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\begin{document}
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\title[Parameter Estimations for  Fractional SPDEs with Jumps]{Parameter Estimations for Linear Parabolic Fractional SPDEs with Jumps}
\author{Wilfried Grecksch}
\address{Martin-Luther-University Halle-Wittenberg, \\ Faculty of Natural Sciences II, 
Institute of Mathematics\\
06099 Halle (Saale)\\
Germany}
\email{wilfried.grecksch@mathematik.uni-halle.de}
%
\author{Hannelore Lisei}
\address{Babe\c{s}-Bolyai University, \\ Faculty of Mathematics and Computer Sciences\\
1, Kog\u{a}lniceanu Street,\\
400084 Cluj-Napoca,\\
Romania}
\email{hanne@math.ubbcluj.ro}
%
\author{Jens Lueddeckens}
\address{Martin-Luther-University Halle-Wittenberg, \\ Faculty of Natural Sciences II, 
Institute of Mathematics\\
06099 Halle (Saale)\\ Germany}
\email{jlueddeckens@web.de}
%
\subjclass{60H15, 62F12, 60G22}
\keywords{Parameter estimation, SPDE, cylindrical fractional Brownian motion, cylindrical Poisson process}
\begin{abstract}
We give an unbiased and consistent estimator for the drift coefficient of a linear parabolic stochastic partial differential equation driven by a multiplicative cylindrical fractional Brownian motion with Hurst index $1/2 <h < 1$ and a cylindrical centered Poisson process, if the observations of the solution process are given in discrete time points. The presented method is based on mean square estimations. 
\end{abstract}
\maketitle

\section{Introduction}

There are lots of papers concerning parameter estimations for stochastic differential equations (SDEs). Results for  SDEs  driven by a fractional Brownian motion (fBm) were  given, for example, by  
{Y. Kozachenko, A. Melnikov, Y. Mishura} \cite{KoMeMi} and {W.L. Xiao, W.G. Zhang, X. Zhang}  \cite{XZZ} (see also the references therein). 
If the driving process is a L{\'e}vy process, then see, for example,  H. Long \cite{L} and the references therein. If the equations are driven by a fBm and a fractional Poisson measure  one can find interesting results and applications in the PhD thesis of {J. Lueddeckens} \cite{JL}. 

Many papers are devoted to the parameter estimation of fractional stochastic partial differential equations (SPDEs). As a representative result we quote here the paper \cite{MT} of B. Maslowski and C.A. Tudor. 

The following estimation criteria are mainly used in constructing estimators for the parameters of SPDEs:
\begin{itemize}
\item maximum likelihood type methods by considering  fundamental  martingales and theorems of Girsanov type;
\item time continuous and time discrete least square criteria; 
\item Kalman-Bucy filters;
\item $L^1$-norm estimations and contrast estimations.
\end{itemize}

Often SPDEs are considered as stochastic evolution equations in Hilbert spaces. For example, a parameter estimation problem for linear diagonalized stochastic partial differential equations driven by a multiplicative fBm is considered by {I. Cialenco} in \cite{C}. 
The stochastic processes defined by the random Fourier coefficients of the solution process describe one dimensional geometric fractional Brownian motions. 
Based on these processes, consistent parameter estimates for the SPDEs are determined using a maximum likelihood type method. So we see, that one dimensional results of parameter estimations are useful for parameter estimations of SPDEs. 

Parameter estimations for diagonal SPDEs are also considered in Chapter 6 in \cite{LR} by S.V. Lototsky, B.L. Rozovsky.

The aim of the present  paper is to give new contributions in the estimation theory of the coefficients of linear homogeneous SPDEs, which are driven by cylindrical fractional Brownian motions and cylindrical Poisson processes.
 The applied estimation criterion uses covariances (as a generalization of the mean squared method), such that the long range dependence property of the fractional Brownian motion with Hurst index $h\in]1/2,1[$ is taken into account.
  Moreover, in this paper weakly, respectively strongly consistent estimators are constructed by using only information  about the underlying process in discrete time points.


