# Skorohod and Stratonovich integrals
for controlled processes

Jian Song and Samy Tindel  Jian Song: Research Center for Mathematics and
Interdisciplinary Sciences, Shandong University, Qingdao, Shandong, 266237,
China and School of Mathematics, Shandong University, Jinan, Shandong, 250100,
China<EMAIL_ADDRESS>Samy Tindel: Department of Mathematics, Purdue
University, 150 N. University Street West Lafayette, Indiana 47907, USA
<EMAIL_ADDRESS>
###### Abstract.

Given a continuous Gaussian process $x$ which gives rise to a $p$-geometric
rough path for $p\in(2,3)$, and a general continuous process $y$ controlled by
$x$, under proper conditions we establish the relationship between the
Skorohod integral $\int_{0}^{t}y_{s}\mathrm{d}^{\diamond}x_{s}$ and the
Stratonovich integral $\int_{0}^{t}y_{s}\mathrm{d}{\bf x}_{s}$. Our strategy
is to employ the tools from rough paths theory and Malliavin calculus to
analyze discrete sums of the integrals.

J.S. is partially supported by Shandong University grant 11140089963041.

S.T. is partially supported by the National Science Foundation under grant
DMS-1952966.

###### Contents

  1. 1 Introduction
    1. 1.1 Background
    2. 1.2 Main result and strategy
    3. 1.3 Notation
  2. 2 Preliminary material
    1. 2.1 Rough path above $\mathbf{x}$
    2. 2.2 Higher dimensional Young integrals
    3. 2.3 The Hilbert space associated to $x$
    4. 2.4 Malliavin calculus for Gaussian processes
    5. 2.5 Discrete rough paths techniques
  3. 3 Correction terms in the case $2\leq p<3$

##  1\. Introduction

For sake of clarity, we will divide this introduction in 3 parts. In Section
1.1 we motivate our problem and recall some previous contributions giving
Stratonovich-Skorohod corrections. Section 1.2 is devoted to a description of
our main result, the strategy employed in the article, and some perspectives
for future works. At the end, some notations used in this article are
introduced in Section 1.3.

###  1.1. Background

In recent decades, two approaches for the analysis of dynamical systems driven
by Gaussian processes have been greatly developed: (i) the “probabilistic”
approach, which invokes stochastic analysis tools and leads to Itô-Skorohod
integration, and (ii) the “pathwise” approach which employs the theory of
rough paths and gives rise to Stratonovich integration. In general, one gets a
more transparent understanding of the system by using the pathwise approach,
while it is more convenient to explore probabilistic properties (e.g. compute
the moments for the solution of a noisy dynamical system driven by a Gaussian
noise) via the probabilistic approach. One key ingredient to understand the
connection between these two approaches is the relationship between Skorohod
and Stratonovich integrals.

For a standard Brownian motion, the relationship between Itô and Stratonovich
integrals is well-known. It is classically obtained by Itô calculus, although
rough paths theory can also be invoked by observing that both Itô and
Stratonovich integrals can be regarded as integrals against rough paths lifted
from a Brownian motion with different second order terms. For general Gaussian
processes (consider fractional Brownian motion as a typical example), however,
it is non-trivial to obtain the relationship. Indeed, for a general Gaussian
process Itô calculus (or martingale calculus) is not available, and moreover
Skorohod integrals cannot be regarded as integrals against rough paths lifted
from the corresponding Gaussian processes. We briefly recall some results
giving Skorohod-Stratonovich corrections below.

Let $x=(x^{1},\dots,x^{d})$ be a $d$-dimensional centered Gaussian process
with i.i.d components giving rise to a $p$-geometric rough path, where $2<p<3$
(see Section 2.1 for more details about geometric rough paths). Denote by $R$
the covariance function of $x$, namely
$R(s,t)=\mathbf{E}[x_{t}^{1}x_{s}^{1}]$. We also set $R_{t}=R(t,t).$ The
correction terms between Skorohod and Stratonovich integrals with respect to
$x$ have been considered in the following cases:

  1. (i)

In [15], the Skorohod-Stratonovich corrections were computed for integrals of
the form $\int_{s}^{t}\nabla f(x_{u})\mathrm{d}^{\diamond}x_{u}$ for a smooth
function $f$ defined on $\mathbb{R}^{d}$. More specifically,

$\int_{0}^{t}\nabla f(x_{r})\mathrm{d}{\bf x}_{r}=\int_{0}^{t}\nabla
f(x_{r})\mathrm{d}^{\diamond}x_{r}+\frac{1}{2}\int_{0}^{t}\Delta
f(x_{r})\mathrm{d}R_{r},$ (1)

where the integral with respect to $x$ on the left-hand side is a Stratonovich
integral while the one on the right-hand side is a Skorohod integral. The
strategy in [15] relied on the fact that $\int_{0}^{t}\nabla
f(x_{u})\mathrm{d}^{\diamond}x_{u}$ is obtained by taking limits of Riemann-
Wick sums of the form:

$S^{\Pi_{st},\diamond}=\sum_{i=0}^{n-1}\sum_{k=1}^{N}\frac{1}{k!}f^{(k)}(x_{t_{i}})\diamond{\big{(}{\bf
x}^{{\bf 1}}_{t_{i}t_{i+1}}\big{)}}^{\diamond k},$ (2)

where $\diamond$ stands for the Wick product. Then the Skorohod-Stratonovich
corrections in [15] were analyzed thanks to a computation of the Wick
corrections in (2). Notice that an extension of this result to the case of a
Gaussian process indexed by $[0,1]^{2}$ with Hölder exponent greater than
$1/3$ was handled in [24]. There some change of variable formulas were derived
for the Stratonovich and Skorohod integrals respectively. As a consequence,
the correction terms were computed explicitly. Also note that some preliminary
cases for a 1-d fractional Brownian motion had been considered in [22].

  2. (ii)

The reference [4] is concerned with solutions of rough differential equations
driven by $x$, where $x$ is again a $d$-dimensional centered Gaussian process
with i.i.d components giving rise to a $p$-geometric rough path (recall that
$2<p<3$). The equation can be written as

$\mathrm{d}y_{r}=\sigma(y_{r})\mathrm{d}{\bf x}_{r},$ (3)

with a smooth enough coefficient $\sigma:\mathbb{R}^{d}\to\mathbb{R}^{d\times
d}$, and we refer to [11] for more details about this object. Below we denote
by $y_{0}$ the initial condition of (3), and $J_{u}^{x}$ is designated as the
Jacobian of the flow map $y_{0}\to y_{u}$. Then the formula for the correction
terms in [4] can be read as

$\int_{0}^{t}y_{r}\mathrm{d}{\bf
x}_{r}=\int_{0}^{t}y_{r}\mathrm{d}^{\diamond}x_{r}+\frac{1}{2}\int_{0}^{t}\text{tr}[\sigma(y_{r})]\mathrm{d}R_{r}\\\
+\int_{[0<r_{1}<r_{2}<t]}\text{tr}\left[J_{r_{2}}^{x}(J_{r_{1}}^{x})^{-1}\sigma(y_{r_{1}})-\sigma(y_{r_{2}})\right]\mathrm{d}R(r_{1},r_{2}).$
(4)

Consider the $i$-th column $\sigma_{i}(x)$ of the coefficient matrix
$\sigma(x)$ as a vector field on $\mathbb{R}^{d}$ for $1\leq i\leq d$. If the
Lie bracket
$[\sigma_{i},\sigma_{j}]=\sigma_{i}\sigma_{j}-\sigma_{j}\sigma_{i}=0$ for
$1\leq i\leq j\leq d$, then the solution $y_{t}$ to (3) is of the form
$y_{t}=\varphi(x_{t},y_{0})$ with
$(\nabla_{x}\cdot\varphi)(x,y_{0})=\text{tr}[\sigma(x,y_{0})]$ and
$J_{t}^{x}=\sigma(y_{t})$ (see [1, Proposition 24]). Clearly in this case, (4)
coincides with (1), noting that the last term on the right-hand side of (4)
now vanishes. Therefore, relation (4) is indeed compatible with (1).

###  1.2. Main result and strategy

In this paper, we consider a $d$-dimensional centered Gaussian process $x$
with i.i.d components. Let $y$ be a _controlled_ process relative to $x$. That
is, the increments $\delta y_{st}:=y_{t}-y_{s}$ can be decomposed along the
increments of $x$ as follows:

$\delta y_{st}^{i}=\sum_{j=1}^{d}y_{s}^{x;ij}\,{\bf x}_{st}^{{\bf
1};j}+r_{st}^{i},~{}\text{ for }i=1,\dots,d,$ (5)

where $y^{x}$ has finite $p$-variation and the remainder $r$ has finite
$\frac{p}{2}$-variation (one can alternatively use Hölder spaces in this
definition). Notice that controlled processes are the natural class of
functions for which a proper rough integration with respect to $x$ can be
constructed (see e.g [13]). The following is the main result of this paper
(see Theorem 3.1 below for a more precise statement). Under proper conditions
on $x$ and $y$,

$\int_{0}^{t}y_{r}\,\mathrm{d}{\bf
x}_{r}=\int_{0}^{t}y_{r}\mathrm{d}^{\diamond}x_{r}+\frac{1}{2}\sum_{i=1}^{d}\int_{0}^{t}y_{r}^{x;ii}\mathrm{d}R_{r}+\sum_{i=1}^{d}\int_{[0<r_{1}<r_{2}<t]}\left(\mathbf{D}^{i}_{r_{1}}y^{i}_{r_{2}}-y_{r_{2}}^{x;ii}\right)\mathrm{d}R(r_{1},r_{2}).$
(6)

On the left-hand side of (6), the integral $\int_{0}^{t}y_{r}\,\mathrm{d}{\bf
x}_{r}$ is understood in the rough path sense (see Proposition 2.20 below for
further details). On the right-hand side of the same equation,
$\int_{0}^{t}y_{r}\mathrm{d}^{\diamond}x_{r}$ stands for the Skorohod
integral, $y^{x}$ is defined by (5), $R$ is the covariance function alluded to
above and $\mathbf{D}$ represents the Malliavin derivative (notions of
Malliavin calculus will be recalled in Section 2.4).

Note that our formula (6) unifies the previous cases (1) and (4). Indeed, we
have argued that (4) can be seen as an extension of (1). Furthermore, note
that the solution $y$ to the rough differential equation (3) with a
sufficiently regular coefficient function $\sigma(y)$ is a controlled process
with $y_{s}^{x;ij}=(\sigma(y_{s}))_{ij}$ and
$\mathbf{D}_{s}y_{t}=J_{t}^{x}(J_{s}^{x})^{-1}\sigma(y_{s})$ for $s\leq t$.
Therefore it is easy to see that (6) is an extension of (4), which is the main
result of [4].

Inspired by [4, 15], our proof of the main result is based on the discrete
sums method combined with tools from rough paths theory and Malliavin
calculus, in which some discrete techniques developed in [17] are also
invoked. We outline the idea as follows.

Consider a controlled process $y$ with a decomposition given by (5),
satisfying some path regularity and Malliavin differentiability conditions.
Let $\pi=\pi^{n}$ denote the uniform partition of $[0,T]$ and $\mathcal{H}$ be
the Hilbert space associated to $x$. Denote

$y^{\pi}(t)=\sum_{k=0}^{n-1}y_{t_{k}}{\bf 1}_{[t_{k},t_{k+1}]}(t).$

We first prove that (see Lemma 3.2)

$\lim_{n\to\infty}y^{\pi}=y\quad\text{in}\quad\mathbb{D}^{1,2}(\mathcal{H}).$

This enables us to show the convergence of the discrete Skorohod integral
$\delta^{\diamond}(y^{\pi})$ to the Skorohod integral
$\delta^{\diamond}(y)=\int_{0}^{T}y_{r}\,\mathrm{d}^{\diamond}x_{r}$ in
$L^{2}(\Omega)$, i.e.,

$\int_{s}^{t}y_{r}\,\mathrm{d}^{\diamond}x_{r}=\lim_{n\to\infty}\sum_{m=0}^{n-1}\left[\sum_{i=1}^{d}y_{t_{m}}^{i}\diamond{\bf
x}^{{\bf 1};i}_{t_{m}t_{m+1}}\right]\text{ in }L^{2}(\Omega),$ (7)

where we have written $\pi=\\{t_{0},\ldots,t_{n}\\}$ with $t_{m}=s+m(t-s)/n$
for $m=0,\ldots,n.$ It is also known from the rough paths theory that the
following holds true almost surely:

$\int_{s}^{t}y_{r}\mathrm{d}{\bf x}_{r}=\lim_{|\pi|\to
0}\sum_{i=1}^{d}\sum_{k=0}^{n-1}\left(y_{t_{k}}^{i}{\bf x}^{{\bf
1};i}_{t_{k}t_{k+1}}+\frac{1}{2}\sum_{j=1}^{d}y_{t_{k}}^{x;ij}{\bf
x}^{2;ji}_{t_{k}t_{k+1}}\right),$ (8)

where the left-hand side above stands for the rough paths integral of $y$ with
respect to $x$.

The Stratonovich-Skorohod correction terms in (6) now can be obtained by
computing the difference between the right-hand sides in (7) and (8). When
computing the difference, one key ingredient will be the forthcoming
Proposition 2.28. This proposition is inspired by the analogous results in
[17] and establishes a general estimate for weighted sums in the second chaos
of the Gaussian process $x$.

To end this subsection, we provide some perspectives for future works. On the
one hand, as an application, some central limit theorems for Skorohod
integrals could be obtained with the help of our main result, generalizing the
results in [17] and [19]. On the other hand, noting that in this article the
Gaussian rough paths with finite $p$-variation for $p\in(1,3)$ are handled, we
believe that our methodology can be carried out for rougher Gaussian paths
with $p\geq 3$. It is also interesting to consider the correction terms for
the processes arising from delay equations ([18]), Volterra equations ([5, 6,
14]), etc.

###  1.3. Notation

Let $\pi:0=t_{0}<t_{1}<\cdots<t_{n}=T$ be a partition on $[0,T]$. Take
$s,t\in[0,T]$. We write $\llbracket s,t\rrbracket$ for the discrete interval
that consists of $t_{k}$’s such that $t_{k}\in[s,t]$. We denote by
$\mathcal{S}_{k}([s,t])$ the simplex
$\\{(t_{1},\dots,t_{k})\in[s,t]^{k};\,t_{1}\leq\cdots\leq t_{k}\\}$. In
contrast, whenever we deal with a discrete interval, we set
$\mathcal{S}_{k}(\llbracket s,t\rrbracket)=\\{(t_{1},\dots,t_{k})\in\llbracket
s,t\rrbracket^{k};\,t_{1}<\cdots<t_{k}\\}$. For $t=t_{k}$ we denote
$t-:=t_{k-1}$, $t+:=t_{k+1}$. We also denote by $\mathcal{D}([s,t])$ the set
of all dissections of $[s,t]$.

For $x=(x^{1},\dots,x^{n})$ and $y=(y^{1},\dots,y^{n})$ in $\mathbb{R}^{n}$,
we write write $xy$ for their dot product $x\cdot y=\sum_{i=1}^{n}x^{i}y^{i}$
and write $|x|$ for the Euclid norm
$\left(\sum_{i=1}^{n}x_{i}^{2}\right)^{1/2}$. The $L^{p}$-norm
$(\mathbb{E}[|\xi|^{p}])^{1/p}$ of a random variable $\xi$ is denoted by
$\|\xi\|_{p}$, for $p\geq 1.$

Generally speaking, we will write $C$ for a generic constant whose exact value
can change from line to line.

