# Pathwise least-squares estimator for linear SPDEs with additive fractional
noise

Pavel Kříž  Charles University, Faculty of Mathematics and Physics, Sokolovská
83, Prague 8, 186 75, Czech Republic.
University of Chemistry and Technology, Prague, Department of Mathematics,
Technická 5, Prague 6, Czech Republic<EMAIL_ADDRESS>and  Jana
Šnupárková  University of Chemistry and Technology, Prague, Department of
Mathematics, Technická 5, Prague 6, Czech Republic<EMAIL_ADDRESS>
###### Abstract.

This paper deals with the drift estimation in linear stochastic evolution
equations (with emphasis on linear SPDEs) with additive fractional noise (with
Hurst index ranging from 0 to 1) via least-squares procedure. Since the least-
squares estimator contains stochastic integrals of divergence type, we address
the problem of its pathwise (and robust to observation errors) evaluation by
comparison with the pathwise integral of Stratonovich type and using its
chain-rule property. The resulting pathwise LSE is then defined implicitly as
a solution to a non-linear equation. We study its numerical properties
(existence and uniqueness of the solution) as well as statistical properties
(strong consistency and the speed of its convergence). The asymptotic
properties are obtained assuming fixed time horizon and increasing number of
the observed Fourier modes (space asymptotics). We also conjecture the
asymptotic normality of the pathwise LSE.

###### Key words and phrases:

Drift estimation,Least squares,Linear SPDEs,Fractional Brownian
motion,Pathwise stochastic integration

###### 2010 Mathematics Subject Classification:

62M09, 60H15, 60G22

The work of PK was supported by the Czech Science Foundation project No.
19-07140S and the work of JŠ was supported by the grant LTAIN19007 Development
of Advanced Computational Algorithms for Evaluating Post-surgery
Rehabilitation.

##  1\. Introduction

Least-squares type estimators of an unknown drift parameter have recently
become very popular in the setting of (semi)linear SPDEs driven by an
infinite-dimensional Brownian motion, because they have advantageous
asymptotic properties (they basically coincide with MLE estimators in this
case) and are relatively easy to implement. For recent theoretical works, the
reader may check [27], [7], [2] or [3] to name just a few. The theory for
models with Brownian motion has reached certain level of maturity so that it
enables several interesting practical applications, such as [1] or [26].

Wide range of randomly evolving phenomena are forced with auto-correlated
noise, which can be effectively modeled by a fractional Brownian motion (fBm).
Stochastic evolution equations driven by a fBm appear in diverse fields such
as biology, neuroscience, hydrology, climatology, finance and many others (see
e.g. monographs [5] or [23] for discussion). Least-squares estimators of the
drift parameter for continuously observed trajectories of solutions to one-
dimensional linear equations with additive fractional noise have been studied
in [13] for Hurst index $H\geq 1/2$ and [14] for general $H\in(0,1)$.

The available literature on parameter estimation for infinite-dimensional
equations driven by an (infinite-dimensional) fBm is rather sparse. A least-
squares estimator for a one-dimensional projection of the solution has been
studied in long-span asymptotic regime in [22] assuming the regular case
$H\geq 1/2$. Drift estimation in spectral setting (Fourier modes are observed)
for infinite-dimensional equations with fractional noise has been studied in
[8] (MLE approach, $H\geq 1/2$) and [17] (ergodic-type estimator,
$H\in(0,1)$). We are not aware of any study on LSE in spectral setting
(Fourier modes are observed) for these models.

LSE-type estimators typically contain stochastic integrals, which makes them
difficult to apply. For equations with Brownian noise, evaluations of Itô
stochastic integrals may be sensitive to small perturbations of trajectories.
A robust version of LSE can be obtained by application of Itô formula (for
simple equations) or rough path theory (more complex equations), see [11]. The
least-squares estimators for equations with fBm incorporate divergence-type
integrals (sometimes called the Skorokhod integrals). These are not defined
pathwise, but rather as the adjoint operators to Malliavin derivatives, which
makes them extremely difficult (if not impossible) to evaluate from the
observed trajectory in practice or in simulations, which prevents them from
wider use. To avoid these complications, the authors in [14] switched to the
ergodic-type estimator for one-dimensional fractional Ornstein-Uhlenbeck
process, which contains only Riemann integrals. Estimators of ergodic type for
more general equations driven by a finite-dimensional fBm were recently
studied in [25]. Application of this ergodic approach in the spectral setting
for infinite-dimensional equations corresponds to the weighted minimum-
contrast estimator (weighted MCE) introduced in [17]. Such estimator is
plausible for stationary processes, but fails for processes at far-from-
stationary state and is difficult to generalize to more complex equations,
because (in contrast to LSE) it requires ergodicity and precise knowledge of
the limiting covariance operator.

In this paper, we start with derivation of a new spectral version of the
least-squares estimator (theoretical LSE) for an unknown drift parameter in
linear SPDEs driven by an additive infinite-dimensional fractional Brownian
motion and prove its strong consistency in space (incl. speed of convergence)
by the tools of Malliavin calculus. We choose to work in spectral setting
(observations in spectral space domain rather than in physical space domain
are available) and to study asymptotic properties under increasing number of
the observed Fourier modes while time horizon is fixed. Such approach makes it
possible to consistently (with error approaching zero) estimate the unknown
parameter based on observations in fixed time horizon, but with sufficient
detail in space (number of Fourier modes). This is particularly useful for
observations having limited time horizon but high space resolution/detail. It
contrasts to more typical time asymptotics, which assumes increasing time
horizon and which is not studied here. Moreover, sever complications with its
numerical evaluation has led us to construct a pathwise version of the LSE,
which is defined (and computable) from the observed trajectory and which is
robust to small perturbations in the observations.

Our construction of the robust pathwise LSE is inspired by the work [11]. It
is based on calculation of pathwise Stratonovich integral, which can be
expressed explicitly in our case, and compensation for difference between
Stratonovich and divergence-type integrals. In our setting with fBm, this
leads to an implicitly defined estimator. We prove its existence and
uniqueness, as well as its strong consistency in space. Surprisingly, if
$H>1/2$, it has slightly better performance than the theoretical LSE. To our
best knowledge, this is the first attempt to define a pathwise robust LSE for
equations driven by fBm. We believe that this pathwise least-squares approach
is applicable also for different models with fBm (such as those in [22] or
[14]) and may offer estimators applicable in practice or in simulations.

Main purpose of this paper is to make the first step in the development of a
theoretical framework necessary for studying spectral asymptotic properties as
well as performing effective enumeration of the least-squares estimators for
stochastic evolution equations with infinite-dimensional auto-correlated
noise. The assumed linearity of the operators ensures reasonable clarity and
simplicity of the exposition. The next step would be to study this problem in
the setting of semilinear equations with non-linear lower order perturbation
with numerous practical applications. This, however, requires fine analysis of
space regularity of solutions and is beyond the scope of this work.

The paper is organized as follows. In section 2 we introduce the studied
problem in more detail. The two new estimators – theoretical and pathwise LSE
– are derived in section 3. Existence and uniqueness of the implicitly defined
pathwise LSE is studied in section 4. Section 5 is devoted to the strong
consistency of the estimators. A simulation study is presented is section 6.
Section 7 summarizes main results and findings of this paper. In Appendix, we
briefly describe some elements of Malliavin calculus, which are useful for the
asymptotic analysis and outline the connection between cylindrical fractional
Brownian motion and Gaussian noise that is white in space and fractional in
time.

##  2\. Problem setup

###  2.1. The model

Consider the evolution equation

(1) $\,{\rm d}X(t)=\lambda AX(t)\,{\rm d}t+BX(t)\,{\rm d}t+\,{\rm
d}B^{H}(t),\quad X(0)=X_{0},$

in a real separable Hilbert space $V$ where $A,B$ are densely defined linear
operators, $B^{H}$ is a cylindrical fractional Brownian motion on $V$,
$X_{0}\in V$ and $\lambda$ is a positive unknown parameter, which we want to
estimate.

Throughout this paper we assume that the equation (1) is diagonalizable, i.e.
there exists an orthonormal basis $\\{e_{k}\\}_{k=1}^{\infty}$ in $V$ such
that for any $k\in\mathbb{N}$ we have

$Ae_{k}=-\alpha_{k}e_{k},\ Be_{k}=-\beta_{k}e_{k},\ \langle
B^{H}(t),e_{k}\rangle_{V}=B^{H}_{k}(t),$

where $\alpha_{k}\geq 0$ and $\beta_{k}\geq 0$. Furthermore, set

$\mu_{k}=\lambda\alpha_{k}+\beta_{k},\ k\in\mathbb{N}.$

######  Remark 2.1.

For existence of the solution to (1) in an appropriate interpolation space,
constructed via spectral decomposition, it suffices that for some
$\gamma\in\mathbb{R}$

(2) $\sum_{k=1}^{\infty}\frac{1}{(1+\mu_{k})^{\gamma}}<\infty$

holds. For more details and proof, see e.g. Theorem 2.1. (and Remark 2.1.) in
[17].

We assume (throughout the paper) that the existence condition (2) is
satisfied. It also implies that

(3) $\lim_{k\to\infty}\mu_{k}=\infty.$

Denote the Fourier modes (projections to eigenvectors) of the solution by
$x_{k}(t)=\langle X(t),e_{k}\rangle_{V}$. By diagonality of the equation (1)
these projections satisfy the system of independent one-dimensional equations

(4) $\begin{array}[]{rcl}{\rm d}x_{k}(t)&=&-\mu_{k}x_{k}(t)\,{\rm d}t+\,{\rm
d}B^{H}_{k}(t),\ t>0,\ k\in\mathbb{N},\\\ x_{k}(0)&=&\langle
X_{0},e_{k}\rangle_{V},\end{array}$

where $\\{B^{H}_{k}(t),t\geq 0\\}$ are mutually independent real-valued
fractional Brownian motions with the same Hurst index $H\in(0,1)$.

The solutions to the equations (4) are mutually independent real-valued
fractional Ornstein-Uhlenbeck processes given by a formula

(5) $x_{k}(t)={\rm e}^{-\mu_{k}t}x_{k}(0)+\int_{0}^{t}{\rm
e}^{-\mu_{k}(t-s)}\,{\rm d}B^{H}_{k}(t),\ t\geq 0.$

for any $k\in\mathbb{N}$.

######  Remark 2.2.

If the operators $A,B$ from (1) are negative definite self-adjoint elliptic
linear differenetial operators of even orders
$2m_{1},2m_{2}\,(m_{1},m_{2}\in\mathbb{N})$, respectively, acting on a compact
$d$-dimensional manifold, then (see [30])

(6) $\alpha_{k}\sim k^{2m_{1}/d},\hbox{ and }\beta_{k}\sim k^{2m_{2}/d},$

where

(7) $a_{k}\sim b_{k}\text{ means that }a_{k}/b_{k}\to C\in(0,\infty)\text{ as
}k\to\infty.$

Such operators are of great interest as they frequently appear in various
stochastic PDEs.

###  2.2. Statistical problem

To estimate the value of $\lambda$, we have continuous sample trajectories of
the first $N$ Fourier modes observed on a fixed time-window $[0,T]$ at our
disposal, i.e. our data are
$\big{\\{}x_{k}(t),t\in[0,T]\big{\\}},\,k=1,\ldots,N$. We aim at deriving an
estimator that would be computable from this data and that would be consistent
with increasing number of Fourier modes $N\to\infty$ (the so-called space
asymptotics), the time horizon $T$ being fixed.

######  Remark 2.3.

In this paper we assume that the Hurst parameter $H$ is known. If not, it can
be consistently determined from a single continuous trajectory of the process
$\big{\\{}x_{1}(t),t\in[0,T]\big{\\}}$ by some of various techniques developed
for estimating $H$ for real-valued processes sampled under in-fill asymptotics
regime (discrete time observation with fixed time horizon and decreasing time
step), cf. [15], [12], [9], or [29]) to name just a few. If, moreover, the
noise processes for individual Fourier modes $B^{H}_{k}$ have different
intensities (volatilities, denote by $\sigma_{k}$), we can simultaneously and
consistently estimate $\sigma_{k}$ and $H$ from the observed modes under in-
fill asymptotics regime using the powers of the second order variations (see
[4], Chapter 3.3) and subsequently eliminate different noise intensities by
appropriate rescaling of the observed processes $x_{k}$.

