# Parameter estimation in rough Bessel model

Yuliya Mishura1,2
<EMAIL_ADDRESS>Anton Yurchenko-Tytarenko3
<EMAIL_ADDRESS>
(1Department of Probability, Statistics and Actuarial Mathematics, Taras
Shevchenko National University of Kyiv
2Division of Mathematics and Physics, Mälardalen University
3Department of Mathematics, University of Oslo )

###### Abstract

In this paper, we construct consistent statistical estimators of the Hurst
index, volatility coefficient, and drift parameter for Bessel processes driven
by fractional Brownian motion with $H<1/2$. As an auxiliary result, we also
prove the continuity of the fractional Bessel process. The results are
illustrated with simulations.

##  1 Introduction

The Bessel process, defined as the square root $X=\sqrt{Z}$ of the solution
$Z=\\{Z(t),~{}t\geq 1\\}$ to the stochastic differential equation (SDE)

$\displaystyle dZ(t)=kdt+2\sqrt{Z(t)}dW(t),\quad Z(0)=x_{0}^{2}>0,$ (1.1)

is a well-known probabilistic model used in a wide range of fields including
physics [6, 21, 24] and finance [11, 12, 13, 23]. If $k\in\mathbb{N}$, $X$ can
be interpreted as the Euclidean norm

$X(t)=\sqrt{B^{2}_{1}(t)+\cdots+B^{2}_{k}(t)}$ (1.2)

of a $k$-dimensional Brownian motion $(B_{1},...,B_{k})$ that is connected
with $W$ in (1.1) via the relation

$W(t)=\sum_{i=1}^{k}\int_{0}^{t}\frac{B_{i}(s)}{X(s)}dB_{i}(s),$

and therefore the parameter $k$ is often referred to as the _dimension_ of the
Bessel process $X$. Moreover, as it is shown in [8, Section 3] (see also
[34]), if $k>1$, the Bessel process $X$ is the unique nonnegative strong
solution to the SDE

$dX(t)=\frac{k-1}{X(t)}dt+dW(t),\quad X(0)=x_{0}>0.$ (1.3)

For more details on various properties of Bessel processes, we refer the
reader to [35, Chapter XI], the book [9] which considers general equations of
the type (1.3) or [3, 4, 30] which deal with the case $0<k<1$.

However, in many applications, standard Brownian motion may not adequately
capture the desired level of complexity observed in real-life phenomena. For
example, a number of empirical studies [5, 7, 10, 15, 16] point out the
presence of memory in financial markets. Other sources [1, 18, 19, 20]
indicate that models exhibiting very low Hölder regularity are better suited
to reflect the behavior of market volatility. Given that Bessel-type processes
are frequently employed in stochastic volatility modeling (see e.g. [26,
Chapter 6]), there is a natural inclination to enhance them by incorporating
the aforementioned memory or roughness. A common way to achieve such an effect
is to replace (in some sense) the standard Brownian driver $W$ with a
fractional Brownian motion $B^{H}=\\{B^{H}(t),~{}t\geq 0\\}$, i.e. a centered
Gaussian process with covariance function

$\mathbb{E}\left[B^{H}(t)B^{H}(s)\right]=\frac{1}{2}\left(t^{2H}+s^{2H}-|t-s|^{2H}\right),\quad
s,t\geq 0.$

For instance, [17, 22, 25] used the property (1.2) as the starting point of
their modification and defined fractional Bessel process $\rho^{H}$ for
$k\in\mathbb{N}$ as

$\rho^{H}(t)=\sqrt{(B^{H}_{1}(t))^{2}+\cdots+(B^{H}_{k}(t))^{2}}$

with $B^{H}_{1}$,…, $B^{H}_{k}$ being $k$ independent fractional Brownian
motions. In particular, they prove that $\rho^{H}$ admits a representation

$d\rho^{H}(t)=H(k-1)\frac{t^{2H-1}}{\rho^{H}(t)}dt+\sum_{i=1}^{k}\frac{B^{H}_{i}(t)}{\rho^{H}(t)}dB^{H}_{i}(t),$
(1.4)

where the integrals w.r.t. fractional Brownian motions are understood in the
divergence sense.

Another possible “fractionalization” of the Bessel process may be achieved by
replacing $W$ with $B^{H}$ directly in (1.3). This approach was discussed in
detail in the series of papers [14, 31, 32, 33]: according to it, the
fractional Bessel process $X^{H}=\\{X^{H}(t),~{}t\geq 0\\}$ is defined as the
a.s. pointwise limit

$X^{H}(t):=\lim_{\varepsilon\downarrow 0}X_{\varepsilon}^{H}(t)$ (1.5)

of stochastic processes given by the SDE

$X^{H}_{\varepsilon}(t)=x_{0}+\int_{0}^{t}\frac{a}{X^{H}_{\varepsilon}(s)\mathbbm{1}_{\\{X^{H}_{\varepsilon}(s)>0\\}}+\varepsilon}ds+\sigma
B^{H}(t).$

In the present paper, we consider the latter notion of the fractional Bessel
process and perform a statistical estimation of parameters $H$, $\sigma$ and
$a$ when $H\in\left(0,\frac{1}{2}\right)$.

  * •

In order to estimate $H$ and $\sigma$, we use the standard technique based on
quadratic variations of fractional Brownian motion. The main challenge arises
in the limit $L^{H}(t):=\lim_{\varepsilon\downarrow
0}\int_{0}^{t}\frac{a}{X^{H}_{\varepsilon}(s)\mathbbm{1}_{\\{X^{H}_{\varepsilon}(s)>0\\}}+\varepsilon}ds$:
when $H<\frac{1}{2}$, [33, Corollary 3.3] only guarantees its continuity in
$t$ almost everywhere w.r.t. the Lebesgue measure. It is not enough for the
analysis of quadratic variations of $L^{H}$, so we prove that $L^{H}$ is
continuous at every $t\geq 0$ and utilize this fact to obtain consistent
estimators of $H$ and $\sigma$.

  * •

Our estimator of $a$ is, in turn, based on the unconventional technique
tailored for our specific model. The problem here is that it is currently not
known whether the fractional Bessel process exhibits any ergodic properties
typically utilized for drift parameter estimation. In our analysis, we exploit
the explicit dynamics of $X^{H}$ instead to compare the behavior of $X^{H}(T)$
and $\int_{0}^{T}\frac{1}{X^{H}(t)}dt$ when $T\to\infty$.