The paper starts with a preliminary section, containing the assumptions needed throughout the paper. A linear SPDE driven by a multiplicative  cylindrical fractional Brownian motion  and a cylindrical Poisson process is introduced in Section 3. In Section 4  the one dimensional stochastic differential equations for the Fourier coefficients  of the solution process of the SPDE is considered and similar to the results from \cite{JL} an estimation criterion of least squares type in discrete time points of the observations for the drift term is formulated. The estimator of the drift coefficient is unbiased. Conditions for choosing the time points are given such that the constructed estimator is unbiased and weakly consistent (Theorem \ref{eigensch1}), respectively strongly consistent (see Theorem \ref{strcons}).  
%%%%%%%%%%%%%%%%
\section{Prelimiaries}

\begin{definition}
A real-valued Gaussian process $(B^h(t))_{t\geq 0}$ with ${E}(B^h(t)) = 0$, for all $t\ge 0$, $B^h(0)=0$ and Hurst index $h\in ]0,1[$ is called  fractional Brownian motion (fBm) if
\[{E}(B^h(t)B^h(s)) = \frac{1}{2}(t^{2h} + s^{2h} - |t-s|^{2h}) \mbox{ for all } s,t\ge 0.\]
\end{definition}

The fBm is not a semimartingale and it is not a Markovian process for $h\neq 1/2$. The fBm is a Wiener process for $h=1/2$. In this paper we consider $h \in ]1/2 , 1[$. Then, the fBm has the so-called long range dependence property.
 



%%%%%%%%%%%%%%%%%
\newpage

{\bf{Assumptions:}}
\begin{itemize}
\item[\bf{(A1)}] Let $(V,H,V^*)$ be a triplet of rigged Hilbert spaces, where $V$ is  compactly embedded into $H$ and $A:V\rightarrow V^*$ is linear and $\langle Av,v\rangle + \alpha_1 \|v\|_V^2 \leq \alpha_2 \|v\|^2_H$ for all $v\in V$ and $\alpha_1 > 0,\, \, \alpha_2 \in \R$ are constants. 
\medskip

\noindent Observe, that $A:D(A) \rightarrow H$ is linear and unbounded with $D(A) = \{v \in V \, : \, Av \in H\}$, which is dense in $H$. The eigenvalues $(\lambda_k)_{k\ge 1}$ of this operator are negative and satisfy $\displaystyle\lim_{k\rightarrow \infty} \lambda_k = -\infty$. 
\medskip

\item[\bf{(A2)}] Let $(h_k)_{k\ge 1}\subset H$ be the complete orthonormal system constructed by the eigenfunctions of $A$.
\item[\bf{(A3)}]  The $C_0$ semigroup $(\T_t)_{t \geq 0}$ defined by 
\[\T_t(x)=\sum^{\infty}_{k=1}\exp\{\lambda_kt\}(x,h_k)h_k, \, \, x \in H\]
is generated by $-A$, where $(\cdot , \cdot)$ denotes the scalar product in $H$.
\item[\bf{(A4)}]  {$\Phi_1, \Phi_2:\,H\rightarrow H$ are Hilbert-Schmidt operators of the type
\[\Phi_i(x)=\sum^{\infty}_{k=1}\mu_{ik}(x,h_k)h_k,\ \  x\in H,\] 
where $\displaystyle\sum^{\infty}_{k=1} \mu_{ik}^2 < \infty$ for $i\in\{1,2\}$.}
\item[\bf{(A5)}]  Let $(B^h_k(t))_{t\geq 0}$, $k\in\{1,2,\ldots\},$ be independent fractional Brownian motions with Hurst index $h \in ]1/2,1[$ and let $$ B^h(t)= \sum_{k=1}^\infty B^h_k(t)h_k,\ \ t\ge 0,$$
denote the cylindrical fBm.
\item[\bf{(A6)}]  Let $(\pi_j(t))_{t \geq 0}$, $j\in\{1,2,\ldots\},$ be independent homogeneous Poisson processes with parameter $\nu$.
\item[\bf{(A7)}]  Consider $\pitil _j(t)=\pi_j(t)- \nu t,\, \, j\in\{1,2,\ldots\}$
and we denote by
$$ \pitil(t)= \sum_{j=1}^\infty \pitil_j(t)h_j,\ \  t\ge 0,$$
the cylindrical centered Poisson process.
\item[\bf{(A8)}] The processes $B^h_k$ and $\pitil_j$ are independent stochastic processes for all $j,k\in\{1,2,...\}$.
\item[\bf{(A9)}]  Let  $X_0 \in H$ be a deterministic initial value.
\item[\bf{(A10)}] All stochastic processes are defined on the same complete filtered probability space $(\Omega,{\mathcal F},({\mathcal F}_t)_{t \geq 0},P)$, where
 ${\mathcal F}_t =\sigma ({\F^{B^h}_t \vee \F^{\pitil}_t }) $ and $\F^{B^h}_t$ and ${\mathcal F}^{\pitil}_t$ denote the $\sigma$-algebras generated by
 $(B^h(s))_{s\in[0,t[}$ and $(\pitil(s))_{s\in[0,t[}$.
\end{itemize}
%%%%%%%%%%%%%%%%
\section{A Linear Fractional Parabolic SPDE with Jumps}
At first we introduce for $k\in\{1,2,\ldots\}$ the one dimensional linear stochastic differential equations
\begin{equation}\label{dYk} dY_k(t)=a\lambda_kY_k(t)dt + \sigma \mu_{1k}Y_k(t)dB^h_k(t) + \eta \mu_{2k}Y_k(t-)d\pitil_k(t),
\end{equation}
where $a,\sigma, \eta$ are positive constants and $Y_k(0)=(X_0,h_k)$.\\ 
The stochastic equation (\ref{dYk}) is defined by
\begin{equation}\label{Yk}
Y_k(t)=Y_k(0) + a\lambda_k\int^t_0\! Y_k(s)ds + \sigma \mu_{1k}\int^t_0 \! Y_k(s)dB^h_k(s) + \eta \mu_{2k}\int^t_0\! Y_k(s-)d\pitil_k(s), 
\end{equation}
for all $t\ge 0$, where the stochastic integral with respect to $B^h_k$ is defined by a divergence integral as in \cite{Nu2} and the stochastic integral with respect to the compensated Poisson process is defined as in \cite{Pr}, Chapter II (or \cite{GiSk}, page 246).
%\begin{lemma} 
%Let $(Z(t))_{t \in [0,T]}$ be a process whose values are Hilbert-Schmidt operators in $H$ with
%\[ {\E}\int^T_0\int^T_0 \left(Z(t),Z(s)\right)_{HS}\phi_h(t-s)dsdt\]
%\[+ {\E}\int^T_0\int^T_0\int^T_0\int^T_0\left(D_pZ(q),D_rZ(s)\right)_{HS}\phi_h(p-s)\phi_h(r-q)dpdqdrds\]
%\[<\,  \infty,\]
%where $\phi(x) = h(2h-1)|x|^{2h-2}$. \\ 
%Then $\displaystyle\int^t_0 Z(s)dB^h(s)$ exists for all $t \in [0,T]$ ({\textcolor[rgb]{1,0,0}{\bf{Duncan, Maslowski, Pasik-Duncan}}}, 2001).
%\end{lemma}
%%%%%%%%%%%%%%%%%%%

