# Pathwise Uniqueness for Multiplicative Young and Rough Differential
Equations Driven by Fractional Brownian Motion

Toyomu Matsuda  Institute of Mathematics, École Polytechnique Fédérale de
Lausanne (EPFL).
<EMAIL_ADDRESS>0000-0002-2422-0863 Avi Mayorcas  Department of
Mathematical Sciences, University of Bath.
<EMAIL_ADDRESS>0000-0003-4133-9740

###### Abstract

We show _pathwise uniqueness_ of multiplicative SDEs, in arbitrary dimensions,
driven by fractional Brownian motion with Hurst parameter $H\in(1/3,1)$ with
volatility coefficient $\sigma$ that is at least $\gamma$-Hölder continuous
for $\gamma>\frac{1}{2H}\vee\frac{1-H}{H}$. This improves upon the long-
standing results of [Lyo94, Lyo98, Dav08] which cover the same regime but
require $\sigma$ to be at least $\frac{1}{H}$-Hölder continuous. Our central
innovation is to combine stochastic averaging estimates with refined versions
of the stochastic sewing lemma, due to [Lê20, Ger22, MP22].

Keywords and phrases. Stochastic differential equations, fractional Brownian
motion, rough paths, stochastic sewing, regularisation by noise.

MSC 2020. 60H10, 60G22, 60L20, 60H50.

###### Contents

  1. 1 Introduction
    1. 1.1 Main Result
    2. 1.2 Overview of the Proof
    3. 1.3 Notation
  2. 2 Preliminaries
    1. 2.1 Properties of the Heat Semi-Group
    2. 2.2 Young Integration
    3. 2.3 Rough Integration
    4. 2.4 Sewing
    5. 2.5 Fractional Brownian Motion
  3. 3 Young Case: $H\in(1/2,1)$
    1. 3.1 Young Integral with Irregular Integrand
    2. 3.2 Pathwise Uniqueness
  4. 4 Rough Case: $H\in(1/3,1/2]$
    1. 4.1 Rough Integral with Irregular Integrand
    2. 4.2 Iterated Integrals with Rough Integrands
    3. 4.3 Pathwise Uniqueness
  5. A Properties of Heat Semi-Groups
  6. B Gaussian and Integral Computations
    1. B.1 Proofs of Lem. 2.11 and Lem. 2.12
    2. B.2 Proof of Lem. 2.13
    3. B.3 Proof of Lem. 4.6

##  1 Introduction

This paper is concerned with _pathwise_ uniqueness of solutions to
multiplicative SDEs driven by fractional Brownian motion (fBm),

$\displaystyle\,\mathrm{d}X_{t}$
$\displaystyle=\sigma(X_{t})\,\,\mathrm{d}B_{t}^{H},\quad
X_{0}=x\in\mathbb{R}^{d_{1}},$ (1)

where $\mathbb{R}^{d_{1}}\ni x\mapsto\sigma(x)$ takes values in the space of
$d_{1}\times d_{2}$ matrices and $B^{H}$ is a $d_{2}$-dimensional fBm with
Hurst parameter $H\in(1/3,1)$. We interpret (1) as either a Young differential
equation, when $H\in(1/2,1)$ or a rough differential equation when
$H\in(1/3,1/2]$. The tools developed herein naturally extend to the even
rougher case $H\in(1/4,1/3)$, however, for the sake of concision they are not
presented, see Rem. 1.4 for a further discussion. Our main result is to obtain
_pathwise_ uniqueness of Young (resp. rough) solutions to (1) for uniformly
elliptic, volatility functions $\sigma$ which are $\gamma$-Hölder continuous
for all

$\gamma>\max\left\\{\frac{1}{2H},\frac{1-H}{H}\right\\}=\begin{cases}\frac{1}{2H},\quad&\text{if
}H\in(1/2,1),\\\ \frac{1-H}{H},&\text{if }H\in(1/3,1/2].\end{cases}$

The ellipticity condition on $\sigma$ is natural since both the Young and
rough interpretation of (1) are natural extensions of the Stratonovich
interpretation, applicable when $H=1/2$, which also requires ellipticity on
$\sigma$ in order to generically guarantee well-posedness. See Rem. 1.3 for a
further discussion of this assumption. We also refer the reader to Rem. 1.7,
Def. 3.5 and Def. 4.5 for further discussion and precise definition of
_pathwise_ uniqueness in this context, for $H\neq 1/2$. At this stage, we
simply note that _pathwise_ uniqueness under the above condition is the first
major improvement on well-posedness theory for multiplicative SDEs driven by
fBm since [Lyo94, Lyo98, Dav08], which together required $\gamma\geq 1/H$.
Hence, our result significantly reduces the regularity requirement on $\sigma$
in arbitrary dimensions. In fact, our present work improves the recent result
[MP22, Thm. 5.2], which treats the regime $H\in(1/2,1)$ and assumes the
technical condition $\gamma>\frac{(1-H)(2-H)}{H(3-H)}$.

Our work fits into the growing literature on the topic of _regularisation by
noise_. This is the phenomenon by which rapidly oscillating and typically
random data lead to improved existence, uniqueness or other desirable
properties of equations which would otherwise be ill-posed or behave
pathologically in some manner, [Fla11, Tzv16, Fla17, GT20, FL21, ALL23].

In recent years there has been significant progress in the study of SDEs with
irregular coefficients and driven by noise lying outside of the Brownian (or
even martingale) class, for which we give a non-exhaustive list of references
[CG16, BM19, ABM20, Lê20, HP21, MP22, Ger22, DG22, GG22, ART23]. More
precisely, our work fits within this latter programme and considers an, as
yet, less studied regime for regularisation by noise; namely the interaction
between noise and irregular multiplicative coefficients. To the best of the
authors’ knowledge, the only other work in this direction is [HTV22] in which
equations with more general noise are studied but under more stringent
dimensional and structural assumptions on the volatility coefficient. Although
not as directly related, we also highlight the work [GH22] which obtained
regularisation by noise for SDEs with multiplicative Brownian noise under the
inclusion of an additive fBm. More widely, we point out that equations of the
type (1) are of interest in a number of applications, most notably in the
study of stochastic volatility [CR96, CCR12, FSW22], turbulent fluids [FR23,
Cri+22a, Cri+22, ACM22] and more generally stochastic processes with memory
[Ber94, BW21, HL20, HL22, LPS23].

When $H=1/2$ there are two natural interpretations of (1), either as a
stochastic, Itô equation or Stranovich equation. These correspond to two
interpretations of the deterministically, ill-posed integration of
$\sigma(X_{t})$ against $\,\mathrm{d}W_{t}\coloneqq\,\mathrm{d}B^{1/2}_{t}$.
Crucially, Itô solutions are martingales while Stratonovich integration
respects the chain rule but does not produce martingale solutions. The notion
of _pathwise_ uniqueness is equally applicable to both, however, since both
interpretations produce solutions adapted to the natural filtration of the
Brownian motion. In Itô’s theory it is readily seen that _pathwise_ uniqueness
holds for (1) (with $H=1/2$) provided $\sigma$ is Lipschitz continuous, see
for example [RY99]. This condition provides a natural sanity check on our
condition as they agree at $H=1/2$. We mention here that the result for Itô
solutions does not require any ellipticity assumption on $\sigma$, while the
same statement holds for Stratonovich solutions, under the additional
requirement of ellipticity on $\sigma$, see Rem. 1.3. As will be remarked
below, this structural condition in the Stratonovich case can be relaxed at
the price of requiring $\sigma$ to be twice differentiable and bounded.

When $H\neq 1/2$ Itô’s theory no longer applies to (1) since the fBm is not a
martingale process, see Sec. 2.5. However, the Stratonovich interpretation
naturally extends beyond the martingale case. Note that for $H>1/2$ no
probabilistic theory is actually required to make sense of the equation,
since, under sufficient regularity assumptions on $\sigma$, one expects
$t\mapsto\sigma(X_{t})$ to carry the same Hölder regularity as the fBm, in
this case larger than $1/2$ and so Young integration theory classically
applies; see Sec. 2.2. In our case the regularity assumption on $\sigma$ is
too weak to naively apply this theory, however, our main contribution in this
regime is to carefully exploit a combination of deterministic analysis and
probabilistic averaging tools to indeed construct a sufficiently regular
integral, see Sec. 3.1. For $H<1/2$ the natural extension of the Stratonovich
integral is provided by [Lyo98, CQ02].

###  1.1 Main Result

We state the main result of our paper below in a somewhat informal manner,
referring to precise definitions in Sec. 3 and Sec. 4 for the Young and rough
cases respectively. We also refer the reader to Sec. 1.3 for the explanation
of any unfamiliar notation.

######  Theorem 1.1 (Pathwise Uniqueness of (1)).

Let $d_{1},d_{2}\in\mathbb{N}$, $B^{H}$ a $d_{2}$-dimensional fractional
Brownian motion with Hurst parameter $H\in(1/3,1)$ and $\gamma\in\mathbb{R}$
satisfy

$\gamma>\max\left\\{\frac{1}{2H},\frac{1-H}{H}\right\\}=\begin{cases}\frac{1}{2H},\quad&\text{if
}H>1/2,\\\ \frac{1-H}{H},&\text{if }H\leq 1/2.\end{cases}$ (2)

Then, given a uniformly elliptic,
$\sigma\in\mathcal{C}^{\gamma}(\mathbb{R}^{d_{1}};\mathbb{R}^{d_{1}\times
d_{2}})$ (i.e. $\sigma\sigma^{\top}$ is uniformly, positive definite, see
(12)) interpreting the SDE (1) as Young differential equation (resp. rough
differential equation with respect to the canonical, geometric, rough path
lift of $B^{H}$) if $H\in(1/2,1)$ (resp. $H\in(1/3,1/2]$), pathwise uniqueness
in the sense of Def. 3.5 (resp. Def. 4.5) of solutions holds for (1).

###### Proof.

See Thm. 3.8 for the case $H\in(1/2,1)$ and Thm. 4.10 for $H\in(1/3,1/2]$. ∎

The parameter range described by (2) is explicitly visualised in Figure 1
below, along with the threshold of $\gamma\geq\frac{1}{H}$, which is higher
than ours in both cases, as required by the standard Young or rough path
approaches, [Lyo94, Lyo98, Dav08].

$1/3$$1/2$$1$$0$$1/2$$1$$2$$3$$H$$(1-H)/H$$1/(2H)$$1/H$ Figure 1: Plots of
admissible parameter regimes for $\gamma$-Hölder continuous volatility to
ensure pathwise uniqueness of (1). Pathwise theory of [Lyo94, Lyo98, Dav08]
requires $\gamma\geq 1/H$ (solid green line). Thm 1.1 allows for
$\gamma>\frac{1}{2H}\vee\frac{1-H}{H}$ (blue and red solid line).

We collect some remarks and ancillary results related to Theorem 1.1. Firstly,
as is the case for SDEs driven by Brownian motion, we have the following
Yamada–Watanabe theorem,

######  Proposition 1.2.

If weak existence, in the Young or rough senses (see Def. 3.2 or Def. 4.2) and
pathwise uniqueness holds (Def. 3.5 or 4.5) for (1), then weak uniqueness and
strong existence, in the appropriate Young or rough senses, hold as well.

###### Proof.

This follows from a generalised Yamada-Watanabe theorem of Kurtz [Kur07]. One
can consult [MP22, Appendix B] for more detailed discussion. ∎

Note that a standard compactness argument (see e.g. [MP22, App. B]) allows one
to construct weak solutions to (1) (in both the Young and rough senses) for
$\sigma\in\mathcal{C}^{\gamma}(\mathbb{R}^{d_{1}};\mathbb{R}^{d_{1}\times
d_{2}})$ with $\gamma>(1-H)/H$, without any assumption of ellipticity. Hence,
due to Prop. 1.2 weak uniqueness and strong existence for (1), under the
additional ellipticity assumption follow under the assumptions of Thm. 1.1.
See Rem. 3.3 and Rem. 4.3 for further discussion of this fact.

######  Remark 1.3 (Ellipticity Assumption).

In Itô’s theory, applicable in the case $H=1/2$, if $\sigma$ is at least
Lipschitz continuous, then no ellipticity is required in order to obtain
existence and uniqueness of solutions. However, if one treats the SDE (1) as a
Stratonovich equation, the elliptic condition becomes necessary to obtain
existence and uniqueness under the same regularity assumption. Alternatively,
one can remove ellipticity at the price of increased regularity on $\sigma$,
this corresponds to the $\gamma>\frac{1}{H}$ threshold discussed earlier.
Since our treatment, beyond the Brownian case, relies on either Young or rough
integration both of which generalise the Stratonovich integral and aims for
improved regularity assumptions, we also rely on an ellipticity assumption for
$\sigma$.

For simplicity of presentation, in Thm. 1.1 (similarly Thm. 3.8 and Thm. 4.10)
this is stated as uniform ellipticity and our proofs only explicitly treat
this case. However, a simple stopping time argument easily allows one to relax
this assumption to obtain well-posedness on any fixed time interval to simple
ellipticity. We refer to the proof of [MP22, Thm. 5.2] for an exposition of
this idea in a similar setting. Since our main focus is the regularity of
$\sigma$ we do not repeat the argument in our paper.

######  Remark 1.4 (Rougher and Smoother Noise Regimes).

It is natural to expect that Thm 1.1 can be extended to the case
$H\in(1/4,1/3]$. However, since this would require the treatment of 3rd order
iterated integrals (as opposed to 2nd order as required in Sec. 4) it would
significantly increase the length technicality of our proofs. Therefore, in
the interests of keeping the paper somewhat concise we do not present this
further extension. It is worth noting that $1/4$ is likely to be a significant
barrier, since even the canonical lift of an fBm to a rough path is not
defined below this threshold, see [CQ02].

Similarly, one could consider the extended Young regime, i.e.
$H\in[1,\infty)\setminus\mathbb{N}$ (see for example [DG22, GG22]). Again, in
order to clarify presentation and avoid additional technicalities we have
chosen not to explicitly consider this case here. However, we do not expect
significant challenge, in this regime the fBm is classically differentiable so
that some arguments may even simplify. We conjecture that in this regime the
threshold, coming from our tools, would become $\gamma>1-\frac{1}{2H}$.

######  Remark 1.5.

The condition of $\gamma$-Hölder regularity for $\sigma$ with
$\gamma>\frac{1-H}{H}$ is necessary to interpret the SDE (1) as a Young
differential equation in the case $H>1/2$ or as rough differential equation
when $H\leq 1/2$. This condition is sufficient for weak existence when
$H\in(1/3,1)$, and also for pathwise uniqueness when $H\in(1/3,1/2]$. When
$H>1/2$, there is a gap between the condition of weak existence and that of
pathwise uniqueness. We conjecture that pathwise uniqueness does not hold in
general for $\gamma<\frac{1}{2H}$, although we are not currently able to
establish a proof.

Due to comparison with Brownian case, we expect that pathwise uniqueness also
holds for $\gamma=\frac{1}{2H}$ when $H>1/2$, however, our current methods
cannot cover this case either.

######  Remark 1.6 (Time Dependence $\sigma$).

As in [GG22], one could naturally consider time dependent $\sigma$, for which
we expect that the shifted stochastic sewing lemma (Lem. 2.4) with controls
would be necessary.

######  Remark 1.7 (_Path-by-Path_ and _Weak_ Uniqueness).

Despite its name the notion of _pathwise_ uniqueness is a purely probabilistic
one, which in this work we combine with truly pathwise notions of existence
(see Def. 3.5 and Def. 4.5). A stronger and more genuinely pathwise notion of
uniqueness was coined by Davie, [Dav08], as _path-by-path_ uniqueness. In our
case, this property would hold if, on a given probability space
$(\Omega,\mathcal{F},\mathbb{P})$ carrying an fBm $B^{H}$, there exists a
$\mathbb{P}$-null set in $\Omega$, which may depend on $\sigma$ and the
initial data such that the Young differential equation (resp. rough
differential equation) for $H>1/2$ (resp. $H\leq 1/2$),

$\displaystyle\mathrm{d}y_{t}=\sigma(y_{t})\,\mathrm{d}B_{t}^{H}(\omega),\quad
y_{0}=x\in\mathbb{R}^{d_{1}}$

has a unique solution. In particular, these solutions are not required to be
adapted to any given filtration. We refer the reader to [SW22] and the
references therein for a more thorough discussion of the distinction between
pathwise and path-by-path uniqueness.

It is not currently clear to the authors what the natural assumptions on
$\sigma$ are to ensure _path-by-path_ uniqueness for solutions to (1). In
[Dav08, Thm. 4.8] an example of (1) is studied for $H=1/2$ with $\sigma$ that
is $2-\varepsilon$-Hölder continuous, for any $0<\varepsilon\ll 1$ which has
infinitely many solutions. However, this is not a counter example to _path-by-
path_ uniqueness since the $\sigma$ here is in fact allowed to be random
(almost surely of the required Hölder continuity) and may depend on the
realisation of the Brownian motion.

Similarly, it is not clear to the authors which assumptions should be required
of $\sigma$ in order to prove weak uniqueness of solutions to (1) (see Def.
3.2 and Def. 4.2).

######  Remark 1.8 (SDEs with Drift).

A natural question following from Thm. 1.1 is to obtain similar results for
fBm driven SDEs with drift and multiplicative noise, of the form

$\displaystyle\mathrm{d}X_{t}=b(X_{t})\,\mathrm{d}t+\sigma(X_{t})\,\mathrm{d}B^{H}_{t}.$
(3)

If $b$ is sufficiently regular, for example Lipschitz continuous, it seems
clear that the arguments of the current paper should adapt without difficulty
to prove pathwise uniqueness, under the assumptions of Thm. 1.1. Conversely,
in [DG22] the authors consider (3) with regular $\sigma$ but irregular drift
term, $b$. Building on the current work, it would be of interest to study the
case when both $\sigma$ and $b$ are irregular. Such situation has been studied
in the Brownian case, e.g. [Kry23].

###  1.2 Overview of the Proof

We give a brief and formal description of our proof strategy for Thm. 1.1. In
both cases for $H\in(1/3,1/2]$ and $H\in(1/2,1)$ our argument is derived from
a strategy given in [Lê20], which relates pathwise uniqueness to construction
of an irregular stochastic integral. In our case, let us consider two (weak)
solutions $X,\,Y$ to (1) defined on the same, filtered, probability space
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},\mathbb{P})$ carrying an fBm
$B^{H}$ with Hurst parameter $H\in(1/4,1)$. Then, letting
$d_{1},\,d_{2}\in\mathbb{N}$ and assuming
$\sigma:\mathbb{R}^{d_{1}}\to\mathbb{R}^{d_{1}\times d_{2}}$ is smooth for
now, the difference $X-Y$ satisfies,

$X_{t}-Y_{t}=\int_{0}^{t}\left(\sigma(X_{r})-\sigma(Y_{r})\right)\,\mathrm{d}B_{r}^{H}.$
(4)

Applying the mean value theorem we see that $\sigma(X_{r})-\sigma(Y_{r})$ is a
$d_{1}\times d_{2}$ matrix whose components are given by

$\sigma_{ij}(X_{r})-\sigma_{ij}(Y_{r})=(X_{r}-Y_{r})\cdot\int_{0}^{1}\nabla\sigma_{ij}(\theta
X_{r}+(1-\theta)Y_{r})\,\mathrm{d}\theta,\quad i=1,\ldots
d_{1},\,j=1,\ldots,d_{2}.$

Thus, we can formally write,

$X_{t}-Y_{t}=\sum_{k=1}^{d_{1}}\int_{0}^{t}(X_{r}^{k}-Y_{r}^{k})\,\mathrm{d}G^{k}_{r},$
(5)

where the components $(G^{k})_{k=1}^{d_{1}}$ are defined by setting

$G^{k}_{t}\coloneqq\int_{0}^{t}\int_{0}^{1}\partial_{k}\sigma(\theta
X_{r}+(1-\theta)Y_{r})\,\mathrm{d}\theta\,\mathrm{d}B^{H}_{r}.$ (6)

Here the product between $\partial_{k}\sigma$ and $\,\mathrm{d}B^{H}$ is
understood in the usual manner as a $d_{1}\times d_{2}$ vector acting on a
vector of length $d_{2}$. Thus, we aim to view the difference $X-Y$ as a
solution to the linear, integral equation

$Z_{t}=\int_{0}^{t}Z_{t}\cdot\,\mathrm{d}G_{t}.$ (7)

Since (7) clearly has unique solutions (if they exist), the question of
pathwise uniqueness of solutions to (1) boils down to rigorous construction of
the integral on the right hand side of (5). Of course, the challenge is to do
so for irregular $\sigma$ and this requires careful use of a priori properties
which we obtain for solutions to (1). Here the technical steps of our proof
differ in the regimes $H\in(1/2,1)$ and $H\in(1/3,1/2]$, however, the spirit
is the same in both cases.

######  Remark 1.9.

It seems reasonable to expect that using similar arguments as ours one could
show the existence of a continuous, adapted, semi-flow
$\Phi:[0,T]^{2}_{\leq}\times\mathbb{R}^{d_{1}}\to\mathbb{R}^{d_{1}}$ defined
such that

$\Phi_{s,t}(x)\coloneqq
X^{s;x}_{t}=x+\int_{s}^{t}\sigma(X^{s;x}_{r})\,\mathrm{d}B^{H}_{r}.$ (8)

Continuity here means weak convergence (as stochastic processes) of
$\Phi_{s,\,\cdot\,}(x)-\Phi_{s,\,\cdot\,}(y)$ to zero as $x\to y$. One
challenge here would be to obtain uniqueness of solutions for all initial data
simultaneously, while relevant null sets in our arguments may (in principle)
depend on the initial data $x$. A common resolution to this issue is to first
obtain stability of the solution in the initial data, see for example [CLL13,
GG22]. It is not currently clear if our tools allow us to obtain similar
stability estimates. This question is also related to stability of the
solution in the coefficient $\sigma$, which in turn would directly allow one
to consider McKean–Vlasov SDEs of the form,

$\,\mathrm{d}X_{t}=(\sigma\ast\mu_{t})(X_{t})\,\mathrm{d}B^{H}_{t},\quad\mu_{t}=\text{Law}(X_{t}),$
(9)

under similar conditions on $\sigma$ as Thm. 1.1. See for example [GHM22,
GG22] for treatments of McKean–Vlasov equations with additive noise and
singular drifts as well as [BCD20, DS23, DS22] for treatments of
multiplicative, rough, McKean–Vlasov equations in the case of smooth
volatilities. Stability estimates with respect to $\sigma$ are also related to
convergence rates of numerical schemes for these and similar equations, see
[FSW22, LL21, GL23].