##  2\. Preliminary material

This section contains some basic tools from rough paths theory and Malliavin
calculus, as well as some analytical results, which are crucial for the
definition and integration of controlled processes.

###  2.1. Rough path above $\mathbf{x}$

In this subsection we shall recall the notion of a rough path above a signal
$x$, and how this applies to Gaussian signals. The interested reader is
referred to [8, 11, 13] for further details.

As mentioned in Section 1.3, for $s<t$ and $m\geq 1$, we consider the simplex
$\mathcal{S}_{m}([s,t])=\\{(u_{1},\ldots,u_{m})\in[s,t]^{m};\,u_{1}<\cdots<u_{m}\\}$.
For notational sake, we just write $\mathcal{S}_{m}$ for
$\mathcal{S}_{m}([0,T])$. The definition of a rough path above a signal $x$
relies on the following notion of increments.

######  Definition 2.1.

Let $k\geq 1$. Then the space of $(k-1)$-increments, denoted by
$\mathcal{C}_{k}([0,T],\mathbb{R}^{d})$ or simply
$\mathcal{C}_{k}(\mathbb{R}^{d})$, is defined as

$\mathcal{C}_{k}(\mathbb{R}^{d})\equiv\left\\{g\in
C(\mathcal{S}_{k};\mathbb{R}^{d});\,\lim_{t_{i}\to t_{i+1}}g_{t_{1}\cdots
t_{k}}=0,\,i\leq k-1\right\\}.$

We now introduce a finite difference operator called $\delta$, which acts on
increments and is useful to split iterated integrals into simpler pieces.

######  Definition 2.2.

Let $g\in\mathcal{C}_{1}(\mathbb{R}^{d})$,
$h\in\mathcal{C}_{2}(\mathbb{R}^{d})$. Then for $(s,u,t)\in\mathcal{S}_{3}$,
we set

$\delta g_{st}=g_{t}-g_{s},\quad\mbox{ and }\quad\delta
h_{sut}=h_{st}-h_{su}-h_{ut}.$

The regularity of increments in $\mathcal{C}_{2}(\mathbb{R}^{d})$ will be
measured in terms of $p$-variation as follows.

######  Definition 2.3.

For $f\in\mathcal{C}_{2}(\mathbb{R}^{d})$ and $p>0$, we define

$\|f\|_{p-{\rm var}}=\|f\|_{p-{\rm
var};[0,T]}=\sup_{(t_{i})\in\mathcal{D}([0,T])}\left(\sum_{i}|f_{t_{i}t_{i+1}}|^{p}\right)^{1/p}.$

The set of increments in $\mathcal{C}_{2}(\mathbb{R}^{d})$ with finite
$p$-variation is denoted by $\mathcal{C}_{2}^{p{\rm-var}}(\mathbb{R}^{d})$.

Note that for a continuous function $g:[0,T]\to\mathbb{R}^{d}$ with finite
$p$-variation, if we set $\|g\|_{p-{\rm var};[0,T]}=\|\delta g\|_{p-{\rm
var};[0,T]}$, then we recover its usual $p$-variation.

With these preliminary definitions in hand, we can now introduce the notion of
a rough path.

######  Definition 2.4.

Let $x$ be a continuous $\mathbb{R}^{d}$-valued path with finite $p$-variation
for some $p\geq 1$. We say that $x$ gives rise to a geometric $p$-rough path
if there exists a family

$\left\\{{\bf
x}^{n;i_{1},\ldots,i_{n}}_{st};\,(s,t)\in\mathcal{S}_{2},\,n\leq\lfloor
p\rfloor,\,i_{1},\ldots,i_{n}\in\\{1,\ldots,d\\}\right\\},$

such that $\mathbf{x}^{\mathbf{1}}_{st}=\delta x_{st}$ and

_(1) Regularity:_ For all $n\leq\lfloor p\rfloor$, each component of
$\mathbf{x}^{n}$ has finite $\frac{p}{n}$-variation in the sense of Definition
2.3.

_(2) Multiplicativity:_ With $\delta\mathbf{x}^{n}$ as in Definition 2.2, we
have

$\delta{\bf x}^{n;i_{1},\ldots,i_{n}}_{sut}=\sum_{n_{1}=1}^{n-1}{\bf
x}^{n_{1};i_{1},\ldots,i_{n_{1}}}_{su}{\bf
x}^{n-n_{1};i_{n_{1}+1},\ldots,i_{n}}_{ut}.$ (9)

_(3) Geometricity:_ Let $x^{\varepsilon}$ be a sequence of piecewise smooth
approximations of $x$. For any $n\leq\lfloor p\rfloor$ and any set of indices
$i_{1},\ldots,i_{n}\in\\{1,\ldots,d\\}$, we assume that ${\bf
x}^{\varepsilon,n;i_{1},\ldots,i_{n}}$ converges in $\frac{p}{n}$-variation to
${\bf x}^{n;i_{1},\ldots,i_{n}}$, where ${\bf
x}^{\varepsilon,n;i_{1},\ldots,i_{n}}_{st}$ is defined for
$(s,t)\in\mathcal{S}_{2}$ by

${\bf
x}^{\varepsilon,n;i_{1},\ldots,i_{n}}_{st}=\int_{(u_{1},\ldots,u_{n})\in\mathcal{S}_{n}([s,t])}dx_{u_{1}}^{\varepsilon,i_{1}}\cdots
dx_{u_{n}}^{\varepsilon,i_{n}}.$

We are now ready to state one of the main assumptions on our standing process
$x$.

######  Hypothesis 2.5.

Throughout the paper, $x$ will designate a continuous $\mathbb{R}^{d}$-valued
path with finite $p$-variation for $p\geq 1$. We assume that $x$ gives rise to
a geometric rough path in the sense of Definition 2.4.

On top of Hypothesis 2.5, we assume that $x_{t}=(x_{t}^{1},\ldots,x_{t}^{d})$
is a continuous centered Gaussian process with i.i.d. components, defined on a
complete probability space $(\Omega,\mathcal{F},\mathbf{P})$. The covariance
function of $x$ is given by

$R(s,t):=\mathbf{E}\left[x_{s}^{j}x_{t}^{j}\right],$ (10)

for any $j\in\\{1,\dots,d\\}$. Throughout the paper, we will also set
$R_{t}:=R(t,t)$.

The information on the path regularity of $x$ is mostly contained in the
rectangular increments $R_{uv}^{st}$ of its covariance function $R$, which are
defined as

$R_{uv}^{st}:=\mathbf{E}\left[(x_{t}^{j}-x_{s}^{j})\,(x_{v}^{j}-x_{u}^{j})\right].$
(11)

The regularity of $R$ is expressed thanks to some $2$d-variation type
quantities. For sake of clarity we first recall the definition of the 2d
$\rho$-variation.

######  Definition 2.6.

Let $\rho\in[1,\infty)$. For a general continuous function
$R:[0,T]^{2}\to\mathbb{R}$, its 2d $\rho$-variation is defined as

$\displaystyle\|R\|_{\rho-{\rm
var};[s,t]\times[u,v]}:=\sup_{\begin{subarray}{c}(t_{i})\in\mathcal{D}([s,t])\\\
(t_{j}^{\prime})\in\mathcal{D}\left(\left[u,v\right]\right)\end{subarray}}\left(\sum_{t^{\prime}_{j}}\sum_{t_{i}}\left|R_{t_{i}t_{i+1}}^{t_{j}^{\prime}t_{j+1}^{\prime}}\right|^{\rho}\right)^{\frac{1}{\rho}}.$
(12)

where

$R_{t_{i}t_{i+1}}^{t_{j}^{\prime}t_{j+1}^{\prime}}=R(t_{i+1},t_{j+1}^{\prime})-R(t_{i+1},t_{j}^{\prime})-R(t_{i},t_{j+1}^{\prime})+R(t_{i},t_{j}^{\prime}).$
(13)

Observe that, whenever the function $R$ in Definition 2.6 is a covariance
function as in (10), the rectangular increment
$R_{t_{i}t_{i+1}}^{t_{j}^{\prime}t_{j+1}^{\prime}}$ can also be written as in
(11).

In the following definition, we consider each element $((s,t),(u,v))$ in
$\mathcal{S}_{2}\times\mathcal{S}_{2}$ as a rectangle and denote it by
$[s,t]\times[u,v]$.

######  Definition 2.7.

A continuous function
$\omega:\mathcal{S}_{2}\times\mathcal{S}_{2}\to\mathbb{R}^{+}$ is called a 2d
control, if it is zero on degenerate rectangles, and super-additive in the
sense that for all rectangles $A,B$ and $C$ contained in $\mathcal{S}_{2}$
satisfying $A\cup B\subset C$ and $A\cap B=\emptyset$,

$\omega(A)+\omega(B)\leq\omega(C).$

With these elementary notions at hand, we next introduce a hypothesis which
allows the use of both rough paths techniques and tools from stochastic
analysis for the underlying process $x$.

######  Hypothesis 2.8.

Let $x$ be a $d$-dimensional continuous and centered Gaussian process with
i.i.d. components, whose initial value is 0 and covariance $R$ is given by
(10). We assume that for some $\rho\in[1,2)$, the function $R$ admits a finite
2d $\rho$-variation.

It is well known that for a continuous function $g:[0,T]\to\mathbb{R}$ with
finite $p$-variation, the function $[a,b]\mapsto\|g\|^{p}_{p-{\rm var};[a,b]}$
is a control. However, for a continuous function $R:[0,T]^{2}\to\mathbb{R}$
with finite 2d $\rho$-variation, the function
$[a,b]\times[c,d]\mapsto\|R\|^{\rho}_{\rho-{\rm var};[a,b]\times[c,d]}$ may
fail to be super-additive for $\rho>1$ (see [9, Theorem 1]). To regain this
property, here we introduce the so-called _controlled 2d $\rho$-variation_ for
$1\leq\rho<\infty$ (this notion is also introduced in [9]).

######  Definition 2.9.

Let $\rho\in[1,\infty)$. For a continuous function $R:[0,T]^{2}\to\mathbb{R}$,
its controlled 2d $\rho$-variation is defined as

$\vvvert R\vvvert_{\rho-{\rm
var};[s,t]\times[u,v]}:=\sup_{\Pi\in\mathcal{P}([s,t]\times[u,v])}\left(\sum_{[t_{i},t_{i+1}]\times[t^{\prime}_{j},t^{\prime}_{j+1}]\in\Pi}\left|R_{t_{i}t_{i+1}}^{t^{\prime}_{j}t^{\prime}_{j+1}}\right|^{\rho}\right)^{\frac{1}{\rho}},$

where $R_{t_{i}t_{i+1}}^{t^{\prime}_{j}t^{\prime}_{j+1}}$ is given in (13),
$\Pi$ is a partition of $[s,t]\times[u,v]$ which is a finite set of
essentially disjoint rectangles whose union is $[s,t]\times[u,v]$, and
$\mathcal{P}([s,t]\times[u,v])$ is the collection of all such partitions.

The norms $\|\cdot\|$ and $\vvvert\cdot\vvvert$ are comparable thanks to the
following property borrowed from [9, Theorem 1]: for all $\rho^{\prime}>\rho$
there exists a constant $C_{\rho,\rho^{\prime}}$ such that

$C_{\rho,\rho^{\prime}}\vvvert f\vvvert_{\rho^{\prime}-{\rm
var};[s_{1},s_{2}]\times[t_{1},t_{2}]}\leq\|f\|_{\rho-{\rm
var};[s_{1},s_{2}]\times[t_{1},t_{2}]}\leq\vvvert f\vvvert_{\rho-{\rm
var};[s_{1},s_{2}]\times[t_{1},t_{2}]}.$ (14)

Moreover, the function $[a,b]\times[c,d]\mapsto\vvvert
f\vvvert^{\rho}_{\rho-{\rm var};[a,b]\times[c,d]}$ is a 2d control ([9,
Theorem 1]).

######  Remark 2.10.

Owing to (14), any continuous function $R:[0,T]^{2}\to\mathbb{R}$ with finite
_2d $\rho$-variation_ also has a finite _controlled 2d
$\rho^{\prime}$-variation_ for all $\rho^{\prime}>\rho.$ Furthermore, for all
$((s,t),(u,v))\in\mathcal{S}_{2}\times\mathcal{S}_{2}$,

$\|R\|^{\rho^{\prime}}_{\rho^{\prime}-{{\rm
var}};[s,t]\times[u,v]}\leq\omega([s,t]\times[u,v]),$

where $\omega$ is the 2d control (as introduced in Definition 2.7) given by

$\omega([s,t]\times[u,v])=\vvvert
R\vvvert^{\rho^{\prime}}_{{\rho^{\prime}}-{{\rm var}};[s,t]\times[u,v]}.$ (15)

######  Remark 2.11.

As an example, if the Gaussian process $x$ is a fractional Brownian motion
with Hurst parameter $H\in(0,\frac{1}{2}]$, the covariance $R$ of $x$ has
finite 2d $\rho$-variation with $\rho=\frac{1}{2H}$ and Hypothesis 2.8 is
satisfied (see [9, Example 1]). If we choose
$\rho^{\prime}>\rho=\frac{1}{2H}$, then the quantity
$\|R\|^{\rho^{\prime}}_{\rho^{\prime}-{{\rm var}};[s,t]\times[u,v]}$ is
controlled by the 2d control $\vvvert
R\vvvert^{\rho^{\prime}}_{\rho^{\prime}-{{\rm var}};[s,t]\times[u,v]}$. Note
that $\vvvert R\vvvert_{\rho-{{\rm var}};[0,T]^{2}}=\infty$ if we choose
$\rho\leq\frac{1}{2H}$ (as shown in [9, Example 2]).

In the sequel we will also request the function $t\mapsto R(t,t)$ to be Hölder
continuous. We now state an additional assumption which guarantees this Hölder
continuity (see e.g [4, 12] for a similar hypothesis).

######  Hypothesis 2.12.

Let $\rho\in[1,2)$ be given in Hypothesis 2.8. We assume that there exists
$C<\infty$ such that for all $s,t\in[0,T]$ the covariance function $R$
satisfies

$\|R(t,\cdot)-R(s,\cdot)\|^{\rho}_{\rho-{\rm{var;}}[0,T]}\leq C|t-s|.$ (16)

######  Remark 2.13.

A direct consequence of Hypothesis 2.12 is that $R_{t}:=R(t,t)$ has finite
$\rho$-variation, by [4, Lemma 2.14]. Moreover, recall that by Hypothesis 2.8,
we have $x_{0}=0$ and hence $R(0,\cdot)=R(\cdot,0)\equiv 0$. This together
with (16) implies that $R(t,\cdot)$ and $R(\cdot,t)$ have finite
$\rho$-variation for each fixed $t\in[0,T].$

######  Remark 2.14.

Given $\rho\in[1,2)$, clearly we have, for $0\leq s_{1}\leq s_{2}\leq T$ and
$0\leq t_{1}\leq t_{2}\leq T$,

$\|R\|_{\rho-\text{var;}[s_{1},s_{2}]\times[t_{1},t_{2}]}^{2}\leq\|R\|_{\rho-\text{var;}[s_{1},s_{2}]\times[0,T]}\|R\|_{\rho-\text{var;}[0,T]\times[t_{1},t_{2}]}.$

Furthermore, it is a direct consequence of (16) that for $0\leq s\leq t\leq
T$,

$\|R\|^{\rho}_{\rho-\text{var;}[s,t]\times[0,T]}\leq C(t-s).$

Combining the two inequalities above, we have the following control on the 2d
$\rho$-variation of $R$: for some positive constant $C$,

$\|R\|_{\rho-\text{var;}[s_{1},s_{2}]\times[t_{1},t_{2}]}^{2\rho}\leq
C(s_{2}-s_{1})(t_{2}-t_{1}).$ (17)

######  Remark 2.15.

Note that for
$1\leq\gamma\leq\gamma^{\prime}<\infty,\|R\|_{\gamma^{\prime}-\text{var;}[s,t]\times[u,v]}\leq\|R\|_{\gamma-\text{var;}[s,t]\times[u,v]}$.
Therefore, under Hypothesis 2.12, inequality (16) and hence (17) hold with
$\rho$ replaced by $\rho^{\prime}\in(\rho,2)$ and $C$ depending on
$(\rho,\rho^{\prime},T)$.