\begin{theorem}\label{solution}
{The process
\[
Y_k(t)\!=\!Y_k(0)\!\left(1+\eta\mu_{2k}\right)^{\pi_k(t)}\cdot \exp\!\left\{\!\sigma\mu_{1k}B^h_k(t)\!-\!\frac{1}{2}\sigma^2 \mu_{1k}^2 t^{2h}\!+a\lambda_kt-\nu \eta \mu_{2k}t\!\right\} 
\]}
\begin{equation}\label{loesformel}
%\left(\prod_{0\leq s \leq t}\left(1+\eta\mu_{2k}\Delta\pi_k(s)\right)\right)
\end{equation}
solves equation \eqref{dYk} for all $t\geq 0$ with probability 1.
\end{theorem}
\begin{proof}
We prove that the process
$Y_k(t)=Y_{1k}(t)Y_{2k}(t)$ with
\[Y_{1k}(t) = \exp\left\{\sigma\mu_{1k}B^h_k(t)-\frac{1}{2}\sigma^2 \mu_{1k}^2 t^{2h}\right\}\]
and
\[Y_{2k}(t) = Y_k(0)\left(1+\eta\mu_{2k}\right)^{\pi_k(t)}\cdot\exp\left\{a\lambda_kt-\nu \eta \mu_{2k}t\right\},\]
is the solution of  \eqref{dYk}. 
Since the fBms and the Poisson processes are independent, we get
\begin{equation}\label{dYProduct}
dY_k(t) = Y_{1k}(t)dY_{2k}(t) + Y_{2k}(t)dY_{1k}(t).
\end{equation}
Obviously it holds
\begin{equation}\label{dY2k}
Y_{2k}(t)=Y_k(0) \exp\left\{a\lambda_kt-\nu \eta \mu_{2k}t + \pi_k(t)\ln\left(1+\eta\mu_{2k}\right)\right\}.
\end{equation}
It follows by a result from \cite{GiSk} (see formula (15) on page 261)
% by specialization 
that (\ref{dY2k}) solves
\[dY_{2k}(t) = Y_{2k}(t)[a\lambda_k-\nu \eta \mu_{2k}]dt + \eta \mu_{2k}Y_{2k}(t-)d\pi_k(t)\]
with $Y_{2k}(0) = Y_k(0)$.
By the definition of $\pitil(t)$ it follows
\[dY_{2k}(t) = a\lambda_k Y_{2k}(t)dt + \eta \mu_{2k}Y_{2k}(t-)d\pitil_k(t)\]
with $Y_{2k}(0) = Y_k(0)$.\\
The fractional It{\^o} formula in \cite{Nu2} (see formula (2.18)) gives that the process $Y_{1k}(t)$ solves
\[dY_{1k}(t) = \mu_{1k}\sigma Y_{1k}(t)dB^h_k(t)\]
with $Y_{1k}(0) =1$.

