text:
\section{Introduction}
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The discrete-time ARCH and GARCH processes are often applied to model financial time series \cite{e1, b1}. Kluppelberg et al. \cite{k1} introduced a continuous-time version of the GARCH(1,1), called COGARCH(1,1), process which has the potential to have a better description of changes by determining the underling continuous structure. The examples of such underling structure are L\'evy processes like Brownian motion and Poisson process. They studied L\'evy driven COGARCH(1,1) processes where noises are considered as increments of some L\'evy process and proved the stationarity and also second order properties, under some regularity conditions. Brockwell et al. \cite{b5} generalized the L\'evy driven COGARCH(1,1) process to the higher order. They defined the COGARCH($p,q$) process $(G_{t})_{t\geq0}$ driven by a L\'evy process $(L_{t})_{t\geq0}$ as the solution to the stochastic differential equation
\begin{align*}
dG_{t}=\sqrt{V_{t}}dL_{t},\qquad t>0,\qquad G_{0}=0,
\end{align*}
where the volatility process $(V_{t})_{t\geq0}$ is a continuous-time ARMA($q,p-1$), called CARMA ($q,p-1$), process. They presented the state-space representation of the volatility process as (see e.g. \cite{b4})
\begin{align}\label{equ1.1}
V_{t}=\alpha_{0}+\mathbf{a}^{\prime}{\mathbf{Y}_{t^{-}}},\qquad t>0,\qquad
V_{0}=\alpha_{0}+\mathbf{a}^{\prime}
{\mathbf{Y}_{0}},
\end{align}
and
\begin{align}\label{equ1.2}
d\mathbf{Y}_{t}=B\mathbf{Y}_{t^{-}}dt+\mathbf{e}(\alpha_{0}+\mathbf{a}^{\prime}\textbf{Y}_{t^{-}})d[L,L]_{t}^{(d)},
\qquad t>0,
\end{align}
where $[L,L]_{t}^{(d)}=\sum_{0<u\leq t}(\Delta L_{u})^{2}$ is the discrete part of the quadratic variation process $([L,L]_{t})_{t\geq0}$ (see Protter \cite[p.66]{p}), $\alpha_{0}>0$ and
\begin{align*}
B = \begin{pmatrix}
0 & 1 & 0 & \cdots & 0\\
0 & 0 & 1 & \cdots & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & 1\\
-\beta_{q} & -\beta_{q-1} & -\beta_{q-2} & \cdots & -\beta_{1}\\
\end{pmatrix},\quad
\mathbf{a}=\begin{pmatrix}
\alpha_{1}\\
\alpha_{2}\\
\vdots\\
\alpha_{q}\\
\end{pmatrix},\quad
\mathbf{e}=\begin{pmatrix}
0\\
0\\
\vdots\\
0\\
1\\
\end{pmatrix},
\end{align*}
with $\alpha_{1},\cdots,\alpha_{p}\in\mathbb{R}, \beta_{1},\cdots, \beta_{q}\in\mathbb{R}, \alpha_{p}\neq0, \beta_{q}\neq0,$  and $\alpha_{p+1}=\cdots=\alpha_{q}=0$. They showed that the state process $(\mathbf{Y}_{t})_{t\geq0}$ can be expressed as a stochastic recurrence equation with the random coefficients. Under some sufficient conditions, they proved that the volatility process $(V_{t})_{t\geq0}$ is strictly stationary which causes the increments with constant length of the process $(G_{t})_{t\geq0}$ to be stationary.