The paper is structured as follows. In Section 2, we provide the definition of
the fractional Bessel process and prove the continuity of the latter. Section
3 is devoted to the estimation of $H$ and $\sigma$. In Section 4, we present
our estimator of the drift and prove its consistency. Section 5 contains
simulations. In Appendix A, we prove a technical result related to the
finiteness of the limit (1.5).

##  2 Rough Bessel processes

Let $B^{H}=\\{B^{H}(t),~{}t\geq 0\\}$ be a fractional Brownian motion with
Hurst index $H\in\left(0,\frac{1}{2}\right)$.

######  Remark 2.1.

Using the Kolmogorov-Chentsov theorem, it is possible to prove that $B^{H}$
has a modification with paths that are locally Hölder continuous up to the
order $H$, i.e. for any $T>0$ and $\lambda\in(0,H)$ there exists a positive
random variable $\Lambda=\Lambda_{T,\lambda}$ such that

$|B^{H}(t_{1})-B^{H}(t_{2})|\leq\Lambda|t_{1}-t_{2}|^{\lambda},\quad
t_{1},t_{2}\in[0,T].$ (2.1)

Moreover, by [2], the random variable $\Lambda$ can be chosen in such a way
that for any $p>0$

$\mathbb{E}[\Lambda^{p}]<\infty.$

In what follows, we always consider this modification of $B^{H}$.

For any $\varepsilon>0$, $x_{0}>0$, $a>0$, $b\geq 0$ and $\sigma>0$, consider
a random process $X^{H}_{\varepsilon}=\\{X^{H}_{\varepsilon}(t),~{}t\geq 0\\}$
given by a stochastic differential equation

$\displaystyle
X^{H}_{\varepsilon}(t)=x_{0}+\int_{0}^{t}\frac{a}{X^{H}_{\varepsilon}(s)\mathbbm{1}_{\\{X^{H}_{\varepsilon}(s)>0\\}}+\varepsilon}ds+\sigma
B^{H}(t).$ (2.2)

Note that the SDE (2.2) has Lipschitz continuous drift and additive noise and
hence, by the standard Picard iteration argument applied pathwisely, (2.2) has
a unique solution for each $\varepsilon>0$. Moreover, by [33, Lemma 2.1] (see
also [14, Lemma 1.6]), for any $\varepsilon_{1}<\varepsilon_{2}$,

$X^{H}_{\varepsilon_{1}}(t)\geq X^{H}_{\varepsilon_{2}}(t),\quad
t\in[0,T],\quad\omega\in\Omega,$ (2.3)

and hence, for any $t\geq 0$, one can define the limit

$X^{H}(t):=\lim_{\varepsilon\downarrow 0}X^{H}_{\varepsilon}(t).$ (2.4)

######  Definition 2.2.

The process $X^{H}$ defined by (2.4) will be called a _fractional_ or _rough
Bessel process_.

######  Remark 2.3.

This construction of rough Bessel processes was introduced and extensively
studied in [33]. In particular, it was shown that

  * 1)

the limit (2.4) is finite, i.e., with probability 1,

$X^{H}(t)<\infty$ (2.5)

for all $t\geq 0$;

  * 2)

with probability 1, $X^{H}(t)\geq 0$ for all $t\geq 0$ and, moreover,
$X^{H}(t)>0$ for almost all $t\geq 0$.

It should be noted that one of the possible cases is missing in Step 2 of the
proof of [33, Theorem 2.1] concerning (2.5). For the reader’s convenience, we
provide the completed version of it in Appendix A.

Since the limit (2.4) is well-defined and finite, the limit

$\displaystyle\lim_{\varepsilon\downarrow
0}\int_{0}^{t}\frac{a}{X^{H}_{\varepsilon}(s)\mathbbm{1}_{\\{X^{H}_{\varepsilon}(s)>0\\}}+\varepsilon}ds$
$\displaystyle=\lim_{\varepsilon\downarrow
0}\left(X^{H}_{\varepsilon}(t)-x_{0}-\sigma B^{H}(t)\right)$ (2.6)
$\displaystyle=X^{H}(t)-x_{0}-\sigma B^{H}(t)$

also exists and is finite. In what follows, we will use the notation

$L^{H}(t):=\lim_{\varepsilon\downarrow
0}\int_{0}^{t}\frac{1}{X^{H}_{\varepsilon}(s)\mathbbm{1}_{\\{X^{H}_{\varepsilon}(s)>0\\}}+\varepsilon}ds,$
(2.7)

i.e. $X^{H}$ satisfies the equation

$X^{H}(t)=x_{0}+aL^{H}(t)+\sigma B^{H}(t),\quad t\geq 0.$ (2.8)

By Fatou’s lemma, for any $t\geq 0$,

$\int_{0}^{t}\frac{1}{X^{H}(s)}ds\leq\liminf_{\varepsilon\downarrow
0}\int_{0}^{t}\frac{1}{X^{H}_{\varepsilon}(s)\mathbbm{1}_{\\{X^{H}_{\varepsilon}(s)>0\\}}+\varepsilon}ds=L^{H}(t)<\infty.$

Denote

$R^{H}(t):=L^{H}(t)-\int_{0}^{t}\frac{1}{X^{H}(s)}ds$

and re-write (2.8) as

$X^{H}(t)=x_{0}+a\int_{0}^{t}\frac{1}{X^{H}(s)}ds+\sigma B^{H}(t)+aR^{H}(t).$
(2.9)

At the moment, it is not clear whether $R^{H}(t)=0$ for all $t\geq 0$.
However, we have the following result.

######  Proposition 2.4.

  * 1)

With probability 1, there exists $\tau>0$ such that for all $t\geq\tau$

$X^{H}(t)>0.$

  * 2)

Let $\tau>0$ be such that $X^{H}(\tau)>0$. Then there exists a neighbourhood
$(\tau_{1},\tau_{2})\ni\tau$ such that for all
$t_{1},t_{2}\in(\tau_{1},\tau_{2})$, $R^{H}(t_{1})-R^{H}(t_{2})=0$, i.e.

