# Level crossings of fractional Brownian motion

Purba Das  King’s College London, England.
<EMAIL_ADDRESS>Rafał Łochowski  Warsaw School of Economics, Poland.
<EMAIL_ADDRESS>Toyomu Matsuda  Institut für Mathematik, Freie Universität
Berlin, Germany.
<EMAIL_ADDRESS>and<EMAIL_ADDRESS>0000-0002-2422-0863 Nicolas Perkowski  Institut für Mathematik, Freie
Universität Berlin, Germany.
<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
###### Abstract

Since the classical work of Lévy, it is known that the local time of Brownian
motion can be characterized through the limit of level crossings. While
subsequent extensions of this characterization have primarily focused on
Markovian or martingale settings, this work presents a highly anticipated
extension to fractional Brownian motion — a prominent non-Markovian and non-
martingale process. Our result is viewed as a fractional analogue of Chacon et
al. [Cha+81]. Consequently, it provides a global path-by-path construction of
fractional Brownian local time.

Due to the absence of conventional probabilistic tools in the fractional
setting, our approach utilizes completely different argument with a flavor of
the subadditive ergodic theorem, combined with the shifted stochastic sewing
lemma recently obtained in [MP22].

Furthermore, we prove an almost-sure convergence of the ($1/H$)-th variation
of fractional Brownian motion with the color=green!60!white, size=, ]RL: I
added the Hurst parameter $H$, along random partitions defined by level
crossings, called Lebesgue partitions. This result raises an interesting
conjecture on the limit, which seems to capture non-Markovian nature of
fractional Brownian motion.

_Keywords and phrases._ Fractional Brownian motion, level crossings, local
time, ($1/H$)-th variation, Lebesgue partition, stochastic sewing lemma.

_MSC 2020._ 60G22, 60J55.

###### Contents

  1. 1 Introduction
    1. 1.1 Main results
    2. 1.2 Conjecture
    3. 1.3 Strategy of the proofs
  2. 2 Variation along uniform Lebesgue partitions
    1. 2.1 Elementary results
    2. 2.2 The shifted stochastic sewing lemma
    3. 2.3 Convergence of variation
  3. 3 Local time via level crossings
    1. 3.1 Heuristics
    2. 3.2 Convergence to local time
    3. 3.3 Uniform convergence
    4. 3.4 Horizontally rough function
  4. A An estimate on log-normal distribution

##  1 Introduction

Level crossings of stochastic processes have been studied since the classical
works of Kac [Kac43] and Rice [Ric45]. Depending on whether the process is
smooth or rough, the study of its level crossings rely on different methods.
As for the smooth case, which is not the scope of this article, the reader can
refer to the survey article [Kra06], the textbook [AW09] and the reference
therein.

By far the most prominent example of color=green!60!white, size=, ]RL: I
changed this sequence slightly a rough stochastic process is a Brownian
motion. The first work on level crossings of Brownian motion is attributed to
Lévy [Lév48], who characterized its local time as color=green!60!white, size=,
]RL: I changed this sequence slightly a limit of normalised numbers of level
crossings. More precisely, for a given process $w$, we set

$U_{s,t}(\varepsilon,w)\vcentcolon=\\#\big{\\{}(u,v):s\leq u<v\leq
t,\,\,w_{u}=0,w_{v}=\varepsilon,\forall
r\in(u,v)\>w_{r}\in(0,\varepsilon)\big{\\}},$

as illustrated in Figure 1. The quantity $U_{s,t}(\varepsilon,w)$ counts the
number of upcrossings from $0$ to $\varepsilon$ in the interval $[s,t]$. For
Brownian motion $W$ and $a\in\mathbb{R}$, we have

$\displaystyle\lim_{\varepsilon\to 0}\varepsilon
U_{0,t}(\varepsilon,W-a)=\frac{1}{2}L_{t}^{W}(a)$

almost surely, where $L_{t}^{W}(a)$ is the local time of $W$ at time $t$ and
at level $a$. The local time is defined as the density of the occupation
measure, see Definition 1.2 below. This result can be found in standard
textbooks such as [IM74, RY99, MP10], and it can be generalized for
semimartingales [El ̵78] and for Markov processes [FT83].

Figure 1: Graphical illustration of $U_{s,t}(\varepsilon,w)$. In the picture,
$U_{s,t}(\varepsilon,w)=2$.

On the other hand, there exist rough stochastic processes that do not fall
under the category of semimartingales or Markovian processes. One such example
is the _fractional Brownian motion_ $B^{H}$, which is a Gaussian process
parameterized by Hurst index $H\in(0,1)$. Precisely, $B^{H}$ is neither a
semimartingale nor Markovian for $H\neq 1/2$, and it becomes Brownian motion
when $H=1/2$.

Fractional Brownian motion possesses a local time color=green!60!white, size=,
]RL: I added the reference to Definition 1.2(defined as in Definition 1.2).
Given Lévy’s result on the local time of Brownian motion, a natural question
arises regarding whether a similar result holds for the local time of
fractional Brownian motion. However, a complete answer to this question has
not been obtained thus far. This is surprising considering the age of Lévy’s
result and that of fractional Brownian motion.

Some works have explored the characterization of the fractional Brownian local
time in relation to level crossings. For instance, the works [Aza90, AW96]
demonstrate that the number of zeros for certain smoothed fractional Brownian
motions converges, in a suitable sense, to the local time. Furthermore, level
crossings of stochastic processes have received attention in the context of
pathwise stochastic calculus [PP15, DOS18, CP19, Łoc+21, Kim22, ACX20], as
well as in some applied literatures [FHW94, Kru98].

Constructing the local time via level crossings is not only a natural problem,
but also it can lead to a significant implication on the path
color=green!60!white, size=, ]RL: I changed this sentence slightly properties
of the process. This was first observed by Chacon et al. [Cha+81] built on
[Per81]. Therein proven is the existence of a measurable set $\Omega_{W}$ such
that $\mathbb{P}(W\in\Omega_{W})=1$ (recall that $W$ is Brownian motion) and
for every $w\in\Omega_{W}$, $a\in\mathbb{R}$ and $t\geq 0$, the limit

$\displaystyle\lim_{\varepsilon\to 0}\varepsilon U_{0,t}(\varepsilon,w-a)$ (1)

exists and is equal to one-half times the local time of $w$ at the level $a$.
Hence, the existence of the limit of level crossings is a path property, and
this provides a path-by-path construction of Brownian local time. Furthermore,
the construction is global in that the convergence in (1) holds for all
$a\in\mathbb{R}$ off a single null set. This result explains why such
construction of the local time receives attention in the pathwise stochastic
calculus. It is worth noting that the master thesis [Lem83] of Lemieux
extended the result for a large class of semimartingales.

The result of Chacon et al. has a remarkable consequence on pathwise quadratic
variation, calculated as a limit of sums of square increments, where the
increments are taken along partitions of a fixed interval with vanishing mesh.
The precise definition of the pathwise quadratic variation is as follows:
given a sequence $\bm{\pi}$ of partitions $\pi_{n}$ ($n\in\mathbb{N}$) with
vanishing mesh, the pathwise quadratic variation $[w]_{\bm{\pi}}$ of a process
$w$ is defined by

$\displaystyle[w]_{\bm{\pi}}\vcentcolon=\lim_{n\to\infty}\sum_{[s,t]\in\pi_{n}}\lvert
w_{t}-w_{s}\rvert^{2}$

whenever the limit exists. A deterministic partition is a partition that does
not depend on the path/process. The classical works [Lév40, Lév48] of Lévy
show that for any refining deterministic partition sequence $\bm{\pi}$ of
$[0,t]$ with vanishing mesh we have

$\displaystyle\mathbb{P}([W]_{\bm{\pi}}=t)=1.$ (2)

Dudley [Dud73] proved that if the deterministic sequence $\bm{\pi}=(\pi_{n})$
satisfies

$\displaystyle\lvert\pi_{n}\rvert\vcentcolon=\max_{[s,t]\in\pi_{n}}\lvert
t-s\rvert=o(1/\log n)$ (3)

then (2) holds, with the optimality of the condition (3) being shown as well.
In general, the pathwise quadratic variation (even when it exists) may depend
on the choice of a sequence of partitions [Fre83, page 47], [DOS18].
Furthermore, the null set in Lévy’s and Dudley’s works depends on the sequence
$\bm{\pi}$ of partitions.

Hence, an obvious question is if, given a stochastic process $X$, there is any
uncountable class $\mathrm{P}$ of partition sequences such that almost surely
for any $\bm{\pi},\bm{\pi}^{\prime}\in\mathrm{P}$ we have
$[X]_{\bm{\pi}}=[X]_{\bm{\pi}^{\prime}}$. Here comes the result of Chacon et
al., which proves that Brownian motion has a measure zero set outside which
any quadratic variation along any sequence of the _Lebesgue partitions_
(defined at the beginning of Section 1.1) of $[0,t]$ with vanishing mesh is
equal to $t$. We remark that, unlike in Dudley’s result, there is
color=green!60!white, size=, ]RL: I changed this sequence no condition on the
decay of meshes of partitions and the null set is uniform over all sequences
of the Lebesgue partitions. color=red!20!white, size=, ]TM: To Purba: can you
cite your works here?

In this paper, we extend Levy’s construction of the local time and the result
of Chacon et al. for fractional Brownian motions with the Hurst parameter
$H<1/2$.

###  1.1 Main results

We write $B^{H}$ for a fractional Brownian motion with $B_{0}^{H}=0$ and with
the Hurst parameter $H\in(0,1)$. Specifically, $B^{H}$ is a one-dimensional
centered Gaussian process satisfying $B^{H}_{0}=0$ and

$\displaystyle\mathbb{E}[(B^{H}_{t}-B^{H}_{s})^{2}]=a_{H}(t-s)^{2H}.$

We choose the constant $a_{H}$ in a manner that the identity (18) below holds.
The specific value of $a_{H}$ is not important for our study.

To introduce the Lebesgue partitions, let $\mathrm{P}$ be a partition of the
space $\mathbb{R}$. That is, we have a strictly increasing sequence
$(y_{n})_{n\in\mathbb{Z}}$ of real numbers such that

$\lim_{n\to-\infty}y_{n}=-\infty,\quad\lim_{n\to+\infty}y_{n}=+\infty,$

and we have $\mathrm{P}=\\{[y_{n-1},y_{n}]:n\in\mathbb{Z}\\}$.
color=green!60!white, size=, ]RL: maybe it is in place to add here the
definition of a partition we use? Let $\Lambda(\mathrm{P})$ be the set of all
endpoints of intervals from $\mathrm{P}$, or
$\Lambda(\mathrm{P})=\\{y_{n}:n\in\mathbb{Z}\\}$. Given a path
$w:[0,\infty)\to\mathbb{R}$, we set $T_{0}(\mathrm{P},w)\vcentcolon=0$ and
recursively define

$\displaystyle
T_{n}(\mathrm{P},w)\vcentcolon=\inf\\{t>T_{n-1}(\mathrm{P},w):w_{t}\in\Lambda(\mathrm{P})\setminus\\{w_{T_{n-1}(\mathrm{P},w)}\\}\\}$
(4)

with $\inf\varnothing=+\infty$. (If $T_{n-1}=+\infty$, we set
$T_{n}\vcentcolon=+\infty$.) Note that we do not assume $w_{0}=0$. The
partition given by

$\\{[T_{n-1}(\mathrm{P},w),T_{n}(\mathrm{P},w)]:n\in\mathbb{N},T_{n}(\mathrm{P},w)\leq
t\\}$ (5)

is called a _Lebesgue partition_. See Figure 2 for a graphical illustration.
Lebesgue partition is also called partition along $y$-axis.

Figure 2: Lebesgue partition

For a uniform partition $\mathrm{P}^{\varepsilon}\vcentcolon=\\{[\varepsilon
k,\varepsilon(k+1)]:k\in\mathbb{Z}\\}$, we simply write
$T_{n}(\varepsilon,w)\vcentcolon=T_{n}(\mathrm{P}^{\varepsilon},w)$. We denote
by $K_{s,t}(\varepsilon,w)$ the number of $\varepsilon$-level crossings in the
interval $[s,t]$, defined precisely as111Our convention is
$\mathbb{N}=\\{1,2,3,\ldots\\}$. In particular $0\notin\mathbb{N}$.

$\displaystyle
K_{s,t}(\varepsilon,w)\vcentcolon=\\#\\{n\in\mathbb{N}\setminus\\{1\\}:T_{n}(\varepsilon,w_{s+\cdot})\leq
t\\}+\mathbf{1}_{\\{w_{s}\in\varepsilon\mathbb{Z}\\}}\mathbf{1}_{\\{T_{1}(\varepsilon,w_{s+\cdot})\leq
t\\}},$ (6)

where $\\#A$ represents the cardinality of the set $A$. We observe

$\sum_{n:T_{n}(\varepsilon,w)\leq t}\lvert
w_{T_{n}(\varepsilon,w)}-w_{T_{n-1}(\varepsilon,w)}\rvert^{\frac{1}{H}}=\varepsilon^{1/H}K_{0,t}(\varepsilon,w)+\mathbf{1}_{\\{w_{0}\notin\varepsilon\mathbb{Z}\\}}\lvert
w_{T_{1}(\varepsilon,w)}-w_{0}\rvert^{\frac{1}{H}}.$

Note that $\lvert w_{T_{1}(\varepsilon,w)}-w_{0}\rvert\leq\varepsilon$.
Therefore, the study of the ($1/H$)-th variation along a sequence of uniform
Lebesgue partitions is equivalent to that of limiting behavior of
$K_{0,t}(\varepsilon,w)$ as $\varepsilon\to 0$.

For $\rho\in\mathbb{R}$ and a process $w$, the process $w+\rho$ is defined by
$(w+\rho)_{t}\vcentcolon=w_{t}+\rho$. Our first main result is on the
($1/H$)-th variation of fractional Brownian motion along uniform Lebesgue
partitions of fractional Brownian motion $B^{H}$.

######  Theorem 1.1 (Convergence of ($1/H$)-th variation along uniform
Lebesgue partitions).

For all $H\in(0,1)$, there exists a positive finite constant
$\mathfrak{c}_{H}$ with the following property. Let $\rho\in\mathbb{R}$,
$t\in(0,\infty)$ and $(\varepsilon_{n})_{n=1}^{\infty}$ be a sequence of
positive numbers such that $\varepsilon_{n}=O(n^{-\eta})$ for some $\eta>0$.
We then have

$\lim_{n\to\infty}\varepsilon_{n}^{1/H}K_{0,t}(\varepsilon_{n},B^{H}+\rho)=\mathfrak{c}_{H}t\quad\text{almost
surely.}$ (7)

The proof of Theorem 1.1 will be given at the end of Section 2.3. A more
explicit representation of the constant $\mathfrak{c}_{H}$ is given by (21).
For $H<1/2$, we can remove the color=green!60!white, size=, ]RL: I removed the
word ’decaying’ condition $\varepsilon_{n}=O(n^{-\eta})$, see Theorem 1.4
below.

Theorem 1.1 concerns level crossings at all levels. We can also consider level
crossings at a specific level, leading to Lévy’s classical work on local time
[Lév48]. For each $\varepsilon\in(0,1)$ and $w\in C([0,T];\mathbb{R})$, we
consider the number of upcrossings color=green!60!white, size=, ]RL: I changed
this sentence by $w$ the space interval $[0,\varepsilon]$ between times $s$
and $t$ by setting

$\displaystyle
U_{s,t}(\varepsilon,w)\vcentcolon=\\#\big{\\{}(u,v)\in[s,t]^{2}:u<v,w_{u}=0,w_{v}=\varepsilon,\forall
r\in(u,v)\>w_{r}\in(0,\varepsilon)\big{\\}}.$ (8)

See Figure 1 for illustration. In the case of Brownian motion ($H=1/2$), it is
well-known (e.g., [IM74, Section 2.2], [RY99, Chapter VI], [MP10, Section 6])
that provided $\varepsilon_{n}\to 0$ and
$\varepsilon_{n}>0$,color=yellow!60!white, size=, ]PD: Just to clarify: does
$\downarrow 0$ means monotonically decreasing? I think we just need $\to 0$
and $\varepsilon_{n}>0$ TM: corrected we have

$\displaystyle\lim_{n\to\infty}\varepsilon_{n}U_{0,t}(\varepsilon_{n},B^{1/2}-a)=\frac{1}{2}L_{t}^{1/2}(a)\quad\text{almost
surely,}$ (9)

where $L^{1/2}$ is the local time of Brownian motion $B^{1/2}$. This
representation of the local time was extended for semimartingales by Karoui
[El ̵78].

We recall that the local time of $B^{H}$, defined just below, exists for all
$H\in(0,1)$.

######  Definition 1.2 (Local time).

We denote by $(L_{t}^{H}(a))_{t\geq 0,a\in\mathbb{R}}$ the _local time_ of
$B^{H}$ with $t$ representing time and $a$ representing level. More precisely,
$L^{H}$ is a unique random field satisfying the following occupation density
formula:

$\displaystyle\int_{0}^{t}f(B^{H}_{r})\mathrm{d}r=\int_{\mathbb{R}}f(a)L_{t}^{H}(a)\mathrm{d}a,\quad\forall
t\geq 0,\forall f\in C(\mathbb{R}).$

As for the existence of $L^{H}$ and its continuity, see e.g., [BG73, GH80].

color=red!20!white, size=, ]TM: cite Berman, S.M.: Local nondeterminism and
local times of Gaussian processes. Bull. Am. Math. Soc. 79(2), 475–477 (1973).
PD:added The following is an extension of Lévy’s result to fractional Brownian
motion.

######  Theorem 1.3 (Local time via level crossings).

Let $H<1/2$, $a\in\mathbb{R}$, $t\in(0,\infty)$ and
$(\varepsilon_{n})_{n=1}^{\infty}$ be a sequence of positive numbers such that
$\varepsilon_{n}\to 0$. We then have

$\displaystyle\lim_{n\to\infty}\varepsilon_{n}^{\frac{1}{H}-1}U_{0,t}(\varepsilon_{n},B^{H}-a)=\frac{\mathfrak{c}_{H}}{2}L_{t}^{H}(a)\quad\text{almost
surely,}$ (10)

where the constant $\mathfrak{c}_{H}$ is the same as in Theorem 1.1 and in
Equation (21).

In principle, the null set of (9) or (10) could potentially depend on the
level $a$ and the sequence $(\varepsilon_{n})_{n=1}^{\infty}$. However, an
intriguing result by Chacon et al. [Cha+81] demonstrates that for Brownian
motion the null set can be chosen uniformly over the level $a$ and the
sequence $(\varepsilon_{n})_{n=1}^{\infty}$. In the next theorem, we extend
this result to fractional Brownian motion.

Before stating the theorem, we introduce some notation. Analogously to
$U_{s,t}$, we denote by $D_{s,t}$ the total number of downcrossings

$\displaystyle
D_{s,t}(\varepsilon,w)\vcentcolon=\\#\big{\\{}(u,v)\in[s,t]^{2}:u<v,w_{v}=0,w_{u}=\varepsilon,\forall
r\in(u,v)\>w_{r}\in(0,\varepsilon)\big{\\}}.$ (11)

Given a partition $\mathrm{P}$ of $\mathbb{R}$, we set

$\displaystyle
V_{s,t}(\mathrm{P},w)\vcentcolon=\sum_{[a,b]\in\mathrm{P}}(b-a)^{\frac{1}{H}}\left(U_{s,t}(b-a,w-a)+D_{s,t}(b-a,w-a)\right).$
(12)

The quantity $V_{s,t}(\mathrm{P},w)$ measures the ($1/H$)-th variation along
the Lebesgue partition defined by $\mathrm{P}$. For the uniform partition
$\mathrm{P}^{\varepsilon}$, we have

$V_{0,t}(\mathrm{P}^{\varepsilon},w)=\varepsilon^{1/H}K_{0,t}(\varepsilon,w)+\mathbf{1}_{\\{w_{0}\notin\varepsilon\mathbb{Z}\\}}\lvert
w_{T_{1}(\varepsilon,w)}-w_{0}\rvert^{\frac{1}{H}}.$

######  Theorem 1.4 (Fractional analogue of Chacon et al.).

Let $H<1/2$ and let $\mathfrak{c}_{H}$ be the constant of Theorem 1.1. Then,
there exists a measurable set $\Omega_{H}\subseteq C([0,\infty);\mathbb{R})$
with the following property.

  * •

$\mathbb{P}(B^{H}\in\Omega_{H})=1$.

  * •

For every $w\in\Omega_{H}$ and $T\in(0,\infty)$, we have

$\displaystyle\lim_{\varepsilon\to
0,\varepsilon>0}\sup_{\begin{subarray}{c}\mathrm{P}:\lvert\mathrm{P}\rvert\leq\varepsilon,\\\
t\leq T\end{subarray}}\lvert V_{0,t}(\mathrm{P},w)-\mathfrak{c}_{H}t\rvert=0.$

  * •

For every $w\in\Omega_{H}$, there exists a continuous map

$\displaystyle[0,\infty)\times\mathbb{R}\ni(t,a)\mapsto
l_{t}(w,a)\in[0,\infty)$

such that for every $T\in(0,\infty)$ we have

$\displaystyle\lim_{\varepsilon\to 0,\varepsilon>0}\sup_{a\in\mathbb{R},t\leq
T}\Big{\lvert}\varepsilon^{\frac{1}{H}-1}U_{0,t}(\varepsilon,w-a)-\frac{\mathfrak{c}_{H}}{2}l_{t}(w,a)\Big{\rvert}=0.$

Furthermore, the occupation density formula holds:

$\displaystyle\int_{0}^{t}f(w_{r})\mathrm{d}r=\int_{\mathbb{R}}f(a)l_{t}(w,a)\mathrm{d}a,\quad\forall
t\geq 0,\forall f\in C(\mathbb{R}).$

###### Proof.

It follows from Theorem 3.23 and Theorem 3.24. ∎

In [Cha+81] the corresponding result for Brownian motion is interpreted as a
_quadratic arc length_. By analogy, we could say that the fractional Brownian
motion of Hurst index $H<1/2$ has _$1/H$ arc length_ $\mathfrak{c}_{H}t$. Note
that the $1/H$ arc length is a purely geometric, path-dependent quantity which
is invariant under translations and time-reparametrizations.

color=yellow!60!white, size=, inline]PD: The main proof uses the SW argument
on $\bar{K}$. So don’t we also have
$\varepsilon_{n}^{\frac{1}{H}}TV^{\varepsilon_{n}}\to[B^{H}]^{(1/H)}$ along
lebesgue partition almost surely? Or am I missing something? TM: It easily
follows from the second item.

