Malliavin derivatives of the Euler scheme (1.2), uniformly in our
approximation parameter $n$. A key observation in this direction is that the
Euler scheme (1.2) is a discrete-time equation driven by the mix of a rough
path (that is the process $x$) and a quadratic Young path (that is, a path
which is a quadratic functional of $x$ and has a Hölder component greater than
$1/2$; see (2.20) for the precise definition). This representation enables us
to adapt the very fruitful idea of _greedy sequence_ put forward in [8], in
order to achieve exponential integrability in a rough paths context. A new
situation for the Euler scheme is that now we have a greedy sequence
corresponding not only to $x$ but also to the quadratic path $q$ introduced in
(2.20). One of our main efforts will then consist in showing a tail estimate
for the greedy sequence via Borell’s inequality. Furthermore, due to the
discrete feature of equation (1.2), a separate estimate will involve the _big_
steps related to our partition of $[0,T]$ (namely the steps for which the
increments $\delta x_{t_{k}t_{k+1}}$ are very large) separately. These
delicate estimates will be developed in Section 3.

The paper is structured as follows. Section 2 contains the preliminary results
on rough paths, Malliavin calculus, and the Euler scheme. In Section 3 we show
that the Malliavin derivatives of the Euler scheme has moments of any order.
After some preparations in Section 4.1-4.5, we prove the weak convergence of
the Euler scheme in Section 4.6.

######  Notation 1.2.

In what follows, we take $n\in\mathbb{N}$ and $\Delta=T/n$, and consider the
uniform partition: $0=t_{0}<t_{1}<\cdots<t_{n}=T$ on $[0,T]$, where
$t_{k}=k\Delta$. We denote by $\llbracket s,t\rrbracket$ the discrete
interval: $\llbracket s,t\rrbracket=\\{t_{k}\in[s,t]:k=0,\dots,n\\}$. For
$u\in[t_{k},t_{k+1})$, we denote $\eta(u)=t_{k}$. For an interval
$[s,t]\subset[0,T]$ we define the continuous- and discrete-time simplexes
${\mathcal{S}}_{2}([s,t])=\\{(u,v):s\leq u\leq v\leq t\\}$ and
${\mathcal{S}}_{2}(\llbracket
s,t\rrbracket)={\mathcal{S}}_{2}([s,t])\cap\llbracket s,t\rrbracket^{2}$. We
use the letters $C$ and $K$ to denote generic constant which can change from
line to line.

##  2\. Preliminary results

In this section we recall some basic notions of rough paths theory and their
application to fractional Brownian motion, which allow a proper definition of
equation (1.1). We also give the necessary elements of Malliavin calculus in
order to quantify the weak convergence rate. Eventually we recall the pathwise
estimates obtained in [27] for the Malliavin derivatives of our Euler scheme.
Notice that this basic presentation can be found in a very similar way in our
companion paper [27].

###  2.1. Elements of rough paths theory

This subsection is devoted to introduce some basic concepts of rough paths
theory. We are going to restrict our analysis to a generic $p$-variation
regularity of the driving path of order $1\leq p<3$, in order to keep
expansions to a reasonable size. We also fix a finite time horizon $T>0$. The
following notation will prevail until the end of the paper: for a finite
dimensional vector space ${\mathcal{V}}$ and two functions $f\in
C([0,T],{\mathcal{V}})$ and $g\in C({\mathcal{S}}_{2}([0,T]),{\mathcal{V}})$
we set

$\displaystyle\delta f_{st}=f_{t}-f_{s},\quad\text{and}\quad\delta
g_{sut}=g_{st}-g_{su}-g_{ut}.$ (2.1)

Let us introduce the analytic requirements in terms of $p$-variation
regularity which will be used in the sequel. Namely consider two paths $x\in
C([0,T],\mathbb{R}^{d})$ and $x^{2}\in
C({\mathcal{S}}_{2}([0,T]),(\mathbb{R}^{d})^{\otimes 2})$. Then we denote

$\|x\|_{p\text{-var},[s,t]}:=\left(\sup_{{\mathcal{P}}}\sum_{(u,v)\in{\mathcal{P}}}{|\delta
x_{uv}|^{p}}\right)^{1/p},\quad\|x^{2}\|_{p/2\text{-var},[s,t]}:=\left(\sup_{{\mathcal{P}}}\sum_{(u,v)\in{\mathcal{P}}}{|x^{2}_{uv}|^{p/2}}\right)^{2/p},$
(2.2)

where the supremum is taken among all partitions of the time interval $[s,t]$,
and for a partition ${\mathcal{P}}$ of $[s,t]$ we write
$(u,v)\in{\mathcal{P}}$ if $u$ and $v$ are two consecutive partition points of
${\mathcal{P}}$. When the semi-norms in (2.2) are finite we say that $x$ and
$x^{2}$ are respectively in $C^{p\text{-var}}([s,t],\mathbb{R}^{d})$ and
$C^{p/2\text{-var}}({\mathcal{S}}_{2}([s,t]),(\mathbb{R}^{d})^{\otimes 2})$.
For convenience, we denote $\|x\|_{p\text{-var}}:=\|x\|_{p\text{-var},[0,T]}$
and $\|x^{2}\|_{p/2\text{-var}}:=\|x^{2}\|_{p/2\text{-var},[0,T]}$. With this
preliminary notation in hand, we can now turn to the definition of rough path.

######  Definition 2.1.

Let $x\in C([0,T],\mathbb{R}^{d})$, $x^{2}\in
C({\mathcal{S}}_{2}([0,T]),(\mathbb{R}^{d})^{\otimes 2})$, and $1\leq p<3$.
Denote $x^{1}_{st}=\delta x_{st}$. We call ${\bf x}:=S_{2}(x):=(x^{1},x^{2})$
a (second-order) $p$-rough path if $\|x\|_{p\text{-var}}<\infty$ and
$\|x^{2}\|_{p/2\text{-var}}<\infty$, and if the following algebraic relation
holds true:

$\displaystyle\delta x^{2}_{sut}=x^{2}_{st}-x^{2}_{su}-x^{2}_{ut}=\delta
x_{su}\otimes\delta x_{ut},$ (2.3)

where we have invoked (2.1) for the definition of $\delta x$ and $\delta
x^{2}$. For a $p$-rough path $S_{2}(x)$, we define a $p$-variation semi-norm
as follows:

$\displaystyle\|S_{2}(x)\|_{p\text{-var}}:=\|x\|_{p\text{-var}}+\|x^{2}\|_{p/2\text{-var}}^{1/2}\,.$
(2.4)

An important subclass of rough paths are the so-called _geometric
$p$-variation rough paths_. A geometric $p$-variation rough path is a
$p$-rough path $(x,x^{2})$ such that there exists a sequence of smooth
$\mathbb{R}^{d}$-valued paths $(x^{n},x^{2,n})$ verifying:

$\displaystyle\lim_{n\to\infty}\left(\|x-x^{n}\|_{p\text{-var}}+\|x^{2}-x^{2,n}\|_{p/2\text{-var}}\right)=0.$
(2.5)

We will mainly consider geometric rough paths in the remainder of the article.

In relation to (2.5), notice that when $x$ is a smooth $\mathbb{R}^{d}$-valued
path, we can choose $x^{2}$ defined as the following iterated Riemann type
integral,

$\displaystyle x^{2}_{st}$ $\displaystyle=$
$\displaystyle\int_{s}^{t}\int_{s}^{u}dx_{v}\otimes dx_{u}.$ (2.6)

It is then easily verified that $S_{2}(x)=(x^{1},x^{2})$, with $x^{2}$ defined
in (2.6), is a $p$-rough path with $p=1$. In fact, this is also the unique way
to lift a smooth path to a $p$-rough path for some $p\geq 1$.

Recall now that we interpret equation (1.1) in the rough paths sense. That is,
we shall consider the following general rough differential equation (RDE):

$\displaystyle y_{t}$ $\displaystyle=$ $\displaystyle
a+\int_{0}^{t}V_{0}(y_{s})ds+\int_{0}^{t}{V}(y_{s})dx_{s}\,,\quad t\in[0,T],$
(2.7)

where $V_{0}$ and $V$ are smooth enough coefficients and $x$ is a rough path
as given in Definition 2.1. We shall interpret equation (2.7) in a way
introduced by Davie in [9], which is conveniently compatible with numerical
approximations.

######  Definition 2.2.

Let $(x,x^{2})$ be a $p$-rough path with $p<3$. We say that $y$ is a solution
of (2.7) on $[0,T]$ if $y_{0}=a$ and there exists a control function $\omega$
on $[0,T]$ (that is, $\omega$ is a two variable function on
${\mathcal{S}}_{2}([0,T])$ which satisfies the super-additivity condition
$\omega(s,t)\geq\omega(s,u)+\omega(u,t)$ for $s,u,t\in[0,T]:s<u<t$), a
constant $K>0$ and $\mu>1$ such that

$\Big{|}\delta y_{st}-\int_{s}^{t}V_{0}(y_{u})\,du-V(y_{s})\delta
x_{st}-\sum_{i,j=1}^{d}\partial V_{i}V_{j}(y_{s})x^{2,ij}_{st}\Big{|}\leq
K\omega(s,t)^{\mu}$ (2.8)

for all $(s,t)\in\mathcal{S}_{2}([0,T])$, where we recall that $\delta y$ is
defined by (2.1) and $\partial V_{i}V_{j}$ is defined as in (1.3).

Notice that if $y$ solves (2.7) according to Definition 2.2, then it is also a
controlled process as defined in [13, 20]. Namely, if $y$ satisfies relation
(2.8), then we also have:

$\displaystyle\delta y_{st}=V(y_{s})\delta x_{st}+r_{st}^{y},$

where $r^{y}\in C^{p/2\text{-var}}({\mathcal{S}}_{2}([0,T]))$. We can thus
define iterated integrals of $y$ with respect to itself thanks to the sewing
map; see Proposition 1 in [20]. This yields the following decomposition:

$\displaystyle\Big{|}\int_{s}^{t}y_{u}^{i}dy^{j}_{u}-y^{i}_{s}\delta
y^{j}_{st}-\sum_{i^{\prime},j^{\prime}=1}^{d}V^{i}_{i^{\prime}}V^{j}_{j^{\prime}}(y_{s})x_{st}^{2,i^{\prime}j^{\prime}}\Big{|}$
$\displaystyle\leq$ $\displaystyle K\omega(s,t)^{3/p},$

for all $(s,t)\in\mathcal{S}_{2}([0,T])$ and $i,j=1,\ldots,m$. In other words,
the signature type path $S_{2}({\bf y})=(y^{1},y^{2})$ defines a rough path
according to Definition 2.1, where $y^{2}$ denotes the iterated integral of
$y$.

We can now state an existence and uniqueness result for rough differential
equations. The reader is referred to e.g. [16, Theorem 10.36] for further
details.

######  Theorem 2.3.

Assume that $V=(V_{j})_{1\leq j\leq d}$ is a collection of $C^{3}_{b}$-vector
fields on $\mathbb{R}^{m}$. Then there exists a unique RDE solution to
equation (2.7), understood as in Definition 2.2. In addition, there exists a
constant $K>0$ such that the unique solution $y$ satisfies the following
estimate:

$\displaystyle|S_{2}(y)_{st}|$ $\displaystyle\leq$ $\displaystyle
K\left(1\vee\|S_{2}(x)\|_{p\text{-var},[s,t]}^{p}\right).$

Whenever $V=(V_{j})_{1\leq j\leq d}$ is a collection of linear vector fields,
existence and uniqueness still hold for equation (2.7). Furthermore, there
exist constants $K_{1},K_{2}>0$ such that we have the estimate:

$\displaystyle|S_{2}(y)_{st}|$ $\displaystyle\leq$ $\displaystyle
K_{1}\|S_{2}(x)\|_{p\text{-var},[s,t]}\,\exp\left(K_{2}\|S_{2}(x)\|_{p\text{-var}}^{p}\right)\,.$

We close this section by recalling a sewing map lemma with respect to discrete
control functions. It is an elaboration of [28, Lemma 2.5] and proves to be
useful in the analysis of the numerical scheme. Let
$\pi:0=t_{0}<t_{1}<\cdots<t_{n-1}<t_{n}=T$ be a generic partition of the
interval $[0,T]$ for $n\in\mathbb{N}$. We denote by $\llbracket s,t\rrbracket$
the discrete interval $\\{t_{k}:s\leq t_{k}\leq t\\}$ for $0\leq s<t\leq T$.

######  Lemma 2.4.

Suppose that $\omega$ is a control on $\llbracket 0,T\rrbracket$. In other
words, $\omega$ is a two variable function on ${\mathcal{S}}_{2}(\llbracket
0,T\rrbracket)$ which satisfies a super-additivity condition:
$\omega(s,t)\geq\omega(s,u)+\omega(u,t)$ for $s,u,t\in\llbracket
0,T\rrbracket:s<u<t$. Consider a Banach space ${\mathcal{B}}$ with norm
$|\cdot|$ and $R:{\mathcal{S}}_{2}(\llbracket 0,T\rrbracket)\to{\mathcal{B}}$,
and denote $\delta R_{sut}=R_{st}-R_{su}-R_{ut}$. Suppose that
$|R_{t_{k}t_{k+1}}|\leq\omega(t_{k},t_{k+1})^{\mu}$ for all
$t_{k}\in\llbracket 0,T\rrbracket$, and that $|\delta
R_{sut}|\leq\omega(s,t)^{\mu}$ with the exponent $\mu>1$. Then the following
relation holds:

$\displaystyle|R_{st}|\leq K_{\mu}\omega(s,t)^{\mu}\,,\quad\text{where}\quad
K_{\mu}=2^{\mu}\,\sum_{l=1}^{\infty}l^{-\mu}.$ (2.9)

The discrete sewing lemma allows to bound discrete sums which are crucial in
our numerical scheme context. As a first application along those lines we
present a probabilistic result below, which combines Proposition 4.1 and
Remark 4.2 in [28].

######  Lemma 2.5.

Consider two processes $f$ and $g$ such that for all $s,t\in\llbracket
0,T\rrbracket$ we have

$\displaystyle\|\delta
f_{st}\|_{L^{2p}}\lesssim|t-s|^{{\alpha}},\qquad\text{and}\qquad\|\delta
g_{st}\|_{L^{2p}}\lesssim|t-s|^{{\beta}},$

for a given $p\geq 1$ and ${\alpha},{\beta}$ such that ${\alpha}+{\beta}>1$.
Let $J_{st}$ be the discrete sum given by

$\displaystyle J_{st}=\sum_{s\leq t_{k}<t}\delta f_{st_{k}}\delta
g_{t_{k}t_{k+1}}.$ (2.10)

Then we have

$\displaystyle\|J_{st}\|_{L^{p}}\lesssim(t-s)^{{\alpha}+{\beta}}.$

###  2.2. Rough path above fractional Brownian motion

We now specialize our setting to a path $x=(x^{1},\dots,x^{d})$ defined as a
standard $d$-dimensional fBm on $[0,T]$ with Hurst parameter
$H\in(\frac{1}{3},\frac{1}{2})$. This fBm is defined on a complete probability
space $(\Omega,{\mathcal{F}},\mathbb{P})$, and we assume that the
$\sigma$-algebra ${\mathcal{F}}$ is generated by $x$. In this situation,
recall that the covariance function of each coordinate of $x$ is defined on
$\mathcal{S}_{2}([0,T])$ by:

$\displaystyle R(s,t)=\frac{1}{2}\left[s^{2H}+t^{2H}-|t-s|^{2H}\right],$
(2.11)

where recall that the simplex ${\mathcal{S}}_{2}([0,T])$ is introduced in
Notation 1.2. We start by reviewing some properties of the covariance function
of $x$ considered as a function on $(\mathcal{S}_{2}([0,T]))^{2}$. Namely,
take $(u,v,s,t)$ in $(\mathcal{S}_{2}([0,T]))^{2}$ and set

$\displaystyle R([u,v],[s,t])$ $\displaystyle=$
$\displaystyle\mathbb{E}\left[\delta x^{j}_{uv}\,\delta
x^{j}_{st}\right],\qquad j=1,\dots,d.$ (2.12)

Then, whenever $H>1/4$, it can be shown that the integral $\int R\,dR$ is
well-defined as a Young integral in the plane (see e.g. [16, Section 6.4]).
Furthermore, if the intervals $[u,v]$ and $[s,t]$ are disjoint, we have

$\displaystyle R([u,v],[s,t])=\int_{u}^{v}\int_{s}^{t}\mu(dr^{\prime}dr).$
(2.13)

Here and in the following, the signed measure $\mu$ is defined as

$\displaystyle\mu(dr^{\prime}dr)$ $\displaystyle=$
$\displaystyle-H(1-2H)|r-r^{\prime}|^{2H-2}dr^{\prime}dr.$ (2.14)

Using the elementary properties above, it is shown in [16, Chapter 15] that
for any piecewise linear or mollifier approximation $x^{n}$ to $x$, the smooth
rough path $S_{2}(x^{n})$ defined by (2.6) converges in the $p$-variation
semi-norm (2.4) to a $p$-geometric rough path $S_{2}(x):=(x^{1},x^{2})$ (given
as in Definition 2.1) for $3>p>1/H$. In addition, for $i\neq j$ the covariance
of $x^{2,ij}$ can be expressed in terms of a $2$-dimensional Young integral:

$\displaystyle\mathbb{E}\left[x^{2,ij}_{uv}x^{2,ij}_{st}\right]$
$\displaystyle=$
$\displaystyle\int_{u}^{v}\int_{s}^{t}R([u,r],[s,r^{\prime}])dR(r^{\prime},r).$
(2.15)

It is also established in [16, Chapter 15] that $S_{2}(x)$ enjoys the
following integrability property.

######  Proposition 2.6.

Let $S_{2}(x):=(x^{1},x^{2})$ be the geometric rough path above $x$ as given
in Definition 2.1, and $p\in(1/H,3)$. Then there exists a random variable
$G_{p}\in\cap_{p\geq 1}L^{p}(\Omega)$ such that
$\|S_{2}(x)\|_{p\text{-var}}\leq G_{p}$, where $\|\cdot\|_{p\text{-var}}$ is
defined by (2.4).

According to Theorem 2.3, given that the vector fields $V\in C_{b}^{3}$,
equation (2.7) driven by a $d$-dimensional fBm $x$ with Hurst parameter
$H>1/3$ admits a unique solution.

###  2.3. Malliavin calculus for $\mathbf{x}$

As mentioned in the introduction, we will analyze the convergence of
distribution for our numerical approximations thanks to Malliavin calculus
tools. We proceed to recall the main concepts which will be used later in the
paper and refer to [30] for further details. We start by labeling a definition
for the Cameron-Martin type space ${\mathcal{H}}$ related to our fractional
Brownian motion $x$.

######  Definition 2.7.

Denote by $\mathcal{E}_{[a,b]}$ the set of step functions on an interval
$[a,b]\subset[0,T]$. We call $\mathcal{H}_{[a,b]}$ the Hilbert space defined
as the closure of $\mathcal{E}_{[a,b]}$ with respect to the scalar product

$\displaystyle\langle\mathbf{1}_{[u,v]},\mathbf{1}_{[s,t]}\rangle_{\mathcal{H}_{[a,b]}}$
$\displaystyle=$ $\displaystyle R([u,v],[s,t]).$

In order to alleviate notations, we will write
$\mathcal{H}=\mathcal{H}_{[a,b]}$ when $[a,b]=[0,T]$. Notice that the mapping
$\mathbf{1}_{[s,t]}\rightarrow\delta x_{st}$ can be extended to an isometry
between $\mathcal{H}_{[a,b]}$ and the Gaussian space associated with
$\\{x_{t},t\in[a,b]\\}$. We denote this isometry by
$h\rightarrow\int_{a}^{b}h\,\delta^{\diamond}x$. The random variable
$\int_{a}^{b}h\,\delta^{\diamond}x$ is called the (first-order) Wiener
integral and is also denoted by $I_{1}(h)$.