$L^{H}(t_{2})-L^{H}(t_{1})=\int_{t_{1}}^{t_{2}}\frac{1}{X^{H}(s)}ds.$ (2.10)

###### Proof.

Item 1) follows directly from [31, Theorem 3.3] so let us prove the claim 2).
Let $\tau>0$ be such that $X^{H}(\tau)>0$. Since
$X^{H}_{\varepsilon}(\tau)\uparrow X^{H}(\tau)$ as $\varepsilon\downarrow 0$,
there exists $\varepsilon_{0}>0$ such that for all
$\varepsilon\leq\varepsilon_{0}$, $X^{H}_{\varepsilon}(\tau)>0$. Moreover, the
process $X^{H}_{\varepsilon_{0}}$ is continuous w.r.t. $t$ hence there exists
a neighbourhood $(\tau_{1},\tau_{2})\ni\tau$ and some positive value $x>0$
such that for all $t\in(\tau_{1},\tau_{2})$

$X^{H}_{\varepsilon_{0}}(t)>x.$

In particular, (2.3) implies that for all $\varepsilon\leq\varepsilon_{0}$

$X^{H}_{\varepsilon}(t)>x,\quad t\in(\tau_{1},\tau_{2}).$

Therefore, for any $\varepsilon\leq\varepsilon_{0}$ and
$t\in(\tau_{1},\tau_{2})$,

$\displaystyle\frac{1}{X^{H}_{\varepsilon}(t)\mathbbm{1}_{\\{X^{H}_{\varepsilon}(t)>0\\}}+\varepsilon}$
$\displaystyle<\frac{1}{x}$

and hence, by the dominated convergence theorem, for any
$t_{1},t_{2}\in(\tau_{1},\tau_{2})$, $t_{1}<t_{2}$,

$\displaystyle L^{H}(t_{2})-L^{H}(t_{1})$
$\displaystyle=\lim_{\varepsilon\downarrow
0}\int_{t_{1}}^{t_{2}}\frac{1}{X^{H}_{\varepsilon}(s)\mathbbm{1}_{\\{X^{H}_{\varepsilon}(s)>0\\}}+\varepsilon}ds$
$\displaystyle=\int_{t_{1}}^{t_{2}}\frac{1}{X^{H}(s)}ds.$

∎

######  Corollary 2.1.

With probability 1, there exists $\tau>0$ such that for all
$t_{2}>t_{1}\geq\tau$

$R^{H}(t_{1})=R^{H}(t_{2})$

and

$L^{H}(t_{2})-L^{H}(t_{1})=\int_{t_{1}}^{t_{2}}\frac{1}{X^{H}(s)}ds.$

Note that [33, Corollary 3.3] establishes only continuity of $L^{H}$ almost
everywhere on $\mathbb{R}_{+}$ w.r.t. the Lebesgue measure. It is not enough
for our purposes: in order to estimate $H$ and $\sigma$, we want to utilize
the behavior of quadratic variations of $X^{H}$ and any possible
discontinuities of $L^{H}$ would create substantial obstacles for our
analysis. It turns out, however, that $L^{H}$ (and hence $X^{H}$) is
continuous at all points $t\geq 0$. The corresponding theorem is provided
below.

######  Theorem 2.5.

  1. 1.

The process $L^{H}$ defined by (2.7) is non-decreasing.

  2. 2.

The processes $X^{H}=\\{X^{H}(t),~{}t\geq 0\\}$ and
$L^{H}=\\{L^{H}(t),~{}t\geq 0\\}$ have continuous paths.

###### Proof.

First of all, observe that for any $0\leq t_{1}<t_{2}\leq T$

$\displaystyle L^{H}(t_{1})$ $\displaystyle=\lim_{\varepsilon\downarrow
0}\int_{0}^{t_{1}}\frac{1}{X^{H}_{\varepsilon}(s)\mathbbm{1}_{\\{X^{H}_{\varepsilon}(s)>0\\}}+\varepsilon}ds$
$\displaystyle\leq\lim_{\varepsilon\downarrow
0}\int_{0}^{t_{2}}\frac{1}{X^{H}_{\varepsilon}(s)\mathbbm{1}_{\\{X^{H}_{\varepsilon}(s)>0\\}}+\varepsilon}ds$
$\displaystyle=L^{H}(t_{2}),$

i.e. the process $L^{H}$ is indeed non-decreasing. Next, fix an arbitrary
deterministic $T>0$, choose an arbitrary $\lambda\in(0,H)$ and let
$\Lambda_{T}$ be such that for all $s,t\in[0,T]$

$|B^{H}(t)-B^{H}(s)|\leq\Lambda|t-s|^{\lambda}.$ (2.11)

Note that the monotonicity of $L^{H}$ implies that its discontinuities can
only take the form of positive jumps and, moreover, left and right limits

$L^{H}(t-):=\lim_{\delta\downarrow 0}L^{H}(t-\delta),\quad
L^{H}(t+):=\lim_{\delta\downarrow 0}L^{H}(t+\delta)$

are well-defined at any point $t\in(0,T)$. Next, observe that the limit
process $X^{H}$ defined by (2.4) satisfies the equation

$X^{H}(t)=x_{0}+aL^{H}(t)-b\int_{0}^{t}X^{H}(s)ds+\sigma B^{H}(t),\quad
t\in[0,T],$

and hence points of discontinuity of $X^{H}$ coincide with the ones of
$L^{H}$, can occur only in the form of positive jumps of the same size as the
corresponding jumps of $L^{H}$ and, finally, one can define limits

$X^{H}(t-):=\lim_{\delta\downarrow 0}X^{H}(t-\delta),\quad
X^{H}(t+):=\lim_{\delta\downarrow 0}X^{H}(t+\delta).$

Assume that $\tau\in[0,T]$ is a point of discontinuity of $L^{H}$ (and hence
$X^{H}$) for some $\omega\in\Omega$ and observe that $X^{H}(\tau)=0$ since
otherwise $\tau$ cannot be a point of discontinuity of $X^{H}$ and $L^{H}$ by
(2.10). Let $\alpha>0$ be such that

$X^{H}(\tau+)=L^{H}(\tau+)=\alpha$

and $\delta>0$ be such that for all $t\in(\tau,\tau+\delta)$

$X^{H}(t)>\frac{9\alpha}{10}.$

Next, choose $\delta_{0}<\delta$ such that

$\frac{4a}{\alpha}\delta_{0}+\sigma\Lambda_{T}\delta_{0}^{\lambda}\leq\frac{\alpha}{4},$

where $\lambda\in(0,H)$ and $\Lambda_{T}>0$ are from (2.11). Since
$X^{H}_{\varepsilon}(\tau+\delta_{0})\uparrow X^{H}(\tau+\delta_{0})$ as
$\varepsilon\downarrow 0$, there exists $\varepsilon_{0}>0$ such that for all
$\varepsilon\in(0,\varepsilon_{0}]$

$X^{H}_{\varepsilon}(\tau+\delta_{0})>\frac{4\alpha}{5},\quad
X^{H}_{\varepsilon}(\tau)\leq 0$