######  Remark 1.5.

In [MP22] the following quantitative estimate was proved. There exists an
$\varepsilon\in(0,1)$ such that for every $m\in[1,\infty)$ we have

$\Big{\|}\sum_{[u,v]\in\pi}|B^{H}_{v}-B^{H}_{u}|^{\frac{1}{H}}-\mathbb{E}\big{[}|B^{H}_{1}|^{\frac{1}{H}}\big{]}t\Big{\|}_{L^{m}(\mathbb{P})}\lesssim_{m,t}|\pi|^{\varepsilon}$

for any deterministic partition $\pi$ of $[0,t]$. Therefore, by the
Borel–Cantelli lemma, for any sequence $(\pi_{n})_{n=1}^{\infty}$ of
deterministic partitions with

$|\pi_{n}|=O(n^{-\delta}),\quad\delta\in(0,\infty),$ (13)

we have
$\lim_{n\to\infty}\sum_{[u,v]\in\pi^{n}}|B^{H}_{v}-B^{H}_{u}|^{\frac{1}{H}}=\mathbb{E}[|B^{H}_{1}|^{\frac{1}{H}}]t$
almost surely. Unlike Theorem 1.4, we need the decaying condition (13). The
work [Dud73] shows that the condition (13) is not optimal for the Brownian
case; finding the optimal condition for the fractional case seems open.

###  1.2 Conjecture

There is an interesting aspect on the constant $\mathfrak{c}_{H}$. For
Brownian motion, the quadratic variation along any deterministic partition
almost surely matches with the quadratic variation along any Lebesgue
partitions. That is,

$\displaystyle\mathfrak{c}_{\frac{1}{2}}=\mathbb{E}[(B^{1/2}_{1})^{2}].$ (14)

It is tempting to guess that such relation holds for $H\neq 1/2$ as well.
Indeed, such conjecture is stated in [CP19, after Lemma 3.5]. However, the
identity (14) is due to the strong Markov property of Brownian motion.
Therefore, for $H\neq 1/2$, there is no reason why $\mathfrak{c}_{H}$ and
$\mathbb{E}[\lvert B^{H}_{1}\rvert^{\frac{1}{H}}]$ should be equal. Motivated
by the simulation shown in Figures 4, we propose the following conjecture.

######  Conjecture 1.6 (The constant $\mathfrak{c}_{H}$).

For fractional Brownian motion with $H\neq 1/2$, we conjecture that the
($1/H$)-th variation of fractional Brownian motion along deterministic
partitions differs from the ($1/H$)-th variation of fractional Brownian motion
along uniform Lebesgue partitions. To be more precise, we conjecture

$\begin{cases}\mathfrak{c}_{H}>\mathbb{E}[\lvert
B_{1}^{H}\rvert^{1/H}]&\text{if }H<1/2,\\\ \mathfrak{c}_{H}<\mathbb{E}[\lvert
B_{1}^{H}\rvert^{1/H}]&\text{if }H>1/2.\end{cases}$

If this is indeed the case, the constant $\mathfrak{c}_{H}$ captures non-
Markovian nature of fractional Brownian motion.

(a)

(b)

Figure 4: Comparison between the variation along a deterministic uniform
partition and that along a Lebesgue partition.  Figure 5: Comparison between
the variation along a deterministic uniform partition and that along a
Lebesgue partition for Brownian motion.

######  Remark 1.7.

We comment on the simulation for Figure 4. The variation, denoted as $V_{t}$,
is given by

$V_{t}\vcentcolon=\sum_{[u,v]\in\pi^{\\#},v\leq t}\lvert
B^{H}_{v}-B^{H}_{u}\rvert^{\frac{1}{H}},$

up to time $T$, where $\\#\in\\{\text{deterministic},\text{Lebesgue}\\}$. We
discretize the fractional Brownian motion with a step size of $T/n$. For the
deterministic partition, we use the partition
$\pi^{\text{deterministic}}=\\{kT/n\\}_{k=1}^{n}$, while for the Lebesgue
partition, we utilize the partition
$\pi^{\text{Lebesgue}}=\\{T_{k}(\varepsilon,B)\\}_{k}$. To simulate the
variation $V$ along a Lebesgue partition, it is crucial to appropriately
choose the parameters $T$, $n$ and $\varepsilon$ to ensure that the partition
is neither too dense nor too sparse. For instance, when $H=0.4$, we simulate
$V$ with $T=0.1$, $n=30000$, and $\varepsilon=0.015$. Similarly, for $H=0.6$,
we simulate $V$ with $T=2$, $n=30000$, and $\varepsilon=0.013$. Figure 5
represents a similar figure for Brownian motion (i.e. $H=1/2$). The figure is
generated with $T=1,n=30000,\varepsilon=0.014$. The figure supports the fact
that for a Brownian motion the quadratic variation along deterministic
partitions is same as that along the Lebesgue partitions.

###  1.3 Strategy of the proofs

Let us outline our strategy to prove Theorem 1.1. The first observation is
that it is easier to deal with an averaged version of $K$. Namely, we define

$\displaystyle\bar{K}_{s,t}(\varepsilon,w)\vcentcolon=\varepsilon^{-1}\int_{-\varepsilon/2}^{\varepsilon/2}K_{s,t}(\varepsilon,w+\rho)\mathrm{d}\rho.$

The quantity $\bar{K}$ is related to _truncated variation_ , see Remark 2.5
below. By scaling (Lemma 2.6), we have

$\displaystyle\bar{K}_{0,1}(\varepsilon,B^{H})=\bar{K}_{0,\varepsilon^{-1/H}}(1,B^{H})\quad\text{in
law.}$

Therefore, at least in law, the limiting behavior of
$\bar{K}_{0,1}(\varepsilon,B^{H})$ as $\varepsilon\to 0$ is the same as that
of $\bar{K}_{0,\varepsilon^{-1/H}}(1,B^{H})$. Rewriting
$T=\varepsilon^{-1/H}$, we thus hope to see the limit of
$T^{-1}\bar{K}_{0,T}(1,B^{H})$.

Regarding this, we are actually in the setting of the superadditive ergodic
theorem. Indeed, it is not difficult to see that the family
$(\bar{K}_{s,t}(1,B^{H}))_{s<t}$ is stationary and superadditive (Lemma 2.6).
Therefore, the limit

$\displaystyle\lim_{T\to\infty}\frac{1}{T}\bar{K}_{0,T}(\varepsilon,B^{H})=\sup_{T>0}\frac{1}{T}\mathbb{E}[\bar{K}_{0,T}(\varepsilon,B^{H})]$

exists almost surely and in $L^{1}(\mathbb{P})$. It turns out that the limit
is the constant $\mathfrak{c}_{H}$ in Theorem 1.1.

To take advantage of this finding on $\bar{K}$, we can naively guess that

$\displaystyle\lvert
K_{0,1}(\varepsilon,B^{H}+\rho)-K_{0,1}(\varepsilon,B^{H}+\rho^{\prime})\rvert$
(15)

is small provided that $\lvert\rho-\rho^{\prime}\rvert$ is small. It is
therefore expected to have

$\displaystyle\lim_{\varepsilon\to
0}\varepsilon^{\frac{1}{H}}K_{0,1}(\varepsilon,B^{H})=\lim_{\varepsilon\to
0}\varepsilon^{\frac{1}{H}}\bar{K}_{0,1}(\varepsilon,B^{H})=\mathfrak{c}_{H}.$
(16)

However, proving (16) requires a non-trivial argument. In fact, the map
$\rho\mapsto K_{0,1}(\varepsilon,w+\rho)$ can be highly discontinuous, and it
is impossible to estimate (15) pathwisely.

This inherent difficulty motivates us to employ a _probabilistic_ argument in
order to prove Theorem 1.1. In addition to superadditivity, a crucial
ingredient is the stochastic sewing lemma introduced by Lê [Lê20], which
provides effective estimates for stochastic Riemann sums. For our specific
problem, we require an extension of this lemma, called the shifted stochastic
sewing lemma, recently obtained by the third and fourth authors [MP22]. This
extension is particularly suitable for capturing asymptotic decorrelation in
stochastic Riemann sums. It is worth mentioning that there are already some
studies that leverage the stochastic sewing lemma for analyzing local times
[HP21, HL22, BLM23].

Our proof of Theorem 1.1 proceeds as follows. Not only is the family
$(K_{s,t}(\varepsilon,B^{H}))_{s<t}$ superadditive, but it is also almost
subadditive:

$\displaystyle K_{s,t}(\varepsilon,B^{H})\leq
K_{s,u}(\varepsilon,B^{H})+K_{u,t}(\varepsilon,B^{H})+1,\quad s<u<t.$

This leads to an approximation:

$\displaystyle\varepsilon^{1/H}K_{0,1}(\varepsilon,B^{H})\approx\sum_{[s,t]\in\pi_{\varepsilon}}\varepsilon^{1/H}K_{s,t}(\varepsilon,B^{H}),$

where $\pi_{\varepsilon}$ is a uniform partition of the interval $[0,1]$ with
$\lvert\pi_{\varepsilon}\rvert\approx\varepsilon^{1/H}$. A similar
approximation holds for $\bar{K}$. Thus, we obtain

$\displaystyle\varepsilon^{1/H}K_{0,1}(\varepsilon,B^{H})-\varepsilon^{1/H}\bar{K}_{0,1}(\varepsilon,B^{H})\approx\sum_{[s,t]\in\pi_{\varepsilon}}\varepsilon^{1/H}[K_{s,t}(\varepsilon,B^{H})-\bar{K}_{s,t}(\varepsilon,B^{H})].$

The Riemann sum on the right-hand side can then be estimated using the shifted
stochastic sewing lemma, which ultimately yields the desired convergence
result.

To establish the convergence to the local time (Theorem 1.3), we follow a
similar line of argument. However, the technical difficulty increases
substantially due to the lack of a counterpart to $\bar{K}$. We postpone an
overview of this technical argument to Section 3.1.

One notable strength of the stochastic sewing lemma is its ability to provide
a quantitative bound. Thanks to this property, we can obtain a quantitative
version of Theorem 1.3, as stated in Theorem 3.1. This quantitative result
enables us to employ the pathwise argument of Chacon et al. [Cha+81], which
elevates the convergence result from Theorem 1.3 to the more refined version
of Theorem 1.4.

#### Outline

The organization of the paper is as follows. Theorem 1.1 is proven in Section
2, while Theorem 1.3 and Theorem 1.4 are proven in Section 3.

In Section 2.1, we derive some elementary results on $K$, in Section 2.2, we
recall our key ingredient called the shifted stochastic sewing lemma, and in
Section 2.3, we give our proof of Theorem 1.1.

In Section 3, after stating a quantitative version of Theorem 1.3 as Theorem
3.1, we review our technical strategy in Section 3.1. Section 3.2 is the most
demanding part of the paper, which proves Theorem 3.1 with the key estimate
being Lemma 3.7. As a consequence of Theorem 3.1, we complete the proof of
Theorem 1.4 in Section 3.3.

#### Notation

Given a path $f\colon[0,T]\to\mathbb{R}^{d}$, we write
$f_{s,t}\vcentcolon=f_{t}-f_{s}$, and we denote by $\dot{f}$ the derivative
$\frac{\mathrm{d}f}{\mathrm{d}t}$. We fix a probability space
$(\Omega,\mathcal{F},\mathbb{P})$ on which fractional Brownian motion is
defined, we write $\mathbb{E}$ for the expectation with respect to
$\mathbb{P}$, and we write

$\lVert F\rVert_{L^{p}(\mathbb{P})}\vcentcolon=\Big{(}\int_{\Omega}\lvert
F\rvert^{p}\mathrm{d}\mathbb{P}\Big{)}^{1/p}$

with usual modification for $p=\infty$. The expression
$X\overset{\operatorname{d}}{=}Y$ means that the random variables $X$ and $Y$
have the same law. We write $A\lesssim A^{\prime}$ if there exists a positive
constant $C$, depending only on some (unimportant) parameters, such that
$A\leq CA^{\prime}$. If we want to emphasize the dependency on parameters
$\alpha,\beta,\ldots$, then we write
$A\lesssim_{\alpha,\beta,\ldots}A^{\prime}$. The following objects appear
throughout the paper.

  * •

We write $B=B^{H}$ for fractional Brownian motion in one dimension. Unlike
Section 1, we mostly suppress the script $H$ (i.e., we simply write $B$ for
fractional Brownian motion). In this paper we will not write down dependency
on $H$. For instance, when we write $A\lesssim A^{\prime}$, the proportional
constant may depend on $H$.

  * •

We denote by $K_{s,t}(\varepsilon,w)$ the total number of $\varepsilon$-level
crossings in the interval $[s,t]$, as precisely defined at (6). We denote by
$U_{s,t}(\varepsilon,w)$ the total number of upcrossings from $0$ to
$\varepsilon$ in the interval $[s,t]$, as precisely defined in (8).

  * •

We write $(L_{t}^{H}(a))_{t\geq 0,a\in\mathbb{R}}=(L_{t}(a))_{t\geq
0,a\in\mathbb{R}}$ for the local time of fractional Brownian motion $B$, see
Definition 1.2.

### Acknowledgement

Our work is aided by the Deutsche Forschungsgemeinschaft (DFG) through the
Berlin–Oxford IRTG 2544 program, which enabled PD to visit Berlin and
financially supports TM. RŁ’s research was partially funded by the National
Science Centre, Poland, under Grant No. 2019/35/B/ST1/04292.
color=red!20!white, size=, ]TM: Maybe comment on exchange program. PD: added

##  2 Variation along uniform Lebesgue partitions

The goal of this section is to prove Theorem 1.1. We begin observing
elementary results on the counting $K$ of level crossings, defined by (6).

###  2.1 Elementary results

Let us first recall the Mandelbrot–Van Ness representation of fractional
Brownian motion [MV68], which will be used throughout.

######  Definition 2.1 (Fractional Brownian motion).

We set

$\displaystyle\mathcal{K}(t,s)\vcentcolon=(t-s)^{H-\frac{1}{2}}-(-s)_{+}^{H-\frac{1}{2}},\quad
s<t.$ (17)

Let $W=(W_{t})_{t\in\mathbb{R}}$ be a two-sided Brownian motion in one
dimension. Throughout the paper, we suppose that fractional Brownian motion
$B=B^{H}$ has the _Mandelbrot–Van Ness representation_

$\displaystyle B_{t}=\int_{-\infty}^{t}\mathcal{K}(t,s)\mathrm{d}W_{s}.$ (18)

Note that we have $\mathbb{E}[(B_{t}-B_{s})^{2}]=a_{H}(t-s)^{2H}$ for some
constant $a_{H}$, whose actual value is irrelevant for us.

######  Lemma 2.2 (Scaling of $K$).

For $\lambda\in(0,\infty)$, we have

$(K_{s,t}(\varepsilon,B+\rho))_{s<t,\;\varepsilon>0,\rho\in\mathbb{R}}\overset{\operatorname{d}}{=}(K_{\lambda^{1/H}s,\lambda^{1/H}t}(\lambda\varepsilon,B+\lambda\rho))_{s<t,\;\varepsilon>0,\rho\in\mathbb{R}}.$

###### Proof.

We set $B^{(\lambda)}_{t}\vcentcolon=\lambda B_{\lambda^{-1/H}t}$. Note that
$B^{(\lambda)}\overset{\operatorname{d}}{=}B$ and observe that

$K_{s,t}(\varepsilon,B_{t}+\rho)=K_{s,t}(\lambda\varepsilon,\lambda(B+\rho))\text{}=K_{\lambda^{1/H}s,\lambda^{1/H}t}(\lambda\varepsilon,B^{(\lambda)}+\lambda\rho).\qed$

######  Lemma 2.3 (Superadditivity of $K$).

Let $r<s<t$ and $w$ be a path. Then,

$K_{r,s}(\varepsilon,w)+K_{s,t}(\varepsilon,w)\leq K_{r,t}(\varepsilon,w)\leq
K_{r,s}(\varepsilon,w)+K_{s,t}(\varepsilon,w)+1.$

###### Proof.

Without loss of generality, we set $r=0$. Recalling the definition of $T_{n}$
from (4), we set

$N\vcentcolon=\max\\{n\in\mathbb{N}\cup\\{0\\}:T_{n}(\varepsilon,w)\leq s\\}.$

Figure 6: Level crossings around the middle time $s$. In case (b), the level
crossing around $s$ is not counted in
$K_{r,s}(\varepsilon,w)+K_{s,t}(\varepsilon,w)$.

As shown in Figure 6, we have the following cases.

  1. (a)

If $w_{T_{N}(\varepsilon,w)}=w_{T_{1}(\varepsilon,s+w_{s+\cdot})}$ or if
$w_{T_{N}(\varepsilon,w)}=w_{s}$, then
$K_{r,t}(\varepsilon,w)=K_{r,s}(\varepsilon,w)+K_{s,t}(\varepsilon,w)$.

  2. (b)

Otherwise, we have
$K_{r,t}(\varepsilon,w)=K_{r,s}(\varepsilon,w)+K_{s,t}(\varepsilon,w)+1$. ∎

For our arguments, the following variants of $K$ will appear.

######  Notation 2.4 (Averaged $K$).

We set

$\bar{K}_{s,t}(\varepsilon,w)\vcentcolon=\varepsilon^{-1}\int_{-\varepsilon/2}^{\varepsilon/2}K_{s,t}(\varepsilon,w+\rho)\mathrm{d}\rho.$

color=yellow!60!white, size=, ]PD: $\bar{K}$ is actually $\varepsilon
TV^{\varepsilon}_{r,s}(\omega)$, which is something know. Should we use
$TV^{\varepsilon}$ notation? color=red!20!white, size=, ]TM: If $\varepsilon$
remains, it seems nicer to use $\bar{K}$, but we should of course mention the
relation.

######  Remark 2.5.

The quantity $\bar{K}$ is related to the so-called _truncated variation_
[Łoc08] defined by

$\displaystyle\operatorname{TV}^{\varepsilon}(w,[s,t])\vcentcolon=\sup_{\pi:\text{partition
of }[s,t]}\sum_{[u,v]\in\pi}\max\\{\lvert w_{v}-w_{u}\rvert-\varepsilon,0\\}.$

Indeed, as shown in [Łoc17, Theorem 1], we have the identity

$\displaystyle\operatorname{TV}^{\varepsilon}(w,[s,t])=\varepsilon^{-1}\bar{K}_{s,t}(\varepsilon,w).$

The advantage of $\bar{K}$ is that in addition to the superadditivity, it is
stationary.

######  Lemma 2.6 (Scaling, superadditivity and stationarity of $\bar{K}$).

Let $r<s<t$ and let $w$ be a process.

  1. (i)

For $\lambda>0$, we have

$\displaystyle(\bar{K}_{s,t}(\varepsilon,B))_{s<t,\varepsilon>0}\overset{\operatorname{d}}{=}(\bar{K}_{\lambda^{1/H}s,\lambda^{1/H}t}(\lambda\varepsilon,B))_{s<t,\varepsilon>0}.$

  2. (ii)

We have

$\displaystyle\bar{K}_{r,s}(\varepsilon,w)+\bar{K}_{s,t}(\varepsilon,w)\leq\bar{K}_{r,t}(\varepsilon,w)\leq\bar{K}_{r,s}(\varepsilon,w)+\bar{K}_{s,t}(\varepsilon,w)+1.$

  3. (iii)

We have
$\bar{K}_{s,t}(\varepsilon,w)=\bar{K}_{0,t-s}(\varepsilon,w_{s+\cdot}-w_{s})$.
In particular,

$\displaystyle\bar{K}_{s,t}(\varepsilon,B)\overset{\operatorname{d}}{=}\bar{K}_{0,t-s}(\varepsilon,B).$

###### Proof.

The claim (a) follows from Lemma 2.2 and the claim (b) follows from Lemma 2.3.
For the claim (c), we observe that for every $\rho\in\mathbb{R}$ we have

$\bar{K}_{s,t}(\varepsilon,w)=\bar{K}_{s,t}(\varepsilon,w+\rho)=\bar{K}_{0,t-s}(\varepsilon,w_{s+\cdot}+\rho)$

In particular, we choose $\rho\vcentcolon=-w_{s}$. ∎

######  Lemma 2.7 (Moments of $K$).

For every $p,t,\varepsilon\in(0,\infty)$ we have
$\mathbb{E}[\sup_{\rho\in\mathbb{R}}K_{0,t}(\varepsilon,B+\rho)^{p}]<\infty$.

###### Proof.

For $\alpha\in(0,H)$, we set

$\llbracket B\rrbracket_{C^{\alpha}([0,t])}\vcentcolon=\sup_{0\leq r<s\leq
t}\frac{|B_{s}-B_{r}|}{(s-r)^{\alpha}}.$

By the Kolmogorov continuity theorem, we have

$\mathbb{E}[\llbracket B\rrbracket_{C^{\alpha}([0,t])}^{p}]<\infty.$ (19)

We set color=green!60!white, size=, ]RL: In subscripts you use somtimes
$\llbracket B\rrbracket_{C^{\alpha}([0,t])}$ and sometimes $\llbracket
B\rrbracket_{C^{\alpha}([0,1])}$. I think we should decide for $t$

$\delta\vcentcolon=\varepsilon^{\frac{1}{\alpha}}(1+\llbracket
B\rrbracket_{C^{\alpha}([0,t])})^{-\frac{1}{\alpha}}.$

To lead to a contradiction, suppose that there exist integers $k$ and $n$ such
that

$k\delta\leq
T_{n}(\varepsilon,B+\rho)<T_{n+1}(\varepsilon,B+\rho)\leq(k+1)\delta\quad\text{with
}T_{n+1}(\varepsilon,B+\rho)\leq t.$

Then,

$\displaystyle\varepsilon=|B_{T_{n+1}(\varepsilon,B+\rho)}-B_{T_{n}(\varepsilon,B+\rho)}|$
$\displaystyle\leq\llbracket B\rrbracket_{C^{\alpha}([0,t])}\delta^{\alpha}$
$\displaystyle=\varepsilon\llbracket
B\rrbracket_{C^{\alpha}([0,t])}(1+\llbracket
B\rrbracket_{C^{\alpha}([0,t])})^{-1}<\varepsilon,$

which is a contradiction. Thus, we must have

$\\#\\{n:k\delta\leq T_{n}(\varepsilon,B+\rho)\leq(k+1)\delta\\}\leq
1\quad\text{for each }k$

and color=green!60!white, size=, ]RL: I think that instead of $\delta^{-1}$ we
should have $t/\delta$

$\sup_{\rho\in\mathbb{R}}K_{0,t}(\varepsilon,B+\rho)\leq
t\delta^{-1}=t\varepsilon^{-\frac{1}{\alpha}}(1+\llbracket
B\rrbracket_{C^{\alpha}([0,t])})^{\frac{1}{\alpha}},$ (20)

which is $L^{p}(\mathbb{P})$-integrable by (19). ∎

In view of Lemma 2.6, the family $(\mathbb{E}[\bar{K}_{0,t}(1,B)])_{t\geq 0}$
satisfies

$\mathbb{E}[\bar{K}_{0,s+t}(1,B)]\geq\mathbb{E}[\bar{K}_{0,s}(1,B)]+\mathbb{E}[\bar{K}_{0,t}(1,B)].$

Therefore, by Fekete’s Lemma, the following limit exists in $[0,\infty]$:

$\mathfrak{c}_{H}\vcentcolon=\lim_{t\to\infty}\frac{1}{t}\mathbb{E}[\bar{K}_{0,t}(1,B)]=\sup_{t>0}\frac{1}{t}\mathbb{E}[\bar{K}_{0,t}(1,B)].$
(21)

The constant $\mathfrak{c}_{H}$ coincides with the one from Theorem 1.1. The
following lemma shows that $\mathfrak{c}_{H}$ is non-trivial.