##  3\. Derivation of the estimator

###  3.1. Theoretical LSE

We derive the estimator for the drift parameter by applying the least-squares
concept to the SDEs understood as a system of linear regression models. In
particular, we find $\lambda$ which minimizes (for fixed time horizont $T$)
the formal sum

$\sum_{k=1}^{N}\int_{0}^{T}\big{(}\dot{x}_{k}(t)+\mu_{k}x_{k}(t)\big{)}^{2}\,{\rm
d}t.$

Note that this is only a heuristic technique, since time derivatives
$\dot{x}_{k}(t)$ do not exist (in the classical sense). However, we can
rewrite the minimizer back in terms of the well-defined stochastic integrals
and Riemann integrals. It reads

(8)
$\hat{\lambda}_{N}=-\dfrac{\sum_{k=1}^{N}\int_{0}^{T}\alpha_{k}x_{k}(t)\,{\rm
d}x_{k}(t)+\sum_{k=1}^{N}\int_{0}^{T}\alpha_{k}\beta_{k}x_{k}^{2}(t)\,{\rm
d}t}{\sum_{k=1}^{N}\int_{0}^{T}\alpha_{k}^{2}x_{k}^{2}(t)\,{\rm d}t},$

where the stochastic integral is understood in the Skorokhod sense (sometimes
referred to as the divergence type integral). Applying the stochastic
differential from the equation (4) we obtain

(9)
$\hat{\lambda}_{N}=\lambda-\dfrac{\sum_{k=1}^{N}\alpha_{k}\int_{0}^{T}x_{k}(t)\,{\rm
d}B^{H}_{k}(t)}{\sum_{k=1}^{N}\alpha_{k}^{2}\int_{0}^{T}x_{k}^{2}(t)\,{\rm
d}t}.$

This form is convenient for proving the asymptotic properties of the estimator
$\hat{\lambda}_{N}$ whereas the form (8) is better for derivation of the
pathwise version of the LSE for $\lambda$.

######  Remark 3.1.

We did not specify the type of stochastic integral with respect to the fBm in
the equations (1) or (4), because all integration concepts coincide in case of
additive noise. However, the choice of the appropriate type of stochastic
integration in (9) (and in (8)) is critical, since the integrands are
stochastic processes here. The choice of the Skorokhod type integral is
motivated by the fact that its expectation is zero, which is advantageous for
the error term in (9) and will be utilized in the sequel. If we choose the
pathwise Stratonovich integral, for example, its nonzero expectation would
generate unwanted bias in (9) and the corresponding estimator would not be
consistent.

###  3.2. Robust pathwise LSE

Although the Skorokhod type integral in (8) has advantageous probabilistic
properties, it is not constructed pathwise, but rather as the adjoint operator
to Malliavin derivative. This imposes sever complications to numerical
evaluation of the estimator $\hat{\lambda}_{N}$ for the observed single
trajectory. To overcome this issue, we derive below its robust pathwise
version.

The strategy, inspired by the paper [11], is to express the estimator in terms
of the pathwise Stratonovich integral (for definition, see [14]) with
appropriate compensation for difference between Stratonovich and Skorokhod
integrals.

$\displaystyle\int_{0}^{T}x_{k}(t)\,{\rm d}x_{k}(t)$
$\displaystyle=\int_{0}^{T}-\mu_{k}x_{k}^{2}(t)\,{\rm
d}t+\int_{0}^{T}x_{k}(t)\,{\rm d}B_{k}^{H}(t)$
$\displaystyle=\int_{0}^{T}-\mu_{k}x_{k}^{2}(t)\,{\rm
d}t+\left(\int_{0}^{T}x_{k}(t)\circ\,{\rm
d}B_{k}^{H}(t)-\Delta(\mu_{k})\right)$
$\displaystyle=\int_{0}^{T}x_{k}(t)\circ\,{\rm d}x_{k}(t)-\Delta(\mu_{k}),$

where $\int_{0}^{T}x_{k}(t)\circ\,{\rm d}x_{k}(t)$ stands for the pathwise
Stratonovich integral and $\Delta(\mu_{k})$ is the compensation. Since $x_{k}$
is in the first Wiener chaos (it is Gaussian), the compensation
$\Delta(\mu_{k})$ takes rather simple form (see [14], formulas (3.4)-(3.6)):

(10) $\displaystyle\Delta(\mu_{k})$
$\displaystyle=\mathbb{E}\left[\int_{0}^{T}x_{k}(t)\circ\,{\rm
d}B_{k}^{H}(t)\right]$
$\displaystyle=\frac{1}{2}T^{2H}\big{(}1-\gamma(1,\mu_{k}T)\big{)}-\mu_{k}^{-2H}\gamma(2H+1,\mu_{k}T)\Big{(}H-\frac{1}{2}\Big{)}$
$\displaystyle\qquad+TH\mu_{k}^{1-2H}\gamma(2H,\mu_{k}T),$

where

$\gamma(h,T)=\int_{0}^{T}{\rm e}^{-x}x^{h-1}\,{\rm d}x.$

Moreover, since Stratonovich integral satisfies the rules of the first order
calculus, we can easily evaluate (without need for rough path lift):

$\int_{0}^{T}x_{k}(t)\circ\,{\rm
d}x_{k}(t)=\frac{x^{2}_{k}(T)-x^{2}_{k}(0)}{2}.$

This provides a simple pathwise formula for evaluation of the Skorokhod-type
integral

$\int_{0}^{T}x_{k}(t)\,{\rm
d}x_{k}(t)=\frac{x^{2}_{k}(T)-x^{2}_{k}(0)}{2}-\Delta(\mu_{k}),$

and the corresponding formula for LSE

$\hat{\lambda}_{N}=-\frac{\sum_{k=1}^{N}\alpha_{k}\left(\frac{x^{2}_{k}(T)-x^{2}_{k}(0)}{2}-\Delta(\mu_{k})\right)+\sum_{k=1}^{N}\alpha_{k}\beta_{k}\int_{0}^{T}x^{2}_{k}(t)\,{\rm
d}t}{\sum_{k=1}^{N}\alpha_{k}^{2}\int_{0}^{T}x^{2}_{k}(t)\,{\rm d}t}.$

Unfortunately, $\mu_{k}=\alpha_{k}\lambda+\beta_{k}$ in this evaluation
formula depend on the unknown value of parameter $\lambda$. A natural
workaround here is to define new estimator (the pathwise LSE) as a solution to
the following equation (in unknown $\Lambda$):

(11)
$\Lambda=-\frac{\sum_{k=1}^{N}\alpha_{k}\left(\frac{x^{2}_{k}(T)-x^{2}_{k}(0)}{2}-\Delta(\alpha_{k}\Lambda+\beta_{k})\right)+\sum_{k=1}^{N}\alpha_{k}\beta_{k}\int_{0}^{T}x^{2}_{k}(t)\,{\rm
d}t}{\sum_{k=1}^{N}\alpha_{k}^{2}\int_{0}^{T}x^{2}_{k}(t)\,{\rm d}t}.$

######  Remark 3.2.

Although the theoretical LSE formula (8) does not explicitly contain the Hurst
parameters $H$, its value is needed to evaluate the stochastic Skorokhod-type
integral. Hence, it is present in defining equation for pathwise (computable)
version of LSE. However, since we consider continuous-time setting, value of
$H$ can be determined exactly from continuous observation of a single
trajectory $\big{\\{}x_{k}(t),t\in[0,T]\big{\\}}$, as discussed above.

######  Remark 3.3.

If $H=1/2$ the pathwise LSE coincides with the theoretical LSE. This follows
from the fact that the right-hand side of (11) is constant (in $\Lambda$) in
this case, which is shown below.

##  4\. Existence and uniqueness of pathwise LSE

We can not expect the existence and uniqueness of the positive solution to
(11) in general. This can be seen on the simplest example of LSE for one-
dimensional Ornstein-Uhlenbeck process (driven by a Wiener process)

$\hat{\lambda}=-\frac{\frac{x^{2}(T)-x^{2}(0)}{2}-\frac{T}{2}}{\int_{0}^{T}x^{2}(t)\,{\rm
d}t},$

which can provide negative estimates. We address this problem below by taking
positive part of the solution and specifying, which one to choose if there are
multiple solutions. To derive the conditions for the existence and uniqueness
of the positive solution to equation (11), we need to understand the
properties of the function on its right-hand side. Set

(12)
$R_{N}(\Lambda)=-\frac{\sum_{k=1}^{N}\alpha_{k}\left(\frac{x^{2}_{k}(T)-x^{2}_{k}(0)}{2}-\Delta(\alpha_{k}\Lambda+\beta_{k})\right)+\sum_{k=1}^{N}\alpha_{k}\beta_{k}\int_{0}^{T}x^{2}_{k}(t)\,{\rm
d}t}{\sum_{k=1}^{N}\alpha_{k}^{2}\int_{0}^{T}x^{2}_{k}(t)\,{\rm d}t},$

with function $\Delta$ defined in (10). Direct calculations lead to the
following formula for its first derivative (after cancelling some terms)

(13) $\displaystyle\frac{\,{\rm d}\Delta}{\,{\rm d}\mu}(\mu)$
$\displaystyle=H(1-2H)\big{(}T\mu^{-2H}\gamma(2H,\mu
T)-\mu^{-2H-1}\gamma(2H+1,\mu T)\big{)}$
$\displaystyle=H(1-2H)\mu^{-2H-1}\int_{0}^{\mu T}{\rm e}^{-s}s^{2H-1}(\mu
T-s)\,{\rm d}s$ $\displaystyle\begin{cases}>0\,\dots\,H<1/2\\\
=0\,\dots\,H=1/2\\\ <0\,\dots\,H>1/2\\\ \end{cases},\quad\forall\mu>0,$

and the second derivative

(14) $\displaystyle\frac{\,{\rm d}^{2}\Delta}{\,{\rm d}\mu^{2}}(\mu)$
$\displaystyle=H(2H-1)\mu^{-2H-2}\int_{0}^{\mu T}{\rm
e}^{-s}s^{2H-1}\big{(}2H\mu T-(2H+1)s\big{)}\,{\rm d}s$
$\displaystyle\begin{cases}<0\,\dots\,H<1/2\\\ =0\,\dots\,H=1/2\\\
>0\,\dots\,H>1/2\\\ \end{cases},\quad\forall\mu>0,$

in view of

$\displaystyle\int_{0}^{\mu T}{\rm e}^{-s}s^{2H-1}\big{(}2H\mu
T-(2H+1)s\big{)}\,{\rm d}s$ $\displaystyle\quad>\int_{0}^{\mu T2H/(2H+1)}{\rm
e}^{-\mu T2H/(2H+1)}s^{2H-1}\big{(}2H\mu T-(2H+1)s\big{)}\,{\rm d}s$
$\displaystyle\quad\quad+\int_{\mu T2H/(2H+1)}^{\mu T}{\rm e}^{-\mu
T2H/(2H+1)}s^{2H-1}\big{(}2H\mu T-(2H+1)s\big{)}\,{\rm d}s=0.$

Derivatives of $R$ are immediate consequence of (13) and (14)

(15) $\displaystyle\frac{\,{\rm d}R_{N}}{\,{\rm d}\Lambda}(\Lambda)$
$\displaystyle=\frac{\sum_{k=1}^{N}\alpha^{2}_{k}\frac{\,{\rm d}\Delta}{\,{\rm
d}\mu}(\alpha_{k}\Lambda+\beta_{k})}{\sum_{k=1}^{N}\alpha_{k}^{2}\int_{0}^{T}x^{2}_{k}(t)\,{\rm
d}t}\begin{cases}>0\,\dots\,H<1/2\\\ =0\,\dots\,H=1/2\\\ <0\,\dots\,H>1/2\\\
\end{cases},\quad\forall\Lambda>0,$

and

$\displaystyle\frac{\,{\rm d}^{2}R_{N}}{\,{\rm d}\Lambda^{2}}(\Lambda)$
$\displaystyle=\frac{\sum_{k=1}^{N}\alpha^{3}_{k}\frac{\,{\rm
d}^{2}\Delta}{\,{\rm
d}\mu^{2}}(\alpha_{k}\Lambda+\beta_{k})}{\sum_{k=1}^{N}\alpha_{k}^{2}\int_{0}^{T}x^{2}_{k}(t)\,{\rm
d}t}\begin{cases}<0\,\dots\,H<1/2\\\ =0\,\dots\,H=1/2\\\ >0\,\dots\,H>1/2\\\
\end{cases},\quad\forall\Lambda>0.$

Moreover, since

$\lim_{\mu\to 0_{+}}\Delta(\mu)=\frac{1}{2}T^{2H},$

we obtain

(16) $\lim_{\Lambda\to 0_{+}}R_{N}(\Lambda)=:R_{N}(0)\in\mathbb{R}.$

Direct calculations yield

$\lim_{\Lambda\to\infty}R_{N}(\Lambda)=\begin{cases}\infty\,\dots\,H<1/2\\\
R_{N}^{\infty}\in\mathbb{R}\,\dots\,H\geq 1/2\end{cases}$

and

$\lim_{\Lambda\to\infty}R_{N}(\Lambda)-\Lambda=-\infty,\quad\forall
H\in(0,1).$

The above properties of $R_{N}$ lead us to the following conclusion:

######  Theorem 4.1.

Fix $N$ and consider the function $R_{N}$ defined in (12) with $R_{N}(0)$
defined in (16) and let $H\in(0,1)$. Sufficient condition for the existence
and uniqueness of the positive solution to equation (11), which can be
rephrased as $R_{N}(\Lambda)=\Lambda$, is

$R_{N}(0)>0.$

If $H\geq 1/2$, this condition is also necessary.