######  Remark 2.16.

Clearly (17) yields the following relations on squares of the form
$[s,t]^{2}$,

$\|R\|_{\rho-\text{var;}[s,t]^{2}}^{\rho}\leq C(t-s).$ (18)

We say that $R$ has finite _Hölder-controlled 2d $\rho$-variation_ if $R$
satisfies both Hypothesis 2.8 and (18). An important consequence of $R$ having
finite Hölder controlled 2d $\rho$-variation is that $x$ has $1/p$-Hölder
continuous sample paths for every $p>2\rho$. It is also readily checked that,
whenever $x$ satisfies (18), we have

$\mathbf{E}\left[\left({\bf x}_{st}^{{\bf 1};i}\right)^{2}\right]\leq
c\,(t-s)^{\frac{1}{\rho}}.$ (19)

######  Remark 2.17.

Similarly to the argument in [3, Remark 2.4], for any process $x$ whose
covariance function $R$ admits a finite $\rho$-variation one can introduce a
deterministic time-change $\tau:[0,T]\to[0,T]$ such that
$\tilde{X}=X\circ\tau$ has finite Hölder-controlled 2d $\rho$-variation. That
is the time changed process $\tilde{X}$ satisfies Hypothesis 2.8 and equation
(18).

The following result (stated e.g. in [11, Theorem 15.33]) relates the 2d
$\rho$-variation of $R$ with the pathwise assumptions allowing to apply the
abstract rough paths theory.

######  Proposition 2.18.

Let $x=(x^{1},\ldots,x^{d})$ be a continuous centered Gaussian process with
i.i.d. components and covariance function $R$ defined by (10). If $R$
satisfies Hypothesis 2.8, then $x$ also satisfies Hypothesis 2.5 provided
$p>2\rho$.

Proposition 2.18 asserts that under Hypothesis 2.8, the Gaussian process $x$
is amenable to rough path analysis. In particular, a rough path integral with
respect to $x$ can be constructed. In this context, the natural class of
integrand one might want to consider is the family of controlled processes.
Its definition is recalled below.

######  Definition 2.19.

Consider a continuous $\mathbb{R}^{d}$-valued path $x$ with finite
$p$-variation for some $p\geq 1$. We say that a continuous
$\mathbb{R}^{d}$-valued path $y$ of finite $p$-variation is controlled by $x$,
if there exist a continuous $\mathbb{R}^{d^{2}}$-valued path $y^{x}$ of finite
$p$-variation and a $1$-increment process
$r\in\mathcal{C}_{2}^{\frac{p}{2}{\rm-var}}(\mathbb{R}^{d})$ as defined in
Definition 2.3, such that

$\delta y_{st}^{i}=\sum_{j=1}^{d}y_{s}^{x;ij}\,{\bf x}_{st}^{{\bf
1};j}+r_{st}^{i},~{}\text{ for }i=1,\dots,d.$ (20)

We are now ready to state the basic integration result for controlled
processes, which can be found e.g. in [8, 11, 13].

######  Proposition 2.20.

Let $T>0$ be fixed. Let ${\bf x}$ be a geometric $p$-rough path lifted from a
continuous $\mathbb{R}^{d}$-valued path with finite $p$-variation for some
$p\in[1,3),$ and let $y$ be a continuous $\mathbb{R}^{d}$-valued path of
finite $p$-variation that is controlled by $x$ in the sense of Definition
2.19. Then for $0\leq s<t\leq T$, one can define the integral
$\int_{s}^{t}y_{r}\mathrm{d}{\bf x}_{r}$ as the limit of the following Riemann
sums,

$\int_{s}^{t}y_{r}d{\bf x}_{r}=\lim_{|\pi_{n}|\to
0}\sum_{k=0}^{n-1}\left(\sum_{i=1}^{d}y^{i}_{t_{k}}{\bf
x}_{t_{k}t_{k+1}}^{1,i}+\sum_{i=1}^{d}\sum_{j=1}^{d}y_{t_{k}}^{x;ij}{\bf
x}_{t_{k}t_{k+1}}^{2;ij}\right),$ (21)

where $\pi_{n}=[s=t_{0}<t_{1}<\dots<t_{n}=t]$ is a partition of $[s,t]$ and
$|\pi_{n}|=\max\limits_{k\in\\{0,\ldots,n-1\\}}|t_{k+1}-t_{k}|.$ In (21),
observe that we have also used the convention on inner products put forward in
Section 1.3. Moreover, there exists a constant $C=C(T,p)$ depending only on
$(T,p)$ such that for all $0\leq s<t\leq T$ we have

$\left|\int_{s}^{t}y_{r}d{\bf x}_{r}-y_{s}{\bf
x}^{1}_{st}-\sum_{i=1}^{d}\sum_{j=1}^{d}y_{s}^{x;ij}{\bf
x}_{st}^{2;ij}\right|\leq C\left(\|{\bf x}^{1}\|_{p-{\rm
var}}\|r\|_{\frac{p}{2}-{\rm var}}+\|{\bf x}^{2}\|_{\frac{p}{2}-{\rm
var}}\|y^{x}\|_{p-{\rm var}}\right)|t-s|^{3/p},$

where we recall that $r$ is the increment introduced in (20).

Recall that our main objective is to compute some Skorohod-Stratonovich
corrections as in [4]. To this aim we will need a more detailed description of
the increments of $y$ than the ones given in (20). Namely we will assume that
$y$ is a second order controlled process as defined below (for the definition
of controlled processes of general order, we refer to [8, Definition 4.17] or
[3, Definition 5.1]).

######  Definition 2.21.

Consider a continuous $\mathbb{R}^{d}$-valued path $x$ with finite
$p$-variation for some $p\geq 1$. We say that a continuous
$\mathbb{R}^{d}$-valued path $y$ of finite $p$-variation is a second-order
controlled process with respect to $x$, if there exist a continuous
$\mathbb{R}^{d^{2}}$-valued path $y^{x}$, a continuous
$\mathbb{R}^{d^{3}}$-valued path $y^{xx}$, both of which are of finite
$p$-variation, and $1$-increment processes
$r\in\mathcal{C}_{2}^{\frac{p}{3}{\rm-
var}}(\mathbb{R}^{d}),r^{x}\in\mathcal{C}_{2}^{\frac{p}{2}{\rm-
var}}(\mathbb{R}^{d^{2}})$ as defined in Definition 2.3, such that for
$i=1,\dots,d$ and $(s,t)\in\mathcal{S}_{2}([0,T])$ we have

$\delta y_{st}^{i}=\sum_{j=1}^{d}y_{s}^{x;ij}\,{\bf x}_{st}^{{\bf
1};j}+\sum_{j,k=1}^{d}y_{s}^{xx;ijk}\,{\bf x}_{st}^{{\bf 2};jk}+r_{st}^{i}.$
(22)

In addition, the increment $y^{x}$ in (22) is a controlled process of order 1,
that is for $i,j=1,\dots,d$ and $(s,t)\in\mathcal{S}_{2}([0,T])$ we have

$\delta y_{st}^{x;ij}=\sum_{k=1}^{d}y_{s}^{xx;ijk}\,{\bf x}_{st}^{{\bf
1};k}+r_{st}^{x;ij}.$ (23)

###  2.2. Higher dimensional Young integrals

In this subsection, we gather some inequalities for Young integrals in
$\mathbb{R}^{n}$ which will feature in our computations throughout the paper.
We start by a relation for integrals in the plane borrowed form [10, 23].

######  Theorem 2.22.

Let $f,R:[0,T]^{2}\to\mathbb{R}$ be continuous functions with finite
$p$-variation and finite $q$-variation respectively for
$\frac{1}{p}+\frac{1}{q}>1$. Specifically recalling our Definition 2.6, we
assume $\|f\|_{p-{\rm var};[0,T]^{2}}<\infty$ and $\|R\|_{q-{\rm
var};[0,T]^{2}}<\infty$. Moreover, assume that for all $s_{1},s_{2}\in[0,T]$,
both $f(s_{1},\cdot)$ and $f(\cdot,s_{2})$ have finite 1-dimensional
$p$-variation as given in Definition 2.3. Then the $2$d Young-Stieltjes
integral of $f$ with respect to $R$ exists and the following Young’s
inequality holds, for
$[\underaccent{\bar}{s}_{1},\bar{s}_{1}]\times[\underaccent{\bar}{s}_{2},\bar{s}_{2}]\subset[0,T]^{2}$,

$\left|\int_{[\underaccent{\bar}{s}_{1},\bar{s}_{1}]\times[\underaccent{\bar}{s}_{2},\bar{s}_{2}]}f(s_{1},s_{2})dR(s_{1},s_{2})\right|\leq
C_{p,q}\Bigg{(}|f(\underaccent{\bar}{s}_{1},\underaccent{\bar}{s}_{2})|+\|f(\underaccent{\bar}{s}_{1},\cdot)\|_{p-{\rm
var};[\underaccent{\bar}{s}_{2},\bar{s}_{2}]}\\\
+\|f(\cdot,\underaccent{\bar}{s}_{2})\|_{p-{\rm
var};[\underaccent{\bar}{s}_{1},\bar{s}_{1}]}+\|f\|_{p-{\rm
var};[\underaccent{\bar}{s}_{1},\bar{s}_{1}]\times[\underaccent{\bar}{s}_{2},\bar{s}_{2}]}\Bigg{)}\|R\|_{q-{\rm
var};[\underaccent{\bar}{s}_{1},\bar{s}_{1}]\times[\underaccent{\bar}{s}_{2},\bar{s}_{2}]}\,.$
(24)

We now state a lemma about integration in $\mathbb{R}^{4}$ which will be
invoked in order to analyze discretization properties for the Malliavin
derivative of a controlled process $y$. Although its proof might be traced
back to [23], we include it here for the sake of clarity since Lemma 2.23 is
tailored for our specific needs.

######  Lemma 2.23.

Let $f,g,R$ be continuous functions defined on $[0,T]^{2}$. Similarly to
Theorem 2.22, we assume that $f,g$ have finite $p$-variation, as well as
$f(s_{1},\cdot),f(\cdot,s_{3}),g(s_{2},\cdot)$ and $g(\cdot,s_{4})$ for fixed
arbitrary $s_{1},s_{2},s_{3},s_{4}\in[0,T]$. We also suppose that $R$ has
finite $q$-variation on $[0,T]^{2}$, with $p,q$ satisfying
$\frac{1}{p}+\frac{1}{q}>1$. Then for
$\underaccent{\bar}{s}_{1},\bar{s}_{1},\ldots,\underaccent{\bar}{s}_{4},\bar{s}_{4}\in[0,T]$
such that $\underaccent{\bar}{s}_{j}<\bar{s}_{j}$ for $j=1,\dots,4$, the
following Young integral in $\mathbb{R}^{4}$ is well defined:

$I^{f,g,R}(\underaccent{\bar}{s}_{1},\bar{s}_{1},\ldots,\underaccent{\bar}{s}_{4},\bar{s}_{4}):=\int_{[\underaccent{\bar}{s}_{1},\bar{s}_{1}]\times[\underaccent{\bar}{s}_{2},\bar{s}_{2}]\times[\underaccent{\bar}{s}_{3},\bar{s}_{3}]\times[\underaccent{\bar}{s}_{4},\bar{s}_{4}]}f(s_{1},s_{3})g(s_{2},s_{4})dR(s_{1},s_{2})\mathrm{d}R(s_{3},s_{4}).$

Moreover,
$I^{f,g,R}(\underaccent{\bar}{s}_{1},\bar{s}_{1},\ldots,\underaccent{\bar}{s}_{4},\bar{s}_{4})$
can be bounded as

$\displaystyle\left|I^{f,g,R}(\underaccent{\bar}{s}_{1},\bar{s}_{1},\ldots,\underaccent{\bar}{s}_{4},\bar{s}_{4})\right|\leq
C_{p,q}~{}\|R\|_{q-{\rm
var};[\underaccent{\bar}{s}_{1},\bar{s}_{1}]\times[\underaccent{\bar}{s}_{2},\bar{s}_{2}]}\|R\|_{q-{\rm
var};[\underaccent{\bar}{s}_{3},\bar{s}_{3}]\times[\underaccent{\bar}{s}_{4},\bar{s}_{4}]}$
$\displaystyle\quad\times\Big{(}|f(\underaccent{\bar}{s}_{1},\underaccent{\bar}{s}_{3})|+\|f(\cdot,\underaccent{\bar}{s}_{3})\|_{p-{\rm
var};[\underaccent{\bar}{s}_{1},\bar{s}_{1}]}+\|f(\underaccent{\bar}{s}_{1},\cdot)\|_{p-{\rm
var};[\underaccent{\bar}{s}_{3},\bar{s}_{3}]}+\|f\|_{p-{\rm
var};[\underaccent{\bar}{s}_{1},\bar{s}_{1}]\times[\underaccent{\bar}{s}_{3},\bar{s}_{3}]}\Big{)}$
$\displaystyle\quad\times\Big{(}|g(\underaccent{\bar}{s}_{2},\underaccent{\bar}{s}_{4})|+\|g(\cdot,\underaccent{\bar}{s}_{4})\|_{p-{\rm
var};[\underaccent{\bar}{s}_{2},\bar{s}_{2}]}+\|g(\underaccent{\bar}{s}_{2},\cdot)\|_{p-{\rm
var};[\underaccent{\bar}{s}_{4},\bar{s}_{4}]}+\|g\|_{p-{\rm
var};[\underaccent{\bar}{s}_{2},\bar{s}_{2}]\times[\underaccent{\bar}{s}_{4},\bar{s}_{4}]}\Big{)}\,.$
(25)

###### Proof.

We will divide this proof in several steps.