######  Lemma 2.8 (Non-triviality of $\mathfrak{c}_{H}$).

We have $\mathfrak{c}_{H}\in(0,\infty)$.

###### Proof.

To see $\mathfrak{c}_{H}>0$, we observe

$\displaystyle\mathfrak{c}_{H}\geq\mathbb{E}[\bar{K}_{0,1}(1,B)]\geq\mathbb{P}(B_{1}\geq
2)>0.$

To see $\mathfrak{c}_{H}<\infty$, we note by Lemma 2.6 that
$(\bar{K}_{s,t}+1)$, $s<t$ is subadditive. Therefore, color=green!60!white,
size=, ]RL: $J$ is undefinied.

$\displaystyle\mathfrak{c}_{H}\leq\mathbb{E}[\bar{K}_{0,1}(1,B)]+1\leq\mathbb{E}[\sup_{\rho}K_{0,1}(1,B-\rho)]+1,$

which is finite by Lemma 2.7. ∎

######  Remark 2.9.

By the subadditivity, we have

$\displaystyle\frac{\mathbb{E}[\bar{K}_{0,t}(1,B)]}{t}\leq\mathfrak{c}_{H}\leq\frac{\mathbb{E}[\bar{K}_{0,t}(1,B)]+1}{t}.$

In particular,

$\displaystyle\Big{\lvert}\mathfrak{c}_{H}-\frac{\mathbb{E}[\bar{K}_{0,t}(1,B)]}{t}\Big{\rvert}\leq
t^{-1}.$ (22)

###  2.2 The shifted stochastic sewing lemma

A key ingredient to the proof of Theorem 1.1 is the following lemma.

######  Lemma 2.10 (Shifted stochastic sewing lemma, [MP22, Theorem 1.1]).

Let $(\mathcal{F}_{t})_{t\in[0,T]}$ be a filtration and let $(A_{s,t})_{0\leq
s<t\leq T}$ be a two-parameter stochastic process such that $A_{s,t}$ is
$\mathcal{F}_{t}$-measurable. Suppose that for some $p\in[2,\infty)$ we have
$A_{s,t}\in L^{p}(\mathbb{P})$ for every $s<t$. Moreover, suppose that for
$v<s<u<t$ and $M\in(0,\infty)$ we have the estimates

$\displaystyle\lVert A_{s,t}-A_{s,u}-A_{u,t}\rVert_{L^{p}(\mathbb{P})}$
$\displaystyle\leq\Gamma_{1}(t-s)^{\beta_{1}},$
$\displaystyle\lVert\mathbb{E}[A_{s,t}-A_{s,u}-A_{u,t}|\mathcal{F}_{v}]\rVert_{L^{p}(\mathbb{P})}$
$\displaystyle\leq\Gamma_{2}(s-v)^{-\alpha}(t-s)^{\beta_{2}},$ (23)

where $t-s\leq M^{-1}(s-v)$ is assumed in (23), with
$\alpha,\beta_{1},\beta_{2}$ satisfying

$\displaystyle\min\\{2\beta_{1},2(\beta_{2}-\alpha),\beta_{2}\\}>1.$

Then there exists a unique $(\mathcal{F}_{t})$-adapted stochastic process
$(\mathcal{A}_{t})_{t\in[0,T]}$ with $\mathcal{A}_{0}=0$ such that

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

for every $v<s<t$, where $t-s\leq M^{-1}(s-v)$ is assumed in the second
estimate. Furthermore, we can find a $\delta>0$, depending only on
$\alpha,\beta_{1},\beta_{2}$, such that

$\displaystyle\Big{\lVert}\mathcal{A}_{T}-\sum_{[s,t]\in\pi}A_{s,t}\Big{\rVert}_{L^{p}(\mathbb{P})}\lesssim_{p,\alpha,\beta_{1},\beta_{2},T}(\Gamma_{1}+\Gamma_{2})\lvert\pi\rvert^{\delta}$

for every partition $\pi$ of $[0,T]$.

######  Remark 2.11.

The stochastic sewing lemma was first obtained in the seminal work [Lê20], and
the first shifted version, where $\alpha=0$, was obtained by [Ger22], which
was extended by [MP22] to handle the case $\alpha>0$.

######  Remark 2.12.

For our problems, the family $(A_{s,t})_{0\leq s<t\leq T}$ satisfies

$\displaystyle\lVert
A_{s,t}\rVert_{L^{p}(\mathbb{P})}\leq\Gamma_{1}(t-s)^{\beta_{1}},\quad\lVert\mathbb{E}[A_{s,t}|\mathcal{F}_{v}]\rVert_{L^{p}(\mathbb{P})}\leq\Gamma_{2}(s-v)^{-\alpha}(t-s)^{\beta_{2}}.$

In this case, by uniqueness of the limit $\mathcal{A}$, we must have
$\mathcal{A}\equiv 0$. In particular, with some
$\delta=\delta(\alpha,\beta_{1},\beta_{2})$ we have

$\displaystyle\Big{\lVert}\sum_{[s,t]\in\pi}A_{s,t}\Big{\rVert}_{L^{p}(\mathbb{P})}\lesssim_{T,p,\alpha,\beta_{1},\beta_{2}}(\Gamma_{1}+\Gamma_{2})\lvert\pi\rvert^{\delta}.$

Let us explain the strength of Lemma 2.10. The key is the estimate (23), which
allows us to bring a _weak_ estimate into a strong estimate. To illustrate an
example, recall the Mandelbrot–Van Ness representation (18), and we write
$(\mathcal{F}_{t})_{t\in\mathbb{R}}$ for the filtration generated by the
Brownian motion $W$ in (18). Suppose that we want to estimate
$\mathbb{E}[F(B)|\mathcal{F}_{v}]$. We then have

$\displaystyle\mathbb{E}[F(B)|\mathcal{F}_{v}]=\mathbb{E}[F(y+\tilde{B})]|_{y=\mathbb{E}[B|\mathcal{F}_{v}]},\quad\tilde{B}\vcentcolon=\int_{v}^{\cdot}\mathcal{K}(\cdot,s)\mathrm{d}W_{s}.$

Then the problem reduces to the estimate of a Gaussian expectation (weak
estimate), which allows us to leverage regularity of the Gaussian density.

Obviously, weak estimates provide better bounds than strong estimates do. As
this point of view is the key to our arguments, let us elaborate on one simple
example. Let $X$ be a one-dimensional standard Gaussian random variable, and
let $F\colon\mathbb{R}\to\mathbb{R}$. If we want to estimate

$\displaystyle\mathbb{E}[\lvert F(X)-F(X+a)\rvert]$

for a small $a\in\mathbb{R}$, then we need to assume some regularity of $F$.
On the other hand, if we want to estimate

$\displaystyle\lvert\mathbb{E}[F(X)]-\mathbb{E}[F(X+a)]\rvert,$

then by the Gaussian change of variable we get

$\displaystyle\mathbb{E}[F(X+a)]=e^{-\frac{a^{2}}{2}}\mathbb{E}[e^{aX}F(X)].$

Hence, only assuming $F$ is bounded, using the Cauchy–Schwarz inequality,
color=red!20!white, size=, ]TM: a bit more detail, CS we get

$\displaystyle\lvert\mathbb{E}[F(X)]-\mathbb{E}[F(X+a)]\rvert\lesssim\lVert
F\rVert_{L^{\infty}}\lvert a\rvert,\quad\forall a\text{ with }\lvert
a\rvert\leq 1.$

Very roughly speaking, we will go through such lines of reasoning to prove
Theorem 1.1. The Gaussian change of variable will be replaced by Girsanov’s
theorem in the spirit of Picard [Pic08], see Lemma 2.16.

###  2.3 Convergence of variation

As already suggested, to prove Theorem 1.1, we will apply the shifted
stochastic sewing, Lemma 2.10. We denote by
$(\mathcal{F}_{t})_{t\in\mathbb{R}}$ the filtration generated by the Brownian
motion $W$ appearing in the Mandelbrot–Van Ness representation (18). The
following is the first observation.

######  Lemma 2.13 (Asymptotic weak estimate of $\bar{K}$).

Let $\zeta\geq 1$ and $v<s<t$. We set
$\varepsilon\vcentcolon=(\frac{t-s}{\zeta})^{H}$. Then, if $\frac{t-s}{s-v}$
is sufficiently small and $p>1$, we have

$\displaystyle\lVert\mathbb{E}[\bar{K}_{s,t}(\varepsilon,B)|\mathcal{F}_{v}]-\mathbb{E}[\bar{K}_{0,\zeta}(1,B)]\rVert_{L^{p}(\mathbb{P})}\lesssim_{p,\zeta}\Big{(}\frac{t-s}{s-v}\Big{)}^{1-H}.$

Lemma 2.13 is an easy consequence of the following result.

######  Lemma 2.14 (Asymptotic independence, [Pic08, Lemma A.1]).

Let $0\leq v<s<t$. Let $F$ and $G$ be measurable with respect to
$\mathcal{F}_{v}$ and

$\sigma(B_{t^{\prime}}-B_{s^{\prime}}:s\leq s^{\prime}<t^{\prime}\leq t)$ (24)

respectively, and suppose that $F,G\in L^{p}(\mathbb{P})$ with
$p\in(1,\infty)$. If $(t-s)(s-v)^{-1}$ is sufficiently small, then we have

$|\mathbb{E}[FG]-\mathbb{E}[F]\mathbb{E}[G]|\lesssim_{p}\Big{(}\frac{t-s}{s-v}\Big{)}^{1-H}\|F\|_{L^{p}(\mathbb{P})}\|G\|_{L^{p}(\mathbb{P})}.$

######  Remark 2.15.

Consequently, we have the following estimate. Let $p\in[2,\infty)$, and we set
$p^{\prime}\vcentcolon=p/(p-1)$. If $G$ is measurable with respect to the
$\sigma$-algebra (24), then for any $\mathcal{F}_{v}$-measurable $F$ we have

$\displaystyle\lvert\mathbb{E}[(G-\mathbb{E}[G])F]\rvert\lesssim_{p}\Big{(}\frac{t-s}{s-v}\Big{)}^{1-H}\lVert
F\rVert_{L^{p^{\prime}}(\mathbb{P})}\lVert
G\rVert_{L^{p^{\prime}}(\mathbb{P})},$

provided that $\frac{t-s}{s-v}$ is sufficiently small. Since
$L^{p^{\prime}}(\mathbb{P})$ is the dual of $L^{p}(\mathbb{P})$ and
$p^{\prime}\leq p$, we have

$\displaystyle\lVert\mathbb{E}[G|\mathcal{F}_{v}]-\mathbb{E}[G]\rVert_{L^{p}(\mathbb{P})}\lesssim_{p}\Big{(}\frac{t-s}{s-v}\Big{)}^{1-H}\lVert
G\rVert_{L^{p}(\mathbb{P})},$ (25)

where we do not need to assume that $\frac{t-s}{s-v}$ is small.

###### Proof of Lemma 2.13.

By Lemma 2.6-(iii) (or by Remark 2.5), color=green!60!white, size=, ]RL:
rather by Remark 2.5 the random variable $\bar{K}_{s,t}(\varepsilon,B)$ is
measurable with respect to $\sigma(B_{r}-B_{s}:s\leq r\leq t)$. The estimate
(25) implies

$\displaystyle\lVert\mathbb{E}[\bar{K}_{s,t}(\varepsilon,B)|\mathcal{F}_{v}]-\mathbb{E}[\bar{K}_{s,t}(\varepsilon,B)]\rVert_{L^{p}(\mathbb{P})}\lesssim\Big{(}\frac{t-s}{s-v}\Big{)}^{1-H}\lVert\bar{K}_{s,t}(\varepsilon,B)\rVert_{L^{p}(\mathbb{P})}.$

By the stationarity and the scaling (Lemma 2.6),

$\displaystyle\bar{K}_{s,t}(\varepsilon,B)\overset{\operatorname{d}}{=}\bar{K}_{0,\zeta}(1,B)$

and the moment $\lVert\bar{K}_{0,\zeta}(1,B)\rVert_{L^{p}(\mathbb{P})}$ is
bounded by Lemma 2.7. The claim now follows. ∎

We recall the Mandelbrot–Van Ness representation (Definition 2.1). The next
lemma is a consequence of Girsanov’s theorem.

######  Lemma 2.16 (Weak estimate on $K$).

Let $v<s<t$, $\varepsilon\in(0,1)$,
$\rho,\rho^{\prime}\in[-\varepsilon/2,\varepsilon/2]$ and
$y:[v,t]\to\mathbb{R}$ be a deterministic continuous path. We set

$\tilde{B}^{v}_{r}\vcentcolon=\int_{v}^{r}(r-u)^{H-1/2}\mathrm{d}W_{u},\quad
v\leq r\leq t$ (26)

color=red!20!white, size=, ]TM: consistent with $\mathcal{B}$ or $\tilde{B}$
and

$b_{H}\vcentcolon=\frac{1}{4(1-H)}\Big{(}\frac{1}{\Gamma(H+1/2)\Gamma(3/2-H)}\Big{)}^{2},$

where $\Gamma$ is the usual Gamma function

$\displaystyle\Gamma(z)\vcentcolon=\int_{0}^{\infty}t^{z-1}e^{-t}\mathrm{d}t.$
(27)

We then have the bound

$|\mathbb{E}[K_{s,t}(\varepsilon,\tilde{B}^{v}+y+\rho)]-\mathbb{E}[K_{s,t}(\varepsilon,\tilde{B}^{v}+y+\rho^{\prime})]|\\\
\lesssim
e^{b_{H}|\rho-\rho^{\prime}|^{2}(s-v)^{-2}(t-v)^{2-2H}}\times\mathbb{E}[K_{s,t}(\varepsilon,\tilde{B}^{v}+y+\rho)^{2}]^{\frac{1}{2}}|\rho-\rho^{\prime}|(s-v)^{-1}(t-v)^{1-H}.$

###### Proof.

The proof is inspired by [Pic08, Theorem A.1]. Let
$\delta\vcentcolon=\rho^{\prime}-\rho$ and

$h_{r}\vcentcolon=\begin{cases}(s-v)^{-1}(r-v)\delta&\text{if }v\leq r\leq
s,\\\ \delta&\text{if }s\leq r.\end{cases}$

Note that the functions $r\mapsto\tilde{B}^{v}_{r}+y_{r}+\rho^{\prime}$ and
$r\mapsto\tilde{B}^{v}_{r}+y_{r}+h_{r}+\rho$ are equal on the interval
$[s,t]$. Thus,

$K_{s,t}(\varepsilon,\tilde{B}^{v}+y+\rho^{\prime})=K_{s,t}(\varepsilon,\tilde{B}^{v}+y+h+\rho).$

We claim

$h_{r}=\int_{v}^{r}(r-u)^{H-1/2}\mathrm{d}g_{u},$

where for $r>v$,

$g_{r}\vcentcolon=\frac{\delta\\{(r-v)^{3/2-H}-(r-r\wedge
s)^{3/2-H}\\}}{\Gamma(H+1/2)\Gamma(3/2-H)(3/2-H)(s-v)}.$

Indeed,

$\displaystyle\dot{g}_{r}$
$\displaystyle\vcentcolon=\frac{\mathrm{d}g_{r}}{\mathrm{d}r}$
$\displaystyle\phantom{\vcentcolon}=\frac{1}{\Gamma(H+1/2)\Gamma(3/2-H)}\frac{\delta}{s-v}\\{(r-v)^{1/2-H}-(r-s)^{1/2-H}\mathbf{1}_{\\{r>s\\}}\\}$

and

$\displaystyle\int_{v}^{r}(r-u)^{H-1/2}(u-v)^{1/2-H}\mathrm{d}u$
$\displaystyle=\int_{0}^{r-v}(r-v-u)^{H-1/2}u^{1/2-H}\mathrm{d}u$
$\displaystyle=(r-v)\int_{0}^{1}(1-u)^{H-1/2}u^{1/2-H}\mathrm{d}u$
$\displaystyle=\Gamma(H+1/2)\Gamma(3/2-H)(r-v),$

where in the last line the relation between the Beta function and the Gamma
function is used. Therefore,

$\int_{v}^{r}(r-u)^{H-1/2}\mathrm{d}g_{u}=\frac{\delta}{s-v}\\{(r-v)-(r-s)\mathbf{1}_{\\{r>s\\}}\\}=h_{r}.$

If we set

$F(w)\vcentcolon=K_{s,t}\Big{(}\varepsilon,\int_{v}^{\cdot}(\cdot-u)^{H-1/2}\mathrm{d}w_{u}+y+\rho\Big{)},$

then $K_{s,t}(\varepsilon,\tilde{B}^{v}+y+\rho^{\prime})=F(W+g)$ and by
Girsanov’s theorem (or the Cameron-Martin theorem)

$\mathbb{E}[F(W+g)]=\mathbb{E}\Big{[}e^{\int_{v}^{t}\dot{g_{r}}\mathrm{d}W_{r}-\frac{1}{2}\int_{v}^{t}|\dot{g}_{r}|^{2}\mathrm{d}r}F(W)\Big{]}.$

Thus,

$\mathbb{E}[K_{s,t}(\varepsilon,\tilde{B}^{v}+y+\rho^{\prime})]-\mathbb{E}[K_{s,t}(\varepsilon,\tilde{B}^{v}+y+\rho)]\\\
=\mathbb{E}\Big{[}\Big{\\{}e^{\int_{v}^{t}\dot{g_{r}}\mathrm{d}W_{r}-\frac{1}{2}\int_{v}^{t}|\dot{g}_{r}|^{2}\mathrm{d}r}-1\Big{\\}}K_{s,t}(\varepsilon,\tilde{B}^{v}+y+\rho)\Big{]}.$

By the Cauchy–Schwarz inequality, it is bounded by

$\mathbb{E}\Big{[}\Big{(}e^{\int_{v}^{t}\dot{g_{r}}\mathrm{d}W_{r}-\frac{1}{2}\int_{v}^{t}|\dot{g}_{r}|^{2}\mathrm{d}r}-1\Big{)}^{2}\Big{]}^{1/2}\mathbb{E}[K_{s,t}(\varepsilon,\tilde{B}^{v}+y+\rho)^{2}]^{\frac{1}{2}}.$

Since $\int_{v}^{t}\dot{g}_{r}\mathrm{d}W_{r}$ is centered Gaussian with
variance

$\displaystyle\int_{v}^{t}\lvert\dot{g}_{r}\rvert^{2}\mathrm{d}r\leq
2b_{H}\delta^{2}(s-v)^{-2}(t-v)^{2-2H},$

color=green!60!white, size=, ]RL: in the formula above I changed $a_{H}$ into
$b_{H}$ we obtain

$\displaystyle\mathbb{E}\Big{[}\Big{(}e^{\int_{v}^{t}\dot{g_{r}}\mathrm{d}W_{r}-\frac{1}{2}\int_{v}^{t}|\dot{g}_{r}|^{2}\mathrm{d}r}-1\Big{)}^{2}\Big{]}$
$\displaystyle=e^{\int_{v}^{t}|\dot{g}_{r}|^{2}\mathrm{d}r}-1$
$\displaystyle\leq\int_{v}^{t}|\dot{g}_{r}|^{2}\mathrm{d}r\times
e^{\int_{v}^{t}|\dot{g}_{r}|^{2}\mathrm{d}r}$ $\displaystyle\lesssim
e^{2b_{H}|\rho-\rho^{\prime}|^{2}(s-v)^{-2}(t-v)^{2-2H}}|\rho-\rho^{\prime}|^{2}(s-v)^{-2}(t-v)^{2-2H},$

(in the second line we used $e^{a}-1\leq a\times e^{a}$ for $a\geq 0$) which
completes the proof. ∎

###### Proof of Theorem 1.1.

In view of the scaling, we may suppose that $T=1$. The proof resembles that of
the subadditive ergodic theorem [Dur19, Theorem 6.4.1].