The L\'evy driven COGARCH processes have the restriction that the relevant underling process has the stationary increments. In contrast with such stationary increment processes, processes with periodically stationary increments have a wider application. For example, the accumulated compensation claims from an insurance company in many cases have periodic increments. Specifically, car insurance companies are dealt with more accidental claims at the starting of the new year and in summer than the other parts of the year. As another example, the precipitation in almost all parts of the world have periodic behavior and so the volume of underground waters are higher in some months and lower in some months in each year. The observations of such time series have significant dependency to the previous periods as well. Motivated by these observations, the underling processes with periodically stationary increments, say semi-L\'evy processes, are more prominent than the L\'evy processes. So, we study semi-L\'evy driven COGARCH, denoted by SLD-COGARCH, process in this paper. The semi-L\'evy processes have been extensively studied by Maejima and sato \cite{m1}. Recognizing such semi-L\'evy processes often can be followed by considering some fixed partitions of the corresponding period where inside successive partitions the process has different L\'evy type structure and has similar structure to the corresponding element of the previous periods.

In this paper, we study the statistical properties of the SLD-COGARCH($p,q$) process. Providing a random recurrence equation with periodic random coefficients for the state process $(\mathbf{Y}_{t})_{t\geq0}$ defined by (\ref{equ1.2}), we show that it converges almost surely under some regular conditions. We establish sufficient conditions for the existence of a strictly periodically stationary solution of the state process $(\mathbf{Y}_{t})_{t\geq0}$ which causes the volatility process $(V_{t})_{t\geq0}$ defined by (\ref{equ1.1}) to be strictly periodically stationary. Furthermore, the increments with constant length of the SLD-COGARCH($p,q$) process are a discrete-time periodically correlated (PC) process. We use some tests to verify the PC behavior of the increments. Finally, we apply the introduced model to some real data set.

This paper is organized as follows. In section 2, we introduce the SLD-COGARCH processes. We also present the structure of the semi-L\'evy process and obtain its characteristic function. Section 3 is devoted to some sufficient conditions that cause the volatility process to be strictly periodically stationary. We also show that the increments with constant length of such SLD-COGARCH process form a PC process. In section 4, we illustrate the results with simulations. In section 5, we apply the model to a set of high-frequency financial data and show that this process has the consistent properties. All proofs are presented in Section 6.