Note that the condition $R_{N}(0)>0$ can be easily verified from the data by
direct evaluation of $R_{N}$ at $\Lambda=0$ (if $\beta_{k}$ are positive) or
by calculating the limit for $\Lambda\to 0_{+}$ (if $\beta_{k}$ are zero). We
are now in a position to define the pathwise LSE correctly:

(17) $\tilde{\lambda}_{N}:=\begin{cases}\text{positive solution to
\eqref{equation modified LSE}, if it exists and is unique};\\\ 0,\text{if
there is no positive solution to \eqref{equation modified LSE}};\\\ \text{the
greater solution to \eqref{equation modified LSE}, if there are two positive
solutions}.\\\ \end{cases}$

######  Remark 4.1.

The choice of the greater solution in the third case ensures that
$\frac{\,{\rm d}R_{N}}{\,{\rm d}\Lambda}(\tilde{\lambda}_{N})<1$, which
corresponds to the limiting behavior of $\frac{\,{\rm d}R_{N}}{\,{\rm
d}\Lambda}$ in the neighborhood of the true value $\lambda$, detailed in Lemma
5.4

######  Remark 4.2.

Monotonicity and convexity (if $H>1/2$) or concavity (if $H<1/2$),
respectively, ensure that the Newton-Raphson numerical method is well suited
for numerical approximation of $\tilde{\lambda}_{N}$.

##  5\. Strong consistency

###  5.1. Strong consistency of theoretical LSE

We start with strong consistency of the theoretical LSE $\hat{\lambda}_{N}$
defined in (8), because it will be helpful to show strong consistency of the
pathwise LSE in the sequel. For simplicity, we assume throughout this section
that

$x_{k}(0)=0,\quad k=1,2,\ldots$

The self-similarity of a fBm enables us to study the effect of different
drifts $\mu_{k}$ on the distributions of the processes $x_{k}$. Its
combination with the (long-span) asymptotic behavior of a real-valued
fractional Ornstein-Uhlenbeck process, studied in [14], forms the strategy of
the proof of the strong consistency (in space) of $\hat{\lambda}_{N}$.

Using integration-by-parts in the formula (5) it is possible to show that the
solutions $\\{x_{k}(t),t\geq 0\\}$ can be expressed as

(18) $x_{k}(t)=\left(B^{H}_{k}(t)-\mu_{k}{\rm e}^{-\mu_{k}t}\int_{0}^{t}{\rm
e}^{\mu_{k}s}B^{H}_{k}(s)\,{\rm d}s\right),\ t\geq 0,$

for any $k\in\mathbb{N}$. This can also be verified by direct substitution of
(18) into the integral form of equation (4).

Let $\\{\tilde{x}_{k}(t),t\geq 0\\}$ be the solution to the equation with unit
drift

(19) $\begin{array}[]{rcl}{\rm d}\tilde{x}_{k}(t)&=&-\tilde{x}_{k}(t)\,{\rm
d}t+\,{\rm d}B^{H}_{k}(t),\quad t>0,\ k\in\mathbb{N},\\\
\tilde{x}_{k}(0)&=&0,\end{array}$

given by the formula

$\tilde{x}_{k}(t)=\int_{0}^{t}{\rm e}^{-(t-s)}\,{\rm d}B^{H}_{k}(t),\quad
t\geq 0,$

or

(20) $\tilde{x}_{k}(t)=\left(B^{H}_{k}(t)-\int_{0}^{t}{\rm
e}^{-(t-s)}B^{H}_{k}(s)\,{\rm d}s\right),\quad t\geq 0,$

for any $k\in\mathbb{N}$.

Using the self-similarity of a fractional Brownian motion in the expression
(20) we get the equality of distributions

(21) $\\{x_{k}(t),t\geq 0\\}\stackrel{{\scriptstyle
d}}{{=}}\left\\{\frac{1}{\mu_{k}^{H}}\tilde{x}_{k}(\mu_{k}\,t),t\geq
0\right\\}$

for any $k\in\mathbb{N}$.

Moreover, let $\\{y(t),t\geq 0\\}$ be the stationary and ergodic solution to
(19) (for $k=1$) given by a formula

$y(t)={\rm e}^{-t}y_{0}+\int_{0}^{t}{\rm e}^{-(t-s)}\,{\rm d}B^{H}_{1}(t),\
t\geq 0.$

Then

$\mathbb{E}\,y(t)=0,\ {\rm var}\,\big{(}y(t)\big{)}=H\Gamma(2H)$

for any $t\geq 0$ and

(22) $\frac{1}{T}\int_{0}^{T}y^{2}(t)\,{\rm
d}t\xrightarrow[T\rightarrow\infty]{}H\Gamma(2H)\quad\hbox{a.s. and in
}L^{1}.$

######  Lemma 5.1.

The following convergence

$\mathbb{E}\,\left[\frac{1}{T}\int_{0}^{T}\big{(}\tilde{x}^{2}_{1}(t)-y^{2}(t)\big{)}\,{\rm
d}t\right]\xrightarrow[T\rightarrow\infty]{}0$

holds.

###### Proof.

We have

$\displaystyle\mathbb{E}\,\left[\frac{1}{T}\int_{0}^{T}\big{(}\tilde{x}^{2}_{1}(t)-y^{2}(t)\big{)}\,{\rm
d}t\right]$
$\displaystyle\qquad=\mathbb{E}\,\left[\frac{1}{T}\Big{(}\int_{0}^{T}2y(t)\big{(}\tilde{x}_{1}(t)-y(t)\big{)}\,{\rm
d}t+\int_{0}^{T}\big{(}\tilde{x}_{1}(t)-y(t)\big{)}^{2}\,{\rm
d}t\Big{)}\right]$ $\displaystyle\qquad=\frac{1}{T}\left(\int_{0}^{T}-2{\rm
e}^{-t}\mathbb{E}\,\big{[}y_{0}\,y(t)\big{]}\,{\rm d}t+\int_{0}^{T}{\rm
e}^{-2t}\mathbb{E}\,[y_{0}^{2}]\,{\rm d}t\right)$
$\displaystyle\qquad=\frac{1}{T}\int_{0}^{1}-2{\rm
e}^{-t}\mathbb{E}\,\big{[}y_{0}\,y(t)\big{]}\,{\rm
d}t+\frac{1}{T}\int_{1}^{T}-2{\rm
e}^{-t}\mathbb{E}\,\big{[}y_{0}\,y(t)\big{]}\,{\rm d}t$
$\displaystyle\qquad\qquad+H\Gamma(2H)\frac{1}{T}\int_{0}^{T}{\rm
e}^{-2t}\,{\rm d}t.$

Obviously, the first and third term tend to zero as $T$ goes to infinity. Also
the second term tends to zero due to [16], Lemma 3.1, and the limit properties
of

$\mathbb{E}\,\big{[}y_{0}\,y(t)\big{]}=\frac{1}{2}2H(2H-1)t^{2H-2}+O(t^{2H-4}),\
t\rightarrow\infty,$

for $H\neq 1/2$ (see [6], Theorem 2.3) and

$\mathbb{E}\,\big{[}y_{0}\,y(t)\big{]}=\frac{1}{2}{\rm e}^{-t},\ t\geq 0,$

for the Wiener case $H=1/2$. ∎

Define the weak asymptotic equivalence $\asymp$ for sequences
$\\{a_{N}\\}_{N\in\mathbb{N}},\\{b_{N}\\}_{N\in\mathbb{N}}$ by the relation

(23) $\biggl{(}a_{N}\asymp
b_{N}\biggr{)}\quad\equiv\quad\biggl{(}\exists\,0<l<L<\infty\text{ such that
}l\,b_{N}\leq a_{N}\leq L\,b_{N},\forall N\in\mathbb{N}\biggr{)}.$

Note that this relation is weaker than the equivalence relation $\sim$ defined
in (7).

Recall the formula (9) (suitable for probabilistic analysis) and denote

$C_{N}=\sum_{k=1}^{N}\alpha_{k}\int_{0}^{T}x_{k}(t)\,{\rm d}B^{H}_{k}(t),$

and

$D_{N}=\sum_{k=1}^{N}\alpha_{k}^{2}\int_{0}^{T}x_{k}^{2}(t)\,{\rm d}t,$

so that

$\hat{\lambda}_{N}=\lambda-\frac{C_{N}}{D_{N}}.$

For the needs of Lemma A.1 we explore the limit behavior of the variance of
$\frac{C_{N}}{\mathbb{E}\,D_{N}}$ and $\frac{D_{N}}{\mathbb{E}\,D_{N}}$.
Denote for a fixed $T$

$T_{k}:=\mu_{k}T,\ k\in\mathbb{N}.$

######  Proposition 5.1.

The expectations of $C_{N}$ and $D_{N}$ satisfy

$\mathbb{E}\,C_{N}=0\quad\hbox{and}\quad\mathbb{E}\,D_{N}\asymp\sum_{k=1}^{N}\frac{\alpha_{k}^{2}}{\mu_{k}^{2H}}.$

###### Proof.

The first part is obvious. By Lemma 5.1

$\mathbb{E}\,\left[\frac{1}{T}\int_{0}^{T}\tilde{x}^{2}_{1}(t)\,{\rm
d}t\right]\xrightarrow[T\rightarrow\infty]{}H\Gamma(2H),$

thus the sequence
$\left\\{1/T_{k}\int_{0}^{T_{k}}\mathbb{E}\,\tilde{x}^{2}_{1}(t)\,{\rm
d}t\right\\}_{k\in\mathbb{N}}$ is positive and bounded. Using (21) and
substitution we get

$\displaystyle\mathbb{E}\,D_{N}$
$\displaystyle=\sum_{k=1}^{N}\alpha_{k}^{2}\int_{0}^{T}\mathbb{E}\,x_{k}^{2}(t)\,{\rm
d}t=\sum_{k=1}^{N}\frac{\alpha_{k}^{2}}{\mu_{k}^{2H}}\int_{0}^{T}\mathbb{E}\,\tilde{x}_{k}^{2}(\mu_{k}t)\,{\rm
d}t$
$\displaystyle=\sum_{k=1}^{N}\frac{\alpha_{k}^{2}}{\mu_{k}^{2H}}\int_{0}^{T}\mathbb{E}\,\tilde{x}_{1}^{2}(\mu_{k}t)\,{\rm
d}t=\sum_{k=1}^{N}\frac{\alpha_{k}^{2}}{\mu_{k}^{2H}}T\frac{1}{T_{k}}\int_{0}^{T_{k}}\mathbb{E}\,\tilde{x}_{1}^{2}(t)\,{\rm
d}t\asymp\sum_{k=1}^{N}\frac{\alpha_{k}^{2}}{\mu_{k}^{2H}}.$

∎

To study the time asymptotics of ${\rm
var}\,\Big{(}\int_{0}^{T}\tilde{x}_{1}^{2}(t)\,{\rm d}t\Big{)}$, we utilize
the results from [14]. Denote

$F_{T}:=\int_{0}^{T}\tilde{x}_{1}(t)\,{\rm
d}B^{H}_{1}(t)=\int_{0}^{T}\Big{(}\int_{0}^{t}{\rm e}^{-(t-s)}\,{\rm
d}B^{H}_{1}(s)\Big{)}\,{\rm d}B^{H}_{1}(t).$

By (3.9) in [14] this equals

(24)
$F_{T}=\left[\Big{(}\frac{\tilde{x}^{2}_{1}(T)}{2}-\mathbb{E}\,\frac{\tilde{x}^{2}_{1}(T)}{2}\Big{)}+\int_{0}^{T}\big{(}\tilde{x}^{2}_{1}(t)-\mathbb{E}\,\tilde{x}^{2}_{1}(t)\big{)}\,{\rm
d}t\right].$

Further define

$Q_{H}(T):=\left\\{\begin{array}[]{ccl}\frac{1}{T}&,&H<\frac{3}{4},\\\
\frac{1}{T\log T}&,&H=\frac{3}{4},\\\
\frac{1}{T^{4H-2}}&,&H>\frac{3}{4}.\end{array}\right.$

Then by [14], Lemma 17, there exists a constant $0<C_{H}<\infty$ such that

(25)
$\mathbb{E}\,\big{[}Q_{H}(T)F^{2}_{T}\big{]}\xrightarrow[T\rightarrow\infty]{}C_{H}.$

As a consequence, we obtain:

######  Lemma 5.2.

$Q_{H}(T)\,{\rm var}\,\Big{(}\int_{0}^{T}\tilde{x}_{1}^{2}(t)\,{\rm
d}t\Big{)}\xrightarrow[T\rightarrow\infty]{}C_{H}$.