Step 1: Decomposition of the integral. We can write

$I^{f,g,R}(\underaccent{\bar}{s}_{1},\bar{s}_{1},\ldots,\underaccent{\bar}{s}_{4},\bar{s}_{4})=\int_{[\underaccent{\bar}{s}_{3},\bar{s}_{3}]\times[\underaccent{\bar}{s}_{4},\bar{s}_{4}]}F(s_{3},s_{4})\mathrm{d}R(s_{3},s_{4})$
(26)

where the function $F$ is defined on $[0,T]^{2}$ by

$F(s_{3},s_{4})=\int_{[\underaccent{\bar}{s}_{1},\bar{s}_{1}]\times[\underaccent{\bar}{s}_{2},\bar{s}_{2}]}f(s_{1},s_{3})g(s_{2},s_{4})\mathrm{d}R(s_{1},s_{2}),$
(27)

and where we observe that the right-hand side of (27) is well defined thanks
to Theorem 2.22. Our strategy in order to estimate $I^{f,g,R}$ will rely on
some succesive applications of (24). Specifically, with (26) in mind, relation
(24) yields

$\left|I^{f,g,R}(\underaccent{\bar}{s}_{1},\bar{s}_{1},\underaccent{\bar}{s}_{2},\bar{s}_{2},\underaccent{\bar}{s}_{3},\bar{s}_{3},\underaccent{\bar}{s}_{4},\bar{s}_{4})\right|\leq
C_{p,q}\Big{(}|F(\underaccent{\bar}{s}_{3},\underaccent{\bar}{s}_{4})|+\|F(\underaccent{\bar}{s}_{3},\cdot)\|_{p-{\rm
var},[\underaccent{\bar}{s}_{4},\bar{s}_{4}]}\\\
+\|F(\cdot,\underaccent{\bar}{s}_{4})\|_{p-{\rm
var},[\underaccent{\bar}{s}_{3},\bar{s}_{3}]}+\|F\|_{p-{\rm
var},[\underaccent{\bar}{s}_{3},\bar{s}_{3}]\times[\underaccent{\bar}{s}_{4},\bar{s}_{4}]}\Big{)}\|R\|_{q-{\rm
var},[\underaccent{\bar}{s}_{3},\bar{s}_{3}]\times[\underaccent{\bar}{s}_{4},\bar{s}_{4}]}.$
(28)

We will now estimate the terms in right-hand side of (28) separately.

Step 2: Upper bound for
$F(\underaccent{\bar}{s}_{3},\underaccent{\bar}{s}_{4})$. Given
$(\underaccent{\bar}{s}_{3},\underaccent{\bar}{s}_{4})\in[0,T]^{2}$ and
recalling the definition (27) of $F$, another application of (24) enables to
write

$|F(\underaccent{\bar}{s}_{3},\underaccent{\bar}{s}_{4})|\leq
C_{p,q}\Big{(}|f(\underaccent{\bar}{s}_{1},\underaccent{\bar}{s}_{3})g(\underaccent{\bar}{s}_{2},\underaccent{\bar}{s}_{4})|+|f(\underaccent{\bar}{s}_{1},\underaccent{\bar}{s}_{3})|\|g(\cdot,\underaccent{\bar}{s}_{4})\|_{p-{\rm
var},[\underaccent{\bar}{s}_{2},\bar{s}_{2}]}\\\
+|g(\underaccent{\bar}{s}_{2},\underaccent{\bar}{s}_{4})|\|f(\cdot,\underaccent{\bar}{s}_{3})\|_{p-{\rm
var},[\underaccent{\bar}{s}_{1},\bar{s}_{1}]}+\|f(\cdot,\underaccent{\bar}{s}_{3})\|_{p-{\rm
var},[\underaccent{\bar}{s}_{1},\bar{s}_{1}]}\|g(\cdot,\underaccent{\bar}{s}_{4})\|_{p-{\rm
var},[\underaccent{\bar}{s}_{2},\bar{s}_{2}]}\Big{)}\|R\|_{q-{\rm
var},[\underaccent{\bar}{s}_{1},\bar{s}_{1}]\times[\underaccent{\bar}{s}_{2},\bar{s}_{2}]}$

and we notice that the above expression can be simplified as

$\displaystyle|F(\underaccent{\bar}{s}_{3},\underaccent{\bar}{s}_{4})|\leq$
$\displaystyle
C_{p,q}\Big{(}|f(\underaccent{\bar}{s}_{1},\underaccent{\bar}{s}_{3})|+\|f(\cdot,\underaccent{\bar}{s}_{3})\|_{p-{\rm
var},[\underaccent{\bar}{s}_{1},\bar{s}_{1}]}\Big{)}$
$\displaystyle\qquad\times\Big{(}|g(\underaccent{\bar}{s}_{2},\underaccent{\bar}{s}_{4})|+\|g(\cdot,\underaccent{\bar}{s}_{4})\|_{p-{\rm
var},[\underaccent{\bar}{s}_{2},\bar{s}_{2}]}\Big{)}\|R\|_{q-{\rm
var},[\underaccent{\bar}{s}_{1},\bar{s}_{1}]\times[\underaccent{\bar}{s}_{2},\bar{s}_{2}]}.$
(29)

Step 3: Upper bound for $\|F(\underaccent{\bar}{s}_{3},\cdot)\|_{p-{\rm
var},[\underaccent{\bar}{s}_{4},\bar{s}_{4}]}$. Recall the Definition 2.3 of
$p$-variation. We thus have

$\|F(\underaccent{\bar}{s}_{3},\cdot)\|_{p-{\rm
var},[\underaccent{\bar}{s}_{4},\bar{s}_{4}]}=\sup_{\pi}\left(\sum_{i}|F(\underaccent{\bar}{s}_{3},v_{i+1})-F(\underaccent{\bar}{s}_{3},v_{i})|^{p}\right)^{1/p}.$

Plugging expression (27) into the above relation, we get

$\|F(\underaccent{\bar}{s}_{3},\cdot)\|_{p-{\rm
var},[\underaccent{\bar}{s}_{4},\bar{s}_{4}]}^{p}=\sup_{\pi}\sum_{i}\left|\int_{[\underaccent{\bar}{s}_{1},\bar{s}_{1}]\times[\underaccent{\bar}{s}_{2},\bar{s}_{2}]}f(s_{1},\underaccent{\bar}{s}_{3})\Big{(}g(s_{2},v_{i+1})-g(s_{2},v_{i})\Big{)}dR(s_{1},s_{2})\right|^{p}.$

We now apply (24) again and we end up with

$\|F(\underaccent{\bar}{s}_{3},\cdot)\|_{p-{\rm
var},[\underaccent{\bar}{s}_{4},\bar{s}_{4}]}\leq C_{p,q}\|R\|_{q-{\rm
var},[\underaccent{\bar}{s}_{1},\bar{s}_{1}]\times[\underaccent{\bar}{s}_{2},\bar{s}_{2}]}\sum_{k=1}^{4}V_{k},$
(30)

where the terms $V_{1},V_{2}$ are respectively defined by

$\displaystyle V_{1}$
$\displaystyle=|f(\underaccent{\bar}{s}_{1},\underaccent{\bar}{s}_{3})|\sup_{\pi}\left(\sum_{i}\left|g(\underaccent{\bar}{s}_{2},v_{i+1})-g(\underaccent{\bar}{s}_{2},v_{i})\right|^{p}\right)^{1/p};$
$\displaystyle V_{2}$
$\displaystyle=|f(\underaccent{\bar}{s}_{1},\underaccent{\bar}{s}_{3})|\sup_{\pi}\left(\sum_{i}\|g(\cdot,v_{i+1})-g(\cdot,v_{i})\|_{p-{\rm
var},[\underaccent{\bar}{s}_{2},\bar{s}_{2}]}^{p}\right)^{1/p},$

and similarly the terms $V_{3},V_{4}$ are expressed as

$\displaystyle V_{3}$
$\displaystyle=\sup_{\pi}\left(\sum_{i}|g(\underaccent{\bar}{s}_{2},v_{i+1})-g(\underaccent{\bar}{s}_{2},v_{i})|^{p}\|f(\cdot,\underaccent{\bar}{s}_{3})\|_{p-{\rm
var},[\underaccent{\bar}{s}_{1},\bar{s}_{1}]}^{p}\right)^{1/p};$
$\displaystyle V_{4}$
$\displaystyle=\sup_{\pi}\left(\sum_{i}\|f(\cdot,\underaccent{\bar}{s}_{3})(g(*,v_{i+1})-g(*,v_{i}))\|_{p-{\rm
var},[\underaccent{\bar}{s}_{1},\bar{s}_{1}]\times[\underaccent{\bar}{s}_{2},\bar{s}_{2}]}^{p}\right)^{1/p}.$

In addition, the terms $V_{1},V_{2},V_{3}$ are easily bounded. Indeed,
resorting again to Definition 2.3, we get

$V_{1}=|f(\underaccent{\bar}{s}_{1},\underaccent{\bar}{s}_{3})|\|g(\underaccent{\bar}{s}_{2},\cdot)\|_{p-{\rm
var},[\underaccent{\bar}{s}_{4},\bar{s}_{4}]},\qquad
V_{2}\leq|f(\underaccent{\bar}{s}_{1},\underaccent{\bar}{s}_{3})|\|g\|_{p-{\rm
var},[\underaccent{\bar}{s}_{2},\bar{s}_{2}]\times[\underaccent{\bar}{s}_{4},\bar{s}_{4}]},$
(31)

and

$V_{3}=\|f(\cdot,\underaccent{\bar}{s}_{3})\|_{p-{\rm
var},[\underaccent{\bar}{s}_{1},\bar{s}_{1}]}\|g(\underaccent{\bar}{s}_{2},\cdot)\|_{p-{\rm
var},[\underaccent{\bar}{s}_{4},\bar{s}_{4}]}\,.$ (32)

For the term $V_{4}$, by Definition 2.6, it is readily checked that

$\displaystyle V_{4}$
$\displaystyle\leq\|f(\cdot,\underaccent{\bar}{s}_{3})\|_{p-{\rm
var},[\underaccent{\bar}{s}_{1},\bar{s}_{1}]}\|g\|_{p-{\rm
var},[\underaccent{\bar}{s}_{2},\bar{s}_{2}]\times[\underaccent{\bar}{s}_{4},\bar{s}_{4}]}\,.$
(33)

Hence, plugging (31), (32) and (33) into (30), we end up with

$\|F(\underaccent{\bar}{s}_{3},\cdot)\|_{p-{\rm
var},[\underaccent{\bar}{s}_{4},\bar{s}_{4}]}\leq
C_{p,q}\Big{(}|f(\underaccent{\bar}{s}_{1},\underaccent{\bar}{s}_{3})|+\|f(\cdot,\underaccent{\bar}{s}_{3})\|_{p-{\rm
var},[\underaccent{\bar}{s}_{1},\bar{s}_{1}]}|\Big{)}\\\
\Big{(}\|g(\underaccent{\bar}{s}_{2},\cdot)\|_{p-{\rm
var},[\underaccent{\bar}{s}_{4},\bar{s}_{4}]}+\|g\|_{p-{\rm
var},[\underaccent{\bar}{s}_{2},\bar{s}_{2}]\times[\underaccent{\bar}{s}_{4},\bar{s}_{4}]}\Big{)}\|R\|_{q-{\rm
var},[\underaccent{\bar}{s}_{1},\bar{s}_{1}]\times[\underaccent{\bar}{s}_{2},\bar{s}_{2}]}.$
(34)

Furthermore, notice that in a similar way we get

$\|F(\cdot,\underaccent{\bar}{s}_{4})\|_{p-{\rm
var},[\underaccent{\bar}{s}_{4},\bar{s}_{4}]}\leq
C_{p,q}\Big{(}|g(\underaccent{\bar}{s}_{2},\underaccent{\bar}{s}_{4})|+\|g(\cdot,\underaccent{\bar}{s}_{4})\|_{p-{\rm
var},[\underaccent{\bar}{s}_{2},\bar{s}_{2}]}|\Big{)}\\\
\Big{(}\|f(\underaccent{\bar}{s}_{1},\cdot)\|_{p-{\rm
var},[\underaccent{\bar}{s}_{3},\bar{s}_{3}]}+\|f\|_{p-{\rm
var},[\underaccent{\bar}{s}_{1},\bar{s}_{1}]\times[\underaccent{\bar}{s}_{3},\bar{s}_{3}]}\Big{)}\|R\|_{q-{\rm
var},[\underaccent{\bar}{s}_{1},\bar{s}_{1}]\times[\underaccent{\bar}{s}_{2},\bar{s}_{2}]}.$
(35)

Step 4: Upper bound for $\|F\|_{p-{\rm
var},[\underaccent{\bar}{s}_{3},\bar{s}_{3}]\times[\underaccent{\bar}{s}_{4},\bar{s}_{4}]}$.
According to Definition 2.6, one can write

$\|F\|^{p}_{p-{\rm
var},[\underaccent{\bar}{s}_{3},\bar{s}_{3}]\times[\underaccent{\bar}{s}_{4},\bar{s}_{4}]}=\sup_{\pi}\sum_{t_{i},t_{j}^{\prime}}\left|F(t_{i},t^{\prime}_{j})+F(t_{i+1},t^{\prime}_{j+1})-F(t_{i},t^{\prime}_{j+1})-F(t_{i+1},t_{j}^{\prime})\right|^{p}\,,$

where we recall that $\pi$ takes the form
$\pi\in\mathcal{D}([\underaccent{\bar}{s}_{3},\bar{s}_{3}])\times\mathcal{D}([\underaccent{\bar}{s}_{4},\bar{s}_{4}])$
and the notation $\mathcal{D}([s,t])$ is introduced in Section 1.3. Hence with
the expression (27) of $F$ in mind we get

$\|F\|^{p}_{p-{\rm
var},[\underaccent{\bar}{s}_{3},\bar{s}_{3}]\times[\underaccent{\bar}{s}_{4},\bar{s}_{4}]}\\\
=\sup_{\pi}\sum_{t_{i},t_{j}^{\prime}}\left|\int_{[\underaccent{\bar}{s}_{1},\bar{s}_{1}]\times[\underaccent{\bar}{s}_{2},\bar{s}_{2}]}(f(s_{1},t_{i+1})-f(s_{1},t_{i}))(g(s_{2},t_{j+1}^{\prime}-g(s_{2},t_{j}^{\prime}))dR(s_{1},s_{2})\right|^{p}.$

In this context, relation (24) can thus be read as

$\|F\|_{p-{\rm
var},[\underaccent{\bar}{s}_{3},\bar{s}_{3}]\times[\underaccent{\bar}{s}_{4},\bar{s}_{4}]}\leq
C\|R\|_{q-{\rm
var},[\underaccent{\bar}{s}_{1},\bar{s}_{1}]\times[\underaccent{\bar}{s}_{2},\bar{s}_{2}]}\sup_{\pi}\Bigg{(}\sum_{t_{i},t_{j}^{\prime}}\left|Q_{ij^{\prime}}\right|^{p}\Bigg{)}^{1/p},$

where the term $Q_{ij^{\prime}}$ is defined by

$\displaystyle
Q_{ij^{\prime}}=|(f(\underaccent{\bar}{s}_{1},t_{i+1})-f(\underaccent{\bar}{s}_{1},t_{i}))(g(\underaccent{\bar}{s}_{2},t_{j+1}^{\prime})-g(\underaccent{\bar}{s}_{2},t_{j}^{\prime}))|$
$\displaystyle\quad+|f(\underaccent{\bar}{s}_{1},t_{i+1})-f(\underaccent{\bar}{s}_{1},t_{i})|\|g(\cdot,t^{\prime}_{j+1})-g(\cdot,t^{\prime}_{j})\|_{p-{\rm
var},[\underaccent{\bar}{s}_{2},\bar{s}_{2}]}$
$\displaystyle\quad+\|f(\cdot,t_{i+1})-f(\cdot,t_{i})\|_{p-{\rm
var},[\underaccent{\bar}{s}_{1},\bar{s}_{1}]}|g(\underaccent{\bar}{s}_{2},t^{\prime}_{j+1})-g(\underaccent{\bar}{s}_{2},t^{\prime}_{j})|$
$\displaystyle\quad+\|f(\cdot,t_{i+1})-f(\cdot,t_{i})\|_{p-{\rm
var},[\underaccent{\bar}{s}_{1},\bar{s}_{1}]}\|g(\cdot,t^{\prime}_{j+1})-g(\cdot,t^{\prime}_{j})\|_{p-{\rm
var},[\underaccent{\bar}{s}_{2},\bar{s}_{2}]},$

and we notice that $Q_{ij^{\prime}}$ can easily be simplified as

$\displaystyle Q_{ij^{\prime}}=$
$\displaystyle\Big{(}|(f(\underaccent{\bar}{s}_{1},t_{i+1})-f(\underaccent{\bar}{s}_{1},t_{i}))|+\|f(\cdot,t_{i+1})-f(\cdot,t_{i})\|_{p-{\rm
var},[\underaccent{\bar}{s}_{1},\bar{s}_{1}]}\Big{)}$
$\displaystyle~{}~{}~{}\times\Big{(}|(g(\underaccent{\bar}{s}_{2},t_{j+1}^{\prime})-g(\underaccent{\bar}{s}_{2},t_{j}^{\prime}))|+\|g(\cdot,t^{\prime}_{j+1})-g(\cdot,t^{\prime}_{j})\|_{p-{\rm
var},[\underaccent{\bar}{s}_{2},\bar{s}_{2}]}\Big{)}\,.$