If we substitute these results in (\ref{dYProduct}), then we get that the process (\ref{loesformel})
solves equation \eqref{dYk}.
\end{proof}
%%%%%%%%%%%%%%%
We will prove the following a priori estimates:  
\begin{lemma}\label{apriori1}
There are constants $C_0>0,\ C_1>0$ and  $C_2>1$ such that for all $t\geq 0$ and $k\in\{1,2,...\}$ it holds
\begin{equation}\label{apr1}
E|Y_k(t)|^2 \leq F(t)|Y_k(0)|^2
\end{equation}
with 
\[F(t) = C_0\exp\{2\sigma^2 C_1 t^{2h} + \nu t(C_2 - 1)\}.\]
\end{lemma}
\begin{proof}
Since the fBms and the Poisson processes are independent, we get
\[E|Y_k(t)|^2 = |Y_k(0)|^2 \left(E\exp\left\{2\sigma \mu_{1k}B^h_k(t)\right\}\right)\left(E\left(\left(1+\eta \mu_{2k}\right)^{2\pi_k(t)}\right)\right)\times\]
\begin{equation}\label{HG1}\times \exp\left\{-\sigma^2 \mu_{1k}^2 t^{2h}+2a\lambda_kt-2\nu \eta \mu_{2k}t\right\}.
\end{equation}
{Since $\lambda_k<0$ for each $k\in\{1,2,..\}$ and $(\mu_{2k})_{k\ge 1}$ is a bounded sequence, we have
$$\exp\left\{-\sigma^2 \mu_{1k}^2 t^{2h}+2a\lambda_kt-2\nu \eta \mu_{2k}t\right\}\le C_0 \mbox{ for each } t\ge 0,$$
where $C_0>0$ is a constant.}

The random variable $Z_1:=\exp\left\{2\sigma \mu_{1k}B^h_k(t)\right\}$ is log-normally distributed, so  we get for its expectation 
\[E(Z_1) = \exp\{2\sigma^2 \mu_{1k}^2 t^{2h}\}.\]
 From the boundedness of $(\mu_{1k})_{k\ge 1}$ it follows the existence of a positive constant $C_1$ with
\[E(Z_1) \leq \exp\{2\sigma^2 C_1 t^{2h}\}.\]
For $Z_2:=(1+ \eta \mu_{2k})^{2\pi _k(t)}$ we compute 
\[E(Z_2)= E\left(\left(1+\eta \mu_{2k}\right)^{2\pi_k(t)}\right) 
= \sum_{j=0}^{\infty}\left(1+\eta \mu_{2k}\right)^{2j}\exp\{-\nu t\}\frac{(\nu t)^j}{j!}.\]
But the sequence $(\mu^2_{2k})_{k\ge 1}$ is bounded, hence there is a  constant $C_2>1$ such that
\[E(Z_2) \leq \exp\{-\nu t\}\sum_{j=0}^{\infty}\frac{(\nu t C_2)^j}{j!} = \exp\{\nu t(C_2-1)\}.\]
Then we get with (\ref{HG1}) the inequality (\ref{apr1}).
\end{proof}
We now consider a solution definition of mild solution type. 

\begin{theorem}\label{mildYk}
Let $k\in\{1,2,\ldots\}$. The process $(Y_k(t))_{t \geq 0}$  defined by \eqref{loesformel} solves \eqref{Yk} if and only if the equation
\[ Y_k(t) = Y_k(0)\exp\{\lambda_kat\} + \int^t_0 \exp\{\lambda_ka(t-s)\} \sigma \mu_{1k}Y_k(s)dB^h_k(s)+\]
\begin{equation} \label{milddYk1} +  \int^t_0 \exp\{\lambda_ka(t-s)\} \eta \mu_{2k}Y_k(s-)d\pitil_k(s), \mbox{ for all } t\ge 0
\end{equation}
holds.
\end{theorem}
\begin{proof}
 If the process $(Y_k(t))_{t \geq 0}$ is the solution of (\ref{Yk}) defined by (\ref{loesformel}), then this process solves (\ref{milddYk1}) too.