#### Young case: $H\in(1/2,1)$

In this regime we expect to have solutions $X,\,Y$ which are $\alpha$-Hölder
continuous for any $\alpha\in(1/2,H)$ and in turn we expect the map $t\mapsto
G_{t}^{k}$ (as defined by (6)) to be at least $1/2$-Hölder continuous. Thus we
expect to be able to make sense of the integral on the right hand side of (5)
as a Young integral, see Sec. 3.1 below. However, the limiting factor is the
regularity of the composition $\partial_{k}\sigma(\theta
X_{r}+(1-\theta)Y_{r})$ inside $G$. If we allow $\sigma$ to be $\gamma$-Hölder
continuous then one expects $r\mapsto\partial_{k}\sigma(\theta
X_{r}+(1-\theta)Y_{r})$ to be $\alpha(\gamma-1)$-Hölder continuous. Thus, in
order to naively define $G$ we would require
$\alpha(\gamma-1)+\alpha>1\,\,\iff\,\,\gamma>\frac{1}{\alpha}>\frac{1}{H}$,
(recall this is the condition of [Lyo94, Dav08]). To go below this threshold
we make more detailed use of the fact that one expects $\theta X+(1-\theta)Y$
to be _controlled_ in a suitable probabilistic sense by $B^{H}$ (see Def.
3.6). This allows us to exploit averaging properties of the fBm in order to
obtain sharper stochastic estimates on the composition
$\partial_{k}\sigma(\theta X_{r}+(1-\theta)Y_{r})$. These estimates are
obtained by Prop. 3.7 giving control (essentially) of the form

$\Big{\|}\int_{s}^{t}f(X_{r})\mathrm{d}B_{r}^{H}\Big{\|}_{L^{p}_{\omega}}\lesssim_{\beta,l,p,\gamma,X,T}\|f\|_{\mathcal{C}_{x}^{\gamma-1}}\big{(}(t-s)^{\gamma
H}+(t-s)^{\left(1+\frac{1}{2H}+(\gamma-1)\right)H}\big{)},$ (10)

for $f:\mathbb{R}^{d_{1}}\to\mathbb{R}^{d_{1}\times d_{2}}$ that is at least
$(\gamma-1)$-Hölder continuous and any $X$ _probabilistically controlled_ by
$B^{H}$ with remainder that is at least $1/2$-Hölder continuous. In order to
apply the sewing lemma (see Lem. 2.3) we therefore require $\gamma
H\wedge\left(\frac{1}{2H}+\gamma\right)H\geq\frac{1}{2}\iff\gamma>\frac{1}{2H}$.
The proof of this estimate is carried out by considering the germ,

$A_{s,t}\vcentcolon=\mathbb{E}_{s-(t-s)}\left[\int_{s}^{t}f(X_{s-(t-s)}+X^{\prime}_{s-(t-s)}B_{s-(t-s),r}^{H})\,\mathrm{d}B^{H}_{r}\Big{|}\mathcal{F}_{s-(t-s)}\right],$

where $X^{\prime}$ denotes the Gubinelli derivative of $X$ and applying
Gerencsér’s shifted, stochastic, sewing lemma [Ger22, Lem. 2.2] (see Sec.
2.4). The significant challenge here is to treat the stochastic integral
inside the germ $A$, this is handled by somewhat involved Gaussian
computations, carried out in App. B. The remainder of Sec. 3 essentially
consists of applying this abstract control to construct $G$ as defined by (6)
by considering a smooth approximation $\sigma_{n}\to\sigma$ converging in any
$\gamma^{\prime}$-Hölder topology for $\frac{1}{2H}<\gamma^{\prime}<\gamma$,
see the proof of Thm. 3.8 in Sec. 3.2. Prior to this final step we also show
that all weak solutions to (1) are _probabilistically controlled_ by $B^{H}$
in a suitable sense, see Lem. 3.9, which allows us to apply Prop. 3.7.

#### Rough case: $H\in(1/3,1/2]$

In the rough case, our strategy is essentially the same, in that we reduce our
problem to constructing the integral (6) and solving the linearised equation
(5). However, in this case we can no longer appeal to Young integration theory
for either task. Concerning the second, since we cannot expect $X,\,Y$ to be
more regular than the noise, i.e. $\alpha$-Hölder continuous for any
$\alpha\in(1/3,H)$, even if we assume $\sigma$ to be smooth, we cannot expect
$r\mapsto G_{r}$ (with $G$ as in (6)) to be more regular than the noise either
and so the integral of $X-Y$ against $\,\mathrm{d}G$ is not defined as a Young
integral since we no longer have $2\alpha>1$. This difficult is relatively
easily circumvented by instead appealing to rough integration. Assuming for
now that we can construct $G$, the natural solution is to define the joint
lift of $(B^{H},G)$ to a rough path

$\left(B^{H},G^{k},\int B^{H}\otimes\,\mathrm{d}B^{H},\int
B^{H}\otimes\,\mathrm{d}G^{k},\int G^{k}\otimes\,\mathrm{d}B^{H},\int
G^{k}\otimes\,\mathrm{d}G^{l}\right)_{k,\,l\,=1}^{d_{1}}.$ (11)

Since $X,\,Y$ can both be shown to be _controlled_ by $B^{H}$ (in the sense of
Gubinelli’s controlled rough paths, see [Gub04] and Def. 2.2) the integration
of $X-Y$ against this joint rough path follows from standard results. The
construction of the joint lift for smooth $\sigma$ is also relatively
standard, the central challenge is to obtain this construction for $\sigma$
satisfying the assumptions of Thm. 1.1.

It turns out that this construction relies on the same estimates that allow us
to construct $G$ in the first place. As in the Young case the core of the
proof is to obtain precise, probabilistic estimates on integrals of singular
functions against fBm, in this case these are of the form

$\int_{s}^{t}f(X_{r})\,\mathrm{d}B^{H}_{r},\quad\int_{s}^{t}f(X_{r})B^{H}_{s,r}\otimes\,\mathrm{d}B^{H}_{r}\quad\text{and}\quad\int_{s}^{t}\int_{s}^{r_{2}}g(Y_{r_{1}})\,\mathrm{d}B^{H}_{r_{1}}\otimes
f(X_{r_{2}})\,\mathrm{d}B^{H}_{r_{2}}.$

As before, we take $f,\,g$ to be $\gamma-1$-Hölder continuous, playing the
role of $\nabla\sigma$, this time, however, we assume $X,\,Y$ are
deterministically _controlled_ by $B^{H}$, see Def. 2.2. Control on these
integrals, similar in spirit to (10) are obtained progressively through Prop.
4.7, Prop. 4.8 and Prop. 4.9. These estimates are also more technical than in
the Young case and we are required to apply the stochastic sewing lemma twice,
first in the fully shifted sense of [MP22] and then in the original form given
by [Lê20]. We refer the reader to the proofs of the above propositions for
details, simply remarking here that the main innovation is to use the
knowledge that $X$ and $Y$ are controlled by $B^{H}$ to again obtain improved
regularising effects through averaging.

Once sufficient control on the above integrals is obtained we are again able
to pass along a smooth approximation $\sigma_{n}$ to $\sigma$, converging in
any $\gamma^{\prime}$-Hölder topology for
$\frac{1-H}{H}<\gamma^{\prime}<\gamma$. As before, we are then in position to
appeal to uniqueness of solutions to the linear equation (7) and conclude.

###  1.3 Notation

Throughout the paper we use the following notations and conventions.

  * •

When stating an inequality we either write
$A\lesssim_{\alpha,\,\beta,\,\ldots}B$ to indicate that the inequality holds
up to a constant depending on the parameters $\alpha,\,\beta,\,\ldots$ or
write that there exists a $C\coloneqq C(\alpha,\beta,\ldots)>0$ such that
$A\leq CB$. If we write $A\lesssim B$ without explicit dependents we mean that
the inequality holds up to a constant depending on unimportant parameters that
we do not keep track of.

  * •

We $\mathbb{N}\coloneqq\\{1,\,2,\,\ldots\\}$ for the strictly positive
integers and set $\mathbb{N}_{0}\coloneqq\mathbb{N}\cup\\{0\\}$. As usual we
use $\mathbb{R}^{d}$, for $d\in\mathbb{N}$, to denote Cartesian products of
the real numbers. Given a positive integer $d$, we write $l\in\mathbb{N}^{d}$
(resp. $l\in\mathbb{N}_{0}^{d}$) to denote a multi-index made up of positive
(resp. non-negative) integers of length $d$ and set
$|l|\coloneqq\sum_{i=1}^{d}l_{i}$. As is standard, we write $\partial_{x}$
(resp. $\partial_{t}$) for the derivatives on $\mathbb{R}^{d}$ (resp.
$[0,\infty)$) and given $l\in\mathbb{N}_{0}^{d}$ we let
$\partial_{x}^{l}\coloneqq(\partial^{l_{1}}_{x_{1}},\ldots,\partial^{l_{d}}_{x_{d}})$.
Where it will not use confusion we sometimes write $\dot{f}(t,x)$ for the
derivative in time of a function on $\mathbb{R}_{+}\times\mathbb{R}^{d}$.

  * •

We fix two positive integers $d_{1},\,d_{2}\in\mathbb{N}$ which will always
denote, respectively, the dimension of the solution process $t\mapsto
X_{t}\in\mathbb{R}^{d_{1}}$ and the dimension of the noise $t\mapsto
B^{H}_{t}\in\mathbb{R}^{d_{2}}$. Note that all results hold even in the case
$d_{2}<d_{1}$.

  * •

Given $S,\,T\in\mathbb{R}$ such that $S<T$, $d\in\mathbb{N}$ and a function
$f:[S,T]\to\mathbb{R}^{d}$, we define the increment, $f_{s,t}\coloneqq
f_{t}-f_{s}$. Furthermore, defining the sets

$\displaystyle[S,T]^{n}_{\leq}$
$\displaystyle:=\\{(s_{1},\ldots,s_{n})\in[S,T]^{n}\,:\,s_{1}\leq
s_{2}\leq\cdots\leq s_{n}\\},\quad\text{for}\,\,n\in\mathbb{N}$

we specify a map $\delta:C([S,T]^{2}_{\leq};\mathbb{R}^{d})\rightarrow
C([S,T]^{3}_{\leq};\mathbb{R}^{d})$ such that $\delta
A_{s,u,t}=A_{s,t}-A_{s,u}-A_{u,t}.$

  * •

Given, $d_{1},\,d_{2}\in\mathbb{N}$, we write $\mathcal{M}(d_{1},d_{2})$ for
the space of $d_{1}\times d_{2}$ matrices and for $d_{1}=d_{2}\eqqcolon d$ we
define the set of positive definite, square matrices,

$\mathcal{M}_{+}(d)\coloneqq\left\\{\Gamma\in\mathcal{M}(d,d)\,:\,z^{\top}\Gamma
z>0,\,\text{for all }z\in\mathbb{R}^{d}\setminus\\{0\\}\right\\}.$ (12)

Letting $I_{d}\in\mathcal{M}(d,d)$ denote the identity matrix, given
$\Gamma\in\mathcal{M}(d,d)$ and $c\in\mathbb{R}$ we say $\Gamma>c$ if and only
if $\Gamma-cI_{d}\in\mathcal{M}_{+}(d)$ and $\Gamma<c$ if and only if
$cI_{d}-\Gamma\in\mathcal{M}_{+}(d)$. By identifying
$\mathcal{M}(d_{1},d_{2})$ with $\mathbb{R}^{d_{1}\times d_{2}}$, we equip
$\mathcal{M}(d_{1},d_{2})$ with the usual Euclidean norm
$\lvert\,\cdot\,\rvert$. Finally, given $\Gamma\in\mathcal{M}(d_{1},d_{2})$ we
use $\Gamma^{\top}\in\mathcal{M}(d_{2},d_{1})$ to denote the transposed
matrix.

  * •

Given $\Gamma\in\mathcal{M}_{+}(d)$ we define the generalised heat kernel
$p_{\Gamma}:\mathbb{R}^{d}\to\mathbb{R}$,

$p_{\Gamma}(x)\vcentcolon=\frac{1}{(2\pi)^{d/2}(\det\Gamma)^{1/2}}\exp\Big{(}-\frac{\langle
x,\Gamma^{-1}x\rangle}{2}\Big{)}$

and for a function $f:\mathbb{R}^{d}\to\mathbb{R}$ we set
$\mathcal{P}_{\Gamma}f\vcentcolon=p_{\Gamma}\ast f$. When $f$ is vector-
valued, $\mathcal{P}_{\Gamma}$ acts component-wise. If $\Gamma=tI_{d}$ for
some $t\in(0,\infty)$ we simply write
$\mathcal{P}_{t}\vcentcolon=\mathcal{P}_{tI_{d}}$.

  * •

Given $k\in\mathbb{N}_{0}$ we write
$C^{k}(\mathbb{R}^{d_{1}};\mathbb{R}^{d_{2}})$ (resp.
$C^{k}(\mathbb{R}^{d_{1}};\mathcal{M}(d_{1},d_{2}))$) for the space of
$k$-times, continuously differentiable mappings from $\mathbb{R}^{d_{1}}$ to
$\mathbb{R}^{d_{2}}$ (resp. $\mathbb{R}^{d_{1}}$ to
$\mathcal{M}(d_{1},d_{2})$). From now on, we typically only specify the domain
of a given function space, leaving the range unstated where it is clear from
context, or does not affect the statement. For
$\gamma\in\mathbb{R}\setminus\mathbb{N}_{0}$ we let
$\mathcal{C}^{\gamma}(\mathbb{R}^{d_{1}})$ denote the set of functions finite
under the norm,

$\lVert
f\rVert_{\mathcal{C}^{\gamma}_{x}}\vcentcolon=\sum_{l\in\mathbb{N}_{0}^{d_{1}},\,\lvert
l\rvert<\lfloor\gamma\rfloor}\lVert\partial^{l}_{x}f\rVert_{L^{\infty}_{x}}+\sum_{l\in\mathbb{N}_{0}^{d_{1}},\,\lvert
l\rvert=\lfloor\gamma\rfloor}\sup_{x\,\neq
y\,\in\mathbb{R}^{d}}\frac{\lvert\partial^{l}_{x}f(x)-\partial^{l}_{x}f(y)\rvert}{\lvert
x-y\rvert^{\gamma-\lfloor\gamma\rfloor}}.$

Similarly, given $0\leq S\leq T<\infty$ and $d\in\mathbb{N}$ we write
$\mathcal{C}^{\gamma}([S,T];\mathbb{R}^{d})$ for the paths
$f:[S,T]\to\mathbb{R}^{d}$ finite under the norm,

$\lVert
f\rVert_{\mathcal{C}^{\gamma}_{[S,T]}}\vcentcolon=\sum_{l=0}^{\lfloor\gamma\rfloor-1}\lVert\partial^{l}_{t}f\rVert_{L^{\infty}_{[S,T]}}+\sup_{(s,t)\in[S,T]^{2}_{\leq}}\frac{\lvert\partial^{l}_{t}f(t)-\partial^{l}_{t}f(s)\rvert}{\lvert
t-s\rvert^{\gamma-\lfloor\gamma\rfloor}}.$

When $S=0$ we use the abbreviations
$\mathcal{C}^{\gamma}_{T}\coloneqq\mathcal{C}^{\gamma}([0,T];\mathbb{R}^{d})$
and $\|f\|_{\mathcal{C}^{\gamma}_{T}}$. As also introduce the family of
seminorms

$\displaystyle\llbracket
f\rrbracket_{\mathcal{C}^{\gamma}_{[S,T]}}\vcentcolon=\sup_{(s,t)\in[S,T]^{2}_{\leq}}\frac{\lvert
f_{t}-f_{s}\rvert}{\lvert t-s\rvert^{\gamma}}.$ (13)

For $k\in\mathbb{N}_{0}$ we let
$\mathcal{C}^{k}(\mathbb{R}^{d_{1}};\mathbb{R}^{d_{1}})$ (in turn
$\mathcal{C}^{k}(\mathbb{R}^{d_{1}};\mathcal{M})$ and
$\mathcal{C}^{k}([S,T];\mathbb{R}^{d})$) denote the set of functions whose
first $k$, weak, derivatives are bounded.

For, $\gamma<0$ we define the norm

$\|f\|_{\mathcal{C}^{\gamma}_{x}}\coloneqq\sup_{t\in(0,1)}t^{-\frac{\gamma}{2}}\|\mathcal{P}_{t}f\|_{L^{\infty}}$
(14)

and declare $\mathcal{C}^{\gamma}(\mathbb{R}^{d_{1}};\mathbb{R}^{d_{2}})$
(resp. $\mathcal{C}^{\gamma}(\mathbb{R}^{d_{1}};\mathcal{M}(d_{1},d_{2}))$) to
be the set of distributions
$f\in\mathcal{S}^{\prime}(\mathbb{R}^{d_{1}};\mathbb{R}^{d_{2}})$ (resp.
$f\in\mathcal{S}^{\prime}(\mathbb{R}^{d_{1}};\mathcal{M}(d_{1},d_{2}))$) which
are finite under the norm of (14). Note that for
$\gamma\in\mathbb{R}\setminus\mathbb{Z}$ the space
$\mathcal{C}^{\gamma}(\mathbb{R}^{d_{1}})$ coincides with the standard
Hölder–Besov spaces defined via Littlewood–Payley blocks, see [Tri92, Sec.
2.6.4].

#### Acknowledgement

Both authors wish to thank K. Daterotis, L. Galeati, M. Gerescér and N.
Perkowski for illuminating discussions during the course of this project. The
main portion of this research was conducted when TM was a PhD student at Freie
Universität Berlin with financial support by the German Science Foundation
(DFG) via the IRTG 2544. AM gratefully acknowledges financial support from DFG
Research Unit FOR2402.

##  2 Preliminaries

We recall some useful facts concerning the heat semi-group, Young and rough
integration (stochastic) sewing as well as the fractional Brownian motion.

###  2.1 Properties of the Heat Semi-Group

Let $\alpha,\,\beta\in\mathbb{R}$ with $\beta\geq 0\vee\alpha$, $t>0$,
$d\in\mathbb{N}$ and $c>1$. Then the following hold;

  * •

given $\Gamma\in\mathcal{M}_{+}(d)$ such that $c^{-1}<\Gamma<c$

$\|\mathcal{P}_{t\Gamma}\|_{\mathcal{C}^{\beta}_{x}}\lesssim_{c,\alpha,\beta}(t\wedge
1)^{-\frac{\beta-\alpha}{2}}\|f\|_{\mathcal{C}^{\alpha}_{x}}.$ (15)

  * •

given $\Gamma_{1},\,\Gamma_{2}\in\mathcal{M}_{+}(d)$ such
$c^{-1}<\Gamma_{i}<c$ for $i=1,\,2$

$\lVert(\mathcal{P}_{t\Gamma_{1}}-\mathcal{P}_{t\Gamma_{2}})f\rVert_{\mathcal{C}^{\beta}_{x}}\lesssim_{c,\alpha,\beta}(t\wedge
1)^{-\frac{\beta-\alpha}{2}}\lvert\Gamma_{1}-\Gamma_{2}\rvert\lVert
f\rVert_{\mathcal{C}^{\alpha}_{x}}.$ (16)

Inequality (15) is formally stated and proved as Lem. A.1 while (16) is
formally stated and proved as Lem. A.2.

###  2.2 Young Integration

We briefly review the core aspects of _Young integration_ [You36], which we
appeal to for $H\in(1/2,1)$. We refer the reader to [Gal23, FV10] for modern
accounts of the theory. Let $T>0$, $\alpha,\beta\in(0,1)$, and let
$g\in\mathcal{C}^{\beta}_{T}(\mathbb{R}^{d_{2}})$. Suppose that
$f\in\mathcal{C}^{\alpha}_{T}(\mathbb{R}^{d_{1}};\mathcal{M}(d_{1},d_{2}))$.
If $\alpha+\beta>1$, then, as shown for example by [FV10, Thm. 6.8], the limit

$\displaystyle\int_{0}^{T}f_{r}\,\mathrm{d}g_{r}\vcentcolon=\lim_{\begin{subarray}{c}\pi\text{
is a partition of }[0,T];\\\ \lvert\pi\rvert\to
0\end{subarray}}\sum_{[s,t]\in\pi}f_{s}g_{s,t}$

exists, and furthermore, one has the following estimate,

$\displaystyle\Big{\lvert}\int_{s}^{t}f_{r}\,\mathrm{d}g_{r}-f_{s}g_{s,t}\Big{\rvert}\lesssim_{\alpha,\beta}\llbracket
f\rrbracket_{\mathcal{C}^{\alpha}_{[s,t]}}\llbracket
g\rrbracket_{\mathcal{C}^{\beta}_{[s,t]}}(t-s)^{\alpha+\beta},$ (17)

where the seminorms
$\llbracket\,\cdot\,\rrbracket_{\mathcal{C}^{\gamma}_{[s,t]}}$ are defined by
(13) (for $\gamma\in\\{\alpha,\beta\\}$ here).

###  2.3 Rough Integration

When the regularity of the driver $g$ is below this threshold, one may still
construct a suitable notion of integral, provided a suitable enhancement to
the path $g$, and the additional constraint that the integrand $f$ must _look
like_ $g$, in a suitable sense, at small scales. These notions are made
precise in the following two definitions.