Summarizing our computations in this step, we have found that

$\|F\|_{p-{\rm
var},[\underaccent{\bar}{s}_{3},\bar{s}_{3}]\times[\underaccent{\bar}{s}_{4},\bar{s}_{4}]}\leq
C\|R\|_{q-{\rm
var},[\underaccent{\bar}{s}_{1},\bar{s}_{1}]\times[\underaccent{\bar}{s}_{2},\bar{s}_{2}]}\Big{(}\|f(\underaccent{\bar}{s}_{1},\cdot)\|_{p-{\rm
var},[\underaccent{\bar}{s}_{3},\bar{s}_{3}]}+\|f\|_{p-{\rm
var},[\underaccent{\bar}{s}_{1},\bar{s}_{1}]\times[\underaccent{\bar}{s}_{3},\bar{s}_{3}]}\Big{)}\\\
\times\Big{(}g(\underaccent{\bar}{s}_{2},\cdot)\|_{p-{\rm
var},[\underaccent{\bar}{s}_{4},\bar{s}_{4}]}+\|g\|_{p-{\rm
var},[\underaccent{\bar}{s}_{2},\bar{s}_{2}]\times[\underaccent{\bar}{s}_{4},\bar{s}_{4}]}\Big{)}\,.$
(36)

Step 5: Conclusion. Let us gather our estimates (29), (34), (35) and (36) into
(28). Then we let the patient reader check that (25) is achieved. This
finishes the proof. ∎

###  2.3. The Hilbert space associated to $x$

Consider a continuous d-dimensional centered Gaussian process $x$ on $[0,T]$
with covariance function $R$ given by (10). Every component of $x$ (say
$x^{1}$) is a 1-dimensional centered Gaussian process with covariance $R$. In
this section we review some basic facts about the related Hilbert space
$\mathcal{H}$ of functions for which Wiener integrals with respect to $x$ (see
e.g. [20]) are well defined.

The Hilbert space $\mathcal{H}$ is the completion of the set of step functions

$\mathcal{E}=\left\\{\sum_{i=1}^{n}a_{i}{\bf
1}_{\left[0,t_{i}\right]}:a_{i}\in\mathbb{R}\text{,
}t_{i}\in\left[0,T\right],i=1,\dots,n\text{ for }n\in\mathbb{N}\right\\},$

with respect to the inner product

$\left\langle\sum_{i=1}^{n}a_{i}{\bf 1}_{[0,t_{i}]},\sum_{j=1}^{m}b_{j}{\bf
1}_{[0,s_{j}]}\right\rangle_{\mathcal{H}}=\sum_{i=1}^{n}\sum_{j=1}^{m}a_{i}b_{j}R\left(t_{i},s_{j}\right).$

Observe that this inner product can also be written as

$\displaystyle\left\langle\sum_{i=1}^{n}a_{i}{\bf
1}_{[0,t_{i}]},\sum_{j=1}^{m}b_{j}{\bf
1}_{[0,s_{j}]}\right\rangle_{\mathcal{H}}=$
$\displaystyle\int_{0}^{T}\int_{0}^{T}\left(\sum_{i=1}^{n}a_{i}{\bf
1}_{[0,t_{i}]}(t)\right)\left(\sum_{j=1}^{m}b_{j}{\bf
1}_{[0,s_{j}]}(s)\right)~{}dR(t,s).$ (37)

One can further relate $\mathcal{H}$ to our driving process $x$ in the
following way: let $\mathbf{H}$ be the closure of the set

${\bf\mathsf{E}}=\left\\{\sum\nolimits_{i=1}^{n}a_{i}x^{1}_{t_{i}}:a_{i}\in\mathbb{R},\text{
}t_{i}\in\left[0,T\right],i=1,\dots,n\text{ for }n\in\mathbb{N}\right\\},$

in $L^{2}(\Omega,\mathcal{F},\mathbf{P})$. Then the linear map
$x^{1}:\mathcal{E}\to\mathsf{E}$ defined by $x^{1}({\bf 1}_{[0,t]})=x^{1}_{t}$
extends to a linear isometry between $\mathcal{H}$ and $\mathbf{H}$. Hence,
$\mathbf{H}=\\{x^{1}(h),h\in\mathcal{H}\\}$ and this family is known as the
isonormal Gaussian process related to $x^{1}$ (see [20, Definition 1.1.1]).
Note that $x^{1}(h)$ for $h\in\mathcal{H}$ is called the Wiener integral of
$h$ with respect to $x^{1}$ and is usually denoted by
$\int_{0}^{T}h(s)dx^{1}_{s}$.

######  Remark 2.24.

Recall that we have assumed $x_{0}=0$ and thus $R(0,0)=0$. Thus relation (37)
suggests

$\displaystyle\langle
h_{1},h_{2}\rangle_{\mathcal{H}}=\int_{0}^{T}\int_{0}^{T}h_{1}(s)h_{2}(t)dR(s,t)~{}~{}\text{
for }h_{1},h_{2}\in\mathcal{H},$ (38)

whenever the 2D Young’s integral on the right-hand side is well-defined (see,
e.g., [2, Proposition 4] for details).

######  Remark 2.25.

Denoting by $\mathcal{E}([a,b])$ the set of step functions in $\mathcal{E}$
restricted on $[a,b]\subset[0,T]$, the closure $\mathcal{H}([a,b])$ of
$\mathcal{E}([a,b])$ with respect to the inner product (37) then coincides
with $\mathcal{H}$ restricted on $[a,b]$, and for $f,g\in\mathcal{H}$,

$\left\langle f\,{\bf 1}_{[a,b]},\,g\,{\bf
1}_{[a,b]}\right\rangle_{\mathcal{H}}=\left\langle
f,\,g\right\rangle_{\mathcal{H}([a,b])}.$ (39)

###  2.4. Malliavin calculus for Gaussian processes

In this subsection, we collect some basic concepts of Malliavin calculus, and
we refer to [20] for more details.

Recall that $x_{t}$ is a continuous centered $d$-dimensional Gaussian process
with i.i.d. components, defined on a complete probability space
$(\Omega,\mathcal{F},\mathbf{P})$. For the sake of simplicity, we assume that
$\mathcal{F}$ coincides with the $\sigma$-algebra generated by
$\\{x_{t};\,t\in[0,T]\\}$. For the $d$-dimensional process $x$, we define an
extension of the Wiener integral defined as follows: let
$\varphi=(\varphi^{1},\ldots,\varphi^{d})$ be an element of $\mathcal{H}^{d}$
where we recall that $\mathcal{H}$ has been introduced in Section 2.3. Then we
set

$x(\varphi)=\sum_{j=1}^{d}x^{j}(\varphi^{j})\,,$ (40)

where each term $x^{j}(\varphi^{j})$ is a 1-d Wiener integral as in Section
2.3.

A smooth functional of $x$ is a random variable of the form
$F=f(x(\varphi_{1}),\ldots,x(\varphi_{n}))$, where $n\geq 1$,
$\\{\varphi_{1},\ldots,\varphi_{n}\\}$ is a family of elements of
$\mathcal{H}^{d}$ and each $x(\varphi_{i})$ is understood as in (40).
Moreover, we assume that the function $f:\mathbb{R}^{n}\to\mathbb{R}$ is
smooth and its partial derivatives grow at most polynomially fast. Then, the
Malliavin derivative $\mathbf{D}F$ of $F$ is the $\mathcal{H}^{d}$-valued
random variable defined by

$\mathbf{D}F=\sum_{k=1}^{n}\frac{\partial f}{\partial
x_{k}}(x(\varphi_{1}),\dots,x(\varphi_{n}))\varphi_{k}.$ (41)

One can show that $\mathbf{D}$ is closable from $L^{2}(\Omega)$ to
$L^{2}(\Omega;\mathcal{H})$, and thus one may span the space of the smooth and
cylindrical random variables under the norm

$\|F\|_{1,2}=\left(\mathbb{E}[F^{2}]+\mathbb{E}[\|\mathbf{D}F\|_{\mathcal{H}}^{2}]\right)^{\frac{1}{2}}\,.$

The resulting closure is called Sobolev space $\mathbb{D}^{1,2}$.

######  Remark 2.26.

As seen in (41), the Malliavin derivative $\mathbf{D}F$ of a functional $F$ is
a $\mathbb{R}^{d}$-valued process. The $i$-th coordinate of $\mathbf{D}F$
corresponds to the Malliavin derivative of $F$ with respect to the randomness
in $x^{i}$ only. It will be denoted by $\mathbf{D}^{i}F$ in the sequel.

The divergence operator $\delta^{\diamond}$ (also known as the Skorohod
integral) is the adjoint operator of the Malliavin derivative operator
$\mathbf{D}$ defined by the duality relation

$\mathbb{E}[F\delta^{\diamond}(u)]=\mathbb{E}[\langle\mathbf{D}F,u\rangle_{\mathcal{H}^{d}}],~{}~{}\text{
for all }F\in\mathbb{D}^{1,2}\text{ and }\text{ for all }u\in\text{Dom
}\delta^{\diamond}.$

Here $\text{Dom }\delta^{\diamond}$ is the domain of the divergence operator
$\delta^{\diamond}$, which is the space of $\mathcal{H}$-valued random
variables $u\in L^{2}(\Omega;\mathcal{H}^{d})$ such that
$|\mathbb{E}[\langle\mathbf{D}F,u\rangle_{\mathcal{H}^{d}}]|\leq
c_{F}\|F\|_{2}$ with some constant $c_{F}$ depending on $F$, for all
$F\in\mathbb{D}^{1,2}$. In particular,
$\mathbb{D}^{1,2}(\mathcal{H}^{d})\subset\text{Dom }\delta^{\diamond}.$ Note
that for $u\in\text{Dom }\delta^{\diamond}$, we have $\delta^{\diamond}(u)\in
L^{2}(\Omega)$ and $\mathbb{E}[\delta^{\diamond}(u)]=0$. By convention, we
also take the following notation, for $u\in\text{Dom }\delta^{\diamond}$,

$\int_{0}^{T}u_{t}\,\mathrm{d}^{\diamond}x_{t}:=\delta^{\diamond}(u).$ (42)

For our main computations below we shall invoke the following relation taken
from [22]: for any $G\in\mathbb{D}^{1,2}(\mathbb{R}^{d})$ and $0\leq a<b\leq
T$ we have

$\delta^{\diamond}(G\,{\bf
1}_{[a,b]})=\sum_{i=1}^{d}\int_{a}^{b}G^{i}\,\mathrm{d}^{\diamond}x_{t}^{i}=\sum_{i}^{d}G^{i}\diamond\delta
x_{ab}^{i},$ (43)

where $\diamond$ stands for the Wick product (see [15] for a brief account on
Wick products). Moreover, according to [16, Proposition 4.7], relation (43)
can be simplified as

$\delta^{\diamond}(G\,{\bf 1}_{[a,b]})=\sum_{i=1}^{d}G^{i}\,\delta
x_{ab}^{i}-\langle\mathbf{D}^{i}G^{i},{\bf 1}_{[a,b]}\rangle_{\mathcal{H}}.$
(44)

###  2.5. Discrete rough paths techniques

In this subsection, we develop some inequalities about discrete sums in a
rough paths context. This kind of sum will feature prominently in the analysis
of our Skorohod-Stratonovich corrections.

We state a crucial lemma about convergence of discrete sums in the second
chaos of $x$. It generalizes [17, Lemma 3.4] to a generic Gaussian process (as
opposed to the fractional Brownian motion case handled in [17]).

######  Proposition 2.27.

Let $x$ be a $\mathbb{R}^{d}$-valued Gaussian process satisfying Hypotheses
2.8 and 2.12. For $n\geq 1$ we consider the uniform partition on $[0,T]$,
namely $t_{k}=\frac{k}{n}T$. We define a process
$F=\\{F_{t}^{ij};t\in\llbracket 0,T\rrbracket,i,j=1,\ldots,d\\}$ by
$F_{0}^{ij}=0$, and for all $t>0$,

$F_{t}^{ij}=\sum\limits_{t_{k}=0}^{t_{-}}\Big{(}{\bf
x}_{t_{k}t_{k+1}}^{2;ij}-\mathbf{E}[{\bf
x}_{t_{k}t_{k+1}}^{2;ij}]\Big{)}=\begin{cases}\sum\limits_{t_{k}=0}^{t_{-}}{\bf
x}_{t_{k}t_{k+1}}^{2;ij}~{},&i\neq j,\\\
\sum\limits_{t_{k}=0}^{t_{-}}\Big{(}{\bf
x}_{t_{k}t_{k+1}}^{2;ii}-\mathbf{E}[{\bf
x}_{t_{k}t_{k+1}}^{2;ii}]\Big{)}~{},&i=j,\end{cases}$ (45)

where we recall the notation $t_{-}$ from Section 1.3 and where ${\bf x}^{2}$
is introduced in Definition 2.4. Then for all $q\geq 1$,
$\rho^{\prime}\in(\rho,2)$, $(s,t)\in\mathcal{S}_{2}(\llbracket
0,T\rrbracket)$ and $n\geq 1$ the following inequality holds true

$\left(\mathbf{E}\left[\left|\delta
F_{st}^{ij}\right|^{q}\right]\right)^{1/q}\leq
C\frac{(t-s)^{\frac{1}{2}}}{n^{\beta-\frac{1}{2}}}\,,$ (46)

where $C=C(q,\rho,T),\beta=\frac{1}{\rho}\in(1/2,1]$ for $i\neq j$, and
$C=C(q,\rho,\rho^{\prime},T),\beta=\frac{1}{\rho^{\prime}}\in(1/2,1)$ for
$i=j$.

###### Proof.

Due to the hyper-contractivity property of the second Wiener chaos, it
suffices to show the case $q=2$. In addition, we assume (without loss of
generality) that $s=t_{m_{1}}<t=t_{m_{2}}$ for $0\leq m_{1}<m_{2}\leq n$.