###### Proof.

In the first step we show that ${\rm
var}\,\big{(}\tilde{x}_{1}^{2}(T)\big{)}\xrightarrow[T\rightarrow\infty]{}{\rm
var}\,y_{0}^{2}$.

Since $\tilde{x}_{1}(T)=y(T)-{\rm e}^{-T}y_{0}$, using the triangle inequality
we obtain

$\displaystyle\left(\sqrt{\mathbb{E}\,y^{2}(T)}-{\rm
e}^{-T}\sqrt{\mathbb{E}\,y_{0}^{2}}\right)^{2}\leq$
$\displaystyle\mathbb{E}\,\tilde{x}_{1}^{2}(T)\leq\left(\sqrt{\mathbb{E}\,y^{2}(T)}+{\rm
e}^{-T}\sqrt{\mathbb{E}\,y_{0}^{2}}\right)^{2}$ $\displaystyle
H\Gamma(2H)(1-{\rm e}^{-T})^{2}\leq$
$\displaystyle\mathbb{E}\,\tilde{x}_{1}^{2}(T)\leq H\Gamma(2H)(1+{\rm
e}^{-T})^{2}.$

Similarly,

$\mathbb{E}\,y_{0}^{4}\,(1-{\rm
e}^{-T})^{4}\leq\mathbb{E}\,\tilde{x}_{1}^{4}(T)\leq\mathbb{E}\,y_{0}^{4}\,(1+{\rm
e}^{-T})^{4}.$

Therefore

$\displaystyle\overbrace{\mathbb{E}\,y_{0}^{4}\,(1-{\rm
e}^{\\!-T})^{4}-\big{(}H\Gamma(2H)(1+{\rm
e}^{\\!-T})^{2}\big{)}^{2}}^{\xrightarrow[T\rightarrow\infty]{}\,{\rm
var}\,y_{0}^{2}}\leq{\rm var}\,\tilde{x}_{1}^{2}(T)$ $\displaystyle\hskip
142.26378pt\leq\underbrace{\mathbb{E}\,y_{0}^{4}\,(1+{\rm
e}^{\\!-T})^{4}-\big{(}H\Gamma(2H)(1-{\rm
e}^{-T})^{2}\big{)}^{2}}_{\xrightarrow[T\rightarrow\infty]{}\,{\rm
var}\,y_{0}^{2}}.$

In the second step we apply (25) and the first step to equality (24) and yield
the statement. ∎

######  Proposition 5.2.

The variances of $C_{N}$ and $D_{N}$ satisfy

$\left\\{\begin{array}[]{lllll}H<\frac{3}{4}&\Rightarrow&{\rm
var}\,C_{N}\asymp\sum_{k=1}^{N}\dfrac{\alpha_{k}^{2}}{\mu_{k}^{4H-1}}&,&{\rm
var}\,D_{N}\asymp\sum_{k=1}^{N}\dfrac{\alpha_{k}^{4}}{\mu_{k}^{4H+1}},\\\
H=\frac{3}{4}&\Rightarrow&{\rm
var}\,C_{N}\asymp\sum_{k=1}^{N}\dfrac{\alpha_{k}^{2}}{\mu_{k}^{2}}\log
T_{k}&,&{\rm
var}\,D_{N}\asymp\sum_{k=1}^{N}\dfrac{\alpha_{k}^{4}}{\mu_{k}^{4}}\log
T_{k},\\\ H>\frac{3}{4}&\Rightarrow&{\rm
var}\,C_{N}\asymp\sum_{k=1}^{N}\dfrac{\alpha_{k}^{2}}{\mu_{k}^{2}}&,&{\rm
var}\,D_{N}\asymp\sum_{k=1}^{N}\dfrac{\alpha_{k}^{4}}{\mu_{k}^{4}}.\end{array}\right.$

###### Proof.

Using again (3.9) in [14], (21) and the fact that $\\{\tilde{x}_{k}(t),t\geq
0\\}$ have the same law for all $k\in\mathbb{N}$, we get equality of laws:

$\int_{0}^{T}x_{k}(t)\,{\rm d}B^{H}_{k}(t)\stackrel{{\scriptstyle
d}}{{=}}\dfrac{1}{\mu_{k}^{2H}}\int_{0}^{T_{k}}\tilde{x}_{1}(t)\,{\rm
d}B^{H}_{1}(t)$

for any $k\in\mathbb{N}$. This implies that

$\displaystyle{\rm var}\,C_{N}$ $\displaystyle={\rm
var}\,\Big{(}\sum_{k=1}^{N}\alpha_{k}\int_{0}^{T}x_{k}(t)\,{\rm
d}B^{H}_{k}(t)\Big{)}=\sum_{k=1}^{N}\alpha_{k}^{2}\,{\rm
var}\,\Big{(}\int_{0}^{T}x_{k}(t)\,{\rm d}B^{H}_{k}(t)\Big{)}$
$\displaystyle=\sum_{k=1}^{N}\frac{\alpha_{k}^{2}}{\mu_{k}^{4H}}\,{\rm
var}\,\Big{(}\int_{0}^{T_{k}}\tilde{x}_{1}(t)\,{\rm
d}B^{H}_{1}(t)\Big{)}=\sum_{k=1}^{N}\frac{\alpha_{k}^{2}}{\mu_{k}^{4H}}\frac{1}{Q_{H}(T_{k})}\mathbb{E}\,\big{[}Q_{H}(T_{k})F^{2}_{T_{k}}\big{]},$

where the last equality follows from the definition of $F_{T}$. By (25) the
sequence
$\left\\{\mathbb{E}\,\big{[}Q_{H}(T_{k})F^{2}_{T_{k}}\big{]}\right\\}_{k\in\mathbb{N}}$
is bounded and together with the definition of $Q_{H}$ the first part of the
statements follows.

For the variance of $D_{N}$, apply (21) to get

$\displaystyle{\rm var}\,D_{N}$ $\displaystyle={\rm
var}\,\Big{(}\sum_{k=1}^{N}\alpha_{k}^{2}\int_{0}^{T}x_{k}^{2}(t)\,{\rm
d}t\Big{)}=\sum_{k=1}^{N}\alpha_{k}^{4}\,{\rm
var}\,\Big{(}\int_{0}^{T}x_{k}^{2}(t)\,{\rm d}t\Big{)}$
$\displaystyle=\sum_{k=1}^{N}\frac{\alpha_{k}^{4}}{\mu_{k}^{4H}}\,{\rm
var}\,\Big{(}\int_{0}^{T}\tilde{x}_{k}^{2}(\mu_{k}t)\,{\rm d}t\Big{)}$
$\displaystyle=\sum_{k=1}^{N}\frac{\alpha_{k}^{4}}{\mu_{k}^{4H+2}}\frac{1}{Q_{H}(T_{k})}Q_{H}(T_{k})\,{\rm
var}\,\Big{(}\int_{0}^{T_{k}}\tilde{x}_{1}^{2}(t)\,{\rm d}t\Big{)},$

and Lemma 5.2 to obtain that the sequence $\left\\{Q_{H}(T_{k})\,{\rm
var}\,\Big{(}\int_{0}^{T_{k}}\tilde{x}_{1}^{2}(t)\,{\rm
d}t\Big{)}\right\\}_{k\in\mathbb{N}}$is bounded. The second parts of the
statements are again concluded by the definition of $Q_{H}$. ∎

Combining Proposition 5.1 and Proposition 5.2 we obtain the asymptotic
formulas for variances of $\frac{C_{N}}{\mathbb{E}\,D_{N}}$ and
$\frac{D_{N}}{\mathbb{E}\,D_{N}}$.

(26) $\left\\{\begin{array}[]{llll}H<\frac{3}{4}:&{\rm
var}\,\dfrac{C_{N}}{\mathbb{E}\,D_{N}}\asymp\dfrac{\sum_{k=1}^{N}\dfrac{\alpha_{k}^{2}}{\mu_{k}^{4H-1}}}{\left(\sum_{k=1}^{N}\frac{\alpha_{k}^{2}}{\mu_{k}^{2H}}\right)^{\\!2}}&,&{\rm
var}\,\dfrac{D_{N}}{\mathbb{E}\,D_{N}}\asymp\dfrac{\sum_{k=1}^{N}\dfrac{\alpha_{k}^{4}}{\mu_{k}^{4H+1}}}{\left(\sum_{k=1}^{N}\frac{\alpha_{k}^{2}}{\mu_{k}^{2H}}\right)^{\\!2}},\\\\[19.91692pt]
H=\frac{3}{4}:&{\rm
var}\,\dfrac{C_{N}}{\mathbb{E}\,D_{N}}\asymp\dfrac{\sum_{k=1}^{N}\dfrac{\alpha_{k}^{2}}{\mu_{k}^{2}}\log
T_{k}}{\left(\sum_{k=1}^{N}\frac{\alpha_{k}^{2}}{\mu_{k}^{3/2}}\right)^{\\!2}}&,&{\rm
var}\,\dfrac{D_{N}}{\mathbb{E}\,D_{N}}\asymp\dfrac{\sum_{k=1}^{N}\dfrac{\alpha_{k}^{4}}{\mu_{k}^{4}}\log
T_{k}}{\left(\sum_{k=1}^{N}\frac{\alpha_{k}^{2}}{\mu_{k}^{3/2}}\right)^{\\!2}},\\\\[19.91692pt]
H>\frac{3}{4}:&{\rm
var}\,\dfrac{C_{N}}{\mathbb{E}\,D_{N}}\asymp\dfrac{\sum_{k=1}^{N}\dfrac{\alpha_{k}^{2}}{\mu_{k}^{2}}}{\left(\sum_{k=1}^{N}\frac{\alpha_{k}^{2}}{\mu_{k}^{2H}}\right)^{\\!2}}&,&{\rm
var}\,\dfrac{D_{N}}{\mathbb{E}\,D_{N}}\asymp\dfrac{\sum_{k=1}^{N}\dfrac{\alpha_{k}^{4}}{\mu_{k}^{4}}}{\left(\sum_{k=1}^{N}\frac{\alpha_{k}^{2}}{\mu_{k}^{2H}}\right)^{\\!2}}.\end{array}\right.$

Note that if for some $c,\delta>0$ and for all $N\in\mathbb{N}$

(27) ${\rm var}\,\frac{C_{N}}{\mathbb{E}\,D_{N}}\leq c\,N^{-\delta}\ \hbox{
and }\ {\rm var}\,\frac{D_{N}}{\mathbb{E}\,D_{N}}\leq c\,N^{-\delta}$

holds true then application of Lemma A.1 ($C_{N}/\mathbb{E}\,D_{N}$ and
$(D_{N}-\mathbb{E}\,D_{N})/\mathbb{E}\,D_{N}$ are in the second Wiener chaos,
see (47) in Appendix) provides us with

$\frac{C_{N}}{\mathbb{E}\,D_{N}}\xrightarrow[N\rightarrow\infty]{}0\quad\hbox{
and }\quad\frac{D_{N}}{\mathbb{E}\,D_{N}}\xrightarrow[N\rightarrow\infty]{}1\
\hbox{ a.s.}$

so that the strong consistency of the estimate

$\hat{\lambda}_{N}=\lambda-\dfrac{\frac{C_{N}}{\mathbb{E}\,D_{N}}}{\frac{D_{N}}{\mathbb{E}\,D_{N}}}\xrightarrow[N\rightarrow\infty]{}\lambda\
\hbox{ a.s.}$

follows.

It is difficult to verify the sufficient conditions (27) in general and it
should be done on case-by-case basis. However, if we adopt assumption (6),
often satisfied for parabolic SPDEs, the calculations of asymptotic variances
simplify substantially.

######  Theorem 5.1.

Let $B\equiv 0$, i.e. $\beta_{k}=0$ for all $k\in\mathbb{N}$, and the power
growth (6) is satisfied for $\alpha_{k}$. Then $\hat{\lambda}_{N}$ is strongly
consistent for any $m_{1},d\in\mathbb{N}$.