######  Definition 2.1 (Rough Paths and Geometric Rough Paths - [Lyo98]).

Let $d\in\mathbb{N}$, $\alpha\in(1/3,1/2]$,
$g\in\mathcal{C}_{T}^{\alpha}(\mathbb{R}^{d})$ and
$\mathbb{G}\colon[0,T]_{\leq}^{2}\to\mathbb{R}^{d}\otimes\mathbb{R}^{d}$. The
pair $(g,\mathbb{G})$ is called an $\alpha$-Hölder _rough path_ if

  * •

the pair satisfies the following Hölder regularity estimate

$\displaystyle\lVert(g,\mathbb{G})\rVert_{\alpha}\vcentcolon=\sup_{[s,t]\in[0,T]_{\leq}}\frac{\lvert
g_{t}-g_{s}\rvert}{\lvert
t-s\rvert^{\alpha}}+\frac{\lvert\mathbb{G}_{s,t}\rvert}{\lvert
t-s\rvert^{2\alpha}}<\infty,$

  * •

and the following algebraic relationship; for $(s,u,t)\in[0,T]_{\leq}^{3}$,

$\displaystyle\mathbb{G}_{s,t}=\mathbb{G}_{s,u}+\mathbb{G}_{u,t}+G_{s,u}\otimes
G_{u,t}.$ (18)

We say that a rough path is _geometric_ if there exists a sequence of smooth
functions $(g^{n})_{n=1}^{\infty}$ such that

$\displaystyle\lim_{n\to\infty}\lVert
g-g^{n}\rVert_{\mathcal{C}^{\alpha}_{T}}=0\quad\text{and}\quad\lim_{n\to\infty}\sup_{[s,t]\in[0,T]_{\leq}^{2}}\frac{\lvert\mathbb{G}_{s,t}-\mathbb{G}^{n}_{s,t}\rvert}{\lvert
t-s\rvert^{2\alpha}}=0,$

where
$\mathbb{G}^{n}_{s,t}\vcentcolon=\int_{s}^{t}g^{n}_{s,r}\otimes\dot{g}^{n}_{r}\,\mathrm{d}r$.

######  Definition 2.2 (Space of Controlled Paths - [Gub04]).

Let $d_{1},\,d_{2}\in\mathbb{N}$, $\alpha\in(0,1)$,
$\beta\in(\alpha\vee(1-\alpha),1)$ and
$g\in\mathcal{C}^{\alpha}_{T}(\mathbb{R}^{d_{2}})$. A path
$f\colon[0,T]\to\mathbb{R}^{d_{1}}$ is said to be _controlled_ by $g$, if
there exists an $f^{\prime}\colon[0,T]\to\mathcal{M}(d_{1},d_{2})$ such that

$\displaystyle\lVert(f,f^{\prime})\rVert_{\mathscr{D}_{g}^{\beta}}\vcentcolon=\lVert
f^{\prime}\rVert_{\mathcal{C}^{\beta-\alpha}_{T}}+\sup_{(s,t)\in[0,T]_{\leq}^{2}}\frac{\lvert
f_{s,t}-f^{\prime}_{s}g_{s,t}\rvert}{\lvert t-s\rvert^{\beta}}<\infty.$ (19)

The map $f^{\prime}$ is called the _Gubinelli derivative_ of $f$. We write
$\mathscr{D}^{\beta}_{g}$ for the space of $\beta$-regular paths controlled by
$g$. Note that for $\beta^{\prime}>\beta$ one has
$\mathscr{D}^{\beta^{\prime}}_{g}\subset\mathscr{D}^{\beta}_{g}$.

The definition of the space of controlled rough paths allows us to give
meaning to the rough integral against a given path. For $\alpha\in(1/3,1)$ and
$\beta\in(1-\alpha,1)$ let $(g,\mathbb{G})$ be an $\alpha$-Hölder rough path
and $f\in\mathscr{D}^{\beta}_{g}$. Then, referring for example to [FH20, Thm.
4.10], one sees that the limit

$\int_{0}^{T}f_{r}\,\mathrm{d}g_{r}\vcentcolon=\lim_{\begin{subarray}{c}\pi\text{
is a partition of }[0,T];\\\ \lvert\pi\rvert\to
0\end{subarray}}\sum_{[s,t]\in\pi}f_{s}g_{s,t}+f^{\prime}_{s}\mathbb{G}_{s,t}$

exists and defines the rough integral of $f$ against $g$. Moreover, similarly
to (17), we have

$\Big{\lvert}\int_{s}^{t}f_{r}\,\mathrm{d}g_{r}-f_{s}g_{s,t}-f^{\prime}_{s}\mathbb{G}_{s,t}\Big{\rvert}\lesssim_{\alpha,\beta}\lVert(f,f^{\prime})\rVert_{\mathscr{D}_{g}^{\beta}}\lVert(g,\mathbb{G})\rVert_{\alpha}(t-s)^{\alpha+\beta}.$
(20)

###  2.4 Sewing

We require a number of versions of the sewing lemma, first introduced by
[Gub04] and [FL06].

######  Lemma 2.3 (Sewing Lemma - [Gub04, FL06]).

Let $E$ be a Banach space, $T>0$ and $A:[0,T]_{\leq}^{2}\to E$ be an
$E$-valued, two parameter process. Then, suppose that for some
$\varepsilon,C\in(0,\infty)$ it holds that

$\displaystyle\lVert\delta A_{s,u,t}\rVert_{E}\leq C\lvert
t-s\rvert^{1+\varepsilon},\quad\forall\,(s,u,t)\in[0,T]^{3}_{\leq},$

then there exists a unique map $\mathcal{A}\colon[0,T]\to E$ such that

$\displaystyle\lVert\mathcal{A}_{t}-\mathcal{A}_{s}-A_{s,t}\rVert_{E}\lesssim_{\varepsilon}C\lvert
t-s\rvert^{1+\varepsilon},\quad\forall\,(s,t)\in[0,T]_{\leq}^{2}.$ (21)

Furthermore, it holds that

$\displaystyle\mathcal{A}_{t}-\mathcal{A}_{s}=\lim_{\begin{subarray}{c}\pi\text{
is a partition of }[s,t],\\\ \lvert\pi\rvert\to
0\end{subarray}}\sum_{[s^{\prime},t^{\prime}]\in\pi}A_{s^{\prime},t^{\prime}}.$

A proof of Lem. 2.3 can be found in [Gub04] as well as the proof of [FH20,
Lem. 4.2].

As is now standard in the field of regularisation by noise, we will need the
stochastic version of Lem. 2.3, obtained first by Lê [Lê20, Thm. 2.1]. For our
purpose, we will require the following _shifted_ version.

######  Lemma 2.4 (Shifted Stochastic Sewing Lemma - [MP22, Thm. 1.1]).

Let $T\in(0,\infty)$,
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\in[0,T]},\mathbb{P})$ be a given
filtered probability space and $(A_{s,t})_{(s,t)\in[0,T]_{\leq}^{2}}$ be an
$\mathbb{R}^{d}$-valued two-parameter stochastic process such that for all
$(s,t)\in[0,T]^{2}_{\leq}$ $A_{s,t}$ is $\mathcal{F}_{t}$-measurable and there
exist constants $\Gamma_{1},\,\Gamma_{2}\geq 0$,
$\varepsilon_{1},\,\varepsilon_{2}\in(0,1)$ and
$\alpha\in[0,1/2+\varepsilon_{2})$ such that for all
$(v,s,u,t)\in[0,T]^{4}_{\leq}$ (with $t-s<s-v$) and $p\in[2,\infty)$, it holds
that

$\begin{split}\lVert\delta
A_{s,u,t}\rVert_{L^{p}_{\omega}}&\leq\Gamma_{1}(t-s)^{\frac{1}{2}+\varepsilon_{1}},\\\
\lVert\mathbb{E}[\delta
A_{s,u,t}|\mathcal{F}_{v}]\rVert_{L^{p}_{\omega}}&\leq\Gamma_{2}(s-v)^{-\alpha}(t-s)^{1+\varepsilon_{2}}.\end{split}$
(22)

Then, there exists a unique (up to modification) stochastic process
$\mathcal{A}$ such that $\mathcal{A}_{0}=0$ and

$\displaystyle\lVert\mathcal{A}_{t}-\mathcal{A}_{s}-A_{s,t}\rVert_{L^{p}_{\omega}}$
$\displaystyle\lesssim_{d,p,\varepsilon_{1},\varepsilon_{2},\alpha}\Gamma_{1}(t-s)^{\frac{1}{2}+\varepsilon_{1}}+\Gamma_{2}(t-s)^{1+\varepsilon_{2}-\alpha},$
$\displaystyle\lVert\mathbb{E}[\mathcal{A}_{t}-\mathcal{A}_{s}-A_{s,t}|\mathcal{F}_{v}]\rVert_{L^{p}_{\omega}}$
$\displaystyle\lesssim_{d,p,\varepsilon_{1},\varepsilon_{2},\alpha}\Gamma_{2}(s-v)^{-\alpha}(t-s)^{1+\varepsilon_{2}}$

for every $(v,s,t)\in[0,T]^{3}_{\leq}$ (with $t-s\leq s-v$ assumed in the
second estimate). Furthermore, there exists an $\varepsilon\in(0,1)$ such
that, for all partitions $\pi$ of $[s,t]\subseteq[0,T]$,

$\displaystyle\Big{\lVert}\mathcal{A}_{t}-\mathcal{A}_{s}-\sum_{[s^{\prime},t^{\prime}]\in\pi}A_{s^{\prime},t^{\prime}}\Big{\rVert}_{L^{p}_{\omega}}\lesssim_{d,p,\varepsilon_{1},\varepsilon_{2},\alpha,T}(\Gamma_{1}+\Gamma_{2})\lvert\pi\rvert^{\varepsilon}.$

We make a number of remarks related to the stochastic sewing lemma.

######  Remark 2.5.

The original version stochastic sewing lemma of [Lê20] is stated for $v=s$ and
$\alpha=0$. The shifted stochastic sewing lemma was first obtained by [Ger22,
Lem. 2.2], where the case $v=s-M(t-s)$ for some $M>0$ and $\alpha=0$ are
treated. Therefore, Lem. 2.4 generalises both the sewing lemmas of [Lê20] and
[Ger22], since it allows for arbitrary shifts.

######  Remark 2.6.

We note that in the setting of [Lê20, Ger22], as remarked in the paragraph
preceding [Ger22, Lem. 2.2], it is in fact enough to check that the condition
analogous to (22), in that setting, holds for $s,u,t\in[0,T]^{3}_{\leq}$ also
satisfying

$\displaystyle\min\\{u-s,t-u\\}\geq(t-s)/3.$

This assumption on $u$ is necessary in the proof of Prop. 3.7 and Prop. 4.7
111It is natural to expect that such assumption on $u$ is also valid to impose
for the fully shifted version [MP22, Thm. 1.1]; however, this fact is not
explicitly shown therein and furthermore, we do not require the restriction in
the cases where we apply the version of [MP22]..

######  Remark 2.7.

We repeatedly make use of the uniqueness statement of the (stochastic) sewing
lemma in the following manner. Given $A^{1}:[0,T]^{2}_{\leq}\to E$ and
$A^{2}:[0,T]^{2}_{\leq}\to E$ such that, for some $\varepsilon>0$,

$\displaystyle\lVert A^{1}_{s,t}-A^{2}_{s,t}\rVert_{E}\lesssim\lvert
t-s\rvert^{1+\varepsilon}\quad\forall(s,t)\in[0,T]_{\leq}^{2}.$ (23)

Then, we may set $A_{s,t}\vcentcolon=A^{1}_{s,t}-A^{2}_{s,t}$ which satisfies

$\lVert\delta A_{s,u,t}\rVert_{E}\leq\lVert
A_{s,t}^{1}-A^{2}_{s,t}\rVert_{E}+\lVert
A^{1}_{s,u}-A^{2}_{s,u}\rVert_{E}+\lVert
A^{1}_{u,t}-A^{2}_{u,t}\rVert_{E}\lesssim\lvert t-s\rvert^{1+\varepsilon}.$

Hence, the assumptions of Lem. 2.3 are satisfied by $A:[0,T]^{2}_{\leq}\to E$.
However, since the constant $\mathcal{A}\equiv 0$ obviously satisfies the
condition (21), it must hold that

$\lim_{\begin{subarray}{c}\pi\text{ is a partition of }[s,t],\\\
\lvert\pi\rvert\to
0\end{subarray}}\sum_{[s^{\prime},t^{\prime}]\in\pi}A^{1}_{s^{\prime},t^{\prime}}=\lim_{\begin{subarray}{c}\pi\text{
is a partition of }[s,t],\\\ \lvert\pi\rvert\to
0\end{subarray}}\sum_{[s^{\prime},t^{\prime}]\in\pi}A^{2}_{s^{\prime},t^{\prime}}$
(24)

under the additional condition that at least one of the Riemann sum
approximations converges.

Similarly, in the setting of the stochastic sewing lemma, if

$\lVert
A^{1}_{s,t}-A^{2}_{s,t}\rVert_{L^{p}_{\omega}}\lesssim(t-s)^{\frac{1}{2}+\varepsilon_{1}},\quad\lVert\mathbb{E}[A^{1}_{s,t}-A^{2}_{s,t}|\mathcal{F}_{v}]\rVert_{L^{p}_{\omega}}\lesssim(s-v)^{-\alpha}(t-s)^{1+\varepsilon_{2}}$

with $\alpha\in[0,1/2+\varepsilon_{2})$, then the identity (24) holds in
$L^{p}(\Omega)$. If the latter estimate also holds with $\alpha=0$, then one
only needs to check the bound for $v=s$ or $v=s-(t-s)$.

We will apply an appropriate version of the result depending on the problem at
hand. In our paper, we use the three versions of the stochastic sewing lemma:
in Sec. 3 we use Gerencsér’s version, [Ger22, Lem. 2.2] (see proof of Prop.
3.7), while in Sec. 4 we use both Lê’s version [Lê20, Thm. 2.1] (see proof of
Prop. 4.7) and the fully shifted version (see proof of Lem. 4.6).

###  2.5 Fractional Brownian Motion

We collect some necessary background on the fractional Brownian motion,
referring to [Nua06, Sec. 5] and [Gal22, App. A.1] for more details. As
specified in the notation sections we fix a $d_{2}\in\mathbb{N}$ and will
consider noise taking values in $\mathbb{R}^{d_{2}}$. We also fix a
probability space $(\Omega,\mathcal{F},\mathbb{P})$.

######  Definition 2.8.

Let $H\in(0,1)$. The _fractional Brownian motion_ (fBm) with Hurst parameter
$H$, started at $0$, is a centred Gaussian process

$B^{H}=(B^{H,i})_{i=1}^{d_{2}}:[0,\infty)\times\Omega\to\mathbb{R}^{d_{2}}$

such that $B^{H}_{0}=0$ and for all $i,\,j=1,\ldots,d_{2}$, it holds that

$\displaystyle\mathbb{E}[B^{H,i}_{t}B^{H,j}_{s}]=\frac{\delta_{ij}}{2}(s^{2H}+t^{2H}-(t-s)^{2H}),\quad\forall\,0\leq
s<t<\infty,$

where $\delta_{ij}$ is the Kronecker delta.

It will be useful to have a more concrete representation for the fBm. Given a
Brownian motion $W$, the formally written process

$\displaystyle
B^{H}_{t}\vcentcolon=(\mathcal{K}_{H}\,\mathrm{d}W)_{t}=\int_{0}^{t}K_{H}(t,s)\,\mathrm{d}W_{s}$
(25)

defines an fBm with Hurst parameter $H$. We refer the reader to [Nua06, Sec.
5.1.3] for more details. In the case $H>1/2$ the kernel $K_{H}$ is explicitly
given by

$\displaystyle
K_{H}(t,s)\vcentcolon=c_{H}s^{\frac{1}{2}-H}\int_{s}^{t}(u-s)^{H-\frac{3}{2}}u^{H-\frac{1}{2}}\,\mathrm{d}u,$
(26)

while for $H<1/2$ one has

$K_{H}(t,s)\vcentcolon=c_{H}\left(\Big{(}\frac{t}{s}\Big{)}^{H-\frac{1}{2}}(t-s)^{H-\frac{1}{2}}-\Big{(}H-\frac{1}{2}\Big{)}s^{\frac{1}{2}-H}\int_{s}^{t}u^{H-\frac{3}{2}}(u-s)^{H-\frac{1}{2}}\,\mathrm{d}u\right).$
(27)

In both cases, the constant $c_{H}$ ensures that $B^{H}$ has the correct
covariance.

The main benefit of this representation is that since the operator
$\mathcal{K}_{H}$ admits an inverse onto its domain of definition, given an
fBm $B^{H}$ we can recover the associated Brownian motion
$W_{t}\vcentcolon=(\mathcal{K}_{H}^{-1}B^{H})_{t}$. An explicit formula for
the inverse is given, for example, by [Gal22, Eq. A.7]. Recall that a Brownian
motion $W$ is called an $(\mathcal{F}_{t})$-Brownian motion if for every $s<t$
the random variable $W_{t}-W_{s}$ is independent of $\mathcal{F}_{s}$.

######  Definition 2.9.

Given a complete filtered space $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq
0},\mathbb{P})$, an adapted process $(B^{H}_{t})_{t\geq 0}$ is called an
$(\mathcal{F}_{t})$-_fractional Brownian motion_ if the process
$\mathcal{K}_{H}^{-1}B^{H}$ is an $(\mathcal{F}_{t})$-Brownian motion.

When $H\in(1/3,1/2]$, the process $B^{H}$ can be lifted to a second order
geometric rough path, we state this fact without proof below, for details see
[FH20, Ch. 10].

######  Lemma 2.10 (Canonical Lift of fBm, [FH20, Ch. 10]).

Let $H\in(1/3,1/2]$ and $(B^{H,(n)})_{n=1}^{\infty}$ be a piecewise linear
approximation of the fBm $B^{H}$. Then the canonical lift of $B^{H,(n)}$

$\left(B^{H,(n)},\mathbb{B}^{H,(n)}\right)\vcentcolon=\left(B^{H,(n)},\left(\int_{s}^{t}B^{H,(n)}_{s,r}\otimes\dot{B}^{H,(n)}_{r}\,\mathrm{d}r\right)_{0\leq
s<t<\infty}\right)$

converges in the space of $\alpha$-Hölder rough paths for any
$\alpha\in(1/3,H)$ to some geometric rough path $(B^{H},\mathbb{B}^{H})$,
called the _canonical lift of the fBm $B^{H}$_.

As the stochastic sewing lemma is a key tool in our proofs, we will need to
understand the behaviour of the fBm under conditioning. Let $B^{H}$ be an
$(\mathcal{F}_{t})$-fBm. Recalling the Volterra representation (25), for $s<t$
we observe the decomposition

$B^{H}_{t}=\mathbb{E}\left[B^{H}_{t}|\mathcal{F}_{s}\right]+\left(B^{H}_{t}-\mathbb{E}\left[B^{H}_{t}|\mathcal{F}_{s}\right]\right)=\int_{0}^{s}K_{H}(t,r)\,\mathrm{d}W_{r}+\int_{s}^{t}K_{H}(t,r)\,\mathrm{d}W_{r},$

where the first term is measurable with respect to $\mathcal{F}_{s}$ and the
second term is independent of $\mathcal{F}_{s}$. In view of this
decomposition, we define the processes $Y^{H,s}=(Y^{H,s,i})_{i=1}^{d_{2}}$ and
$\tilde{B}^{H,s}=(\tilde{B}^{H,s,i})_{i=1}^{d_{2}}$ by

$\displaystyle Y^{H,s}_{t}$
$\displaystyle\vcentcolon=\int_{0}^{s}K_{H}(t,r)\,\mathrm{d}W_{r},$ (28)
$\displaystyle\tilde{B}^{H,s}_{t}$
$\displaystyle\vcentcolon=\int_{s}^{t}K_{H}(t,r)\,\mathrm{d}W_{r}.$ (29)

Informally, we refer to $Y$ as the _history process_ and $\tilde{B}$ as the
_update process_. When, $H=1/2$, by convention we set $Y^{H,s}\vcentcolon=0$
and $\tilde{B}^{H,s}\vcentcolon=B^{H}$. We additionally set

$\displaystyle\rho^{2}_{H}(s,t)\vcentcolon=\mathbb{E}[(\tilde{B}^{H,s,i}_{t})^{2}]=\int_{s}^{t}K_{H}(t,r)^{2}\,\mathrm{d}r$
(30)

and have the following, two sided bound, for all $H\in(0,1)$

$\displaystyle(t-s)^{2H}\lesssim\rho^{2}_{H}(s,t)\leq(t-s)^{2H}.$ (31)

The upper bound of (31) follows trivially from (26) and (27). In particular
one readily checks

$\int_{s}^{t}K_{H}(t,r)^{2}\,\mathrm{d}r\leq\int_{0}^{s}\left(K_{H}(t,r)-K_{H}(s,r)\right)^{2}\,\mathrm{d}r+\int_{s}^{t}K_{H}(t,r)^{2}\,\mathrm{d}r=(t-s)^{2H}.$

The lower bound of (31), whose simple proof can be found for example as
[Gal22, App. A.1], expresses the so-called _local nondeterminism_ property of
fBm, see also [Ber73].

The function $t\mapsto\rho_{H}(s,t)$ is increasing, whence we can consider the
Riemann-Stieltjes integral $\int_{a}^{b}f(r)\rho_{H}(s,\mathrm{d}r)$. Since
$\rho_{H}(s,\cdot)$ is differentiable, one actually has
$\rho_{H}(s,\mathrm{d}r)=\frac{\mathrm{d}\rho_{H}(s,r)}{\mathrm{d}r}\,\mathrm{d}r$.
In particular,

$\int_{u}^{t}\rho_{H}^{\gamma}(s,r)\rho_{H}(s,\mathrm{d}r)=\frac{1}{\gamma+1}\int_{u}^{t}\frac{\mathrm{d}\rho_{H}^{\gamma+1}(s,r)}{\mathrm{d}r}\,\mathrm{d}r\leq\frac{\max\\{\rho_{H}^{\gamma+1}(s,t),\rho_{H}^{\gamma+1}(s,u)\\}}{\lvert\gamma+1\rvert}.$
(32)

A similar remark holds for $\rho^{2}_{H}(s,\mathrm{d}r)$.

We record the following lemmas on $Y^{H,s}$ and $\tilde{B}^{H,s}$, whose
proofs will be given in App. B.

######  Lemma 2.11.

Let $H\in(1/3,1)$ and we define $Y^{H,s}$ by (28). The process $Y^{H,s}$ is
smooth in $(s,\infty)$. Furthermore, for every $p\in[1,\infty)$ and
$(u,t)\in[s,\infty)_{\leq}^{2}$, we have

$\displaystyle\lVert\dot{Y}^{H,s}_{t}\rVert_{L^{p}_{\omega}}\lesssim_{p}(t-s)^{H-1},\quad\lVert
Y^{H,s}_{u,t}\rVert_{L^{p}_{\omega}}\lesssim_{p}(t-s)^{H}-(u-s)^{H}.$

###### Proof.

The proof is given in App, B.1. ∎

The following crucial lemma, shows that for $H\in(1/3,1/2)$, the _update
process_ of the Volterra, fBm can be lifted to a second order rough path.