The space ${\mathcal{H}}$ is very useful in order to define Wiener integrals
with respect to $x$. In this paper we also need to introduce another Cameron-
Martin type space $\bar{{\mathcal{H}}}$. The space $\bar{{\mathcal{H}}}$
allows to identify pathwise derivatives with respect to $x$ and the Malliavin
derivatives. In order to construct $\bar{{\mathcal{H}}}$, let ${\mathcal{R}}$
be the linear operator such that ${\mathcal{R}}:h\in{\mathcal{H}}\to\langle
h,\mathbf{1}_{[0,t]}\rangle_{{\mathcal{H}}}$. Then the space
$\bar{{\mathcal{H}}}$ is defined as the Hilbert space
$\bar{{\mathcal{H}}}={\mathcal{R}}({\mathcal{H}})$ equipped with the inner
product

$\displaystyle\langle{\mathcal{R}}(g),{\mathcal{R}}(h)\rangle_{\bar{{\mathcal{H}}}}=\langle
g,h\rangle_{{\mathcal{H}}}.$

We refer to [19, 31] for more details about the spaces ${\mathcal{H}}$ and
$\bar{{\mathcal{H}}}$.

For the sake of conciseness, we refer to [30] for a proper definition of
Malliavin derivatives and related Sobolev spaces in Gaussian analysis. Let us
just mention that we will denote the Malliavin derivative by ${D}F$, the
Sobolev spaces by $\mathbb{D}^{k,p}$ and the corresponding norms by
$\|F\|_{k,p}$. We denote by $D^{k}F$ the $k$th iteration of the Malliavin
derivative $D$ applied on $F$. The $n$-th order chaos of $x$ is denoted by
${\mathcal{K}}^{x}_{n}$. Also notice that we are considering a $d$-dimensional
fBm $x=(x^{1},\dots,x^{d})$. Therefore, we shall consider partial Malliavin
derivatives with respect to each coordinate $x^{i}$ in the sequel. Those
partial derivatives will be denoted by $D^{(i)}$. Then for
$h=(h^{1},\dots,h^{d})\in{\mathcal{H}}^{d}$ we write
$D_{h}F=\sum_{i=1}^{d}\langle D^{(i)}F,h^{i}\rangle_{{\mathcal{H}}}$. For
$L\geq 2$ we also denote by $D^{L}_{h}$ the iterated versions of $D_{h}$.
Namely we set

$\displaystyle D^{L}_{h}F=D_{h}\circ\cdots\circ D_{h}F.$ (2.16)

The Sobolev spaces related to the Malliavin derivatives are denoted by
$\mathbb{D}^{k,p}$ and the corresponding norms are written $\|\cdot\|_{k,p}$.
The dual of the Malliavin derivative is the Skorohod integral, for which we
use the notation $\delta^{\diamond}$. Its domain includes the space
$\mathbb{D}^{1,2}({\mathcal{H}}^{d})$, and the integration by parts formula
can be read as

$\displaystyle\mathbb{E}[F\delta^{\diamond}(u)]=\mathbb{E}[\langle
DF,u\rangle_{{\mathcal{H}}^{d}}],$ (2.17)

valid for $F\in\mathbb{D}^{1,2}$ and
$u\in\mathbb{D}^{1,2}({\mathcal{H}}^{d})$.

####  2.3.1. Differentiability

As we will see below, under the condition that $V\in C_{b}^{\lfloor
1/{\gamma}\rfloor+1}$ the solution $y$ to (2.7) is infinitely differentiable
in the Malliavin calculus sense. We shall express its Malliavin derivative in
terms of the Jacobian $\Phi$ of the equation, which is defined by the relation
$\Phi_{t}^{ij}=\partial_{a_{j}}y_{t}^{(i)}$, where recall that
$a=(a_{1},\dots,a_{m})$ is the initial value of the system (2.7). Setting
$\partial V_{j}$ for the Jacobian of $V_{j}$ seen as a function from
$\mathbb{R}^{m}$ to $\mathbb{R}^{m}$, let us recall that $\Phi$ is the unique
solution to the linear equation

$\Phi_{t}=\text{Id}_{m}+\int_{0}^{t}\partial
V_{0}(y_{s})\,\Phi_{s}\,ds+\sum_{j=1}^{d}\int_{0}^{t}\partial
V_{j}(y_{s})\,\Phi_{s}\,dx^{j}_{s}.$ (2.18)

Moreover, the following results hold true:

######  Proposition 2.8.

Let $y$ be the solution to equation (2.7) and suppose
$(V_{0},V_{1},\dots,V_{d})$ is a collection of vector fields in $C_{b}^{3}$.
Then for every $i=1,\ldots,m$, $t>0$, and $a\in\mathbb{R}^{m}$, we have
$y_{t}^{(i)}\in\mathbb{D}^{2,p}({\mathcal{H}})$ for $p\geq 1$ and

${D}^{(j)}_{s}y_{t}=\Phi_{s,t}V_{j}(y_{s}),\quad j=1,\ldots,d,\quad 0\leq
s\leq t,$

where ${D}^{(j)}_{s}y^{(i)}_{t}$ is the $j$-th component of
${D}_{s}y^{(i)}_{t}$, $\Phi_{t}=\partial_{a}y_{t}$ solves equation (2.18) and
$\Phi_{s,t}=\Phi_{t}\Phi_{s}^{-1}$.

Let us now quote the result [8], which gives a useful estimate for moments of
the Jacobian of rough differential equations driven by Gaussian processes.
Note that this result is expressed in terms of $p$-variations, for which we
refer to [16].

######  Proposition 2.9.

Consider a fractional Brownian motion $x$ with Hurst parameter $H\in(1/4,1/2]$
and $p>1/H$. Then for any $\eta\geq 1$, there exists a finite constant
$c_{\eta}$ such that the Jacobian $\Phi$ defined by (2.18) satisfies:

$\mathbb{E}\left[\|\Phi\|^{\eta}_{p\text{-}{\rm var};[0,1]}\right]=c_{\eta}.$
(2.19)

###  2.4. Path-wise estimate of Euler scheme and its derivatives

In the remaining of the section we state a path-wise upper-bound estimate of
the Malliavin derivatives of $y^{n}$ obtained in our companion paper [27]. We
first introduce some notations.

Let $b$ be a fBm independent of $x$. Recall that the rough paths above $x$ and
$b$ are denoted by $(x^{1},x^{2})$ and $(b^{1},b^{2})$, respectively (see
Definition 2.1). We introduce some second chaos processes which play a
prominent role in the analysis of Euler schemes (see [28]). Namely for
$[s,t]\in\llbracket 0,T\rrbracket$ we set

$\displaystyle q_{st}^{ij}=\sum_{s\leq
t_{k}<t}\left(x_{t_{k}t_{k+1}}^{2,ij}-\frac{1}{2}\Delta^{2H}\mathbf{1}_{\\{i=j\\}}\right)\,,\quad\text{and}\quad
q^{b,ij}_{st}=\sum_{s\leq
t_{k}<t}\left(b_{t_{k}t_{k+1}}^{2,ij}-\frac{1}{2}\Delta^{2H}\mathbf{1}_{\\{i=j\\}}\right).$
(2.20)

We recall a basic inequality taken from [28, Lemma 3.4]: for
$(s,t)\in{\mathcal{S}}_{2}(\llbracket 0,T\rrbracket)$ we have

$\displaystyle\left(\mathbb{E}[\|q_{st}\|^{2}]\right)^{1/2}\lesssim\frac{(t-s)^{1/2}}{n^{2H-1/2}}\,.$
(2.21)

We also introduce a Gaussian process w which encompasses the coordinates of
both the driving noise $x$ and the extra noise $b$. Specifically we define

$\displaystyle\delta\text{w}_{st}:=(\delta\text{w}_{st}^{1},\dots,\delta\text{w}_{st}^{2d}):=(\delta
x_{st}^{1},\dots,\delta x_{st}^{d},\delta b_{st}^{1},\dots,\delta
b_{st}^{d}).$ (2.22)

Furthermore, we define a control $\omega$ by

$\displaystyle\omega(s,t)=\|{\bf w}\|_{p\text{-var};\llbracket
s,t\rrbracket}^{p}+\|q\|_{p/2\text{-var};\llbracket
s,t\rrbracket}^{p/2}+\|q^{b}\|_{p/2\text{-var};\llbracket
s,t\rrbracket}^{p/2},\qquad(s,t)\in{\mathcal{S}}_{2}(\llbracket
0,T\rrbracket),$ (2.23)

where $q$ is defined in (2.20), $\text{w}=(x,b)$ according to (2.22) and ${\bf
w}=S_{2}(\text{w})$ is the $p$-rough path above w (see Definition 2.1). Let
${\alpha}>0$ be a positive constant. Denote $s_{0}=0$. Then given $s_{j}$, we
define $s_{j+1}$ recursively as

$s_{j+1}=\begin{cases}s_{j}+\Delta\,,&\textnormal{if
}\omega(s_{j},s_{j}+\Delta)>{\alpha}\\\ \max\\{u\in\llbracket
0,T\rrbracket:\,u>s_{j}\text{ and
}\omega(s_{j},u)\leq{\alpha}\\}\,,&\textnormal{if
}\omega(s_{j},s_{j}+\Delta)\leq{\alpha}\end{cases}$ (2.24)

Next we split the set of $s_{j}$’s as

$\displaystyle
S_{0}=\\{s_{j}:{\alpha}/2\leq\omega(s_{j},s_{j+1})\leq{\alpha}\\};\quad
S_{1}=\\{s_{j}:\omega(s_{j},s_{j+1})<{\alpha}/2\\};$ (2.25) $\displaystyle
S_{2}=\\{s_{j}:\omega(s_{j},s_{j+1})>{\alpha}\\}.$ (2.26)

We set

$\displaystyle{\mathcal{M}}_{0}=\prod_{s_{j}\in
S_{0}}\left(K\omega(s_{j},s_{j+1})^{1/p}+1\right),\qquad{\mathcal{M}}_{1}=\prod_{s_{j}\in
S_{1}}\left(K\omega(s_{j},s_{j+1})^{1/p}+1\right),$
$\displaystyle{\mathcal{M}}_{2}=\prod_{s_{j}\in
S_{2}}\left(K|\delta{\text{w}}_{s_{j}s_{j+1}}|+K\Delta^{2H}+1\right),$ (2.27)

and $K$ is a constant independent of $n$.

We now recall the following path-wise estimates for the Euler scheme in [27,
Theorem 4.13].

######  Theorem 2.10.

Take $r,r^{\prime}\geq 0$ and let $k_{0},k_{0}^{\prime}\in\mathbb{N}$ be such
that $r\in(t_{k_{0}},t_{k_{0}+1}]$ and
$r^{\prime}\in(t_{k_{0}^{\prime}},t_{k_{0}^{\prime}+1}]$. Suppose that $V\in
C^{4}_{b}$ and $p>1/H$. Define a Malliavin derivative vector $\xi^{n}$ as

$\xi^{n}_{t}=(y^{n}_{t},D_{r}y^{n}_{t},D_{r}D_{r^{\prime}}y^{n}_{t}):=(\xi_{t}^{n,0},\xi_{t}^{n,1},\xi_{t}^{n,2})$
(2.28)

Then for $L=0,1,2$ and all $(s,t)\in{\mathcal{S}}_{2}\llbracket 0,T\rrbracket$
we have the estimate

$\displaystyle\|\xi^{n,L}\|_{p\text{-var},\llbracket s,t\rrbracket}\leq
K\cdot\omega(s,t)^{1/p}\cdot{\mathcal{G}}\,,$ (2.29)

where the random variable ${\mathcal{G}}$ is defined by

$\displaystyle{\mathcal{G}}=|S_{0}\cup S_{1}\cup
S_{2}|\cdot({\mathcal{M}}_{0}\cdot{\mathcal{M}}_{1}\cdot{\mathcal{M}}_{2})^{L}\,,$
(2.30)

and where the quantities $S_{i}$, ${\mathcal{M}}_{i}$ are respectively defined
for $i=0,1,2$ in (2.25)-(2.26) and (2.4). Moreover, for
$(s,t)\in{\mathcal{S}}_{2}(\llbracket s_{j},s_{j+1}\rrbracket)$ such that
$s_{j}\in S_{0}\cup S_{1}$ we have

$\displaystyle|\delta\xi^{n,L}_{st}-{\mathcal{L}}^{L}V(y^{n}_{s})\delta
x_{st}|\leq K\omega(s,t)^{2/p}\cdot{\mathcal{G}}^{2},\qquad L=0,1,2,$ (2.31)

where we have set

${\mathcal{L}}^{0}V(y^{n}_{s})=V(y^{n}_{s}),\qquad{\mathcal{L}}^{1}V(y^{n}_{s})=\partial
V(y^{n}_{s})D_{r}y^{n}_{s},$

and where ${\mathcal{L}}^{2}V(y^{n}_{s})$ is defined by

$\displaystyle{\mathcal{L}}^{2}V(y^{n}_{s})=\partial^{2}V(y^{n}_{s})D_{r}y^{n}_{s}D_{r^{\prime}}y^{n}_{s}+\partial
V(y^{n}_{s})D_{r^{\prime}}D_{r}y^{n}_{s}.$

(The reader is referred to [27, equation (3.17)] for the general definition of
the operator ${\mathcal{L}}^{L}$.) In both estimates (2.29) and (2.31), $K$ is
a constant independent of $r$, $r^{\prime}$, $j$ and $n$.

##  3\. Uniform Integrability for Malliavin derivatives of the Euler scheme

In this section we tackle the integrability issue for the Malliavin
derivatives of the Euler scheme. Before proceeding to our main considerations,
some remarks about our global strategy are in order. Recall that
${\mathcal{M}}_{0}$, ${\mathcal{M}}_{1}$, ${\mathcal{M}}_{2}$ are defined in
(2.4), $b$ is an independent copy of $x$, and $q^{b}$ are defined in (2.20).
Recall that the signature ${\bf x}=S_{2}(x)=(x^{1},x^{2})$ of $x$ is defined
in Definition 2.1.

  1. (1)

Due to the bound (2.29), the integrability of ${\mathcal{M}}_{0}$,
${\mathcal{M}}_{1}$, ${\mathcal{M}}_{2}$ is our main task towards a uniform
bound for the Malliavin derivatives as a function of $n$. We mostly focus on
this problem in the sequel.

  2. (2)

Once the bounds in Theorem 2.10 are established, the fractional Brownian
motion $b$ does not play any particular role in our estimates. Hence for
notational sake we perform our computations below in the case $b\equiv
q^{b}\equiv 0$. Modifications to cover the $b\neq 0$ case are trivial.

We now turn our attention to the integrability of the random variables
${\mathcal{M}}_{i}$.

###  3.1. Uniform Integrability of ${\mathcal{M}}_{1}$ and ${\mathcal{M}}_{2}$

In this subsection, we consider the uniform integrability of
${\mathcal{M}}_{1}$ and ${\mathcal{M}}_{2}$ in (2.4). The proof is achieved
thanks to a tail analysis of the cardinality of the large size steps, that is
steps with size $>\\!\\!>{\alpha}$.

######  Theorem 3.1.

Let ${\mathcal{M}}_{2}={\mathcal{M}}_{2}(n)$ be the random variable defined by
(2.4). Specifically, since we are assuming that $b=0$, ${\mathcal{M}}_{2}$ is
given by

$\displaystyle{\mathcal{M}}_{2}=\prod_{s_{j}\in S_{2}}\left(K|\delta
x_{s_{i}s_{i+1}}|+K\Delta^{2H}+1\right),$ (3.1)

where $S_{2}$ is the subset of $\llbracket 0,T\rrbracket$ displayed in (2.26).
Recall that $x$ is a fBm with Hurst parameter $H>1/3$. Then we have
$\sup_{n\in\mathbb{N}}\mathbb{E}[|{\mathcal{M}}_{2}|^{\nu}]<\infty$ for all
$\nu\geq 1$.

###### Proof.

Recall that $\omega$ is defined by (2.23). Since we assume $b=0$ and thus
$q^{b}=0$, the control $\omega$ is reduced to

$\displaystyle\omega(t_{k},t_{k+1})=|{\bf
x}_{t_{k}t_{k+1}}|^{p}+|q_{t_{k}t_{k+1}}|^{p/2},$

so that definition (2.26) yields the following relation:

$\displaystyle S_{2}=\\{t_{k}:\omega(t_{k},t_{k+1})>{\alpha}\\}=$
$\displaystyle\\{t_{k}:|{\bf
x}_{t_{k}t_{k+1}}|^{p}+|q_{t_{k}t_{k+1}}|^{p/2}>{\alpha}\\}.$ (3.2)

Moreover, it is readily checked that

$\displaystyle|{\bf
x}_{t_{k}t_{k+1}}|^{p}+|q_{t_{k}t_{k+1}}|^{p/2}\lesssim|\delta
x_{t_{k}t_{k+1}}|^{p}+|x^{2}_{t_{k}t_{k+1}}|+|\Delta|^{2Hp}.$

If we choose $n$ large enough so that $|\Delta|^{2Hp}<\\!\\!<{\alpha}$, from
the expression (3.2) we get

$\displaystyle S_{2}$ $\displaystyle\subset\\{t_{k}:|{\bf
x}_{t_{k}t_{k+1}}|^{2}>K_{p}{\alpha}^{2/p}\\}=:S_{3}.$

This inclusion implies that

$\displaystyle{\mathcal{M}}_{2}=$ $\displaystyle\prod_{s_{j}\in
S_{2}}\left(K|{\bf x}_{s_{j}s_{j+1}}|+1\right)\leq\prod_{s_{j}\in
S_{3}}\left(K|{\bf x}_{s_{j}s_{j+1}}|+1\right)\leq C_{{\alpha}}\prod_{s_{j}\in
S_{3}}K|{\bf x}_{s_{j}s_{j+1}}|,$

where we have invoked the fact that $|{\bf
x}_{s_{j}s_{j+1}}|>K_{p}^{1/2}{\alpha}^{1/p}$ whenever $s_{j}\in S_{3}$ for
the last inequality. We now divide the proof in several steps.

_Step 1: Some pathwise bounds._ Recall again that $x$ is a fBm with Hurst
parameter $H>1/3$. Pick then ${\beta}<H$ such that

$\displaystyle{\mathcal{G}}\equiv\|{\bf x}\|_{[0,T],{\beta}}=\|\delta
x\|_{[0,T],{\beta}}+\|x^{2}\|_{[0,T],2{\beta}}^{1/2}\,,$ (3.3)

as defined in (2.4), is almost surely finite. Then since $|{\bf
x}_{s_{j}s_{j+1}}|\leq{\mathcal{G}}n^{-\beta}$ we have

$\displaystyle{\mathcal{M}}_{2}\leq\prod_{t_{k}\in
S_{3}}K\left({\mathcal{G}}n^{-{\beta}}\right)\leq(K{\mathcal{G}}n^{-{\beta}})^{|S_{3}|}.$

In addition, according to (3.3) we have

$\displaystyle S_{3}\subset\bigcup_{i,j}\left(S^{i}_{31}\cup S^{ij}_{32}\cup
S^{ij}_{33}\right),$

where the sets $S_{31}$, $S_{32}$, $S_{33}$ are defined by

$\displaystyle S_{31}^{i}=\\{t_{k}:(\delta
x_{t_{k}t_{k+1}}^{i})^{2}>{\alpha}_{p}\\},\quad
S_{32}^{ij}=\\{t_{k}:x_{t_{k}t_{k+1}}^{2,i,j}>{\alpha}_{p}\\},\quad
S_{33}^{ij}=\\{t_{k}:-x_{t_{k}t_{k+1}}^{2,i,j}>{\alpha}_{p}\\},$

where ${\alpha}_{p}$ is some constant depending on ${\alpha}$. Therefore
recalling that $K$ designates a generic constant, we have obtained the
following upper bound for the random variable ${\mathcal{M}}_{2}$:

$\displaystyle{\mathcal{M}}_{2}\leq\left(\prod_{i=1}^{d}(K{\mathcal{G}}n^{-\beta})^{|S^{i}_{31}|}\right)\cdot\left(\prod_{i,j=1}^{d}(K{\mathcal{G}}n^{-\beta})^{|S^{ij}_{32}|}\right)\cdot\left(\prod_{i,j=1}^{d}(K{\mathcal{G}}n^{-\beta})^{|S^{ij}_{33}|}\right).$
(3.4)

In the following we consider the integrability of the random variable
$(K{\mathcal{G}}n^{-\beta})^{|S^{i}_{31}|}$ for all $i=1,\dots,d$ which appear
in the right-hand side of (3.4). The other terms in (3.4) can be handled very
similarly.