Furthermore, $X^{H}_{\varepsilon}(\tau)\uparrow X^{H}(\tau)=0$ as
$\varepsilon\downarrow 0$ hence $X^{H}_{\varepsilon}(\tau)\leq 0$ for all
$\varepsilon>0$ and therefore for any $\varepsilon\in(0,\varepsilon_{0}]$ one
can define

$\tau_{\varepsilon}^{-}:=\sup\left\\{s\in(\tau,\tau+\delta_{0}):~{}X^{H}_{\varepsilon}(s)=\frac{\alpha}{4}\right\\}$

and

$\tau_{\varepsilon}^{+}:=\sup\left\\{s\in(\tau_{\varepsilon}^{-},\tau+\delta_{0}):~{}X^{H}_{\varepsilon}(s)=\frac{3\alpha}{4}\right\\}.$

By continuity, it is evident that
$X_{\varepsilon}^{H}(\tau_{\varepsilon}^{-})=\frac{\alpha}{4}$,
$X_{\varepsilon}^{H}(\tau_{\varepsilon}^{+})=\frac{3\alpha}{4}$ and
$X_{\varepsilon}^{H}(t)\geq\frac{\alpha}{4}$ for any
$t\in[\tau_{\varepsilon}^{-},\tau_{\varepsilon}^{+}]$. Therefore for any
$\varepsilon\in(0,\varepsilon_{0}]$

$\displaystyle\frac{\alpha}{2}$
$\displaystyle=X_{\varepsilon}^{H}(\tau_{\varepsilon}^{+})-X_{\varepsilon}^{H}(\tau_{\varepsilon}^{-})$
$\displaystyle=\int_{\tau_{\varepsilon}^{-}}^{\tau_{\varepsilon}^{+}}\frac{a}{X^{H}_{\varepsilon}(s)\mathbbm{1}_{\\{X^{H}_{\varepsilon}(s)>0\\}}+\varepsilon}ds-b\int_{\tau_{\varepsilon}^{-}}^{\tau_{\varepsilon}^{+}}X_{\varepsilon}^{H}(s)ds+\sigma\left(B^{H}(\tau_{\varepsilon}^{+})-B^{H}(\tau_{\varepsilon}^{-})\right)$
$\displaystyle\leq\frac{4a}{\alpha}(\tau_{\varepsilon}^{+}-\tau_{\varepsilon}^{-})+\sigma\Lambda_{T}(\tau_{\varepsilon}^{+}-\tau_{\varepsilon}^{-})^{\lambda}$
$\displaystyle\leq\frac{4a}{\alpha}\delta_{0}+\sigma\Lambda_{T}\delta_{0}^{\lambda}$
$\displaystyle<\frac{\alpha}{4},$

which gives a contradiction. Therefore, $\alpha=0$ and
$X^{H}(\tau)=X^{H}(\tau+)$, i.e. $X^{H}$ (and hence $L^{H}$) is right-
continuous.

It remains to notice that $X^{H}(t-)=X^{H}(t)$ for any $t\in(0,T]$. Indeed, as
mentioned above, $X^{H}$ is continuous at $t$ if $X^{H}(t)>0$. If
$X^{H}(t)=0$, since $X^{H}(t)\geq 0$ can only have positive jumps,

$X^{H}(t)-X^{H}(t-)=-X^{H}(t-)$

and, since $X^{H}$ can potentially have only positive jumps, $X^{H}(t-)=0$. ∎

Note that the pre-limit processes $X^{H}_{\varepsilon}$ given by (2.2) are
monotonically non-decreasing w.r.t. $\varepsilon$. Moreover, with probability
1, their pointwise limit is continuous function by Theorem 2.5. Therefore, by
Dini’s theorem, we immediately obtain uniform convergence which is summarized
in the following corollary.

######  Corollary 2.2.

For any $T>0$,

$\sup_{t\in[0,T]}|X^{H}(t)-X^{H}_{\varepsilon}(t)|\to
0\quad\text{a.s.},\quad\varepsilon\downarrow 0.$

##  3 Estimation of Hurst index and volatility coefficient

Let us now move on to the parameter estimation for the rough Bessel processes.
Our first goal is to estimate the Hurst index $H\in\left(0,\frac{1}{2}\right)$
and the volatility parameter $\sigma>0$. Note that, by Theorem 2.5, the
process $L^{H}$ is continuous and non-decreasing and hence has zero quadratic
variation. Therefore, we can utilize the standard estimation technique based
on power variations. For more details on this method, we refer the reader to
[28].

Throughout this section, we assume that the rough Bessel process
$X^{H}=\\{X^{H}(t),~{}t\geq 0\\}$ satisfies equation (2.9) with (unknown)
parameters $a,\sigma>0$, $H<1/2$ and is observed on a discrete uniform
partition $0=t_{0}<t_{1}<...<t_{n}=T$ of a fixed compact $[0,T]$,
$t_{k}:=\frac{kT}{n}$.

###  3.1 Estimation of $H$

Let us start with some useful notation for quadratic variations.

######  Notation 3.1.

For a stochastic process $\xi=\\{\xi(t),~{}t\in[0,T]\\}$ observed on a
discrete uniform partition $0=t_{0}<t_{1}<...<t_{n}=T$, $t_{k}:=\frac{kT}{n}$,
denote

$V^{n}_{1,2}(\xi):=\sum_{k=0}^{n-1}(\xi(t_{k+1})-\xi(t_{k}))^{2}$ (3.1)

and

$V^{n}_{2,2}(\xi):=\sum_{k=0}^{n-2}(\xi(t_{k+2})-2\xi(t_{k+1})+\xi(t_{k}))^{2}.$
(3.2)

Note that the zero-mean Gaussian sequence $\\{B^{H}(n)-B^{H}(n-1),n\geq 1\\}$,
is stationary and it is easy to check that

$\displaystyle\mathbb{E}\left[B^{H}(1)\left(B^{H}(n)-B^{H}(n-1)\right)\right]$
$\displaystyle=\frac{1}{2}\left(n^{2H}-2(n-1)^{2H}+(n-2)^{2H}\right)$
$\displaystyle=O(n^{2H-2})\to 0$

as $n\to\infty$. Hence $\\{B^{H}(n)-B^{H}(n-1),k\geq 1\\}$ is ergodic and, by
the ergodic theorem, with probability 1,