\section{Semi-L\'evy driven COGARCH process}
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In this section, we describe the structure of semi-L\'evy compound Poisson process in subsection 2.1. Following the method of Brockwell et al. \cite{b5}, we present the structure of corresponding semi-L\'evy compound Poisson process as the underling process of the COGARCH process in subsection 2.2.
\subsection{Semi-L\'evy process}
We present the definition of semi-L\'evy Poisson and semi-L\'evy compound Poisson processes. Then, we drive the characteristic function of the semi-L\'evy compound Poisson process.
\begin{definition} \textnormal{\textbf{(Semi-L\'evy Poisson process)}}\\
Let $A_{j}=(s_{_{j-1}},s_{_{j}}]$  where $0=s_{_{0}}<s_{_{1}}<\cdots$ constitutes a partition of positive real line
and $|A_{j}|=|A_{j+d}|=s_{_{j}}-s_{_{j-1}}$
 for some integer $d$
and all $j\in\mathbb{N}$
 and also $\tau=\sum_{j=1}^{d}|A_{j}|$.
 Then, a non-homogeneous Poisson process $\big(N(t)\big)_{t\geq0}$
 is called a semi-L\'evy Poisson process with period $\tau>0$ and intensity function $\lambda(u)$
if $E\big(N(t)\big)=\Lambda(t)$
where
\begin{align}\label{equ2.1}
\Lambda(t)=\int_{0}^{t}\lambda(u)du
\end{align}
and $\lambda(t)=\sum_{j=1}^{d}\lambda_{j}I_{\mathfrak{D}_{j}}(t)$ for $t,\lambda_{j}\geq0$ and $\mathfrak{D}_{j}=\bigcup_{k=0}^{\infty}A_{j+kd}$.
\end{definition}
One can easily verify that a semi-L\'evy Poisson process has periodically stationary increments.
\begin{definition} \textnormal{\textbf{(Semi-L\'evy compound Poisson process)}}\\
Let $\big(N(t)\big)_{t\geq0}$ be a semi-L\'evy Poisson process with some period $\tau>0$. Then, the semi-L\'evy compound Poisson, denoted by SLCP, process $(S_{t})_{t\geq0}$ is defined as
\begin{align}\label{equ2.2}
S_{t}=\delta t+\sum_{n=1}^{N(t)}Z_{n}
\end{align}
where $\delta\in\mathbb{R}$ and $Z_{n}=\sum_{j=1}^{d}Z_{n}^{(j)}I_{\lbrace \Upsilon_{_{n}}\in\mathfrak{D}_{j}\rbrace}$ in which $\Upsilon_{_{n}}$ is the arrival time of $n^{th}$ jump $Z_{n}$, $\mathfrak{D}_{j}=\bigcup_{k=0}^{\infty}A_{j+kd}$ and $Z_{n}^{(j)}$ are independent and identically distributed (i.i.d.) random variables with distribution $F_{j}, j=1, \cdots, d$, such that $\int_{\mathbb{R}}z^{2}F_{j}(dz)<\infty$.
\end{definition}
The characteristic function of this process is derived from the following Lemma.
\begin{lemma}
Let $(S_{t})_{t\geq0}$ be the SLCP process introduced by Definition 2.2. Then its characteristic function has the following L\'evy-Khintchine representation
\begin{align}\label{equ2.3}
E[e^{iuS_{t}}]=exp\Big[iu\gamma(t)+\int_{\mathbb{R}}(e^{iu\mathit{z}}-1-iu\mathit{z}
I_{\lbrace|z|\leq1\rbrace})\nu_{t}(dz)\Big],
\end{align}
where
\begin{align*}
\gamma(t)=\delta t+\int_{|z|\leq1}z\nu_{t}(dz)
\end{align*}
and
\begin{align*}
\nu_{t}(dz)=\sum_{j=1}^{r-1}(m+1)\lambda_{j}|A_{j}|F_{j}(dz)+\sum_{j=r}^{d}m\lambda_{j}|A_{j}|F_{j}(dz)+
\lambda_{r}(t-s_{_{r-1+md}})F_{r}(dz),
\end{align*}
in which $m=[\frac{t}{\tau}]$ and $r\in\lbrace1,\cdots, d\rbrace$.
\end{lemma}
Proof: see Appendix, P1.
\begin{remark}
(i) By the fact that the numbers of jumps of the SLCP process $(S_{t})_{t\geq0}$ in the time interval $[0,t]$ are finite and the size of each jump is finite as well, we deduce that the process $(S_{t})_{t\geq0}$ is a finite variation process, see \cite[p.14]{r1}. Hence, the process $(S_{t})_{t\geq0}$ is a semimartingale \cite[p.55]{p}.

(ii) By Definition 2.2, we have that if there is a jump at point $u$, $0\leq u\leq t$, then
\begin{align}\label{equ2.4}
\Delta S_{u}:=S_{u}-S_{u^{-}}&=\big[\delta u+\sum_{n=1}^{N(u)}Z_{n}\big]-\big[\delta
u^{-}+\sum_{n=1}^{N(u^{-})}Z_{n}\big]=Z_{N(u)},
\end{align}
otherwise $\Delta S_{u}=0$.  Thus, by the Theorem 22 of Protter \cite[p.66]{p}, if there is a jump at point $u$ then $\Delta[S,S]_{u}=|Z_{N(u)}|^{2}$ and otherwise $\Delta[S,S]_{u}=0$. So, the quadratic variation of the process $(S_{t})_{t\geq0}$ is given by
\begin{align}\label{equ2.5}
[S,S]_{t}=\sum_{0\leq u\leq t}\Delta[S,S]_{u}=\sum_{0\leq u\leq t}|Z_{N(u)}|^{2}=\sum_{n=1}^{N(t)}
|Z_{n}|^{2}.
\end{align}
\end{remark}