Case 1: $i=j$. In this case, due to the definition (45) of $F$ and the
geometric nature of ${\bf x}$ assumed in Definition 2.4, we have

$\mathbf{E}\left[\left(\delta
F_{st}^{ii}\right)^{2}\right]=\mathbf{E}\left(\sum_{k=m_{1}}^{m_{2}-1}\left[({\bf
x}^{{\bf 1};i}_{t_{k}t_{k+1}})^{2}-\mathbf{E}[({\bf x}^{{\bf
1};i}_{t_{k}t_{k+1}})^{2}]\right]\right)^{2},$

and expanding the square on the right hand side above we get

$\mathbf{E}\left[\left(\delta
F_{st}^{ii}\right)^{2}\right]=\sum_{k,l=m_{1}}^{m_{2}-1}\left\\{\mathbf{E}[({\bf
x}^{{\bf 1};i}_{t_{k}t_{k+1}})^{2}({\bf x}^{{\bf
1};i}_{t_{l}t_{l+1}})^{2}]-\mathbf{E}[({\bf x}^{{\bf
1};i}_{t_{k}t_{k+1}})^{2}]\mathbf{E}[({\bf x}^{{\bf
1};i}_{t_{l}t_{l+1}})^{2}]\right\\}.$ (47)

In order to evaluate the right-hand side of (47) we apply a particular case of
Wick’s formula for centered Gaussian random variables $X$ and $Y$, which can
be stated as:

$\mathbf{E}[X^{2}Y^{2}]-\mathbf{E}[X^{2}]\,\mathbf{E}[Y^{2}]=2\left(\mathbf{E}[X\,Y]\right)^{2}.$

Plugging this result into (47) and recalling the definition (11) of
$R_{st}^{uv}$ we obtain

$\mathbf{E}\left[\left(\delta
F_{st}^{ii}\right)^{2}\right]=2\sum_{k,l=m_{1}}^{m_{2}-1}\left(\mathbf{E}\left[{\bf
x}^{{\bf 1};i}_{t_{k}t_{k+1}}{\bf x}^{{\bf
1};i}_{t_{l}t_{l+1}}\right]\right)^{2}=2\sum_{k,l=m_{1}}^{m_{2}-1}\left(R^{t_{k}t_{k+1}}_{t_{l}t_{l+1}}\right)^{2}.$
(48)

Therefore invoking elementary properties of $p$-variations we end up with

$\displaystyle\mathbf{E}\left[\left(\delta F_{st}^{ii}\right)^{2}\right]\leq$
$\displaystyle
2\sup_{k,l}\left(R^{t_{k}t_{k+1}}_{t_{l}t_{l+1}}\right)^{2-\rho}\|R\|^{\rho}_{\rho-\text{var};[s,t]^{2}}.$
(49)

On the right-hand side of (49), notice that under Hypothesis 2.12,
$\|R\|^{\rho}_{\rho-\text{var};[s,t]^{2}}$ can be upper bounded by $C(t-s)$
thanks to (18). Moreover, a simple use of Cauchy-Schwarz inequality, together
with (19), shows that

$\displaystyle|R_{t_{l}t_{l+1}}^{t_{k}t_{k+1}}|=\left|\mathbb{E}[{\bf
x}^{1,i}_{t_{l}t_{l+1}}{\bf
x}^{1,i}_{t_{k}t_{k+1}}]\right|\leq\left(\mathbb{E}\left[\left|{\bf
x}^{1,i}_{t_{l}t_{l+1}}\right|^{2}\right]\mathbb{E}\left[\left|{\bf
x}^{1,i}_{t_{k}t_{k+1}}\right|^{2}\right]\right)^{\frac{1}{2}}\leq\frac{C_{T}}{n^{\frac{1}{\rho}}}.$

Reporting this information into (49) and recalling that
$\beta=\frac{1}{\rho}$, it is seen that

$\mathbf{E}\left[\left(\delta F_{st}^{ii}\right)^{2}\right]\leq
CT^{2\beta-1}\frac{(t-s)}{n^{2\beta-1}}.$ (50)

This ends our proof for the case $i=j$.

Case 2: $i\neq j$. According to our definition (45), if $i\neq j$ we have

$\mathbf{E}\left[\left(\delta
F_{st}^{ij}\right)^{2}\right]=\mathbf{E}\left[\left(\sum\limits_{k=m_{1}}^{m_{2}-1}{\bf
x}_{t_{k}t_{k+1}}^{{\bf
2};ij}\right)^{2}\right]=\sum\limits_{k,l=m_{1}}^{m_{2}-1}\mathbf{E}\left[{\bf
x}_{t_{k}t_{k+1}}^{{\bf 2};ij}{\bf x}_{t_{l}t_{l+1}}^{{\bf 2};ij}\right].$

Therefore, invoking the proofs of [11, Theorem 15.33 and Proposition 15.28]
for the computation of $\mathbf{E}[{\bf x}_{t_{k}t_{k+1}}^{{\bf 2};ij}{\bf
x}_{t_{l}t_{l+1}}^{{\bf 2};ij}]$, we end up with

$\mathbf{E}\left[\left(\delta
F_{st}^{ij}\right)^{2}\right]=\sum_{k,l=m_{1}}^{m_{2}-1}\int_{t_{k}}^{t_{k+1}}\int_{t_{l}}^{t_{l+1}}R_{t_{k}v_{1}}^{t_{l}v_{2}}\,\mathrm{d}R(v_{1},v_{2}).$
(51)

We now fix $(k,l)$ and denote $G(v_{1},v_{2})=R_{t_{k}v_{1}}^{t_{l}v_{2}}$.
Then $G(t_{k},\cdot)=G(\cdot,t_{l})=0$. For any $\rho^{\prime}\in(\rho,2)$,
Hypothesis 2.8 implies $R$ has finite 2d $\rho^{\prime}$-variation, and
Hypothesis 2.12 implies both $R(t,\cdot)$ and $R(\cdot,t)$ have finite
$\rho^{\prime}$-variation for all $t\in[0,T]$. Hence resorting to Theorem
2.22, we have for some fixed $\rho^{\prime}\in(\rho,2),$

$\left|\int_{t_{k}}^{t_{k+1}}\int_{t_{l}}^{t_{l+1}}R_{t_{k}v_{1}}^{t_{l}v_{2}}\,\mathrm{d}R(v_{1},v_{2})\right|\leq
C\|R\|^{2}_{\rho^{\prime}\text{-var};[t_{k},t_{k+1}]\times[t_{l},t_{l+1}]},$

for some constant $C=C(\rho^{\prime},T)$ depending on $(\rho^{\prime},T)$
only. Plugging this inequality into (51) we obtain

$\displaystyle\mathbf{E}\left[\left(\delta F_{st}^{ij}\right)^{2}\right]\leq$
$\displaystyle
C(\rho^{\prime},T)\sum_{k,l=m_{1}}^{m_{2}-1}\|R\|^{2}_{\rho^{\prime}\text{-var};[t_{k},t_{k+1}]\times[t_{l},t_{l+1}]}$
$\displaystyle\leq
C(\rho^{\prime},T)\sup_{k,l}\|R\|^{2-\rho^{\prime}}_{\rho\text{-var};[t_{k},t_{k+1}]\times[t_{l},t_{l+1}]}\sum_{k,l=m_{1}}^{m_{2}-1}\|R\|^{\rho^{\prime}}_{\rho^{\prime}\text{-var};[t_{k},t_{k+1}]\times[t_{l},t_{l+1}]}.$

Therefore, thanks to Remark 2.10, we have

$\mathbf{E}\left[\left(\delta F_{st}^{ij}\right)^{2}\right]\leq
C(\rho^{\prime},T)\sup_{k,l}\|R\|^{2-\rho^{\prime}}_{\rho^{\prime}\text{-var};[t_{k},t_{k+1}]\times[t_{l},t_{l+1}]}\sum_{k,l=m_{1}}^{m_{2}-1}\omega([t_{k},t_{k+1}]\times[t_{l},t_{l+1}]),$

where $\omega$ is a control given in (15). Furthermore, the super-additivity
of $\omega$ yields

$\displaystyle\mathbf{E}\left[\left(\delta F_{st}^{ij}\right)^{2}\right]$
$\displaystyle\leq
C(\rho^{\prime},T)\sup_{k,l}\|R\|^{2-\rho^{\prime}}_{\rho^{\prime}\text{-var};[t_{k},t_{k+1}]\times[t_{l},t_{l+1}]}~{}\omega([s,t]^{2})$
$\displaystyle\leq
C(\rho,\rho^{\prime},T)\sup_{k,l}\|R\|^{2-\rho^{\prime}}_{\rho^{\prime}\text{-var};[t_{k},t_{k+1}]\times[t_{l},t_{l+1}]}~{}(t-s),$
(52)

where the last inequality is due to (15), (14), and (17). Finally, by
Hypothesis 2.12 (and Remark 2.15), we have

$\|R\|^{2\rho^{\prime}}_{\rho^{\prime}\text{-var};[s,t]\times[u,v]}\leq
C(\rho,\rho^{\prime},T)(t-s)(u-v),$

and therefore setting $\beta=1/\rho^{\prime}$, inequality (52) becomes

$\displaystyle\mathbf{E}\left[\left(\delta F_{st}^{ij}\right)^{2}\right]\leq$
$\displaystyle
C(\rho,\rho^{\prime},T)\left(\frac{T}{n}\right)^{2\beta-1}(t-s).$ (53)

With (50) and (53) in hand our claim (46) is now easily achieved, which
concludes the proof. ∎

Note that (46) is still valid for both cases of $i=j$ and $i\neq j$, if we
choose $\beta=\frac{1}{\rho^{\prime}}$ for any $\rho^{\prime}\in(\rho,2)$. We
now give a weighted version of Proposition 2.27, which plays an important role
in our correction computations.

######  Proposition 2.28.

Let $x$ be a $\mathbb{R}^{d}$-valued Gaussian process satisfying Hypotheses
2.8 and 2.12. Let $\rho^{\prime}\in(\rho,2)$ be fixed. For $n\geq 1$ we
consider the uniform partition on $[0,T]$, namely $t_{k}=\frac{k}{n}T$, as
well as the process $F$ defined by (45). Let now $f$ be a controlled process
in the $L^{q}(\Omega)$ sense, namely such that there exists a process $g$
fulfilling (in the matrix sense), for some
$\gamma\in(\frac{1}{4},\frac{1}{2\rho})$ and for all $q\geq 1$,

$\|f_{t}\|_{q}+\|g_{t}\|_{q}\leq C,\quad\|\delta f_{st}-g_{s}\,{\bf x}^{{\bf
1}}_{st}\|_{q}\leq C(t-s)^{2\gamma},\quad\|\delta g_{st}\|_{q}\leq
C(t-s)^{\gamma}.$ (54)

Then the following estimate holds true for $(s,t)\in\mathcal{S}_{2}(\llbracket
0,T\rrbracket)$:

$\displaystyle\Big{\|}\sum_{t_{k}=s}^{t-}f_{t_{k}}\otimes\delta
F_{t_{k}t_{k+1}}\Big{\|}_{q}$ $\displaystyle\leq$ $\displaystyle
C\frac{(t-s)^{\frac{1}{2}}}{n^{\beta-\frac{1}{2}}},$

where $C=C(q,\rho,\rho^{\prime},T)$ and
$\beta=\frac{1}{\rho^{\prime}}\in(1/2,1)$.

###### Proof.

This proposition was proved in [17, Corollary 4.9] when $x$ is a fractional
Brownian motion. Although we generalize this result to a wider class of
Gaussian processes, our proof goes along the same lines. Therefore we shall
omit the details for sake of conciseness. ∎

##  3\. Correction terms in the case $2\leq p<3$

In this section we derive a correction formula for controlled processes which
are also in the domain of the Skorohod integral. As mentioned in the
introduction, we have restricted our analysis to the case $p<3$. Although we
believe that our methodology could be extended to $p<4$, this generalisation
would require a cumbersome study of third order integrals and related weighted
sums.

######  Theorem 3.1.

Let $x$ be a Gaussian rough path with covariance given by (10) satisfying
Hypotheses 2.8 and 2.12 with $\rho\in[1,\frac{3}{2})$. This implies that $x$
has finite $p$-variation for $p>2\rho$. We can assume
$\frac{1}{p}+\frac{1}{\rho}>1$, noting that $\rho<\frac{3}{2}.$

Let $y$ be a second-order controlled process in the sense of Definition 2.21,
and we assume $\mathbb{E}[\|y\|^{2}_{p-{\rm var};[0,T]}]<\infty$. In
particular, the rough integral $\int_{0}^{t}y_{r}d{\bf x}_{r}$ is defined as
in Proposition 2.20, resorting to the convention on inner products of Section
1.3. We also assume that $y\in\mathbb{D}^{1,2}(\mathcal{H}^{d})$, so that the
Skorohod integral of $y$ given in (42) is well defined. Furthermore, we
suppose that $\mathbf{D}_{0}y$ has finite $p$-variation with
$\mathbb{E}[\|\mathbf{D}_{0}y\|^{2}_{p-{\rm var};[0,T]}]<\infty$, and
$\mathbf{D}y$ has finite 2d $p$-variation with
$\mathbb{E}[\|\mathbf{D}y\|^{2}_{p-{\rm var};[0,T]^{2}}]<\infty.$ Then for all
$t\in[0,T]$ we have almost surely

$\int_{0}^{t}y_{r}\,\mathrm{d}{\bf
x}_{r}=\int_{0}^{t}y_{r}\mathrm{d}^{\diamond}x_{r}+\frac{1}{2}\sum_{i=1}^{d}\int_{0}^{t}y_{r}^{x;ii}\mathrm{d}R_{r}+\sum_{i=1}^{d}\int_{\mathcal{S}_{2}([0,t])}\left(\mathbf{D}^{i}_{r_{1}}y^{i}_{r_{2}}-y_{r_{2}}^{x;ii}\right)\mathrm{d}R(r_{1},r_{2}),$
(55)

where we recall from Section 2.1 that $R_{r}:=R(r,r)$ and where the Malliavin
derivative $\mathbf{D}^{i}$ is introduced in Remark 2.26.