Let $(\tilde{Y}_k(t))_{t \geq 0} $ be a solution process of (\ref{milddYk1}).
Then we get
\[\tilde{Y}_k(t)= Y_k(0)\exp\{a\lambda_kt\} + \exp\{a\lambda_kt\}\int^t_0 \exp\{-a\lambda_ks\}\sigma\mu_{1k}\tilde{Y}_k(s) dB^h_k(s)+\]
\begin{equation}\label{product}+ \exp\{a\lambda_kt\}\int^t_0 \exp\{-a\lambda_ks\}\eta \mu_{2k}\tilde{Y}_k(s-)d\pitil_k(s).
\end{equation}
Obviously  $\exp\{a\lambda_k t\}$  is deterministic and differentiable and the stochastic differentials of the  stochastic integrals in formula (\ref{product}) exist. If we use the stochastic product formula to the two last terms of the sum in formula (\ref{product}), then we get
\[d\tilde{Y}_k(t)= a \lambda_k \tilde{Y}_k(t)dt´+ \sigma \mu_{1k}\tilde{Y}_k(t) dB^h_k(t) + \eta \mu_{2k}\tilde{Y}_k(t-)d\pitil_k(t).\]
It follows from formula (\ref{product}) that $\tilde{Y}_k(0)=Y_k(0)$. That is, $\tilde{Y}_k(t) = Y_k(t)$ for all $t\geq 0$ with probability 1. 
\end{proof}

We introduce {for $n\ge 1$ and $t\ge 0$}
\[X_n(t):= \sum^n_{k=1}Y_k(t)h_k\]
By (\ref{apr1}) and $Y_k(0) =(X_0,h_k)$  we get
\begin{equation}\label{apriori3}
{E}\|X_n(t)\|^2_H = \sum^n_{k=1} {E} |Y_k(t)|^2 \leq \|X_0\|_H^2F(t)
\end{equation}
for every $t>0$. It follows also from this inequality and the definition of $F(t)$ that
there is for all $T>0$ a positive constant $C_T$ such that 
\begin{equation}\label{apriori4}
{E}\|X_n(t)\|_H^2 \leq C_T \|X_0\|^2_H
\end{equation}
and
\begin{equation}\label{apriori5}
{E}\int^t_0 \|X_n(s)\|_H^2ds  \leq TC_T  \|X_0\|^2_H
\end{equation}
for all $t\in [0,T]$ and all $n\ge 1$. \\

Consequently for $t>0$ {there exists} in $L^2(\Omega\,;\,H)$ and in $L^2(\Omega \times [0,T]\, ; H)$ the process
\begin{equation} \label{Xt} 
X(t): = \sum^{\infty}_{k=1}Y_k(t)h_k
\end{equation}
and the a priori estimates (\ref{apriori4}) and (\ref{apriori5}) hold also for $X(t)$. \\ \\
Since $Y_k(0) = (X_0,h_k)$ holds, we obtain $X_0 = \displaystyle \sum ^{\infty}_{k=1} (Y_k(0),h_k)h_k$.\\
It holds for $X_n(t)$
\[X_n(t) = \sum^n_{k=1}\exp\{\lambda_kat\}(X_0,h_k)h_k\]
\[+ \sigma \int^t_0 \sum^n_{k=1}\exp\{\lambda_ka(t-s)\} \mu_{1k}(X_n(s),h_k)h_kdB^h_k(s)\]
\[ +\eta  \int^t_0 \sum^n_{k=1}\exp\{\lambda_ka(t-s)\} \mu_{2k}(X_n(s-),h_k)h_kd\pitil_k(s).\]

Consequently, we get by the definition of the semigroup ${\T}_t$ and the operators $\Phi_1,\, \Phi_2$
\begin{equation}\label{mildXn} 
dX_n(t)= \T_t X_n(0) +\sigma \int^t_0\T_{t-s}\Phi_1X_n(s)dB^h(s) + \eta\int^t_0 \T_{t-s}\Phi_2X_n(s-)d\pitil(s),
\end{equation}
where $B^h$ and $\pitil$   are the cylindrical fractional Brownian motion and the cylindrical centered Poisson process defined by the sequences $(B^h_k(t))_{t\geq 0}, \, \, k\in\{1,2,\ldots\}$, and 
$(\pitil _j(t))_{t\geq 0}, \, \, j\in\{1,2,\ldots\}$.