###### Proof.

Denote the growth exponent of $\alpha_{k}$ by $M_{1}:=2m_{1}/d$. Then using
formula (26) we obtain

$\displaystyle{\rm var}\,\frac{C_{N}}{\mathbb{E}\,D_{N}}$
$\displaystyle\asymp{\rm var}\,\frac{D_{N}}{\mathbb{E}\,D_{N}}$ (31)
$\displaystyle\asymp\left\\{\begin{array}[]{lll}\dfrac{\sum_{k=1}^{N}k^{M_{1}(3-4H)}}{\left(\sum_{k=1}^{N}k^{2M_{1}(1-H)}\right)^{\\!2}}\asymp
N^{-(M_{1}+1)}&,&\\!H<\frac{3}{4},\\\\[11.38109pt]
\dfrac{\sum_{k=1}^{N}\log(k^{M_{1}}T)}{\left(\sum_{k=1}^{N}k^{1/2M_{1}}\right)^{\\!2}}\asymp
N^{-(M_{1}+1)}\log N&,&\\!H=\frac{3}{4},\\\
\dfrac{N}{\left(\sum_{k=1}^{N}k^{2M_{1}(1-H)}\right)^{\\!2}}\asymp
N^{-\big{(}4M_{1}(1-H)+1\big{)}}&,&\\!H>\frac{3}{4}.\end{array}\right.$

Thus (27) is satisfied for any $m_{1},d$, so the conclusion follows from Lemma
A.1. ∎

######  Theorem 5.2.

Assume the power growth of eigenvalues $\alpha_{k}$ and $\beta_{k}$ given in
(6):

$\alpha_{k}\sim k^{2m_{1}/d}\hbox{ and }\beta_{k}\sim k^{2m_{2}/d}.$

If

  * (i)

$m_{1}>m_{2}$ then $\hat{\lambda}_{N}$ is strongly consistent for any
$m_{1},m_{2},d\in\mathbb{N}$,

  * (ii)

$m_{1}<m_{2}$ and

$m_{1}>\left\\{\begin{array}[]{lll}-\frac{d}{4}+\frac{1}{2}m_{2}&,&H<\frac{3}{4},\\\\[5.69054pt]
-\frac{d}{4}+m_{2}(2H-1)&,&H\geq\frac{3}{4},\end{array}\right.$

then $\hat{\lambda}_{N}$ is strongly consistent.

###### Proof.

  * (i)

Applying again formula (26) we obtain the same estimates as in the proof of
Theorem 5.1 thus the conclusion follows.

  * (ii)

For conciseness, we consider the canonical case (all the involved series have
power growth) only. The special cases, where some series in the expectation of
$D_{N}$ or the variances of $C_{N},D_{N}$ have logarithmic growth, have to be
analyzed separately. Nevertheless, they are covered by (ii) as well.

For ${\rm var}\,\frac{D_{N}}{\mathbb{E}\,D_{N}}$ we get the same estimates as
in the proof of Theorem 5.1 but with $M_{1}$ substituted by $M_{2}:=2m_{2}/d$.
For ${\rm var}\,\frac{C_{N}}{\mathbb{E}\,D_{N}}$ formula (26) yields

$\displaystyle{\rm var}\,\dfrac{C_{N}}{\mathbb{E}\,D_{N}}$ (35)
$\displaystyle\quad\asymp\left\\{\begin{array}[]{l@{\\!}ll}\dfrac{\sum_{k=1}^{N}k^{2M_{1}-M_{2}(4H-1)}}{\left(\sum_{k=1}^{N}k^{2(M_{1}-HM_{2})}\right)^{\\!2}}\asymp
N^{-2M_{1}+M_{2}-1}&,&H<\frac{3}{4},\\\\[11.38109pt]
\dfrac{\sum_{k=1}^{N}k^{2M_{1}+2M_{2}}\log(k^{M_{2}}T)}{\left(\sum_{k=1}^{N}k^{2M_{1}-3/2M_{2}}\right)^{\\!2}}\asymp
N^{-2M_{1}+M_{2}-1}\log N&,&H=\frac{3}{4},\\\\[11.38109pt]
\dfrac{\sum_{k=1}^{N}k^{2(M_{1}-M_{2})}}{\left(\sum_{k=1}^{N}k^{2(M_{1}-HM_{2})}\right)^{\\!2}}\asymp
N^{-2\big{(}M_{1}+M_{2}(1-2H)\big{)}-1}&,&H>\frac{3}{4}.\end{array}\right.$

These estimates give us the required relationship between $m_{1},m_{2}$ and
$d$ to meet (27). The rest follows from Lemma A.1.

∎

######  Remark 5.1.

The case $m_{1}=m_{2}$ is reduced to the case of Theorem 5.1.

######  Remark 5.2.

The expressions for ${\rm var}\,\frac{C_{N}}{\mathbb{E}\,D_{N}}$ given in
formulas (31) and (35) show the speed of the convergence of
$\hat{\lambda}_{N}$ to $\lambda$.

###  5.2. Strong consistency of pathwise LSE

In this subsection, strong consistency (as $N\to\infty$) of
$\tilde{\lambda}_{N}$ is demonstrated. We continue to denote the true value of
the unknown parameter by $\lambda$. Recall that the theoretical LSE satisfies
$\hat{\lambda}_{N}=R_{N}(\lambda)$. Hence, if $\hat{\lambda}_{N}$ is strongly
consistent (e.g. if the conditions of Theorem 5.1 or Theorem 5.2 are
satisfied), we have:

$\lim_{N\to\infty}R_{N}(\lambda)=\lambda.$

To show that the pathwise LSE $\tilde{\lambda}_{N}=R_{N}(\tilde{\lambda}_{N})$
is arbitrarily close to $\lambda$ for sufficiently large $N$, the limiting
behavior of derivatives of $R_{N}$ is studied. The following elementary
observation will be useful in the sequel:

######  Lemma 5.3.

Let $a_{k},b_{k}\geq 0$ for $k=1,2,\dots$,
$\lim_{k\to\infty}b_{k}=b\in(0,\infty)$, and
$\sum_{k=1}^{\infty}a_{k}=\infty$. Then

$\lim_{N\to\infty}\frac{\sum_{k=1}^{N}a_{k}b_{k}}{\sum_{k=1}^{N}a_{k}b}=1.$

###### Proof.

It is a simple consequence of the Stolz–Cesaro theorem. ∎

Next lemma bounds the derivatives of $R_{N}$ on a compact neighborhood of
$\lambda$ from above by a constant smaller than one (for sufficiently large
N):

######  Lemma 5.4.

Let the power growth condition (6) be satisfied and $H\neq 1/2$. Further
assume:

(36) $\lim_{k\to\infty}\frac{\beta_{k}}{\alpha_{k}}=c\in[0,\infty],$ (37)
$\sum_{k=1}^{\infty}\frac{\alpha_{k}^{2}}{(\alpha_{k}+\beta_{k})^{2H}}=\infty.$

Then, almost surely, if we take arbitrary $0<\Lambda_{L}<\Lambda_{U}<\infty$
we have

(38)
$\displaystyle\lim_{N\to\infty}\sup_{\Lambda\in[\Lambda_{L},\Lambda_{U}]}\left[\frac{1}{1-2H}\frac{\,{\rm
d}R_{N}}{\,{\rm
d}\Lambda}(\Lambda)\right]=\frac{(\lambda+c)^{2H}}{(\Lambda_{L}+c)^{2H}},$
$\displaystyle\lim_{N\to\infty}\inf_{\Lambda\in[\Lambda_{L},\Lambda_{U}]}\left[\frac{1}{1-2H}\frac{\,{\rm
d}R_{N}}{\,{\rm
d}\Lambda}(\Lambda)\right]=\frac{(\lambda+c)^{2H}}{(\Lambda_{U}+c)^{2H}},$

where the right-hand sides equal one if $c=\infty$.

###### Proof.

Recall the formula for the derivative of $R_{N}$ in (15) and the notation from
previous subsection
$D_{N}=\sum_{k=1}^{N}\alpha_{k}^{2}\int_{0}^{T}x^{2}_{k}(t)\,{\rm d}t$. It
follows from the proof of Proposition 5.1 that

$\mathbb{E}D_{N}=\sum_{k=1}^{N}\frac{\alpha_{k}^{2}}{\mu_{k}^{2H}}T\varphi(\mu_{k}T),$

where

$\varphi(\tau)=\frac{1}{\tau}\int_{0}^{\tau}\mathbb{E}\tilde{x}^{2}_{1}(t)\,{\rm
d}t\xrightarrow[\tau\to\infty]{}H\Gamma(2H).$

Use (15) to write (for any $\Lambda>0$):

(39) $\displaystyle\frac{\,{\rm d}R_{N}}{\,{\rm
d}\Lambda}(\Lambda)=\frac{\sum_{k=1}^{N}\alpha^{2}_{k}\frac{\,{\rm
d}\Delta}{\,{\rm
d}\mu}(\alpha_{k}\Lambda+\beta_{k})}{D_{N}}=\frac{\mathbb{E}D_{N}}{D_{N}}\frac{\sum_{k=1}^{N}\alpha^{2}_{k}\frac{\,{\rm
d}\Delta}{\,{\rm
d}\mu}(\alpha_{k}\Lambda+\beta_{k})}{\sum_{k=1}^{N}\frac{\alpha_{k}^{2}}{\mu_{k}^{2H}}T\varphi(\mu_{k}T)}.$

Formula (13) together with $\lim_{T\to\infty}\gamma(h,T)=\Gamma(h)$ yield

(40) $\lim_{\mu\to\infty}\mu^{2H}\frac{\,{\rm d}\Delta}{\,{\rm
d}\mu}(\mu)=TH(1-2H)\Gamma(2H).$

Consequently, we can factorize

$\displaystyle\frac{\,{\rm d}R_{N}}{\,{\rm
d}\Lambda}(\Lambda)=\frac{\mathbb{E}D_{N}}{D_{N}}U_{N}V_{N}(\Lambda)W_{N}(\Lambda),$

with

$\displaystyle U_{N}$
$\displaystyle=\frac{\sum_{k=1}^{N}\frac{\alpha_{k}^{2}}{\mu_{k}^{2H}}TH\Gamma(2H)}{\sum_{k=1}^{N}\frac{\alpha_{k}^{2}}{\mu_{k}^{2H}}T\varphi(\mu_{k}T)},$
$\displaystyle V_{N}(\Lambda)$
$\displaystyle=\frac{\sum_{k=1}^{N}\alpha^{2}_{k}\frac{\,{\rm d}\Delta}{\,{\rm
d}\mu}(\alpha_{k}\Lambda+\beta_{k})}{\sum_{k=1}^{N}\frac{\alpha^{2}_{k}}{(\alpha_{k}\Lambda+\beta_{k})^{2H}}TH(1-2H)\Gamma(2H)},$
$\displaystyle W_{N}(\Lambda)$
$\displaystyle=\frac{\sum_{k=1}^{N}\frac{\alpha^{2}_{k}}{(\alpha_{k}\Lambda+\beta_{k})^{2H}}TH(1-2H)\Gamma(2H)}{\sum_{k=1}^{N}\frac{\alpha_{k}^{2}}{\mu_{k}^{2H}}TH\Gamma(2H)}.$

Recall that condition (6) implies (cf. proofs of Theorem 5.1 and Theorem 5.2)

$\lim_{N\to\infty}\frac{D_{N}}{\mathbb{E}D_{N}}=1\quad\text{almost surely}.$

Assumptions of Lemma 5.3 with

$a_{k}=\frac{\alpha_{k}^{2}}{\mu_{k}^{2H}},\quad
b_{k}=T\varphi(\mu_{k}T),\quad\text{and }b=TH\Gamma(2H),$

are guaranteed by conditions (37) and (3). In result,

$\lim_{N\to\infty}U_{N}=1.$

Denote (motivated by Lemma 5.3)

$a_{k}(\Lambda)=\frac{\alpha^{2}_{k}}{(\alpha_{k}\Lambda+\beta_{k})^{2H}},\quad
b_{k}(\Lambda)=(\alpha_{k}\Lambda+\beta_{k})^{2H}\frac{\,{\rm d}\Delta}{\,{\rm
d}\mu}(\alpha_{k}\Lambda+\beta_{k})$

and

$b=TH(1-2H)\Gamma(2H).$

Fix arbitrary $0<\Lambda_{L}<\Lambda_{U}<\infty$. Obviously, due to (3) and
(40),

$\sup_{\Lambda\in[\Lambda_{L},\Lambda_{U}]}\big{|}b_{k}(\Lambda)-b\big{|}\xrightarrow[k\to\infty]{}0,$

and, due to (37)

$\inf_{\Lambda\in[\Lambda_{L},\Lambda_{U}]}\sum_{k=1}^{N}a_{k}(\Lambda)=\sum_{k=1}^{N}a_{k}(\Lambda_{U})\xrightarrow[N\to\infty]{}\infty.$