###### Proof.

Let $\pi=\pi_{n}$ be the uniform partition of order $n$ of $[0,t]$, whose
generic element is still denoted by $t_{k}=\frac{k}{n}t.$ A natural
discretization of $y$ along $\pi$ is given by

$y^{\pi}(r)=\sum_{k=0}^{n-1}y_{t_{k}}{\bf 1}_{[t_{k},t_{k+1})}(r),\quad
r\in[0,t].$ (56)

Notice that we have assumed that $y\in\mathbb{D}^{1,2}(\mathcal{H}^{d})$.
Hence both divergence integrals $\delta^{\diamond}(y^{\pi})$ and
$\delta^{\diamond}(y)$, as given in (42), are well defined. Moreover,
according to (43), we have

$\int_{0}^{t}y_{r}^{\pi}\mathrm{d}^{\diamond}x_{r}=\sum_{i=1}^{d}\sum_{k=0}^{n-1}y_{t_{k}}^{i}\diamond\mathbf{x}_{t_{k}t_{k+1}}^{{\bf
1};i},$

and owing to (44) this can be recast as

$\int_{0}^{t}y_{r}^{\pi}\mathrm{d}^{\diamond}x_{r}=\sum_{i=1}^{d}\sum_{k=0}^{n-1}y_{t_{k}}^{i}\mathbf{x}^{1;i}_{t_{k}t_{k+1}}-\langle\mathbf{D}^{i}y_{t_{k}}^{i},{\bf
1}_{[t_{k},t_{k+1}]}\rangle_{\mathcal{H}}\,.$ (57)

In addition, we will prove in the forthcoming Lemma 3.2 that
$\delta^{\diamond}(y^{\pi})$ converges in $L^{2}(\Omega)$ to
$\delta^{\diamond}(y)$. Otherwise stated, for $t\in[0,T]$ we have

$\int_{0}^{t}y_{r}\,\mathrm{d}^{\diamond}x_{r}=\lim_{n\to\infty}\int_{0}^{t}y_{r}^{\pi}\,\mathrm{d}^{\diamond}x_{r}.$
(58)

Therefore combining (57) and (58), we get the following limit in
$L^{2}(\Omega)$:

$\displaystyle\int_{0}^{t}y_{r}\,\mathrm{d}^{\diamond}x_{r}$ $\displaystyle=$
$\displaystyle\lim_{n\to\infty}\sum_{i=1}^{d}\sum_{k=0}^{n-1}\left(y_{t_{k}}^{i}{\bf
x}^{{\bf 1};i}_{t_{k}t_{k+1}}-\langle{\mathbf{D}}^{i}y^{i}_{t_{k}},{\bf
1}_{[t_{k},t_{k+1}]}\rangle_{\mathcal{H}}\right).$ (59)

On the other hand, owing to the fact that $y$ is a controlled process in the
sense of Definition 2.19, Proposition 2.20 asserts that
$\int_{0}^{t}y_{r}\mathrm{d}{\bf x}_{r}$ is defined as a rough paths integral
and hence almost surely we have

$\int_{0}^{t}y_{r}\mathrm{d}{\bf
x}_{r}=\lim_{n\to\infty}\left(\sum_{i=1}^{d}\sum_{k=0}^{n-1}y_{t_{k}}^{i}{\bf
x}^{{\bf
1};i}_{t_{k}t_{k+1}}+\sum_{i,j=1}^{d}\sum_{k=0}^{n-1}y_{t_{k}}^{x;ij}{\bf
x}^{{\bf 2};ij}_{t_{k}t_{k+1}}\right).$ (60)

Gathering relations (59) and (60), we get the following expression for the
Stratonovich-Skorohod correction term:

$\displaystyle\int_{0}^{t}y_{r}\mathrm{d}{\bf
x}_{r}-\int_{0}^{t}y_{r}\,\mathrm{d}^{\diamond}x_{r}=\lim_{n\to\infty}\sum_{i=1}^{d}\sum_{k=0}^{n-1}\left(\sum_{j=1}^{d}y_{t_{k}}^{x;ij}{\bf
x}^{2;ij}_{t_{k}t_{k+1}}+\langle{\mathbf{D}}^{i}y^{i}_{t_{k}},{\bf
1}_{[t_{k},t_{k+1}]}\rangle_{\mathcal{H}}\right),$ (61)

where the limit on the right-hand side above is understood in probability. In
(61), notice that the left-hand side is well defined thanks to the standing
assumptions of our Theorem. Hence the right-hand side of (61) also makes
sense, and we will now identify the limits therein.

In order to compute the limit for the terms $y_{t_{k}}^{x;ij}{\bf
x}^{2;ij}_{t_{k}t_{k+1}}$ in (61), observe that $y$ is a second order
controlled process according to Definition 2.21. Hence $y^{x}$ is a controlled
process satisfying relation (23). Since we have assumed that Hypotheses 2.8
and 2.12 are fulfilled, Proposition 2.28 for the increment $F$ can be applied
with $f=y^{x}$. Recalling (see (45)) that

$\delta F_{t_{k}t_{k+1}}^{ij}={\bf x}_{t_{k}t_{k+1}}^{2;ij}-\mathbb{E}[{\bf
x}_{t_{k}t_{k+1}}^{2;ij}],$

we end up with the following relation, valid for $i,j=1,\dots,d$, where the
limit has to be considered in the $L^{1}(\Omega)$ sense:

$\lim\limits_{n\to\infty}\sum_{k=0}^{n-1}y_{t_{k}}^{x;ij}\Big{(}{\bf
x}^{2;ij}_{t_{k}t_{k+1}}-\mathbf{E}\left[{\bf
x}^{2;ij}_{t_{k}t_{k+1}}\right]\Big{)}=0.$ (62)

In particular, going back to (61), we get that for $i\neq j$ we have

$\lim\limits_{n\to\infty}\sum\limits_{i\neq
j}^{d}\sum_{k=0}^{n-1}y_{t_{k}}^{x;ij}{\bf x}^{2;ij}_{t_{k}t_{k+1}}=0.$ (63)

Let us deal with the left-hand side of (62) when $i=j$. Specifically, we will
express the limit of the sums
$\sum_{k=0}^{n-1}y_{t_{k}}^{x;ii}\mathbf{E}\left[{\bf
x}^{2;ii}_{t_{k}t_{k+1}}\right]$ as a Young integral. To this aim, notice that
$2{\bf x}_{t_{k}t_{k+1}}^{2,ii}=\left({\bf x}_{t_{k}t_{k+1}}^{1,i}\right)^{2}$
due to the geometric assumption in Definition 2.4. Hence invoking the fact
that $R_{t_{k}}=R(t_{k},t_{k})$ we have

$\displaystyle 2\mathbf{E}\left[{\bf x}^{2;ii}_{t_{k}t_{k+1}}\right]$
$\displaystyle=$
$\displaystyle\mathbf{E}\left[(x_{t_{k+1}}^{i}-x_{t_{k}}^{i})^{2}\right]=R_{t_{k+1}}-2R(t_{k+1},t_{k})+R_{t_{k}}$
$\displaystyle=$
$\displaystyle\left(R_{t_{k+1}}-R_{t_{k}}\right)-2\left(R(t_{k+1},t_{k})-R(t_{k},t_{k})\right).$

Therefore for all $i=1,\dots,d$, we obtain a decomposition of the form

$\sum_{k=0}^{n-1}y_{t_{k}}^{x;ii}\mathbf{E}\left[{\bf
x}^{2;ii}_{t_{k}t_{k+1}}\right]=\frac{1}{2}I_{n}^{i}-J_{n}^{i}$ (64)

where $I_{n}^{i},J_{n}^{i}$ are respectively defined by

$I_{n}^{i}=\sum_{k=0}^{n-1}y_{t_{k}}^{x;ii}\delta
R_{t_{k}t_{k+1}},\quad\text{and }\quad
J_{n}^{i}=\sum_{k=0}^{n-1}y_{t_{k}}^{x;ii}\big{(}R(t_{k+1},t_{k})-R(t_{k},t_{k})\big{)}.$
(65)

The limit of for the term $I^{i}_{n}$ in (64) can be computed easily. Indeed,
thanks to Remark 2.13 we know that $t\to R_{t}$ has finite $\rho$-variation.
Furthermore, since $y$ is a second order controlled process, Definition 2.21
entails that $y^{x}$ has finite $p$-variation. We have also mentioned in
Theorem 3.1 that $p^{-1}+\rho^{-1}>1$. Hence classical Young integration
arguments reveal that for $i=1,\dots,d$ we have almost surely,

$\lim_{n\to\infty}I_{n}^{i}=\int_{0}^{t}y_{r}^{x;ii}dR_{r}.$ (66)

As far as the term $J_{n}^{i}$ in (65) is concerned, let us recast this
expression in terms of a 2-d Riemann sum. Namely we define another uniform
partition $\\{v_{l};0\leq l\leq n-1\\}$ of $[0,t]$, with $v_{l}=\frac{l}{n}t.$
Then we start by writing

$J_{n}^{i}=\sum_{k=0}^{n-1}y_{t_{k}}^{x;ii}\big{(}R(t_{k+1},v_{k})-R(t_{k},v_{k})\big{)}.$
(67)

In addition, notice that thanks to Remark 2.13 we have $R(\cdot,0)=0$. Thus an
immediate telescoping sum argument yields the following relation, valid for
$k=0,\dots,n-1:$

$R(t_{k+1},v_{k})-R(t_{k},v_{k})=\sum_{l=0}^{k-1}R_{v_{l}v_{l+1}}^{t_{k}t_{k+1}}.$

Reporting this identity into (67), we get

$J_{n}^{i}=\sum_{k=0}^{n-1}y_{t_{k}}^{x;ii}\sum_{l=0}^{k-1}R_{v_{l}v_{l+1}}^{t_{k}t_{k+1}}=\sum_{0\leq
l<k\leq n-1}y_{t_{k}}^{x;ii}R_{v_{l}v_{l+1}}^{t_{k}t_{k+1}}.$ (68)

This decomposition prompts us to define a degenerate function $f$ in the plane
as $f^{i}(u,v)=y_{u}^{x;ii}{\bf 1}_{[0<v<u<t]}$. With this notation in hand,
relation (68) reads

$J_{n}^{i}=\sum_{k,l=0}^{n-1}f^{i}(t_{k},v_{l})R_{v_{l}v_{l+1}}^{t_{k}t_{k+1}}.$

In order to analyze the convergence of $J_{n}^{i}$, we now argue as follows:
first $R$ has a finite 2-dimensional $\rho$-variation. The function
$f^{i}(u,v)=y_{u}^{x,ii}{\bf 1}_{[0<v<u<t]}$ is also easily seen to have a
finite 2-dimensional $p$-variation (owing to the fact that $y^{x,ii}$ has
finite $p$-variation), and recall that $p^{-1}+\rho^{-1}>1$. Hence standard
convergence procedures for 2d-Young integrals show that almost surely

$\lim_{n\to\infty}J_{n}^{i}=\int_{0}^{t}\int_{0}^{t}f^{i}(u,v)\,\mathrm{d}R(u,v)=\int_{\mathcal{S}_{2}([0,t])}y_{r_{2}}^{x;ii}\,\mathrm{d}R(r_{1},r_{2}).$
(69)

Summarizing our considerations for the case $i=j$, we gather (66) and (69)
into the decomposition (64). We conclude that almost surely,

$\lim\limits_{n\to\infty}\sum\limits_{i=1}^{d}\sum_{k=0}^{n-1}y_{t_{k}}^{x;ii}\mathbb{E}[{\bf
x}^{2;ii}_{t_{k}t_{k+1}}]=\frac{1}{2}\sum_{i=1}^{d}\int_{0}^{t}y_{r}^{x;ii}\mathrm{d}R_{r}-\sum_{i=1}^{d}\int_{\mathcal{S}_{2}([0,t])}y_{r_{2}}^{x;ii}\,\mathrm{d}R(r_{1},r_{2}).$
(70)

We now go back to (61), and handle the terms
$\langle\mathbf{D}^{i}y^{i}_{t_{k}},{\bf
1}_{[t_{k},t_{k+1}]}\rangle_{\mathcal{H}}$ therein. We write the inner product
in $\mathcal{H}$ in an explicit way thanks to (38), which yields

$\langle\mathbf{D}^{i}y^{i}_{t_{k}},{\bf
1}_{[t_{k},t_{k+1}]}\rangle_{\mathcal{H}}=\int_{0}^{t}\int_{0}^{t}\mathbf{D}_{r_{1}}^{i}y^{i}_{t_{k}}{\bf
1}_{[0,t_{k}]}(r_{1}){\bf
1}_{[t_{k},t_{k+1}]}(r_{2})\,\mathrm{d}R(r_{1},r_{2}).$

We thus have

$\lim_{n\to\infty}\sum_{i=1}^{d}\sum_{k=0}^{n-1}\langle\mathbf{D}^{i}y^{i}_{t_{k}},{\bf
1}_{[t_{k},t_{k+1}]}\rangle_{\mathcal{H}}=\lim_{n\to\infty}\sum_{i=1}^{d}\sum_{k=0}^{n-1}\int_{0}^{t}\int_{0}^{t}\mathbf{D}_{r_{1}}^{i}y^{i}_{t_{k}}{\bf
1}_{[0,t_{k}]}(r_{1}){\bf
1}_{[t_{k},t_{k+1}]}(r_{2})\,\mathrm{d}R(r_{1},r_{2}).$

We now argue similarly to what we did for (69). Namely one of our standing
assumptions is that $(r_{1},r_{2})\to\mathbf{D}^{i}_{r_{1}}y_{r_{2}}^{i}{\bf
1}_{\mathcal{S}_{2}}(r_{1},r_{2})$ has a finite 2-dimensional $p$-variation.
Since $R$ admits a finite $\rho$-variation and $p^{-1}+\rho^{-1}>1$, standard
results concerning convergence of Riemann sums to Young integrals show that
almost surely we have

$\lim_{n\to\infty}\sum_{i=1}^{d}\sum_{k=0}^{n-1}\langle\mathbf{D}^{i}y^{i}_{t_{k}},{\bf
1}_{[t_{k},t_{k+1}]}\rangle_{\mathcal{H}}=\sum_{i=1}^{d}\int_{\mathcal{S}_{2}([0,t])}\mathbf{D}^{i}_{r_{1}}y^{i}_{r_{2}}\,\mathrm{d}R(r_{1},r_{2}).$
(71)

We can now conclude our proof easily. That is plugging (62), (63), (70) and
(71) into (61), we end up with, almost surely,

$\int_{0}^{t}y_{r}\mathrm{d}{\bf
x}_{r}-\int_{0}^{t}y_{r}\mathrm{d}^{\diamond}x_{r}\\\
=\frac{1}{2}\sum_{i=1}^{d}\int_{0}^{t}y_{r}^{x;ii}\mathrm{d}R_{r}-\sum_{i=1}^{d}\int_{\mathcal{S}_{2}([0,t])}y_{r_{2}}^{x;ii}\mathrm{d}R(r_{1},r_{2})+\sum_{i=1}^{d}\int_{\mathcal{S}_{2}([0,t])}\mathbf{D}_{r_{1}}^{i}y_{r_{2}}^{i}\mathrm{d}R(r_{1},r_{2}),$

from which the claim (55) is immediately deduced. This concludes the proof. ∎

We close this section by proving a technical result which has been used in
order to derive relation (58).

######  Lemma 3.2.

Assume the same conditions as in Theorem 3.1. Then $y^{\pi}$ defined in (56)
converges to $y$ in $\mathbb{D}^{1,2}(\mathcal{H}^{d})$, i.e.
$\lim\limits_{|\pi|\to
0}\mathbb{E}[\|y^{\pi}-y\|^{2}_{\mathcal{H}^{d}}+\|\mathbf{D}y^{\pi}-\mathbf{D}y\|^{2}_{(\mathcal{H}^{d})^{\otimes
2}}]=0$.

###### Proof.

According to (38), we have

$\|y^{\pi}-y\|_{\mathcal{H}^{d}}^{2}=\sum_{i,j=0}^{n-1}\int_{[t_{i},t_{i+1}]\times[t_{j},t_{j+1}]}\langle
y_{t_{i}}-y_{s},y_{t_{j}}-y_{t}\rangle~{}\mathrm{d}R(s,t),$

where we recall that $\pi=\\{0=t_{0}<t_{1}<\dots<t_{n}=t\\}.$ On each
rectangle $[t_{i},t_{i+1}]\times[t_{j},t_{j+1}]$ we apply Theorem 2.22 to the
function

$f_{ij}(s,t)=\langle y_{s}-y_{t_{i}},y_{t}-y_{t_{j}}\rangle,$

which is allowed since $f_{ij}$ is easily seen to be a function in
$\mathcal{C}_{2}^{p-\text{var}}$.