With the definition of $X(t)$ from formula (\ref{Xt}) and the a priori estimates \eqref{apriori4}, \eqref{apriori5}, with (\ref{mildXn}), by the definition of $Y_k(t)$ and the definitions of the stochastic integrals it is easy to prove, that the following result holds:
 
\begin{theorem}\label{mildX} 
The process $(X(t))_{t\in [0,T]}$ with $X(t)=\displaystyle\sum^{\infty}_{k=1} Y_k(t)h_k$, $t\in[0,T]$, solves
\begin{equation}\label{mildsolX}
X(t)= \T_t X_0 +\sigma \int^t_0\T_{t-s}\Phi_1X(s)dB^h(s) + \eta\int^t_0 \T_{t-s}\Phi_2X(s-)d\pitil(s)
\end{equation}
for all $t\in [0,T].$
\end{theorem}
\begin{remark} The last theorem shows, that $(X(t))_{t\in[0,T]}$ is the \textit{ mild solution} of 
\[dX(t) = aAX(t)dt + \sigma \Phi_1X(t)dB^h(t) + \eta \Phi_2X(t-)d\pitil(t)\, \, , X(0)=X_0.\]
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Parameter Estimation}
%%%%%%%%%%%%%
In what follows we assume $Y_k(0)>0$ and $1+\eta\mu_{2k}>0$ for all $k\in\{1,2,\dots\}$. Let  $k\in\{1,2,\dots\}$ be arbitrary. We introduce a method to estimate the parameter $a$ in equation \eqref{Yk} for the process $(Y_k(t))_{t\in [0,T]}$. By construction we can interpret this process as the process corresponding to the $k$-th Fourier coefficient of $(X(t))_{t\in[0,T]}$ with respect to $h_k$.  We introduce the process
\[\xi_k(t):=\ln(Y_k(t))=\ln(Y_k(0))+\sigma \mu_{1k}B^h_k(t)-\frac{1}{2}\sigma^2 \mu_{1k}^2 t^{2h}\]
\begin{equation}\label{ln}
+a\lambda_kt-\nu \eta \mu_{2k} t
+\ln(1+\eta\mu_{2k})\pi_k(t).
\end{equation}
Then,\newpage
\[{E}(\xi_k(t))={E}(\ln(Y_k(t)))=\ln(Y_k(0))-\frac{1}{2}\sigma^2 \mu_{1k}^2 t^{2h}\]
\begin{equation}\label{Eln}
+a\lambda_kt-\nu \eta \mu_{2k}t
+(1+\eta\mu_{2k})\nu t.
\end{equation}
\begin{remark}
Parameter estimation problems involving maximum likelihood  methods for equations of type (\ref{ln}) without Poisson processes were considered in \cite{BeToTu}.
\end{remark}

%%%%%%%%%%%%%%%%%%%%%
 
We consider for $n\in\{1,2,\ldots\}$ and for $\beta >1,$ such that $n^\beta\in {\mathbb N}$, the partitions 
\[t_1=\frac{1}{n}<t_2= \frac{2}{n}< \ldots < t_{n^{\beta}}= \frac{n^{\beta}}{n}=:T(n^{\beta})\,.\]
{Having statistical observations for} $X(\cdot)$ in this time points, we can calculate $\xi_k(t_1),\ldots,\xi_k(T(n^{\beta}))$.\\ 
We introduce the following estimation criterion analogous to the one given in \cite{JL}:
\begin{equation}\label{a1}
\min\left\{\sum^{n^{\beta}}_{i,j=1} {{\rm cov}}\left(\xi_k(t_i),\xi_k(t_j)\right)\, \, : \, \,a>0 \right\} \,.
\end{equation}