######  Lemma 2.12.

Given $H\in(1/3,1/2)$, then the pair
$(\tilde{B}^{H,s},\tilde{\mathbb{B}}^{H,s})$ is a geometric rough path of
order $\alpha$ for any $\alpha<H$, where

$\tilde{\mathbb{B}}^{H,s}_{u,t}\vcentcolon=\mathbb{B}^{H}_{u,t}-\int_{u}^{t}Y^{H,s}_{u,r}\otimes\,\mathrm{d}B^{H}_{r}-\int_{u}^{t}B^{H}_{u,r}\otimes\dot{Y}^{H,s}_{r}\,\mathrm{d}r+\int_{u}^{t}Y^{H,s}_{u,r}\otimes\dot{Y}^{H,s}_{r}\,\mathrm{d}r.$
(33)

The second integral is well-defined as Riemann–Stieltjes integral, since
$Y^{H,s}$ is smooth in $(s,\infty)$.

###### Proof.

The proof is given in App. B.1. ∎

Below and throughout the text, the integral with respect to $\tilde{B}^{H,s}$
is understood as Young integral when $H>1/2$ and as rough integral with
respect to the geometric rough path
$(\tilde{B}^{H,s},\tilde{\mathbb{B}}^{H,s})$ when $H\leq 1/2$.

######  Lemma 2.13.

Let $H\in(1/3,1)$, $0\leq s\leq u<t$, $g\in
C^{3}(\mathbb{R}^{d_{1}},\mathbb{R})$ and $a\in\mathcal{M}(d_{1},d_{2})$.
Then, given $\tilde{B}^{H,s}$ and $\rho_{H}(s,r)$ as defined by (29) and (30)
respectively, one has the identity,

$\mathbb{E}\left[\int_{u}^{t}g(a\tilde{B}^{H,s}_{r})\,\mathrm{d}\tilde{B}_{r}^{H,s,i}\right]=\frac{1}{2}\sum_{j=1}^{d_{1}}a^{ji}\int_{u}^{t}[\partial_{j}\mathcal{P}_{\rho^{2}_{H}(s,r)aa^{\top}}g](0)\,\rho^{2}_{H}(s,\mathrm{d}r),$

where the integral on the left-hand side is understood as Young integral when
$H>1/2$ and as rough integral with respect to the geometric rough path
$(\tilde{B}^{H,s},\tilde{\mathbb{B}}^{H,s})$ (see Lem. 2.12) when $H\leq 1/2$.
The integral in the right-hand side is understood as Riemann–Stieltjes
integral with respect to the increasing function $r\mapsto\rho^{2}_{H}(s,r)$.

###### Proof.

The proof is given in App. B.2. ∎

##  3 Young Case: $H\in(1/2,1)$

In this section we prove Thm. 1.1 for the regime $H\in(1/2,1)$. In this
parameter range we are able to appeal to Young integration theory in order to
give a pathwise definition of solutions to the equation.

$\,\mathrm{d}X_{t}=\sigma(X_{t})\,\mathrm{d}B^{H}_{t},\quad
X_{0}=x\in\mathbb{R}^{d_{1}}$ (34)

To make this precise we recall the following definition.

######  Definition 3.1.

Given $T>0$, $d_{1},\,d_{2}\in\mathbb{N}$ a path
$g\in\mathcal{C}^{\alpha}([0,T];\mathbb{R}^{d_{2}})$ and a map
$\sigma:\mathbb{R}^{d_{1}}\to\mathcal{M}(d_{1},d_{2})$. We say that a path
$y:[0,T]\to\mathbb{R}^{d_{1}}$ is a _solution to the Young differential
equation_

$\displaystyle\mathrm{d}y_{t}=\sigma(y_{t})\,\mathrm{d}g_{t},\quad
y_{0}=x\in\mathbb{R}^{d_{1}}$ (35)

on $[0,T]$ if $y$ is such that
$\sigma(y)\in\mathcal{C}^{\beta}([0,T];\mathcal{M}(d_{1},d_{2}))$ for some
$\beta>0$ such that $\beta>1-\alpha$ and for all $t\in[0,T]$ one has

$\displaystyle y_{t}=x+\int_{0}^{t}\sigma(y_{r})\,\mathrm{d}g_{r},$

where the integral on the right hand side is understood as a Young integral as
described in Sec. 2.2.

Since we are interested in probabilistic notions of uniqueness we develop the
following series of probabilistic definitions for solutions to (34).

######  Definition 3.2 (Existence and Uniqueness of Weak-Young Solutions).

We say that a tuple

$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},\mathbb{P},B^{H},X),$

of a filtered probability space, an fBm of Hurst parameter $H$ and an
$(\mathcal{F}_{t})$-adapted process $X$ is a weak-Young solution to (34) if
$B^{H}$ is an $(\mathcal{F}_{t})_{t\geq 0}$ fBm (in the sense of Def. 2.9) and
for $\mathbb{P}$-a.e $\omega\in\Omega$, the process $X(\omega)$ solves the
differential equation (34) in the sense of Young (Def. 3.1).

We say that weak uniqueness holds for (34) in the Young sense, if for any two
weak solutions $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq
0},\mathbb{P},B^{H},X)$ and
$(\tilde{\Omega},\tilde{\mathcal{F}},(\tilde{\mathcal{F}_{t}})_{t\geq
0},\tilde{\mathbb{P}},\tilde{B}^{H},\tilde{X})$ as above, the processes
$X,\,\tilde{X}$ are equal in law, i.e.

$X_{\\#}\mathbb{P}=\tilde{X}_{\\#}\tilde{\mathbb{P}}\in\mathcal{P}(C([0,+\infty);\mathbb{R})).$

######  Remark 3.3.

As mentioned in the text following Prop. 1.2, a standard compactness argument
(given in [MP22, App. B]) allows for the construction of weak solutions in the
case $H\in(1/2,1)$ and
$\sigma\in\mathcal{C}^{\gamma}(\mathbb{R}^{d_{1}};\mathcal{M}(d_{1},d_{2}))$
for $\gamma>\frac{1-H}{H}$.

Note that since a posteriori one has
$y\in\mathcal{C}^{\alpha}([0,T];\mathbb{R}^{d_{1}})$ for any $\alpha\in(0,H)$,
we have
$\sigma(y)\in\mathcal{C}^{\gamma\alpha}([0,T];\mathcal{M}(d_{1},d_{2}))$.
Hence, the conditions $\alpha<H$ and $\gamma>\frac{1-H}{H}$ imply that
$\alpha\gamma+\alpha=(1+\gamma)\alpha>\frac{\alpha}{H}>1$, hence the Young
integral in (35) is well-defined.

In the regime $H\in(1/2,1)$ our main result (Thm. 1.1) obtains pathwise
uniqueness for $\sigma$ that is $\gamma$-Hölder continuous for
$\gamma>\frac{1}{2H}$ and are uniformly elliptic; in this case note that
$\frac{1}{2H}>\frac{1-H}{H}$, so that a fortiori we obtain pathwise uniqueness
(under the assumption of uniform ellipticity) for these weak solutions and
hence strong existence due to Prop. 1.2.

######  Definition 3.4 (Existence of Strong-Young Solutions).

Given a probability space, $(\Omega,\mathcal{F},\mathbb{P})$ carrying an fBm
$B^{H}$, we say that $X$ is a strong-Young solution to (34) if for
$\mathbb{P}$-a.e. $\omega\in\Omega$, the process $X(\omega)$ solves (34) in
the sense of Young (Def. 3.1) and $X$ is adapted to the natural filtration
generated by $B^{H}$.

######  Definition 3.5 (Pathwise Uniqueness of Young Solutions).

We say that _pathwise-uniqueness_ holds for (34), in the Young sense, if for
any two, weak-Young solutions, $X$ and $\tilde{X}$ (in the sense of Def 3.2),
on the same probability space $(\Omega,\mathcal{F},\mathbb{P})$, it holds that
$X(\omega)=\tilde{X}(\omega)$ for $\mathbb{P}$-a.e. $\omega\in\Omega$.

Pathwise uniqueness for (34) is proven in Sec. 3.2.

###  3.1 Young Integral with Irregular Integrand

As explained in Sec. 1.2, the key to obtaining pathwise uniqueness is to
obtain suitable, stochastic estimates on integrals of the form,

$\displaystyle\int_{s}^{t}f(X_{r})\,\mathrm{d}B^{H}_{r}$ (36)

for an irregular $f$ and a path $X$ in a suitable class. Since we will
eventually take $X$ to be any weak solution to the SDE (34), a natural class
to take are those paths which are stochastically controlled by $B^{H}$. It
turns out that this class is also sufficient for our purposes. The following
definition is given in the full range $H\in(0,1)$ since we also make use of it
in Sec. 4 where we treat the case $H\in(1/3,1/2]$.

######  Definition 3.6.

Let $T>0$, $d_{1},\,d_{2}\in\mathbb{N}$,
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\in[0,T]},\mathbb{P})$ be a filtered
probability space and $B^{H}$ be an $\mathbb{R}^{d_{2}}$ dimensional,
$(\mathcal{F}_{t})$-fBm with Hurst parameter $H\in(1/3,1)$. Then, for
$\beta\in(0,1]$, $l\in(1,\infty)$ and a pair of ($\mathcal{F}_{t}$)-adapted
processes
$(X,X^{\prime}):[0,T]\to\mathbb{R}^{d_{1}}\times\mathcal{M}(d_{1},d_{2})$, we
say that $X$ is _controlled_ by $B^{H}$ and write
$(X,X^{\prime})\in\mathfrak{D}^{\beta,l}_{B^{H}}$ if

  * •

the _derivative process_ , $X^{\prime}$ satisfies the analytic bound,

$\sup_{0\leq s<t\leq
T}\frac{||X^{\prime}_{s,t}||_{L^{p}_{\omega}}}{(t-s)^{\beta
H}}<\infty\quad\text{for all }\,p\in[1,\infty);$

  * •

the remainder

$\displaystyle R_{s,t}\vcentcolon=X_{s,t}-X^{\prime}_{s}B^{H}_{s,t},$ (37)

defined for all $0\leq s<t\leq T$, satisfies the bound

$\sup_{0\leq s<t\leq
T}\frac{||R_{s,t}||_{L^{p}_{\omega}}}{(t-s)^{(1+\beta)H}}<\infty,\quad\text{for
all }\,p\in[1,\infty);$

  * •

and for all $t\geq 0$,
$X^{\prime}(X^{\prime})^{\top}\in\mathcal{M}_{+}(d_{2})$, i.e. there exists an
$l>0$ such that $l^{-1}\leq X^{\prime}_{t}(X^{\prime}_{t})^{\top}\leq l$.

Given $(X,X^{\prime})$ satisfying these conditions and $p\in[1,\infty)$, we
set

$\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{p,\beta}\vcentcolon=\sup_{0\leq
t\leq T}\|X^{\prime}_{t}\|_{L^{p}_{\omega}}+\sup_{0\leq s<t\leq
T}\frac{||X^{\prime}_{s,t}||_{L^{p}_{\omega}}}{(t-s)^{\beta H}}+\sup_{0\leq
s<t\leq T}\frac{||R_{s,t}||_{L^{p}_{\omega}}}{(t-s)^{(1+\beta)H}}.$ (38)

The main result of this subsection is the following proposition, which gives
an analogue of the fundamental estimate of Young integration for integrals of
the form (36).

######  Proposition 3.7.

Let $T>0$, $d_{1},\,d_{2}\in\mathbb{N}$, $B^{H}$ be an
$\mathbb{R}^{d_{2}}$-fBm, with Hurst parameter $H\in(1/2,1)$, adapted to a
given filtration $(\mathcal{F}_{t})_{t\in[0,T]}$, $\beta\in(\frac{1}{2H},1]$
and $l\in[1,\infty)$. Then, given
$(X,X^{\prime})\in\mathfrak{D}^{\beta,l}_{B^{H}}$, $f\in
C^{1}(\mathbb{R}^{d_{1}},\mathbb{R})$, $p\in[2,\infty)$,
$\tilde{\gamma}\in(\frac{1}{2H}-1,0)$ and $(s,t)\in[0,T]^{2}_{\leq}$, it holds
that

$\Big{\|}\int_{s}^{t}f(X_{r})\mathrm{d}B_{r}^{H}\Big{\|}_{L^{p}_{\omega}}\lesssim_{\beta,l,p,\tilde{\gamma},T}\|f\|_{\mathcal{C}_{x}^{\tilde{\gamma}}}\big{(}(t-s)^{(1+\tilde{\gamma})H}+\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{2p,\beta}(t-s)^{(1+\beta+\tilde{\gamma})H}\big{)}.$

###### Proof.

For simplicity, in the remainder of the proof we write $B\vcentcolon=B^{H}$
and do not keep track of dependencies on $\beta,l,p,\tilde{\gamma},T$. Our
strategy of proof is to apply Gerencsér’s shifted stochastic sewing, [Ger22,
Lem. 2.2] (equivalently Lem. 2.4 with $\alpha=0$ and $v=s-(t-s)$) to the germ

$A_{s,t}\vcentcolon=\mathbb{E}_{s-(t-s)}\left[\int_{s}^{t}f(X_{s-(t-s)}+X_{s-(t-s)}^{\prime}B_{s-(t-s),r})\,\mathrm{d}B_{r}\right].$
(39)

As we demonstrate below, the shifted conditional, expectation and
coefficients, along with the shifted stochastic sewing lemma, allow us to
obtain the claimed estimate. We note that the shift is necessary, for instance
to bound (54) below. We decompose the proof into six steps. First demonstrate
that the integral $\int_{s}^{t}f(X_{r})\,\mathrm{d}B^{H}_{r}$ can be realised
as the (shifted) stochastic sewing of (39).

Step 1. This is equivalent to the claim that the following limit holds in
$L^{p}(\Omega;\mathbb{R})$,

$\displaystyle\int_{s}^{t}f(X_{r})\,\mathrm{d}B_{r}=\lim_{\begin{subarray}{c}\pi\text{
is a partition of }[s,t],\\\ \lvert\pi\rvert\to
0\end{subarray}}\sum_{[s^{\prime},t^{\prime}]\in\pi}A_{s^{\prime},t^{\prime}}.$
(40)

Namely, this shows that the germ $(A_{s,t})$ provides a correct approximation
to the integral, (36). To prove this identity, we first set
$v\vcentcolon=s-(t-s)$ and define the auxiliary germ

$\displaystyle
A^{0}_{s,t}\vcentcolon=\int_{s}^{t}f(X_{v}+X_{v}^{\prime}B_{v,r})\,\mathrm{d}B^{H}_{r}.$
(41)

It is directly clear that the Riemann sums of $A^{0}$ and $A$ coincide. To
wit, appealing to the uniqueness element of the stochastic sewing lemma, as in
Rem. 2.7, it suffices to check that

$\displaystyle\lVert A_{s,t}-A^{0}_{s,t}\rVert_{L^{p}_{\omega}}\lesssim\lvert
t-s\rvert^{\frac{1}{2}+\varepsilon_{1}}\quad\forall(s,t)\in[0,T]_{\leq}^{2},$
(42)

and

$\left\|\mathbb{E}_{s-(t-s)}\left[A_{s,t}-A^{0}_{s,t}\right]\right\|_{L^{p}_{\omega}}\lesssim|t-s|^{1+\varepsilon_{2}},\quad\forall(s,t)\in[0,T]_{\leq}^{2}.$

The latter is immediate since
$\mathbb{E}_{s-(t-s)}\left[A_{s,t}-A^{0}_{s,t}\right]=0$. The former also
follows rather straightforwardly, we apply the triangle inequality, the
fundamental estimate of Young integration (17) and the assumptions on
$f,\,(X,X^{\prime})$ to find,

$\|A_{s,t}-A^{0}_{s,t}\|_{L^{p}_{\omega}}\leq\|A_{s,t}\|_{L^{p}_{\omega}}+\|A^{0}_{s,t}\|_{L^{p}_{\omega}}\lesssim_{p,f,(X,X^{\prime}),T}|t-s|^{H}.$
(43)

Hence, (42) follows since we here we only consider $H>1/2$. Therefore, in
order to obtain (40) it is enough to show that

$\displaystyle\int_{s}^{t}f(X_{r})\,\mathrm{d}B_{r}=\lim_{\begin{subarray}{c}\pi\text{
is a partition of }[s,t],\\\ \lvert\pi\rvert\to
0\end{subarray}}\sum_{[s^{\prime},t^{\prime}]\in\pi}A^{0}_{s^{\prime},t^{\prime}}.$
(44)

We achieve this by applying the vanilla sewing lemma, (Lem. 2.3) with
$E=L^{p}(\Omega)$. Indeed, with $v=s-(t-s)$, we have

$\Big{\lvert}\int_{s}^{t}f(X_{r})\,\mathrm{d}B_{r}-A^{0}_{s,t}\Big{\rvert}\leq\Big{\lvert}\int_{s}^{t}f(X_{r})\,\mathrm{d}B_{r}-f(X_{s})B_{s,t}\Big{\rvert}+\lvert
A^{0}_{s,t}-f(X_{v}+X^{\prime}_{v}B_{v,s})B_{s,t}\rvert\\\ +\lvert
f(X_{s})B_{s,t}-f(X_{v}+X^{\prime}_{v}B_{v,s})B_{s,t}\rvert.$

By the remainder estimate of Young integration (17), we obtain

$\displaystyle\Big{\lVert}\int_{s}^{t}f(X_{r})\,\mathrm{d}B_{r}-f(X_{s})B_{s,t}\Big{\rVert}_{L^{p}_{\omega}}$
$\displaystyle\lesssim\lVert
f\rVert_{C^{1}}\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{2p,\beta}(t-s)^{(1+\beta)H},$
$\displaystyle\lVert
A^{0}_{s,t}-f(X_{v}+X^{\prime}_{v}B_{v,s})B_{s,t}\rVert_{L^{p}_{\omega}}$
$\displaystyle\lesssim\lVert
f\rVert_{C^{1}}\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{2p,\beta}(t-s)^{(1+\beta)H}.$

In addition, since

$\displaystyle\lvert f(X_{s})-f(X_{v}+X^{\prime}_{v}B_{v,s})\rvert\leq\lVert
f\rVert_{C^{1}}\lvert R_{v,s}\rvert,$

where $R$ is defined by (37), we get

$\displaystyle\lVert
f(X_{s})B_{s,t}-f(X_{v}+X^{\prime}_{v}B_{v,s})B_{s,t}\rVert_{L^{p}_{\omega}}\lesssim\lVert
f\rVert_{C^{1}}\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{2p,\beta}(t-s)^{(2+\beta)H}.$

Combining these estimates, we obtain

$\displaystyle\Big{\lVert}\int_{s}^{t}f(X_{r})\,\mathrm{d}B_{r}-A^{0}_{s,t}\Big{\rVert}_{L^{p}_{\omega}}\lesssim\lVert
f\rVert_{C^{1}}\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{2p,\beta}(t-s)^{(1+\beta)H}.$

As $(1+\beta)H>1$, appealing this time to the uniqueness element of the
deterministic sewing lemma (see Rem. 2.7), the identity (44) is established.

We are now in a position to exploit our choice of germ, given by (39) to
obtain the claimed estimate. We first recall the Volterra representation of
the fBm, given by (25); i.e. for $v\geq 0$ we set $Y^{v}\vcentcolon=Y^{H,v}$
and $\tilde{B}^{v}\vcentcolon=\tilde{B}^{H,v}$ where the right hand sides are
given by (28) and (29) respectively. By Lem. 2.13, given any matrix
$a\in\mathcal{M}(d_{1},d_{2})$ and $v\leq s\leq t$, $g\in
C^{1}(\mathbb{R}^{d_{2}},\mathbb{R})$ it holds that

$\mathbb{E}\left[\int_{s}^{t}g(a\tilde{B}_{r}^{v})\,\mathrm{d}\tilde{B}_{r}^{v,i}\right]=\frac{1}{2}\sum_{j=1}^{d_{1}}a^{ji}\int_{s}^{t}\left(\partial_{j}\mathcal{P}_{\rho^{2}(v,r)aa^{\top}}g\right)(0)\,\rho^{2}(v,\mathrm{d}r),$
(45)

with $\rho^{2}(v,r)\vcentcolon=\rho_{H}^{2}(v,r)$ defined by (30). We also
note that

$\displaystyle\text{$Y^{v}$ is measurable with respect to $\mathcal{F}_{v}$
while $\tilde{B}^{v}$ is independent of $\mathcal{F}_{v}$}.$ (46)

Recall the germ $A_{s,t}$ from (39) we are now in a position to check the
analytic bounds (22). The first is checked by showing that

$\displaystyle\lVert A_{s,t}\rVert_{L^{p}_{\omega}}\lesssim\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}(t-s)^{(1+\tilde{\gamma})H},$ (47)

which is done in Step 2. below. The second estimate of (22) is shown through
Steps 2 - 5.