Similarly to proof of Lemma 5.3 we can show

$\lim_{N\to\infty}\sup_{\Lambda\in[\Lambda_{L},\Lambda_{U}]}\big{|}V_{N}(\Lambda)-1\big{|}=\lim_{N\to\infty}\sup_{\Lambda\in[\Lambda_{L},\Lambda_{U}]}\left|\frac{\sum_{k=1}^{N}a_{k}(\Lambda)b_{k}(\Lambda)}{\sum_{k=1}^{N}a_{k}(\Lambda)b}-1\right|=0.$

For the last term we can use the upper bound:

$\sup_{\Lambda\in[\Lambda_{L},\Lambda_{U}]}\left[\frac{W_{N}(\Lambda)}{1-2H}\right]=\frac{\sum_{k=1}^{N}\frac{\alpha^{2}_{k}}{(\alpha_{k}\Lambda_{L}+\beta_{k})^{2H}}}{\sum_{k=1}^{N}\frac{\alpha_{k}^{2}}{(\alpha_{k}\lambda+\beta_{k})^{2H}}}.$

Assume first $\lim_{k\to\infty}\frac{\beta_{k}}{\alpha_{k}}=c\in[0,\infty)$.
Then

$\frac{\sum_{k=1}^{N}\frac{\alpha^{2}_{k}}{(\alpha_{k}\Lambda_{L}+\beta_{k})^{2H}}}{\sum_{k=1}^{N}\frac{\alpha_{k}^{2}}{(\alpha_{k}\lambda+\beta_{k})^{2H}}}=\frac{\sum_{k=1}^{N}\alpha^{2-2H}_{k}\big{(}\Lambda_{L}+\frac{\beta_{k}}{\alpha_{k}}\big{)}^{-2H}}{\sum_{k=1}^{N}\alpha^{2-2H}_{k}\big{(}\lambda+\frac{\beta_{k}}{\alpha_{k}}\big{)}^{-2H}}.$

Application of Lemma 5.3 with

$a_{k}=\alpha^{2-2H}_{k},\quad
b_{k}=\Big{(}\Lambda+\frac{\beta_{k}}{\alpha_{k}}\Big{)}^{\\!-2H},\quad
b=(\Lambda+c)^{-2H},$

and $\Lambda=\Lambda_{L}$ or $\Lambda=\lambda$, respectively, and where

$\sum_{k=1}^{\infty}\alpha^{2-2H}_{k}\geq\sum_{k=1}^{\infty}\frac{\alpha_{k}^{2}}{(\alpha_{k}+\beta_{k})^{2H}}=\infty,$

results in

$\lim_{N\to\infty}\frac{\sum_{k=1}^{N}\frac{\alpha^{2}_{k}}{(\alpha_{k}\Lambda_{L}+\beta_{k})^{2H}}}{\sum_{k=1}^{N}\frac{\alpha_{k}^{2}}{(\alpha_{k}\lambda+\beta_{k})^{2H}}}=\lim_{N\to\infty}\frac{\sum_{k=1}^{N}\alpha^{2-2H}_{k}(\Lambda_{L}+c)^{-2H}}{\sum_{k=1}^{N}\alpha^{2-2H}_{k}(\lambda+c)^{-2H}}=\frac{(\lambda+c)^{2H}}{(\Lambda_{L}+c)^{2H}}.$

Next, assume instead $c=\infty$, i.e.
$\lim_{k\to\infty}\frac{\alpha_{k}}{\beta_{k}}=0$. We can proceed similarly to
previous case

$\lim_{N\to\infty}\frac{\sum_{k=1}^{N}\frac{\alpha^{2}_{k}}{(\alpha_{k}\Lambda_{L}+\beta_{k})^{2H}}}{\sum_{k=1}^{N}\frac{\alpha_{k}^{2}}{(\alpha_{k}\lambda+\beta_{k})^{2H}}}=\lim_{N\to\infty}\frac{\sum_{k=1}^{N}\frac{\alpha^{2}_{k}}{\beta_{k}^{2H}}\big{(}\frac{\alpha_{k}}{\beta_{k}}\Lambda_{L}+1\big{)}^{-2H}}{\sum_{k=1}^{N}\frac{\alpha^{2}_{k}}{\beta_{k}^{2H}}\big{(}\frac{\alpha_{k}}{\beta_{k}}\lambda+1\big{)}^{-2H}}=1.$

To summarize, we have

$\lim_{N\to\infty}\sup_{\Lambda\in[\Lambda_{L},\Lambda_{U}]}\left[\frac{W_{N}(\Lambda)}{1-2H}\right]=\begin{cases}\frac{(\lambda+c)^{2H}}{(\Lambda_{L}+c)^{2H}}\quad&\text{if
}c\in[0,\infty),\\\ 1\quad&\text{if }c=\infty.\\\ \end{cases}$

The limit for infimum can be obtained accordingly, since

$\inf_{\Lambda\in[\Lambda_{L},\Lambda_{U}]}\left[\frac{W_{N}(\Lambda)}{1-2H}\right]=\frac{\sum_{k=1}^{N}\frac{\alpha^{2}_{k}}{(\alpha_{k}\Lambda_{U}+\beta_{k})^{2H}}}{\sum_{k=1}^{N}\frac{\alpha_{k}^{2}}{(\alpha_{k}\lambda+\beta_{k})^{2H}}}.$

∎

######  Theorem 5.3.

Assume that the eigenvalues $\alpha_{k}$ and $\beta_{k}$ satisfy the power
growth conditions (6) and the asymptotic conditions (36), (37) and (3).
Further assume that the theoretical LSE $\hat{\lambda}_{N}$ is strongly
consistent (see e.g. Theorem 5.1 and Theorem 5.2). Then the pathwise LSE is
strongly consistent as well:

(41) $\lim_{N\to\infty}\tilde{\lambda}_{N}=\lambda\quad\text{almost surely}.$

Moreover, $\tilde{\lambda}_{N}$ and $\hat{\lambda}_{N}$ have the same speed of
convergence (up to a constant):

(42)
$\lim_{N\to\infty}\frac{|\lambda-\hat{\lambda}_{N}|}{|\lambda-\tilde{\lambda}_{N}|}=2H\quad\text{almost
surely}.$

###### Proof.

Recall that if $H=1/2$, the theorem holds true trivially, because
$\tilde{\lambda}_{N}=\hat{\lambda}_{N}$ in this case. In the rest of the
proof, consider $H\neq 1/2$. First, take $\Lambda_{L}<\lambda$ so that

$(1-2H)\frac{(\lambda+c)^{2H}}{(\Lambda_{L}+c)^{2H}}<1.$

and arbitrary $\Lambda_{U}>\lambda$. Lemma 5.4 guarantees that, for almost all
$\omega$ and some $\Psi>0$,

$\inf_{\Lambda\in[\Lambda_{L},\Lambda_{U}]}\left[1-\frac{\,{\rm
d}R_{N}}{\,{\rm d}\Lambda}(\Lambda)\right]>\Psi,$

for sufficiently large $N$. Fix $\epsilon>0$ such that
$\Lambda_{L}<\lambda-\epsilon<\lambda+\epsilon<\Lambda_{U}$ and denote
$\delta=\epsilon\Psi$. The almost sure convergence
$\lim_{N\to\infty}R_{N}(\lambda)=\lambda$, which is guaranteed by strong
consistency of the theoretical LSE, enables us to take $N$ large enough in
order to

$\big{|}\lambda-R_{N}(\lambda)\big{|}<\delta.$

Combination of the lower bound on derivative of $\Lambda-R_{N}(\Lambda)$ and
its closeness to zero at $\lambda$ implies existence of its root inside
$(\lambda-\epsilon,\lambda+\epsilon)$, which is unique therein and it equals
$\tilde{\lambda}_{N}$. This proves (41).

To demonstrate (42), note that for sufficiently large $N$

$\frac{|\lambda-\hat{\lambda}_{N}|}{|\lambda-\tilde{\lambda}_{N}|}=\left|1-\frac{\,{\rm
d}R_{N}}{\,{\rm d}\Lambda}(\Lambda^{0}_{N})\right|,$

where $\Lambda^{0}_{N}$ is between $\tilde{\lambda}_{N}$ and $\lambda$ and
therefore converges to $\lambda$ almost surely. Next, fix arbitrary
$\epsilon>0$ and take $\Lambda_{L}<\lambda<\Lambda_{U}$ so that the right-hand
sides of (38) are close enough to one and

$\sup_{\Lambda\in[\Lambda_{L},\Lambda_{U}]}\left|\frac{\,{\rm d}R_{N}}{\,{\rm
d}\Lambda}(\Lambda)-(1-2H)\right|<\epsilon,$

for sufficiently large $N$. In particular, for $N$ large enough we have almost
surely

$\left|\frac{\,{\rm d}R_{N}}{\,{\rm
d}\Lambda}(\Lambda^{0}_{N})-(1-2H)\right|<\epsilon.$

This proves the desired almost sure convergence

$\lim_{N\to\infty}\frac{|\lambda-\hat{\lambda}_{N}|}{|\lambda-\tilde{\lambda}_{N}|}=\lim_{N\to\infty}\left|1-\frac{\,{\rm
d}R_{N}}{\,{\rm d}\Lambda}(\Lambda^{0}_{N})\right|=\big{|}1-(1-2H)\big{|}=2H.$

∎

######  Remark 5.3.

Immediate, and possibly surprising, consequence of (42) is that the pathwise
LSE performs asymptotically better than the theoretical LSE whenever $H>1/2$.

###  5.3. Example

As an example of the use of Theorem 5.3, consider a stochastic heat equation
with additional mean reversion, with a distributed noise autocorrelated in
time and with the Dirichlet boundary condition. Formally, we can write this
equation as

(43) $\displaystyle\frac{\partial u}{\partial t}(t,\xi)$
$\displaystyle=\lambda_{1}\;\Delta_{\xi}u(t,\xi)-\lambda_{2}\;u(t,\xi)+\eta^{H}(t,\xi),\quad\text{
for }(t,\xi)\in\mathbb{R}_{+}\times\mathcal{O},$ $\displaystyle u(t,\xi)$
$\displaystyle=0,\quad\text{ for
}(t,\xi)\in\mathbb{R}_{+}\times\partial\mathcal{O},$ $\displaystyle u(0,\xi)$
$\displaystyle=0,\quad\text{ for }\xi\in\mathcal{O},$

where $\Delta_{\xi}$ is Laplace operator in variable $\xi$,
$\mathcal{O}\subset\mathbb{R}^{d}$ is a $d$-dimensional bounded domain with
smooth boundary $\partial\mathcal{O}$, $\lambda_{1}\geq 0$ is the diffusivity
parameter determining the speed of diffusion of a modeled substance along the
domain $\mathcal{O}$, $\lambda_{2}\geq 0$ is the rate of mean reversion
related to the speed of reversion of the amount of substance to the zero mean.
The process $\\{\eta^{H}(t,\xi);\,t\geq 0,\xi\in\mathcal{O}\\}$ is a Gaussian
noise, which is fractional (autocorrelated) in time with Hurst parameter
$H\in(0,1)$ and white in space. Formally it can be modelled as the
differential of a cylindrical fractional Brownian motion $\,{\rm d}B^{H}(t)$
(see Appendix B for details).

This formal equation can be rewritten rigorously as a stochastic evolution
equation (1) in $V=L^{2}(\mathcal{O})$, with $A=\Delta|_{Dom(A)}$, where
$Dom(A)=H^{2}(\mathcal{O})\cap H^{1}_{0}(\mathcal{O})$, being Dirichlet
Laplace operator defined on a standard Sobolev space (cf. [31]) and with $B$
being the identity operator on $L^{2}(\mathcal{O})$, should $\lambda_{1}$ be
the estimated parameter (or vice-versa, should we estimate $\lambda_{2}$).
Consequently, this equation is diagonalizable with eigenvalues satisfying the
power growth condition (6).