%%%%%%%%%%%%%

Equation (\ref{a1}) is a quadratic function with respect to $a$. The factor in front of $a^2$ is given by the positive term
$\lambda ^2_k\Big(\displaystyle \sum_{i=1}^{n^\beta}t_i\Big)^2$. Consequently, there is a unique {estimator} ${\hat{a}}(n^{\beta})$.
\begin{theorem}\label{eigensch1}
 For every $k\in\{1,2,\dots\}$ the estimator \\ \\
${\hat{a}}(n^{\beta})=$
\begin{equation}\label{hata}
-\frac{\displaystyle\sum_{i=1}^{n^\beta}\left(\ln(Y_k(0))-\eta\mu_{2k}\nu t_i+\ln(1+\eta\mu_{2k})\nu t_i-\frac{1}{2}\sigma^2\mu^2_{1k}t_i^{2h}-\ln(Y_k(t_i))\right)}{\displaystyle\lambda_k\sum_{i=1}^{n^\beta} t_i}
\end{equation}
is unbiased and weakly consistent for the parameter $a$.
\end{theorem}
%%%%%%%%%%%%%
\begin{proof}
If we substitute $\ln(Y_k(t_i))$ in the right hand side of ${\hat{a}}(n^{\beta})$, then we get
\begin{equation}\label{positiv}
{\hat{a}}(n^{\beta})=a+\frac{\displaystyle \sum_{i=1}^{n^\beta}(\sigma B^h_k(t_i)+\ln(1+\eta\mu_{2k})\pitil_k(t_i))}{\displaystyle \lambda_k\sum_{i=1}^{n^\beta}t_i}.
\end{equation}
Consequently, the estimator is unbiased. \\ \\
We get
\begin{align*}
%\label{Parameter8}
&{E}[\hat{a}(n^{\beta})-a]^2 \nonumber\\
&={E}\left[\frac{\displaystyle\sum_{i=1}^{n^\beta}\left(\sigma \mu_{1k}B^h_k(t_i)+\ln(1+\eta \mu_{2k})\pitil_k(t_i)\right)}{\displaystyle\lambda_k\sum_{i=1}^{n^\beta}t_i}\right]^2 \nonumber\\
&=\frac{\displaystyle \sum_{i,j=1}^{n^\beta}\left(\frac{\sigma^2\mu^2_{1k}}{2}\left(t_i^{2h}+t_j^{2h}-|t_i-t_j|^{2h}\right)+(\ln(1+\eta\mu_{2k}))^2\nu\min\{t_i,t_j\}\right)}{\displaystyle \lambda_k^2\sum_{i,j=1}^{n^{\beta}}t_it_j}.
\end{align*}
Further we have
\begin{align*}
&\sum_{i,j=1}^{n^\beta}\frac{\sigma^2\mu_{1k}^2}{2}\left(t_i^{2h}+t_j^{2h}-|t_i-t_j|^{2h}\right)\le\sum_{i,j=1}^{n^\beta}\frac{\sigma^2\mu_{1k}^2}{2}\left(t_i^{2h}+t_j^{2H}\right) \nonumber\\
&\le\sigma^2\mu_{1k}^2\sum_{i,j=1}^{n^\beta}\left(\frac{n^{\beta}}{n}\right)^{2h}=\sigma^2{\mu^2_{1k}}\frac{n^{\beta(2+2h)}}{n^{2h}}, \nonumber
\end{align*}
\begin{align*}
\sum_{i,j=1}^{n^\beta}(\ln(1+\eta \mu_{2k}))^2\nu \min\{t_i,t_j\}&\le (\ln(1+\eta\mu{2k}))^2\nu\sum_{i,j=1}^{n^\beta}\left(\frac{n^{\beta}}{n}\right) \nonumber\\
&=(\ln(1+\eta\mu_{2k}))^2 \nu \frac{n^{3\beta}}{n}
\end{align*}   
and
$$
\sum_{i,j=1}^{n^\beta}t_it_j=\frac{1}{n^2}\left(\sum_{i=1}^{n^{\beta}}i\right)^2=\frac{n^{2\beta}(n^{\beta}+1)^2}{4n^2} \ge\frac{n^{4\beta}}{4n^2}.
$$      
Consequently,  
\begin{align*}\label{9}
&\lim_{n\rightarrow\infty}E[\hat{a}(n^{\beta})-a]^2 \nonumber\\
&\le\lim_{n\rightarrow\infty}\frac{4}{\lambda^2_k}\left(\sigma^2\mu^2_{1k}\cdot n^{(2-2h)+\beta(2+2h-4)}+(\ln(1+\eta\mu_{2k}))^2\nu\cdot n^{(2-1)+\beta(3-4)}\right) \nonumber\\
&\le\lim_{n\rightarrow\infty}\frac{4}{\lambda^2_k}\left(\sigma^2\mu^2_{1k}\cdot n^{(2-2h)-\beta(2-2h)}+(\ln(1+\eta\mu_{2k}))^2\nu\cdot n^{1-\beta}\right)=0 \nonumber
\end{align*}
and the weak consistency follows for $\beta > 1$.
\end{proof}
\begin{remark}
\begin{enumerate}
\item In a similar manner we can calculate  estimates for $\sigma^2$ and $\nu$. Then, we need the condition $\beta > 4h -1$ to prove the weak consistency.
\item The estimation of $\eta$ is difficult. A possibility consists in the application of an approximation of $\pitil_k$ by Brownian motions $(B_{k})_{k\ge1}$  with ${E}(B^2_k(t) )= \nu t$ by using the Central Limit Theorem. 
\item If $\eta = 0$, then the random variable
\[\frac{\lambda^2_k}{4}\cdot\frac{\hat{a}(n^{\beta})-a}{\sigma^2\mu^2_{1k}\cdot n^{(2-2h)-\beta(2-2h)}}\]
is asymptotically $N(0,1)$ distributed.  
%\item If $\eta=0$, then we get an estimation for $h$ too.
\end{enumerate}
\end{remark}