Step 2. For $v\vcentcolon=s-(t-s)$, by (46) and the trivial fact that
$X_{v},\,X^{\prime}_{v}$ are measurable with respect to $\mathcal{F}_{v}$ we
observe that

$\displaystyle A_{s,t}$
$\displaystyle=\int_{s}^{t}\mathbb{E}\left[f(x+x^{\prime}(y_{v,r}+\tilde{B}^{v}_{r}))\right]\Big{|}_{x=X_{v},x^{\prime}=X^{\prime}_{v},y=Y^{v}}\dot{Y}_{r}^{v}\,\mathrm{d}r$
$\displaystyle\phantom{=}+\mathbb{E}\left[\int_{s}^{t}f(x+x^{\prime}(y_{v,r}+\tilde{B}^{v}_{r}))\,\mathrm{d}\tilde{B}^{v}_{r}\right]\Big{|}_{x=X_{v},x^{\prime}=X^{\prime}_{v},y=Y^{v}}$
$\displaystyle=\vcentcolon I_{1}+I_{2}.$

Since $x^{\prime}\tilde{B}^{v}_{r}$ is a centred Gaussian random variable with
variance $\rho^{2}(v,r)x^{\prime}(x^{\prime})^{\mathrm{T}}$, we find

$\displaystyle
I_{1}=\int_{s}^{t}\mathcal{P}_{\rho^{2}(v,r)X^{\prime}_{v}(X^{\prime}_{v})^{\top}}f(X_{v}+X^{\prime}_{v}Y_{v,r})\dot{Y}^{v}_{r}\,\mathrm{d}r,$

and by Lem. A.1 and (31)

$\displaystyle\lVert\mathcal{P}_{\rho^{2}(v,r)X^{\prime}_{v}(X^{\prime}_{v})^{\top}}f\rVert_{L^{\infty}}\lesssim\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}\rho^{\tilde{\gamma}}(v,r)\lesssim\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}(r-v)^{\tilde{\gamma}H}.$

By Lem. 2.11 we have
$\lVert\dot{Y}^{v}_{r}\rVert_{L^{p}_{\omega}}\lesssim_{p}(r-v)^{H-1}$.
Therefore,

$\displaystyle\lVert I_{1}\rVert_{L^{p}_{\omega}}\lesssim\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}\int_{s}^{t}(r-v)^{\tilde{\gamma}H+H-1}\,\mathrm{d}r\lesssim\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}(t-s)^{(1+\tilde{\gamma})H}.$

To estimate $I_{2}$, we note that by (45), applied with
$g(z)=f(x+x^{\prime}y_{v,r}+x^{\prime}z)$ we find

$\displaystyle I_{2}$
$\displaystyle=\mathbb{E}\left[\int_{s}^{t}f(x+x^{\prime}(y_{v,r}+\tilde{B}^{v}_{r}))\,\mathrm{d}\tilde{B}^{v}_{r}\right]\Big{|}_{x=X_{v},x^{\prime}=X^{\prime}_{v},y=Y^{v}}$
$\displaystyle=\frac{1}{2}\sum_{j=1}^{d_{1}}(X^{\prime}_{v})^{ji}\int_{s}^{t}\left(\partial_{j}\mathcal{P}_{\rho^{2}(v,r)X^{\prime}_{v}(X^{\prime}_{v})^{\top}}f\right)(X_{v}+X_{v}^{\prime}Y^{v}_{v,r})\,\rho^{2}(v,\mathrm{d}r).$

By Lem. A.1 we have

$\displaystyle\lVert\mathcal{P}_{\rho^{2}(v,r)X^{\prime}_{v}(X^{\prime}_{v})^{\top}}f(X_{v}+X_{v}^{\prime}Y^{v}_{v,r})\rVert_{C^{1}}\lesssim(r-v)^{H(\tilde{\gamma}-1)}\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}},$

so that making use of (32) and (31), we obtain

$\displaystyle\lVert I_{2}\rVert_{L^{p}_{\omega}}$
$\displaystyle\lesssim\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}\int_{s}^{t}(r-v)^{H(\tilde{\gamma}-1)}\,\rho^{2}(v,\mathrm{d}r)\leq\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}\int_{s}^{t}\rho(v,r)^{\tilde{\gamma}-1}\,\rho^{2}(v,\mathrm{d}r)$
$\displaystyle\lesssim\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}\rho(v,t)^{\tilde{\gamma}+1}\lesssim\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}(t-s)^{(\tilde{\gamma}+1)H}.$

This concludes Step 2 and the proof of (49). In the remaining steps we show
that for $u\in(s,t)$ with

$\displaystyle\min\\{t-u,u-s\\}\geq\frac{t-s}{3},$ (48)

we have

$\displaystyle\lVert\mathbb{E}_{s-(t-s)}\left[\delta
A_{s,u,t}\right]\rVert_{L^{p}_{\omega}}\lesssim\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{2p,\beta}(t-s)^{(1+\beta+\tilde{\gamma})H},$
(49)

which is the final estimate needed to show that $A_{s,t}$ satisfies the
analytic bounds of the stochastic sewing lemma, (22). We recall that imposing
the additional condition of (48) does not alter the applicability of the
stochastic sewing lemma, see Rem. 2.6.

Step 3. We first perform some simple algebraic manipulations. Let us set

$\displaystyle v_{1}\vcentcolon=s-(t-s),\quad v_{2}\vcentcolon=s-(u-s),\quad
v_{3}\vcentcolon=u-(t-u).$ (50)

It is clear that,

$\displaystyle v_{1}<v_{2}<\min\\{v_{3},s\\}\leq\max\\{v_{3},s\\}<u<t.$

One may then check that $\mathbb{E}_{v_{1}}\left[\delta
A_{s,u,t}\right]=\mathbb{E}_{v_{1}}[I_{3}+I_{4}]$, where

$\displaystyle I_{3}$
$\displaystyle\vcentcolon=\mathbb{E}_{v_{2}}\left[\int_{s}^{u}\left(f(X_{v_{1}}+X^{\prime}_{v_{1}}B_{v_{1},r})-f(X_{v_{2}}+X^{\prime}_{v_{2}}B_{v_{2},r})\right)\,\mathrm{d}B_{r}\right],$
$\displaystyle I_{4}$
$\displaystyle\vcentcolon=\mathbb{E}_{v_{3}}\left[\int_{u}^{t}\left(f(X_{v_{1}}+X^{\prime}_{v_{1}}B_{v_{1},r})-f(X_{v_{3}}+X^{\prime}_{v_{3}}B_{v_{3},r})\right)\,\mathrm{d}B_{r}\right].$

Since the estimate for $I_{3}$ and $I_{4}$ are similar (the latter requires
exchanging $(s,u)$ for $(u,t)$), we only present the estimates for $I_{3}$.
Given $j=1,2$, we observe

$\displaystyle\mathbb{E}_{v_{2}}\left[\int_{s}^{u}f(X_{v_{j}}+X^{\prime}_{v_{j}}B_{v_{j},r})\,\mathrm{d}B_{r}\right]=$
$\displaystyle\,\mathbb{E}_{v_{2}}\left[\int_{s}^{u}f(X_{v_{j}}+X^{\prime}_{v_{j}}B_{v_{j},r})\dot{Y}_{r}^{v_{2}}\,\mathrm{d}r\right]$
$\displaystyle+\mathbb{E}_{v_{2}}\left[\int_{s}^{u}f(X_{v_{j}}+X^{\prime}_{v_{j}}B_{v_{j},r})\,\mathrm{d}\tilde{B}_{r}^{v_{2}}\right].$

By (46), the first term is equal to

$\displaystyle\int_{s}^{u}\left(\mathcal{P}_{\rho^{2}(v_{2},r)X^{\prime}_{v_{j}}(X^{\prime}_{v_{j}})^{\top}}f\right)(X_{v_{j}}+X^{\prime}_{v_{j}}(B_{v_{j},v_{2}}+Y^{v_{2}}_{v_{2},r}))\dot{Y}^{v_{2}}_{r}\,\mathrm{d}r$

and by (45) the second term is equal to

$\displaystyle\frac{1}{2}X^{\prime}_{v_{j}}\int_{s}^{u}\left(\nabla\mathcal{P}_{\rho^{2}(v_{2},r)X^{\prime}_{v_{j}}(X^{\prime}_{v_{j}})^{\top}}f\right)(X_{v_{j}}+X^{\prime}_{v_{j}}(B_{v_{j},v_{2}}+Y^{v_{2}}_{v_{2},r}))\,\rho^{2}(v_{2},\mathrm{d}r).$

Hence, by addition and subtraction, $I_{3}=I_{3,1}+I_{3,2}$, where

$\displaystyle I_{3,1}\vcentcolon=$
$\displaystyle\int_{s}^{u}\big{(}\mathcal{P}_{\rho^{2}(v_{2},r)X^{\prime}_{v_{1}}(X^{\prime}_{v_{1}})^{\top}}f(X_{v_{1}}+X^{\prime}_{v_{1}}(B_{v_{1},v_{2}}+Y^{v_{2}}_{v_{2},r}))$
$\displaystyle-\mathcal{P}_{\rho^{2}(v_{2},r)X^{\prime}_{v_{2}}(X^{\prime}_{v_{2}})^{\top}}f(X_{v_{2}}+X^{\prime}_{v_{2}}Y^{v_{2}}_{v_{2},r})\big{)}\dot{Y}^{v_{2}}_{r}\,\mathrm{d}r,$
$\displaystyle I_{3,2}\vcentcolon=$
$\displaystyle\frac{1}{2}\int_{s}^{u}\big{(}X^{\prime}_{v_{1}}\nabla\mathcal{P}_{\rho^{2}(v_{2},r)X^{\prime}_{v_{1}}(X^{\prime}_{v_{1}})^{\top}}f(X_{v_{1}}+X^{\prime}_{v_{1}}(B_{v_{1},v_{2}}+Y^{v_{2}}_{v_{2},r}))$
$\displaystyle-X^{\prime}_{v_{2}}\nabla\mathcal{P}_{\rho^{2}(v_{2},r)X^{\prime}_{v_{2}}(X^{\prime}_{v_{2}})^{\top}}f(X_{v_{2}}+X^{\prime}_{v_{2}}Y^{v_{2}}_{v_{2},r})\big{)}\,\rho^{2}(v_{2},\mathrm{d}r).$

Step 4. In view of Step 3, in order to prove (49), it remains to show

$\displaystyle\lVert I_{3,1}\rVert_{L^{p}_{\omega}}$
$\displaystyle\lesssim\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{2p,\beta}(t-s)^{(1+\beta+\tilde{\gamma})H},$
(51) $\displaystyle\lVert I_{3,2}\rVert_{L^{p}_{\omega}}$
$\displaystyle\lesssim\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{2p,\beta}(t-s)^{(1+\beta+\tilde{\gamma})H},$
(52)

along with similar arguments for $I_{4}$ which we do not present. In this
step, we estimate $I_{3,1}$.

We decompose $I_{3,1}=I_{3,1,1}+I_{3,1,2}$, where

$\displaystyle I_{3,1,1}\vcentcolon=$
$\displaystyle\int_{s}^{u}\big{(}\mathcal{P}_{\rho^{2}(v_{2},r)X^{\prime}_{v_{1}}(X^{\prime}_{v_{1}})^{\top}}f(X_{v_{1}}+X^{\prime}_{v_{1}}(B_{v_{1},v_{2}}+Y^{v_{2}}_{v_{2},r}))$
$\displaystyle-\mathcal{P}_{\rho^{2}(v_{2},r)X^{\prime}_{v_{1}}(X^{\prime}_{v_{1}})^{\top}}f(X_{v_{2}}+X^{\prime}_{v_{2}}Y^{v_{2}}_{v_{2},r})\big{)}\dot{Y}^{v_{2}}_{r}\,\mathrm{d}r$

and

$I_{3,1,2}\vcentcolon=\int_{s}^{u}\big{(}\mathcal{P}_{\rho^{2}(v_{2},r)X^{\prime}_{v_{1}}(X^{\prime}_{v_{1}})^{\top}}-\mathcal{P}_{\rho^{2}(v_{2},r)X^{\prime}_{v_{2}}(X^{\prime}_{v_{2}})^{\top}}\big{)}f(X_{v_{2}}+X^{\prime}_{v_{2}}Y^{v_{2}}_{v_{2},r})\dot{Y}^{v_{2}}_{r}\,\mathrm{d}r.$

To estimate $I_{3,1,1}$, we apply Lem. A.1 to obtain the Lipschitz estimate

$\displaystyle\lVert\mathcal{P}_{\rho^{2}(v_{2},r)X^{\prime}_{v_{1}}(X^{\prime}_{v_{1}})^{\top}}f\rVert_{C^{1}}\lesssim\rho^{\tilde{\gamma}-1}(v_{2},r)\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}},$ (53)

which yields

$\lvert I_{3,1,1}\rvert\lesssim\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}\int_{s}^{u}\rho^{\tilde{\gamma}-1}(v_{2},r)\times\lvert
X_{v_{1}}+X^{\prime}_{v_{1}}(B_{v_{1},v_{2}}+Y^{v_{2}}_{v_{2},r})-(X_{v_{2}}+X^{\prime}_{v_{2}}Y^{v_{2}}_{v_{2},r})\rvert\lvert\dot{Y}^{v_{2}}_{r}\rvert\,\mathrm{d}r.$

Recalling the definition of the remainder $R$ of a controlled path from (37),
we observe that
$X_{v_{1}}+X^{\prime}_{v_{1}}B_{v_{1},v_{2}}-X_{v_{2}}=-R_{v_{1},v_{2}}$.
Therefore,

$\lVert I_{3,1,1}\rVert_{L^{p}_{\omega}}\lesssim\lVert
f\rVert_{\mathcal{C}^{\tilde{\gamma}}_{x}}\int_{s}^{u}\rho^{\tilde{\gamma}-1}(v_{2},r)\times\left(\lVert
R_{v_{1},v_{2}}\dot{Y}^{v_{2}}_{r}\rVert_{L^{p}_{\omega}}+\lVert
X_{v_{1},v_{2}}^{\prime}Y_{v_{2},r}^{v_{2}}\dot{Y}^{v_{2}}_{r}\rVert_{L^{p}_{\omega}}\right)\,\mathrm{d}r.$

By Hölder’s inequality and recalling the norm
$\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{p,\beta}$,
defined by (38), we find

$\displaystyle\lVert I_{3,1,1}\rVert_{L^{p}_{\omega}}\lesssim$
$\displaystyle\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{2p,\beta}\int_{s}^{u}\rho^{\tilde{\gamma}-1}(v_{2},r)\Big{(}\lVert\dot{Y}^{v_{2}}_{r}\rVert_{L^{2p}_{\omega}}(v_{2}-v_{1})^{(1+\beta)H}$
$\displaystyle\hskip 150.00023pt+\lVert
Y_{v_{2},r}^{v_{2}}\rVert_{L^{4p}_{\omega}}\lVert\dot{Y}^{v_{2}}_{r}\rVert_{L^{4p}_{\omega}}(v_{2}-v_{1})^{\beta
H}\Big{)}\,\mathrm{d}r.$

By Lem. 2.11 we have
$\lVert\dot{Y}^{v_{2}}_{r}\rVert_{L^{4p}_{\omega}}\lesssim(r-v_{2})^{H-1}$ and
$\lVert Y_{v_{2},r}^{v_{2}}\rVert_{L^{4p}_{\omega}}\lesssim(r-v_{2})^{H}$.
Hence, also using the bound (31) to estimate $\rho(v_{2},r)$, we have

$\displaystyle\int_{s}^{u}\rho^{\tilde{\gamma}-1}(v_{2},r)\lVert\dot{Y}^{v_{2}}_{r}\rVert_{L^{2p}_{\omega}}(v_{2}-v_{1})^{(1+\beta)H}\,\mathrm{d}r$
$\displaystyle\lesssim(v_{2}-v_{1})^{(1+\beta)H}\int_{s}^{u}(r-v_{2})^{\tilde{\gamma}H-1}\,\mathrm{d}r$
(54)
$\displaystyle\lesssim(v_{2}-v_{1})^{(1+\beta)H}(s-v_{2})^{\tilde{\gamma}H}$

and

$\displaystyle\int_{s}^{u}\rho^{\tilde{\gamma}-1}(v_{2},r)\lVert
Y_{v_{2},r}^{v_{2}}\rVert_{L^{4p}_{\omega}}\lVert\dot{Y}^{v_{2}}_{r}\rVert_{L^{4p}_{\omega}}(v_{2}-v_{1})^{\beta
H}\,\mathrm{d}r$ $\displaystyle\lesssim(v_{2}-v_{1})^{\beta
H}\int_{s}^{u}(r-v_{2})^{\tilde{\gamma}H+H-1}\,\mathrm{d}r$
$\displaystyle\lesssim(v_{2}-v_{1})^{\beta H}(u-v_{2})^{(1+\tilde{\gamma})H}.$

Recalling $v_{1}$, $v_{2}$ from (50) and the condition (48) on $u$, we obtain

$\displaystyle\lVert I_{3,1,1}\rVert_{L^{p}_{\omega}}\lesssim\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{2p,\beta}(t-s)^{(1+\beta+\tilde{\gamma})H},$

an estimate compatible with (52).

We move to estimate $I_{3,1,2}$. By Lem. A.2,

$\left\|\left(\mathcal{P}_{\rho^{2}(v_{2},r)X^{\prime}_{v_{1}}(X^{\prime}_{v_{1}})^{\top}}-\mathcal{P}_{\rho^{2}(v_{2},r)X^{\prime}_{v_{2}}(X^{\prime}_{v_{2}})^{\top}}\right)f\,\right\|_{L^{\infty}}\\\
\lesssim\left|X^{\prime}_{v_{1}}(X^{\prime}_{v_{1}})^{\top}-X^{\prime}_{v_{2}}(X^{\prime}_{v_{2}})^{\top}\right|\,\rho^{\tilde{\gamma}}(v_{2},r)\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}.$

Thus, by Hölder’s inequality,

$\displaystyle\lVert I_{3,1,2}\rVert_{L^{p}_{\omega}}\lesssim\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}\int_{s}^{u}\rho^{\tilde{\gamma}}(v_{2},r)\lVert
X^{\prime}_{v_{1},v_{2}}\rVert_{L^{2p}_{\omega}}\lVert\dot{Y}^{v_{2}}_{r}\rVert_{L^{2p}_{\omega}}\,\mathrm{d}r.$

Recalling the definition of
$\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{2p,\beta}$
from (38), applying Lem. 2.11 and (31) we then find

$\displaystyle\lVert I_{3,1,2}\rVert_{L^{p}_{\omega}}$
$\displaystyle\lesssim\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{2p,\beta}(v_{2}-v_{1})^{\beta
H}\int_{s}^{u}(r-v_{2})^{\tilde{\gamma}H+H-1}\,\mathrm{d}r$
$\displaystyle\lesssim\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{2p,\beta}(t-s)^{(1+\beta+\tilde{\gamma})H},$

which, combined with the estimate of $I_{3,1,1}$ in the previous paragraph,
yields (52).

Step 5. In this last step, we estimate $I_{3,2}$ to obtain (52). By adding and
subtracting, we write $I_{3,2}=I_{3,2,1}+I_{3,2,2}+I_{3,2,3}$, where

$\displaystyle I_{3,2,1}\vcentcolon=$
$\displaystyle\frac{1}{2}\int_{s}^{u}X^{\prime}_{v_{1}}\big{(}\nabla\mathcal{P}_{\rho^{2}(v_{2},r)X^{\prime}_{v_{1}}(X^{\prime}_{v_{1}})^{\top}}f(X_{v_{1}}+X^{\prime}_{v_{1}}(B_{v_{1},v_{2}}+Y^{v_{2}}_{v_{2},r}))$
$\displaystyle-\nabla\mathcal{P}_{\rho^{2}(v_{2},r)X^{\prime}_{v_{1}}(X^{\prime}_{v_{1}})^{\top}}f(X_{v_{2}}+X^{\prime}_{v_{2}}Y^{v_{2}}_{v_{2},r})\big{)}\,\rho^{2}(v_{2},\mathrm{d}r),$
$I_{3,2,2}\vcentcolon=\int_{s}^{u}X_{v_{1}}^{\prime}\left(\nabla\mathcal{P}_{\rho^{2}(v_{2},r)X^{\prime}_{v_{1}}(X^{\prime}_{v_{1}})^{\top}}-\nabla\mathcal{P}_{\rho^{2}(v_{2},r)X^{\prime}_{v_{2}}(X^{\prime}_{v_{2}})^{\top}}\right)\\\
\times
f(X_{v_{2}}+X^{\prime}_{v_{2}}Y^{v_{2}}_{v_{2},r})\,\rho^{2}(v_{2},\mathrm{d}r),$

and

$I_{3,2,3}\vcentcolon=\frac{1}{2}\int_{s}^{u}(X^{\prime}_{v_{1}}-X^{\prime}_{v_{2}})\nabla\mathcal{P}_{\rho^{2}(v_{2},r)X^{\prime}_{v_{2}}(X^{\prime}_{v_{2}})^{\top}}f(X_{v_{2}}+X^{\prime}_{v_{2}}Y^{v_{2}}_{v_{2},r})\,\rho^{2}(v_{2},\mathrm{d}r).$

The estimate of $I_{3,2,1}$ and that of $I_{3,2,2}$ are similar to that of
$I_{3,1,1}$ and that of $I_{3,1,2}$ respectively; we only demonstrate how to
estimate $I_{3,2,1}$. By Lem. A.1,

$\displaystyle\lVert\mathcal{P}_{\rho^{2}(v_{2},r)X^{\prime}_{v_{1}}(X^{\prime}_{v_{1}})^{\top}}f\rVert_{C^{2}}\lesssim\rho^{\tilde{\gamma}-2}(v_{2},r)\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}.$

Therefore,

$\displaystyle\lvert I_{3,2,1}\rvert\lesssim\lVert
f\rVert_{\mathcal{C}^{\tilde{\gamma}}_{x}}\int_{s}^{u}\rho^{\tilde{\gamma}-1}(v_{2},r)\left(\lvert
R_{v_{1},v_{2}}\rvert+\lvert
X_{v_{1},v_{2}}^{\prime}Y^{v_{2}}_{v_{2},r}\rvert\right)\,\rho(v_{2},\mathrm{d}r).$

As above, recalling the definition of
$\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{p,\beta}$,
the estimate on $Y^{v_{2}}$ from Lem. 2.11 and the bound (32), we get

$\displaystyle\lVert I_{3,2,1}\rVert$ $\displaystyle\lesssim\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{p,\beta}\Big{(}(v_{2}-v_{1})^{(1+\beta)H}\int_{s}^{u}\rho^{\tilde{\gamma}-1}(v_{2},r)\,\rho(v_{2},\mathrm{d}r)$
$\displaystyle\phantom{\lesssim\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{p,\beta}\Big{(}}\,\,+(v_{2}-v_{1})^{\beta
H}\int_{s}^{u}\rho^{\tilde{\gamma}}(v_{2},r)\,\rho(v_{2},\mathrm{d}r)\Big{)}$
$\displaystyle\lesssim\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{p,\beta}\left[(v_{2}-v_{1})^{(1+\beta)H}\rho^{\tilde{\gamma}}(v_{2},s)+(v_{2}-v_{1})^{\beta
H}\rho^{\tilde{\gamma}+1}(v_{2},u)\right]$

Applying the estimate (31) for $\rho(v_{2},r)$, we get

$\displaystyle\lVert I_{3,2,1}\rVert\lesssim\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{p,\beta}(t-s)^{(1+\beta+\tilde{\gamma})H}.$

We finally estimate $I_{3,2,3}$. Using the bounds (53) and (32),

$\displaystyle\lvert I_{3,2,3}\rvert\lesssim\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}\left|X_{v_{1},v_{2}}^{\prime}\right|\int_{s}^{u}\rho^{\tilde{\gamma}}(v_{2},r)\,\rho(v_{2},\mathrm{d}r)\lesssim\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}\left|X_{v_{1},v_{2}}^{\prime}\right|\rho^{\tilde{\gamma}+1}(v_{2},u).$

Therefore,

$\displaystyle\lVert I_{3,2,3}\rVert_{L^{p}_{\omega}}\lesssim\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{p,\beta}(t-s)^{(1+\beta+\tilde{\gamma})H},$

yielding the estimate (52), which in turn gives us (49), completing the proof.
∎

###  3.2 Pathwise Uniqueness

Thanks to Prop. 3.7, we are ready to prove our main result in the case
$H\in(1/2,1)$.