Step 1, lower bound. We fix a parameter $\zeta\geq 1$, which will go to
infinity at the end. (The parameter $\zeta$ corresponds to the parameter $m$
in [Dur19, Theorem 6.4.1].) Let $\pi_{\varepsilon,\zeta}$ be the partition of
$[0,1]$ with identical mesh size $\zeta\varepsilon^{\frac{1}{H}}$. By the
superadditivity (Lemma 2.3), using the relation $t-s=\zeta\varepsilon^{1/H}$
for $[s,t]\in\pi_{\varepsilon,\zeta}$, we obtain

$\displaystyle\varepsilon^{\frac{1}{H}}K_{0,1}(\varepsilon,B+\rho)\geq\sum_{[s,t]\in\pi_{\varepsilon,\zeta}}\varepsilon^{\frac{1}{H}}K_{s,t}(\varepsilon,B+\rho)=\zeta^{-1}\sum_{[s,t]\in\pi_{\varepsilon,\zeta}}A_{s,t}^{1},$

where $A^{1}_{s,t}\vcentcolon=K_{s,t}((\frac{t-s}{\zeta})^{H},B+\rho)(t-s)$.
Furthermore, we set

$\displaystyle
A^{2}_{s,t}\vcentcolon=\bar{K}_{s,t}\Big{(}\Big{(}\frac{t-s}{\zeta}\Big{)}^{H},B\Big{)}(t-s),\quad
A^{3}_{s,t}\vcentcolon=\mathbb{E}[\bar{K}_{0,\zeta}(1,B)](t-s).$

We see that $A_{s,t}\vcentcolon=A^{1}_{s,t}-A^{3}_{s,t}$ satisfies the
condition of Lemma 2.10. Indeed, by scaling we have

$\displaystyle\lVert
K_{s,t}(\varepsilon,B+\rho)\rVert_{L^{p}(\mathbb{P})}+\lVert\bar{K}_{s,t}(\varepsilon,B+\rho)\rVert_{L^{p}(\mathbb{P})}\lesssim_{p,\zeta}1$

and hence

$\displaystyle\lVert A_{s,t}\rVert_{L^{p}(\mathbb{P})}\leq\lVert
A^{1}_{s,t}\rVert_{L^{p}(\mathbb{P})}+\lVert
A^{3}_{s,t}\rVert_{L^{p}(\mathbb{P})}\lesssim_{p,\zeta}(t-s).$

To estimate the conditional expectation, let $(t-s)/(s-v)$ be so small that
the claim of Lemma 2.13 holds. Since

$\displaystyle
K_{s,t}(\varepsilon,B+\rho)-\bar{K}_{s,t}(\varepsilon,B)=\varepsilon^{-1}\int_{-\varepsilon/2}^{\varepsilon/2}\\{K_{s,t}(\varepsilon,B+\rho)-K_{s,t}(\varepsilon,B+\rho+\rho^{\prime})\\}\mathrm{d}\rho^{\prime},$

by Lemma 2.16, using $|\rho-\rho^{\prime}|\leq\varepsilon$,
$(s-v)^{-1}(t-v)^{1-H}\leq\sqrt{2}(s-v)^{-H}$ (this holds for $(t-s)/(s-v)$
sufficiently small) and $\varepsilon(s-v)^{-H}=((t-s)/(s-v))^{H}/\zeta^{H}$
color=green!60!white, size=, ]RL: I added few explanations here we have

$\displaystyle|\mathbb{E}[K_{s,t}(\varepsilon,B+\rho)-\bar{K}_{s,t}(\varepsilon,B)|\mathcal{F}_{v}]|$
$\displaystyle\lesssim\frac{\mathbb{E}[K_{s,t}(\varepsilon,B+\rho)^{2}|\mathcal{F}_{v}]^{\frac{1}{2}}}{\varepsilon}\int_{-\varepsilon/2}^{\varepsilon/2}e^{2b_{H}\varepsilon^{2}(s-v)^{-2H}}\varepsilon(s-v)^{-H}\mathrm{d}\rho^{\prime}$
$\displaystyle\lesssim_{\zeta}\mathbb{E}[K_{s,t}(\varepsilon,B+\rho)^{2}|\mathcal{F}_{v}]^{\frac{1}{2}}\Big{(}\frac{t-s}{s-v}\Big{)}^{H},$

which readily yields

$\displaystyle\lVert\mathbb{E}[A^{1}_{s,t}-A^{2}_{s,t}|\mathcal{F}_{v}]\rVert_{L^{p}(\mathbb{P})}\lesssim_{p,\zeta}\Big{(}\frac{t-s}{s-v}\Big{)}^{H}(t-s).$

By Lemma 2.13,

$\displaystyle\lVert\mathbb{E}[A^{2}_{s,t}-A^{3}_{s,t}|\mathcal{F}_{v}]\rVert_{L^{p}(\mathbb{P})}\lesssim_{p,\zeta}\Big{(}\frac{t-s}{s-v}\Big{)}^{1-H}(t-s).$

Therefore,

$\displaystyle\lVert\mathbb{E}[A_{s,t}|\mathcal{F}_{v}]\rVert_{L^{p}(\mathbb{P})}\lesssim_{p,\zeta}\Big{(}\frac{t-s}{s-v}\Big{)}^{\min\\{H,1-H\\}}(t-s),$

and we indeed see that $(A_{s,t})_{s<t}$ satisfies the conditions of Lemma
2.10.

Consequently, recalling Remark 2.12, we obtain

$\displaystyle\varepsilon^{\frac{1}{H}}K_{0,1}(\varepsilon,B)\geq\frac{\mathbb{E}[\bar{K}_{0,\zeta}(1,B)]}{\zeta}-R_{\varepsilon,\zeta},$
(28)

where

$\displaystyle\lVert
R_{\varepsilon,\zeta}\rVert_{L^{p}(\mathbb{P})}\lesssim_{p,\zeta}\varepsilon^{\delta}$

for some $\delta$ depending only on $H$. By the Borel–Cantelli lemma, if
$\varepsilon_{n}=O(n^{-\eta})$ for some $\eta>0$, then
$R_{\varepsilon_{n},\zeta}\to 0$ a.s. This implies

$\displaystyle\liminf_{n\to\infty}\varepsilon_{n}^{\frac{1}{H}}K_{0,1}(\varepsilon_{n},B+\rho)\geq\frac{\mathbb{E}[\bar{K}_{0,\zeta}(1,B)]}{\zeta}\quad\text{a.s.}$

Since $\zeta$ is an arbitrary real no smaller than $1$, the lower bound is
obtained.

Step 2, upper bound. Since $(K_{s,t}(\varepsilon,B+\rho)+1)_{s<t}$ is
subadditive, we obtain

$\displaystyle\varepsilon^{\frac{1}{H}}K_{0,1}(\varepsilon,B)\leq\frac{\mathbb{E}[\bar{K}_{0,\zeta}(1,B)]}{\zeta}+\frac{1}{\zeta}+R_{\varepsilon,\zeta},$
(29)

and we similarly obtain the upper bound. ∎

##  3 Local time via level crossings

In this section, we are interested in level crossings at a specific level. Our
goal of this section is to prove Theorem 1.4, fractional analogue of Chacon et
al. [Cha+81]. The key is to obtain a more quantitative version of Theorem 1.3,
as stated just below. Recall the definition of $U_{s,t}(\varepsilon,w)$ from
(8), which counts the total number of upcrossings from $0$ to $\varepsilon$ in
the interval $[s,t]$.

######  Theorem 3.1 (Quantitative bound on number of upcrossing $U$).

Let $H\in(0,1/2)$, $T\in(0,\infty)$ and $a\in\mathbb{R}$. The constant
$\mathfrak{c}_{H}$ is defined by (21). Almost surely, we have the following
quantitative bound:

$\Big{|}\varepsilon^{\frac{1}{H}-1}U_{0,T}(\varepsilon,B-a)-\frac{\mathfrak{c}_{H}}{2}L_{T}(a)\Big{|}\leq\zeta^{-1}L_{T}(a)+\mathcal{R}_{\varepsilon,\zeta,T,a},$

for all $\varepsilon\in(0,\infty)$ and $\zeta\in(1,\infty)$, where there
exists a positive $\kappa$ such that for every $p\in(0,\infty)$ we have

$\|\mathcal{R}_{\varepsilon,\zeta,T,a}\|_{L^{p}(\mathbb{P})}\leq
C_{p,\zeta}T^{1-H}\varepsilon^{\kappa}$

with $C_{p,\zeta}$ independent of $\varepsilon$, $T$ and $a$.

The proof of Theorem 3.1 is somewhat similar to that of Theorem 1.1,
especially the bounds (28) and (29). Indeed, it is based on the
super(sub)-additivity, Girsanov’s theorem and shifted stochastic sewing lemma.
However, a major difficulty here is that we cannot find a counterpart to
$\bar{K}$. This leads to more involved technical arguments. Therefore, instead
of directly going to the proof, in the next section we heuristically explain
our strategy.

###  3.1 Heuristics

Herein we explain our heuristic strategy to prove Theorem 3.1. Let
$(\mathcal{F}_{t})_{t\in\mathbb{R}}$ be the filtration generated by $W$ in the
Mandelbrot–Van Ness representation (18). We set

$\displaystyle A_{s,t}\vcentcolon=U_{s,t}((t-s)^{H},B-a)(t-s)^{1-H}.$

In view of Lemma 2.10, our goal is to show

$\displaystyle\mathbb{E}[A_{s,t}|\mathcal{F}_{v}]\approx\frac{\mathfrak{c}_{H}}{2}\mathbb{E}[L_{s,t}(a)|\mathcal{F}_{v}].$
(30)

Indeed, once the estimate (30) is proven, the rest of the argument is similar
to the proof of Theorem 1.1.

We thus explain heuristically how to prove (30). For simplicity, we set $a=0$,
and we write $\varepsilon\vcentcolon=(t-s)^{H}$. (Strictly speaking, we
actually introduce another parameter $\zeta$ going to infinity and set
$\varepsilon\vcentcolon=(\frac{t-s}{\zeta})^{H}$, but for simplicity here we
set $\zeta=1$.) Let us introduce another parameter $u\in(v,s)$ (in mind
$t-s\ll s-u\ll u-v$), and, recalling the Mandelbrot–Van Ness representation
from Definition 2.1, for $r\in[s,t]$ we decompose

$\displaystyle B_{r}$
$\displaystyle=\int_{-\infty}^{v}\mathcal{K}(r,\theta)\mathrm{d}W_{\theta}+\int_{v}^{u}\mathcal{K}(r,\theta)\mathrm{d}W_{\theta}+\int_{u}^{r}\mathcal{K}(r,\theta)\mathrm{d}W_{\theta}$
$\displaystyle=\vcentcolon X_{r}+Y_{r}+Z_{r}.$

In the interval $[s,t]$ the smooth processes $X$ and $Y$ do not change much
compared to $Z$. Therefore, we can freeze time of $X$ and $Y$ (Lemma 3.9):

$\mathbb{E}[U_{s,t}(\varepsilon,B)|\mathcal{F}_{v}]\approx\mathbb{E}[U_{s,t}(\varepsilon,X_{s}+Y_{s}+Z)|\mathcal{F}_{v}].$

But we see

$\mathbb{E}[U_{s,t}(\varepsilon,X_{s}+Y_{s}+Z)|\mathcal{F}_{v}]=\mathbb{E}[U_{s,t}(\varepsilon,x+Y_{s}+Z)]|_{x=X_{s}},$

and the Gaussian change of variable to $Y$ yields

$\mathbb{E}[U_{s,t}(\varepsilon,x+Y_{s}+Z)]=e^{-\frac{1}{2}(\frac{x}{\sigma_{Y}})^{2}}\mathbb{E}\Big{[}e^{\frac{xY_{s}}{\sigma_{Y}^{2}}}U_{s,t}(\varepsilon,Y_{s}+Z)\Big{]},$

where $\sigma_{Y}$ is the variance of $Y_{s}$ (Lemma 3.11).

For $U_{s,t}(\varepsilon,Y_{s}+Z)$ to be positive, $Y_{s}$ must be around $0$
with high probability. (In other words, if $Y_{s}$ is far away from $0$, the
process $Z$ must move quite a lot, which is costly.) Therefore (Lemma 3.12),

$\displaystyle\mathbb{E}\Big{[}e^{\frac{xY_{s}}{\sigma_{Y}^{2}}}U_{s,t}(\varepsilon,Y_{s}+Z)\Big{]}\approx\mathbb{E}[U_{s,t}(\varepsilon,Y_{s}+Z)]\approx\mathbb{E}\Big{[}U_{s,t}(\varepsilon,Y+Z)\Big{]}.$

As $v\ll u\ll s$, we have $\sigma_{Y}\approx\sigma_{Y+Z}$, with $\sigma_{Y+Z}$
being the variance of $Y+Z$ (Lemma 3.15). In the end, we have (Lemma 3.7)
color=green!60!white, size=, ]RL: I think that instead of $U_{s,t}(1,Y+Z)$ on
the rhs of the equation below we should have $U_{s,t}(\varepsilon,Y+Z)$ there

$\mathbb{E}[U_{s,t}(\varepsilon,B)|\mathcal{F}_{v}]\approx\mathbb{E}\Big{[}U_{s,t}(\varepsilon,Y+Z)\Big{]}e^{-\frac{1}{2}(\frac{X_{s}}{\sigma_{Y+Z}})^{2}}.$

It is well-known that the local time is heuristically represented as integral
of Dirac’s delta function along $B$ (see Lemma 3.19). We then observe (Lemma
3.19) color=green!60!white, size=, ]RL: Maybe it is in place to here to recall
a relationship between the local time and the integral below. TM: sentence
added.

$\displaystyle\mathbb{E}\Big{[}\int_{s}^{t}\delta_{0}(B_{r})\mathrm{d}r\Big{|}\mathcal{F}_{v}\Big{]}$
$\displaystyle\approx\int_{s}^{t}\mathbb{E}[\delta_{0}(B_{s})|\mathcal{F}_{v}]\mathrm{d}r$
$\displaystyle=\frac{1}{\sqrt{2\pi}\sigma_{Y+Z}}e^{-\frac{X_{s}^{2}}{2\sigma_{Y+Z}^{2}}}(t-s).$

It is not obvious, but in Lemma 3.17 we prove

$\displaystyle\sqrt{2\pi}\sigma_{Y+Z}(t-s)^{-H}\mathbb{E}[U_{s,t}(\varepsilon,Y+Z)]\approx\frac{\mathfrak{c}_{H}}{2}.$

Now we see (30). With this heuristic argument in mind, we move to a rigorous
proof in the next section.

###  3.2 Convergence to local time

####  3.2.1 Estimates on level crossings

The following process will appear in our argument.

######  Definition 3.2.

The kernel $\mathcal{K}$ is defined by (17). We denote by
$\tilde{B}=\tilde{B}^{H}$ the _Riemann-Liouville process_

$\tilde{B}_{t}\vcentcolon=\int_{0}^{t}\mathcal{K}(t,r)\mathrm{d}W_{r}.$

In view of the Mandelbrot–Van Ness representation (Definition 2.1), we have

$\displaystyle
B_{t}=\int_{-\infty}^{0}\mathcal{K}(t,r)\mathrm{d}W_{r}+\tilde{B}_{t}.$ (31)

We begin with three elementary lemmas.

######  Lemma 3.3 (Scaling of $U$).

We have the following scaling property: for $\lambda>0$,

$(U_{s,t}(\varepsilon,B+\rho))_{s<t,\varepsilon>0,\rho\in\mathbb{R}}\overset{\mathrm{d}}{=}(U_{\lambda^{1/H}s,\lambda^{1/H}t}(\lambda\varepsilon,B+\lambda\rho))_{s<t,\varepsilon>0,\rho\in\mathbb{R}}.$

A similar result holds with $B$ replaced by $\tilde{B}$.

###### Proof.

Similarly to Lemma 2.2, it follows from the scaling property of $B$ and
$\tilde{B}$. ∎

######  Notation 3.4.

color=yellow!60!white, size=, ]PD: maybe we want this as a noration instead of
definition TM: corrected We set

$\bar{U}_{s,t}(\varepsilon,w)\vcentcolon=U_{s,t}(\varepsilon,w)+\mathbf{1}_{\\{w_{s}\in(0,\varepsilon)\\}}.$

######  Lemma 3.5 (Sub/super-additivity of $U$).

For $s<u<t$ we have

$U_{s,t}(\varepsilon,w)\geq
U_{s,u}(\varepsilon,w)+U_{u,t}(\varepsilon,w),\quad\bar{U}_{s,t}(\varepsilon,w)\leq\bar{U}_{s,u}(\varepsilon,w)+\bar{U}_{u,t}(\varepsilon,w).$

###### Proof.

We have

$\displaystyle
U_{s,t}(\varepsilon,w)=U_{s,u}(\varepsilon,w)+U_{u,t}(\varepsilon,w)+1$

if there exist $a$ and $b$ such that $s\leq a<u<b\leq t$,
$w_{a}=0,w_{b}=\varepsilon$ and $w_{r}\in(0,\varepsilon)$ for all $r\in(a,b)$,
and otherwise

$U_{s,t}(\varepsilon,w)=U_{s,u}(\varepsilon,w)+U_{u,t}(\varepsilon,w).\qed$

######  Lemma 3.6 (Moment bound on $U$).

There exists a positive constant $\beta=\beta(H)$ color=green!60!white, size=,
]RL: maybe better to use $\beta$ instead of $\kappa$ here, since it is later
used where $\kappa$ appears in quite different context such that for
$a\in\mathbb{R}$, $\varepsilon\in(0,\infty)$, $s<t$ and $p\in(0,\infty)$ we
have

$\displaystyle\lVert
U_{s,t}(\varepsilon,B-a)\rVert_{L^{p}(\mathbb{P})}\lesssim_{H,p,\varepsilon}1+(t-s)^{\beta}.$

A similar estimate holds with $B$ replaced by $\tilde{B}$.

###### Proof.

Regarding $B$, the claim follows from the obvious inequality
$U_{s,t}(\varepsilon,B-a)\leq K_{s,t}(\varepsilon,B-a)$ and the estimate (20).
color=green!60!white, size=, ]RL: Regarding $\tilde{B}$, it seems for me too
fast. Maybe some more detailed argument like scaling and Kolmogorov’s lemma
too? Regarding $\tilde{B}$, by (31) we observe that

$\displaystyle\lVert\tilde{B}_{s,t}\rVert_{L^{p}(\mathbb{P})}\leq\lVert
B_{s,t}\rVert_{L^{p}(\mathbb{P})}.$

This yields an estimate on Hölder norm of $\tilde{B}$ in $L^{m}(\mathbb{P})$,
with which we can proceed as in $B$. ∎

color=red!20!white, size=, ]TM: $\zeta$ is defined here; in particular it is
independent of $\varepsilon$. We introduce some notation that will be used
throughout Section 3.2.1. As in the proof of Theorem 1.1, we fix $\zeta\geq
1$, and at the very end we let $\zeta\to\infty$. We fix $v<u<s<t$ with $t-s\ll
s-u\ll u-v$ and set

$\varepsilon\vcentcolon=\left(\frac{t-s}{\zeta}\right)^{H},$ (32)

as shown in Figure 7.

Figure 7: Parameters for Lemma 3.7

We set

$X_{r}\vcentcolon=\int_{-\infty}^{v}\mathcal{K}(r,\theta)\mathrm{d}W_{\theta}-a,\quad
Y_{r}\vcentcolon=\int_{v}^{u}\mathcal{K}(r,\theta)\mathrm{d}W_{\theta},\quad
Z_{r}\vcentcolon=\int_{u}^{r}\mathcal{K}(r,\theta)\mathrm{d}W_{\theta}$ (33)

for $r\in[s,t]$. Let $(\mathcal{F}_{t})_{t\in\mathbb{R}}$ be the filtration
generated by $W$ in the Mandelbrot–Van Ness representation (18). We have the
identity

$\mathbb{E}[U_{s,t}(\varepsilon,B-a)|\mathcal{F}_{v}]=\mathbb{E}[U_{s,t}(\varepsilon,x+Y+Z)]|_{x=X}.$

Finally, we write

$\displaystyle\sigma_{Y}^{2}$
$\displaystyle\vcentcolon=\mathbb{E}[Y_{s}^{2}]=\frac{1}{2H}\\{(s-v)^{2H}-(s-u)^{2H}\\},$
(34) $\displaystyle\sigma_{Y+Z}^{2}$
$\displaystyle\vcentcolon=\mathbb{E}[(Y_{s}+Z_{s})^{2}]=\frac{1}{2H}(s-v)^{2H}.$
(35)

In the spirit of the shifted stochastic sewing (Lemma 2.10), we will estimate

$\mathbb{E}[U_{s,t}(\varepsilon,B-a)|\mathcal{F}_{v}].$

The most crucial ingredient to Theorem 3.1 is the following.
color=red!20!white, size=, ]TM: This is the most technical but most important
lemma.

######  Lemma 3.7 (Asymptotic weak estimate on $U$).

Let $H\in(0,1/2)$ and $p\in(1,\infty)$. We further let $0\leq v<s<t\leq 1$ and
$\zeta\in[1,\infty)$, and set $\varepsilon$ as in (32). Let
$(\mathcal{F}_{t})_{t\in\mathbb{R}}$ be the filtration generated by $W$ in the
Mandelbrot–Van Ness representation (18). We define $X$ by (33) and
$\sigma_{Y+Z}$ by (35). For every $\kappa\in(0,1)$, if $\frac{t-s}{s-v}$ is
sufficiently small, we have color=green!60!white, size=, ]RL: Maybe instead of
$\Big{(}\frac{t-s}{\zeta}\Big{)}^{H}$ it is better to write $\varepsilon$
here? so we get:

$\mathbb{E}[U_{s,t}(\varepsilon,B-a)|\mathcal{F}_{v}]=\frac{\mathbb{E}[\bar{K}_{0,\zeta}(1,B)]}{2\sqrt{2\pi}\sigma_{Y+Z}}e^{-\frac{1}{2}(\frac{X_{s}}{\sigma_{Y+Z}})^{2}}\varepsilon+R_{v,s,t},$
(36)

where (we omit the $a$-dependence of $R_{v,s,t}$ from the notation and the
following estimate holds uniformly in $a$):

$\displaystyle\lVert
R_{v,s,t}\rVert_{L^{p}(\mathbb{P})}\lesssim_{p,\zeta,\kappa}\Big{(}\frac{t-s}{s-v}\Big{)}^{(2-\kappa)H(1-H)}(t-s)^{-\kappa
H}.$ (37)

######  Remark 3.8.

Due to scaling, it suffices to prove Theorem 3.1 with $T=1$. Therefore, in
Lemma 3.7 we assume $t\leq 1$. We then do not have to keep track of dependency
of constants on the final time $T$.

The proof of Lemma 3.7, to which the rest of Section 3.2.1 is devoted, will be
built on several technical lemmas. For the sake of the next lemma, we recall
the Riemann–Liouville operator (e.g., [Pic11])

$I_{\alpha}f(r)\vcentcolon=\frac{1}{\Gamma(\alpha)}\int_{s}^{r}(r-\theta)^{\alpha-1}f(\theta)\mathrm{d}\theta,\quad
r>s,$

where $\Gamma$ is the Gamma function (27) and $\alpha>0$. If $f$ is Lipschitz
with $f_{s}=0$ and $\alpha\in(-1,0]$, we set

$I_{\alpha}f(r)\vcentcolon=\frac{1}{\Gamma(1+\alpha)}\int_{s}^{r}(r-\theta)^{\alpha}\dot{f}(\theta)\mathrm{d}\theta.$

The family $(I_{\alpha})_{\alpha>-1}$ has the semigroup property
$I_{\alpha}I_{\beta}=I_{\alpha+\beta}$.