\subsection{Semi-L\'evy driven COGARCH process}
The representation of the L\'evy driven COGARCH($p,q$) process is based on the structure of the GARCH($p,q$) process $(\xi_{n})_{n\in\mathbb{N}^{0}}$ as
\begin{align*}
\xi_{n}&=\sqrt{V_{n}}\varepsilon_{n}\\
V_{n}&=\alpha_{0}+\alpha_{1}\xi_{n-1}^{2}+\cdots+\alpha_{p}\xi_{n-
p}^{2}+\beta_{1}V_{n-1}+\cdots+\beta_{q}V_{n-q},\quad n\geq s,
\end{align*}
where $s=max\lbrace p, q\rbrace$ and $(\varepsilon_{n})_{n\in\mathbb{N}^{0}}$ is a sequence of i.i.d. random variables. The process $(\xi_{n})_{n\in\mathbb{N}^{0}}$ can be interpreted as the log return of the asset price process $(G_{n})_{n\in\mathbb{N}^{0}}$, say $ln(G_{n+1}/G_{n})$. The volatility process $(V_{n})_{n\in\mathbb{N}^{0}}$ is as an ARMA($q,p-1$) process driven by the noise sequence $(V_{n-1}\varepsilon_{n-1}^{2})_{n\in\mathbb{N}}$. Motivated by this, the dynamic of the L\'evy driven COGARCH($p,q$) process for the logarithm of the asset price process $(G_{t})_{t\geq0}$ is defined as
\begin{align*}
dG_{t}=\sqrt{V_{t}}dL_{t}, \qquad t>0, \qquad G_{0}=0,
\end{align*}
where $(L_{t})_{t\geq0}$ is a L\'evy process and $(V_{t})_{t\geq0}$ is a continuous-time analog of the self-exciting ARMA($q,p-1$) process $(V_{n})_{n\in\mathbb{N}^{0}}$, see Brockwell et al. \cite{b5}.

We generalize the L\'evy driven COGARCH($p,q$) process by replacing the L\'evy process $(L_{t})_{t\geq0}$ with the SLCP process $(S_{t})_{t\geq0}$.

\begin{definition}\textnormal{\textbf{\big(SLD-COGARCH($p,q$)\big)}}\\
Let $(S_{t})_{t\geq0}$ be the SLCP process with some period $\tau>0$ defined by (\ref{equ2.2}). Then the left-continuous volatility process $(V_{t})_{t\geq0}$ is defined as
\begin{align}\label{equ2.6}
V_{t}=\alpha_{0}+\mathbf{a}^{\prime}{\mathbf{Y}_{t^{-}}},\qquad t>0,\qquad
V_{0}=\alpha_{0}+\mathbf{a}^{\prime}
{\mathbf{Y}_{0}},
\end{align}
where the state process $(\mathbf{Y}_{t})_{t\geq0}$ is the unique c\`adl\`ag solution of the stochastic differential equation
\begin{align}\label{equ2.7}
d\mathbf{Y}_{t}=B\mathbf{Y}_{t^{-}}dt+\mathbf{e}(\alpha_{0}+\mathbf{a}^{\prime}\textbf{Y}_{t^{-}})d[S,S]_{t},
\qquad t>0,
\end{align}
with an initial value $\mathbf{Y}_{0}$ which is independent of the driving process $(S_{t})_{t\geq0}$, in which $[S,S]_{t}$ is the quadratic variation of the process $(S_{t})_{t\geq0}$ and the ($q\times q$)-matrix $B$ and vectors $\mathbf{a}$ and $\mathbf{e}$ are defined as
\begin{align}\label{equ2.8}
B = \begin{pmatrix}
0 & 1 & 0 & \cdots & 0\\
0 & 0 & 1 & \cdots & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & 1\\
-\beta_{q} & -\beta_{q-1} & -\beta_{q-2} & \cdots & -\beta_{1}\\
\end{pmatrix},\quad
\mathbf{a}=\begin{pmatrix}
\alpha_{1}\\
\alpha_{2}\\
\vdots\\
\alpha_{q}\\
\end{pmatrix},\quad
\mathbf{e}=\begin{pmatrix}
0\\
0\\
\vdots\\
0\\
1\\
\end{pmatrix},
\end{align}
where $p$ and $q$ are integers such that $q\geq p\geq1, \alpha_{0}>0, \alpha_{1},\cdots,\alpha_{p}\in\mathbb{R}, \beta_{1},\cdots,\beta_{q}\in\mathbb{R}, \alpha_{p}\neq0, \beta_{q}\neq0,$ and $\alpha_{p+1}=\cdots=\alpha_{q}=0$.\\
If the volatility process $(V_{t})_{t\geq0}$ defined by (\ref{equ2.6}) is almost surely non-negative, then $(G_{t})_{t\geq0}$ given by
\begin{align}\label{equ2.9}
dG_{t}=\sqrt{V_{t}}dS_{t},\qquad t>0,\quad G_{0}=0,
\end{align}
is called the SLCP driven COGARCH(p,q), denoted by SLD-COGARCH(p,q), process with parameters $B$, $\mathbf{a}$, $\alpha_{0}$ and the driving process $(S_{t})_{t\geq0}$.
\end{definition}