Firstly, assume we want to estimate the diffusivity parameter $\lambda_{1}$.
If $\lambda_{2}=0$, we have a stochastic heat equation with $\alpha_{k}\sim
k^{2/d}$, and $\beta_{k}=0$. Theorem 5.1 guarantees strong consistency of the
theoretical LSE $\hat{\lambda}_{N}$. If $\lambda_{2}>0$, there is an
additional mean reversion in the process and we have $\alpha_{k}\sim k^{2/d}$,
and $\beta_{k}\sim 1$. We can apply Theorem 5.2 (i) to get strong consistency
of the theoretical LSE $\hat{\lambda}_{N}$. In both cases, Theorem 5.3 implies
strong consistency of the pathwise LSE $\tilde{\lambda}_{N}$ with the speed of
convergence (in terms of RMSE) given by

(44)
$\sqrt{\mathbb{E}\,(\tilde{\lambda}_{N}-\lambda)^{2}}\asymp\left\\{\begin{array}[]{lll}N^{-(1/d+1/2)}&,&H<\frac{3}{4},\\\\[2.84526pt]
N^{-(1/d+1/2)}\sqrt{\log N}&,&H=\frac{3}{4},\\\
N^{-\big{(}4(1-H)/d+1/2\big{)}}&,&H>\frac{3}{4}.\end{array}\right.$

Secondly, let $\lambda_{2}$ be the parameter of interest. If $\lambda_{1}=0$
we have $\alpha_{k}\sim 1$ and $\beta_{k}=0$, and the existence condition (2)
is violated. Hence, we consider the case $\lambda_{1}>0$, when the process
combines diffusion and mean reversion. For the eigenvalues, we have
$\alpha_{k}\sim 1$, and $\beta_{k}\sim k^{2/d}$. Theorem 5.2 (ii) provides us
with strong consistency of $\hat{\lambda}_{N}$ if $d>2$ for $H<3/4$ and
$d>4(2H-1)$ otherwise. To show strong consistency of the pathwise LSE by
Theorem 5.3, we need in addition that $d\geq 4H$ so that (37) is satisfied. If
the conditions for consistency are satisfied, the speed of convergence is

(45)
$\sqrt{\mathbb{E}\,(\tilde{\lambda}_{N}-\lambda)^{2}}\asymp\left\\{\begin{array}[]{lll}N^{1/d-1/2}&,&H<\frac{3}{4},\\\\[2.84526pt]
N^{1/d-1/2}\sqrt{\log N}&,&H=\frac{3}{4},\\\\[2.84526pt]
N^{(4H-2)/d-1/2}&,&H>\frac{3}{4}.\end{array}\right.$

##  6\. Simulation study

To illustrate the actual performance of the studied pathwise LSE we perform a
Monte Carlo analysis for two specific equations – a 1D heat equation (the
simplest setting) and a 2D heat equation with coupling with surroundings (a
model popular e.g. in oceanography). All simulations were performed in
statistical software R.

###  6.1. 1D stochastic heat equation

This equation has very simple structure and serves as an easy-to-undersand
toy-model suitable for illustration of the theory above. We simulate a
stochastic heat equation with zero boundary condition from subsection 5.3 on
the line segment (0,1), with true diffusivity $\lambda_{1}=1$ being a
parameter to be estimated ($\lambda=\lambda_{1}$) and without additional mean
reversion ($\lambda_{2}=0$). In particular, we set

  * •

$d=1$, $\mathcal{O}=(0,1)$ and time horizon $T=1$,

  * •

$\bigl{(}Af\bigr{)}(\xi)=\frac{\partial^{2}f}{\partial\xi^{2}},\
\xi\in(0,1),\quad B\equiv 0$,

  * •

$e_{k}(\xi)=\sin(k\pi\xi),\quad\alpha_{k}=(k\pi)^{2},\quad\beta_{k}=0\quad\forall
k\in\mathbb{N}$,

  * •

two scenarios for Hurst index: $H\in\\{0.2,0.8\\}$,

  * •

two scenarios for initial condition: (i) $x_{k}(0)=0$ for all $k\geq 1$ (zero
IC), and (ii) $x_{1}(0)=10,\ x_{2}(0)=5,\ x_{3}(0)=2,\ x_{k}(0)=0,\ k\geq 4$
(non-zero IC).

The observed Fourier modes take the following form

$x_{k}(t)=\int_{0}^{1}u(t,\xi)\sin(k\pi\xi)\,{\rm d}\xi.$

For simulations we used the Spectral Galerkin Method (see [20], chapter
10.7.), which is very efficient in our setting (space discretization provides
system of independent linear stochastic ODEs, which, in addition, can be
solved explicitly). Heat maps of four simulated sample solutions for various
scenarios are shown in Figure 1.

Figure 1. Heat maps of sample solutions of 1D stochastic heat equation with
zero (the upper row) and non-zero (the lower row) initial conditions.
Horizontal axis represents the physical space domain and vertical axis the
time domain.

We test performance of the pathwise LSE, defined in (17), and compare it with
the weighted MCE, an alternative estimator defined and studied in [17], cf.
formulas (3.21) – (3.23) and Theorem 3.2 therein. Recall that the weighted MCE
is strongly consistent and asymptotically normal with increasing number of
observed modes.

Simulation results were obtained from 200 runs for each scenarios producing
sample of 200 estimates for each setting. Figure 2 illustrates the convergence
of the estimates to the true value of the parameter with increasing number of
Fourier modes by showing selected sample quantiles. It confirms the
convergence for all settings. In addition, non-zero initial condition
significantly improves the performance of the pathwise LSE. Figure 3 shows
convergence of the root mean square error (RMSE) for samples of the two types
of estimators (pathwise LSE and weighted MCE) and its comparison with the
theoretical speed of convergence for RMSE from (44) (applicable only for
solutions with zero initial condition). In case of zero initial condition,
RMSE for the pathwise LSE and the weighted MCE practically coincide and their
speed of convergence is consistent with the theoretical one. For solutions
with far-from-zero initial conditions RMSE for the pathwise LSE is close to
zero even for small number of Fourier modes, whereas the weighted MCE
converges significantly slower. Figure 4 shows comparison of sample quantiles
of pathwise LSE (40 Fourier modes considered) with quantiles of normal
distribution including 95% confidence envelope. Although not proved
theoretically, these Q-Q plots suggest asymptotic normality of the pathwise
LSE for all settings considered.

Figure 2. Sample quantile ranges (green areas) and median (red line) of the
simulated estimates (200 samples), true value is 1.
Left column: Hurst index = 0.2 (rough case); right column: Hurst index = 0.8
(smooth case).
First row: solutions with zero init. cond.; second row: solutions with non-
zero init. cond.

Figure 3. Root mean square error (RMSE) for pathwise LSE and weighted MCE and
its theoretical speed of convergence from (44) (applicable only for zero
initial condition).
Left column: Hurst index = 0.2 (rough case); right column: Hurst index = 0.8
(smooth case).
First row: solutions with zero init. cond.; second row: solutions with non-
zero init. cond.

Figure 4. Q-Q plot for sample of 200 pathwise LSE estimates (calculated from
40 Fourier modes) including 95% confidence envelope for normal distribution.
Left column: Hurst index = 0.2 (rough case); right column: Hurst index = 0.8
(smooth case).
First row: solutions with zero init. cond.; second row: solutions with non-
zero init. cond.

###  6.2. 2D stochastic heat equation with coupling with surroundings

Equations of this type are popular (not only) in physical oceanography, where
they model fluctuations of the temperature of the top layer of the ocean
around its long-time average, see [28] or [21]. These equations fall into the
setting of Example in 5.3 with $d=2$ (a two-dimensional domain being the
surface of the ocean), where the 2D Laplace operator models thermal conduction
along the water surface, the mean-reversion term reflects atmosphere and ocean
coupling and the noise term represents random forcing (e.g. vertical heat
fluxes due to turbulent environment) – for more details see [28]. The authors
in [28] and [21] (Section 6.1.1) discuss estimation of parameters in these
equations based on observations in spectral domain under the assumption that
the noise process is white in time and space (generated by a standard
cylindrical Brownian motion). In contrast, we study parameter estimation
assuming more general noise, which can be autocorrelated in time (generated by
a cylindrical fractional Brownian motion).

In particular, we simulate equation (43) with the following setting:

  * •

$d=2$, $\mathcal{O}=(0,1)^{2}$ and time horizon $T=1$,

  * •

$\lambda_{1}=1$ (the parameter to be estimated), $\lambda_{2}=1$,

  * •

Hurst index $H=0.3$, zero initial condition.

We used again the Spectral Galerkin Method with eigenfuctions
$\sin(k_{1}\pi\xi_{1})\times\sin(k_{2}\pi\xi_{2})$ and corresponding
eigenvalues $1+(k_{1}^{2}+k_{2}^{2})\pi^{2}$, where $k_{1},k_{2}=0,1,2,\dots$
and $(\xi_{1},\xi_{2})\in(0,1)^{2}$.

The simulated distribution of the heat along the water surface at specific
time instants is shown in Figure 5. Results of the Monte Carlo study of the
behavior of the pathwise LSE obtained from 200 runs are depicted in Figure 6.
They confirm our theoretical findings on the almost-sure convergence of the
pathwise LSE and its speed of convergence (in terms of RMSE). Moreover, Q-Q
plot indicate its asymptotic normality, although we do not provide theoretical
reasoning (for conciseness).

Figure 5. Heat maps of snapshots of a sample solution of 2D stochastic heat
equation with mean reversion on a rectangular physical space domain taken at
four time instants (from left to right: t=0.2, 0.4, 0.6 and 0.8).

Figure 6. Characteristics of the convergence of pathwise LSE for 2D heat
equation with mean reversion. 200 runs, Zero IC and H=0.3.
Top-left: Four trajectories of estimates calculated from four different
samples (realizations).
Top-right: Sample quantile ranges (green areas) and median (red line), true
value is 1.
Bottom-left: Convergence of sample RMSE and theoretical RMSE, see (44).
Bottom-right: Q-Q plot for sample of estimates (from 40 Fourier modes)
including 95% confidence envelope for normal distribution.

###  6.3. Discussion

Results of the simulations confirm our findings on strong consistency of the
pathwise LSE and on the speed of the convergence. In case of zero initial
condition, it behaves similarly to the weighted MCE. However, for non-zero
initial condition, pathwise LSE significantly outperforms the weighted MCE.
This is in accordance with similar findings from finite-dimensional models
(cf. [18] or [19]), where LSE-type estimators improve with far-from-stationary
initial conditions (the drift part dominates the noise part in the dynamics of
the process in such case), whereas the ergodic-type estimators (such as the
weighted MCE) reflecting long-term stationary behavior are ruined.

The theory for the weighted MCE, developed in [17], covers only single-
operator equations, so it is not directly applicable for the second example
(2D stochastic heat equation with coupling with surroundings). To our best
knowledge, the pathwise LSE is the first efficient tool for parameter
estimation in this equation (important in oceanography, for example) when the
noise is generated by a cylindrical fractional Brownian motion with general
Hurst parameter $0<H<1$.

Although we did not address the asymptotic normality of the pathwise LSE in
this paper (to keep it concise), we conjecture that it holds. Our conjecture
is based on simulation results and asymptotic similarity with the weighted
MCE, whose asymptotic normality has been proven in [17].

##  7\. Conclusions

Using the least-squares formalism we have derived a spectral version of the
least-squares estimator of the drift parameter in linear SPDEs with additive
fractional noise for arbitrary $H\in(0,1)$. This estimator is strongly
consistent in space, however, it can not be directly implemented pathwise,
since it contains divergence-type stochastic integral w.r.t. a fractional
process. This fact prevents its simulations and its use in practice. We
addressed this issue by eliminating the stochastic integral getting a novel
pathwise LSE. This estimator is constructed trully pathwise and, moreover, is
robust to minor perturbations of the observed process (it does not contain
differentiation of the observed trajectory), which makes it particularly
useful in practice.

Using the standard tools from classical probability and analysis, as well as
modern tools from Malliavin calculus, we demonstrated that the newly developed
pathwise LSE is strongly consistent in space and we conjecture that it is also
asymptotically normal. Simulations reveal that it can efficiently utilize
information about the unknown parameter from both non-stationary and
stationary parts to the observed process.

The presented pathwise least-squares estimation procedure appears to be
promising approach for various equations driven by fractional Brownian motion.
The potentially interesting directions of further research may include (but
are not limited to) proving asymptotic normality, studying different
observation/sampling schemes, different asymptotic regimes, or generalization
to semilinear stochastic equations driven by fractional Brownian motion.

##  Appendix A Elements from Malliavin calculus

In the Appendix, we describe some basic constructions from Malliavin calculus
in the setting of our problem. These constructions form a key ingredient for
the study of asymptotic behavior of our estimators.

###  A.1. Construction of an infinite-dimensional isonormal Gaussian process

Let $W^{(k)}=\big{\\{}W^{(k)}(h^{(k)}),\,h^{(k)}\in\mathcal{H}^{(k)}\big{\\}}$
be an isonormal Gaussian process generated by a fractional Brownian motion
$\\{B_{k}^{H}(t),t\geq 0\\}$ for any $k\in\mathbb{N}$ and defined over a
separable Hilbert space $\mathcal{H}^{(k)}$. The processes $W^{(k)}$ are
mutually independent.

Define

$\mathcal{H}=\bigoplus_{n=1}^{\infty}\mathcal{H}^{(n)}$

as a direct sum of $\mathcal{H}^{(n)}$’s, i.e.