######  Theorem 3.8 (Pathwise Uniqueness in the Young regime).

Let $T>0$, $d_{1},\,d_{2}\in\mathbb{N}$, $B^{H}$ be an fBm with Hurst
parameter $H\in(1/2,1)$, $\gamma\in(\frac{1}{2H},1)$ and
$\sigma\in\mathcal{C}^{\gamma}(\mathbb{R}^{d_{1}};\mathcal{M}(d_{1},d_{2}))$
be such that $\sigma\sigma^{\top}$ is uniformly elliptic, i.e. there exists an
$l\in[1,\infty)$ such that $l^{-1}\leq\sigma(x)\sigma^{\top}(x)\leq l$ for all
$x\in\mathbb{R}^{d_{1}}$. Then, pathwise uniqueness holds for (34) in the
sense of Def. 3.5.

Before giving the proof of Thm. 3.8, we prepare the following lemma.

######  Lemma 3.9.

In the setting of Thm. 3.8, let
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\in[0,T]},\mathbb{P},X,B^{H})$ be a
weak-Young solution to (34) (see Def. 3.2). Then, we have
$(X,\sigma(X))\in\mathfrak{D}^{\gamma,l}_{B^{H}}$ as given by Def. 2.2.

###### Proof.

Let $\alpha<H$, but sufficiently close to $H$. We recall the seminorm
$\llbracket\,\cdot\,\rrbracket_{\mathcal{C}^{\alpha}_{[s,t]}}$ from (13).
Since

$\displaystyle X_{s,t}=\int_{s}^{t}\sigma(X_{r})\,\mathrm{d}B^{H}_{r},$

the remainder estimate of Young integration, estimate (17), gives

$\displaystyle\lvert
X_{s,t}-\sigma(X_{s})B^{H}_{s,t}\rvert\lesssim_{\alpha,\gamma}\lVert\sigma\rVert_{\mathcal{C}^{\gamma}_{x}}\llbracket
X\rrbracket_{\mathcal{C}^{\alpha}_{[s,t]}}^{\gamma}\left\|B^{H}\right\|_{\mathcal{C}^{\alpha}_{[s,t]}}(t-s)^{(1+\gamma)\alpha}.$
(55)

By [FV10, Lem. 10.7], we have

$\displaystyle\llbracket
X\rrbracket_{\mathcal{C}^{\alpha}_{T}}\lesssim_{\alpha,\gamma,T}(1+\lVert\sigma\rVert_{\mathcal{C}^{\gamma}_{x}}\lVert
B^{H}\rVert_{\mathcal{C}^{\alpha}_{T}})^{\frac{1}{\alpha}}.$ (56)

Now, due to Gaussianity of $B^{H}$ and as a consequence of the Kolmogorov
continuity criterion, for all $p<\infty$, there exists a $C\coloneqq
C(p,\alpha,T,\lVert\sigma\rVert_{\mathcal{C}_{x}^{\gamma}})>0$ such that

$\displaystyle\left\|\llbracket
X\rrbracket_{\mathcal{C}_{T}^{\alpha}}\right\|_{L^{p}_{\omega}}\leq C.$ (57)

Combining (55) and (57), we get $\lVert
X_{s,t}\rVert_{L^{p}_{\omega}}\lesssim(t-s)^{H}$. In particular, we have
$\lVert\sigma(X)_{s,t}\rVert_{L^{p}_{\omega}}\lesssim(t-s)^{\gamma H}$. To get
an estimate of the remainder $R_{s,t}=X_{s,t}-\sigma(X_{s})B_{s,t}^{H}$, it
suffices to apply the sewing lemma in $L^{p}_{\omega}$ with germ
$A_{s,t}\vcentcolon=\sigma(X_{s})B_{s,t}^{H}$, since

$\lVert\delta
A_{s,u,t}\rVert_{L^{p}_{\omega}}=\left\|\sigma(X)_{s,u}B^{H}_{u,t}\right\|_{L^{p}_{\omega}}\lesssim(t-s)^{(1+\gamma)H}.$

∎

We are now ready to give the proof of Thm. 3.8.

###### Proof of Thm. 3.8.

Let $\alpha\in(1/2,H)$ be arbitrarily close to $H$ and $(X,B^{H})$ and
$(Y,B^{H})$ be two weak solutions defined on the same filtered probability
space $(\Omega,\mathcal{F},(\mathcal{F}_{t}),\mathbb{P})$ and driven by the
same $(\mathcal{F}_{t})$-fBm, $B^{H}$. Since the requirement on
$\gamma\in(\frac{1}{2H},1)$ is given by an open interval it is always possible
to find a $\gamma^{\prime}\in(\frac{1}{2H},\gamma)$. We then let
$(\sigma_{n})_{n=1}^{\infty}$ be a smooth approximation to $\sigma$ converging
in
$\mathcal{C}^{\gamma^{\prime}}(\mathbb{R}^{d_{1}};\mathcal{M}(d_{1},d_{2}))$
(which exists by the compact embedding between Hölder–Besov spaces). The
remainder estimate of Young integration (17) gives

$\displaystyle\Big{\lvert}\int_{s}^{t}\sigma(Z_{r})\,\mathrm{d}B^{H}_{r}-\sigma(Z_{s})B^{H}_{s,t}\Big{\rvert}\lesssim_{\alpha,\gamma^{\prime}}\lVert\sigma\rVert_{\mathcal{C}_{x}^{\gamma^{\prime}}}\lVert
Z\rVert_{\mathcal{C}_{T}^{\alpha}}^{\gamma^{\prime}}\left\|B^{H}\right\|_{\mathcal{C}^{\alpha}_{T}}(t-s)^{(1+\gamma^{\prime})\alpha}$
(58)

for $Z\in\\{X,Y\\}$.

From the estimate (58) and convergence of the $\sigma_{n}$ in
$\mathcal{C}^{\gamma^{\prime}}(\mathbb{R}^{d_{1}};\mathcal{M}(d_{1},d_{2}))$,
it follows that

$\displaystyle\lim_{n\to\infty}\int_{0}^{\cdot}\sigma_{n}(Z_{r})\,\mathrm{d}B^{H}_{r}=\int_{0}^{\cdot}\sigma(Z_{r})\,\mathrm{d}B^{H}_{r}$

uniformly in $[0,T]$. As the computation (5) shows, we have

$\displaystyle\int_{0}^{t}\left(\sigma_{n}(X_{r})-\sigma_{n}(Y_{r})\right)\,\mathrm{d}B^{H}_{r}=\sum_{k}\int_{0}^{t}(X^{k}_{r}-Y^{k}_{r})\,\mathrm{d}V^{k}_{n}(r),$

where

$\displaystyle
V^{k}_{n}(r)\vcentcolon=\int_{0}^{1}\int_{0}^{r}\partial_{k}\sigma_{n}(\theta
X_{u}+(1-\theta)Y_{u})\,\mathrm{d}B^{H}_{u}\,\mathrm{d}\theta.$

Hence, we obtain

$\displaystyle
X_{t}-Y_{t}=\lim_{n\to\infty}\sum_{k}\int_{0}^{t}(X^{k}_{r}-Y^{k}_{r})\,\mathrm{d}V^{k}_{n}(r).$
(59)

Now our task is to prove the convergence of $V_{n}$. By Lem. 3.9, for
$(s,t)\in[0,T]^{2}_{\leq}$ we have

$\displaystyle\theta
X_{s,t}+(1-\theta)Y_{s,t}=(\theta\sigma(X_{s})+(1-\theta)\sigma(Y_{s}))B_{s,t}+R^{\theta}_{s,t}$

with

$\displaystyle\lVert
R_{s,t}^{\theta}\rVert_{L^{p}_{\omega}}\lesssim_{p,\gamma,x,\sigma}(t-s)^{(1+\gamma)H}.$

One technical problem is that the Gubinelli derivative
$\theta\sigma(X)+(1-\theta)\sigma(Y)$ might be degenerate, that is, the matrix

$\displaystyle\left[\theta\sigma(X)+(1-\theta)\sigma(Y)\right]\left[\theta\sigma(X)+(1-\theta)\sigma(Y)\right]^{\top}$

could have an eigenvalue $0$. (In the end, we will show that $X=Y$, hence it
must be non-degenerate; but a priori we do not know this.) Because of this
technical problem, we proceed as follows.

Step 1. To clarify our argument, we first assume that

$\displaystyle\sup_{x,\,y\in\mathbb{R}^{d}}\lvert\sigma(x)-\sigma(y)\rvert\leq\frac{l^{-2}}{4}.$
(60)

This condition implies that for all $x,y\in\mathbb{R}^{d_{1}}$ and
$\theta\in(0,1)$ we have

$\displaystyle\left[\theta\sigma(x)+(1-\theta)\sigma(y)\right]$
$\displaystyle\left[\theta\sigma(x)+(1-\theta)\sigma(y)\right]^{\top}$
$\displaystyle=[\sigma(x)+(1-\theta)(\sigma(y)-\sigma(x))][\sigma(x)+(1-\theta)(\sigma(y)-\sigma(x))]^{\top}$
$\displaystyle\geq
l^{-1}-\frac{l^{-1}}{2}-\frac{l^{-2}}{4}\geq\frac{l^{-1}}{4}.$

Therefore, in combination of Lem. 3.9, we see that

$\displaystyle(\theta
X+(1-\theta)Y,\theta\sigma(X)+(1-\theta)\sigma(Y))\in\mathfrak{D}^{\gamma,l/4}_{B^{H}}.$

Prop. 3.7 and standard embedding of Hölder–Besov spaces, yields

$\Big{\lVert}\int_{s}^{t}\partial_{k}(\sigma_{n}-\sigma_{m})(\theta
X_{r}^{1}+(1-\theta)X_{r}^{2})\,\mathrm{d}B^{H}_{r}\Big{\rVert}_{L^{p}_{\omega}}\\\
\lesssim_{p,\gamma^{\prime},x,\sigma}\lVert\sigma_{n}-\sigma_{m}\rVert_{\mathcal{C}_{x}^{\gamma^{\prime}}}(t-s)^{\gamma^{\prime}H}.$

By Kolmogorov’s continuity theorem, we see that there exists a process $V^{k}$
such that for any $\beta<\gamma H$,

$\displaystyle\lim_{n\to\infty}\lVert\lVert
V^{k}-V^{k}_{n}\rVert_{\mathcal{C}^{\beta}_{T}}\rVert_{L^{p}_{\omega}}=0.$

Since $\gamma^{\prime}H>1/2$, we may take $\beta>1/2$. Therefore, recalling
(59) we observe

$\displaystyle
X_{t}-Y_{t}=\sum_{k}\int_{0}^{t}(X^{k}_{r}-Y^{k}_{r})\,\mathrm{d}V^{k}_{r}\quad\text{a.s.}$

As this shows that $X-Y$ solves the linear Young differential equation

$\displaystyle\mathrm{d}x_{t}=\sum_{k}x^{k}_{t}\,\mathrm{d}V^{k}_{t},\quad
x_{0}=0,$

we see that $X-Y=0$ a.s.

Step 2. Now we do not assume (60). The new ingredient is a stopping time
argument. We choose $\varepsilon>0$ so that

$\displaystyle\sup_{x,y:\lvert
x-y\rvert\leq\varepsilon}\lvert\sigma(x)-\sigma(y)\rvert\leq\frac{l^{-2}}{5},$

and we set $T^{(0)}\vcentcolon=0$ and inductively

$\displaystyle T^{(i)}\vcentcolon=\inf\\{\,t\geq T^{(i-1)}:\lvert
X_{t}-X_{T^{(i-1)}}\rvert\geq\varepsilon/2\text{ or }\lvert
Y_{t}-Y_{T^{(i-1)}}\rvert\geq\varepsilon/2\,\\}.$

Using the notation $\llbracket X\rrbracket_{\mathcal{C}^{\alpha}_{T}}$ from
(13), if $T^{(i)}\leq T$,

$\frac{\varepsilon}{2}=\max\left\\{\lvert
X_{T^{(i)}}-X_{T^{(i-1)}}\rvert,\lvert
Y_{T^{(i)}}-Y_{T^{(i-1)}}\rvert\right\\}\leq\max\left\\{\llbracket
X\rrbracket_{\mathcal{C}_{T}^{\alpha}},\llbracket
Y\rrbracket_{\mathcal{C}_{T}^{\alpha}}\right\\}(T^{(i)}-T^{(i-1)})^{\alpha}.$

Hence, the a priori estimate (56) implies that

$\displaystyle
T^{(i)}-T^{(i-1)}\gtrsim_{\alpha,\gamma,T}\varepsilon^{-\frac{1}{\alpha}}(1+\lVert\sigma\rVert_{\mathcal{C}^{\gamma}_{x}}\|B^{H}\|_{\mathcal{C}^{\alpha}_{T}})^{\frac{1}{\alpha^{2}}}$
(61)

uniformly over $i$, as long as $T^{(i)}\leq T$.

To see that $X=Y$ a.s. up to time $T^{(1)}$, let $\sigma^{(1)}$ be a
$\gamma$-Hölder map such that $\sigma^{(1)}=\sigma$ in an
$\varepsilon$-neighborhood of the initial condition $x$ and such that

$\displaystyle\sup_{x,y\in\mathbb{R}^{d_{1}}}\lvert\sigma^{(1)}(x)-\sigma^{(1)}(y)\rvert\leq\frac{l^{-2}}{4}.$

We set

$\displaystyle
X^{(1)}_{t}\vcentcolon=x+\int_{0}^{t}\sigma^{(1)}(X_{r})\,\mathrm{d}r,\quad
Y^{(1)}_{t}\vcentcolon=x+\int_{0}^{t}\sigma^{(1)}(Y_{r})\,\mathrm{d}r$

and

$\displaystyle
V^{(1),k}_{n}(t)\vcentcolon=\int_{0}^{t}\int_{0}^{1}\partial_{k}\sigma_{n}(\theta
X^{(1)}_{r}+(1-\theta)Y^{(1)}_{r})\,\mathrm{d}\theta\,\mathrm{d}B^{H}_{r}.$

Up to time $T^{(1)}$, we have

$\displaystyle
X_{t}-Y_{t}=\lim_{n\to\infty}\sum_{k}\int_{0}^{t}(X^{k}_{r}-Y^{k}_{r})\,\mathrm{d}V^{(1),k}_{n}(r).$

Due to our choice of $\sigma^{(1)}$, the argument of Step 1 shows that
$V^{(1),k}_{n}$ converges to some $V^{(1),k}$ in $\mathcal{C}_{T}^{\beta}$,
and that $X=Y$ up to time $T^{(1)}$.

Step 3. Obviously we want to repeat the operation of Step 2, but now there is
a small problem that $X_{T^{(1)}}$ is random. For this sake, let
$(x_{m})_{m\in\mathbb{N}}$ be a countable dense set of $\mathbb{R}^{d_{1}}$,
and let $\sigma^{m}$ be a $\gamma$-Hölder map such that $\sigma^{m}=\sigma$ in
an $\varepsilon$-neighborhood of $x_{m}$ and such that

$\displaystyle\sup_{x,y\in\mathbb{R}^{d_{1}}}\lvert\sigma^{m}(x)-\sigma^{m}(y)\rvert\leq\frac{l^{-2}}{4}.$
(62)

For each $m$ and $Z\in\\{X,Y\\}$, we define the approximation,

$\Sigma^{Z,m}(t)\vcentcolon=\begin{cases}\sigma(Z_{t})\mathbf{1}_{\\{t\leq
T^{(1)}\\}}+\sigma^{m}(Z_{t})\mathbf{1}_{\\{t>T^{(1)}\\}},&\text{if }\lvert
x_{m}-X_{T^{(1)}}\rvert=\lvert x_{m}-Y_{T^{(1)}}\rvert\leq\varepsilon/2,\\\
\sigma(Z_{\min\\{t,T^{(1)}\\}}),&\text{otherwise },\end{cases}$

along with the cut-off equations,

$\displaystyle
X^{(2,m)}_{t}\vcentcolon=x+\int_{0}^{t}\Sigma^{X,m}(r)\,\mathrm{d}B^{H}_{r},\qquad
Y^{(2,m)}_{t}\vcentcolon=x+\int_{0}^{t}\Sigma^{Y,m}(r)\,\mathrm{d}B^{H}_{r}$

and approximate linearisation,

$\displaystyle
V^{(2,m),k}_{n}(t)\vcentcolon=\int_{0}^{t}\int_{0}^{1}\partial_{k}\sigma_{n}(\theta
X^{(2,m)}_{r}+(1-\theta)Y^{(2,m)}_{r})\,\mathrm{d}\theta\,\mathrm{d}B^{H}_{r}.$

Notice that $X^{(2,m)}$, $Y^{(2,m)}$ are both adapted. We claim that, for
$Z\in\\{X,Y\\}$,

$\displaystyle\lvert
Z_{s,t}^{(2,m)}-\Sigma^{Z,m}(s)B^{H}_{s,t}\rvert\lesssim(\lVert\sigma\rVert_{\mathcal{C}_{x}^{\gamma}}+\lVert\sigma^{m}\rVert_{\mathcal{C}_{x}^{\gamma}})\lVert
Z\rVert_{\mathcal{C}_{T}^{\alpha}}^{\gamma}\left\|B^{H}\right\|_{\mathcal{C}_{T}^{\alpha}}(t-s)^{(1+\gamma)\alpha}.$
(63)

Indeed, by the remainder estimate of Young’s integral (17), the estimate (63)
is obvious if $t\leq T^{(1)}$ or if $s\geq T^{(1)}$. Suppose that
$s<T^{(1)}<t$. If $\lvert x_{m}-X_{T^{(1)}}\rvert\leq\varepsilon/2$, then

$\displaystyle\lvert
Z_{s,T^{(1)}}^{(2,m)}-\sigma(Z_{s})B^{H}_{s,T^{(1)}}\rvert$
$\displaystyle\lesssim\lVert\sigma\rVert_{\mathcal{C}_{x}^{\gamma}}\lVert
Z\rVert_{\mathcal{C}_{T}^{\alpha}}^{\gamma}\left\|B^{H}\right\|_{\mathcal{C}_{T}^{\alpha}}(t-s)^{(1+\gamma)\alpha},$
$\displaystyle\lvert
Z_{T^{(1)},t}^{(2,m)}-\sigma^{m}(Z_{T^{(1)}})B^{H}_{T^{(1)},t}\rvert$
$\displaystyle\lesssim\lVert\sigma^{m}\rVert_{\mathcal{C}_{x}^{\gamma}}\lVert
Z\rVert_{\mathcal{C}_{T}^{\alpha}}^{\gamma}\left\|B^{H}\right\|_{\mathcal{C}_{T}^{\alpha}}(t-s)^{(1+\gamma)\alpha}.$

Since $\lvert x_{m}-X_{T^{(1)}}\rvert=\lvert
x_{m}-Y_{T^{(1)}}\rvert\leq\varepsilon/2$, we have

$\displaystyle\lvert\sigma^{m}(Z_{T^{(1)}})-\sigma(Z_{s})\rvert=\lvert\sigma(Z_{T^{(1)}})-\sigma(Z_{s})\rvert\leq\lVert\sigma\rVert_{\mathcal{C}_{x}^{\gamma}}\lVert
Z\rVert_{\mathcal{C}_{T}^{\alpha}}^{\gamma}(T^{(1)}-s)^{\gamma\alpha},$

and the estimate (63) follows. The case where $\lvert
x_{m}-X_{T^{(1)}}\rvert>\varepsilon/2$ is similar.

Now we check that the Gubinelli derivative
$\Sigma^{\theta,m}\vcentcolon=\theta\Sigma^{X,m}+(1-\theta)\Sigma^{Y,m}$ is
non-degenerate. Indeed, for $t\leq T^{(1)}$ we have

$\displaystyle\Sigma^{\theta,m}(t)\Sigma^{\theta,m}(t)^{\top}=\sigma(X_{t})\sigma(X_{t})^{\top}\geq
l^{-1},$

and for $t>T^{(1)}$ the condition (62) implies that
$\Sigma^{\theta,m}(t)\Sigma^{\theta,m}(t)^{\top}\geq l^{-1}/4$. Therefore,
Prop. 3.7 and the Kolmogorov continuity theorem show that $V^{(2,m)}_{n}$
converges to some $V^{(2,m)}$ in $L^{p}_{\omega}$. By the diagonalization
argument, we may suppose that almost surely for every $m\in\mathbb{N}$ and
$\delta\in(0,\gamma H)$ we have

$\displaystyle\lim_{n\to\infty}\lVert
V^{(2,m)}-V^{(2,m)}_{n}\rVert_{\mathcal{C}_{T}^{\gamma H-\delta}}=0.$

Since $(x_{m})$ is dense, we can find a random $m=m(\omega)$ so that $\lvert
x_{m}-X_{T^{(1)}}\rvert<\varepsilon/2$. We then have

$\displaystyle
X_{t}-Y_{t}=\sum_{k}\int_{0}^{t}(X^{k}_{r}-Y^{k}_{r})\,\mathrm{d}V^{(2,m),k}(r)$

up to $t\leq T^{(2)}$, hence $X=Y$ on $[0,T^{(2)}]$. It is now clear that this
algorithm can be continued, and at some point we must have $T^{(i)}\geq T$ due
to (61). ∎

##  4 Rough Case: $H\in(1/3,1/2]$

In this section we prove Thm. 1.1 for the regime $H\in(1/3,1/2]$. As discussed
at the opening of Sec. 2.3, for Hurst parameters below $1/2$ we can no longer
appeal to Young integration theory to make sense of
$\int_{0}^{t}\sigma(X_{s})\,\mathrm{d}B^{H}_{s}$ in (1). Here we recall the
fact that for $H\in(1/3,1/2]$ we can $\mathbb{P}$-a.s. canonically lift
$B^{H}$ to a geometric rough path $(B^{H},\mathbb{B}^{H})$ (see Lem. 2.10) as
well as the definition of $\mathscr{D}^{\beta}_{B^{H}}$; the space of
$\beta$-regular, rough paths, controlled by $B^{H}$ (see Def. 2.2). These
preparations are enough to give a pathwise definition of solutions to rough
differential equations (RDEs).