_Step 2: Tail estimates for $|S_{31}|$._ For each subset
$\mathbf{u}\equiv\\{u_{j};j=1,\dots,n^{\prime}\\}$, $n^{\prime}\leq n$ of the
set of discrete instants $\\{t_{k},k=0,1,\dots,n\\}$ we denote

$\displaystyle{\mathcal{X}}^{i}(\mathbf{u})=\sum_{t_{k}\in\\{u_{j};\,j=1,\dots,n^{\prime}\\}}(|\delta
x^{i}_{t_{k}t_{k+1}}|^{2}-\Delta^{2H}).$

Then notice that if $|S_{31}^{i}|=n^{\prime}$, there exists a set
$\mathbf{u}=\\{u_{j};\,j=1,\dots,n^{\prime}\\}$ such that for all
$j=1,\dots,n^{\prime}$ we have $(\delta
x^{i}_{u_{j}u_{j+1}})^{2}>{\alpha}_{p}$. Hence if we take a constant
$K_{{\alpha},p}$ such that $K_{{\alpha},p}<{\alpha}_{p}$ and take $n$ large
enough, we have

$\displaystyle{\mathcal{X}}^{i}(\mathbf{u})>n^{\prime}K_{{\alpha},p}.$

We have thus proved that

$\displaystyle\\{|S_{31}^{i}|=n^{\prime}\\}$
$\displaystyle\subset\bigcup_{\mathbf{u}\subset\\{t_{k}\\}}\left\\{{\mathcal{X}}^{i}(\mathbf{u})>K_{p,{\alpha}}\cdot
n^{\prime}\right\\}.$

As a consequence of the above relation, we trivially get

$\displaystyle\mathbb{P}\\{|S_{31}^{i}|=n^{\prime}\\}\leq$
$\displaystyle\,\sum_{\mathbf{u}\subset\\{t_{k}\\}}\mathbb{P}\left\\{{\mathcal{X}}^{i}(\mathbf{u})>K_{p,{\alpha}}\cdot
n^{\prime}\right\\}.$ (3.5)

Next set
$\sigma_{\mathbf{u}}^{2}=(\mathbb{E}|{\mathcal{X}}^{i}(\mathbf{u})|^{2})$.
Owing to a slight variation of (2.21) we have $\sigma_{\mathbf{u}}^{2}\lesssim
1/n^{4H-1}$. Therefore starting from the right-hand side of (3.5) we get

$\displaystyle\mathbb{P}(|S^{i}_{31}|=n^{\prime})\leq$
$\displaystyle\,\sum_{\\{u_{j};\,j=1,\dots,n^{\prime}\\}\subset\\{t_{k}\\}}\mathbb{P}\left\\{\frac{{\mathcal{X}}^{i}(\mathbf{u})}{\sigma_{\mathbf{u}}}>\frac{K_{p,{\alpha}}\cdot
n^{\prime}}{\sigma_{\mathbf{u}}}\right\\}$ $\displaystyle\leq$
$\displaystyle\,\sum_{\\{u_{j};\,j=1,\dots,n^{\prime}\\}\subset\\{t_{k}\\}}\mathbb{P}\left\\{\frac{{\mathcal{X}}^{i}(\mathbf{u})}{\sigma_{\mathbf{u}}}>\frac{K_{p,{\alpha}}\cdot
n^{\prime}}{1/n^{2H-1/2}}\right\\}.$ (3.6)

The right-hand side of (3.1) is handled in the following way: taking into
account the fact that ${\mathcal{X}}^{i}(\mathbf{u})/\sigma_{\mathbf{u}}$ is a
normalized random variable in the second chaos of $x$, we apply Borell’s
inequality (see e.g. [21, Theorem 5.12]). In addition the number of sets of
the form $\mathbf{u}=\\{u_{j};j=1,\dots,n^{\prime}\\}$ is
${n}\choose{n^{\prime}}$. Hence we end up with

$\displaystyle\mathbb{P}(|S^{i}_{31}|=n^{\prime})\leq$
$\displaystyle\,\frac{n!}{n^{\prime}!(n-n^{\prime})!}\exp\left(-n^{2H-1/2}\cdot
K_{p,{\alpha}}\cdot n^{\prime}\right)$ $\displaystyle\leq$
$\displaystyle\,n^{n^{\prime}}\exp\left(-n^{2H-1/2}\cdot K_{p,{\alpha}}\cdot
n^{\prime}\right).$ (3.7)

_Step 3: Computations involving ${\mathcal{G}}$._ Let us now turn our
attention to the term ${\mathcal{G}}$ in (3.3). Since our fBm $x$ is a
Gaussian process, Fernique’s lemma asserts that
$\mathbb{P}({\mathcal{G}}>x)\leq e^{-Kx^{2}}$ for a given constant $K$ and
$x\geq 1$. This sub-Gaussian bound is sufficient to claim that for all
$n^{\prime}\geq 1$ we have

$\displaystyle\mathbb{E}{\mathcal{G}}^{n^{\prime}}\leq
K(n^{\prime})^{n^{\prime}}=Ke^{n^{\prime}\ln n^{\prime}}.$ (3.8)

We are now ready to go back to the study of the random variable
$(K{\mathcal{G}}n^{-{\beta}})^{\nu|S_{31}^{i}|}$, $K>0$. Namely we apply
Hölder’s inequality with two conjugates $p,q>1$, and we combine this with
(3.1) and (3.8). We get

$\displaystyle\mathbb{E}\left((K{\mathcal{G}}n^{-\beta})^{\nu
n^{\prime}}\mathbf{1}_{\\{|S_{31}^{i}|=n^{\prime}\\}}\right)$ $\displaystyle=$
$\displaystyle K^{\nu
n^{\prime}}n^{-{\beta}Kn^{\prime}}(\mathbb{E}[{\mathcal{G}}^{pKn^{\prime}}])^{1/p}\mathbb{P}(|S_{31}^{i}|=n^{\prime})^{1/q}$
(3.9) $\displaystyle\leq$ $\displaystyle e^{\nu n^{\prime}\ln
n^{\prime}}n^{-\beta Kn^{\prime}}\mathbb{P}(|S_{31}^{i}|=n^{\prime})^{1/q}$
$\displaystyle\leq$ $\displaystyle\exp\left(f(n^{\prime})\right),$

where the function $f$ is defined by

$\displaystyle f(n^{\prime}):=K_{1}n^{\prime}\ln
n^{\prime}-{\beta}Kn^{\prime}\ln n+K_{2}n^{\prime}\ln
n-K_{3}n^{2H-1/2}n^{\prime},$ (3.10)

for three positive constants $K_{1}$, $K_{2}$, $K_{3}$ whose exact value is
irrelevant.

We now compute the maximum of the function $f$ thanks to elementary
considerations. First we calculate

$\displaystyle f^{\prime\prime}(n^{\prime})=K/n^{\prime}\geq 0.$

Therefore $f$ is upward convex and

$\displaystyle\sup_{2\leq n^{\prime}\leq n}f(n^{\prime})\leq f(2)\vee f(n)\leq
f(2)+f(n).$ (3.11)

Moreover one can explicitly compute $f(2)$ and $f(n)$ thanks to the expression
(3.10). We obtain

$\displaystyle f(n)=(K_{1}+K_{2}-\beta K)n\ln n-Kn^{2H+1/2},\qquad f(2)=K-K\ln
n-Kn^{2H-1/2}.$

Reporting this expression into (3.9), we discover that

$\displaystyle\mathbb{E}(K{\mathcal{G}}n^{-\beta})^{\nu|S_{31}^{i}|}$
$\displaystyle=\sum_{n^{\prime}=0}^{n}\mathbb{E}\left((K{\mathcal{G}}n^{-\beta})^{\nu
n^{\prime}}\mathbf{1}_{\\{|S_{31}|=n^{\prime}\\}}\right)\leq\sum_{n^{\prime}=2}^{n}\exp\left(f(n^{\prime})\right)$
$\displaystyle\leq n\exp\left(f(n)+f(2)\right)\leq n\exp(C_{1}n\ln
n-C_{2}n^{2H+1/2}).$ (3.12)

_Step 4: Conclusion._ Since we have assumed $H>1/3$, it is readily checked
that the right-hand side of (3.1) is dominated by a constant. Therefore, we
end up with the inequality $\sup_{n\geq
1}\mathbb{E}[(K{\mathcal{G}}n^{-{\beta}})^{\nu|S_{31}^{i}|}]\equiv M<\infty$.
This concludes the uniform (in $n$) integrability of
$(K{\mathcal{G}}n^{-\beta})^{\nu|S^{i}_{31}|}$, for all indices $i=1,\dots,d$.
The integrability of the other two quantities
$(K{\mathcal{G}}n^{-\beta})^{\nu|S^{ij}_{32}|}$ and
$({\mathcal{G}}n^{-\beta})^{\nu|S^{ij}_{33}|}$ can be shown in a similar way.
Combining these integrability results with relation (3.4) and with Hölder’s
inequality, we obtain the uniform integrability of ${\mathcal{M}}_{2}^{\nu}$.
Our proof is complete. ∎

Once the bound of ${\mathcal{M}}_{2}$ is established, we can link the expected
value of ${\mathcal{M}}_{1}$ to that of ${\mathcal{M}}_{2}$ by the observation
that there are less steps with a small size ($<\\!\\!<{\alpha}$) than with a
large size ($>\\!\\!>{\alpha}$). This is the content of the following result.

######  Corollary 3.2.

Let ${\mathcal{M}}_{1}$ be defined in (2.4) and we are still working with a
fBm $x$ with Hurst parameter $H>1/3$. Then
$\sup_{n\in\mathbb{N}}\mathbb{E}[|{\mathcal{M}}_{1}|^{\nu}]<\infty$ for all
$\nu\geq 1$.

###### Proof.

In order to consider the integrability of ${\mathcal{M}}_{1}$ we observe that
by the definition of $S_{1}$, for $s_{j}\in S_{1}$ we have
$\omega(s_{j+1},s_{j+1}+\Delta)\geq{\alpha}/2$. Therefore,

$\displaystyle
S_{1}\subset\\{t_{k}:\omega(t_{k},t_{k+1})>{\alpha}/2\\}=:S_{3}^{\prime},$

and so

$\displaystyle{\mathcal{M}}_{1}=\prod_{s_{j}\in
S_{1}}(K_{1}\omega(s_{j},s_{j+1})^{1/p}+1)\leq(K_{1}{\alpha}+1)^{|S_{1}|}\leq
K^{|S_{3}^{\prime}|}.$

Observe that $K^{|S_{3}^{\prime}|}$ is in the form similar to
${\mathcal{M}}_{2}$ in (2.4). So in a similar way as in Theorem 3.1, we can
show that $K^{\nu|S_{3}^{\prime}|}$ and thus ${\mathcal{M}}_{1}^{\nu}$ is
uniformly integrable. ∎

###  3.2. Integrability of ${\mathcal{M}}_{0}$

In this section, we will take care of the products in (2.4) involving small
increments of $\omega$. Now recall that those increments, defined by (2.23),
involve the Gaussian process w and the second chaos process $q$. The presence
of $q$ will require a specific translation procedure on the Wiener space,
which is carried out in Section 3.2.1. Then a weighted sum argument is invoked
in Section 3.2.2.

####  3.2.1. Translation of the fBm and some functionals

Let us recall that $x$ is a fBm with $H>1/3$ and $q$ is defined in (2.20). In
this subsection, we consider an upper-bound estimate for the translation of
the fBm $x$ and the process $q$. Notice that in the sequel our generic random
element in the space $\Omega$ will be denoted by $\phi$.

######  Lemma 3.3.

Take $3>p>1/H$ and $p^{\prime}>1$ such that $1/p+1/p^{\prime}>1$. Let $h$ be a
path in $C^{p^{\prime}\text{-var}}([0,T],\mathbb{R}^{m})$, and let $T_{h}$
denote the translation operator: $T_{h}\phi=\phi+h$ on the Wiener space
related to our fBm. Then the following translation inequality holds:

$\displaystyle\|T_{h}q\|_{p/2\text{-var},\llbracket
s,t\rrbracket}^{p/2}+\|T_{h}{\bf x}\|_{p\text{-var},\llbracket
s,t\rrbracket}^{p}\leq K_{p}\left(\|q\|_{p/2\text{-var},\llbracket
s,t\rrbracket}^{p/2}+\|{\bf
x}\|_{p\text{-var},[s,t]}^{p}+\|h\|_{p^{\prime}\text{-var},[s,t]}^{p}\right),$
(3.13)

where $K_{p}$ is a constant depending only on $p$.

###### Proof.

The estimate of $\|T_{h}{\bf x}\|_{p\text{-var},\llbracket s,t\rrbracket}^{p}$
is shown in Lemma 3.1 [8]. In the following we consider the estimate of
$\|T_{h}q\|_{p/2\text{-var},\llbracket s,t\rrbracket}^{p/2}$. Specifically,
consider an element $u,v\in{\mathcal{S}}_{2}(\llbracket s,t\rrbracket)$. By
definition we can write

$\displaystyle T_{h}q_{uv}=q_{uv}+A^{1}_{uv}+A^{2}_{uv}+A^{3}_{uv}\,,$ (3.14)

where

$\displaystyle A^{1}_{uv}=\sum_{u\leq t_{k}<v}\int_{t_{k}}^{t_{k+1}}\delta
h_{t_{k}r}\otimes dx_{r}\,,\quad A^{2}_{uv}=\sum_{u\leq
t_{k}<v}\int_{t_{k}}^{t_{k+1}}\delta x_{t_{k}r}\otimes dh_{r}\,,$ (3.15)
$\displaystyle A^{3}_{uv}=\sum_{u\leq t_{k}<v}\int_{t_{k}}^{t_{k+1}}\delta
h_{t_{k}r}\otimes dh_{r}\,.$ (3.16)

Next we further decompose the term $A^{1}$ into

$\displaystyle A_{uv}^{1}=A_{uv}^{11}+A_{uv}^{12},$ (3.17)

where $A^{11}$ and $A^{12}$ are respectively defined by

$\displaystyle A^{11}_{uv}=\int_{u}^{v}\delta h_{ur}\otimes
dx_{r}\,,\quad\text{and}\quad A^{12}_{uv}=-{\mathcal{J}}_{u}^{v}(h,x),$ (3.18)

and where the term ${\mathcal{J}}$ above is given as

$\displaystyle{\mathcal{J}}_{u}^{v}(h,x)=\sum_{u\leq t_{k}<v}\delta
h_{ut_{k}}\otimes\delta x_{t_{k}t_{k+1}}\,.$ (3.19)

In the following, we bound the terms on the right side of (3.14).

First, by a direct computation for all $(u,r,v)\in{\mathcal{S}}_{3}(\llbracket
s,t\rrbracket)$ we have

$\displaystyle\delta A^{12}_{urv}=\delta h_{ur}\otimes\delta x_{rv}.$ (3.20)

In order to bound $\delta A^{12}$, we consider the function

$\displaystyle\omega(u,v)=:\|h\|_{p^{\prime}\text{-var},[u,v]}\|{\bf
x}\|_{p\text{-var},[u,v]}.$ (3.21)

It is well known that since $1/p+1/p^{\prime}>1$, $\omega$ is a control
function. In fact, it is easy to show that $\omega_{1}=:\omega^{1/\mu}$ is a
control function for $\mu$ such that $1/p+1/p^{\prime}>\mu>1$. It follows from
(3.20) and the definition of $\omega_{1}$ that

$\displaystyle|\delta A^{12}_{urv}|\leq\omega_{1}(u,v)^{\mu}.$

In addition, it is readily checked from our definition (3.19) that
$A^{12}_{t_{k}t_{k+1}}=0$ for all $t_{k}\in\llbracket s,t\rrbracket$.
Therefore a direct application of Lemma 2.4 yields:

$\displaystyle|A^{12}_{uv}|\leq
K_{\mu}\,\omega_{1}(u,v)^{\mu}=K_{\mu}\,\omega(u,v).$ (3.22)

Let us turn to the estimate of $A^{11}$ defined by (3.18). In that case, due
to the fact that $A^{11}$ can be interpreted as a Young integral, some
elementary estimates (see e.g. [33]) reveal that

$\displaystyle|A^{11}_{uv}|\leq\omega(u,v).$ (3.23)

Hence reporting (3.22) and (3.23) into (3.17) we end up with

$\displaystyle|A^{1}_{uv}|\leq(K_{\mu}+1)\omega(u,v)\leq(K_{\mu}+1)(\|h\|_{p^{\prime}\text{-var},[u,v]}^{2}+\|{\bf
x}\|_{p\text{-var},[u,v]}^{2}),$ (3.24)

where we recall that the control $\omega$ is given by (3.21). The term $A^{2}$
in (3.15) can be bounded in a similar way as for $A^{1}$, and we obtain the
same estimate as in (3.24). The details are thus omitted.

In order to bound $A^{3}$ defined by (3.16), we apply Young’s inequality again
and also the super-additivity of the control
$\omega_{2}(u,v)=:\|h\|_{p^{\prime}\text{-var},[u,v]}^{2}$ (Notice that
$\omega_{2}$ is a control owing to the fact that $p^{\prime}<2$). We get

$\displaystyle|A^{3}_{uv}|\leq\sum_{u\leq
t_{k}<v}\|h\|_{p^{\prime}\text{-var},[t_{k},t_{k+1}]}^{2}\leq\|h\|_{p^{\prime}\text{-var},[u,v]}^{2}.$

Putting together the estimates of $A^{1}$, $A^{2}$ and $A^{3}$ and equation
(3.14), we obtain

$\displaystyle|T_{h}q_{uv}|\leq|q_{uv}|+2(K_{\mu}+1)\|{\bf
x}\|_{p\text{-var},[u,v]}^{2}+(2K_{\mu}+3)\|h\|_{p^{\prime}\text{-var},[u,v]}^{2}.$
(3.25)

Now consider a generic partition $\pi=\\{u_{j}\\}$ of $\llbracket
s,t\rrbracket$. Thanks to (3.25) and super-additivity properties we have

$\displaystyle\sum_{\\{u_{j}\\}}|T_{h}q_{u_{j}u_{j+1}}|^{p/2}$
$\displaystyle\leq
K_{p}\left(\sum_{\\{u_{j}\\}}|q_{u_{j}u_{j+1}}|^{p/2}+\sum_{\\{u_{j}\\}}\|{\bf
x}\|_{p\text{-var},[u_{j},u_{j+1}]}^{p}+\sum_{\\{u_{j}\\}}\|h\|_{p^{\prime}\text{-var},[u_{j},u_{j+1}]}^{p}\right)$
$\displaystyle\leq K_{p}\left(\|q\|_{p/2\text{-var},[s,t]}^{p/2}+\|{\bf
x}\|_{p\text{-var},[s,t]}^{p}+\|h\|_{p^{\prime}\text{-var},[s,t]}^{p}\right).$

Finally, taking the sup over all partitions of $[s,t]$ on the left side we
obtain the desired estimate (3.13). ∎

####  3.2.2. Integrability of ${\mathcal{M}}_{0}$

This section is devoted to a study of the intermediate sized increments of
$\omega$. Otherwise stated, we are ready to show the uniform integrability of
${\mathcal{M}}_{0}$.

######  Theorem 3.4.

Let $S_{0}$ and ${\mathcal{M}}_{0}$ be defined in (2.4), for a fBm $x$ with
Hurst parameter $H>1/3$ and a threshold ${\alpha}>0$. Then for any given
${\gamma}<2H+1$ there exists $K=K_{{\gamma}}$ such that for all $a\geq 1$ we
have

$\displaystyle\mathbb{P}(|S_{0}|>a)\leq Ke^{-Ka^{{\gamma}}}.$ (3.26)

In particular,
$\sup_{n\in\mathbb{N}}\mathbb{E}[{\mathcal{M}}_{0}^{\nu}]<\infty$ for all
$\nu\geq 1$.

###### Proof.

The proof will be done in several steps.

Step 1. Preparations. Let us go back to inequality (3.13). Remember that
$p>1/H$ therein. Since $H>1/4$, it is easily checked that one can pick
$p^{\prime}>(H+1/2)^{-1}$ such that $p$, $p^{\prime}$ still satisfy
$\frac{1}{p}+\frac{1}{p^{\prime}}>1$. This pair of $p$, $p^{\prime}$ will be
fixed for the remainder of the proof. Recalling the constant $K_{p}$ featuring
in (3.13) and our threshold ${\alpha}$, we also choose ${\beta}>0$ small
enough so that ${\alpha}/2-K_{p}{\beta}>0$. Since $p>1/H$, according to [28,
Remark 3.6] there exists an almost surely finite random variable $G_{p}$ such
that $\sup_{n\in\mathbb{N}}\|q\|_{p/2\text{-var},\llbracket
s,t\rrbracket}^{p/2}\leq G_{p}$. Related to those quantities, we define the
following two sets:

$\displaystyle A^{n}$ $\displaystyle=$
$\displaystyle\\{\phi\in\Omega:\|q\|_{p/2\text{-var},\llbracket
s,t\rrbracket}^{p/2}+\|{\bf x}\|_{p\text{-var},[s,t]}^{p}<{\beta}\\},$
$\displaystyle A$ $\displaystyle=$ $\displaystyle\\{\phi\in\Omega:G_{p}+\|{\bf
x}\|_{p\text{-var},[s,t]}^{p}<{\beta}\\},$

where we recall that the typical element of
$(\Omega,{\mathcal{F}},\mathbb{P})$ is denoted by $\phi$. It is clear that
$A\subset A^{n}$.