######  Lemma 3.9 (Fix time of $X$ and $Y$).

For $p\in(1,\infty)$, $\zeta\in[1,\infty)$, $\varepsilon$ given by (32) and
$\kappa\in(1-1/p,1)$, there exists a positive constant $c$, depending on $H$
only, color=yellow!60!white, size=, ]PD: what is the conditions on $c$?
color=red!20!white, size=, ]TM: $c$ is $c_{2}$ in the proof. such that if
$(t-s)(s-u)^{-1}$ is sufficiently small, color=red!20!white, size=, ]TM: how
small depends on $p$ and $\kappa$ we have

$\|\mathbb{E}[U_{s,t}(\varepsilon,B-a)|\mathcal{F}_{u}]-\mathbb{E}[U_{s,t}(\varepsilon,X_{s}+Y_{s}+Z)|\mathcal{F}_{u}]\|_{L^{p}(\mathbb{P})}\\\
\lesssim_{p,\kappa,\zeta}\lVert
U_{s,t}(\varepsilon,B-a)\rVert_{L^{1}(\mathbb{P})}^{1-\kappa}e^{-ca^{2}}\Big{(}\frac{t-s}{s-u}\Big{)}^{1-H}.$

color=yellow!60!white, size=, ]PD: $\zeta$ is not in the statement. I think
$\zeta=\varepsilon^{-1/H}$

###### Proof.

The proof is similar to [Pic08, Lemma A.1]. We have

$\mathbb{E}[U_{s,t}(\varepsilon,X_{s}+Y_{s}+Z)|\mathcal{F}_{u}]=\mathbb{E}[U_{s,t}(\varepsilon,x_{s}+y_{s}+Z)]|_{x=X,y=Y}$

and

$\mathbb{E}[U_{s,t}(\varepsilon,x_{s}+y_{s}+Z)]=\mathbb{E}[U_{s,t}(\varepsilon,x+y+w+Z)]$

where

$w_{r}\vcentcolon=-(x_{r}+y_{r}-x_{s}-y_{s}),\quad r\in[s,t].$

Since $X$ and $Y$ are smooth on $[s,t]$, we may suppose that the realizations
$x$ and $y$ are smooth as well. As $w_{s}=0$,

$w=I_{1}\dot{w}=I_{H-\frac{1}{2}}I_{\frac{3}{2}-H}\dot{w}$

and

$w_{r}+\int_{s}^{r}\mathcal{K}(r,\theta)\mathrm{d}W_{\theta}=\int_{s}^{r}\mathcal{K}(r,\theta)\mathrm{d}\Big{(}W_{\theta}+c_{1}\big{(}I_{\frac{3}{2}-H}\dot{w}\big{)}_{\theta}\Big{)}$

for some constant $c_{1}$ depending only on $H$.

By Girsanov’s theorem,

$\mathbb{E}[U_{s,t}(\varepsilon,x+y+w+Z)]=\mathbb{E}\Big{[}U_{s,t}(\varepsilon,x+y+Z)\\\
\times\exp\Big{(}c_{1}\int_{s}^{t}\frac{\mathrm{d}}{\mathrm{d}\theta}I_{\frac{3}{2}-H}\dot{w}\,\mathrm{d}W_{\theta}-\frac{c_{1}^{2}}{2}\int_{s}^{t}\Big{|}\frac{\mathrm{d}}{\mathrm{d}\theta}I_{\frac{3}{2}-H}\dot{w}\Big{|}^{2}\mathrm{d}\theta\Big{)}\Big{]}.$

Therefore, if $p^{-1}+q^{-1}=1$, by Hölder’s inequality,

$|\mathbb{E}[U_{s,t}(\varepsilon,x+y+Z)]-\mathbb{E}[U_{s,t}(\varepsilon,x_{s}+y_{s}+Z)]|\\\
\lesssim\mathbb{E}\Big{[}\Big{|}\exp\Big{(}c_{1}\int_{s}^{t}\frac{\mathrm{d}}{\mathrm{d}\theta}I_{\frac{3}{2}-H}\dot{w}\mathrm{d}W_{\theta}-\frac{c^{2}_{1}}{2}\int_{s}^{t}\Big{|}\frac{\mathrm{d}}{\mathrm{d}\theta}I_{\frac{3}{2}-H}\dot{w}\Big{|}^{2}\mathrm{d}\theta\Big{)}-1\Big{|}^{q}\Big{]}^{\frac{1}{q}}\\\
\times\mathbb{E}[U_{s,t}(\varepsilon,x+y+Z)^{p}]^{\frac{1}{p}}.$

Since the random variable

$\int_{s}^{t}\frac{\mathrm{d}}{\mathrm{d}\theta}I_{\frac{3}{2}-H}\dot{w}(\theta)\mathrm{d}W_{\theta}$

is Gaussian, by Lemma A.1,

$\mathbb{E}\Big{[}\Big{|}\exp\Big{(}c_{1}\int_{s}^{t}\frac{\mathrm{d}}{\mathrm{d}\theta}I_{\frac{3}{2}-H}\dot{w}\mathrm{d}W_{\theta}-\frac{c^{2}_{1}}{2}\int_{s}^{t}\Big{|}\frac{\mathrm{d}}{\mathrm{d}\theta}I_{\frac{3}{2}-H}\dot{w}\Big{|}^{2}\mathrm{d}\theta\Big{)}-1\Big{|}^{q}\Big{]}^{\frac{1}{q}}\\\
\lesssim_{q}\Big{(}\int_{s}^{t}\Big{|}\frac{\mathrm{d}}{\mathrm{d}\theta}I_{\frac{3}{2}-H}\dot{w}\Big{|}^{2}\mathrm{d}\theta\Big{)}^{\frac{1}{2}}\exp\Big{(}C_{q}\int_{s}^{t}\Big{|}\frac{\mathrm{d}}{\mathrm{d}\theta}I_{\frac{3}{2}-H}\dot{w}\Big{|}^{2}\mathrm{d}\theta\Big{)}.$

Hence, by setting

$\displaystyle
S_{s,t}\vcentcolon=\int_{s}^{t}\Big{|}\frac{\mathrm{d}}{\mathrm{d}\theta}I_{\frac{3}{2}-H}(\dot{X}+\dot{Y})\Big{|}^{2}\mathrm{d}\theta,$

we have color=green!60!white, size=, ]RL: What is $\mathbb{E}_{u}...$ here? I
guess it is just $\mathbb{E}(...|{\cal F}_{u})$

$\displaystyle\|\mathbb{E}[U_{s,t}(\varepsilon,B-a)|\mathcal{F}_{u}]-\mathbb{E}[U_{s,t}(\varepsilon,X_{s}+Y_{s}+Z)|\mathcal{F}_{u}]\|_{L^{p}(\mathbb{P})}$
$\displaystyle\lesssim\Big{[}\mathbb{E}\left(\mathbb{E}[U_{s,t}(\varepsilon,B-a)^{p}|\mathcal{F}_{u}]S_{s,t}^{\frac{p}{2}}e^{pC_{q}S_{s,t}}\right)\Big{]}^{\frac{1}{p}}$
$\displaystyle\leq\|U_{s,t}(\varepsilon,B-a)\|_{L^{p_{1}}(\mathbb{P})}\Big{\|}S_{s,t}^{\frac{1}{2}}e^{C_{q}S_{s,t}}\Big{\|}_{L^{q_{1}}(\mathbb{P})},$

where $p_{1}^{-1}+q_{1}^{-1}=p^{-1}$. Choose $p_{2}$ so that
$p_{1}^{-1}=(1-\kappa)+\kappa p_{2}^{-1}$ (since $\kappa>1-p^{-1}$, this is
possible by choosing $p_{1}$ close to $p$). By the log-convexity of $L^{p}$
norms,

$\displaystyle\|U_{s,t}(\varepsilon,B-a)\|_{L^{p_{1}}(\mathbb{P})}\leq\|U_{s,t}(\varepsilon,B-a)\|_{L^{1}(\mathbb{P})}^{1-\kappa}\|U_{s,t}(\varepsilon,B-a)\|_{L^{p_{2}}(\mathbb{P})}^{\kappa}.$

We also have (by the Cauchy–Schwarz inequality)

$\displaystyle\|U_{s,t}(\varepsilon,B-a)\|_{L^{p_{2}}(\mathbb{P})}$
$\displaystyle\leq\mathbb{P}(\lVert B\rVert_{L^{\infty}([0,1])}\geq
a)^{\frac{1}{2}}\lVert
U_{s,t}(\varepsilon,B-a)\rVert_{L^{2p_{2}}(\mathbb{P})}$
$\displaystyle\lesssim e^{-c_{2}a^{2}}\lVert
U_{s,t}(\varepsilon,B-a)\rVert_{L^{2p_{2}}(\mathbb{P})}.$

The scaling property (Lemma 3.3) gives

$\|U_{s,t}(\varepsilon,B-a)\|_{L^{2p_{2}}(\mathbb{P})}=\|U_{s/(t-s),t/(t-s)}(\zeta^{-H},B-(t-s)^{-H}a)\|_{L^{2p_{2}}(\mathbb{P})}.$

By Lemma 3.6,

$\displaystyle\|U_{s/(t-s),t/(t-s)}(\zeta^{-H},B-(t-s)^{-H}a)\|_{L^{2p_{2}}(\mathbb{P})}\lesssim_{p_{2},\zeta}1.$

color=yellow!60!white, size=, ]PD: $\zeta$ depends on $H$ and $\varepsilon$
color=green!60!white, size=, ]RL: opposite, $\varepsilon$ depends on $\zeta$,
$t-s$ and $H$ It remains to see

$\displaystyle\Big{\|}S_{s,t}^{\frac{1}{2}}e^{C_{q}S_{s,t}}\Big{\|}_{L^{q_{1}}(\mathbb{P})}\lesssim\Big{(}\frac{t-s}{s-u}\Big{)}^{1-H},\quad\text{if
}\frac{t-s}{s-u}\text{ is sufficiently small.}$

This was essentially proven in [Pic08, Lemma A.1] (our $S_{s,t}$ corresponds
to $L$ therein). color=red!20!white, size=, inline]TM: Easy way to see this:
we have

$\displaystyle\lVert I_{1/2-H}(\dot{X}+\dot{Y})\rVert_{L^{2}([0,T])}=\lVert
I_{-1/2-H}(X+Y)\rVert_{L^{2}([0,T])}\sim\lVert
X+Y\rVert_{W^{H+1/2,2}([0,T])}.$

Therefore,

$\displaystyle\lVert
S_{s,t}\rVert_{L^{1}(\mathbb{P})}^{2}\lesssim\int_{s}^{t}\int_{s}^{t}\frac{\lVert(X+Y)_{r_{1},r_{2}}\rVert_{L^{2}(\mathbb{P})}^{2}}{\lvert
r_{1}-r_{2}\rvert^{2(H+1/2)+1}}\mathrm{d}r_{1}\mathrm{d}r_{2}.$

Since

$\displaystyle\lVert(X+Y)_{r_{1},r_{2}}\rVert_{L^{2}(\mathbb{P})}^{2}\lesssim(s-u)^{2H-2}\lvert
r_{1}-r_{2}\rvert^{2},$

we see

$\displaystyle\lVert
S_{s,t}\rVert_{L^{1}(\mathbb{P})}^{2}\lesssim\int_{s}^{t}\int_{s}^{t}\frac{(s-u)^{2H-2}(r_{2}-r_{1})^{2}}{\lvert
r_{2}-r_{1}\rvert^{2H+2}}\mathrm{d}r_{1}\mathrm{d}r_{2}\lesssim(s-u)^{2H-2}(t-s)^{2-2H}.$

Now apply Fernique’s theorem. color=green!60!white, size=, inline]RL: Maybe it
is worth to write this remark in the text, with the explanation what does the
notation $W^{H+1/2,2}([0,T])$ and $\lVert(X+Y)_{r_{1},r_{2}}\rVert$ mean … ∎

######  Remark 3.10.

We note that a similar reasoning shows that for $p<p_{1}<\infty$ if
$\frac{t-s}{s-u}$ is sufficiently small, we have

$\displaystyle\lVert
U_{s,t}(\varepsilon,Y_{s}+Z)\rVert_{L^{p}(\mathbb{P})}\lesssim_{\zeta,p,p_{1}}\lVert
U_{s,t}(\varepsilon,Y+Z)\rVert_{L^{p_{1}}(\mathbb{P})}.$

color=yellow!60!white, size=, ]PD: $\lesssim_{p,\varepsilon,H}$. Do the bound
also depend on choice of $p_{1}$? color=red!20!white, size=, inline]TM: Yes,
it depends on $p_{1}$. I think it is better not to write dependency on $H$, as
it is not important. I will comment in Notation section.

######  Lemma 3.11 (Gaussian change of variable).

Recall $\sigma_{Y}$ from (34). We have the estimate

$\mathbb{E}[U_{s,t}(\varepsilon,X_{s}+Y_{s}+Z)|\mathcal{F}_{v}]=e^{-\frac{1}{2}(\frac{X_{s}}{\sigma_{Y}})^{2}}\mathbb{E}\Big{[}\Big{.}e^{\frac{X_{s}Y_{s}}{\sigma_{Y}^{2}}}U_{s,t}(\varepsilon,Y_{s}+Z)\Big{|}\mathcal{F}_{v}\Big{]}.$

###### Proof.

We set $F(\eta)\vcentcolon=\mathbb{E}[U_{s,t}(\varepsilon,\eta+Z)]$ for
$\eta\in\mathbb{R}$. Since $X,Y$ and $Z$ are independent and $X$ is
$\mathcal{F}_{v}$-measurable, color=green!60!white, size=, inline]RL: Maybe it
is easier to omit the middle term and just write

$\mathbb{E}[U_{s,t}(\varepsilon,X_{s}+Y_{s}+Z)|\mathcal{F}_{v}]=\mathbb{E}[F(x_{s}+Y_{s})]|_{x=X}.$

Since $Y_{s}$ is Gaussian with the variance $\sigma_{Y}^{2}$ and the mean $0$,
we observe

$\displaystyle\mathbb{E}[F(x_{s}+Y_{s})]$
$\displaystyle=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}F(x_{s}+\sigma_{Y}\eta)e^{-\frac{\eta^{2}}{2}}\mathrm{d}\eta$
$\displaystyle=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}F(\sigma_{Y}\eta)e^{-\frac{1}{2}(\eta-\sigma_{Y}^{-1}x_{s})^{2}}\mathrm{d}\eta$
$\displaystyle=e^{-\frac{1}{2}(\frac{x_{s}}{\sigma_{Y}})^{2}}\mathbb{E}\Big{[}e^{\frac{Y_{s}x_{s}}{\sigma_{Y}^{2}}}F(Y_{s})\Big{]}.$

color=yellow!60!white, size=, ]PD: I am strugeling to understand the fist
equation with $\mathbb{E}[F(x_{s}+Y_{s})]=$ color=red!20!white, size=,
inline]TM: $Y_{s}$ is Gaussian with variance $\sigma_{Y}^{2}$. The claim thus
follows. ∎

######  Lemma 3.12 ($Y_{s}$ must be near $0$).

For every $p_{1}\in(1,\infty)$, if $\frac{t-s}{s-u}$ is sufficiently small,
then

$\Big{|}\mathbb{E}\Big{[}e^{\frac{x_{s}Y_{s}}{\sigma_{Y}^{2}}}U_{s,t}(\varepsilon,Y_{s}+Z)\Big{]}-\mathbb{E}[U_{s,t}(\varepsilon,Y_{s}+Z)]\Big{|}\\\
\lesssim_{H,p_{1}}\frac{|x_{s}|(t-u)^{H}}{\sigma_{Y}^{2}}e^{c(\frac{|x_{s}|(t-u)^{H}}{\sigma_{Y}^{2}})^{2}}\|U_{s,t}(\varepsilon,Y+Z)\|_{L^{p_{1}}(\mathbb{P})}$

with $c$ depending only on $H$ and $p_{1}$.

###### Proof.

For $U_{s,t}(\varepsilon,Y_{s}+Z)$ to be non-zero, we must have
$\inf_{r\in[s,t]}|Y_{s}+Z_{r}|=0$. Therefore,

$\mathbb{E}\Big{[}e^{\frac{x_{s}Y_{s}}{\sigma_{Y}^{2}}}U_{s,t}(\varepsilon,Y_{s}+Z)\Big{]}-\mathbb{E}[U_{s,t}(\varepsilon,Y_{s}+Z)]\\\
=\mathbb{E}\Big{[}\Big{(}e^{\frac{x_{s}Y_{s}}{\sigma_{Y}^{2}}}-1\Big{)}U_{s,t}(\varepsilon,Y_{s}+Z)\mathbf{1}_{\\{\|Z\|_{L^{\infty}([s,t])}\geq|Y_{s}|\\}}\Big{]}.$

Using the inequality

$|e^{\lambda}-1|\leq e^{|\lambda|}|\lambda|,\quad\lambda\in\mathbb{R},$

we estimate

$\Big{|}\mathbb{E}\Big{[}\Big{(}e^{\frac{x_{s}Y_{s}}{\sigma_{Y}^{2}}}-1\Big{)}U_{s,t}(\varepsilon,Y_{s}+Z)\mathbf{1}_{\\{\|Z\|_{L^{\infty}([s,t])}\geq|Y_{s}|\\}}\Big{]}\Big{|}\\\
\leq\frac{|x_{s}|}{\sigma_{Y}^{2}}\mathbb{E}\Big{[}e^{\frac{|x_{s}|\|Z\|_{L^{\infty}([s,t])}}{\sigma_{Y}^{2}}}\|Z\|_{L^{\infty}([s,t])}U_{s,t}(\varepsilon,Y_{s}+Z)\Big{]},$

and, by Hölder’s inequality, the expectation on the right hand side is bounded
by color=green!60!white, size=, ]RL: I changed the order of apperance of
square brackets

$\left[\mathbb{E}U_{s,t}(\varepsilon,Y_{s}+Z)^{p_{1}}\right]^{\frac{1}{p_{1}}}\left[\mathbb{E}\|Z\|_{L^{\infty}([s,t])}^{p_{2}}\right]^{\frac{1}{p_{2}}}\left[\mathbb{E}e^{\frac{p_{3}|x_{s}|\|Z\|_{L^{\infty}([s,t])}}{\sigma_{Y}^{2}}}\right]^{\frac{1}{p_{3}}},$

where $p_{1},p_{2},p_{3}\in(1,\infty)$ satisfy

$\frac{1}{p_{1}}+\frac{1}{p_{2}}+\frac{1}{p_{3}}=1.$

By Remark 3.10, if $\frac{t-s}{s-u}$ is sufficiently small, we have

$\left[\mathbb{E}U_{s,t}(\varepsilon,Y_{s}+Z)^{p_{1}}\right]^{\frac{1}{p_{1}}}\lesssim_{p_{1}}\left[\mathbb{E}[U_{s,t}(\varepsilon,Y+Z)^{p_{4}}\right]^{\frac{1}{p_{4}}},\quad
p_{4}\vcentcolon=p_{1}^{2}.$

Recalling $\tilde{B}$ from Definition 3.2, the scaling property yields

$\mathbb{E}\|Z\|_{L^{\infty}([s,t])}^{p_{2}}\leq\mathbb{E}\|Z\|_{L^{\infty}([u,t])}^{p_{2}}=(t-u)^{p_{2}H}\mathbb{E}\|\tilde{B}\|_{L^{\infty}([0,1])}^{p_{2}}$

and similarly

$\mathbb{E}\Big{[}e^{\frac{p_{3}|x_{s}|\|Z\|_{L^{\infty}([s,t])}}{\sigma_{Y}^{2}}}\Big{]}\leq\mathbb{E}\Big{[}e^{\frac{p_{3}|x_{s}|(t-u)^{H}}{\sigma_{Y}^{2}}\|\tilde{B}\|_{L^{\infty}([0,1])}}\Big{]}.$

Since $\|\tilde{B}\|_{L^{\infty}([0,1])}$ has a Gaussian tail by Fernique’s
theorem, there exists a constant $c$ depending only on $H$ such that

$\mathbb{E}\Big{[}e^{\frac{p_{3}|x_{s}|(t-u)^{H}}{\sigma_{Y}^{2}}\|\tilde{B}\|_{L^{\infty}([0,1])}}\Big{]}\lesssim
e^{c(\frac{p_{3}|x_{s}|(t-u)^{H}}{\sigma_{Y}^{2}})^{2}}.$

Now the claim is proved. ∎

######  Lemma 3.13 (Sharp bound on $U$).

For every $p_{1},p_{2}\in(1,\infty)$ we have

$\|U_{s,t}(\varepsilon,Y+Z)\|_{L^{p_{1}}(\mathbb{P})}\lesssim_{\zeta,p_{1},p_{2}}\Big{(}\frac{t-s}{s-v}\Big{)}^{\frac{H}{p_{1}p_{2}}}.$

###### Proof.

By the scaling, color=yellow!60!white, size=, ]PD: takeing
$\zeta=\varepsilon^{1/H}$

$\|U_{s,t}(\varepsilon,Y+Z)\|_{L^{p_{1}}(\mathbb{P})}=\|U_{s-v,t-v}(\varepsilon,\tilde{B})\|_{L^{p_{1}}(\mathbb{P})}=\|U_{\frac{s-v}{t-s},\frac{t-v}{t-s}}(\zeta^{-H},\tilde{B})\|_{L^{p_{1}}(\mathbb{P})}.$

We set $k_{1}\vcentcolon=\frac{s-v}{t-s}$ and
$k_{2}\vcentcolon=\frac{t-v}{t-s}$. color=green!60!white, size=, ]RL: I moved
the remark that $k_{2}-k_{1}=1$ just before the application of Fernique We
observe