$\frac{1}{n}\sum_{k=0}^{n-1}(B^{H}(k+1)-B^{H}(k))^{2}\to\mathbb{E}[(B^{H}(1))^{2}]=1$

and

$\frac{1}{n}\sum_{k=0}^{n-2}(B^{H}(k+2)-2B^{H}(k+1)+B^{H}(k))^{2}\to\mathbb{E}[(B^{H}(2)-2B^{H}(1))^{2}]=4-2^{2H}$

as $n\to\infty$. Furthermore, by the self-similarity property of $B^{H}$,

$\frac{n^{-1+2H}}{T^{2H}}\sum_{k=0}^{n-1}\left(B^{H}(t_{k+1})-B^{H}(t_{k})\right)^{2}\stackrel{{\scriptstyle\text{law}}}{{=}}\frac{1}{n}\sum_{k=0}^{n-1}(B^{H}(k+1)-B^{H}(k))^{2},$
$\displaystyle\frac{n^{-1+2H}}{T^{2H}}$
$\displaystyle\sum_{k=0}^{n-2}(B^{H}(t_{k+2})-2B^{H}(t_{k+1})+B^{H}(t_{k}))^{2}$
$\displaystyle\stackrel{{\scriptstyle\text{law}}}{{=}}\frac{1}{n}\sum_{k=0}^{n-2}(B^{H}(k+2)-2B^{H}(k+1)+B^{H}(k))^{2},$

and we have the following result.

######  Theorem 3.2.

Let $H\in(0,1)$ and $\sigma>0$. Then, as $n\to\infty$,

$\left(\frac{n}{T}\right)^{-1+2H}V^{n}_{1,2}(\sigma
B^{H})\xrightarrow{\mathbb{P}}\sigma^{2}T$

and

$\left(\frac{n}{T}\right)^{-1+2H}V^{n}_{2,2}(\sigma
B^{H})\xrightarrow{\mathbb{P}}(4-2^{2H})\sigma^{2}T.$

######  Remark 3.3.

A more general result related to pathwise integrals w.r.t. $B^{H}$ can be
found in [29, Theorem 1]. More detailed information on the convergence rates
can be found in [28, Chapter 2].

As established in Theorem 2.5, the process $aL^{H}$ in the right-hand side of
(2.9) is a continuous process of bounded variation and hence, with probability
1,

$V^{n}_{1,2}(aL^{H})\to 0,\quad V^{n}_{2,2}(aL^{H})\to 0,\quad n\to\infty.$

Therefore, we immediately get the following corollary.

######  Corollary 3.1.

Let $X^{H}$ be a rough Bessel process given by (2.9) observed on a discrete
uniform partition $0=t_{0}<t_{1}<...<t_{n}=T$, $t_{k}=\frac{kT}{n}$. Then, as
$n\to\infty$,

$\left(\frac{n}{T}\right)^{-1+2H}V^{n}_{1,2}(X^{H})\xrightarrow{\mathbb{P}}\sigma^{2}T$

and

$\left(\frac{n}{T}\right)^{-1+2H}V^{n}_{2,2}(X^{H})\xrightarrow{\mathbb{P}}(4-2^{2H})\sigma^{2}T.$

Next, define

$\widehat{H}:=\frac{\log\left(4-\frac{V^{n}_{2,2}(X^{H})}{V^{n}_{1,2}(X^{H})}\right)}{2\log
2}.$ (3.3)

######  Theorem 3.4.

The estimator $\widehat{H}$ given by (3.3) is (weakly) consistent estimator of
the Hurst index $H$, i.e.

$\widehat{H}\xrightarrow{\mathbb{P}}H,\quad n\to\infty.$

###### Proof.

The statement follows immediately from Corollary 3.1 and the continuous
mapping theorem. ∎

######  Remark 3.5.

By Corollary 3.1,

$4-\frac{V^{n}_{2,2}(X^{H})}{V^{n}_{1,2}(X^{H})}\xrightarrow{\mathbb{P}}2^{2H}>1,\quad
n\to\infty,$

and hence, in practice, the logarithm in (3.3) is well-defined for large
enough values of $n$. Moreover,

$\displaystyle V^{n}_{2,2}(X^{H})$
$\displaystyle=\sum_{k=0}^{n-2}(X^{H}(t_{k+2})-2X^{H}(t_{k+1})+X^{H}(t_{k}))^{2}$
(3.4)
$\displaystyle=\sum_{k=0}^{n-2}\left((X^{H}(t_{k+2})-X^{H}(t_{k+1}))-(X^{H}(t_{k+1})-X^{H}(t_{k}))\right)^{2}$
$\displaystyle\leq
2\sum_{k=0}^{n-2}(X^{H}(t_{k+2})-X^{H}(t_{k+1}))^{2}+2\sum_{k=0}^{n-2}(X^{H}(t_{k+1})-X^{H}(t_{k}))^{2}$
$\displaystyle\leq 4V^{n}_{1,2}(X^{H}),$

i.e.

$4-\frac{V^{n}_{2,2}(X^{H})}{V^{n}_{1,2}(X^{H})}\geq 0,$

and the equality in (3.4) occurs only if
$x_{0}=X^{H}(t_{0})=X^{H}(t_{1})=...=X^{H}(t_{n})$. It remains an open
question whether such an event can occur with positive probability but, in
practice, it is never the case.

###  3.2 Estimation of $\sigma$

Next, assume that the Hurst index $H$ is known and the goal is to estimate the
volatility coefficient $\sigma$. Using Corollary 3.1, it can be done in a
straightforward manner. Namely, we have the following result.

######  Theorem 3.6.

Let the Hurst index $H$ be known. Then the estimator

$\widehat{\sigma}=\frac{1}{T^{H}}\sqrt{n^{-1+2H}V^{n}_{1,2}(X^{H})}$ (3.5)

is (weakly) consistent estimator of the volatility coefficient $\sigma$, i.e.

$\widehat{\sigma}\xrightarrow{\mathbb{P}}\sigma,\quad n\to\infty.$

###### Proof.

The statement follows immediately from Corollary 3.1 and the continuous
mapping theorem. ∎

##  4 Drift parameter estimation

Let us move to the estimation of the drift parameter $a$ in (2.9). In contrast
to the high-frequency setting considered in Section 3, we will now assume that
we observe a continuous path of $X^{H}$ on $[0,T]$ with $T\to\infty$. We study
the following estimator of the parameter $a$

$\widehat{a}(T):=\frac{X^{H}(T)}{\int_{0}^{T}\frac{1}{X^{H}(t)}dt}.$ (4.1)

######  Remark 4.1.

Note that we do not assume any prior knowledge of the parameters $H$ or
$\sigma$.

Let us start by presenting a well-known result related to the growth of
fractional Brownian motion (for more details, see [27], [28, Section B 3.5] or
[31, Corollary 2.2]).