Step 2. Tail inclusion relations. Let $a\geq 1$ be our generic threshold.
Having the notation of Step 1 in mind we define a constant $\kappa$ as
follows:

$\displaystyle\kappa=\left(\frac{{\alpha}/2-K_{p}{\beta}}{K_{p}}\right)^{1/p}a^{1/p^{\prime}}.$
(3.27)

Recall that the Cameron-Martin type space $\bar{{\mathcal{H}}}$ is defined in
Section 2.3. Let us call $B_{\bar{{\mathcal{H}}}}$ the unit ball in
$\bar{{\mathcal{H}}}$, namely:
$B_{\bar{{\mathcal{H}}}}=\\{h\in\bar{{\mathcal{H}}};\|h\|_{\bar{{\mathcal{H}}}}\leq
1\\}$. Our first aim is to show that

$\displaystyle A+\kappa B_{\bar{{\mathcal{H}}}}\subset A^{n}+\kappa
B_{\bar{{\mathcal{H}}}}\subset\\{|S_{0}|\leq a\\}.$ (3.28)

Suppose that $\phi\in A^{n}+\kappa B_{\bar{{\mathcal{H}}}}$. In the following,
we show that $|S_{0}|\leq a$ for such $\phi$, which then implies the relation
(3.28). First, for $\phi\in A^{n}+\kappa B_{\bar{{\mathcal{H}}}}$ we have
$\phi-h\in A^{n}$ for some $h\in\kappa B_{\bar{{\mathcal{H}}}}$, and thus

$\displaystyle\|q(\phi-h)\|_{p/2\text{-var},\llbracket
s,t\rrbracket}^{p/2}+\|{\bf x}(\phi-h)\|_{p\text{-var},[s,t]}^{p}<{\beta}.$

Recall that $T_{h}{\bf x}(\phi)={\bf x}(\phi+h)$ for any
$h\in\bar{{\mathcal{H}}}$ almost surely. Hence the above relation becomes

$\displaystyle\|T_{-h}q(\phi)\|_{p/2\text{-var},\llbracket
s,t\rrbracket}^{p/2}+\|T_{-h}{\bf
x}(\phi)\|_{p\text{-var},[s,t]}^{p}<{\beta}.$ (3.29)

We now consider the control $\omega$ defined by
$\omega(s,t)=\|q\|_{p/2\text{-var},\llbracket s,t\rrbracket}^{p/2}+\|{\bf
x}\|_{p\text{-var},\llbracket s,t\rrbracket}^{p}$. For a generic element
$\phi\in A^{n}+\kappa B_{\bar{{\mathcal{H}}}}$ we have

$\displaystyle\omega(s,t)(\phi)$ $\displaystyle=$
$\displaystyle\|T_{h}T_{-h}q(\phi)\|_{p/2\text{-var},\llbracket
s,t\rrbracket}^{p/2}+\|T_{h}T_{-h}{\bf x}(\phi)\|_{p\text{-var},\llbracket
s,t\rrbracket}^{p}.$

Hence invoking Lemma 3.3 we get

$\displaystyle\omega(s,t)(\phi)$ $\displaystyle\leq$ $\displaystyle
K_{p}\left(\|T_{-h}q(\phi)\|_{p/2\text{-var},\llbracket
s,t\rrbracket}^{p/2}+\|T_{-h}{\bf
x}(\phi)\|_{p\text{-var},[s,t]}^{p}+\|h\|^{p}_{p^{\prime}\text{-var},[s,t]}\right),$

and owing to (3.29) one ends up with the following relation valid for all
$\phi\in A^{n}+\kappa B_{\bar{{\mathcal{H}}}}$:

$\displaystyle\omega(s,t)(\phi)$ $\displaystyle\leq$ $\displaystyle
K_{p}{\beta}+K_{p}\|h\|^{p}_{p^{\prime}\text{-var},[s,t]}.$

In particular, when $s=s_{j}$ and $t=s_{j+1}$ for $s_{j}\in S_{0}$ we obtain

$\displaystyle{\alpha}/2\leq\omega(s_{j},s_{j+1})\leq
K_{p}{\beta}+K_{p}\|h\|^{p}_{p^{\prime}\text{-var},[s_{j},s_{j+1}]},$

and thus

$\displaystyle\|h\|^{p^{\prime}}_{p^{\prime}\text{-var},[s_{j},s_{j+1}]}\geq\left(\frac{{\alpha}/2-K_{p}{\beta}}{K_{p}}\right)^{p^{\prime}/p}.$
(3.30)

Since $\omega_{1}(s,t)\equiv\|h\|_{p^{\prime},[s,t]}^{p^{\prime}}$ is a
control it follows from (3.30) that

$\displaystyle\|h\|^{p^{\prime}}_{p^{\prime}\text{-var},[0,T]}\geq\sum_{s_{j}\in
S_{0}}\|h\|^{p^{\prime}}_{p^{\prime}\text{-var},[s_{j},s_{j+1}]}\geq\left(\frac{{\alpha}/2-K_{p}{\beta}}{K_{p}}\right)^{p^{\prime}/p}|S_{0}|.$
(3.31)

We now specify the left-hand side of (3.31). First since we have chosen
$p^{\prime}>(H+1/2)^{-1}$, the reference [7, Page 14] asserts that
$|h|_{\bar{{\mathcal{H}}}}\geq\|h\|_{p^{\prime}\text{-var},[0,T]}$. Moreover
we have assumed that $h\in\kappa B_{\bar{{\mathcal{H}}}}$. We thus obtain

$\displaystyle\kappa^{p^{\prime}}\geq|h|_{\bar{{\mathcal{H}}}}^{p^{\prime}}\geq\|h\|^{p^{\prime}}_{p^{\prime}\text{-var},[0,T]}.$

Plugging this inequality into (3.31), we obtain that if $\phi\in A^{n}+\kappa
B_{\bar{{\mathcal{H}}}}$ then

$\displaystyle|S_{0}|\leq\kappa^{p^{\prime}}\left(\frac{{\alpha}/2-K_{p}{\beta}}{K_{p}}\right)^{-p^{\prime}/p}=a,$

where the last identity stems from the definition (3.27) of $\kappa$. We have
thus proved that if $\phi\in A^{n}+\kappa B_{\bar{{\mathcal{H}}}}$, then
$|S_{0}|\leq a$. This concludes the proof of (3.28).

Step 3. Tail estimates. Let us introduce some extra bits of notation. Namely
we write $\Phi$ for the standard Gaussian CDF. For a set $A\subset\Omega$ we
also define $a_{A}\in\mathbb{R}$ as the number such that
$\Phi(a_{A})=\mathbb{P}(A)$. Then the isoperimetric type inequality in [26,
Theorem 4.3], together with (3.28), yield

$\displaystyle\mathbb{P}(|S_{0}|>a)\leq\mathbb{P}((A+\kappa
B_{\bar{{\mathcal{H}}}})^{c})\leq
e^{-K(a_{A}+\kappa)^{2}}=e^{K\frac{\kappa^{2}}{2}-K(a_{A}+\kappa)^{2}}e^{-K\frac{\kappa^{2}}{2}}.$

Let $K_{A}>0$ be an upper bound of the quadratic function
$f(\kappa)=K\frac{\kappa^{2}}{2}-K(a_{A}+\kappa)^{2}$ on $\mathbb{R}$. Then
considering a constant $K$ which can change from line to line and recalling
the definition (3.27) of $\kappa$, we get

$\displaystyle\mathbb{P}(|S_{0}|>a)\leq
K_{A}\,e^{-K\frac{\kappa^{2}}{2}}=K_{A}e^{-Ka^{2/p^{\prime}}}.$ (3.32)

Recall again that $p^{\prime}$ can be chosen arbitrarily close to
$(H+1/2)^{-1}$. Hence $2/p^{\prime}$ is of the form $2H+1-\varepsilon$ for a
small $\varepsilon>0$. This conclude the tail estimate (3.26). It follows
immediately from (3.26) that $|S_{0}|^{\nu}$ and thus
${\mathcal{M}}_{0}^{\nu}$ is uniformly integrable for any $\nu\geq 1$. ∎

###  3.3. Integrability of Malliavin derivatives

With the preliminary results of Sections 3.1 and 3.2 in hand, we can now turn
to the integrability result for the Malliavin derivatives of the Euler scheme.
Notice that we restrict our analysis here to the first 2 Malliavin derivatives
of $y^{n}$. However, it is clear that our estimates could be extended to
arbitrary Malliavin derivatives.

######  Theorem 3.5.

Let $y^{n}$ be the Euler scheme defined by (1.2). The first and second
Malliavin derivatives of $y^{n}$ are contained in the vector $\xi^{n}$
introduced in (2.28). We assume that the vector field $V$ is $C^{4}_{b}$ and
that $x$ is a fBm with Hurst parameter $H>1/3$. Then for all $\nu\geq 1$ we
have

$\mathbb{E}\left[\|\xi^{n}\|_{p\text{-var}}^{\nu}\right]<\infty.$ (3.33)

In particular, the following sup-norm inequality holds true:

$\mathbb{E}\Big{[}\sup_{n\in\mathbb{N},\,r,r^{\prime},t\in[0,T]}|\xi^{n}_{t}|^{\nu}\Big{]}<\infty.$
(3.34)

###### Proof.

Inequality (3.33) follows by showing that all terms in the right-hand side of
(2.29) have moments of all orders. Applying Theorem 3.4, Corollary 3.2 and
Theorem 3.1 respectively we obtain the integrability of ${\mathcal{M}}_{0}$,
${\mathcal{M}}_{1}$, ${\mathcal{M}}_{2}$. The integrability of $|S_{0}|$
follows from (3.26). The integrability of $|S_{1}|$ and $|S_{2}|$ are implied
by the relation $|S_{i}|\lesssim{\mathcal{M}}_{i}$, $i=1,2$, respectively. The
upper bound (3.34) is an easy consequence of (3.33). ∎

##  4\. Weak convergence

With our Malliavin derivative and integrability estimates in hand, in this
section we consider the weak convergence of the Euler scheme. The first
sections are preparations of the main result.

###  4.1. Estimation of an inner product in ${\mathcal{H}}^{\otimes 2}$

In this subsection, we derive a useful upper-bound estimate for an inner
product of the form
$\langle\varphi,\mathbf{1}_{[u,v]}\otimes\mathbf{1}_{[s,t]}\rangle_{{\mathcal{H}}^{\otimes
2}}$, involving some indicator functions. We first need a positivity result
for the rectangular increment function $R$ of the fBm.

######  Lemma 4.1.

Recall that the covariance $R$ is defined in (2.11), with rectangular
increments $R([u,v],[s,t])$ introduced in (2.12). Then for any
$u,v,s,t\in\mathbb{R}$ such that $s\leq u\leq v\leq t$ we have

$\displaystyle R([u,v],[s,t])\geq 0.$

###### Proof.

We first write

$\displaystyle R([u,v],[s,t])=R([u,v],[u,v])+R([u,v],[u,v]^{C}),$

where we denoted $[u,v]^{C}=[s,t]\setminus[u,v]$. Since
$R([u,v],[u,v])=(v-u)^{2H}$ it suffices to show that
$R([u,v],[u,v]^{C})\geq-(v-u)^{2H}$.

By definition (2.11)-(2.12) of $R$ we can write

$\displaystyle R([u,v],[u,v]^{C})=R([u,v],[s,u])+R([u,v],[v,t])$ (4.1)
$\displaystyle=$
$\displaystyle\frac{1}{2}(|v-s|^{2H}-|u-s|^{2H}-|v-u|^{2H})+\frac{1}{2}(|t-u|^{2H}-|t-v|^{2H}-|v-u|^{2H}).$

Note that $|v-s|^{2H}-|u-s|^{2H}$ and $|t-u|^{2H}-|t-v|^{2H}$ are nonnegative.
We thus obtain

$\displaystyle R([u,v],[u,v]^{C})\geq-|v-u|^{2H}.$ (4.2)

The proof is complete. ∎

The above positivity result leads to a surprisingly easy bound on products in
${\mathcal{H}}^{\otimes 2}$.

######  Lemma 4.2.

Let $\varphi\in{\mathcal{H}}^{\otimes 2}$. For each $s\in[0,T]$ we assume that
$\varphi(s,\cdot)\in C^{p\text{-var}}([0,T])$. Let
$f_{s}=\|\varphi(s,\cdot)\|_{p\text{-var}}$ for $s\in[0,T]$. We also assume
that $f\in C^{p\text{-var}}([0,T])$. For $s,t,u,v\in[0,T]$: $s<t$, $u<v$ we
define
${\alpha}(\eta,\zeta)=\mathbf{1}_{[u,v]}(\eta)\mathbf{1}_{[s,t]}(\zeta)$ for
$\eta,\zeta\in[0,T]$. Then the following relation holds

$\displaystyle\Big{|}\langle\varphi,{\alpha}\rangle_{{\mathcal{H}}^{\otimes
2}}\Big{|}\leq 4(t-s)^{2H}(v-u)^{2H}\|\varphi\|_{\infty}.$ (4.3)

###### Proof.

Starting from Definition 2.7 and taking limits on indicator functions of
rectangles (similarly to [16, Lemma 15.39]), one can prove that the norm in
${\mathcal{H}}^{\otimes 2}$ can be expressed as a double 2D Young integral of
the form

$\displaystyle\langle\varphi,{\alpha}\rangle_{{\mathcal{H}}^{\otimes
2}}=\langle\varphi,\mathbf{1}_{[u,v]}\otimes\mathbf{1}_{[s,t]}\rangle_{{\mathcal{H}}^{\otimes
2}}=\int_{[0,T]^{4}}\varphi(\eta,\zeta)\,\mathbf{1}_{[u,v]}(\eta^{\prime})\mathbf{1}_{[s,t]}(\zeta^{\prime})\,dR(\eta,\eta^{\prime})dR(\zeta,\zeta^{\prime}).$

One can then integrate out the $\eta^{\prime}$ and $\zeta^{\prime}$ variables
in order to get

$\langle\varphi,{\alpha}\rangle_{{\mathcal{H}}^{\otimes
2}}=\int_{[0,T]^{2}}\varphi(\eta,\zeta)\,dR(\eta,[u,v])dR(\zeta,[s,t]).$ (4.4)

We further decompose the inner product
$\langle\varphi,{\alpha}\rangle_{{\mathcal{H}}^{\otimes 2}}$ using the
identity:

$\displaystyle{\bf 1}={\alpha}_{1}+{\alpha}_{2}+{\alpha}_{3}+{\alpha}_{4},$

where the functions ${\alpha}_{1},\ldots,{\alpha}_{4}$ are given by

$\displaystyle{\alpha}_{1}(\eta,\zeta)=\mathbf{1}_{[u,v]}(\eta)\mathbf{1}_{[s,t]}(\zeta),\qquad{\alpha}_{2}(\eta,\zeta)=\mathbf{1}_{[u,v]^{C}}(\eta)\mathbf{1}_{[s,t]}(\zeta),$
$\displaystyle{\alpha}_{3}(\eta,\zeta)=\mathbf{1}_{[u,v]}(\eta)\mathbf{1}_{[s,t]^{C}}(\zeta),\qquad{\alpha}_{4}(\eta,\zeta)=\mathbf{1}_{[u,v]^{C}}(\eta)\mathbf{1}_{[s,t]^{C}}(\zeta),$

and where similarly to what we wrote in Lemma 4.1, we have set
$[s,t]^{C}=[0,T]\setminus[s,t]$ and $[u,v]^{C}=[0,T]\setminus[u,v]$. Otherwise
stated, we recast (4.4) as

$\displaystyle\langle\varphi,{\alpha}\rangle_{{\mathcal{H}}^{\otimes
2}}=\int_{[0,T]^{2}}\varphi(\eta,\zeta)\,dR(\eta,[u,v])dR(\zeta,[s,t])=\sum_{i=1}^{4}J_{T}^{i},$
(4.5)

where the terms $J_{T}^{i}$ are respectively defined by

$\displaystyle
J_{T}^{i}=\int_{[0,T]^{2}}\varphi(\eta,\zeta){\alpha}_{i}(\eta,\zeta)\,dR(\eta,[u,v])dR(\zeta,[s,t]).$

Those four terms will be handled with slightly different arguments. That is
for $J^{1}_{T}$, owing to Lemma 4.1 we have that both $dR(\zeta,[s,t])$ and
$dR(\eta,[u,v])$ are positive when $\eta\in[u,v]$ and $\zeta\in[s,t]$.
Therefore, we have

$\displaystyle|J^{1}_{T}|\leq\|\varphi\|_{\infty}R([s,t],[s,t])R([u,v],[u,v])\leq(t-s)^{2H}(v-u)^{2H}\|\varphi\|_{\infty}.$
(4.6)

For the second term $J_{T}^{2}$ in (4.5) we observe that $dR(\zeta,[s,t])$ is
positive and $dR(\eta,[u,v])$ is negative. Therefore the product
$dR(\eta,[s,t])\cdot dR(\eta,[u,v])$ does not change sign and we get

$\displaystyle|J^{2}_{T}|\leq\|\varphi\|_{\infty}|R([u,v],[u,v]^{C})R([s,t],[s,t])|.$

Hence thanks to an elementary computation similar to (4.1)-(4.2) we discover
that

$\displaystyle|J^{2}_{T}|\leq(t-s)^{2H}(v-u)^{2H}\|\varphi\|_{\infty}.$ (4.7)

In conclusion, gathering (4.6), (4.7) and similar bounds for $J^{3}_{T}$,
$J^{4}_{T}$ into (4.5), we get the desired estimate (4.3). This concludes the
proof. ∎

We now extend the previous lemma to the indicator of a simplex in $[0,T]^{2}$.

######  Lemma 4.3.

Let $\varphi\in{\mathcal{H}}^{\otimes 2}$ be as in Lemma 4.2. Let
${\beta}\in{\mathcal{H}}^{\otimes 2}$ be of the form

$\displaystyle\beta_{st}(u,v)=\mathbf{1}_{{\mathcal{S}}_{2}([s,t])}(u,v),$
(4.8)

where we recall that the simplex ${\mathcal{S}}_{2}([s,t])$ is defined in
Notation 1.2. Then there exists a constant $C_{H}$ such that the following
relation holds

$\displaystyle\Big{|}\langle\varphi,\beta_{st}\rangle_{{\mathcal{H}}^{\otimes
2}}\Big{|}\leq C_{H}(t-s)^{4H}\|\varphi\|_{\infty}.$ (4.9)

###### Proof.

We will use a dyadic partition of the function ${\beta}$. Namely for $n\geq 0$
and $0\leq i\leq 2^{n}$ we set $u_{i,n}=s+2^{-n}(t-s)i$. Next for $\ell\geq 1$
we define

$\displaystyle{\beta}_{st}^{\ell}=\sum_{n=1}^{\ell}\sum_{i=0}^{2^{n-1}-1}\mathbf{1}_{[u_{2i,n},u_{2i+1,n}]\times[u_{2i+1,n},u_{2i+2,n}]}.$

Then it can be shown that
$\|{\beta}^{\ell}_{st}-{\beta}_{st}\|_{{\mathcal{H}}^{\otimes 2}}\to 0$. In
order to prove the lemma it thus suffices to show that for all $\ell\geq 1$ we
have

$\displaystyle\Big{|}\langle\varphi,\beta_{st}^{\ell}\rangle_{{\mathcal{H}}^{\otimes
2}}\Big{|}\leq(2^{4H}-2)^{-1}(t-s)^{4H}\|\varphi\|_{\infty}.$

In the following we prove this relation with the help of Lemma 4.2. We first
observe that by the definition of ${\beta}^{\ell}$

$\displaystyle\Big{|}\langle\varphi,\beta_{st}^{\ell}\rangle_{{\mathcal{H}}^{\otimes
2}}\Big{|}\leq\sum_{n=1}^{\ell}\sum_{i=0}^{2^{n-1}-1}\Big{|}\langle\varphi,\mathbf{1}_{[u_{2i,n},u_{2i+1,n}]\times[u_{2i+1,n},u_{2i+2,n}]}\rangle_{{\mathcal{H}}^{\otimes
2}}\Big{|}.$

Applying Lemma 4.2 with
$(s,u,v,t)=(u_{2i,n},u_{2i+1,n},u_{2i+1,n},u_{2i+2,n})$, we obtain

$\displaystyle\Big{|}\langle\varphi,\beta_{st}^{\ell}\rangle_{{\mathcal{H}}^{\otimes
2}}\Big{|}\leq\sum_{n=1}^{\ell}\sum_{i=0}^{2^{n-1}-1}(2^{-n}(t-s))^{2H}(2^{-n}(t-s))^{2H}\|\varphi\|_{\infty}$
$\displaystyle=\frac{1}{2}(t-s)^{4H}\|\varphi\|_{\infty}\sum_{n=1}^{\ell}(2^{n})^{1-4H}\leq\frac{1}{2^{4H}-2}(t-s)^{4H}\|\varphi\|_{\infty}.$

This completes the proof of our claim (4.9). ∎

In the sequel we will also need an inequality for products in ${\mathcal{H}}$.
Its proof is similar to the proof of Lemma 4.3 and is omitted for sake of
conciseness.