Moreover, we prove the following result:

\begin{theorem}\label{strcons}
Consider  $\beta> \frac{3-2h}{2-2h}$ such that $n^\beta\in {\mathbb N}$. Then the estimate \eqref{hata} for the parameter $a$ is strongly consistent for all $k\in\{1,2,\ldots\}$.
\end{theorem}
\begin{proof}
It is well known that, if for all $\varepsilon > 0$ the relation 
\[\sum^{\infty}_{n=1} P(|{\hat{a}}(n^{\beta}) - a| > \varepsilon  ) < \infty\] holds, then $\displaystyle \lim_{n \rightarrow \infty}{\hat{a}}(n^{\beta}) = a $ with probability 1.

Let $k\in\{1,2,\ldots\}$ be arbitrary.
We know from the end of the proof of the last theorem that for the variance of ${\hat{a}}(n^{\beta})$ we can write
\[V({\hat{a}}(n^{\beta})) \le \frac{4}{\lambda^2_k}\left(\sigma^2\mu^2_{1k}\cdot n^{(2-2h)-\beta(2-2h)}+(\ln(1+\eta\mu_{2k}))^2\nu\cdot n^{1-\beta}\right).\]
Then, by using Chebyshev's inequality we obtain
\begin{align*}
&\sum^{\infty}_{n=1} P(|{\hat{a}}(n^{\beta}) - a| > \varepsilon ) \le \sum^{\infty}_{n=1} \frac{1}{\varepsilon ^2}V({\hat{a}}(n^{\beta}))\\
&\le \frac{4}{\lambda^2_k}\sum^{\infty}_{n=1}\sigma^2\mu^2_{1k}\cdot n^{(2-2h)-\beta(2-2h)}+ \frac{4}{\lambda^2_k}\sum^{\infty}_{n=1} (\ln(1+\eta\mu_{2k}))^2\nu\cdot n^{1-\beta}.
\end{align*}
Obviously, the first, respectively, the second sum on the right hand side of the last inequality are convergent, if 
\[\beta > \frac{3-2h}{2-2h},\, \, {\rm{respectively,}} \, \, \beta > 2.\]
Hence, we get the statement  for $\beta >\frac{3-2h}{2-2h}>2$ (since $1/2<h<1$). 
\end{proof} 


%%%%%%%%%%%%%
%{\bf{Outlook}}
%\begin{itemize}
%\item We substitute in the above SPDE the cylindrical Poisson process by a cylindrical fractional Poisson process 
%(fPp), where fPp is defined with respect to the kernel, which defines the fBm. We get also unbiased and weak consistent estimates.
%\item If we account a mean reversion term in the SPDE, then for the proof of the consistency  of the estimated mean reversion term, the application of an ergodic theorem is necessary. 
%\item Does strong consistency hold?
%\item \textcolor[rgb]{1,0.41,0.13}{{\bf{Maslowski}} and {\bf{Tudor}}} considered an operator $\Phi$  with a diagonal and non-diagonal part for the case of an infinite dimensional Ornstein-Uhlenbeck process of the type \[dX(t)=AX(t)dt + \Phi B^h(t)\] and discussed in the proofs the two parts separately. Using this idea we will discuss the influence of (fractional) jump processes.
%\item How can we treat the case $h \in ]0,1/2[$?
%\item Can we study nonlinear equations of this type?
%\end{itemize}

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