######  Theorem 4.2.

For any $\delta>0$ there exist random time $\tau_{\delta}$ and positive random
variable $\xi$ such that, with probability 1,

$\max_{s\in[0,T]}|B^{H}(t)|\leq\xi T^{H+\delta}$ (4.2)

for all $T>\tau_{\delta}$.

######  Lemma 4.3.

Let $X^{H}$ be a rough Bessel process satisfying (2.9). Then, with probability
1,

$\liminf_{T\to\infty}\frac{1}{\sqrt{T}}\int_{0}^{T}\frac{1}{X^{H}(t)}dt>0.$
(4.3)

###### Proof.

Take an arbitrary $\delta\in\left(0,\frac{1}{2}-H\right)$ and fix an
$\omega\in\Omega$ such that the corresponding path of $B^{H}$ is continuous,
(4.2) holds and $R^{H}$ becomes constant after some random time point $\tau$
as described in Corollary 2.1.

Assume that (4.3) does not hold, i.e. there exists an increasing sequence
$\\{T_{n},~{}n\geq 1\\}$ such that $T_{n}\to\infty$ as $n\to\infty$, $T_{1}$
exceeds $\tau_{\delta}$ from Theorem 4.2, and

$\frac{1}{\sqrt{T_{n}}}\int_{0}^{T_{n}}\frac{1}{X^{H}(t)}dt\to 0,\quad
n\to\infty.$

Integrating both sides of (2.9) from $0$ to $T_{n}$, we obtain that for all
$n\geq 1$

$\displaystyle\int_{0}^{T_{n}}X^{H}(t)dt$
$\displaystyle=x_{0}T_{n}+a\int_{0}^{T_{n}}\int_{0}^{t}\frac{1}{X^{H}(s)}dsdt$
(4.4)
$\displaystyle\quad+\sigma\int_{0}^{T_{n}}B^{H}(t)dt+a\int_{0}^{T_{n}}R^{H}(t)dt.$

Let us prove that the equality (4.4) cannot hold by comparing the asymptotics
of its left- and right-hand sides.

On the one hand, using the Cauchy-Schwartz inequality, it is easy to see that
for all $T>0$

$\int_{0}^{T}X^{H}(t)dt\int_{0}^{T}\frac{1}{X^{H}(t)}dt\geq T^{2},$

so

$\frac{1}{T^{3/2}_{n}}\int_{0}^{T_{n}}X^{H}(s)ds\to\infty,\quad n\to\infty.$

On the other hand,

$\frac{x_{0}T_{n}}{T^{3/2}_{n}}\to 0,\quad n\to\infty,$
$\displaystyle\frac{1}{T^{3/2}_{n}}\int_{0}^{T_{n}}\int_{0}^{t}\frac{1}{X^{H}(s)}dsdt$
$\displaystyle=\frac{1}{T^{3/2}_{n}}\int_{0}^{T_{n}}\frac{1}{X^{H}(t)}(T_{n}-t)dt$
$\displaystyle\leq\frac{T_{n}}{T^{3/2}_{n}}\int_{0}^{T_{n}}\frac{1}{X^{H}(t)}dt$
$\displaystyle\to 0,\quad n\to\infty,$

and, by (4.2),

$\displaystyle\frac{1}{T^{3/2}_{n}}\left|\int_{0}^{T_{n}}B^{H}(t)dt\right|$
$\displaystyle\leq\frac{T^{1+H+\delta}}{T^{3/2}_{n}}\frac{\xi}{1+H+\delta}\to
0,\quad n\to\infty.$

Finally, since $R^{H}$ is constant after some time point $\tau$,

$\frac{1}{T^{3/2}_{n}}\int_{0}^{T_{n}}R^{H}(t)dt\to 0,\quad n\to\infty.$

In other words, if such sequence $\\{T_{n},~{}n\geq 1\\}$ exists, the left-
hand side of (4.4) divided by $T_{n}^{3/2}$ converges to $\infty$ whereas the
right-hand side divided by $T_{n}^{3/2}$ converges to zero. We get a
contradiction that proves (4.3). ∎

We are now ready to move to the main result of this section.

######  Theorem 4.4.

With probability 1,

$\widehat{a}(T)\to a,\quad T\to\infty.$

###### Proof.

First of all, note that, by Lemma 4.3,

$\frac{x_{0}}{\int_{0}^{T}\frac{1}{X^{H}(s)}ds}\to 0\quad\text{a.s.}$

when $T\to\infty$. Next, since there exists $\tau>0$ such that $R^{H}(t)=0$
for all $t\geq\tau$, Lemma 4.3 also implies that

$\frac{aR^{H}(t)}{\int_{0}^{T}\frac{1}{X^{H}(s)}ds}\to 0\quad\text{a.s.}$

when $T\to\infty$. Finally, fixing $\delta\in\left(0,\frac{1}{2}-H\right)$ and
using (4.2), we can deduce that

$\displaystyle\frac{|B^{H}(T)|}{\int_{0}^{T}\frac{1}{X^{H}(s)}ds}$
$\displaystyle\leq\frac{1}{T^{\frac{1}{2}-H-\delta}}\frac{\xi}{\frac{1}{\sqrt{T}}\int_{0}^{T}\frac{1}{X^{H}(s)}ds}\to
0\quad\text{a.s.}$

as $T\to\infty$. Therefore, with probability 1,

$\displaystyle\widehat{a}_{1}(T)$
$\displaystyle=\frac{x_{0}+a\int_{0}^{t}\frac{1}{X^{H}(s)}ds+\sigma
B^{H}(t)+aR^{H}(t)}{\int_{0}^{T}\frac{1}{X^{H}(s)}ds}$
$\displaystyle=a+\frac{x_{0}}{\int_{0}^{T}\frac{1}{X^{H}(s)}ds}+\frac{\sigma
B^{H}(t)}{\int_{0}^{T}\frac{1}{X^{H}(s)}ds}+\frac{aR^{H}(t)}{\int_{0}^{T}\frac{1}{X^{H}(s)}ds}$
$\displaystyle\to a,\quad T\to\infty.$

∎

##  5 Simulations

In this section, we illustrate our results with simulations. In all cases, we
use the standard Euler scheme to simulate the pre-limit processes
$X^{H}_{\varepsilon}$ given by (2.2) with $\varepsilon=0.0001$. Each estimator
was tested on 1000 samples. All simulations were performed using R programming
language; in order to simulate trajectories of fractional Brownian motion, the
package somebm was used.

###  5.1 Simulation of $\widehat{H}$ and $\widehat{\sigma}$

We start by testing the estimators $\widehat{H}$ and $\widehat{\sigma}$ given
by (3.3) and (3.5) respectively. The real values of the parameters are chosen
to be $H=0.3$ and $\sigma=1$ whereas the time horizon is $T=1$.