######  Lemma 4.4.

Let $\varphi\in{\mathcal{H}}$ be a function in $C^{p\text{-var}}([0,T])$. Then
the following relation holds

$\displaystyle\Big{|}\langle\varphi,\mathbf{1}_{[s,t]}\rangle_{{\mathcal{H}}}\Big{|}\leq(t-s)^{2H}\|\varphi\|_{\infty}$

for all $(s,t)\in{\mathcal{S}}_{2}(0,T)$.

###  4.2. An extension of the sewing lemma

In this section we extend Lemma 2.4 to the integral of two controlled
processes. Our findings are summarized in the following lemma.

######  Lemma 4.5.

Let $S_{2}(x):=(x^{1},x^{2})$ be the geometric rough path above $x$ as given
in Definition 2.1, and $p<3$. We consider two couples of paths
$(z,z^{\prime})$ and $(\tilde{z},\tilde{z}^{\prime})$ with $z,\tilde{z}\in
C([s,t],\mathbb{R}^{m})$ and $z^{\prime},\tilde{z}^{\prime}\in
C([s,t],\mathbb{R}^{m\times d})$. Let
$\omega^{x}(u,v)=\|x\|_{p\text{-var},[u,v]}^{p}$ for
$(u,v)\in{\mathcal{S}}_{2}([s,t])$. We assume the existence of two controlled
functions $\omega^{z}$, $\omega_{1}^{z}$ and $\omega^{z^{\prime}}$ on
$\llbracket s,t\rrbracket$ such that for all
$(u,v)\in{\mathcal{S}}_{2}(\llbracket s,t\rrbracket)$ we have

$\displaystyle|\delta
z_{uv}-z_{u}^{\prime}x^{1}_{uv}|\leq\omega^{z}(u,v)^{2/p},\qquad|\delta
z_{uv}|\leq\omega_{1}^{z}(u,v)^{1/p},\qquad|\delta
z_{uv}^{\prime}|\leq\omega^{z^{\prime}}(u,v)^{1/p}.$ (4.10)

We also assume that the relations in (4.10) hold for $\tilde{z}$, with related
increments $\tilde{z}^{\prime},\omega^{z^{\prime}}$. Next we introduce some
new control functions:

$\displaystyle\omega^{x,z,z^{\prime}}=\omega^{x}+\omega^{z}+\omega^{z^{\prime}},\qquad\text{and}\qquad\omega^{x,z,z^{\prime}}_{1}=\omega^{x,z,z^{\prime}}+\omega_{1}^{z}\,,$
(4.11)

and similarly for $(\tilde{z},\tilde{z}^{\prime})$. We now define some
remainder terms in the integrals of $z$ with respect to $\tilde{z}$ or $x$.
Namely for $(u,v)\in{\mathcal{S}}_{2}(\llbracket s,t\rrbracket)$ we set

$\displaystyle R_{uv}^{z\tilde{z}}=\int_{u}^{v}(\delta
z_{ur}-z_{u}^{\prime}x^{1}_{ur})\otimes d\tilde{z}_{r}\qquad\text{and}\qquad
R_{uv}^{\tilde{z}x}=\int_{u}^{v}(\delta\tilde{z}_{ur}-\tilde{z}_{u}^{\prime}x^{1}_{ur})\otimes
dx_{r},$ (4.12)

where the above integrals are understood in the rough path sense. We suppose
that the increments $R$ are such that for any point $t_{k}$ in our generic
partition of $[s,t]$ we have

$\displaystyle|R^{z\tilde{z}}_{t_{k}t_{k+1}}|\leq\omega^{x}(t_{k},t_{k+1})^{3/p}\,\qquad\text{and}\qquad|R^{\tilde{z}x}_{t_{k}t_{k+1}}|\leq\omega^{x}(t_{k},t_{k+1})^{3/p}.$
(4.13)

Then the following relation holds for all
$(u,v)\in{\mathcal{S}}_{2}(\llbracket s,t\rrbracket)$:

$\displaystyle\Big{|}R^{z\tilde{z}}_{uv}\Big{|}\leq
K_{p}[\omega^{R}(u,v)]^{\mu},$ (4.14)

where $\mu>1$ is a given constant and where $K_{p}>0$ is a constant depending
on $p$. In (4.14), the control $\omega^{R}$ is also defined by the relation

$[\omega^{R}(u,v)]^{\mu}:=\left(\omega^{z}(u,v)^{2/p}+\omega^{z^{\prime}}(u,v)^{1/p}\omega^{x}(u,v)^{1/p}\right)\omega_{1}^{x,\tilde{z},\tilde{z}^{\prime}}(u,v)^{1/p}\\\
\times\left(\omega^{x,\tilde{z},\tilde{z}^{\prime}}(u,v)^{1/p}+\|\tilde{z}^{\prime}\|_{\infty,[u,v]}+1\right)\,.$
(4.15)

###### Proof.

The proof of the lemma is an application of Lemma 2.4. Namely the existence of
$R$ as a rough integral is ensured by general rough paths considerations (see
e.g. [20]). Then some elementary manipulations starting from the definition
(4.12) of $R$ show that for $(u,s,v)\in{\mathcal{S}}_{3}(\llbracket
0,T\rrbracket)$ we have

$\displaystyle\delta R^{z\tilde{z}}_{usv}=(\delta
z_{us}-z_{u}^{\prime}x^{1}_{us})\otimes\delta\tilde{z}_{sv}+z_{us}^{\prime}\int_{s}^{v}x^{1}_{sr}\otimes
d\tilde{z}_{r}\,,$ (4.16)

where we recall that $\delta$ is defined as in (2.3).

We first consider the case when $(z,\tilde{z})=(\tilde{z},x)$, that is the
remainder $R^{\tilde{z}x}$ defined in (4.12). In this case one can recast
(4.16) as

$\displaystyle\delta
R^{\tilde{z}x}_{usv}=(\delta\tilde{z}_{us}-\tilde{z}_{u}^{\prime}x^{1}_{us})\otimes
x_{sv}^{1}+\tilde{z}_{us}^{\prime}\,x^{2}_{sv}.$ (4.17)

Applying the conditions in (4.10) we thus get

$\displaystyle|\delta
R^{\tilde{z}x}_{usv}|\leq\omega^{\tilde{z}}(u,v)^{2/p}\cdot\omega^{x}(u,v)^{1/p}+\omega^{\tilde{z}^{\prime}}(u,v)^{1/p}\cdot\omega^{x}(u,v)^{2/p}.$
(4.18)

Moreover, we have assumed that (4.13) holds true for the increments
$R^{\tilde{z}x}_{t_{k}t_{k+1}}$. Hence a direct application of Lemma 2.4
implies that (4.18) also holds when $\delta R^{\tilde{z}x}_{usv}$ is replaced
by $R^{\tilde{z}x}_{uv}$, that is relation (4.14) holds for $R^{\tilde{z}x}$
defined by (4.12).

In order to prove (4.14) for a general $\tilde{z}$, let us first bound the
integral $\int_{s}^{v}x^{1}_{sr}\otimes d\tilde{z}_{r}$ in (4.16). To this
aim, we observe that a simple integration by parts (valid for rough integrals
thanks to a limiting procedure on smooth approximations) yields the relation

$\int_{s}^{v}x^{1}_{sr}\otimes
d\tilde{z}_{r}=x^{1}_{sv}\otimes\delta\tilde{z}_{sv}-\int_{s}^{v}\delta\tilde{z}_{sr}\otimes
dx_{r}\,.$ (4.19)

Next owing to our conditions (4.10) for $\tilde{z}$, the first term in the
right hand side of (4.19) is bounded by

$\displaystyle|x^{1}_{sv}\otimes\delta\tilde{z}_{sv}|\leq\omega^{x}(s,v)^{1/p}\omega_{1}^{\tilde{z}}(s,v)^{1/p}.$
(4.20)

For the second term in the right-hand side of (4.19), let us write

$\displaystyle\int_{s}^{v}\delta\tilde{z}_{sr}\otimes
dx_{r}=\tilde{z}_{s}^{\prime}\otimes x^{2}_{sv}+R^{\tilde{z}x}_{sv}.$

Since we have obtained that (4.14) holds for $R^{\tilde{z}x}$, for every
$s\leq u<v\leq t$ we end up with

$\Big{|}\int_{u}^{v}\delta\tilde{z}_{ur}\otimes
dx_{r}\Big{|}\leq|R^{\tilde{z}x}_{uv}|+|\tilde{z}_{u}^{\prime}|\omega^{x}(u,v)^{2/p}\\\
\leq\omega^{\tilde{z}}(u,v)^{2/p}\cdot\omega^{x}(u,v)^{1/p}+\omega^{\tilde{z}^{\prime}}(u,v)^{1/p}\cdot\omega^{x}(u,v)^{2/p}+\|\tilde{z}^{\prime}\|_{\infty,[u,v]}\,\omega^{x}(u,v)^{2/p}.$
(4.21)

We can now safely plug (4.20) and (4.21) (with $u$ replaced by $s$) into
relation (4.19). This yields the estimate

$\Big{|}\int_{s}^{v}x^{1}_{sr}\otimes
d\tilde{z}_{r}\Big{|}\leq\omega^{\tilde{z}}(u,v)^{2/p}\cdot\omega^{x}(u,v)^{1/p}+\omega^{\tilde{z}^{\prime}}(u,v)^{1/p}\cdot\omega^{x}(u,v)^{2/p}\\\
+\|\tilde{z}^{\prime}\|_{\infty,[u,v]}\cdot\omega^{x}(u,v)^{2/p}+\omega^{x}(s,v)^{1/p}\omega_{1}^{\tilde{z}}(s,v)^{1/p}.$
(4.22)

Let us now return to relation (4.16). By a simple application of (4.10) one
discovers that

$\displaystyle|\delta
R^{z\tilde{z}}_{usv}|\leq\omega^{z}(u,v)^{2/p}\cdot\omega_{1}^{\tilde{z}}(u,v)^{1/p}+\omega^{z^{\prime}}(u,v)^{1/p}\cdot\Big{|}\int_{s}^{v}x^{1}_{sr}\otimes
d\tilde{z}_{r}\Big{|}.$

Inserting (4.22) into this relation, we thus obtain

$\Big{|}\delta R^{z\tilde{z}}_{usv}\Big{|}\leq
K_{p}\left(\omega^{z}(u,v)^{2/p}+\omega^{z^{\prime}}(u,v)^{1/p}\omega^{x}(u,v)^{1/p}\right)\omega_{1}^{x,\tilde{z},\tilde{z}^{\prime}}(u,v)^{1/p}\\\
\times\left(\omega^{x,\tilde{z},\tilde{z}^{\prime}}(u,v)^{1/p}+\|\tilde{z}^{\prime}\|_{\infty,[u,v]}+1\right).$
(4.23)

Note that the right-hand side of (4.23) is equal to $K_{p}\omega^{R}(u,v)$
defined in (4.15). Taking (4.13) into account, another use of Lemma 2.4
(together with an application of [16, Exercise 1.9 (iii)] to show that
${\omega}^{R}$ is a control), proves our claim (4.14). ∎

###  4.3. Interpolation of the Euler method

In this section we shall extend our Euler scheme to a process in continuous
time and obtain some uniform bounds. Specifically, recall that the Euler
approximation $y^{n}$ is defined on $\llbracket 0,T\rrbracket$ by (1.2) and
for convenience we will take $V_{0}\equiv 0$. For $t$ in the continuous
interval $[0,T]$ we shall use the following interpolation:

$\displaystyle\delta y^{n}_{t_{k}t}=$ $\displaystyle V(y^{n}_{t_{k}})\delta
x_{t_{k}t}+\frac{1}{2}\sum_{j=1}^{d}\partial
V_{j}V_{j}(y^{n}_{t_{k}})(t-t_{k})^{2H},\qquad t\in[t_{k},t_{k+1}].$ (4.24)

Recall that ${\bf x}=S_{2}(x)=(x^{1},x^{2})$ denotes the rough path above $x$
(as given in Definition 2.1).

In the sequel we will also need some continuous interpolations of the
processes $q$ and $q^{b}$, which had been defined on the grid in (2.20).
Namely for $(s,t)\in{\mathcal{S}}_{2}([0,T])$ such that $t_{k_{1}}\leq
s<t_{k_{1}+1}<\cdots<t_{k_{2}-1}\leq t<t_{k_{2}}$, we define

$\displaystyle q_{st}^{ij}=\sum_{k_{1}\leq k<k_{2}}\left(x_{t_{k}\vee
s,t_{k+1}\wedge t}^{2,ij}-\frac{1}{2}(t_{k+1}\wedge t-t_{k}\vee
s)^{2H}\mathbf{1}_{\\{i=j\\}}\right)$ (4.25) $\displaystyle
q^{b,ij}_{st}=\sum_{k_{1}\leq k<k_{2}}\left(b_{t_{k}\vee s,t_{k+1}\wedge
t}^{2,ij}-\frac{1}{2}(t_{k+1}\wedge t-t_{k}\vee
s)^{2H}\mathbf{1}_{\\{i=j\\}}\right).$ (4.26)

With the above definition (4.25) in hand, we will also extend the definition
of $\omega$ from the grid to $[0,T]$.

######  Lemma 4.6.

Recall that $\omega$ has been defined on ${\mathcal{S}}_{2}(\llbracket
0,T\rrbracket)$ by (2.23) for a fixed partition length parameter $n$, and that
we are considering $p<3$. We now extend $\omega$ to ${\mathcal{S}}([0,T])$ by
setting:

$\displaystyle\omega(s,t)=\|{\bf
w}\|_{p\text{-var};[s,t]}^{p}+\|q\|_{p/2\text{-var};[s,t]}^{p/2}+\|q^{b}\|_{p/2\text{-var};[s,t]}^{p/2}+|t-s|.$
(4.27)

Then $\omega$ is a control on $[0,T]$. In other words, $\omega$ is super-
additive, continuous and vanishes on the diagonal.

###### Proof.

We first note that

$\displaystyle\|q\|_{p/2\text{-var};[s,t]}^{p/2}\leq\|q\|_{p/2\text{-var};\llbracket
s,t\rrbracket}^{p/2}+\|{\bf w}\|_{p\text{-var};[s,t]}^{p}+|t-s|,$ (4.28)

and the same kind of inequality holds true for $q^{b}$. Therefore
$\omega(s,t)$ is finite almost surely. Thanks to the definition (4.27) of
$\omega$ it is also readily checked that the superadditivity and zero on the
diagonal properties hold true.

It remains to show the continuity of $\omega$. To this aim, taking into
account the definition (4.27) of $\omega$, it is easily seen that we only have
to focus on the increments $q$ and $q^{b}$ in (4.25)-(4.26). Moreover $q$ and
$q^{b}$ are handled exactly in the same way. Hence we will just focus our
attention on $\tilde{\omega}$ given by

$\displaystyle\tilde{\omega}(s,t):=\|q\|_{p/2\text{-var};[s,t]}^{p/2}\,.$
(4.29)

Take $s<u<t<\eta(u)+\Delta$, where we recall from Notation 1.2 that $\eta(u)$
is the largest $t_{k}\in\llbracket 0,T\rrbracket$ such that $t_{k}\leq u$. In
the following we show that $\tilde{\omega}(s,t)-\tilde{\omega}(s,u)\to 0$ as
$t-u\to 0$, which is one of the main steps towards the continuity of
$\tilde{\omega}$.

Owing to the definition (4.29) of $\tilde{\omega}$, for any $\varepsilon>0$,
we can find a partition of $[s,t]$, denoted by $s=v_{0}<\cdots<v_{N}=t$ such
that

$\displaystyle\tilde{\omega}(s,t)\leq\sum_{i=0}^{N-1}|q_{v_{i}v_{i+1}}|^{p/2}+\varepsilon\,.$
(4.30)

Suppose that $u\in[v_{i_{0}},v_{i_{0}+1}]:=[v,v^{\prime}]$. Then we can bound
the summation in (4.30) by the following

$\displaystyle\tilde{\omega}(s,t)$ $\displaystyle\leq$
$\displaystyle\sum_{i=0}^{i_{0}-1}|q_{v_{i}v_{i+1}}|^{p/2}+|q_{vv^{\prime}}|^{p/2}+\sum_{i=i_{0}+1}^{N-1}|q_{v_{i}v_{i+1}}|^{p/2}+\varepsilon$
(4.31) $\displaystyle\leq$
$\displaystyle\tilde{\omega}(s,v)+|q_{vv^{\prime}}|^{p/2}+\tilde{\omega}(v^{\prime},t)+\varepsilon.$

Now go back to (4.28) for $v^{\prime}$ and $t$ and pick $u$ close enough to
$t$ so that $v^{\prime}\in[u,t]$ satisfies $\|q\|_{p/2\text{-var};\llbracket
v^{\prime},t\rrbracket}=0$ (since our grid $\llbracket u^{\prime},t\rrbracket$
has fixed mesh $T/n$, this is easily seen when $u\to t$, owing to our
expression (4.25)). One can thus recast (4.28) as

$\displaystyle\tilde{\omega}(v^{\prime},t):=\|q\|_{p/2\text{-var};[v^{\prime},t]}^{p/2}\leq\|{\bf
w}\|_{p\text{-var};[v^{\prime},t]}^{p}+|t-v^{\prime}|.$ (4.32)

It is then easily seen from (4.32) that $\lim_{v^{\prime}\to
t}\tilde{\omega}(v^{\prime},t)=0$. Since $u<v^{\prime}<t$, we will pick $u$
close enough to $t$ so that $\tilde{\omega}(v^{\prime},t)\leq\varepsilon$.
Plugging this information into (4.31), we obtain

$\displaystyle\tilde{\omega}(s,t)\leq\tilde{\omega}(s,v)+|q_{vv^{\prime}}|^{p/2}+2\varepsilon.$
(4.33)

In addition, if $u\to t$ we also have $|v^{\prime}-u|\to 0$. Therefore, basic
continuity properties of $q$ ensure that
$\big{|}|q_{vv^{\prime}}|^{p/2}-|q_{vu}|^{p/2}\big{|}\leq\varepsilon$ if $u$
is close enough to $t$. Hence (4.33) becomes

$\tilde{\omega}(s,t)\leq\tilde{\omega}(s,v)+|q_{vu}|^{p/2}+3\varepsilon\leq\tilde{\omega}(s,u)+3\varepsilon,$

where we have used the super-additivity property of $\tilde{\omega}$ for the
second inequality. Since $\tilde{\omega}(s,t)\geq\tilde{\omega}(s,u)$ by
monotonicity properties, we have obtained

$\displaystyle|\tilde{\omega}(s,t)-\tilde{\omega}(s,u)|\leq 3\varepsilon,$
(4.34)

for all $(s,u,t)\in{\mathcal{S}}_{3}([0,T])$ such that $|t-u|$ is sufficiently
small. Since $\varepsilon$ in (4.34) can be arbitrarily small, this proves
that $\lim_{u\to t}\tilde{\omega}(s,u)=\tilde{\omega}(s,t)$. The same kind of
arguments also show that $\lim_{u\to
s}\tilde{\omega}(u,t)=\tilde{\omega}(s,t)$, which completes our proof. ∎

We now go back to the interpolated version of our Euler scheme $y^{n}$. In the
following we show that $(y^{n},x,b)$ is a rough path, which is an important
step in the convergence analysis.