$\displaystyle\|U_{k_{1},k_{2}}(\zeta^{-H},\tilde{B})\|_{L^{p_{1}}(\mathbb{P})}^{p_{1}}$
$\displaystyle=\sum_{a\in\mathbb{Z}}\mathbb{E}\Big{[}U_{k_{1},k_{2}}(\zeta^{-H},\tilde{B})^{p_{1}}\mathbf{1}_{\\{\tilde{B}_{k_{1}}\in(a-1,a]\\}}\Big{]}$
$\displaystyle=\sum_{a\in\mathbb{Z}}\mathbb{E}\Big{[}U_{k_{1},k_{2}}(\zeta^{-H},\tilde{B})^{p_{1}}\mathbf{1}_{\\{\tilde{B}_{k_{1}}\in(a-1,a]\\}}\mathbf{1}_{\\{\max_{r\in[k_{1},k_{2}]}|\tilde{B}_{r}-\tilde{B}_{k_{1}}|\geq|a|-1\\}}\Big{]}$
$\displaystyle\leq\sum_{a\in\mathbb{Z}}\mathbb{E}[U_{k_{1},k_{2}}(\zeta^{-H},\tilde{B})^{p_{1}q_{2}}]^{\frac{1}{q_{2}}}\mathbb{P}(\max_{r\in[k_{1},k_{2}]}|\tilde{B}_{r}-\tilde{B}_{k_{1}}|\geq|a|-1)^{\frac{1}{q_{2}}}$
$\displaystyle\hskip
170.71652pt\times\mathbb{P}(\tilde{B}_{k_{1}}\in(a-1,a])^{\frac{1}{p_{2}}},$

where $p_{2}^{-1}+(2q_{2})^{-1}=1$ color=green!60!white, size=, ]RL: I guess
it should be $p_{2}^{-1}+2q_{2}^{-1}=1$. By Lemma 3.6,

$\displaystyle\mathbb{E}[U_{k_{1},k_{2}}(\zeta^{-H},\tilde{B})^{p_{1}q_{2}}]^{\frac{1}{q_{2}}}\lesssim_{p_{1},p_{2},\zeta}1.$

Since

$\displaystyle\lVert\tilde{B}_{r_{1}}-\tilde{B}_{r_{2}}\rVert_{L^{2}(\mathbb{P})}^{2}\leq\lVert
B_{r_{1}}-B_{r_{2}}\rVert_{L^{2}(\mathbb{P})}^{2}\lesssim\lvert
r_{1}-r_{2}\rvert^{2H},$

by Kolmogorov’s continuity theorem and Fernique’s theorem (note that
$k_{2}-k_{1}=1$) we obtain

$\displaystyle\mathbb{P}(\max_{r\in[k_{1},k_{2}]}|\tilde{B}_{r}-\tilde{B}_{k_{1}}|\geq|a|-1)\lesssim
e^{-ca^{2}}.$

Finally, considering the density of $\tilde{B}_{k_{1}}$ we see
$\mathbb{P}(\tilde{B}_{k_{1}}\in(a-1,a])\lesssim k_{1}^{-H}$,
color=red!20!white, size=, ]TM: Since the density of $\tilde{B}_{k_{1}}$ is
bounded by $k_{1}^{-H}$ and the claim follows. ∎

######  Lemma 3.14 (Fix time of $Y$).

For every $p_{1}\in(1,\infty)$, if $\frac{t-s}{s-u}$ is sufficiently small, we
have

$|\mathbb{E}[U_{s,t}(\varepsilon,Y_{s}+Z)]-\mathbb{E}[U_{s,t}(\varepsilon,Y+Z)]|\lesssim_{\zeta,p_{1}}\Big{(}\frac{t-s}{s-v}\Big{)}^{H/p_{1}}\Big{(}\frac{t-s}{s-u}\Big{)}^{1-H}.$

###### Proof.

As in Lemma 3.9, we get

$\displaystyle|\mathbb{E}[U_{s,t}(\varepsilon,Y_{s}+Z)]-\mathbb{E}[U_{s,t}(\varepsilon,Y+Z)]|\lesssim_{\zeta,p_{1}}\lVert
U_{s,t}(\varepsilon,Y+Z)\rVert_{L^{p_{1}}}\Big{(}\frac{t-s}{s-u}\Big{)}^{1-H}.$

We then apply Lemma 3.13. ∎

######  Lemma 3.15 ($\sigma_{Y}$ vs $\sigma_{Y+Z}$).

If $\frac{s-u}{s-v}\leq\frac{1}{2}$, for every $p_{1}\in(1,\infty)$, we have

$\Big{|}e^{-\frac{1}{2}(\frac{X_{s}}{\sigma_{Y+Z}})^{2}}-e^{-\frac{1}{2}(\frac{X_{s}}{\sigma_{Y}})^{2}}\Big{|}\lesssim_{p_{1}}e^{-\frac{1}{2p_{1}}(\frac{X_{s}}{\sigma_{Y+Z}})^{2}}\Big{(}\frac{s-u}{s-v}\Big{)}^{2H}.$

###### Proof.

Recalling (34), we have

$\displaystyle|\sigma_{Y+Z}^{-2}-\sigma_{Y}^{-2}|=\sigma_{Y+Z}^{-2}\sigma_{Y}^{-2}\frac{(s-u)^{2H}}{2H}\lesssim\sigma_{Y+Z}^{-2}\Big{(}\frac{s-u}{s-v}\Big{)}^{2H}.$

color=green!60!white, size=, inline]RL: I think that $2H$ should be moved
above the fraction bar:

Using the inequality $1-e^{-\lambda}\leq\lambda$ for $\lambda\geq 0$, we
observe

$\displaystyle
e^{-\frac{1}{2}(\frac{X_{s}}{\sigma_{Y+Z}})^{2}}-e^{-\frac{1}{2}(\frac{X_{s}}{\sigma_{Y}})^{2}}$
$\displaystyle=e^{-\frac{1}{2}(\frac{X_{s}}{\sigma_{Y+Z}})^{2}}\Big{\\{}1-e^{-\frac{X_{s}^{2}}{2}(\sigma_{Y}^{-2}-\sigma_{Y+Z}^{-2})}\Big{\\}}$
$\displaystyle\lesssim
e^{-\frac{1}{2}(\frac{X_{s}}{\sigma_{Y+Z}})^{2}}X_{s}^{2}(\sigma_{Y}^{-2}-\sigma_{Y+Z}^{-2})$
$\displaystyle\lesssim
e^{-\frac{1}{2}(\frac{X_{s}}{\sigma_{Y+Z}})^{2}}\sigma_{Y+Z}^{-2}X_{s}^{2}\Big{(}\frac{s-u}{s-v}\Big{)}^{2H}.$

Since $\sup_{\lambda\geq 0}\lambda
e^{-(\frac{1}{2}-\frac{1}{2p_{1}})\lambda}<\infty$, we obtain the claimed
estimate. ∎

######  Lemma 3.16 (Combining estimates obtained so far).

For every $p\in(1,\infty)$, $\kappa\in(1-p^{-1},1)$ and $p_{1}\in(1,2)$, if
$\frac{t-s}{s-u}$ and $\frac{s-u}{u-v}$ are sufficiently small, we have

$\mathbb{E}[U_{s,t}(\varepsilon,B-a)|\mathcal{F}_{v}]=\mathbb{E}[U_{s-v,t-v}(\varepsilon,\tilde{B})]e^{-\frac{1}{2}(\frac{X_{s}}{\sigma_{Y+Z}})^{2}}+R^{1}_{v,u,s,t}(a)+R^{2}_{v,u,s,t}(a),$
(38)

where

$\displaystyle\lVert
R^{1}_{v,u,s,t}(a)\rVert_{L^{p}(\mathbb{P})}\lesssim_{p,\zeta,\kappa}\mathbb{E}[U_{s,t}(\varepsilon,B-a)]^{1-\kappa}e^{-ca^{2}}\Big{(}\frac{t-s}{s-u}\Big{)}^{1-H}$
(39)

with $c$ being a constant depending only on $H,\kappa,p$, and almost surely

$\lvert
R^{2}_{v,u,s,t}(a)\rvert\lesssim_{p_{1},\zeta}e^{-\frac{1}{2p_{1}}(\frac{X_{s}}{\sigma_{Y+Z}})^{2}}\frac{(t-u)^{H}}{\sigma_{Y+Z}}\Big{(}\frac{t-s}{s-v}\Big{)}^{H/p_{1}}+e^{-\frac{1}{2}(\frac{X_{s}}{\sigma_{Y}})^{2}}\Big{(}\frac{t-s}{s-v}\Big{)}^{H/p_{1}}\Big{(}\frac{t-s}{s-u}\Big{)}^{1-H}.$
(40)

In particular,

$\Big{\lVert}\mathbb{E}[U_{s,t}(\varepsilon,B-a)|\mathcal{F}_{v}]-\mathbb{E}[U_{s-v,t-v}(\varepsilon,\tilde{B})]e^{-\frac{1}{2}(\frac{X_{s}}{\sigma_{Y+Z}})^{2}}\Big{\rVert}_{L^{p}(\mathbb{P})}\\\
\lesssim_{p,\zeta,\kappa}\Big{(}\frac{t-s}{s-u}\Big{)}^{1-H}+\Big{(}\frac{t-s}{s-u}\Big{)}^{-H}\Big{(}\frac{t-s}{s-v}\Big{)}^{(2-\kappa)H}.$
(41)

color=green!60!white, size=, inline]RL: I think that in the expression for
$R^{2}$, $(t-u)^{H}$ should be replaced by $(t-u)^{2H}\vee(t-u)^{H}$
($(t-u)^{2H}$ appears in the estimate for $R_{4}$) and $p_{1}$ by $p_{1}^{2}$:
TM: we have $t-u\leq 1$ and we can replace $p_{1}^{2}$ by $p_{1}$. RL: OK.

###### Proof.

In view of Lemma 3.11, we decompose

$\mathbb{E}[U_{s,t}(\varepsilon,B-a)|\mathcal{F}_{v}]=R_{1}+R_{2}+R_{3}+R_{4}+R_{5},$

where

$\displaystyle R_{1}$
$\displaystyle\vcentcolon=\mathbb{E}[U_{s,t}(\varepsilon,B-a)|\mathcal{F}_{v}]-\mathbb{E}[U_{s,t}(\varepsilon,X_{s}+Y_{s}+Z)|\mathcal{F}_{v}],$
$\displaystyle R_{2}$
$\displaystyle\vcentcolon=e^{-\frac{1}{2}(\frac{X_{s}}{\sigma_{Y}})^{2}}\mathbb{E}\Big{[}\Big{.}e^{\frac{X_{s}Y_{s}}{\sigma_{Y}^{2}}}U_{s,t}(\varepsilon,Y_{s}+Z)\Big{|}\mathcal{F}_{v}\Big{]}-e^{-\frac{1}{2}(\frac{X_{s}}{\sigma_{Y}})^{2}}\mathbb{E}[U_{s,t}(\varepsilon,Y_{s}+Z)],$
$\displaystyle R_{3}$
$\displaystyle\vcentcolon=e^{-\frac{1}{2}(\frac{X_{s}}{\sigma_{Y}})^{2}}\mathbb{E}[U_{s,t}(\varepsilon,Y_{s}+Z)]-e^{-\frac{1}{2}(\frac{X_{s}}{\sigma_{Y}})^{2}}\mathbb{E}[U_{s,t}(\varepsilon,Y+Z)],$
$\displaystyle R_{4}$
$\displaystyle\vcentcolon=e^{-\frac{1}{2}(\frac{X_{s}}{\sigma_{Y}})^{2}}\mathbb{E}[U_{s,t}(\varepsilon,Y+Z)]-e^{-\frac{1}{2}(\frac{X_{s}}{\sigma_{Y+Z}})^{2}}\mathbb{E}[U_{s,t}(\varepsilon,Y+Z)],$
$\displaystyle R_{5}$
$\displaystyle\vcentcolon=e^{-\frac{1}{2}(\frac{X_{s}}{\sigma_{Y+Z}})^{2}}\mathbb{E}[U_{s,t}(\varepsilon,Y+Z)]=\mathbb{E}[U_{s-v,t-v}(\varepsilon,\tilde{B})]e^{-\frac{1}{2}(\frac{X_{s}}{\sigma_{Y+Z}})^{2}}.$

By Lemma 3.9,

$\|R_{1}\|_{L^{p}(\mathbb{P})}\lesssim_{H,p,\zeta}\mathbb{E}[U_{s,t}(\varepsilon,B-a)]^{1-\kappa}e^{-ca^{2}}\Big{(}\frac{t-s}{s-u}\Big{)}^{1-H}.$

By Lemma 3.14,

$\displaystyle\lvert
R_{3}\rvert\lesssim_{\zeta,p_{1}}e^{-\frac{1}{2}(\frac{X_{s}}{\sigma_{Y}})^{2}}\Big{(}\frac{t-s}{s-v}\Big{)}^{H/p_{1}}\Big{(}\frac{t-s}{s-u}\Big{)}^{1-H}.$

To estimate $R_{2}$, by Lemma 3.12,

$\displaystyle\lvert R_{2}\rvert\lesssim
e^{-\frac{1}{2}(\frac{X_{s}}{\sigma_{Y}})^{2}}\frac{|X_{s}|(t-u)^{H}}{\sigma_{Y}^{2}}e^{c(\frac{|X_{s}|(t-u)^{H}}{\sigma_{Y}^{2}})^{2}}\|U_{s,t}(\varepsilon,Y+Z)\|_{L^{p_{1}}(\mathbb{P})}.$

If $\frac{t-u}{u-v}$ is sufficiently small, we have

$\displaystyle
c(\frac{(t-u)^{H}}{\sigma_{Y}})^{2}\leq\frac{1}{2}-\frac{1}{2p_{1}},$

hence

$\displaystyle\lvert R_{2}\rvert\lesssim
e^{-\frac{1}{2p_{1}}(\frac{X_{s}}{\sigma_{Y}})^{2}}\frac{|X_{s}|(t-u)^{H}}{\sigma_{Y}^{2}}\|U_{s,t}(\varepsilon,Y+Z)\|_{L^{p_{1}}(\mathbb{P})}.$

Using the estimate $\sup_{\lambda\geq 0}\lambda
e^{-(\frac{1}{2p_{1}}-\frac{1}{2p_{1}^{2}})\lambda^{2}}<\infty$ and Lemma
3.13, we get

$\displaystyle\lvert R_{2}\rvert$ $\displaystyle\lesssim
e^{-\frac{1}{2p_{1}^{2}}(\frac{X_{s}}{\sigma_{Y}})^{2}}\frac{(t-u)^{H}}{\sigma_{Y}}\Big{(}\frac{t-s}{s-v}\Big{)}^{H/p_{1}^{2}}$
$\displaystyle\lesssim
e^{-\frac{1}{2p_{1}^{2}}(\frac{X_{s}}{\sigma_{Y+Z}})^{2}}\frac{(t-u)^{H}}{\sigma_{Y+Z}}\Big{(}\frac{t-s}{s-v}\Big{)}^{H/p_{1}^{2}}.$

Finally, we estimate $R_{4}$. By Lemma 3.15, we get

$\displaystyle\lvert
R_{4}\rvert\lesssim_{p_{1}}\|U_{s,t}(\varepsilon,Y+Z)\|_{L^{1}(\mathbb{P})}e^{-\frac{1}{2p_{1}}(\frac{X_{s}}{\sigma_{Y+Z}})^{2}}\Big{(}\frac{s-u}{s-v}\Big{)}^{2H}.$

By Lemma 3.13, we obtain

$\displaystyle\lvert R_{4}\rvert$ $\displaystyle\lesssim
e^{-\frac{1}{2p_{1}}(\frac{X_{s}}{\sigma_{Y+Z}})^{2}}\Big{(}\frac{s-u}{s-v}\Big{)}^{2H}\Big{(}\frac{t-s}{s-v}\Big{)}^{H/p_{1}^{2}}$
$\displaystyle\lesssim
e^{-\frac{1}{2p_{1}^{2}}(\frac{X_{s}}{\sigma_{Y+Z}})^{2}}\frac{(t-u)^{2H}}{\sigma_{Y+Z}}\Big{(}\frac{t-s}{s-v}\Big{)}^{H/p_{1}^{2}}.$

Setting $R_{v,u,s,t}^{1}(a)\vcentcolon=R_{1}$ and
$R_{v,u,s,t}^{2}(a)\vcentcolon=R_{2}+R_{3}+R_{4}$, we get the estimates (39)
and (40). In particular, by the trivial bound on the exponential function and
Lemma 3.6,

$\begin{gathered}\lVert
R^{1}_{v,u,s,t}(a)\rVert_{L^{p}(\mathbb{P})}\lesssim_{p,\zeta}\Big{(}\frac{t-s}{s-u}\Big{)}^{1-H},\\\
\lvert
R^{2}_{v,u,s,t}(a)\rvert\lesssim_{\zeta,\kappa}\Big{(}\frac{s-u}{t-s}\Big{)}^{H}\Big{(}\frac{t-s}{s-v}\Big{)}^{(2-\kappa)H}+\Big{(}\frac{t-s}{s-u}\Big{)}^{1-H}\Big{(}\frac{t-s}{s-v}\Big{)}^{(1-\kappa)H}.\end{gathered}$

Since $\frac{t-s}{s-v}\leq 1$, these bounds lead to (41). ∎

As a final ingredient in the proof of Lemma 3.7, we estimate
$\mathbb{E}[U_{s-v,t-v}(\varepsilon,\tilde{B})]$.

######  Lemma 3.17 (Asymptotics on constants).

For every $\kappa\in(0,1)$, if $\frac{t-s}{s-u}$ and $\frac{s-u}{u-v}$ are
sufficiently small, we have the estimate color=green!60!white, size=, ]RL:
notation $\bar{K}_{0}^{\zeta}(1,B)$ or $\bar{K}_{0,{\zeta}}(1,B)$

$\Big{\lvert}\sqrt{2\pi}\varepsilon^{-1}\sigma_{Y+Z}\mathbb{E}[U_{s-v,t-v}(\varepsilon,\tilde{B})]-\frac{1}{2}\mathbb{E}[\bar{K}_{0,\zeta}(1,B)]\Big{\rvert}\\\
\lesssim_{\kappa,\zeta}\Big{(}\frac{t-s}{s-u}\Big{)}^{-H}\Big{(}\frac{t-s}{s-v}\Big{)}^{(1-\kappa)H}+(t-s)^{-\kappa
H}\Big{(}\frac{t-s}{s-u}\Big{)}^{1-H}.$

###### Proof.

By integrating (38) color=green!60!white, size=, ]RL: I changed this, since in
the previous version there was a reference to estimate in Lemma 3.7 which we
still need to prove over $\mathbb{R}$ with respect to $a$, we get

$\int_{\mathbb{R}}\mathbb{E}[U_{s,t}(\varepsilon,B-a)]\mathrm{d}a=\mathbb{E}[U_{s-v,t-v}(\varepsilon,\tilde{B})]\mathbb{E}\Big{[}\int_{\mathbb{R}}e^{-\frac{1}{2}(\frac{X_{s}}{\sigma_{Y+Z}})^{2}}\mathrm{d}a\Big{]}\\\
+\mathbb{E}\Big{[}\int_{\mathbb{R}}R^{1}_{v,u,s,t}(a)\mathrm{d}a\Big{]}+\mathbb{E}\Big{[}\int_{\mathbb{R}}R^{2}_{v,u,s,t}(a)\mathrm{d}a\Big{]}.$
(42)

We will estimate each term of (42).

By the scaling (Lemma 3.3),

$\displaystyle\int_{\mathbb{R}}\mathbb{E}[U_{s,t}(\varepsilon,B-a)]\mathrm{d}a$
$\displaystyle=\int_{\mathbb{R}}\mathbb{E}[U_{\frac{s}{t-s}\zeta,\frac{t}{t-s}\zeta}(1,B-\zeta^{H}(t-s)^{-H}a)]\mathrm{d}a$
$\displaystyle=\zeta^{-H}(t-s)^{H}\int_{\mathbb{R}}\mathbb{E}[U_{\frac{s}{t-s}\zeta,\frac{t}{t-s}\zeta}(1,B-a)]\mathrm{d}a$
$\displaystyle=\varepsilon\int_{\mathbb{R}}\mathbb{E}[U_{\frac{s}{t-s}\zeta,\frac{t}{t-s}\zeta}(1,B-a)]\mathrm{d}a.$

We recall the downcrossing $D_{s,t}(\varepsilon,w)$ from (11). By definition,
we have

$\displaystyle\bar{K}_{s,t}(\varepsilon,w)=\int_{\mathbb{R}}\\{U_{s,t}(\varepsilon,w-a)+D_{s,t}(\varepsilon,w-a)\\}\mathrm{d}a$

Since $D_{s,t}(\varepsilon,w)=U_{s,t}(\varepsilon,-w-\varepsilon)$ and
$B\overset{\operatorname{d}}{=}-B$, we obtain

$\displaystyle\int_{\mathbb{R}}\mathbb{E}[U_{\frac{s}{t-s}\zeta,\frac{t}{t-s}\zeta}(1,B-a)]\mathrm{d}a=\int_{\mathbb{R}}\mathbb{E}[D_{\frac{s}{t-s}\zeta,\frac{t}{t-s}\zeta}(1,B-a)]\mathrm{d}a=\frac{1}{2}\mathbb{E}[\bar{K}_{\frac{s}{t-s}\zeta,\frac{t}{t-s}\zeta}(1,B)].$

By the stationarity of $\bar{K}$ (Lemma 2.6),

$\displaystyle\mathbb{E}[\bar{K}_{\frac{s}{t-s}\zeta,\frac{t}{t-s}\zeta}(1,B)]=\mathbb{E}[\bar{K}_{0,\zeta}(1,B)].$

Therefore,

$\displaystyle\int_{\mathbb{R}}\mathbb{E}[U_{s,t}(\varepsilon,B-a)]\mathrm{d}a=\frac{\mathbb{E}[\bar{K}_{0,\zeta}(1,B)]}{2}\varepsilon.$
(43)

Recalling how $X$ depends on $a$ from (33), for any $\sigma>0$

$\int_{\mathbb{R}}e^{-\frac{1}{2}(\frac{X_{s}}{\sigma})^{2}}\mathrm{d}a=\sqrt{2\pi}\sigma,$

in particular, we have

$\displaystyle\int_{\mathbb{R}}e^{-\frac{1}{2}(\frac{X_{s}}{\sigma_{Y+Z}})^{2}}\mathrm{d}a=\sqrt{2\pi}\sigma_{Y+Z},\quad\int_{\mathbb{R}}e^{-\frac{1}{2}(\frac{X_{s}}{\sigma_{Y}})^{2}}\mathrm{d}a=\sqrt{2\pi}\sigma_{Y}.$
(44)

Combining (42), (43) and (44), we obtain

$\displaystyle\Big{\lvert}\sqrt{2\pi}\varepsilon^{-1}\sigma_{Y+Z}\mathbb{E}[U_{s-v,t-v}(\varepsilon,\tilde{B})]-\frac{\mathbb{E}[\bar{K}_{0,\zeta}(1,B)]}{2}\Big{\rvert}\leq\varepsilon^{-1}\sum_{i=1,2}\int_{\mathbb{R}}\lVert
R^{i}_{v,u,s,t}(a)\rVert_{L^{1}(\mathbb{P})}\mathrm{d}a.$

It remains to estimate the right-hand side.