At first, we analyze $\widehat{H}$ for different sizes $n$ of the partition.
For each $n$, 1000 trajectories of $X^{H}$ were generated and the value of
$\widehat{H}$ was computed for each of the generated paths. Table 1 contains
mean, variance and coefficient of variation (CV, i.e. the ratio of the
standard deviation to the mean) for $\widehat{H}$ and Figure 1 depicts the
corresponding box-and-whisker plots.

| $n=100$ | $n=1000$ | $n=10000$ | $n=100000$
---|---|---|---|---
Mean | 0.3074612 | 0.3018724 | 0.299684 | 0.2999879
Variance | 0.007608759 | 0.0006979747 | 0.00007438351 | 0.000007596678
CV | 0.2837047 | 0.08751781 | 0.02877894 | 0.009187727
Table 1: Performance of the estimator $\widehat{H}$ given by (3.3) for
different sizes of the partition. The real value is $H=0.3$. Figure 1:
Performance of the estimator $\widehat{H}$ given by (3.3) for different sizes
of the partition. The real value $H=0.3$ is depicted with a grey dashed line.

Next, we perform the same procedure for the estimator $\widehat{\sigma}$ given
by (3.5) under assumption that the real value of $H$ is known. The results are
given in Table 2 and Figure 2

| $n=100$ | $n=1000$ | $n=10000$ | $n=100000$
---|---|---|---|---
Mean | 0.9964649 | 0.9999765 | 0.9996666 | 0.9999794
Variance | 0.005895953 | 0.0006022594 | 0.00005529735 | 0.000005777412
CV | 0.07705752 | 0.02454155 | 0.007438699 | 0.002403674
Table 2: Performance of the estimator $\widehat{\sigma}$ given by (3.5) for
different sizes of the partition under assumption that $H$ is known. The real
value is $\sigma=1$. Figure 2: Performance of the estimator $\widehat{\sigma}$
given by (3.5) for different sizes of the partition under assumption that $H$
is known. The real value $\sigma=1$ is depicted with a grey dashed line.

Finally, we consider a more realistic situation when neither $H$ nor $\sigma$
are known. In this case, we first estimate $H$ using the estimator
$\widehat{H}$ given by (3.3) and then plug in the result in the estimator
$\widehat{\sigma}$ defined by (3.5). The results of this estimation of
$\sigma$ are given in Table 3 and Figure 3

| $n=100$ | $n=1000$ | $n=10000$ | $n=100000$
---|---|---|---|---
Mean | 1.098972 | 1.021549 | 0.9991251 | 1.001703
Variance | 0.1783677 | 0.03501052 | 0.006181774 | 0.000860674
CV | 0.3843009 | 0.183164 | 0.0786931 | 0.02928738
Table 3: Performance of the estimator $\widehat{\sigma}$ given by (3.5) for
different sizes of the partition; $H$ is unknown and estimated using (3.3).
The real value is $\sigma=1$. Figure 3: Performance of the estimator
$\widehat{\sigma}$ given by (3.5) for different sizes of the partition; $H$ is
unknown and estimated using (3.3). The real value $\sigma=1$ is depicted with
a grey dashed line.

Overall, in all cases, simulations show convergence of the estimators to the
true values. This also applies to the least favorable situation of estimation
of $\sigma$ when $H$ is unknown.

###  5.2 Simulation of $\widehat{a}(T)$

Next, we test the performance of the estimator $\widehat{a}(T)$ for different
time horizons $T$ using the same numerical techniques as in Subsection 5.1.
The real values of the parameters are once again chosen to be $H=0.3$,
$\sigma=1$ and the integral $\int_{0}^{T}\frac{1}{X^{H}(t)}dt$ is approximated
by the integral sum

$\frac{1}{n}\sum_{i=0}^{[Tn]-1}\frac{1}{\max\left\\{X^{H}\left(\frac{i}{n}\right),0.001\right\\}}$

with $n=10000$. For each $T$, 1000 simulations of $\widehat{a}(T)$ were
performed. The results are given in Table 4 and Figure 4

| $T=1$ | $T=10$ | $T=100$ | $T=1000$
---|---|---|---|---
Mean | 3.872642 | 2.409692 | 2.109389 | 2.029289
Variance | 4.09064 | 0.5914294 | 0.1749705 | 0.06310528
CV | 0.5222618 | 0.3191463 | 0.1983014 | 0.1237909
Table 4: Performance of the estimator $\widehat{a}(T)$ for different time
horizons $T$. The real value is $a=2$. Figure 4: Performance of the estimator
$\widehat{a}(T)$ for different time horizons $T$. The real value $a=2$ is
depicted with a grey dashed line.

Simulations confirm the convergence of $\widehat{a}(T)$ to $a$. However, they
also indicate that relatively high values of $T$ are required to guarantee
reasonable variance of the estimator.

##  6 Acknowledgements

The present research is carried out within the frame and support of the
ToppForsk project nr. 274410 of the Research Council of Norway with title
STORM: Stochastics for Time-Space Risk Models. The first author is supported
by The Swedish Foundation for Strategic Research, grant Nr. UKR22-0017 and by
Japan Science and Technology Agency CREST, project reference number
JPMJCR2115.

##  Appendix A Finiteness of rough Bessel processes

As mentioned in Remark 2.3, the goal of this appendix is to prove the
following result.

######  Theorem A.1.

Let $B^{H}$ be a fractional Brownian motion with $H<1/2$ and $X^{H}$ be
defined by (2.4). Then, with probability 1, for any $t\geq 0$

$X^{H}(t)<\infty.$ (A.1)

Before moving to the proof of Theorem A.1, let us first give some auxiliary
results.