######  Lemma 4.7.

Consider the interpolated Euler scheme introduced in (4.24). Recall that $x$
is our driving fBm and $b$ is another fBm with parameter $H>1/3$, independent
of $x$. Also recall that the augmented process $\emph{w}=(x,b)$ has been
introduced in (2.22). We assume that the vector field $V$ sits in $C^{4}_{b}$.
Then the following holds true.

_(i)_ Denote by $Z$ the couple $Z=(y^{n},\emph{w})$. Then $Z$ admits a lift
$S_{2}(Z)$ according to Definition 2.1. Moreover, recalling the sets $S_{0}$,
$S_{1}$ in (2.25), consider $s_{j}\in S_{0}\cup S_{1}$. Then for all
$(s,t)\subset{\mathcal{S}}_{2}([s_{j},s_{j+1}])$ we have the following uniform
bound in $n$:

$\displaystyle\|S_{2}(Z)\|_{p\text{-var},[s,t]}\leq K\cdot\omega(s,t)^{1/p},$
(4.35)

where $p>1/H$, $K$ is a constant depending on $V$ and where the control
$\omega$ is defined in (4.27).

_(ii)_ Under the setting of Theorem 2.10, we now consider the vector-valued
stochastic process
$\tilde{Z}=(y^{n},D_{r}y^{n},D_{r^{\prime}}D_{r}y^{n},\emph{w})$ for
$r,r^{\prime}\in[0,T]$. We still assume that $(s,t)\subset(s_{j},s_{j+1})$
with $s_{j}\in S_{0}\cup S_{1}$. Then we have (still uniformly in $n$):

$\displaystyle\|S_{2}(\tilde{Z})\|_{p\text{-var},[s,t]}\leq
K\cdot\omega(s,t)^{1/p}\cdot{\mathcal{G}}^{2},$ (4.36)

where the quantities ${\mathcal{G}}$ have been introduced in (2.30).

###### Proof.

We first prove (i). Take $(s,t)\subset(s_{j},s_{j+1})$ such that $s_{j}\in
S_{0}\cup S_{1}$. Theorem 2.10, applied for $L=0$, shows that for
$(s,t)\in{\mathcal{S}}_{2}(\llbracket s_{j},s_{j+1}\rrbracket)$ we have

$\displaystyle|\delta y_{st}^{n}|\leq K\omega(s,t)^{1/p}|S_{0}\cup S_{1}\cup
S_{2}|\quad\text{and}\quad|\delta y_{st}^{n}-V(y_{s}^{n})\delta x_{st}|\leq
K\omega(s,t)^{2/p}\,,$ (4.37)

where $\omega$ is the control given in (4.27). Using standard interpolation
methods, it can be shown in a straightforward way that the relation for
$|\delta y^{n}_{st}|$ in (4.37) still holds if
$(s,t)\in{\mathcal{S}}_{2}([s_{j},s_{j+1}])$. So in order to prove (4.35) it
remains to show that $\int_{s}^{t}y^{n}_{su}\otimes d\text{w}_{u}$ and
$\int_{s}^{t}y^{n}_{su}\otimes dy^{n}_{u}$ are bounded by $\omega(s,t)^{2/p}$.

Consider the following remainder process for $(s,t)\subset(s_{j},s_{j+1})$
such that $s_{j}\in S_{0}\cup S_{1}$:

$\displaystyle R_{st}=\int_{s}^{t}\left(\delta y^{n}_{su}-V(y^{n}_{s})\delta
x_{su}\right)\otimes d\text{w}_{u}.$ (4.38)

Note that $R$ is a remainder of the form $R^{y^{n}w}$, defined as in (4.12).
In order to apply Lemma 4.5 to this remainder, we need to check that
$R_{t_{k}t_{k+1}}\leq\omega(t_{k},t_{k+1})^{3/p}$ as in (4.13). Now according
to (4.38) we have

$\displaystyle R_{t_{k}t_{k+1}}=\int_{t_{k}}^{t_{k+1}}\left(\delta
y^{n}_{t_{k}u}-V(y^{n}_{t_{k}})\delta x_{t_{k}u}\right)\otimes d\text{w}_{u}.$
(4.39)

Furthermore, owing to our interpolation formula (4.24), for all
$u\in[t_{k},t_{k+1}]$ we have

$\displaystyle\delta y^{n}_{t_{k}u}-V(y^{n}_{t_{k}})\delta x_{t_{k}u}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\sum_{j=1}^{d}\partial
V_{j}V_{j}(y^{n}_{t_{k}})(u-t_{k})^{2H}.$

Reporting this identity into (4.39), we end up with

$\displaystyle R_{t_{k}t_{k+1}}=\frac{1}{2}\sum_{j=1}^{d}\partial
V_{j}V_{j}(y^{n}_{t_{k}})\otimes\int_{t_{k}}^{t_{k+1}}(u-t_{k})^{2H}d\text{w}_{u}.$
(4.40)

The stochastic integral in the right-hand side of (4.40) can be interpreted in
the Young sense, and it is easy to see that
$|R_{t_{k}t_{k+1}}|\lesssim\omega(t_{k},t_{k+1})^{3/p}$, where $\omega$ is
still the control introduced in (4.27). This proves (4.13) for the remainder
$R$, and therefore one can safely apply Lemma 4.5 in order to get that,
provided $s_{j}\in S_{0}\cup S_{1}$,

$\displaystyle|R_{st}|\leq K\omega(s,t)^{3/p}\qquad\text{ for
}(s,t)\in{\mathcal{S}}_{2}(\llbracket s_{j},s_{j+1}\rrbracket).$ (4.41)

Going back to (4.38), observe that we have

$\displaystyle\int_{s}^{t}\delta y^{n}_{su}\otimes
d\text{w}_{u}=V(y^{n}_{s})\int_{s}^{t}\delta x_{su}\otimes
d\text{w}_{u}+R_{st}.$

Invoking (4.41) and since $x$ is part of the rough path w, we easily get

$\displaystyle\Big{|}\int_{s}^{t}\delta y^{n}_{su}\otimes
d\text{w}_{u}\Big{|}\leq K\omega(s,t)^{2/p},\qquad\text{ for
}(s,t)\in{\mathcal{S}}_{2}(\llbracket s_{j},s_{j+1}\rrbracket).$ (4.42)

In the following we extend the above estimate for $\int_{s}^{t}\delta
y^{n}_{su}d\text{w}_{u}$ to any $(s,t)\in{\mathcal{S}}_{2}([s_{j},s_{j+1}])$.
That is we take $(s,t)$ such that $s\in[t_{k},t_{k+1}]$ and
$t\in[t_{k^{\prime}},t_{k^{\prime}+1}]$. We write

$\displaystyle\int_{s}^{t}\delta y^{n}_{su}\otimes d\text{w}_{u}$
$\displaystyle=$ $\displaystyle\int_{s}^{t_{k+1}}\delta y^{n}_{su}\otimes
d\text{w}_{u}+\int_{t_{k+1}}^{t_{k^{\prime}}}\delta y^{n}_{su}\otimes
d\text{w}_{u}+\int_{t_{k^{\prime}}}^{t}\delta y^{n}_{su}\otimes d\text{w}_{u}$
$\displaystyle=$ $\displaystyle\int_{s}^{t_{k+1}}\delta y^{n}_{su}\otimes
d\text{w}_{u}+\delta
y^{n}_{st_{k+1}}\otimes\delta\text{w}_{t_{k+1}t_{k^{\prime}}}+\int_{t_{k+1}}^{t_{k^{\prime}}}\delta
y^{n}_{t_{k+1}u}\otimes d\text{w}_{u}$ $\displaystyle\quad+\delta
y^{n}_{st_{k^{\prime}}}\otimes\delta\text{w}_{t_{k^{\prime}}t}+\int_{t_{k^{\prime}}}^{t}\delta
y^{n}_{t_{k^{\prime}}u}\otimes d\text{w}_{u}=:\sum_{i=1}^{5}I_{i}.$

Let us bound the terms $I_{1},\ldots,I_{5}$ above. First it follows from
(4.42) that $|I_{3}|\leq K\omega(s,t)^{2/p}$. It is also clear that $|I_{2}|$
and $|I_{4}|$ are bounded by the same estimate $\omega(s,t)^{2/p}$. In order
to bound $I_{1}$, observe that according to our interpolation formula (4.24)
we have

$I_{1}=V(y^{n}_{t_{k}})\int_{s}^{t_{k+1}}\delta x_{su}\otimes d\text{w}_{u}\\\
+\frac{1}{2}\sum_{j=1}^{d}\partial
V_{j}V_{j}(y^{n}_{t_{k}})\otimes\int_{s}^{t_{k+1}}\left[(u-t_{k})^{2H}-(s-t_{k})^{2H}\right]d\text{w}_{u}.$
(4.43)

It is clear that the first integral in (4.43) is bounded by
$\omega(s,t_{k+1})^{2/p}$. Note also that
$(u-t_{k})^{2H}-(s-t_{k})^{2H}\leq(u-s)^{2H}$. So applying Young’s inequality
we obtain that the second integral in (4.43) is bounded by
$(t_{k+1}-s)^{2H}\omega(s,t_{k+1})^{1/p}$. Combining these two estimates we
obtain that $|I_{1}|\lesssim\omega(s,t_{k+1})^{2/p}$. The term $I_{5}$ is
bounded in the similar way. Putting together our upper bounds on $I_{1}$, …,
$I_{5}$, we have thus obtained that (4.42) holds for any
$(s,t)\in{\mathcal{S}}_{2}([s_{j},s_{j+1}])$. Summarizing our considerations
so far, we have proved that

$\displaystyle\|M^{1}\|_{p\text{-var};[s,t]}\leq K\omega(s,t)\qquad\text{with
}M_{uv}^{1}:=\int_{u}^{v}\delta y^{n}_{ur}\otimes d\text{w}_{r}.$ (4.44)

The proof is now finished along the same arguments. Namely the increment
$M_{uv}^{2}:=\int_{u}^{v}\delta y^{n}_{ur}\otimes dy^{n}_{ur}$ can be bounded
similarly to $M^{1}$, leading to the same inequality as (4.44). This proves
our claim (4.35).

Eventually (4.36) is also obtained along the same lines. We just apply Theorem
2.10 with $L=1,2$ in the arguments, and also write
${\mathcal{G}}^{2}\cdot\omega(s,t)$ instead of $\omega(s,t)$ in all
inequalities for the remainders. This completes our proof. ∎

###  4.4. Integrability of some linear equations

Our convergence estimates are based on linearization procedures. In this
section we bound some related linear differential equations. We start by
defining the objects we wish to study.

######  Definition 4.8.

Recall that every $V^{i}$ has to be seen, for $i=1,\dots,m$, as a smooth
vector field on $\mathbb{R}^{m}$. Let $y^{n}$ be the interpolated scheme
(4.24). Then for $i=1,\dots,m$ we define an averaged $\mathbb{R}^{d\times
m}$-valued process
$\tilde{V}^{i}(t)=\\{\tilde{V}_{ji^{\prime}}^{i}(t);\,j=1,\dots,d,\,i^{\prime}=1,\dots,m\\}$.
This process is indexed by $t\in[0,T]$ and is given by

$\displaystyle\tilde{V}_{ji^{\prime}}^{i}(t)=\int_{0}^{1}\partial_{i^{\prime}}V^{i}_{j}(\theta
y_{t}+(1-\theta)y^{n}_{t})d\theta.$

We also define a $(\mathbb{R}^{d\times m})^{m}$-valued process as
$\tilde{V}(t)=(\tilde{V}^{1}(t),\dots,\tilde{V}^{m}(t))$.

We are now ready to define the linear equation we wish to analyze.

######  Definition 4.9.

Let $\tilde{V}$ be the $(\mathbb{R}^{d\times m})^{m}$-valued process
introduced in Definition 4.8. We will call ${\Gamma}$ the $\mathbb{R}^{m\times
m}$-valued solution to the following systems of equations on $[0,T]$:

$\displaystyle{\Gamma}_{t}^{ii^{\prime}}=\text{Id}_{ii^{\prime}}+\sum_{j=1}^{d}\sum_{i^{\prime\prime}=1}^{m}\int_{0}^{t}\tilde{V}^{i}_{ji^{\prime\prime}}(s){\Gamma}_{s}^{i^{\prime\prime}i^{\prime}}dx_{s}^{j},\quad\text{for}\quad
i,i^{\prime}\in\\{1,\ldots,m\\}.$ (4.45)

For conciseness we will simply write (4.45) as
${\Gamma}_{t}=\text{Id}+\int_{0}^{t}\tilde{V}(s){\Gamma}_{s}\,dx_{s}$. We also
denote by ${\Lambda}$ the inverse of ${\Gamma}$, namely ${\Lambda}$ is defined
by the relation ${\Lambda}_{t}{\Gamma}_{t}\equiv\text{Id}$.

Our next lemma presents an important estimate for the processes ${\Gamma}$ and
${\Lambda}$ defined above.

######  Lemma 4.10.

Let the assumption be as in Theorem 2.10. Let $p>\frac{1}{H}$ and $q$ be such
that $\frac{1}{p}+\frac{1}{q}>1$. Let ${\Gamma}$ and ${\Lambda}$ be defined in
(4.45). Then

_(a)_ For all $(s,t)\in{\mathcal{S}}_{2}([0,T])$ we have

$|\delta{\Gamma}_{st}|+\sup_{r\in[0,T]}|D_{r}(\delta{\Gamma}_{st})|+\sup_{r,r^{\prime}\in[0,T]}|D_{rr^{\prime}}^{2}(\delta{\Gamma}_{st})|\\\
\leq K\cdot\omega(s,t)^{1/p}\cdot|S_{0}\cup S_{1}\cup
S_{2}|\cdot{\mathcal{M}}_{0}\cdot{\mathcal{M}}_{1}\cdot\exp({\mathcal{N}}),$
(4.46)

where $\omega$ is defined in (4.27) and ${\mathcal{N}}$ is some random
variable such that ${\mathcal{N}}^{1/q}$ has a Gaussian tail. The relation
still holds when ${\Gamma}$ is replaced by ${\Lambda}$.

_(b)_ Both processes ${\Gamma}$ and ${\Lambda}$ and their Malliavin
derivatives are uniformly integrable. Precisely, for all $p\geq 1$ we have we
have

$\displaystyle\mathbb{E}\left[\sup_{n\in\mathbb{N},r,r^{\prime},t\in[0,T]}\left(|{\Gamma}_{t}|^{p}+|D_{r}({\Gamma}_{t})|^{p}+|D_{rr^{\prime}}^{2}({\Gamma}_{t})|^{p}\right)\right]<\infty.$
(4.47)

###### Proof.

Applying Corollary 3.2 and Theorem 3.4 to the right-hand side of (4.46) we
conclude the integrability relation in (4.47). It thus remains to prove
relation (4.46).

In the following we prove the estimate (4.46) for ${\Gamma}$. Note that due to
the fact that the initial condition Id in (4.45) is nondegenerate, the process
${\Lambda}$ is well-defined and satisfies a differential equation which is
very similar to ${\Gamma}$ (see e.g. [7, 22] for more details). Therefore the
estimate of ${\Lambda}$ in (4.46) can be obtained by following the same steps
as for ${\Gamma}$. The proof for ${\Lambda}$ and its derivatives is thus
omitted.

With our Definition 4.8 in mind, let us first introduce an auxiliary process
$\xi$ given, for $i,i^{\prime\prime}\in\\{1,\dots,m\\}$ and $t\in[0,T]$, by

$\displaystyle\xi_{t}^{ii^{\prime\prime}}=\sum_{j=1}^{d}\int_{0}^{t}\tilde{V}^{i}_{ji^{\prime\prime}}(s)dx_{s}^{j}.$

We now separate the estimates for $\xi$ into two different cases

  1. (i)

_Case $(s,t)\subset(s_{j},s_{j+1})$ such that $s_{j}\in S_{0}\cup S_{1}$._ In
this situation, since $(y^{n},x,b)$ is a rough path (see Lemma 4.7 (i)) and
$y$ is a process controlled by $x$, it is readily checked that for all
$(s,t)\subset(s_{j},s_{j+1})$ such that $s_{j}\in S_{0}\cup S_{1}$ we have:

$\displaystyle\|S_{2}(\xi)\|_{p\text{-var},[s,t]}\leq
K\cdot\omega(s,t)^{1/p}.$ (4.48)

Furthermore, note that one can recast equation (4.45) as a linear system of
the form

$\displaystyle
d{\Gamma}_{t}^{ii^{\prime}}=\sum_{i^{\prime\prime}=1}^{m}{\Gamma}_{t}^{i^{\prime\prime}i^{\prime}}d\xi_{t}^{ii^{\prime\prime}}.$
(4.49)

Observe that the path $\xi$ is a functional of the process $Z$ introduced in
Lemma 4.7. Now we recall from [16, Theorem 10.53] that for a linear equation
like (4.49), there exist two constants $C_{1}$, $C_{2}$ such that

$\displaystyle|S_{2}({\Gamma},Z)_{st}|$ $\displaystyle\leq$ $\displaystyle
C_{1}|{\Gamma}_{s}|\cdot\|S_{2}(\xi)\|_{p\text{-var},[s,t]}\cdot\exp(C_{2}\|S_{2}(\xi)\|_{p\text{-var},[s,t]})$
(4.50) $\displaystyle\leq$ $\displaystyle
C_{1}|{\Gamma}_{s}|\cdot\omega(s,t)^{1/p}\exp(C_{2}\omega(s,t)^{1/p}),$

where the second relation stems from (4.48). In addition, we have chosen
$s_{j}\in S_{0}\cup S_{1}$. Therefore, one can simplify (4.50) and obtain that
for any $(s,t)\in{\mathcal{S}}_{2}([s_{j},s_{j+1}])$,

$\displaystyle\|S_{2}({\Gamma},Z)\|_{p\text{-var},[s,t]}\leq
K\omega(s,t)^{1/p}\cdot|{\Gamma}_{s}|\,.$ (4.51)

  2. (ii)

_Case $(s,t)\subset(s_{j},s_{j+1})$ such that $s_{j}\in S_{2}$._ For $s_{j}\in
S_{2}$ equation (4.45) is a linear equation driven by $x$ and so we can apply
the integrability result in [8] to get

$\displaystyle\|S_{2}({\Gamma})\|_{p\text{-var},[s,t]}\leq
K|{\Gamma}_{s}|\cdot\|{\bf
x}\|_{p\text{-var},[s,t]}\cdot\exp({\mathcal{N}}_{s_{j}s_{j+1}}),$ (4.52)

where ${\mathcal{N}}_{s_{j}s_{j+1}}$ is a random variable such that, denoting
${\mathcal{N}}:=\sum_{s_{j}\in S_{2}}{\mathcal{N}}_{s_{j}s_{j+1}}$, the random
variable $\exp(K\cdot{\mathcal{N}})$ is integrable for any constant $K>0$.

Our estimate (4.46) for $\delta{\Gamma}_{st}$ is now easily obtained. Namely
we iterate (4.51)-(4.52). This yields

$\displaystyle|\delta{\Gamma}_{st}|\leq K\left(\sum_{s_{j}\in S_{0}\cup
S_{1}}\omega(s_{j},s_{j+1})^{1/p}+\sum_{s_{j}\in S_{2}}\|{\bf
x}\|_{p\text{-var};[s_{j},s_{j+1}]}\right)\cdot{\mathcal{M}}_{0}\cdot{\mathcal{M}}_{1}\cdot\exp({\mathcal{N}}).$

Finally taking into account the definition of $S_{i}$, $i=0,1,2$ it follows
that

$\displaystyle|\delta{\Gamma}_{st}|\leq K\omega(s,t)^{1/p}\cdot|S_{0}\cup
S_{1}\cup
S_{2}|\cdot{\mathcal{M}}_{0}\cdot{\mathcal{M}}_{1}\cdot\exp({\mathcal{N}}).$

Namely we have proved (4.46) for ${\Gamma}$. It remains to upper bound the
Malliavin derivatives of ${\Gamma}$.