By (39),

$\displaystyle\lVert
R^{1}_{v,u,s,t}(a)\rVert_{L^{1}(\mathbb{P})}\lesssim_{\zeta,\kappa}\mathbb{E}[U_{s,t}(\varepsilon,B-a)]^{1-\kappa}e^{-ca^{2}}\Big{(}\frac{t-s}{s-u}\Big{)}^{1-H}.$

By Jensen’s inequality and (43),

$\displaystyle\int_{\mathbb{R}}\mathbb{E}[U_{s,t}(\varepsilon,B-a)]^{1-\kappa}e^{-ca^{2}}\mathrm{d}a\lesssim_{\kappa}\Big{(}\int_{\mathbb{R}}\mathbb{E}[U_{s,t}(\varepsilon,B-a)]e^{-ca^{2}}\mathrm{d}a\Big{)}^{1-\kappa}\lesssim_{\zeta,\kappa}\varepsilon^{1-\kappa}.$

This gives an estimate for $R^{1}$. To estimate $R^{2}$ we use (40) and (44),
and obtain

$\displaystyle\int_{\mathbb{R}}\mathbb{E}\left|R^{2}_{v,u,s,t}(a)\right|\mathrm{d}a$
$\displaystyle=\mathbb{E}\int_{\mathbb{R}}\left|R^{2}_{v,u,s,t}(a)\right|\mathrm{d}a$
$\displaystyle\lesssim_{\zeta,\kappa}(t-u)^{H}\Big{(}\frac{t-s}{s-v}\Big{)}^{(1-\kappa)H}+\sigma_{Y}\Big{(}\frac{t-s}{s-v}\Big{)}^{(1-\kappa)H}\Big{(}\frac{t-s}{s-u}\Big{)}^{1-H}.$

Recalling (34), we have $\sigma_{Y}\lesssim(s-v)^{H}$. Using also $t-u\leq
2(s-u)$ and $\varepsilon=(\frac{t-s}{\zeta})^{H}$, we get

$\displaystyle\varepsilon^{-1}\int_{\mathbb{R}}\lVert
R^{2}_{v,u,s,t}(a)\rVert_{L^{1}(\mathbb{P})}\mathrm{d}a\lesssim_{\zeta,\kappa}\Big{(}\frac{t-s}{s-u}\Big{)}^{-H}\Big{(}\frac{t-s}{s-v}\Big{)}^{(1-\kappa)H}+\Big{(}\frac{t-s}{s-v}\Big{)}^{-\kappa
H}\Big{(}\frac{t-s}{s-u}\Big{)}^{1-H}.$

Noting that $(\frac{t-s}{s-v})^{-\kappa H}\leq(t-s)^{-\kappa H}$ (recall that
we assume that $0\leq v<s<t\leq 1$), we conclude the proof. ∎

###### Proof of Lemma 3.7.

By the bound (41) in Lemma 3.16, we have

$\Big{\lVert}U_{s,t}(\varepsilon,B-a)-\frac{\mathbb{E}[\bar{K}_{0,\zeta}(1,B)]\varepsilon}{2\sqrt{2\pi}\sigma_{Y+Z}}e^{-\frac{1}{2}(\frac{X_{s}}{\sigma_{Y+Z}})^{2}}\Big{\rVert}_{L^{p}(\mathbb{P})}\\\
\lesssim_{p,\zeta,\kappa}\Big{\lVert}\Big{(}\mathbb{E}[U_{s-v,t-v}(\varepsilon,\tilde{B})]-\frac{\mathbb{E}[\bar{K}_{0,\zeta}(1,B)]}{2\sqrt{2\pi}\sigma_{Y+Z}}\varepsilon\Big{)}e^{-\frac{1}{2}(\frac{X_{s}}{\sigma_{Y+Z}})^{2}}\Big{\rVert}_{L^{p}(\mathbb{P})}\\\
+\Big{(}\frac{t-s}{s-u}\Big{)}^{1-H}+\Big{(}\frac{t-s}{s-u}\Big{)}^{-H}\Big{(}\frac{t-s}{s-v}\Big{)}^{(2-\kappa)H}$

if $\frac{t-s}{s-u}$ and $\frac{s-u}{u-v}$ are sufficiently small. By the
bound
$\frac{\varepsilon}{\sigma_{Y+Z}}\lesssim_{\zeta}(\frac{t-s}{s-v})^{H}\leq 1$
and Lemma 3.17 we have

$\Big{\lvert}\mathbb{E}[U_{s-v,t-v}(\varepsilon,\tilde{B})]-\frac{\mathbb{E}[\bar{K}_{0,\zeta}(1,B)]}{2\sqrt{2\pi}\sigma_{Y+Z}}\varepsilon\Big{\rvert}\leq\Big{\lvert}{\sqrt{2\pi}\varepsilon^{-1}\sigma_{Y+Z}}\mathbb{E}[U_{s-v,t-v}(\varepsilon,\tilde{B})]-\frac{1}{2}\mathbb{E}[\bar{K}_{0,\zeta}(1,B)]\Big{\rvert}\\\
\lesssim_{\zeta,\kappa}(t-s)^{-\kappa
H}\Big{(}\frac{t-s}{s-u}\Big{)}^{1-H}+\Big{(}\frac{t-s}{s-u}\Big{)}^{-H}\Big{(}\frac{t-s}{s-v}\Big{)}^{(2-\kappa)H}.$

Therefore,

$\Big{\lVert}U_{s,t}(\varepsilon,B-a)-\frac{\mathbb{E}[\bar{K}_{0,\zeta}(1,B)]\varepsilon}{2\sqrt{2\pi}\sigma_{Y+Z}}e^{-\frac{1}{2}(\frac{X_{s}}{\sigma_{Y+Z}})^{2}}\Big{\rVert}_{L^{p}(\mathbb{P})}\\\
\lesssim_{p,\zeta,\kappa}(t-s)^{-\kappa
H}\Big{(}\frac{t-s}{s-u}\Big{)}^{1-H}+\Big{(}\frac{t-s}{s-u}\Big{)}^{-H}\Big{(}\frac{t-s}{s-v}\Big{)}^{(2-\kappa)H}$

if $\frac{t-s}{s-u}$ and $\frac{s-u}{u-v}$ are sufficiently small. To
optimize, we choose $u$ so that

$\displaystyle\frac{t-s}{s-u}=\Big{(}\frac{t-s}{s-v}\Big{)}^{(2-\kappa)H}.$

Note that, as $H<1/2$, the exponent $(2-\kappa)H$ is less than $1$. Therefore,
if $\frac{t-s}{s-v}$ is sufficiently small, then $\frac{t-s}{s-u}$ and
$\frac{s-u}{u-v}$ are sufficiently small as well. This gives the claimed
bound. ∎

######  Remark 3.18.

Recall $\bar{U}$ from Notation 3.4. Since

$\displaystyle\int_{\mathbb{R}}\mathbb{E}[\bar{U}_{s,t}(\varepsilon,B-a)]\mathrm{d}a=\frac{\mathbb{E}[\bar{K}_{0,\zeta}(1,B)]+1}{2}\varepsilon,$

we similarly obtain

$\displaystyle\mathbb{E}[\bar{U}_{s,t}(\varepsilon,B-a)|\mathcal{F}_{v}]=\frac{\mathbb{E}[\bar{K}_{0,\zeta}(1,B)]+1}{2\sqrt{2\pi}\sigma_{Y+Z}}e^{-\frac{1}{2}(\frac{X_{s}}{\sigma_{Y+Z}})^{2}}\varepsilon+\bar{R}_{v,s,t}$

with, provided that $\frac{t-s}{s-v}$ is sufficiently small,

$\displaystyle\lVert\bar{R}_{v,s,t}\rVert_{L^{p}(\mathbb{P})}\lesssim_{p,\zeta,\kappa}\Big{(}\frac{t-s}{s-v}\Big{)}^{(2-\kappa)H(1-H)}(t-s)^{-\kappa
H}.$

####  3.2.2 Estimates on the local time

The following is the last technical ingredient for Theorem 3.1.

######  Lemma 3.19 (Local time approximation).

Let $H\in(0,1/2)$. We set

$\displaystyle\tilde{A}_{s,t}$
$\displaystyle\vcentcolon=\mathbb{E}[\delta_{0}(B_{s}-a)|\mathcal{F}_{s-(t-s)}](t-s)$
(45)
$\displaystyle\phantom{\vcentcolon}=\sqrt{\frac{H}{\pi}}e^{-\frac{H}{(t-s)^{2H}}\mathbb{E}[B_{s}-a|\mathcal{F}_{s-(t-s)}]^{2}}(t-s)^{1-H}.$

Then, there exists a $\delta>0$ such that for any $p<\infty$ and for any
partition $\pi$ of $[0,1]$,

$\Big{\|}L_{1}(a)-\sum_{[s,t]\in\pi}\tilde{A}_{s,t}\Big{\|}_{L^{p}(\mathbb{P})}\lesssim_{p}|\pi|^{\delta}.$

###### Proof.

We write $L_{s,t}(a)\vcentcolon=L_{t}(a)-L_{s}(a)$. We use the shifted
stochastic sewing (Lemma 2.10). To this end, it suffices to check

$\|L_{s,t}(a)\|_{L^{p}(\mathbb{P})}\lesssim_{p}(t-s)^{1-H},\quad\|\tilde{A}_{s,t}\|_{L^{p}(\mathbb{P})}\lesssim_{p}(t-s)^{1-H}$
(46)

and

$\|\mathbb{E}[L_{s,t}(a)-\tilde{A}_{s,t}|\mathcal{F}_{v}]\|_{L^{p}(\mathbb{P})}\lesssim_{p}(s-v)^{-1-H}(t-s)^{2},\quad
t-s\leq s-v.$ (47)

color=green!60!white, size=, inline]RL: I am not sure if in the above
relations one should not have $\sim_{p}$ instead of $\sim$. TM: Yes, it
depends on $p$ The estimate for $L$ in (46) is well known and can be shown for
example by (non-shifted) stochastic sewing with
$\Xi_{u,v}=\mathbb{E}[\int_{u}^{v}\delta(B_{r}-a)\mathrm{d}r|\mathcal{F}_{u}]$,
and the estimate for $\tilde{A}$ in (46) is not difficult to show. Hence, we
focus on the estimate (47). In [MP22, Lemma 4.7], an estimate similar to (47)
is obtained, but the exponents therein depend on $p$. We slightly improve the
argument.

We have color=green!60!white, size=, ]RL: Some reference or explanation where
does these relationships (also (45)) come from would be desirable, $L_{s,t}$
needs to be defined

$\mathbb{E}[L_{s,t}(a)-\tilde{A}_{s,t}|\mathcal{F}_{v}]=\sqrt{\frac{H}{\pi}}\int_{s}^{t}\Big{\\{}e^{-\frac{H}{(r-v)^{2H}}\mathbb{E}[B_{r}-a|\mathcal{F}_{v}]^{2}}(r-v)^{-H}\\\
-e^{-\frac{H}{(s-v)^{2H}}\mathbb{E}[B_{s}-a|\mathcal{F}_{v}]^{2}}(s-v)^{-H}\Big{\\}}\mathrm{d}r.$

For simplification, we replace $B-a$ by $B$. We decompose the integrand as
$R_{1}+R_{2}+R_{3}$, where

$R_{1}\vcentcolon=e^{-\frac{H}{(r-v)^{2H}}\mathbb{E}[B_{r}|\mathcal{F}_{v}]^{2}}(r-v)^{-H}-e^{-\frac{H}{(r-v)^{2H}}\mathbb{E}[B_{r}|\mathcal{F}_{v}]^{2}}(s-v)^{-H},$
$R_{2}\vcentcolon=e^{-\frac{H}{(r-v)^{2H}}\mathbb{E}[B_{r}|\mathcal{F}_{v}]^{2}}(s-v)^{-H}-e^{-\frac{H}{(s-v)^{2H}}\mathbb{E}[B_{r}|\mathcal{F}_{v}]^{2}}(s-v)^{-H},$
$R_{3}\vcentcolon=e^{-\frac{H}{(s-v)^{2H}}\mathbb{E}[B_{r}|\mathcal{F}_{v}]^{2}}(s-v)^{-H}-e^{-\frac{H}{(s-v)^{2H}}\mathbb{E}[B_{s}|\mathcal{F}_{v}]^{2}}(s-v)^{-H}.$

To obtain (47), it suffices to show

$\displaystyle\lVert R_{1}\rVert_{L^{p}(\mathbb{P})}+\lVert
R_{2}\rVert_{L^{p}(\mathbb{P})}+\lVert
R_{3}\rVert_{L^{p}(\mathbb{P})}\lesssim_{p}(s-v)^{-1-H}(t-s).$

Since

$0\leq(s-v)^{-H}-(r-v)^{-H}\lesssim(s-v)^{-H-1}(r-s),$

we have

$|R_{1}|\lesssim(s-v)^{-H-1}(t-s).$

We observe

$\displaystyle
e^{-\frac{H}{(r-v)^{2H}}\mathbb{E}[B_{r}|\mathcal{F}_{v}]^{2}}-e^{-\frac{H}{(s-v)^{2H}}\mathbb{E}[B_{r}|\mathcal{F}_{v}]^{2}}$
$\displaystyle=e^{-\frac{H}{(r-v)^{2H}}\mathbb{E}[B_{r}|\mathcal{F}_{v}]^{2}}(1-e^{-H((s-v)^{-2H}-(r-v)^{-2H})\mathbb{E}[B_{r}|\mathcal{F}_{v}]^{2}})$
$\displaystyle\lesssim
e^{-\frac{H}{(r-v)^{2H}}\mathbb{E}[B_{r}|\mathcal{F}_{v}]^{2}}\mathbb{E}[B_{r}|\mathcal{F}_{v}]^{2}((s-v)^{-2H}-(r-v)^{-2H})$
$\displaystyle\lesssim
e^{-\frac{H}{(r-v)^{2H}}\mathbb{E}[B_{r}|\mathcal{F}_{v}]^{2}}\mathbb{E}[B_{r}|\mathcal{F}_{v}]^{2}(s-v)^{-2H-1}(r-s)$
$\displaystyle\lesssim(r-v)^{2H}(s-v)^{-2H-1}(r-s)$
$\displaystyle\lesssim(s-v)^{-1}(t-s),$

where in the last step we used that $(r-v)\leq(t-s)+(s-v)\leq 2(s-v)$. Hence,

$|R_{2}|\lesssim(s-v)^{-1-H}(t-s).$

Finally, we estimate $R_{3}$. Suppose that
$\mathbb{E}[B_{r}|\mathcal{F}_{v}]^{2}\leq\mathbb{E}[B_{s}|\mathcal{F}_{v}]^{2}$.
Then,

$\Big{|}e^{-\frac{H}{(s-v)^{2H}}\mathbb{E}[B_{r}|\mathcal{F}_{v}]^{2}}-e^{-\frac{H}{(s-v)^{2H}}\mathbb{E}[B_{s}|\mathcal{F}_{v}]^{2}}\Big{|}\\\
\leq
e^{-\frac{H}{(s-v)^{2H}}\mathbb{E}[B_{r}|\mathcal{F}_{v}]^{2}}\frac{H}{(s-v)^{2H}}(\mathbb{E}[B_{s}|\mathcal{F}_{v}]^{2}-\mathbb{E}[B_{r}|\mathcal{F}_{v}]^{2}).$

Since

$\displaystyle\mathbb{E}[B_{s}|\mathcal{F}_{v}]^{2}-\mathbb{E}[B_{r}|\mathcal{F}_{v}]^{2}=2\mathbb{E}[B_{r}|\mathcal{F}_{v}](\mathbb{E}[B_{s}|\mathcal{F}_{v}]-\mathbb{E}[B_{r}|\mathcal{F}_{v}])+(\mathbb{E}[B_{s}|\mathcal{F}_{v}]-\mathbb{E}[B_{r}|\mathcal{F}_{v}])^{2}$

and

$e^{-\frac{H}{(s-v)^{2H}}\mathbb{E}[B_{r}|\mathcal{F}_{v}]^{2}}(s-v)^{-H}|\mathbb{E}[B_{r}|\mathcal{F}_{v}]|\lesssim
1,$

we obtain

$\Big{|}e^{-\frac{H}{(s-v)^{2H}}\mathbb{E}[B_{r}|\mathcal{F}_{v}]^{2}}-e^{-\frac{H}{(s-v)^{2H}}\mathbb{E}[B_{s}|\mathcal{F}_{v}]^{2}}\Big{|}\\\
\lesssim(s-v)^{-H}|\mathbb{E}[B_{s}|\mathcal{F}_{v}]-\mathbb{E}[B_{r}|\mathcal{F}_{v}]|+(s-v)^{-2H}|\mathbb{E}[B_{s}|\mathcal{F}_{v}]-\mathbb{E}[B_{r}|\mathcal{F}_{v}]|^{2}.$

A similar estimate holds if
$\mathbb{E}[B_{r}|\mathcal{F}_{v}]^{2}\geq\mathbb{E}[B_{s}|\mathcal{F}_{v}]^{2}$.
Therefore, it remains to note

$\|\mathbb{E}[B_{s}|\mathcal{F}_{v}]-\mathbb{E}[B_{r}|\mathcal{F}_{v}]\|_{L^{p}(\mathbb{P})}\lesssim_{p}(s-v)^{H-1}(t-s).\qed$

####  3.2.3 Concluding estimates

Now we can finish the proof of Theorem 3.1. Recall from Remark 3.8 that we can
set $T=1$. Let $\pi$ be a partition of $[0,1]$. By Lemma 3.5,

$\varepsilon^{\frac{1}{H}-1}U_{0,1}(\varepsilon,B-a)\geq\sum_{[s,t]\in\pi}\varepsilon^{\frac{1}{H}-1}U_{s,t}(\varepsilon,B-a),$
(48)
$\varepsilon^{\frac{1}{H}-1}U_{0,1}(\varepsilon,B-a)\leq\sum_{[s,t]\in\pi}\varepsilon^{\frac{1}{H}-1}\bar{U}_{s,t}(\varepsilon,B-a).$
(49)

Here and henceforth, $\varepsilon$ is an independent parameter; unlike
Subsection 3.2.1, we do not assume the relation (32).

######  Lemma 3.20 (Lower bound on $U$).

Let $H\in(0,1/2)$, $p\in[2,\infty)$, $\varepsilon\in(0,1)$ and
$\zeta\in[1,\infty)$. Then, we have

$\varepsilon^{\frac{1}{H}-1}U_{0,1}(\varepsilon,B-a)\geq\frac{1}{2\zeta}\mathbb{E}[\bar{K}_{0,\zeta}(1,B)]L_{1}(a)-R_{\varepsilon},$

where for some $\delta$ depending only on $H$ we have

$\|R_{\varepsilon}\|_{L^{p}(\mathbb{P})}\lesssim_{p,\zeta}\varepsilon^{\delta}.$

###### Proof.

We define $\tilde{A}$ by (45), and we set

$\displaystyle\hat{A}_{s,t}\vcentcolon=U_{s,t}(\zeta^{-H}(t-s)^{H},B-a)\Big{(}\frac{t-s}{\zeta}\Big{)}^{1-H}.$

By Lemma 3.6, we have

$\displaystyle\lVert\hat{A}_{s,t}\rVert_{L^{p}(\mathbb{P})}\lesssim(t-s)^{1-H}.$

By Lemma 3.7,

$\displaystyle\mathbb{E}[\hat{A}_{s,t}|\mathcal{F}_{v}]$
$\displaystyle=\frac{1}{2\zeta}\mathbb{E}[\bar{K}_{0,\zeta}(1,B)]\sqrt{\frac{H}{\pi(s-v)^{2H}}}e^{-\frac{H\mathbb{E}[B_{s}|\mathcal{F}_{v}]^{2}}{(s-v)^{2H}}}(t-s)+R_{v,s,t}$
$\displaystyle=\frac{1}{2\zeta}\mathbb{E}[\bar{K}_{0,\zeta}(1,B)]\mathbb{E}[\tilde{A}_{s,t}|\mathcal{F}_{v}]+R_{v,s,t},$

where

$\displaystyle\lVert
R_{v,s,t}\rVert_{L^{p}(\mathbb{P})}\lesssim_{p,\zeta,\kappa}\Big{(}\frac{t-s}{s-v}\Big{)}^{(2-\kappa)H(1-H)}(t-s)^{1-(1+\kappa)H}.$

for any $\kappa\in(0,1)$. Since $H<1/2$, choosing $\kappa$ sufficiently small,
we can suppose that

$\displaystyle 1-(1+\kappa)H>\frac{1}{2},\quad
1-(1+\kappa)H+(2-\kappa)H(1-H)>1.$

Hence, by Lemma 2.10, with some $\delta=\delta(H)$,

$\displaystyle\Big{\lVert}\sum_{[s,t]\in\pi}\Big{(}\hat{A}_{s,t}-\frac{1}{2\zeta}\mathbb{E}[\bar{K}_{0,\zeta}(1,B)]\tilde{A}_{s,t}\Big{)}\Big{\rVert}_{L^{p}(\mathbb{P})}\lesssim_{p,\zeta}\lvert\pi\rvert^{\delta}.$

In particular, considering a partition of size
$\zeta\varepsilon^{\frac{1}{H}}$, the claim follows in view of (48) and Lemma
3.19. ∎

######  Lemma 3.21 (Upper bound on $U$).

Let $H\in(0,1/2)$, $p\in[2,\infty)$, $\varepsilon\in(0,1)$ and
$\zeta\in[1,\infty)$. Then, we have

$\varepsilon^{\frac{1}{H}-1}U_{0,1}(\varepsilon,B-a)\leq\frac{1}{2\zeta}(\mathbb{E}[\bar{K}_{0,\zeta}(1,B)]+1)L_{1}(a)+\bar{R}_{\varepsilon},$

where for some $\delta$ depending only on $H$ we have

$\|\bar{R}_{\varepsilon}\|_{L^{p}(\mathbb{P})}\lesssim_{p,\zeta}\varepsilon^{\delta}.$

###### Proof.

In view of Remark 3.18 and (49), the proof is similar to Lemma 3.20. ∎

###### Proof of Theorem 3.1.

It readily follows from Lemma 3.20, Lemma 3.21 and the estimate (22). ∎

###  3.3 Uniform convergence

A naive application of Theorem 3.1 yields that, by the Borel–Cantelli lemma,
for any $a\in\mathbb{R}$ and for any
$\bm{\varepsilon}=(\varepsilon_{n})_{n=1}^{\infty}$ with polynomial decay,
there exists a measurable set $\Omega_{a,\bm{\varepsilon}}$ such that
$\mathbb{P}(B\in\Omega_{a,\bm{\varepsilon}})=1$ and for every
$w\in\Omega_{a,\bm{\varepsilon}}$ the limit

$\displaystyle\lim_{n\to\infty}\varepsilon_{n}^{\frac{1}{H}-1}U_{0,t}(\varepsilon_{n},w-a)$

exists for every $t\geq 0$. However, as observed by Chacon et al. [Cha+81],
the quantitative estimate in Theorem 3.1 implies more strongly that we can
take $\Omega_{a,\bm{\varepsilon}}$ uniformly over $a$ and $\bm{\varepsilon}$.
Furthermore, we can remove the polynomial decaying condition.