$\mathcal{H}=\big{\\{}(h^{(1)},h^{(2)},\ldots),\
h^{(n)}\in\mathcal{H}^{(n)}\wedge\sum_{n=1}^{\infty}\|h^{(n)}\|^{2}_{\mathcal{H}^{(n)}}<\infty\big{\\}},$

with the scalar product

$\big{\langle}(h^{(1)},h^{(2)},\ldots),(g^{(1)},g^{(2)},\ldots)\big{\rangle}_{\mathcal{H}}=\sum_{n=1}^{\infty}\langle
h^{(n)},g^{(n)}\rangle_{\mathcal{H}^{(n)}}.$

Then $\mathcal{H}$ is a separable Hilbert space.

For $h=(h^{(1)},h^{(2)},\ldots)\in\mathcal{H}$ define

$W(h)=\lim_{N\rightarrow\infty}\sum_{n=1}^{N}W^{(n)}(h^{(n)}).$

Note that this limit, understood in $L^{2}$-sense, is well-defined because

$\mathbb{E}\,\Big{(}\sum_{n=N}^{\infty}W^{(n)}(h^{(n)})\Big{)}^{2}=\sum_{n=N}^{\infty}\mathbb{E}\big{(}W^{(n)}(h^{(n)})\big{)}^{2}=\sum_{n=N}^{\infty}\|h^{(n)}\|^{2}_{\mathcal{H}^{(n)}}\xrightarrow[N\rightarrow\infty]{}0.$

Since

$\displaystyle\mathbb{E}\,\big{(}W(h)W(g)\big{)}$
$\displaystyle=\mathbb{E}\,\Big{(}\sum_{n=1}^{\infty}W^{(n)}(h^{(n)})\Big{)}\Big{(}\sum_{n=1}^{\infty}W^{(n)}(g^{(n)})\Big{)}$
$\displaystyle=\sum_{n=1}^{\infty}\mathbb{E}\,\big{(}W^{(n)}(h^{(n)})W^{(n)}(g^{(n)})\big{)}=\sum_{n=1}^{\infty}\langle
h^{(n)},g^{(n)}\rangle_{\mathcal{H}^{(n)}}=\langle h,g\rangle_{\mathcal{H}}$

for any $h=(h^{(1)},h^{(2)},\ldots),g=(g^{(1)},g^{(2)},\ldots)\in\mathcal{H}$
the process $\big{\\{}W(h),h\in\mathcal{H}\big{\\}}$ is an isonormal Gaussian.

Denote by $C\\!H_{n}^{(k)}$ the $n$-th Wiener chaos associated with $W^{(k)}$
for any $k\in\mathbb{N}$, i.e. $C\\!H_{n}^{(k)}$ is the closed (in
$L^{2}(\Omega)$) linear subspace generated by

$\big{\\{}H_{n}\big{(}W^{(k)}(h^{(k)})\big{)},\|h^{(k)}\|_{\mathcal{H}^{(k)}}=1\big{\\}},$

where $H_{n}$ is the $n$-th Hermite polynomial. Similarly, let $C\\!H_{n}$ be
the $n$-th Wiener chaos associated with $W$, i.e. $C\\!H_{n}$ is the closed
linear subspace generated by

$\big{\\{}H_{n}\big{(}W(h)\big{)},\|h\|_{\mathcal{H}}=1\big{\\}}.$

Fix $k\in\mathbb{N}$, take arbitrary $h^{(k)}\in\mathcal{H}^{(k)}$ such that
$\|h^{(k)}\|_{\mathcal{H}^{(k)}}=1$ and define

$h=(0,\ldots,0,\underset{\quad\uparrow k\mathrm{th}}{h^{(k)}},0,\ldots,0).$

Then

$\|h\|^{2}_{\mathcal{H}}=\|h^{(k)}\|^{2}_{\mathcal{H}^{(k)}}=1$

and

$W(h)=\sum_{n=1}^{\infty}W^{(n)}(h^{(n)})=W^{(k)}(h^{(k)}).$

Consider $X=H_{n}\big{(}W^{(k)}(h^{(k)})\big{)}\in C\\!H_{n}^{(k)}$. Then
$X=H_{n}\big{(}W(h)\big{)}$ and $X\in C\\!H_{n}$. By the linearity and
$L^{2}(\Omega)$-closedness of $C\\!H_{n}$ it follows that

(46) $C\\!H_{n}^{(k)}\subset C\\!H_{n}\quad\forall k\in\mathbb{N}.$

###  A.2. Selected functionals of the solution in the second chaos

It can be shown (see e.g. [14]) that

$\int_{0}^{T}x_{k}(t)\,{\rm d}B^{H}_{k}(t)\in
C\\!H_{2}^{(k)},\quad\int_{0}^{T}x_{k}^{2}(t)\,{\rm
d}t-\mathbb{E}\,\left[\int_{0}^{T}x_{k}^{2}(t)\,{\rm d}t\right]\in
C\\!H_{2}^{(k)}.$

Using the construction of the infinite-dimensional isonormal Gaussian process
above and the chaos embedding (46), we easily get for

$C_{N}=\sum_{k=1}^{N}\alpha_{k}\int_{0}^{T}x_{k}(t)\,{\rm d}B^{H}_{k}(t),$

and

$D_{N}=\sum_{k=1}^{N}\alpha_{k}^{2}\int_{0}^{T}x_{k}^{2}(t)\,{\rm d}t,$

that

(47) $C_{N}\in C\\!H_{2}\quad\text{ and }\quad D_{N}-\mathbb{E}\,D_{N}\in
C\\!H_{2}\quad\forall N\in\mathbb{N}.$

###  A.3. Almost sure convergence on a fixed chaos

The following lemma is an important tool to demonstrate strong consistency of
our estimators. It turns the $L^{2}$-convergence of a sequence of variables on
a fixed chaos into the almost sure convergence. This idea, based on the
combination of hypercontractivity with Borel–Cantelli 0-1 law, has already
been used before, but we decided to formulate it as a separate lemma, because
we believe it is interesting on its own.

######  Lemma A.1.

Let $F_{N},N\in\mathbb{N},$ be random variables in a fixed Wiener chaos of an
isonormal Gaussian process such that for some constants $c>0$ and $\delta>0$
the inequality

$\mathbb{E}\,F^{2}_{N}\leq\frac{c}{N^{\delta}}$

holds for any $N\in\mathbb{N}$. Then
$F_{N}\xrightarrow[N\rightarrow\infty]{}0$ a.s.

###### Proof.

Consider parameters $\zeta$ and $\eta$ so that $0<\zeta<\delta/2$ and
$\eta>\frac{1}{\delta/2-\zeta}$. Denote by $C$ a positive constant
(independent of $N$), which may change from line to line and calculate

$\mathbb{P}\,\big{(}|F_{N}|>N^{-\zeta}\big{)}\leq\frac{\mathbb{E}\,|F_{N}|^{\eta}}{N^{-\zeta\eta}}\leq
C\frac{(\mathbb{E}\,F^{2}_{N})^{\eta/2}}{N^{\zeta\eta}}\leq
C\frac{1}{N^{\eta(\delta/2-\zeta)}},$

where Chebyshev’s inequality and hypercontractivity property on the fixed
Wiener chaos (see e.g. [24], Theorem 2.7.2) were used. Application of
Borel–Cantelli lemma yields the almost sure convergence.

∎

##  Appendix B Cylindrical fractional Brownian motion and fractional Gaussian
noise

The use of cylindrical fractional Brownian motion as a rigorous model for
fractional Gaussian noise $\\{\eta^{H}(t,\xi);\,t>0,\xi\in\mathcal{O}\\}$ from
equation (43) can be found in several publications. More detailed exposition
can be found e.g. in [10]. In this Appendix, we briefly outline the connection
between the two objects.

Let us begin with white noise in spatial dimension
$\mathcal{O}\subset\mathbb{R}^{d}$ (no time coordinate considered). A natural
space associated with (43) is $V=L^{2}(\mathcal{O})$ with an orthonormal basis
$\\{e_{k}\\}_{k=1}^{\infty}$. Take a sequence of i.i.d. real-valued random
variables $\\{U_{k}\\}_{k=1}^{\infty}$ having standard Gaussian distribution.
Now we can formally model the white noise in space as

(48) $\eta(\xi)=\sum_{k=1}^{\infty}e_{k}(\xi)U_{k},\quad\xi\in\mathcal{O}.$

Indeed, take arbitrary $f,g\in L^{2}(\mathcal{O})$ and calculate (again
formally)

$\int_{\mathcal{O}}f(\xi)\eta(\xi)\,{\rm
d}\xi=\sum_{k=1}^{\infty}U_{k}\biggl{(}\int_{\mathcal{O}}f(\xi)e_{k}(\xi)\,{\rm
d}\xi\biggr{)},$

which is a centered Gaussian random variable with variance $||f||^{2}_{V}$,
and

$\displaystyle\mathbb{E}\,$
$\displaystyle\biggl{(}\int_{\mathcal{O}}f(\xi)\eta(\xi)\,{\rm
d}\xi\biggr{)}\biggl{(}\int_{\mathcal{O}}g(\xi)\eta(\xi)\,{\rm
d}\xi\biggr{)}=\mathbb{E}\,\biggl{(}\sum_{k=1}^{\infty}U_{k}\left\langle
f,e_{k}\right\rangle_{V}\biggr{)}\biggl{(}\sum_{k=1}^{\infty}U_{k}\left\langle
g,e_{k}\right\rangle_{V}\biggr{)}$
$\displaystyle=\sum_{k=1}^{\infty}\left\langle
f,e_{k}\right\rangle_{V}\left\langle
g,e_{k}\right\rangle_{V}=\int_{\mathcal{O}}f(\xi)g(\xi)\,{\rm d}\xi.$

In particular, by taking $f$ and $g$ the indicator functions, we get that the
noise is uncorrelated on arbitrary two disjoint subsets of $\mathcal{O}$ and
it has constant intensity (variance is proportional to the volume of the
subset).

######  Remark B.1.

Although the series in (48) does not converge in the $L^{2}(\Omega)$ sense in
the space $V$, it does converge in some larger Hilbert space $\tilde{V}$ so
that there exits a Hilbert-Schmidt embedding of $V$ into $\tilde{V}$ (this
corresponds to the fact that white noise can not be considered as a proper
function, but rather as a distribution).

Continue with modeling of the fractional (autocorrelated) noise in time
domain. Recall that real-valued fractional Brownian motion $\\{b^{H}(t),t\geq
0\\}$ with Hurst index $H\in(0,1)$ is a centered Gaussian process with
continuous trajectories starting from zero and with autocovariance function

$\mathbb{E}\,b^{H}(t)b^{H}(s)=\frac{1}{2}\bigl{(}t^{2H}+s^{2H}-|t-s|^{2H}\bigr{)}.$

As a consequence, it has stationary Gaussian increments. If $H=1/2$ we obtain
standard Wiener process (with independent increments), if $H<1/2$ we have
negatively correlated increments and for $H>1/2$ we have positively correlated
increments. Formal derivative of fBm (which exists only in distributional
sense) thus represents a good model for fractional noise in time.

To conclude, take the family of independent real-valued fractional Brownian
motions $\\{B^{H}_{k}(t),t\geq 0\\},k=1,2,\ldots$ and construct the
cylindrical fBm on $V=L^{2}(\mathcal{O})$ using an orthonormal basis
$\\{e_{k}\\}_{k=1}^{\infty}$ as a formal sum

$B^{H}(t)=\sum_{k=1}^{\infty}e_{k}B^{H}_{k}(t).$

Increments (differentials) of this cylindrical fBm can serve as a model for
the noise that is white in space and fractional in time. Indeed, for a fixed
$0<s<t$ we have

$B^{H}(t)-B^{H}(s)=\sum_{k=1}^{\infty}e_{k}\bigl{(}B^{H}_{k}(t)-B^{H}_{k}(s)\bigr{)},$

which is the (scaled) white-in-space noise. Next, for a fixed measurable
$D\subset\mathcal{O}$ with the finite volume take

$\displaystyle\left\langle B^{H}(t),1_{D}\right\rangle_{V}$
$\displaystyle=\sum_{k=1}^{\infty}\biggl{(}\int_{\mathcal{O}}1_{D}(\xi)e_{k}(\xi)\,{\rm
d}\xi\biggr{)}B^{H}_{k}(t)$
$\displaystyle=\sum_{k=1}^{\infty}w_{k}B^{H}_{k}(t)=\beta^{H}(t),$

which itself is a (scaled) fractional Brownian motion, since
$\sum_{k=1}^{\infty}w^{2}_{k}=\|1_{D}\|^{2}_{V}<\infty$ and so it represents
an integrated fractional-in-time noise.

## Acknowledgments

We would like to thank to the anonymous referee for valuable comments and
suggestions that helped to improve clarity and readability this paper.

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