Recall that ${\Gamma}$ satisfies equation (4.45), with $\tilde{V}$ given in
Definition 4.8. For sake of clarity, the remainder of our computations will be
done assuming that all our quantities are real-valued (we will therefore drop
the indices from our next equations). Moreover according to our standing
assumptions, the process $\tilde{V}$ is Malliavin differentiable. Hence using
standard arguments for the differentiation of rough differential equations
(see [6, 7, 25, 31]) we get that $D_{r}{\Gamma}_{t}$ satisfies the linear
equation:

$\displaystyle D_{r}{\Gamma}_{t}$ $\displaystyle=$ $\displaystyle
D_{r}\int_{0}^{t}\tilde{V}(s){\Gamma}_{s}dx_{s}$ (4.53) $\displaystyle=$
$\displaystyle\tilde{V}(r){\Gamma}_{r}+\int_{r}^{t}\partial\tilde{V}(s)(\theta
D_{r}y_{s}+(1-\theta)D_{r}y^{n}_{s}){\Gamma}_{s}dx_{s}+\int_{r}^{t}\tilde{V}(s)D_{r}{\Gamma}_{s}dx_{s}.$

Therefore one can use the variation of constant method, similarly to [25,
equation (2.7)], in order to get the following representation for
$D_{r}{\Gamma}_{t}$:

$\displaystyle
D_{r}{\Gamma}_{t}={\Gamma}_{t}^{r}\tilde{V}(r){\Gamma}_{r}+{\Gamma}_{t}^{r}\int_{r}^{t}{\Lambda}_{s}^{r}\partial\tilde{V}(s)(\theta
D_{r}y_{s}+(1-\theta)D_{r}y^{n}_{s}){\Gamma}_{s}dx_{s},$ (4.54)

where $\\{{\Gamma}^{r}_{t};\,t\in[r,T]\\}$ is the solution of equation (4.49)
such that ${\Gamma}_{r}^{r}=\text{Id}$ and ${\Lambda}_{t}^{r}$ is the inverse
of ${\Gamma}^{r}_{t}$. Note that because ${\Gamma}$ and ${\Gamma}^{r}$ satisfy
the same equation with different initials, the estimate of ${\Gamma}$ in
(4.52) also holds for ${\Gamma}^{r}$. In order to estimate
$D_{r}{\Gamma}_{t}$, it thus remains to get the estimate (4.46) for the
integral

$\displaystyle\int_{r}^{t}{\Lambda}_{s}^{r}\partial\tilde{V}(s)(\theta
D_{r}y_{s}+(1-\theta)D_{r}y^{n}_{s}){\Gamma}_{s}dx_{s}$ (4.55)

in (4.54). Recall that ${\Gamma}$ is the solution of the linear system (4.49)
driven by $Z$, where we recall that $Z=(y^{n},\text{w})$. According to (4.36)
$(D_{r}y,D_{r}y^{n},Z)$ is a rough path. So ${\Gamma}$ can also be considered
as the solution of a linear system driven by $(D_{r}y,D_{r}y^{n},Z)$. Hence
along the same line as for (4.50) we can estimate the quantity (4.55), and
thus we obtain the bound (4.46) for $D_{r}{\Gamma}_{t}$.

We turn to the equation satisfied by $D^{2}_{rr^{\prime}}{\Gamma}$.
Differentiating (4.53), we let the patient reader check that the second
derivative verifies a linear equation of the form

$\displaystyle
D_{rr^{\prime}}^{2}{\Gamma}_{t}=D_{r^{\prime}}[\tilde{V}(r){\Gamma}_{r}]+D_{r}[\tilde{V}(r^{\prime}){\Gamma}_{r^{\prime}}]+{\mathcal{E}}_{rr^{\prime}}(t)+\int_{r\vee
r^{\prime}}^{t}\tilde{V}(s)D^{2}_{rr^{\prime}}{\Gamma}_{s}dx_{s},$ (4.56)

where the term ${\mathcal{E}}_{rr^{\prime}}(t)$ is defined by

$\displaystyle{\mathcal{E}}_{rr^{\prime}}(t)$ $\displaystyle=$
$\displaystyle\int_{r\vee r^{\prime}}^{t}\partial\tilde{V}(s)(\theta
D_{r}y_{s}+(1-\theta)D_{r}y^{n}_{s})D_{r^{\prime}}{\Gamma}_{s}dx_{s}$
$\displaystyle+\int_{r\vee r^{\prime}}^{t}\partial\tilde{V}(s)(\theta
D_{r^{\prime}}y_{s}+(1-\theta)D_{r^{\prime}}y^{n}_{s})D_{r}{\Gamma}_{s}dx_{s}$
$\displaystyle+\int_{r\vee r^{\prime}}^{t}\partial^{2}\tilde{V}(s)(\theta
D_{r^{\prime}}y_{s}+(1-\theta)D_{r^{\prime}}y^{n}_{s})(\theta
D_{r}y_{s}+(1-\theta)D_{r}y^{n}_{s}){\Gamma}_{s}dx_{s}.$

It is clear that the process $D_{rr^{\prime}}^{2}{\Gamma}$ satisfies a linear
equation system analogous to (4.53). The estimate can thus be obtained by
following the same arguments as above, invoking again [25]. This completes the
proof of (4.46). ∎

###  4.5. A decomposition of the error process

In [28, equations (6.14) and (7.6)], we have decomposed the error process
$y_{t}-y^{n}_{t}$ according to the Jacobian of the equation and some remainder
terms. In the following proposition we get a similar decomposition, adapted to
our needs for the weak convergence estimates. Notice that similarly to what we
did in Section 4.4, we will drop the indices from our formulae below for sake
of readability.

######  Lemma 4.11.

We work under the conditions of Lemma 4.7. Let $y$ and $y^{n}$ be the
solutions of equation (2.7) and the Euler scheme (4.24), respectively. Let
${\Gamma}$ and ${\Lambda}$ be respectively the solution of equation (4.45) and
its inverse ${\Lambda}={\Gamma}^{-1}$. We set $\eta(s)=t_{k}$ for
$s\in[t_{k},t_{k+1})$. For $t\in[0,T]$ we also define

$\displaystyle I_{t}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{0}^{t}\partial V\partial
VV(y^{n}_{\eta(s)})\cdot(s-\eta(s))^{2H}dx_{s}+\int_{0}^{t}\left(\int_{\eta(s)}^{s}\int_{\eta(s)}^{u}\partial^{2}V(y^{n}_{v})dy^{n}_{v}dy^{n}_{u}\right)dx_{s}$
(4.57) $\displaystyle=:$ $\displaystyle I_{t}^{1}+I_{t}^{2}.$

Then the difference $y_{t}^{n}-y_{t}$ can be decomposed as

$\displaystyle y_{t}-y^{n}_{t}$ $\displaystyle=$
$\displaystyle\sum_{e=1}^{5}J_{t}^{e},$ (4.58)

where the processes $J^{1}_{t},J^{2}_{t},J^{3}_{t}$ are respectively defined
by

$\displaystyle J_{t}^{1}$ $\displaystyle=$
$\displaystyle{\Gamma}_{t}\int_{0}^{t}{\Lambda}_{\eta(s)}\partial
VV(y^{n}_{\eta(s)})\delta x_{\eta(s),s}\,dx_{s}$ $\displaystyle J_{t}^{2}$
$\displaystyle=$
$\displaystyle{\Gamma}_{t}\int_{0}^{t}({\Lambda}_{s}-{\Lambda}_{\eta(s)})\partial
VV(y^{n}_{\eta(s)})\delta x_{\eta(s),s}\,dx_{s}$ $\displaystyle J_{t}^{3}$
$\displaystyle=$
$\displaystyle{\Gamma}_{t}\int_{0}^{t}{\Lambda}_{s}\,dI_{s}\,,$

and where $J^{4}_{t},J^{5}_{t}$ are given by

$\displaystyle J_{t}^{4}$ $\displaystyle=$
$\displaystyle-H\cdot{\Gamma}_{t}\int_{0}^{t}({\Lambda}_{s}-{\Lambda}_{\eta(s)})\partial
VV(y^{n}_{\eta(s)})\cdot(s-\eta(s))^{2H-1}\,ds$ $\displaystyle J_{t}^{5}$
$\displaystyle=$
$\displaystyle-H\cdot{\Gamma}_{t}\int_{0}^{t}{\Lambda}_{\eta(s)}\partial
VV(y^{n}_{\eta(s)})\cdot(s-\eta(s))^{2H-1}\,ds.$

###### Proof.

We first recall that the continuous time Euler scheme defined in (4.24) can be
written, for $s\in[0,T]$, as

$\displaystyle\delta y^{n}_{\eta(s),s}=V(y^{n}_{\eta(s)})\delta
x_{\eta(s),s}+\frac{1}{2}\partial VV(y^{n}_{\eta(s)})\cdot(s-\eta(s))^{2H},$
(4.59)

where we $\eta_{s}=t_{k}$ for $s\in[t_{k},t_{k+1})$. One can also write
equation (4.59) in integral form, which yields an expression of the form:

$\displaystyle
y^{n}_{t}=y_{0}+\int_{0}^{t}V(y^{n}_{\eta(s)})dx_{s}-H\int_{0}^{t}\partial
VV(y^{n}_{\eta(s)})(s-\eta(s))^{2H-1}ds.$ (4.60)

Gathering (4.60) with equation (2.7) for which we omit the drift term, we get

$y_{t}-y^{n}_{t}=\int_{0}^{t}(V(y_{s})-V(y^{n}_{s}))\,dx_{s}+\int_{0}^{t}(V(y^{n}_{s})-V(y^{n}_{\eta(s)}))\,dx_{s}\\\
-H\cdot\int_{0}^{t}\partial VV(y^{n}_{\eta(s)})\cdot(s-\eta(s))^{2H-1}\,ds.$
(4.61)

Next we will consider a decomposition of the quantity
$V(y^{n}_{s})-V(y^{n}_{\eta(s)})$ in (4.61). Namely we apply the chain rule
twice to obtain

$V(y^{n}_{s})-V(y^{n}_{\eta(s)})=\int_{\eta(s)}^{s}\partial
V(y^{n}_{u})dy^{n}_{u}=\partial V(y^{n}_{\eta(s)})\delta
y^{n}_{\eta(s),s}+\int_{\eta(s)}^{s}\int_{\eta(s)}^{u}\partial^{2}V(y^{n}_{v})dy^{n}_{v}dy^{n}_{u}.$
(4.62)

Plugging (4.59) into (4.62) and then integrating in $x$ we thus get

$\displaystyle\int_{0}^{t}(V(y^{n}_{s})-V(y^{n}_{\eta(s)}))dx_{s}$
$\displaystyle=$ $\displaystyle\int_{0}^{t}\partial
V(y^{n}_{\eta(s)})\left(V(y^{n}_{\eta(s)})\delta
x_{\eta(s),s}+\frac{1}{2}\partial
VV(y^{n}_{\eta(s)})\cdot(s-\eta(s))^{2H}\right)dx_{s}$ (4.63)
$\displaystyle+\int_{0}^{t}\left(\int_{\eta(s)}^{s}\int_{\eta(s)}^{u}\partial^{2}V(y^{n}_{v})dy^{n}_{v}dy^{n}_{u}\right)dx_{s},$

Recalling the definition of $I_{t}^{1}$, $I_{t}^{2}$ in (4.57), equation
(4.63) can also be read as

$\int_{0}^{t}(V(y^{n}_{s})-V(y^{n}_{\eta(s)}))dx_{s}=\int_{0}^{t}\partial
VV(y^{n}_{\eta(s)})\delta x_{\eta(s),s}dx_{s}+I_{t}^{1}+I_{t}^{2}.$ (4.64)

We now decompose the quantity $V(y_{s})-V(y^{n}_{s})$ in (4.61). Specifically
we write

$\displaystyle V(y_{t})-V(y^{n}_{t})=\int_{0}^{1}\partial V(\theta
y_{t}+(1-\theta)y^{n}_{t})d\theta\cdot(y_{t}-y^{n}_{t})=\tilde{V}(t)\cdot(y_{t}-y^{n}_{t}),$
(4.65)

where we recall that the process $\tilde{V}$ has been introduced in Definition
4.8.

We are ready to plug (4.64) and (4.65) into (4.61) in order to get the
following linear equation for $y-y^{n}$:

$\displaystyle
y_{t}-y^{n}_{t}=\int_{0}^{t}\tilde{V}(s)\cdot(y_{s}-y^{n}_{s})dx_{s}+K_{t},$
(4.66)

where the process $K_{t}$ is given by

$\displaystyle K_{t}=\int_{0}^{t}\partial VV(y^{n}_{\eta(s)})\delta
x_{\eta(s)s}dx_{s}+I^{1}_{t}+I^{2}_{t}-H\cdot\int_{0}^{t}\partial
VV(y^{n}_{\eta(s)})\cdot(s-\eta(s))^{2H-1}ds.$ (4.67)

Eventually we recall that ${\Gamma}$ solves the Jacobian type equation (4.45)
and that ${\Lambda}_{t}={\Gamma}_{t}^{-1}$. Hence applying Duhamel’s principle
in order solve (4.66), we get

$\displaystyle y_{t}-y^{n}_{t}={\Gamma}_{t}\int_{0}^{t}{\Lambda}_{s}\,dK_{s}.$

Thanks to our expression (4.67), the above equation can be written more
explicitly as

$\displaystyle y_{t}-y^{n}_{t}={\Gamma}_{t}\int_{0}^{t}{\Lambda}_{s}\partial
VV(y^{n}_{\eta(s)})\delta
x_{\eta(s)s}dx_{s}+{\Gamma}_{t}\int_{0}^{t}{\Lambda}_{s}d(I^{1}_{s}+I^{2}_{s})$
$\displaystyle-H\cdot{\Gamma}_{t}\int_{0}^{t}{\Lambda}_{s}\partial
VV(y^{n}_{\eta(s)})\cdot(s-\eta(s))^{2H-1}ds.$ (4.68)

With relation (4.5) in hand, we can now easily identify the terms in (4.58).
Indeed, the second term on the right-hand side of equation (4.5) is exactly
$J^{3}_{t}$. Also, in the same equation, by plugging the decomposition
${\Lambda}_{s}={\Lambda}_{\eta(s)}+({\Lambda}_{s}-{\Lambda}_{\eta(s)})$ into
the first and third terms we identify the first and third terms as
$J_{t}^{1}+J_{t}^{2}$ and $J_{t}^{4}+J_{t}^{5}$, respectively. We thus
conclude the identity (4.58). The proof is complete. ∎

###  4.6. The weak convergence of the Euler scheme

We can now gather all the previous preliminary estimates in order to obtain
our main result. This is summarized in the theorem below.

######  Theorem 4.12.

Consider a vector field $V\in C^{4}_{b}$ and a driving fBm $x$ with Hurst
parameter $H>1/3$. Let $y$ be the solution of the rough differential equation
(2.7). The corresponding interpolated Euler scheme is $y^{n}$, displayed in
(4.24). Then for any $f\in C^{4}_{b}(\mathbb{R}^{d})$ and $t\in[0,T]$ there is
a constant $C>0$ independent of $n$ such that

$\displaystyle\left|\mathbb{E}f(y^{n}_{t})-\mathbb{E}f(y_{t})\right|\leq\frac{C}{n^{4H-1-\varepsilon}}.$
(4.69)

###### Proof.

For conciseness we will prove the theorem for the case $V_{0}\equiv 0$ only.
The general case can be considered in the similar way and is left to the
patient reader.

Let $f$ be a generic $C_{b}^{4}$ function. For $t\in[0,T]$ we define an
interpolated process

$\displaystyle f_{1}(t)=\int_{0}^{1}\partial
f({\lambda}y_{t}+(1-{\lambda})y^{n}_{t})d{\lambda}.$ (4.70)

Here note that in order to alleviate notations, we still drop indices and
perform our computation as if our quantities were real-valued. Next a simple
application of the fundamental theorem of calculus plus Lemma 4.11 reveal that

$\displaystyle f(y_{t})-f(y^{n}_{t})$ $\displaystyle=$ $\displaystyle
f_{1}(t)\cdot(y_{t}-y^{n}_{t})=\sum_{e=1}^{5}f_{1}(t)J_{t}^{e}.$ (4.71)

The remainder of the proof is dedicated to estimate the five terms in the
right-hand side of (4.71). For sake of conciseness we prove (4.69) for
$t\in\llbracket 0,T\rrbracket$ only. The proof for $t\in[0,T]$ follows the
same lines and is left to patient reader.

_Step 1: Estimating $J_{t}^{1}$ and $J_{t}^{5}$._ In this step, we consider
the first and fifth term in (4.71). Note that the integrals in the expressions
for $J_{t}^{1}$ and $J_{t}^{5}$ are in fact discrete sums. We can thus combine
those two terms in order to get:

$\displaystyle f_{1}(t)(J_{t}^{1}+J_{t}^{5})$ $\displaystyle=$
$\displaystyle\sum_{t_{k}<t}f_{1}(t){\Gamma}_{t}\Lambda_{t_{k}}\partial
VV(y^{n}_{t_{k}})\left(x^{2}_{t_{k}t_{k+1}}-\frac{1}{2}\Delta^{2H}\right).$
(4.72)

Let us say a few words about the term $\psi_{k}\equiv
x^{2}_{t_{k}t_{k+1}}-\frac{1}{2}\Delta^{2H}$ in the right-hand side of (4.72).
First we highlight again the fact that we are performing 1-d type computations
in order to simplify notation. In a $d$-dim setting we would consider random
variables of the form

$\displaystyle\psi^{ij}_{k}=x^{2,ij}_{t_{k}t_{k+1}}-\frac{1}{2}\Delta^{2H}\mathbf{1}_{\\{i=j\\}},\quad\text{for}\quad
i,j\in\\{1,\dots,d\\}.$

Here we will just focus on the terms $\psi_{k}\equiv\psi^{ii}_{k}$, which are
the most demanding ones. We leave the off-diagonal terms $\psi^{ij}_{k}$ to
the patient reader for sake of conciseness. Next we should also have in mind
the fact that $\psi_{k}$ can be written as

$\displaystyle\psi_{k}=\frac{1}{2}(\delta
x_{t_{k}t_{k+1}})^{2}-\frac{1}{2}\Delta^{2H}=\frac{1}{2}\Delta^{2H}H_{2}\left(\frac{(\delta
x_{t_{k}t_{k+1}})^{2}}{\Delta^{2H}}\right),$

where $H_{2}$ stands for the Hermite polynomial $H_{2}(x)=x^{2}-1$. Invoking
[30, Page 23] we thus get

$\displaystyle\psi_{k}=\delta^{\diamond,2}({\beta}_{t_{k}t_{k+1}}),$

where ${\beta}_{t_{k}t_{k+1}}$ is defined by (4.8) and $\delta^{\diamond,2}$
stands for a double Skorohod integral (see Section 2.3 for Malliavin calculus
notation). Hence applying twice the integration by parts (2.17), we end up
with

$\displaystyle\mathbb{E}\Big{[}f_{1}(t)(J_{t}^{1}+J_{t}^{5})\Big{]}=\sum_{t_{k}<t}\mathbb{E}\left[\left\langle
D^{2}\big{[}f_{1}(t){\Gamma}_{t}\Lambda_{t_{k}}\partial
VV(y^{n}_{t_{k}})\big{]},{\beta}_{t_{k}t_{k+1}}\right\rangle_{{\mathcal{H}}^{\otimes
2}}\right]\,,$

where recall that ${\beta}$ is defined in (4.8). Applying Lemma 4.3 with
$\varphi$ given by

$\varphi=D^{2}\big{[}f_{1}(t){\Gamma}_{t}\Lambda_{t_{k}}\partial
VV(y^{n}_{t_{k}})\big{]}\,,$

and recalling that $f_{1}$ is the process in (4.70), we obtain

$\displaystyle\Big{|}\mathbb{E}\Big{[}f_{1}(t)(J_{t}^{1}+J_{t}^{5})\Big{]}\Big{|}\leq\sum_{t_{k}<t}n^{-4H}\mathbb{E}\left[\left\|D^{2}\big{[}f_{1}(t){\Gamma}_{t}\Lambda_{t_{k}}\partial
VV(y^{n}_{t_{k}})\big{]}\right\|_{\infty}\right]\,.$ (4.73)

The integrability results Theorem 3.5 and Lemma 4.10 (b) guarantee the uniform
integrability in $n$ of the sup-norm in the inequality (4.73). Therefore, we
have the estimate

$\displaystyle\Big{|}\mathbb{E}\Big{[}f_{1}(t)(J_{t}^{1}+J_{t}^{5})\Big{]}\Big{|}\leq
C\sum_{t_{k}<t}n^{-4H}=Cn^{1-4H}.$ (4.74)

_Step 2: Estimating $J_{t}^{2}$._ We turn to the estimate of $J_{t}^{2}$ in
(4.71) and Lemma 4.11. Observe that according to the fact that
${\Lambda}={\Gamma}^{-1}$ and $\Lambda$ solves (4.45), we have
${\Lambda}_{s}-{\Lambda}_{\eta(s)}=-\int_{\eta(s)}^{s}\Lambda_{u}\tilde{V}(u)dx_{u}$.
Substituting this into $J_{t}^{2}$ we obtain

$\displaystyle J_{t}^{2}$ $\displaystyle=$
$\displaystyle-{\Gamma}_{t}\int_{0}^{t}\Big{(}\int_{\eta(s)}^{s}\Lambda_{u}\tilde{V}(u)dx_{u}\Big{)}\partial
VV(y^{n}_{\eta(s)})\delta x_{\eta(s)s}dx_{s}.$ (4.75)

Now let us write

$\displaystyle\Lambda_{u}\tilde{V}(u)=\big{(}\Lambda_{u}\tilde{V}(u)-\Lambda_{\eta(s)}\tilde{V}(\eta(s))\big{)}+\Lambda_{\eta(s)}\tilde{V}(\eta(s)).$

Reporting this relation into our expression (4.75) for $J_{t}^{2}$ yields the
decomposition:

$\displaystyle J_{t}^{2}$ $\displaystyle=$
$\displaystyle-{\Gamma}_{t}\int_{0}^{t}\int_{\eta(s)}^{s}\big{(}\Lambda_{u}\tilde{V}(u)-\Lambda_{\eta(s)}\tilde{V}(\eta(s))\big{)}dx_{u}\cdot\partial
VV(y^{n}_{\eta(s)})\delta x_{\eta(s)s}dx_{s}$ (4.76)
$\displaystyle-\sum_{t_{k}<t}{\Gamma}_{t}\int_{t_{k}}^{t_{k+1}}\Lambda_{t_{k}}\tilde{V}(t_{k})\delta
x_{t_{k}s}\partial VV(y^{n}_{t_{k}})\delta
x_{t_{k}s}dx_{s}=:J_{t}^{21}+J_{t}^{22}.$

We now proceed to the analysis of $J^{21}_{t}$ and $J^{22}_{t}$ above.