The arguments below are essentially given in [Cha+81] and [Lem83], but we
repeat them for the reader’s convenience. We begin with the following lemma.

######  Lemma 3.22 (Uniform convergence over grids).

Let $H\in(0,1/2)$ and $t\in(0,\infty)$. We define the grid

$\displaystyle G_{k}:=\\{ik^{-7}:i\in\mathbb{Z},|i|\leq k^{8}\\},\quad
k\in\mathbb{N}\cup\\{0\\}.$

We then have

$\displaystyle\lim_{k\to\infty}\max_{x\in
G_{k}}\big{\lvert}k^{-6(\frac{1}{H}-1)}U_{0,t}(k^{-6},B-x)-\frac{\mathfrak{c}_{H}}{2}L_{t}(x)\big{\rvert}=0\quad\text{almost
surely.}$

###### Proof.

In the notation of Theorem 3.1, we have

$\displaystyle\max_{x\in
G_{k}}\big{\lvert}k^{-6(\frac{1}{H}-1)}U_{0,t}(k^{-6},B-x)-\frac{\mathfrak{c}_{H}}{2}L_{t}(x)\big{\rvert}\leq\zeta^{-1}\sup_{x\in\mathbb{R}}L_{t}(x)+\max_{x\in
G_{k}}R_{k,\zeta,x}.$

Since $x\mapsto L_{t}(x)$ is continuous and $L_{t}(\cdot)$ is supported on

$\displaystyle\\{x\in\mathbb{R}:\lvert x\rvert\leq\lVert
B\rVert_{L^{\infty}([0,t])}\\},$

we see that $\sup_{x\in\mathbb{R}}L_{t}(x)<\infty$ a.s. By Theorem 3.1,

$\displaystyle\lVert\max_{x\in G_{k}}R_{k,\zeta,x}\rVert_{p}^{p}\leq\sum_{x\in
G_{k}}\lVert R_{k,\zeta,x}\rVert_{p}^{p}\lesssim_{p,\zeta}k^{-p\delta+8},$

where $\delta$ is independent of $p$. Since $p$ can be arbitrarily large, the
Borel–Cantelli lemma implies that almost surely we have

$\displaystyle\lim_{k\to\infty}\max_{x\in G_{k}}R_{k,\zeta,x}=0$

and

$\displaystyle\limsup_{k\to\infty}\max_{x\in
G_{k}}\big{\lvert}k^{-6(\frac{1}{H}-1)}U_{0,t}(k^{-6},B-x)-\frac{\mathfrak{c}_{H}}{2}L_{t}(x)\big{\rvert}\leq\zeta^{-1}\sup_{x\in\mathbb{R}}L_{t}(x).$

Since $\zeta$ is arbitrary, we complete the proof. ∎

######  Theorem 3.23 (Uniform convergence to local time, [Lem83, Theorem
II.2.4]).

Let $H\in(0,1/2)$ and $T\in(0,\infty)$. Almost surely, we have

$\lim_{\varepsilon\to 0}\sup_{t\leq
T}\sup_{a\in\mathbb{R}}\big{\lvert}\varepsilon^{\frac{1}{H}-1}U_{0,t}(\varepsilon,B-a)-\frac{\mathfrak{c}_{H}}{2}L_{t}(a)\big{\rvert}=0.$

###### Proof.

Firstly, by an elementary argument using monotonicity of $U$ and continuity of
$L$ [Lem83, Note after Theorem II.2.4], it suffices to prove that for each
$t\in(0,\infty)$ we have

$\displaystyle\lim_{\varepsilon\to
0}\sup_{a\in\mathbb{R}}\big{\lvert}\varepsilon^{\frac{1}{H}-1}U_{0,t}(\varepsilon,B-a)-\frac{\mathfrak{c}_{H}}{2}L_{t}(a)\big{\rvert}=0\quad\text{a.s.}$

By Lemma 3.22, we can find an $\Omega_{1}\subseteq\Omega$ with
$\mathbb{P}(\Omega_{1})=1$ such that for any $\delta\in(0,1)$ and
$\omega\in\Omega_{1}$ there exists an $N=N(\delta,\omega)$ with the following
inequalities:

$\displaystyle(k-1)^{-6}-(k-1)^{-7}>k^{-6}\quad\forall k\geq N,$ (50)
$\displaystyle\lVert B(\omega)\rVert_{L^{\infty}([0,t])}<N-1,$ (51)
$\displaystyle\sup_{k\geq N}\max_{x\in
G_{k}}\big{\lvert}k^{-6(\frac{1}{H}-1)}U_{0,t}(k^{-6},B(\omega)-x)-\frac{\mathfrak{c}_{H}}{2}L_{t}(x)(\omega)\big{\rvert}<\delta.$
(52)

The argument below holds on the event $\Omega_{1}$. For
$\varepsilon\leq(N+1)^{-6}$, there exists a unique $m=m_{\varepsilon}\geq N+1$
such that

$\displaystyle(m+1)^{-6}<\varepsilon\leq m^{-6}.$

If $|x|\geq N-1$, then by (51) we have $L^{H}_{t}(x)=0$. On the other hand, if
$|x|<N-1$, then we define

$\displaystyle x_{k}\vcentcolon=\max_{y\in G_{k}}\\{y\leq x\\}$

for all $k\geq N$. Since $x<x_{m-1}+(m-1)^{-7}$, we have

  * •

$x_{m-1}\leq x<x+\varepsilon<x_{m-1}+(m-1)^{-7}+m^{-6}\leq x_{m-1}+(m-1)^{-6}$
and

  * •

$x<x_{m+2}+(m+2)^{-7}<x_{m+2}+(m+2)^{-7}+(m+2)^{-6}<x+\varepsilon$,

where (50) are applied in both items. Hence, defining the two sets $I_{m-1}$
and $\bar{I}_{m+2}$ as

$I_{m-1}:=\left[x_{m-1},x_{m-1}+(m-1)^{-6}\right],\quad\bar{I}_{m+2}:=\left[\bar{x}_{m+2},\bar{x}_{m+2}+(m+2)^{-6}\right],$

where $\bar{x}_{m+2}\vcentcolon=x_{m+2}+(m+2)^{-7}$, we have the inclusions

$\displaystyle\bar{I}_{m+2}\subseteq[x,x+\varepsilon]\subseteq I_{m-1}.$ (53)

Now we move to the bound on $U$. We first observe the monotonicity of $U$:

$\displaystyle U_{0,t}(\varepsilon_{1},B-x_{1})\leq
U_{0,t}(\varepsilon_{2},B-x_{2})$

provided that
$[x_{2},x_{2}+\varepsilon_{2}]\subseteq[x_{1},x_{1}+\varepsilon_{1}]$. The
relation (53) thus yields

$\displaystyle U_{0,t}((m-1)^{-6},B-x_{m-1})\leq U_{0,t}(\varepsilon,B-x)\leq
U_{0,t}((m+2)^{-6},B-\bar{x}_{m+2}).$

Hence,

$\sup_{x\in\mathbb{R}}\big{\lvert}\varepsilon^{\frac{1}{H}-1}U_{0,t}(\varepsilon,B-x)-\frac{\mathfrak{c}_{H}}{2}L_{t}(x)\big{\rvert}\leq
A_{\varepsilon}+\bar{A}_{\varepsilon}+\frac{\mathfrak{c}_{H}}{2}\sup_{x,y:\lvert
x-y\rvert\leq 2\varepsilon}\lvert L_{t}(x)-L_{t}(y)\rvert,$ (54)

where

$\displaystyle A_{\varepsilon}$ $\displaystyle\vcentcolon=\sup_{x\in
G_{m_{\varepsilon}-1}}\big{\lvert}\varepsilon^{\frac{1}{H}-1}U_{0,t}((m_{\varepsilon}-1)^{-6},B-x)-\frac{\mathfrak{c}_{H}}{2}L_{t}(x)\big{\rvert},$
$\displaystyle\bar{A}_{\varepsilon}$ $\displaystyle\vcentcolon=\sup_{x\in
G_{m_{\varepsilon}+2}}\big{\lvert}\varepsilon^{\frac{1}{H}-1}U_{0,t}((m_{\varepsilon}+2)^{-6},B-x)-\frac{\mathfrak{c}_{H}}{2}L_{t}(x)\big{\rvert}.$

Due to the uniform continuity of $L_{t}(\cdot)$, the last term of (54)
converges to $0$. To estimate $A_{\varepsilon}$, we observe the bound

$A_{\varepsilon}\leq\sup_{x\in
G_{m_{\varepsilon}-1}}\big{\lvert}(m_{\varepsilon}-1)^{-6(\frac{1}{H}-1)}U_{0,t}((m_{\varepsilon}-1)^{-6},B-x)-\frac{\mathfrak{c}_{H}}{2}L_{t}(x)\big{\rvert}\\\
+\sup_{x\in
G_{m_{\varepsilon}-1}}\big{\lvert}\\{(m_{\varepsilon}-1)^{-6(\frac{1}{H}-1)}-\varepsilon^{\frac{1}{H}-1}\\}U_{0,t}((m_{\varepsilon}-1)^{-6},B-x)\big{\rvert}.$

By (52),

$\displaystyle\limsup_{\varepsilon\to 0}\sup_{x\in
G_{m_{\varepsilon}-1}}\big{\lvert}(m_{\varepsilon}-1)^{-6(\frac{1}{H}-1)}U_{0,t}((m_{\varepsilon}-1)^{-6},B-x)-\frac{\mathfrak{c}_{H}}{2}L_{t}(x)\big{\rvert}\leq\delta.$

On the other hand,

$\displaystyle\lvert(m_{\varepsilon}-1)^{-6(\frac{1}{H}-1)}-\varepsilon^{\frac{1}{H}-1}\rvert$
$\displaystyle\lesssim\lvert\varepsilon-(m_{\varepsilon}-1)^{-6}\rvert^{\frac{1}{H}-1}$
$\displaystyle\lesssim(m_{\varepsilon}-1)^{-7(\frac{1}{H}-1)}.$

As (52) implies

$\displaystyle\sup_{\varepsilon\in(0,1)}\sup_{x\in
G_{m_{\varepsilon}-1}}(m_{\varepsilon}-1)^{-6(\frac{1}{H}-1)}U_{0,t}((m_{\varepsilon}-1)^{-6},B-x)<\infty,$

we obtain

$\displaystyle\lim_{\varepsilon\to 0}\sup_{x\in
G_{m_{\varepsilon}-1}}\big{\lvert}\\{(m_{\varepsilon}-1)^{-6(\frac{1}{H}-1)}-\varepsilon^{\frac{1}{H}-1}\\}U_{0,t}((m_{\varepsilon}-1)^{-6},B-x)\big{\rvert}=0.$

Hence, we get $\limsup_{\varepsilon\to 0}A_{\varepsilon}\leq\delta$, and we
get a similar estimate for $\bar{A}_{\varepsilon}$. Recalling (54), this
implies

$\displaystyle\lim_{\varepsilon\to
0}\sup_{x\in\mathbb{R}}\big{\lvert}\varepsilon^{\frac{1}{H}-1}U_{0,t}(\varepsilon,B-x)-\frac{\mathfrak{c}_{H}}{2}L_{t}(x)\big{\rvert}\leq
2\delta.$

Since $\delta$ is arbitrary, we conclude the proof. ∎

Recall the total number $D_{s,t}(\varepsilon,w)$ of downcrossings from (11)
and the variation $V_{s,t}(\mathrm{P},w)$ along Lebesgue partition
$\mathrm{P}$ from (12). Since the total number of upcrossings and that of
downcrossings can differ by at most $1$, almost surely we have

$\displaystyle\lim_{\varepsilon\to 0}\sup_{t\leq
T}\sup_{a\in\mathbb{R}}\big{\lvert}\varepsilon^{\frac{1}{H}-1}D_{0,t}(\varepsilon,B-a)-\frac{\mathfrak{c}_{H}}{2}L_{t}(a)\big{\rvert}=0\quad\forall
T\geq 0,$

or

$\displaystyle\lim_{\varepsilon\to 0}\sup_{t\leq
T}\sup_{a\in\mathbb{R}}\big{\lvert}\varepsilon^{\frac{1}{H}-1}(U_{0,t}(\varepsilon,B-a)+D_{0,t}(\varepsilon,B-a))-\mathfrak{c}_{H}L_{t}(a)\big{\rvert}=0\quad\forall
T\geq 0.$ (55)

######  Theorem 3.24 (Uniform convergence of variation, [Lem83, Proposition
III.2.1]).

Let $H\in(0,1/2)$ and $T\in(0,\infty)$. Almost surely, we have

$\lim_{\varepsilon\to 0}\sup_{t\leq
T}\sup_{\begin{subarray}{c}\mathrm{P}:\text{ \emph{partition of}
$\mathbb{R}$},\\\ \lvert\mathrm{P}\rvert\leq\varepsilon\end{subarray}}\lvert
V_{0,t}(\mathrm{P},B)-\mathfrak{c}_{H}t\rvert=0.$

###### Proof.

We have the identity

$\displaystyle
V_{0,t}(\mathrm{P},B)=\int_{\mathbb{R}}\sum_{[a,b]\in\mathrm{P}}(b-a)^{\frac{1}{H}-1}\\{U_{0,t}(b-a,B-a)+D_{0,t}(b-a,B-a)\\}\mathbf{1}_{(a,b]}(x)\mathrm{d}x.$

Setting $I\vcentcolon=[-\lVert B\rVert_{L^{\infty}([0,T])},\lVert
B\rVert_{L^{\infty}([0,T])}]$, the occupation density formula yields

$\lvert V_{0,t}(\mathrm{P},B)-\mathfrak{c}_{H}t\rvert\\\
\leq\int_{I}\sum_{[a,b]\in\mathrm{P}}(b-a)^{\frac{1}{H}-1}\lvert
U_{0,t}(b-a,B-a)+D_{0,t}(b-a,B-a)-\mathfrak{c}_{H}L_{t}(x)\rvert\mathbf{1}_{(a,b]}(x)\mathrm{d}x.$

For $x\in[a,b]$, we have the bound

$\lvert U_{0,t}(b-a,B-a)+D_{0,t}(b-a,B-a)-\mathfrak{c}_{H}L_{t}(x)\rvert\\\
\leq\lvert U_{0,t}(b-a,B-a)+D_{0,t}(b-a,B-a)-\mathfrak{c}_{H}L_{t}(a)\rvert\\\
+\mathfrak{c}_{H}\sup_{a_{1},a_{2}:\lvert
a_{1}-a_{2}\rvert\leq\lvert\mathrm{P}\rvert}\lvert
L_{t}(a_{1})-L_{t}(a_{2})\rvert.$

Therefore,

$\lvert V_{0,t}(\mathrm{P},B)-\mathfrak{c}_{H}t\rvert\leq\lvert
I\rvert\times\Big{\\{}\sup_{\delta\leq\lvert\mathrm{P}\rvert}\sup_{a\in\mathbb{R}}\lvert
U_{0,t}(\delta,B-a)+D_{0,t}(\delta,B-a)-\mathfrak{c}_{H}L_{t}(a)\rvert\\\
+\mathfrak{c}_{H}\sup_{\lvert
a_{1}-a_{2}\rvert\leq\lvert\mathrm{P}\rvert}\lvert
L_{t}(a_{1})-L_{t}(a_{2})\rvert\Big{\\}}.$

In view of (55) and the uniform continuity of $L$, the claim follows. ∎

###  3.4 Horizontally rough function

In a recent work [CD23], a concept of _quadratic roughness_ has been
introduced. This pathwise quadratic roughness property ensures an invariant
notion of quadratic variation, i.e., given two (appropriate) partition
sequences $\bm{\pi}$ and $\bm{\sigma}$, quadratic roughness of function $x$
implies $[x]_{\bm{\pi}}=[x]_{\bm{\sigma}}$. As expected, Brownian motion
satisfies this quadratic roughness property. In fact, for any _deterministic_
partition sequence $\bm{\pi}=(\pi^{n})$ with $|\pi^{n}|\log n\to 0$, we have
$[B^{1/2}]_{\bm{\pi}}(t)=t$ almost surely. That is there exists
$\Omega_{\bm{\pi}}\subset\Omega$ of full $\mathbb{P}$-measure such that for
all $\omega\in\Omega_{\bm{\pi}},\;[\omega]_{\bm{\pi}}(t)=t$. On the other
hand, by [Dud73] there exists for each $\omega\in\Omega$ a partition
$\pi=\pi(\omega)$ such that $[\omega]_{\pi(\omega)}(t)=0$, and therefore
$\cap_{\bm{\pi}}\Omega_{\bm{\pi}}=\varnothing$. So even for Brownian motion
quadratic roughness does not ensure an almost sure invariance of quadratic
variation across all deterministic partitions (partitions purely on time
variable). So an obvious question is: are there any notion of roughness which
ensures almost sure invariance of quadratic variation across a large
(uncountable) class of partition sequence? To answer this question we define
the notion of _horizontal roughness_ (the word ‘horizontal’ represents path
dependent Lebesgue type partitions constructed from level crossings).

######  Definition 3.25 (Horizontally rough: an invariance notion for $p$-th
variation).

A function $x\in C^{0}([0,T],\mathbb{R})$ is called _horizontally rough_ if
for any $t\in[0,T]$, $\rho\in\mathbb{R}$ and
$\varepsilon=\\{\varepsilon_{n}\\}$ with $\varepsilon_{n}\downarrow 0$,

$\lim_{n\to\infty}\frac{K_{0,t}(\varepsilon_{n},x+\rho)}{K_{0,t}(\varepsilon_{n},x)}=1.$

This notion of horizontal roughness is completely pathwise and scale
invariant. If a continuous function $x$ has the horizontally rough property
and $x$ also has $p$-th variation along a uniform Lebesgue partition, then $x$
has $p$-th variation along all uniform Lebesgue partition and the $p$-th
variation is the same across different uniform Lebesgue partitions. Unlike the
notion of quadratic roughness defined in [CD23], this notion of horizontally
rough ensures an invariant notion of $p$-th variation across a large class of
Lebesgue type partitions almost surely i.e. there is a common measure zero set
outside which the $p$-th variation along any uniform Lebesgue type partition
is the same.

######  Example 3.26.

The following are examples of horizontally rough functions.

  * •

From the definition any linear function is horizontally rough.

  * •

Using results from [Cha+81, Lem83], we can show that Brownian motion and more
generally continuous semimartingales are horizontally rough almost surely.

  * •

Theorem 1.4 shows that fractional Brownian motion with Hurst index $H<1/2$ is
horizontally rough almost surely.

It is interesting to construct horizontally rough functions sampled from a
non-Gaussian, non-semimartingale process. We leave this and further properties
of such functions as a future work.

##  Appendix A An estimate on log-normal distribution

######  Lemma A.1.

Let $Z$ be a standard normal distribution, and let $q\in[2,\infty)$. We then
have

$\displaystyle\lVert e^{\lambda
Z-\frac{\lambda^{2}}{2}}-1\rVert_{L^{q}(\mathbb{P})}\leq\sqrt{2\log
2}\sqrt{q-1}\lambda e^{(q-1)\lambda^{2}},\quad\forall\lambda\geq 0.$

###### Proof.

By [Nua06, Equation (1.1)], we have

$\displaystyle e^{\lambda
Z-\frac{\lambda^{2}}{2}}-1=\sum_{n=1}^{\infty}\lambda^{n}H_{n}(Z),$

where $H_{n}$ is $n$th Hermite polynomial. By the triangle inequality,

$\displaystyle\lVert e^{\lambda
Z-\frac{\lambda^{2}}{2}}-1\rVert_{L^{q}(\mathbb{P})}\leq\sum_{n=1}^{\infty}\lambda^{n}\lVert
H_{n}(Z)\rVert_{L^{q}(\mathbb{P})}.$

The hypercontractivity [Nua06, Theorem 1.4.1] implies that

$\displaystyle\lVert
H_{n}(Z)\rVert_{L^{q}(\mathbb{P})}\leq(q-1)^{\frac{n}{2}}\lVert
H_{n}(Z)\rVert_{L^{2}(\mathbb{P})}.$

Furthermore, by [Nua06, Lemma 1.1.1],

$\displaystyle\lVert H_{n}(Z)\rVert_{L^{2}(\mathbb{P})}=\frac{1}{\sqrt{n!}}.$

Therefore,

$\displaystyle\lVert e^{\lambda
Z-\frac{\lambda^{2}}{2}}-1\rVert_{L^{q}(\mathbb{P})}$
$\displaystyle\leq\sqrt{q-1}\lambda\sum_{n=0}^{\infty}\frac{\lambda^{n}(q-1)^{\frac{n}{2}}}{\sqrt{n!}}$
$\displaystyle\leq\sqrt{q-1}\lambda\Big{(}\sum_{n=0}^{\infty}\frac{2^{-n}}{n+1}\Big{)}^{\frac{1}{2}}\Big{(}\sum_{n=0}^{\infty}\frac{2^{n}\lambda^{2n}(q-1)^{n}}{n!}\Big{)}^{\frac{1}{2}}$
$\displaystyle\leq\sqrt{q-1}\lambda\sqrt{2\log 2}e^{(q-1)\lambda^{2}}.\qed$

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