######  Definition 4.1 (Solutions to RDEs).

Given $T>0$, $d_{1},\,d_{2}\in\mathbb{N}$, $H\in(1/3,1/2]$,
$(B^{H}(\omega),\mathbb{B}^{H}(\omega))$ an almost sure realisation of the
canonical, geometric rough path lift for the fBm with Hurst parameter $H$ (see
Lem. 2.10) and a map $\sigma:\mathbb{R}^{d_{1}}\to\mathcal{M}(d_{1},d_{2})$,
we say that a path $(y,y^{\prime})$ controlled by
$(B^{H}(\omega),\,\mathbb{B}^{H}(\omega))$ is a _solution to the rough
differential equation_

$\mathrm{d}y_{t}=\sigma(y_{t})\,\mathrm{d}B_{t}^{H}(\omega),\quad y_{0}=x,$
(64)

on $[0,T]$, if the path
$(\sigma(y),\nabla\sigma(y)y^{\prime})\in\mathscr{D}^{\gamma\alpha}_{B^{H}(\omega)}$
for some $\alpha\in(1/3,H)$ and $\gamma>\frac{1-\alpha}{\alpha}$ and if for
every $t\in[0,T]$ one has

$\displaystyle
y_{t}=x+\int_{0}^{t}\sigma(y_{r})\,\mathrm{d}B_{r}^{H}(\omega),$

where the integral is understood as a rough integral as described in Sec. 2.3.

Note that the assumption
$(\sigma(y),\nabla\sigma(y)y^{\prime})\in\mathscr{D}^{\gamma\alpha}_{B^{H}(\omega)}$
for $\gamma>\frac{1-\alpha}{\alpha}>1$ is the minimum required to make sense
of the integral in (64) as a rough integral. If $(y,y^{\prime})$ is a solution
to (64), the remainder estimate of rough integrals (20) shows that the map
$t\mapsto\int_{0}^{t}\sigma(y_{r})\,\mathrm{d}B^{H}_{r}(\omega)$ is again
controlled by $B^{H}(\omega)$ with Gubinelli derivative $\sigma(y)$. Due to
this observation, it in fact holds that
$(y,\sigma(y))\in\mathscr{D}^{2\alpha}_{B^{H}(\omega)}$, hence appealing to
[FH20, Prop. 6.4] and _true roughness_ of $B^{H}$ one sees that for all
solutions in the sense of Def. 4.1, it holds that $y^{\prime}=\sigma(y)$.

As in the Young case, for an fBm on a given probability space
$(\Omega,\mathcal{F},\mathbb{P})$, Def. 4.1 is an $\omega$-wise definition. We
therefore give the following probabilistic definitions, which are analogues of
Def. 3.2, 3.4 and 3.5 for solutions to

$\displaystyle\mathrm{d}X_{t}=\sigma(X_{t})\,\mathrm{d}B^{H}_{t},\quad
X_{0}=x.$ (65)

######  Definition 4.2 (Weak-Rough Solutions).

Given $T>0$, say that a tuple

$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\in[0,T]},\mathbb{P},B^{H},X,X^{\prime}),$

of a filtered probability space, an fBm of Hurst parameter $H$ and an
$(\mathcal{F}_{t})$-adapted, controlled rough path, $(X,X^{\prime})$ is a
weak-rough solution to (65), if $B^{H}$ is an $(\mathcal{F}_{t})_{t\geq 0}$
fBm (in the sense of Def. 2.9) and for $\mathbb{P}$-a.e $\omega\in\Omega$, the
process $(X(\omega),X^{\prime}(\omega))$ is a rough solution to (65)in the
sense of Def. 4.1.

We say that weak uniqueness holds for (65) in the rough sense, if for any two
weak solutions $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq
0},\mathbb{P},B^{H},X,X^{\prime})$ and
$(\tilde{\Omega},\tilde{\mathcal{F}},(\tilde{\mathcal{F}_{t}})_{t\geq
0},\tilde{\mathbb{P}},\tilde{B}^{H},\tilde{X},\tilde{X}^{\prime})$ as above,
the processes $(X,X^{\prime}),\,(\tilde{X},\tilde{X}^{\prime})$ are equal in
law, i.e. one has

$(X,X^{\prime})_{\\#}\mathbb{P}=(\tilde{X},\tilde{X}^{\prime})_{\\#}\tilde{\mathbb{P}}\quad\text{as
elements of}\quad\mathcal{P}\left(C([0,T];\mathbb{R}^{d_{1}})\times
C\left([0,T];\mathcal{M}(d_{1},d_{2})\right)\right)$

######  Remark 4.3.

As mentioned in the Young case (Rem. 3.3) a standard compactness argument (a
la [MP22, Appendix B]) allows one to construct weak-rough solutions in the
case
$\sigma\in\mathcal{C}^{\gamma}(\mathbb{R}^{d_{1}};\mathcal{M}(d_{1},d_{2}))$
for $\gamma>\frac{1-H}{H}$. Note that in contrast to Def. 4.1 this is
explicitly an assumption on the coefficient $\sigma$ which in turn ensures the
required regularity of the controlled path
$(\sigma(X),\nabla\sigma(X),X^{\prime})$ for some $\alpha\in(1/3,H)$
sufficiently close to $H$.

In the regime $H\in(1/3,1/2)$ our main result (Thm. 1.1) obtains pathwise
uniqueness for $\sigma$ that is $\gamma$-Hölder continuous for
$\gamma>\frac{1-H}{H}$ and uniformly elliptic so that we obtain pathwise
uniqueness for these weak solutions (under the assumption of uniform
ellipticity) and hence strong existence due to Prop. 1.2.

######  Definition 4.4 (Strong-Rough Solutions).

Given $T>0$, a probability space, $(\Omega,\mathcal{F},\mathbb{P})$ carrying
an fBm $B^{H}$, we say that $(X,X^{\prime})$ is a strong-rough solution to
(65) on $[0,T]$ if for $\mathbb{P}$-a.e. $\omega\in\Omega$, the process
$(X(\omega),X^{\prime}(\omega)$ is a rough solution to (65) in the sense of
Def. 4.1, on $[0,T]$ and $X$ is adapted to the natural filtration generated by
$B^{H}$.

######  Definition 4.5 (Pathwise Uniqueness of Rough Solutions).

We say that _pathwise-uniqueness_ holds for (65) in the rough sense, if for
any two, weak-rough solutions, $(X,X^{\prime})$ and
$(\tilde{X},\tilde{X}^{\prime})$ (in the sense of Def. 4.2), on the same
probability space $(\Omega,\mathcal{F},\mathbb{P})$, it holds that
$(X(\omega),X^{\prime}(\omega))=(\tilde{X}(\omega),\tilde{X}^{\prime}(\omega))$
for $\mathbb{P}$-a.e. $\omega\in\Omega$.

The goal of this section is to prove pathwise uniqueness for (65) under the
assumption that $\sigma$ is uniformly elliptic and $\gamma>(1-H)/H$. Namely,
we prove Thm. 1.1 in the rough case, $H\in(1/3,1/2]$, which is restated as
Thm. 4.10 below.

###  4.1 Rough Integral with Irregular Integrand

As in Sec. 3, the key to prove pathwise uniqueness is to obtain a sharp
stochastic estimate on stochastic integral of the form (36). Differently from
the Young case, however, we also require suitable estimates on iterated
integrals of the form (11). The iterated integrals are treated in Sec. 4.2
below.

To treat the first order integrals we apply the stochastic sewing lemma (Lem.
2.4). We will need the following technical estimate, whose proof will be given
in App. B.

######  Lemma 4.6.

Let $T>0$, $H\in(1/3,1/2]$ and $B^{H}$ be an fBm adapted to a given filtration
$(\mathcal{F}_{t})_{t\in[0,T]}$. We also fix a $\tau\in[0,T]$, define
$\tilde{B}^{H,\tau}$ by (29) and let $\xi$ be an
$\mathcal{F}_{\tau}$-measurable random variable valued in a measurable space
$E$. Then, given a map $F\colon E\times\mathbb{R}^{d_{2}}\to\mathbb{R}$ such
that $\sup_{x}\lVert F(x,\cdot)\rVert_{C^{2}_{y}}<\infty$, one has, for any
$\gamma\in(\frac{1}{2H}-1,1)$ and $(s,t)\in[\tau,T]_{\leq}^{2}$ that

$\Big{\|}\int_{s}^{t}F(\xi,B^{H}_{\tau,r})\,\mathrm{d}\tilde{B}^{H,\tau}_{r}-\frac{F(\xi,B_{\tau,s}^{H})+F(\xi,B_{\tau,t}^{H})}{2}\tilde{B}^{H,\tau}_{s,t}\Big{\|}_{L^{p}_{\omega}}\\\
\lesssim_{p,\gamma}\lVert\lVert
F(\xi,\cdot)\rVert_{\mathcal{C}^{\gamma}_{y}}\rVert_{L^{2p}_{\omega}}(t-s)^{(1+\gamma)H}.$

The integral with respect to $\tilde{B}^{H,\tau}$ is understood as rough
integral with respect to the geometric rough path constructed in Lem. 2.12.

###### Proof.

See App. B.3. ∎

It is worth mentioning that the proof of Lem. 4.6 is based on the fully
shifted stochastic sewing lemma, in the spirit of [MP22, Prop. 3.5]. The rest
of this subsection is devoted to proving the following proposition, which is
the analogue of Prop. 3.7. We recall, again, the definition of the
stochastically controlled, path space, $\mathfrak{D}^{1,l}_{B^{H}}$, and
associated norm,
$\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}\,\cdot\,\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{p,\beta}$,
from Def. 3.6.

######  Proposition 4.7.

Let $T>0$, $d_{1},\,d_{2}\in\mathbb{N}$, $B^{H}$ be an
$\mathbb{R}^{d_{2}}$-fBm with Hurst parameter $H\in(1/3,1/2]$, adapted to a
given filtration $(\mathcal{F}_{t})_{t\in[0,T]}$, $p\in[2,\infty)$,
$l\in[1,\infty)$, $(X,X^{\prime})\in\mathfrak{D}^{1,l}_{B^{H}}$, $f\in
C^{3}(\mathbb{R}^{d_{1}},\mathbb{R})$ and
$\tilde{\gamma}\in\left(\frac{1-H}{H}-1,1\right)$. Then, for
$(s,t)\in[0,T]_{\leq}^{2}$ it holds that

$\Big{\lVert}\int_{s}^{t}f(X_{r})\,\mathrm{d}B_{r}^{H}-f(X_{s})B_{s,t}^{H}\Big{\rVert}_{L^{p}_{\omega}}\lesssim_{p,l,\tilde{\gamma},T}\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}(1+\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{2p,1})(t-s)^{(1+\tilde{\gamma})H}.$

###### Proof.

For concision, throughout the proof we simply write $B\vcentcolon=B^{H}$. Let
us set

$\displaystyle
A_{s,t}\vcentcolon=\int_{s}^{t}f(X_{s}+X_{s}^{\prime}B_{s,r})\,\mathrm{d}B_{r}.$
(66)

The integral is understood as rough integral: the path $r\mapsto
f(X_{s}+X_{s}^{\prime}B_{s,r})$ is controlled by $B$ with Gubinelli derivative
$\nabla f(X_{s}+X_{s}^{\prime}B_{s,r})X^{\prime}_{s}$. In particular, the
estimate (20) yields

$\displaystyle\lvert A_{s,t}-f(X_{s})B_{s,t}-\nabla
f(X_{s})X^{\prime}_{s}\mathbb{B}_{s,t}\rvert\lesssim(t-s)^{3\alpha}$

for $\alpha\in(1/3,H)$. Hence, the uniqueness part of the sewing lemma (see
Rem. 2.7) implies that

$\displaystyle\int_{s}^{t}f(X_{r})\,\mathrm{d}B_{r}=\lim_{\lvert\pi\rvert\to
0}\sum_{[s^{\prime},t^{\prime}]\in\pi}A_{s^{\prime},t^{\prime}},$

where $\pi$ is a partition of $[s,t]$. As in Prop. 3.7, we apply the
stochastic sewing to $A_{s,t}$. This time, Lê’s original version [Lê20] is
sufficient (that is $\alpha=0$ and $v=s$ in Lem. 2.4). Precisely, we prove the
estimates

$\displaystyle\lVert\delta
A_{s,u,t}\rVert_{L^{p}_{\omega}}\leq\Gamma_{1}(t-s)^{\frac{1}{2}+\varepsilon_{1}},\quad\lVert\mathbb{E}[\delta
A_{s,u,t}|\mathcal{F}_{s}]\rVert_{L^{p}_{\omega}}\leq\Gamma_{2}(t-s)^{1+\varepsilon_{2}}$

for $s<u<t$ with appropriate $\varepsilon_{1},\varepsilon_{2}>0$ and
proportional constants $\Gamma_{1},\Gamma_{2}$. We split the rest of the proof
into three steps.

Step 1. We estimate $\lVert\delta A_{s,u,t}\rVert_{L^{p}_{\omega}}$. Writing
$Y\vcentcolon=Y^{H,u}$ and $\tilde{B}\vcentcolon=\tilde{B}^{H,u}$ as in (28)
and (29) respectively, we observe

$\displaystyle\delta A_{s,u,t}=$
$\displaystyle\int_{u}^{t}\left(f(X_{s}+X_{s}^{\prime}B_{s,r})-f(X_{u}+X_{u}^{\prime}B_{u,r})\right)\,\mathrm{d}B_{r}$
$\displaystyle=$
$\displaystyle\int_{u}^{t}\left(f(X_{s}+X_{s}^{\prime}B_{s,r})-f(X_{u}+X_{u}^{\prime}B_{u,r})\right)\dot{Y_{r}}\,\mathrm{d}r$
(67)
$\displaystyle+\int_{u}^{t}\left(f(X_{s}+X_{s}^{\prime}B_{s,r})-f(X_{u}+X_{u}^{\prime}B_{u,r})\right)\,\mathrm{d}\tilde{B}_{r},$
(68)

The integral (67), by the triangle inequality, is bounded by

$\displaystyle\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}\int_{u}^{t}\lvert
X_{s}+X_{s}^{\prime}B_{s,r}-(X_{u}+X_{u}^{\prime}B_{u,r})\rvert^{\tilde{\gamma}}\lvert\dot{Y}_{r}\rvert\,\mathrm{d}r.$

Defining the remainder $R$ as in Def. 3.6, we have

$\displaystyle
X_{s}+X_{s}^{\prime}B_{s,r}-(X_{u}+X_{u}^{\prime}B_{u,r})=-R_{s,u}-X^{\prime}_{s,u}B_{u,r},$

and since, due to Lem. 2.11 we have
$\lVert\dot{Y}_{r}\rVert_{L^{2p}_{\omega}}\lesssim_{p}(r-u)^{H-1}$, it follows
that

$\Bigg{\|}\int_{u}^{t}\left(f(X_{s}+X_{s}^{\prime}B_{s,r})-f(X_{u}+X_{u}^{\prime}B_{s,r})\right)\dot{Y_{r}}\,\mathrm{d}r\,\Bigg{\|}_{L^{p}_{\omega}}\\\
\leq\lVert f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}\int_{u}^{t}\lVert\lvert
R_{s,u}+X^{\prime}_{s,u}B_{u,r}\rvert^{\tilde{\gamma}}\rVert_{L^{2p}_{\omega}}\lVert\dot{Y}_{r}\rVert_{L^{2p}_{\omega}}\,\mathrm{d}r\lesssim_{p}\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{2p,1}^{\tilde{\gamma}}(t-s)^{(1+2\tilde{\gamma})H}.$

To estimate (68), for $v\in\\{s,u\\}$ we apply Lem. 4.6 with

$\displaystyle\tau=u,\quad\xi=(X_{v}+X_{v}^{\prime}B_{v,u},X_{v}^{\prime}),\quad
F((x,x^{\prime}),y)=f(x+x^{\prime}y)$

to obtain the estimate

$\Big{\lVert}\int_{u}^{t}f(X_{v}+X_{v}^{\prime}B_{v,r})\,\mathrm{d}\tilde{B}_{r}-\frac{f(X_{v}+X^{\prime}_{v}B_{v,u})+f(X_{v}+X^{\prime}_{v}B_{v,t})}{2}\tilde{B}_{u,t}\Big{\rVert}_{L^{p}_{\omega}}\\\
\lesssim_{p,l,\tilde{\gamma}}\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}(t-s)^{(1+\tilde{\gamma})H}.$

Since

$\Big{\lvert}\frac{f(X_{s}+X^{\prime}_{s}B_{s,u})+f(X_{s}+X^{\prime}_{s}B_{s,t})}{2}-\frac{f(X_{u})+f(X_{u}+X^{\prime}_{u}B_{u,t})}{2}\Big{\rvert}\\\
\lesssim\lVert f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}\big{(}\lvert
X_{s,u}-X_{s}^{\prime}B_{s,u}\rvert^{\tilde{\gamma}}+\lvert
X^{\prime}_{s,u}B_{u,t}\rvert^{\tilde{\gamma}}\big{)},$

we see that the $L^{p}_{\omega}$-norm of the integral (68) is bounded by (up
to constant)

$\displaystyle\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}(t-s)^{(1+\tilde{\gamma})H}+\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{2p,1}^{\tilde{\gamma}}(t-s)^{(1+2\tilde{\gamma})H}.$

Hence,

$\displaystyle\lVert\delta
A_{s,u,t}\rVert_{L^{p}_{\omega}}\lesssim_{p,l,\tilde{\gamma}}\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}(t-s)^{(1+\tilde{\gamma})H}+\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{2p,1}^{\tilde{\gamma}}(t-s)^{(1+2\tilde{\gamma})H}.$
(69)

Step 2. Next we estimate $\mathbb{E}[\delta A_{s,u,t}|\mathcal{F}_{s}]$, which
is essentially done in Prop. 3.7. We define $Y$ and $\tilde{B}$ as in Step 1,
and we write
$\mathbb{E}_{u}[\,\cdot\,]\vcentcolon=\mathbb{E}[\,\cdot\,|\mathcal{F}_{u}]$.
We have

$\displaystyle\mathbb{E}_{u}\left[\delta A_{s,u,t}\right]=$
$\displaystyle\mathbb{E}_{u}\left[\int_{u}^{t}\left(f(X_{s}+X_{s}^{\prime}B_{s,r})-f(X_{u}+X_{u}^{\prime}B_{s,r})\right)\dot{Y_{r}}\,\mathrm{d}r\right]$
$\displaystyle+\mathbb{E}_{u}\left[\int_{u}^{t}\left(f(X_{s}+X_{s}^{\prime}B_{s,r})-f(X_{u}+X_{u}^{\prime}B_{s,r})\right)\,\mathrm{d}\tilde{B}_{r}\right].$

For $v\in\\{s,u\\}$, we observe

$\mathbb{E}_{u}\left[\int_{u}^{t}f(X_{v}+X_{v}^{\prime}B_{v,r})\dot{Y_{r}}\,\mathrm{d}r\right]\\\
=\int_{u}^{t}\mathcal{P}_{\rho^{2}(u,r)X^{\prime}_{v}(X^{\prime}_{v})^{\top}}f(X_{v}+X^{\prime}_{v}(B_{v,u}+Y_{u,r}))\dot{Y}_{r}\,\mathrm{d}r$

with $\rho^{2}(u,r)\vcentcolon=\rho^{2}_{H}(u,r)$ defined by (30). As in the
proof of Prop. 3.7 (especially (45)), Lem. 2.13 implies that

$\mathbb{E}_{u}\left[\int_{u}^{t}f(X_{v}+X_{v}^{\prime}B_{v,r})\,\mathrm{d}\tilde{B}_{r}\right]\\\
=\frac{1}{2}X^{\prime}_{v}\int_{u}^{t}\nabla\mathcal{P}_{\rho^{2}(u,r)X^{\prime}_{v}(X^{\prime}_{v})^{\top}}f(X_{v}+X^{\prime}_{v}(B_{v,u}+Y_{u,r}))\,\rho^{2}(u,\mathrm{d}r).$

Hence, $\mathbb{E}_{u}\left[\delta A_{s,u,t}\right]=J_{1}+J_{2}$, where

$J_{1}\vcentcolon=\int_{u}^{t}\big{(}\mathcal{P}_{\rho^{2}(u,r)X^{\prime}_{s}(X^{\prime}_{s})^{\top}}f(X_{s}+X^{\prime}_{s}(B_{s,u}+Y_{u,r}))\\\
-\mathcal{P}_{\rho^{2}(u,r)X^{\prime}_{u}(X^{\prime}_{u})^{\top}}f(X_{u}+X^{\prime}_{u}Y_{u,r})\big{)}\dot{Y}_{r}\,\mathrm{d}r,$
$J_{2}\vcentcolon=\frac{1}{2}\int_{u}^{t}\big{(}X^{\prime}_{s}\nabla\mathcal{P}_{\rho^{2}(u,r)X^{\prime}_{s}(X^{\prime}_{s})^{\top}}f(X_{s}+X^{\prime}_{s}(B_{s,u}+Y_{u,r}))\\\
-X^{\prime}_{u}\nabla\mathcal{P}_{\rho^{2}(u,r)X^{\prime}_{u}(X^{\prime}_{u})^{\top}}f(X_{u}+X^{\prime}_{u}Y_{u,r})\big{)}\,\rho^{2}(u,\mathrm{d}r).$