In order to bound the term $J_{t}^{22}$ in our decomposition (4.76), observe
that this term is of the form $\sum_{t_{k}<t}f_{t_{k}}\delta g_{t_{k}t_{k+1}}$
as in Lemma 2.5. Precisely, we have

$\displaystyle
J_{t}^{22}=-\sum_{t_{k}<t}\underbrace{{\Gamma}_{t}\Lambda_{t_{k}}\tilde{V}(t_{k})\partial
VV(y^{n}_{t_{k}})}_{=f_{t_{k}}}\cdot\underbrace{\int_{t_{k}}^{t_{k+1}}\delta
x_{t_{k}s}\delta x_{t_{k}s}dx_{s}}_{=\delta g_{t_{k}t_{k+1}}}.$ (4.77)

Moreover, according to (4.46) and the $L^{p}$-estimates for
${\mathcal{M}}_{0}$, ${\mathcal{M}}_{1}$, it is readily checked that for all
$p\geq 1$ and $(u,v)\in{\mathcal{S}}_{2}(\llbracket 0,T\rrbracket)$ we have

$\displaystyle\left(\mathbb{E}\left[|\delta
f_{uv}|^{2p}\right]\right)^{\frac{1}{2p}}\lesssim|v-u|^{H-\varepsilon}.$
(4.78)

In addition $g$ has to be seen as a triple iterated integral of $x$. It has
been shown in [28, Lemma 4.3] that for all
$(u,v)\in{\mathcal{S}}_{2}(\llbracket 0,T\rrbracket)$ we have

$\displaystyle\left(\mathbb{E}\left[|\delta
g_{uv}|^{2p}\right]\right)^{\frac{1}{2p}}\lesssim\frac{|v-u|^{1/2}}{n^{3H-1/2}}.$
(4.79)

Since we are considering points $u,v$ on the grid $\llbracket 0,T\rrbracket$,
it is readily checked that $v-u\geq T/n$. Hence one can play with the
exponents in (4.79) and write

$\displaystyle\left(\mathbb{E}\left[|\delta
g_{uv}|^{2p}\right]\right)^{\frac{1}{2p}}\lesssim\frac{|v-u|^{1-H+2\varepsilon}}{n^{4H-1-2\varepsilon}}.$
(4.80)

It follows that gathering (4.78) and (4.80) one can apply Lemma 2.5 to (4.77)
and get

$\displaystyle|\mathbb{E}[f_{1}(t)J_{t}^{22}]|\leq\frac{C}{n^{4H-1-\varepsilon}}.$
(4.81)

In order to bound $\mathbb{E}[f_{1}(t)J_{t}^{21}]$, where $J_{t}^{21}$ is
defined in (4.76), we need to make a further decomposition. Using the product
rule plus equation (4.45) for ${\Lambda}$, Definition 4.8 for $\tilde{V}$, as
well as relation (2.7) and (4.24) for $y$ and $y^{n}$, we can write

$\Lambda_{u}\tilde{V}(u)-\Lambda_{\eta(s)}\tilde{V}(\eta(s))=\int_{\eta(s)}^{u}f_{2}(v)dx_{v}+\int_{\eta(s)}^{u}f_{3}(v)d(v-\eta(v))^{2H}\,,$
(4.82)

where we have set

$\displaystyle f_{2}(v)$ $\displaystyle=$
$\displaystyle-{\Lambda}_{v}\tilde{V}(v)\tilde{V}(v)+{\Lambda}_{v}\partial\tilde{V}(v)$
(4.83) $\displaystyle f_{3}(v)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\Lambda_{v}\int_{0}^{1}\partial\partial V(\theta
y_{v}+(1-\theta)y^{n}_{v})(1-\theta)\partial VV(y_{\eta(s)})\,,$ (4.84)

and where we denote

$\displaystyle\partial\tilde{V}(v)=\int_{0}^{1}\partial\partial V(\theta
y_{v}+(1-\theta)y^{n}_{v})(\theta
V(y_{v})+(1-\theta)V(y^{n}_{\eta(v)}))d\theta.$

Then we write

$\displaystyle\int_{\eta(s)}^{u}f_{2}(v)dx_{v}=f_{2}(\eta(s))\delta
x_{\eta(s),u}+\int_{\eta(s)}^{u}(f_{2}(v)-f_{2}(\eta(s)))dx_{v}.$

Substituting the above into $J_{t}^{21}$ we obtain a weighted sum of two 4th
and one 5th order multiple integral in the form $\sum_{0\leq t_{k}<t}h_{k}$.
Precisely, we have $J_{t}^{21}=-(J_{t}^{211}+J_{t}^{212}+J_{t}^{213})$, where

$\displaystyle J^{211}_{t}$ $\displaystyle=$ $\displaystyle\sum_{0\leq
t_{k}<t}{\Gamma}_{t}\int_{t_{k}}^{t_{k+1}\wedge
t}\int_{t_{k}}^{s}f_{2}(t_{k})\delta x_{t_{k},u}dx_{u}\cdot\partial
VV(y^{n}_{t_{k}})\delta x_{t_{k}s}dx_{s}\equiv\sum_{0\leq
t_{k}<t}{\Gamma}_{t}h^{211}_{t}$ $\displaystyle J_{t}^{212}$ $\displaystyle=$
$\displaystyle\sum_{0\leq t_{k}<t}{\Gamma}_{t}\int_{t_{k}}^{t_{k+1}\wedge
t}\int_{t_{k}}^{s}\int_{t_{k}}^{u}(f_{2}(v)-f_{2}(t_{k}))dx_{v}dx_{u}\cdot\partial
VV(y^{n}_{t_{k}})\delta x_{t_{k}s}dx_{s}$ $\displaystyle\equiv$
$\displaystyle\sum_{0\leq t_{k}<t}{\Gamma}_{t}h^{212}_{k}$ $\displaystyle
J^{213}_{t}$ $\displaystyle=$
$\displaystyle-{\Gamma}_{t}\int_{0}^{t}\int_{\eta(s)}^{s}\int_{\eta(s)}^{u}f_{3}(v)d(v-\eta(v))^{2H}dx_{u}\cdot\partial
VV(y^{n}_{\eta(s)})\delta x_{\eta(s)s}dx_{s}.$ (4.85)

We now proceed to estimate the terms $J^{211}_{t}$, $J^{212}_{t}$ and
$J^{213}_{t}$.

One can easily analyze the term $J^{211}_{t}$ by writing

$\displaystyle h^{211}_{k}=f_{2}(t_{k})\partial
VV(y^{n}_{t_{k}})x^{4}_{t_{k},t_{k+1}\wedge t},$

where $x^{4}_{st}$ denotes the 4th order iterated integral over the interval
$[s,t]$. It follows that $\|h^{211}_{k}\|_{L^{p}}\lesssim\frac{1}{n^{4H}}$ for
$p\geq 1$. This implies that

$\displaystyle\mathbb{E}[f_{1}(t)J_{t}^{211}]\leq\sum_{t_{k}<t}C\cdot
n^{-4H}\leq\frac{C}{n^{4H-1}}.$ (4.86)

In the same way we can show that the bound (4.86) also holds for
$J^{213}_{t}$.

As far as $J_{t}^{212}$ is concerned, one can recast the term $h^{212}_{k}$ as
$h^{212}_{k}=\partial VV(y^{n}_{t_{k}})\hat{h}^{212}_{k}$, with

$\displaystyle\hat{h}^{212}_{k}=\int_{t_{k}}^{t_{k+1}\wedge
t}\int_{t_{k}}^{s}\int_{t_{k}}^{u}(f_{2}(v)-f_{2}(t_{k}))dx_{v}dx_{u}\delta
x_{t_{k}s}dx_{s}.$ (4.87)

The quantity $\hat{h}^{212}_{k}$ has to be seen as a 5th order iterated
integral. One way to quantify this is to resort to Fubini’s theorem and write

$\hat{h}^{212}_{k}=\int_{t_{k}}^{t_{k+1}}z^{t_{k}t_{k+1}}_{v}dx_{v},\quad\text{with}\quad
z_{v}^{t_{k}t_{k+1}}=(f_{2}(v)-f_{2}(t_{k}))\int_{v}^{t_{k+1}}dx_{u}\int_{u}^{t_{k+1}}\delta
x_{t_{k}s}dx_{s}.$

Using the rough path property of $x$ recalled in Section 2.2 and the
definition of $f_{2}$ in (4.82), it is readily checked that $z^{t_{k}t_{k+1}}$
is of order $(1/n)^{4H-\varepsilon}$ for any $\varepsilon>0$. Reporting this
information in (4.87), one gets the almost sure relation

$\displaystyle|\hat{h}_{k}^{212}|\leq\frac{G}{n^{5(H-\varepsilon)}},$

where $G\in\cap_{p\geq 1}L^{p}(\Omega)$. With relation (4.6) in mind and
taking into account the definition (4.70) of $f_{1}$, we discover that

$\displaystyle\Big{|}\mathbb{E}[f_{1}(t)J_{t}^{212}]\Big{|}\leq\frac{C}{n^{-1+5(H-\varepsilon)}}\leq\frac{C}{n^{4H-1}}.$
(4.88)

Summarizing our considerations for the term $J^{2}_{t}$, we gather our
estimates (4.86) and (4.88). This yields the desired estimate

$\displaystyle\Big{|}\mathbb{E}[f_{1}(t)J_{t}^{2}]\Big{|}\leq\frac{C}{n^{4H-1-\varepsilon}}.$
(4.89)

_Step 3: Estimating $J_{t}^{3}$._ In this step, we consider the term
$J_{t}^{3}$ defined in Lemma 4.11. Also recall that $I_{t}$ has been
decomposed into $I^{1}_{t}+I^{2}_{t}$ in (4.57). Accordingly we shall write

$\displaystyle
J^{3}_{t}=J^{31}_{t}+J^{32}_{t}\equiv{\Gamma}_{t}\int_{0}^{t}{\Lambda}_{s}dI^{1}_{s}+{\Gamma}_{t}\int_{0}^{t}{\Lambda}_{s}dI^{2}_{s},$

and estimate $J^{31}_{t}$, $J^{32}_{t}$ separately. Resorting to expression
(4.57) for $I^{2}$, let us write $J^{32}_{t}$ as

$\displaystyle
J^{32}_{t}={\Gamma}_{t}\int_{0}^{t}\int_{\eta(s)}^{s}\int_{\eta(s)}^{u}{\Lambda}_{s}\partial^{2}V(y^{n}_{v})dy^{n}_{v}dy^{n}_{u}dx_{s}.$

In this way, it is readily checked that $J^{32}_{t}$ exhibits the same type of
regularity as $J^{2}_{t}$ defined by (4.75). The complete analysis of
$J^{32}_{t}$ thus follows the same steps as $J^{2}_{t}$. It relies on another
discretization procedure, similar to (4.76). Namely one writes
$J^{32}_{t}=J^{321}_{t}+J^{322}_{t}$, with

$\displaystyle J^{321}_{t}$ $\displaystyle=$
$\displaystyle{\Gamma}_{t}\int_{0}^{t}\int_{\eta(s)}^{s}\int_{\eta(s)}^{u}\left({\Lambda}_{s}\partial^{2}V(y^{n}_{v})-{\Lambda}_{\eta(s)}\partial^{2}V(y_{\eta(v)})\right)dy^{n}_{v}dy^{n}_{u}dx_{s},$
$\displaystyle J^{322}_{t}$ $\displaystyle=$
$\displaystyle{\Gamma}_{t}\sum_{t_{k}<t}{\Lambda}_{t_{k}}\partial^{2}V(y_{t_{k}})\int_{t_{k}}^{t_{k+1}}\int_{\eta(s)}^{s}\int_{\eta(s)}^{u}dy^{n}_{v}dy^{n}_{u}dx_{s}.$

In addition, along the same lines as for (4.77) and resorting to the discrete
dynamics (4.24) of $y^{n}$, one can express $J^{321}_{t}$ as a weighted sum of
triple integrals of $x$. We can thus proceed as in the estimation of
$J^{2}_{t}$ and get the same inequalities as in (4.81), (4.86) and (4.88).
Details are left to the reader for sake of conciseness. We obtain

$\displaystyle\Big{|}\mathbb{E}[f_{1}(t)J^{32}_{t}]\Big{|}\leq\frac{C}{n^{4H-1-\varepsilon}}.$
(4.90)

In order to bound the term $J^{31}_{t}$, we first use another step of
discretization. That is we decompose $J^{31}_{t}$ as $J^{311}_{t}+J^{312}_{t}$
with

$\displaystyle J_{t}^{311}$ $\displaystyle=$
$\displaystyle\frac{1}{2}{\Gamma}_{t}\int_{0}^{t}{\Lambda}_{\eta(s)}\partial
V\partial VV(y^{n}_{\eta(s)})\cdot(s-\eta(s))^{2H}dx_{s}$ $\displaystyle
J_{t}^{312}$ $\displaystyle=$
$\displaystyle\frac{1}{2}{\Gamma}_{t}\int_{0}^{t}({\Lambda}_{s}-{\Lambda}_{\eta(s)})\partial
V\partial VV(y^{n}_{\eta(s)})\cdot(s-\eta(s))^{2H}dx_{s}\,.$

Note that by applying Lemma 2.5 we can bound $\mathbb{E}[f_{1}(t)J_{t}^{311}]$
by $\frac{1}{n^{4H-1-\varepsilon}}$. Indeed, we can write

$\displaystyle f_{1}(t)J_{t}^{311}$ $\displaystyle=$
$\displaystyle\frac{1}{2}f_{1}(t){\Gamma}_{t}\sum_{0\leq
t_{k}<t}{\Lambda}_{t_{k}}\partial V\partial
VV(y^{n}_{t_{k}})\cdot\nu_{t_{k},t_{k+1}\wedge t},$ (4.91)

where the increment $\nu$ is defined by

$\displaystyle\nu_{uv}=\int_{u}^{v}(s-\eta(s))^{2H}dx_{s}.$ (4.92)

Next recall the following result from Lemma 4.6 in [28]: For a fBm $x$ with
Hurst parameter $H$ and $f$ such that $\|f\|_{{\gamma}}\in L^{p}$ for all
${\gamma}<H$ and $p\geq 1$, we have

$\displaystyle\left\\{\mathbb{E}\left[\Big{|}\sum_{0\leq
t_{k}<t}f_{t_{k}}\nu_{t_{k},t_{k+1}\wedge
t}\Big{|}^{p}\right]\right\\}^{1/p}\leq\frac{CT}{n^{4H-1-\varepsilon}}.$
(4.93)

One can apply directly this estimate to (4.91) in order to get

$\displaystyle\mathbb{E}[f_{1}(t)J_{t}^{311}]\leq\frac{C}{n^{4H-1-\varepsilon}}.$
(4.94)

The term $J^{312}_{t}$ has to be compared to $J^{21}_{t}$ in (4.76). We can
thus follow some computations which are very similar to (4.82)-(4.6). We end
up with 2nd and 3rd integrals involving $x$ and the increment $\nu$ in (4.92).
Having the regularity (4.93) of $\nu$ into account we let the reader check
that

$\displaystyle\mathbb{E}[f_{1}(t)J_{t}^{312}]\leq\frac{C}{n^{4H-1-\varepsilon}},$
(4.95)

similarly to (4.86) and (4.88). We can thus conclude this step by gathering
(4.94) and (4.95). This yields

$\displaystyle|\mathbb{E}[f_{1}(t)J_{t}^{3}]|\leq Cn^{1-4H+\varepsilon}.$
(4.96)

_Step 4: Estimating $J_{t}^{4}$._ We now turn to an estimate of the term
$J^{4}_{t}$ in Lemma 4.11. According to the expression therein and equation
(4.45) for ${\Lambda}$, observe that

$\displaystyle\mathbb{E}[f_{1}(t)J_{t}^{4}]$ $\displaystyle=$
$\displaystyle-H\cdot\int_{0}^{t}Q^{t}_{s}\cdot(s-\eta(s))^{2H-1}ds,$ (4.97)

where the quantity $Q^{t}_{s}$ is given by

$\displaystyle
Q_{s}^{t}=\mathbb{E}\left[f_{1}(t)\left(\int_{\eta(s)}^{s}{\Gamma}_{t}{\Lambda}_{u}\tilde{V}(u)^{T}dx_{u}\right)\partial
VV(y^{n}_{\eta(s)})\right].$

As for $J_{t}^{31}$ we can show that

$\displaystyle\left|\mathbb{E}\left[f_{1}(t){\Gamma}_{t}\left(\int_{\eta(s)}^{s}{\Lambda}_{u}\tilde{V}(u)^{T}dx_{u}\right)\partial
VV(y^{n}_{\eta(s)})\right]\right|\leq\frac{C}{n^{2H-\varepsilon}}.$ (4.98)

Indeed, by writing and substituting

${\Lambda}_{u}\tilde{V}(u)^{T}={\Lambda}_{\eta(s)}\tilde{V}(\eta(s))^{T}+({\Lambda}_{u}\tilde{V}(u)^{T}-{\Lambda}_{\eta(s)}\tilde{V}(\eta(s))^{T})$

into (4.98), we decompose (4.98) into two components. The second component
obtained is a double integral over the interval $[\eta(s),s]$, which is
bounded by $\frac{1}{n^{2H-\varepsilon}}$. On the other hand, the first
component is of the form $\mathbb{E}[F\delta x_{\eta(s)s}]$, where

$\displaystyle
F=f_{1}(t){\Gamma}_{t}\left({\Lambda}_{\eta(s)}\tilde{V}(\eta(s))^{T}\right)\partial
VV(y^{n}_{\eta(s)}).$

Note that $F$ is an integrable variable whose Malliavin derivative $DF$ is
also integrable. So applying integration by parts to $\mathbb{E}[F\delta
x_{\eta(s)s}]$ and then Lemma 4.4 with $\varphi=DF$, together with the upper-
bound estimates in Lemma 4.10 and Theorem 2.10, we obtain the bound
$\frac{1}{n^{2H}}$. Gathering those consideration and (4.98) into (4.97), we
end up with

$\displaystyle\mathbb{E}[f_{1}(t)J_{t}^{4}]\leq\frac{C}{n^{4H-1-\varepsilon}}.$
(4.99)

Step 5: Conclusion. Taking expectations on both sides of (4.71) and reporting
(4.74), (4.89), (4.96), and (4.99) we discover that (4.69) holds true. This
finishes the proof. ∎

## Acknowledgements

Y. Liu is partially supported by PSC-CUNY Award # 66385-00 54. S. Tindel is
partially supported by NSF grants DMS-1952966 and DMS-2153915.

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