######  Proposition A.2.

Let $X^{H}_{1}=\\{X^{H}_{1}(t),~{}t\geq 0\\}$ be a stochastic process defined
by

$X^{H}_{1}(t)=x_{0}+\int_{0}^{t}\frac{a}{X^{H}_{1}(s)\mathbbm{1}_{\\{X^{H}_{1}(s)>0\\}}+1}ds+\sigma
B^{H}(t).$

Fix $T>0$, $\lambda\in(0,H)$ and let $\Lambda=\Lambda_{T,\lambda}$ be a random
variable such that (2.1) holds. Then, with probability 1,

$\sup_{t\in[0,T]}|X^{H}_{1}(t)|\leq\left(x_{0}+aT\right)+\sigma
T^{\lambda}\Lambda.$

###### Proof.

Fix $\omega\in\Omega$ such that the corresponding path $t\mapsto
B^{H}(\omega,t)$ is Hölder continuous. In particular,

$\max_{t\in[0,T]}|B^{H}(\omega,t)|\leq\Lambda T^{\lambda}.$

Taking into account that $\frac{a}{x\mathbbm{1}_{\\{x>0\\}}+1}\leq a$ for any
$x\in\mathbb{R}$, we can write

$\displaystyle|X^{H}_{1}(t)|$ $\displaystyle\leq x_{0}+\int_{0}^{t}ads+\sigma
T^{\lambda}\Lambda$ $\displaystyle\leq\left(x_{0}+aT\right)+\sigma
T^{\lambda}\Lambda.$

∎

######  Proposition A.3.

Let $X^{H}_{\varepsilon}$ be defined by (2.2). Fix $T>0$, $\lambda\in(0,H)$
and let $\Lambda=\Lambda_{T,\lambda}$ be a random variable such that (2.1)
holds. Then, for any $\varepsilon\in(0,1]$ and $t\in[0,T]$,

$|X^{H}_{\varepsilon}(t)|<x_{0}+aT\left(\frac{2}{x_{0}}\vee 1\right)+\sigma
T^{\lambda}\Lambda\quad\text{a.s.}$ (A.2)

###### Proof.

Fix $\varepsilon\in(0,1]$ and take $\omega\in\Omega$ such that the
corresponding path $t\mapsto B^{H}(\omega,t)$ is Hölder continuous. In
particular,

$\max_{t\in[0,T]}|B^{H}(\omega,t)|\leq\Lambda T^{\lambda},$

Denote also

$\tau_{1,\varepsilon}:=\sup\left\\{t\in[0,T]~{}\Big{|}~{}\forall
s\in[0,t]:~{}X^{H}_{\varepsilon}(s)\geq\frac{x_{0}}{2}\right\\}.$

Our goal is to prove (A.2) separately for $t\in[0,\tau_{1,\varepsilon}]$ and
$t\in[\tau_{1,\varepsilon},T]$.

Step 1: $t\in[0,\tau_{1,\varepsilon}]$. In this case,
$X^{H}_{\varepsilon}(s)\geq\frac{x_{0}}{2}$ for all $s\in[0,t]$ and hence we
can write

$\displaystyle|X^{H}_{\varepsilon}(t)|$
$\displaystyle=\left|x_{0}+\int_{0}^{t}\frac{a}{X^{H}_{\varepsilon}(s)\mathbbm{1}_{\\{X^{H}_{\varepsilon}(s)>0\\}}+\varepsilon}ds+\sigma
B^{H}(t)\right|$ $\displaystyle\leq
x_{0}+\int_{0}^{t}\frac{a}{X^{H}_{\varepsilon}(s)\mathbbm{1}_{\\{X^{H}_{\varepsilon}(s)>0\\}}+\varepsilon}ds+\sigma
T^{\lambda}\Lambda$ $\displaystyle\leq
x_{0}+a\int_{0}^{t}\frac{2}{x_{0}}ds+\sigma T^{\lambda}\Lambda$
$\displaystyle\leq\left(x_{0}+\frac{2aT}{x_{0}}\right)+\sigma
T^{\lambda}\Lambda$ $\displaystyle\leq\left(x_{0}+aT\left(\frac{2}{x_{0}}\vee
1\right)\right)+\sigma T^{\lambda}\Lambda$

as required.

Step 2: $t\in[\tau_{1,\varepsilon},T]$. Consider

$\tau_{2,\varepsilon}(t):=\sup\left\\{s\in[\tau_{1,\varepsilon},t]~{}\Big{|}~{}|X^{H}_{\varepsilon}(s)|<\frac{x_{0}}{2}\right\\}.$

If $\tau_{2,\varepsilon}(t)=t$, (A.2) holds automatically, so let us assume
that $\tau_{2,\varepsilon}(t)<t$. In this case, by continuity,
$X^{H}_{\varepsilon}(\tau_{2,\varepsilon}(t))=\frac{x_{0}}{2}$ and, moreover,
$|X^{H}_{\varepsilon}(s)|\geq\frac{x_{0}}{2}$ for all
$s\in[\tau_{2,\varepsilon}(t),t]$ and we have two possibilities.

  * 1)

If $X^{H}_{\varepsilon}(s)\leq-\frac{x_{0}}{2}$ for all
$s\in[\tau_{2,\varepsilon}(t),t]$, then

$X^{H}_{1}(s)\leq X^{H}_{\varepsilon}(s)<0$

and hence, by Proposition A.2,

$|X^{H}_{\varepsilon}(s)|\leq|X^{H}_{1}(s)|\leq\left(x_{0}+aT\right)+\sigma
T^{\lambda}\Lambda.$ (A.3)

  * 2)

If $X^{H}_{\varepsilon}(s)\geq\frac{x_{0}}{2}$ for all
$s\in[\tau_{2,\varepsilon}(t),t]$, then, just like in Case 1 above,

$\displaystyle|X^{H}_{\varepsilon}(t)|$
$\displaystyle\leq\left|X^{H}_{\varepsilon}(\tau_{2,\varepsilon}(t))\right|+\left|\int_{\tau_{2,\varepsilon}(t)}^{t}\frac{a}{X^{H}_{\varepsilon}(s)\mathbbm{1}_{\\{X^{H}_{\varepsilon}(s)>0\\}}+\varepsilon}ds\right|$
(A.4)
$\displaystyle\qquad+\sigma\left|B^{H}(t)-B^{H}(\tau_{2,\varepsilon}(t))\right|$
$\displaystyle\leq\frac{x_{0}}{2}+a\int_{\tau_{2,\varepsilon}(t)}^{t}\frac{2}{x_{0}}ds+\sigma
T^{\lambda}\Lambda$
$\displaystyle\leq\left(\frac{x_{0}}{2}+\frac{2aT}{x_{0}}\right)+\sigma
T^{\lambda}\Lambda.$

Hence, summarizing (A.3) and (A.4), for any $t\in[\tau_{1,\varepsilon},T]$

$X^{H}(t)\leq x_{0}+aT\left(\frac{2}{x_{0}}\vee 1\right)+\sigma
T^{\lambda}\Lambda,$

which ends the proof. ∎

###### Proof of Theorem A.1.

By definition, for any $t\geq 0$

$X_{\varepsilon}(t)\uparrow X^{H}(t)\quad a.s.,\quad\varepsilon\downarrow 0,$

so (A.1) follows immediately from Proposition A.3. ∎

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