Recall that we have assumed $p^{-1}+\rho^{-1}>1$. Throughout the proof, we
choose $p^{\prime}>p$ and $\rho^{\prime\prime}>\rho^{\prime}>\rho$ satisfying

$(p^{\prime})^{-1}+(\rho^{\prime})^{-1}>1~{}\text{ and
}~{}(p^{\prime})^{-1}+(\rho^{\prime\prime})^{-1}>1.$

Since we also have $f_{ij}(t_{i},\cdot)=0$ and $f_{ij}(\cdot,t_{j})=0$, we get

$\displaystyle\|y^{\pi}-y\|_{\mathcal{H}^{d}}^{2}\leq$ $\displaystyle
C\sum_{i,j=0}^{n-1}\Big{(}\|y\|_{p^{\prime}-{\rm
var};[t_{i},t_{i+1}]}\|y\|_{p^{\prime}-{\rm
var};[t_{j},t_{j+1}]}\Big{)}\|R\|_{\rho^{\prime}-{\rm
var};[t_{i},t_{i+1}]\times[t_{j},t_{j+1}]}.$ (72)

In order to bound the right-hand side of (72), we introduce a new function
$\omega_{1}$, defined by

$\omega_{1}([a,b]\times[c,d])=\|y\|^{p^{\prime}}_{p^{\prime}-{\rm
var};[a,b]}\|y\|^{p^{\prime}}_{p^{\prime}-{\rm var};[c,d]}.$ (73)

Then it is readily checked that $\omega_{1}$ is also a 2d-control in the sense
of Definition 2.7. The following is easily deduced from (72):

$\displaystyle\|y^{\pi}-y\|_{\mathcal{H}^{d}}^{2}\leq$ $\displaystyle
C\sup_{i,j}\Big{(}\omega([t_{i},t_{i+1}]\times[t_{j},t_{j+1}])\Big{)}^{\frac{1}{\rho^{\prime}}-\frac{1}{\rho^{\prime\prime}}}$
$\displaystyle\times\sum_{i,j=0}^{n-1}\Big{(}\omega_{1}([t_{i},t_{i+1}]\times[t_{j},t_{j+1}])\Big{)}^{\frac{1}{p^{\prime}}}\Big{(}\omega([t_{i},t_{i+1}]\times[t_{j},t_{j+1}])\Big{)}^{\frac{1}{\rho^{\prime\prime}}},$
(74)

where $\omega$ is the control defined in (15). Now both $\omega_{1}$ and
$\omega$ above are 2d-controls. Hence an easy extension of [11, Exercise 1.9]
to a 2d setting shows that
$\omega_{1}^{1/p^{\prime}}\omega^{1/{\rho^{\prime\prime}}}$ is also a
2d-control. Hence one can resort to the super-additivity property of
$\omega_{1}^{1/p^{\prime}}\omega^{1/{\rho^{\prime\prime}}}$ in order to deduce
the following from (74):

$\|y^{\pi}-y\|_{\mathcal{H}^{d}}^{2}\leq
C\sup_{i,j}\Big{(}\omega([t_{i},t_{i+1}]\times[t_{j},t_{j+1}])\Big{)}^{\frac{1}{\rho^{\prime}}-\frac{1}{\rho^{\prime\prime}}}\Big{(}\omega_{1}([0,t]^{2})\Big{)}^{\frac{1}{p^{\prime}}}\Big{(}\omega([0,t]^{2})\Big{)}^{\frac{1}{\rho^{\prime\prime}}}.$
(75)

We now turn to an upper bound on
$\|\mathbf{D}y^{\pi}-\mathbf{D}y\|_{(\mathcal{H}^{d})^{\otimes 2}}$. To this
aim we first express this quantity using the norm in
$(\mathcal{H}^{d})^{\otimes 2}$ induced by (38). This yields

$\displaystyle\|\mathbf{D}y^{\pi}-\mathbf{D}y\|^{2}_{(\mathcal{H}^{d})^{\otimes
2}}$ $\displaystyle=$
$\displaystyle\sum_{i,j=0}^{n-1}\int_{[0,t_{i+1}]\times[0,t_{j+1}]\times[t_{i},t_{i+1}]\times[t_{j},t_{j+1}]}\langle\mathbf{D}_{u}y_{t_{i}}-\mathbf{D}_{u}y_{s},\mathbf{D}_{v}y_{t_{j}}-\mathbf{D}_{v}y_{t}\rangle~{}\mathrm{d}R(u,v)\mathrm{d}R(s,t).$
(76)

We apply Lemma 2.23 to the right-hand side of (76) and get

$\displaystyle\|\mathbf{D}y^{\pi}-\mathbf{D}y\|^{2}_{(\mathcal{H}^{d})^{\otimes
2}}\leq C\sum_{i,j=0}^{n-1}\|R\|_{\rho^{\prime}-{\rm
var};[0,t_{i+1}]\times[0,t_{j+1}]}\|R\|_{\rho^{\prime}-{\rm
var};[t_{i},t_{i+1}]\times[t_{j},t_{j+1}]}$
$\displaystyle\qquad\qquad\qquad\times\Big{(}\|\mathbf{D}_{0}y_{t_{i}}-\mathbf{D}_{0}y_{\cdot}\|_{p^{\prime}-{\rm
var};[t_{i},t_{i+1}]}+\|\mathbf{D}_{\cdot}y_{t_{i}}-\mathbf{D}_{\cdot}y_{\cdot}\|_{p^{\prime}-{\rm
var};[0,t_{i+1}]\times[t_{i},t_{i+1}]}\Big{)}$
$\displaystyle\qquad\qquad\qquad\times\Big{(}\|\mathbf{D}_{0}y_{t_{j}}-\mathbf{D}_{0}y_{\cdot}\|_{p^{\prime}-{\rm
var};[t_{j},t_{j+1}]}+\|\mathbf{D}_{\cdot}y_{t_{j}}-\mathbf{D}_{\cdot}y_{\cdot}\|_{p^{\prime}-{\rm
var};[0,t_{j+1}]\times[t_{j},t_{j+1}]}\Big{)}.$

As a preliminary step, we also bound the variations on intervals of the form
$[0,t_{j}]$ by variations on $[0,T]$. Thus one can bound
$\|\mathbf{D}y^{\pi}-\mathbf{D}y\|^{2}_{(\mathcal{H}^{d})^{\otimes 2}}$ by

$C\|R\|_{\rho^{\prime}-{\rm
var};[0,T]^{2}}\sum_{i,j=0}^{n-1}\|R\|_{\rho^{\prime}-{\rm
var};[t_{i},t_{i+1}]\times[t_{j},t_{j+1}]}\Big{(}\|\mathbf{D}_{0}y\|_{p^{\prime}-{\rm
var};[t_{i},t_{i+1}]}+\|\mathbf{D}y\|_{p^{\prime}-{\rm
var};[0,T]\times[t_{i},t_{i+1}]}\Big{)}\\\
\times\Big{(}\|\mathbf{D}_{0}y\|_{p^{\prime}-{\rm
var};[t_{j},t_{j+1}]}+\|\mathbf{D}y\|_{p^{\prime}-{\rm
var};[0,T]\times[t_{j},t_{j+1}]}\Big{)}.$ (77)

We now wish to apply super-additivity properties of the $p$-variations, as we
did for (75). However, note that the function
$[a,b]\times[c,d]\mapsto\|\mathbf{D}y\|^{p^{\prime}}_{p^{\prime}-{\rm
var};[a,b]\times[c,d]}$ may fail to be super-additive (see [9, Theorem 1]).
Hence we need to resort to the controlled 2d variation as introduced in
Definition 2.9. Specifically, it follows from (77) that
$\|\mathbf{D}y^{\pi}-\mathbf{D}y\|^{2}_{(\mathcal{H}^{d})^{\otimes 2}}$ can be
upper bounded by

$C\|R\|_{\rho^{\prime}-{\rm
var};[0,T]^{2}}\sum_{i,j=0}^{n-1}\|R\|_{\rho^{\prime}-{\rm
var};[t_{i},t_{i+1}]\times[t_{j},t_{j+1}]}\Big{(}\|\mathbf{D}_{0}y\|_{p^{\prime}-{\rm
var};[t_{i},t_{i+1}]}+\vvvert\mathbf{D}y\vvvert_{p^{\prime}-{\rm
var};[0,T]\times[t_{i},t_{i+1}]}\Big{)}\\\
\times\Big{(}\|\mathbf{D}_{0}y\|_{p^{\prime}-{\rm
var};[t_{j},t_{j+1}]}+\vvvert\mathbf{D}y\vvvert_{p^{\prime}-{\rm
var};[0,T]\times[t_{j},t_{j+1}]}\Big{)}.$ (78)

Notice that the right-hand side of (78) is finite, noting that $p<p^{\prime}$
and owing to (14). Furthermore, noting that the function
$[c,d]\mapsto\vvvert\mathbf{D}y\vvvert^{p^{\prime}}_{p^{\prime}-{\rm
var};[0,T]\times[c,d]}$ is a control, we can define the following 2d controls
(where we use [11, Exercise 1.9] again):

$\displaystyle\omega_{2}([a,b]\times[c,d])$ $\displaystyle=$
$\displaystyle\|\mathbf{D}_{0}y\|^{p^{\prime}}_{p^{\prime}-{\rm
var};[a,b]}\|\mathbf{D}_{0}y\|^{p^{\prime}}_{p^{\prime}-{\rm var};[c,d]}$ (79)
$\displaystyle\omega_{3}([a,b]\times[c,d])$ $\displaystyle=$
$\displaystyle\vvvert\mathbf{D}y\vvvert^{p^{\prime}}_{p^{\prime}-{\rm
var};[0,T]\times[a,b]}\vvvert\mathbf{D}y\vvvert^{p^{\prime}}_{p^{\prime}-{\rm
var};[0,T]\times[c,d]}$ (80) $\displaystyle\omega_{4}([a,b]\times[c,d])$
$\displaystyle=$
$\displaystyle\|\mathbf{D}_{0}y\|^{p^{\prime}}_{p^{\prime}-{\rm
var};[a,b]}\vvvert\mathbf{D}y\vvvert^{p^{\prime}}_{p^{\prime}-{\rm
var};[0,T]\times[c,d]}.$ (81)

Now, similarly to (75), relation (78) entails

$\displaystyle\|\mathbf{D}y^{\pi}-\mathbf{D}y\|^{2}_{(\mathcal{H}^{d})^{\otimes
2}}$ (82) $\displaystyle\leq C\|R\|_{\rho^{\prime}-{\rm
var};[0,T]^{2}}\sup_{i,j}\Big{(}\omega([t_{i},t_{i+1}]\times[t_{j},t_{j+1}])\Big{)}^{\frac{1}{\rho^{\prime}}-\frac{1}{\rho^{\prime\prime}}}\Big{(}\omega([0,t]^{2})\Big{)}^{\frac{1}{\rho^{\prime\prime}}}\sum_{k=2}^{4}\Big{(}\omega_{k}([0,t]^{2})\Big{)}^{\frac{1}{p^{\prime}}},$

where $\omega$ is the control given in (15). This is our desired bound for the
difference $\mathbf{D}y^{\pi}-\mathbf{D}y$.

Let us summarize our considerations so far. Gathering inequalities (75) and
(82), we have proved that

$\displaystyle\|y^{\pi}-y\|^{2}_{\mathcal{H}^{d}}+\|\mathbf{D}y^{\pi}-\mathbf{D}y\|^{2}_{(\mathcal{H})^{\otimes
2}}$ (83) $\displaystyle\leq$ $\displaystyle
C\left(1+\|R\|_{\rho^{\prime}-{\rm
var};[0,T]^{2}}\right)\Big{(}\omega([0,t]^{2})\Big{)}^{\frac{1}{\rho^{\prime\prime}}}\sum_{k=1}^{4}\Big{(}\omega_{k}([0,t]^{2})\Big{)}^{\frac{1}{p^{\prime}}}\sup_{i,j}\Big{(}\omega([t_{i},t_{i+1}]\times[t_{j},t_{j+1}])\Big{)}^{\frac{1}{\rho^{\prime}}-\frac{1}{\rho^{\prime\prime}}},$

where the controls $\omega,\omega_{1},\omega_{2},\omega_{3},\omega_{4}$ are
respectively defined by (15), (73), (79), (80) and (81). We can now argue as
follows: first, according to (17), (14) and (15) we have

$\lim_{n\to\infty}\sup_{i,j}\omega([t_{i},t_{i+1}]\times[t_{j},t_{j+1}])=0.$

Next we have assumed in Theorem 3.1 that

$\mathbb{E}\left[\|y\|^{2}_{p-{\rm var};[0,T]}+\|\mathbf{D}_{0}y\|^{2}_{p-{\rm
var};[0,T]}+\|\mathbf{D}y\|^{2}_{p-{\rm var};[0,T]^{2}}\right]<\infty.$

Therefore one can take expected valued in (83) in order to get

$\lim_{n\to\infty}\mathbb{E}\left[\|y^{\pi}-y\|_{\mathcal{H}^{d}}^{2}+\|\mathbf{D}y^{\pi}-\mathbf{D}y\|^{2}_{(\mathcal{H}^{d})^{\otimes
2}}\right]=0,$

which is our claim. This concludes our proof. ∎

We close this section by showing that our main Theorem 3.1 generalizes
previous Skorohod-Stratonovich integral correction formulae.

######  Remark 3.3 (A comparison with [15]).

In [15], the relationship (1) is obtained for a $\gamma$-Hölder Gaussian
process $x$ with $\gamma\in(0,1)$. The process $y$ considered in [15] is of
the special form $y=f(x)$ for $f\in C^{2N}$ with
$N=\lfloor\frac{1}{\gamma}\rfloor$.

Our main result Theorem 3.1 holds for Gaussian processes possessing finite
$p$-variation with $p\in(2,3)$ (or $\rho\in[1,\frac{3}{2})$). Noting that
Propositions 2.27 and 2.28 hold under Hypotheses 2.8 and 2.12 for $p\in(2,4)$,
we believe that our approach could be extended to $p\in(2,4)$. A key
difference between $p\in(2,3)$ and $p\in[3,4)$ is that a weighted sum in the
third chaos of $x$ will be involved in the rough integral (60) for
$p\in[3,4)$. Thus for $p\in(2,4)$, to calculate the Skorohod-Stratonovich
correction term, we also need develop some estimation for the weighted sum in
third chaos of $x$ which is parallel to Proposition 2.28.

Note that the condition $p\in(2,3)$ ($\rho\in[1,\frac{3}{2})$) is also used to
define the Young integrals appearing in (66), (69) and (71). However, if we
further assume that $R_{\cdot}$ and $R(\cdot,t)$ for each $t\in[0,T]$ are
absolutely continuous as in [15, Hypothesis 3.1], which is satisfied by
fractional Brownian motion $B^{H}$ with Hurst parameter $H\in(0,1)$, then the
integrals in (66), (69) and (71) are automatically well-defined as Riemann
integrals.

######  Remark 3.4 (A comparison with [4]).

In [4], the relationship between $\int_{0}^{t}y_{r}\mathrm{d}{\bf x}_{r}$ and
$\int_{0}^{t}y_{r}\mathrm{d}^{\diamond}x_{r}$ is studied, where $x$ satisfies
Hypotheses 2.8 and 2.12, and $y$ is the solution to (3) with $\sigma$ being
sufficiently regular. Indeed, under the conditions assumed in the main Theorem
in [4], our main result Theorem 3.1 also holds. More specifically,
$\mathbb{E}[\|y\|^{2}_{p-{\rm var};[0,T]}]<\infty$ is a consequence of [4,
Theorem 2.25]; $\mathbb{E}[\|\mathbf{D}_{0}y\|^{2}_{p-{\rm
var};[0,T]}]+\mathbb{E}[\|\mathbf{D}y\|^{2}_{p-{\rm var};[0,T]^{2}}]<\infty$
follows from $\mathbf{D}_{s}y_{t}={\bf 1}_{[0,t)}(s)J_{t}^{\bf X}(J_{x}^{\bf
X})^{-1}\sigma(Y_{s})$ and Theorem [4, Theorem 2.27] (see also the end of the
proof [4, Proposition 4.10]). With those relations in mind, our main Theorem
3.1 also covers the analysis performed in [4].

#### Acknowledgment

We would like to thank Tom Cass for some interesting discussions about this
project.

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