Then the rest of the argument is identical to the proof of Prop. 3.7. ($J_{1}$
corresponds to $I_{3,1}$ and $J_{2}$ to $I_{3,2}$.) In particular, we obtain

$\displaystyle\lVert\mathbb{E}_{s}[\delta
A_{s,u,t}]\rVert_{L^{p}_{\omega}}\lesssim_{p,l,\tilde{\gamma},T}\lVert
f\rVert_{\mathcal{C}^{\tilde{\gamma}}}\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{2p,1}(t-s)^{(2+\tilde{\gamma})H}.$
(70)

Step 3. Thanks to the estimates (69) and (70), we are in the setting of Lem.
2.4 with $\alpha=0$ and $v=s$. Therefore, the remainder estimate of Lem. 2.4
yields

$\Big{\lVert}\int_{s}^{t}f(X_{r})\,\mathrm{d}B_{r}-A_{s,t}\Big{\rVert}_{L^{p}_{\omega}}\lesssim_{p,l,\tilde{\gamma},T}\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}(1+\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{2p,1})(t-s)^{(1+\tilde{\gamma})H}.$
(71)

On the other hand, we have

$A_{s,t}-\frac{f(X_{s})+f(X_{s}+X_{s}^{\prime}B_{s,t})}{2}B_{s,t}\\\
=\int_{s}^{t}\left(f(X_{s}+X_{s}^{\prime}B_{s,r})-\frac{f(X_{s})+f\left(X_{s}+X_{s}^{\prime}B_{s,t}\right)}{2}\right)\dot{Y}_{r}^{s}\,\mathrm{d}r\\\
+\int_{s}^{t}f(X_{s}+X_{s}^{\prime}B_{s,r})\,\mathrm{d}\tilde{B}_{r}^{s}-\frac{f(X_{s})+f(X_{s}+X_{s}^{\prime}B_{s,r})}{2}\tilde{B}_{s,t}^{s}.$

By Lem. 2.11, the first term is bounded by $\lVert
f\rVert_{\mathcal{C}^{\tilde{\gamma}}_{x}}(t-s)^{2H}$ up to constant, while by
Lem. 4.6 the second term is bounded by $\lVert
f\rVert_{\mathcal{C}^{\tilde{\gamma}}_{x}}(t-s)^{(1+\tilde{\gamma})H}$ up to
constant. Therefore,

$\displaystyle\Big{\lVert}A_{s,t}-\frac{f(X_{s})+f(X_{s}+X_{s}^{\prime}B_{s,t})}{2}B_{s,t}\Big{\rVert}_{L^{p}_{\omega}}\lesssim_{p,l,\tilde{\gamma},T}\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}(t-s)^{(1+\tilde{\gamma})H}.$ (72)

Finally, we observe

$\Big{\lVert}\frac{f(X_{s})+f(X_{s}+X_{s}^{\prime}B_{s,t})}{2}B_{s,t}-f(X_{s})B_{s,t}\Big{\rVert}_{L^{p}_{\omega}}\lesssim_{p,l,\tilde{\gamma},T}\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}(t-s)^{(1+\tilde{\gamma})H}.$ (73)

Combining (71), (72) and (73), we obtain the desired estimate. ∎

###  4.2 Iterated Integrals with Rough Integrands

Referring back to the strategy described for the rough case in Sec. 1.2, as
compared to the Young case, we also require sufficient control on iterated
integrals of the form described by (11). In this section (over the following
two propositions) we treat the case when the $(G^{k})_{k=1}^{d_{2}}$ invoked
therein is a generic path of the form

$G^{k}_{t}\coloneqq\int_{0}^{t}f(X_{r})\,\mathrm{d}B^{H}_{r},\quad
k=1,\ldots,d_{2},$

for $f$ as in Prop. 4.7. In Prop. 4.8, we treat iterated integrals of the form
$\int G^{k}\otimes\,\mathrm{d}B^{H}$ (which in turn gives the estimate on
$\int B^{H}\otimes\,\mathrm{d}G^{k}$ by integration by parts), and in Prop.
4.9 we treat iterated integrals of the form $\int
G^{k}\otimes\,\mathrm{d}G^{l}$. These two propositions as well as Prop. 4.7
will be used in Sec. 4.3 below to construct the rough path lift (11) by
proving the convergence of rough paths lifts associated to smooth
approximations of $\sigma$.

Since we already have access to Prop. 4.7, the estimates on iterated integrals
follows from the deterministic sewing lemma (Lem. 2.3). We recall again the
notations $\mathfrak{D}^{1,l}_{B^{H}}$ and
$\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}\,\cdot\,\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{p,1}$
from Def. 3.6.

######  Proposition 4.8.

Let $H\in(1/3,1/2]$, $p\in[2,\infty)$, $l\in[1,\infty)$,
$(X,X^{\prime})\in\mathfrak{D}^{1,l}_{B^{H}}$, $f\in
C^{3}(\mathbb{R}^{d_{1}})$ and
$\tilde{\gamma}\in\left(\frac{1-H}{H}-1,1\right)$. Then, for
$(s,t)\in[0,T]_{\leq}^{2}$ and $i,j\in\\{1,2,\ldots,d_{2}\\}$ we have

$\Big{\lVert}\int_{s}^{t}f(X_{r})B^{H,i}_{s,r}\,\mathrm{d}B^{H,j}_{r}-f(X_{s})\mathbb{B}^{H,i,j}_{s,t}\Big{\rVert}_{L^{p}_{\omega}}\lesssim_{p,l,\tilde{\gamma},T}\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}(1+\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{4p,1})(t-s)^{(2+\tilde{\gamma})H}.$

###### Proof.

For concision, we write $B\vcentcolon=B^{H}$ and
$\mathbb{B}\vcentcolon=\mathbb{B}^{H}$. We fix a $\tau\leq s$, and we set

$\bar{A}_{s,t}\vcentcolon=B^{i}_{\tau,s}\int_{s}^{t}f(X_{r})\,\mathrm{d}B^{j}_{r}+f(X_{s})\mathbb{B}_{s,t}^{i,j}.$

Step 1. We show that $\bar{A}_{s,t}$ is a correct approximation to the
iterated integral $\int_{s}^{t}f(X_{r})B_{\tau,r}^{i}\,\mathrm{d}B_{r}^{j}$.
For this sake, we set

$\bar{A}^{\prime}_{s,t}\vcentcolon=f(X_{s})B^{i}_{\tau,s}B^{j}_{s,t}\\\
+f(X_{s})\mathbb{B}^{i,j}_{s,t}+\sum_{k,l}B^{i}_{\tau,s}\partial_{k}f(X_{s})(X^{\prime}_{s})^{kl}\mathbb{B}_{s,t}^{lj}.$

By definition of the rough integral, we have

$\displaystyle\int_{s}^{t}f(X_{r})B^{i}_{\tau,r}\,\mathrm{d}B^{j}_{r}=\lim_{\begin{subarray}{c}\pi\text{
is a partition of }[s,t],\\\ \lvert\pi\rvert\to
0\end{subarray}}\sum_{[u,v]\in\pi}\bar{A}^{\prime}_{u,v}.$

In view of the uniqueness part of the sewing lemma (see Rem. 2.7), our goal is
to prove
$\lvert\bar{A}_{s,t}-\bar{A}^{\prime}_{s,t}\rvert\lesssim_{f,X,X^{\prime},\omega}(t-s)^{3H}$.
We compute

$\displaystyle\bar{A}_{s,t}-\bar{A}^{\prime}_{s,t}=B^{i}_{\tau,s}\Big{(}\int_{s}^{t}f(X_{r})\,\mathrm{d}B_{r}-f(X_{s})B^{j}_{s,t}-\sum_{k,l}\partial_{k}f(X_{s})(X^{\prime}_{s})^{kl}\mathbb{B}_{s,t}^{lj}\Big{)}$

By the remainder estimate of the rough integral (20), we have

$\displaystyle\Big{\lvert}\int_{s}^{t}f(X_{r})\,\mathrm{d}B_{r}-f(X_{s})B^{j}_{s,t}-\sum_{k,l}\partial_{k}f(X_{s})(X^{\prime}_{s})^{kl}\mathbb{B}_{s,t}^{lj}\Big{\rvert}\lesssim_{f,X,X^{\prime},\omega}(t-s)^{3H},$

whence $\bar{A}$ is a correct approximation.

Step 2. This time we do not need the stochastic sewing; the deterministic
sewing (Lem. 2.3) in $L^{p}_{\omega}$ suffices. We compute

$\displaystyle\delta\bar{A}_{s,u,t}=-B^{i}_{s,u}\int_{u}^{t}f(X_{r})\,\mathrm{d}B^{j}_{r}+f(X_{s})(\mathbb{B}_{s,t}^{i,j}-\mathbb{B}_{s,u}^{i,j})-f(X_{u})\mathbb{B}_{u,t}^{i,j}.$

Using Chen’s relation
$\mathbb{B}^{i,j}_{u,t}=\mathbb{B}^{i,j}_{s,t}-\mathbb{B}^{i,j}_{s,u}-B^{i}_{s,u}B^{j}_{u,t}$,
we obtain

$\displaystyle\delta
A_{s,u,t}=-B^{i}_{s,u}\Big{(}\int_{u}^{t}f(X_{r})\,\mathrm{d}B_{r}^{j}-f(X_{u})B^{j}_{u,t}\Big{)}-f(X)_{s,u}(\mathbb{B}^{i,j}_{s,t}-\mathbb{B}^{i,j}_{s,u}).$

Since

$\lVert
B^{i}_{s,u}\rVert_{L^{p}_{\omega}}\lesssim_{p}(t-s)^{H},\,\,\lVert\mathbb{B}^{i,j}_{s,t}\rVert_{L^{p}_{\omega}}\lesssim_{p}(t-s)^{2H},\,\,\lVert
f(X)_{s,u}\rVert_{L^{p}_{\omega}}\lesssim_{p,\tilde{\gamma}}\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{p\tilde{\gamma},1}^{\tilde{\gamma}}(t-s)^{\tilde{\gamma}H},$

and by Prop. 4.7

$\displaystyle\Big{\lVert}\int_{u}^{t}f(X_{r})\,\mathrm{d}B_{r}^{j}-f(X_{u})B^{j}_{u,t}\Big{\rVert}_{L^{2p}_{\omega}}\lesssim_{p,l,\tilde{\gamma},T}\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}(1+\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{4p,1})(t-s)^{(1+\tilde{\gamma})H},$

using Cauchy–Schwarz inequality we obtain

$\displaystyle\lVert\delta
A_{s,u,t}\rVert_{L^{p}_{\omega}}\lesssim_{p,l,\tilde{\gamma},T}\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}(1+\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{4p,1})(t-s)^{(2+\tilde{\gamma})H}.$

As $(2+\tilde{\gamma})H>1$, the sewing lemma in $L^{p}_{\omega}$ gives

$\Big{\lVert}\int_{s}^{t}f(X_{r})B_{\tau,r}^{i}\,\mathrm{d}B_{r}^{j}-\bar{A}_{s,t}\Big{\rVert}_{L^{p}_{\omega}}\lesssim_{p,l,\tilde{\gamma},T}\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}(1+\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{4p,1})(t-s)^{(2+\tilde{\gamma})H}.$

It remains to set $\tau=s$. ∎

######  Proposition 4.9.

Let $H\in(1/3,1/2]$, $p\in[2,\infty)$, $l\in[1,\infty)$,
$(X,X^{\prime}),(Y,Y^{\prime})\in\mathfrak{D}^{1,l}_{B^{H}}$, $f,g\in
C^{3}(\mathbb{R}^{d})$ and $\tilde{\gamma}\in\left(\frac{1-H}{H}-1,1\right)$.
Then, for every $(s,t)\in[0,T]_{\leq}^{2}$ and $i,j\in\\{1,2,\ldots,d_{2}\\}$
we have

$\Big{\lVert}\int_{s}^{t}\Big{(}\int_{s}^{r_{2}}g(Y_{r_{1}})\,\mathrm{d}B_{r_{1}}^{H,i}\Big{)}f(X_{r_{2}})\,\mathrm{d}B^{H,j}_{r_{2}}-g(Y_{s})f(X_{s})\mathbb{B}_{s,t}^{H,i,j}\Big{\rVert}_{L^{p}_{\omega}}\\\
\lesssim_{p,l,\tilde{\gamma},T}\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}\lVert
g\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}(1+\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{8p,1})(1+\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}Y\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{8p,1})(t-s)^{(2+\tilde{\gamma})H}.$

###### Proof.

For concision, throughout the proof we write $B\vcentcolon=B^{H}$ and
$\mathbb{B}\vcentcolon=\mathbb{B}^{H}$. We fix a $\tau\in[0,s]$. To estimate
the iterated integral

$\displaystyle\int_{s}^{t}\Big{(}\int_{\tau}^{r_{2}}g(Y_{r_{1}})\,\mathrm{d}B_{r_{1}}^{i}\Big{)}f(X_{r_{2}})\,\mathrm{d}B^{j}_{r_{2}}$

our choice of the germ is

$\tilde{A}_{s,t}\vcentcolon=\Big{(}\int_{\tau}^{s}g(Y_{r})\,\mathrm{d}B^{i}_{r}\Big{)}\int_{s}^{t}f(X_{r})\,\mathrm{d}B^{j}_{r}+g(Y_{s})\int_{s}^{t}B^{i}_{s,r}f(X_{r})\,\mathrm{d}B^{j}_{r}.$

Step 1. To see that $\tilde{A}_{s,t}$ is a correct approximation, we set

$\tilde{A}^{\prime}_{s,t}\vcentcolon=\Big{(}\int_{\tau}^{s}g(Y_{r})\,\mathrm{d}B_{r}^{i}\Big{)}f(X_{s})B^{j}_{s,t}+g(Y_{s})f(X_{s})\mathbb{B}^{i,j}_{s,t}\\\
+\sum_{k,l}\Big{(}\int_{\tau}^{s}g(Y_{r})\,\mathrm{d}B^{i}_{r}\Big{)}\partial_{k}f(Y_{s})(Y^{\prime}_{s})^{kl}\mathbb{B}_{s,t}^{l,j}.$

By definition of the rough integral, we have

$\displaystyle\int_{s}^{t}\Big{(}\int_{\tau}^{r_{2}}g(Y_{r_{1}})\,\mathrm{d}B_{r_{1}}^{i}\Big{)}f(X_{r_{2}})\,\mathrm{d}B^{j}_{r_{2}}=\lim_{\begin{subarray}{c}\pi\text{
is a partition of }[s,t],\\\ \lvert\pi\rvert\to
0\end{subarray}}\sum_{[u,v]\in\pi}\tilde{A}^{\prime}_{u,v}.$

In view of the uniqueness part of the sewing lemma (see Rem. 2.7), our goal is
to prove
$\lvert\tilde{A}_{s,t}-\tilde{A}^{\prime}_{s,t}\rvert\lesssim(t-s)^{3H}$. We
observe

$\tilde{A}_{s,t}-\tilde{A}^{\prime}_{s,t}=\Big{(}\int_{\tau}^{s}g(Y_{r})\,\mathrm{d}B_{r}^{i}\Big{)}\Big{(}\int_{s}^{t}f(X_{r})\,\mathrm{d}B^{j}_{r}-f(X_{s})B^{j}_{s,t}-\sum_{k,l}\partial_{k}f(Y_{s})(Y^{\prime}_{s})^{kl}\mathbb{B}_{s,t}^{l,j}\Big{)}\\\
+g(Y_{s})\Big{(}\int_{s}^{t}B^{i}_{s,r}f(X_{r})\,\mathrm{d}B^{j}_{r}-f(X_{s})\mathbb{B}_{s,r}^{i,j}\Big{)}.$

By the remainder estimate of the rough integral (20), we have

$\displaystyle\Big{\lvert}\int_{s}^{t}f(X_{r})\,\mathrm{d}B^{j}_{r}-f(X_{s})B^{j}_{s,t}-\sum_{k,l}\partial_{k}f(Y_{s})(Y^{\prime}_{s})^{kl}\mathbb{B}_{s,t}^{l,j}\Big{\rvert}\lesssim(t-s)^{3H},$
$\displaystyle\Big{\lvert}\int_{s}^{t}B^{i}_{s,r}f(X_{r})\,\mathrm{d}B^{j}_{r}-f(X_{s})\mathbb{B}_{s,r}^{i,j}\Big{\rvert}\lesssim(t-s)^{3H}.$

Hence, $\tilde{A}_{s,t}$ is a correct approximation.

Step 2. Again we do not need the stochastic sewing; the deterministic sewing
lemma (Lem. 2.3) in $L^{p}_{\omega}$ suffices. We compute

$\delta\tilde{A}_{s,u,t}=-\Big{(}\int_{s}^{u}g(Y_{r})\,\mathrm{d}B^{i}_{r}-g(Y_{s})B^{i}_{s,u}\Big{)}\int_{u}^{t}f(X_{r})\,\mathrm{d}B^{j}_{r}-g(X)_{s,u}\int_{u}^{t}f(X_{r})B^{i}_{u,r}\,\mathrm{d}B^{j}_{r}.$

By Prop. 4.7 and Prop. 4.8,

$\displaystyle\Big{\lVert}\int_{s}^{u}g(Y_{r})\,\mathrm{d}B^{i}_{r}-g(Y_{s})B^{i}_{s,u}\Big{\rVert}_{L^{2p}_{\omega}}$
$\displaystyle\lesssim\lVert
g\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}(1+\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{4p,1})(t-s)^{(1+\tilde{\gamma})H},$
$\displaystyle\Big{\lVert}\int_{u}^{t}f(X_{r})\,\mathrm{d}B^{j}_{r}\Big{\rVert}_{L^{2p}_{\omega}}$
$\displaystyle\lesssim\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}(1+\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}Y\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{4p,1})(t-s)^{H},$
$\displaystyle\lVert g(X)_{s,u}\rVert_{L^{2p}_{\omega}}$
$\displaystyle\lesssim\lVert
g\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{4p,1}(t-s)^{\tilde{\gamma}H},$
$\displaystyle\Big{\lVert}\int_{u}^{t}f(X_{r})B^{i}_{u,r}\,\mathrm{d}B^{j}_{r}\Big{\rVert}_{L^{2p}_{\omega}}$
$\displaystyle\lesssim\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}(1+\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}Y\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{8p,1})(t-s)^{2H}.$

Since $(2+\tilde{\gamma})H>1$, we are in the setting of the sewing lemma (Lem.
2.3). The remainder estimate yields

$\Big{\lVert}\int_{s}^{t}\Big{(}\int_{s}^{r_{2}}g(Y_{r_{1}})\,\mathrm{d}B_{r_{1}}^{i}\Big{)}f(X_{r_{2}})\,\mathrm{d}B^{j}_{r_{2}}-\tilde{A}_{s,t}\Big{\rVert}_{L^{p}_{\omega}}\\\
\lesssim\lVert g\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}(1+\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{8p,1})(1+\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}Y\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{8p,1})(t-s)^{(2+\tilde{\gamma})H}.$

Setting $\tau=s$, we get

$\Big{\lVert}\int_{s}^{t}\Big{(}\int_{s}^{r_{2}}g(Y_{r_{1}})\,\mathrm{d}B_{r_{1}}^{i}\Big{)}f(X_{r_{2}})\,\mathrm{d}B^{j}_{r_{2}}-g(Y_{s})\int_{s}^{t}f(X_{r})B^{i}_{s,r}\,\mathrm{d}B^{j}_{r}\Big{\rVert}_{L^{p}_{\omega}}\\\
\lesssim\lVert g\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}(1+\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}X\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{8p,1})(1+\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}Y\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{8p,1})(t-s)^{(2+\tilde{\gamma})H}.$

By Prop. 4.8,

$\Big{\lVert}\int_{s}^{t}f(X_{r})B^{i}_{s,r}\,\mathrm{d}B^{j}_{r}-f(X_{s})\mathbb{B}^{i,j}_{s,t}\Big{\rVert}_{L^{p}_{\omega}}\lesssim\lVert
f\rVert_{\mathcal{C}_{x}^{\tilde{\gamma}}}(1+\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}Y\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{4p,1})(t-s)^{(2+\tilde{\gamma})H},$

and the proof is complete. ∎

###  4.3 Pathwise Uniqueness

Our final result is the following, which is a restatement of Thm. 1.1 in the
regime $(1/3,1/2]$.

######  Theorem 4.10 (Pathwise Uniqueness in the Rough Regime).

Let $T>0$, $d_{1},\,d_{2}\in\mathbb{N}$, $B^{H}$ be an fBm with Hurst
parameter $H\in(1/3,1/2]$, $\gamma\in((1-H)/H,2)$ and
$\sigma\in\mathcal{C}^{\gamma}(\mathbb{R}^{d_{1}};\mathcal{M}(d_{1},d_{2}))$
be such that $\sigma\sigma^{\top}$ is uniformly elliptic, i.e. there exists an
$l\in[1,\infty)$ such that $l^{-1}\leq\sigma(x)\sigma^{\top}(x)\leq l$ for all
$x\in\mathbb{R}^{d_{1}}$. Then, pathwise uniqueness holds for (65) in the
sense of Def. 4.5.

Before giving the proof of Thm. 4.10, we require some preparations. Let
$\gamma\in(\frac{1-H}{H},2)$ be as above and fix an $\alpha\in(1/3,H)$ such
that $\gamma+\gamma\alpha>1$. Let $\rho\in
C^{\infty}(\mathbb{R}^{d_{1}};\mathcal{M}(d_{1},d_{2}))$ (which we will
eventually think of as a smooth approximation to $\sigma$) and
$(X,X^{\prime}),(Y,Y^{\prime})\in\mathscr{D}^{\gamma\alpha}_{B^{H}}$ (see Def.
2.2). Then, we set

$\displaystyle
G_{t}^{k}\vcentcolon=G[\rho]^{k}_{t}\vcentcolon=\int_{0}^{t}\Big{(}\int_{0}^{1}\partial_{k}\rho(\theta
X_{r}+(1-\theta)Y_{r})\,\mathrm{d}\theta\Big{)}\,\mathrm{d}B_{r}^{H},\quad\text{for}\,k=1,\ldots,d.$
(74)