We present the definition of matrix norms and some notation that will be used throughout the paper.
\begin{definition}
Let $||\cdot||_{r}$ denote the vector $L^{r}$-norm in $\mathbb{C}^{q}$, for $r\in[1,\infty]$. Then the matrix $L^{r}$-norm of the $(q\times q)-$matrix $C$ is defined as
\begin{align}\label{equ2.10}
||C||_{r}=\sup_{\mathbf{c}\in\mathbb{C}^{q}\setminus\lbrace0\rbrace}\frac{||C\mathbf{c}||_{r}}
{||\mathbf{c}||_{r}}.
\end{align}
\end{definition}
The matrix $L^{r}$-norms of the $L^{1}$, $L^{2}$ and $L^{\infty}$  are called the maximum absolute column sum norm, spectral norm and maximum absolute row sum norm, respectively.
\begin{definition}
Let $P$ be a matrix such that $P^{-1}BP$ is a diagonal matrix, where B is defined by (\ref{equ2.8}). Then the natural matrix norm of  $C$ is
\begin{align*}
||C||_{P,r}=||P^{-1}CP|| _{r}.
\end{align*}
\end{definition}
The natural matrix norm $||\cdot||_{P,r}$  corresponding to the natural vector norm is defined as
\begin{align}\label{equ2.11}
||\mathbf{c}||_{P,r}=||P^{-1}\mathbf{c}|| _{r}, \qquad \mathbf{c}\in\mathbb{C}^{q}.
\end{align}
The eigenvalues of the matrix B are denoted by $\eta_{1}, \cdots, \eta_{q}$ and $\eta:=\max_{i=1,\cdots,q}\mathcal{R}e(\eta_{i})$, where $\mathcal{R}e(\eta_{i})$ is the real part of the eigenvalue $\eta_{i}$. The ($q\times q$)-identity matrix is denoted by $I_{q}$ or simply $I$.
\section{Periodic stationarity of the increments}
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In this section, we show that under some conditions, the volatility process $(V_{t})_{t\geq0}$ defined by (\ref{equ2.6}) is strictly periodically stationary with period $\tau$. The increments with constant length of the SLD-COGARCH process are a periodically correlated (PC) process. Furthermore, under a necessary and sufficient condition, the volatility process $(V_{t})_{t\geq0}$ is non-negative.

The following theorem shows that the state process of the SLD-COGARCH process defined by (\ref{equ2.7}) satisfies a multivariate random recurrence equation with periodic random coefficients.
\begin{theorem}
Let $(\mathbf{Y}_{t})_{t\geq0}$ be the state process of the SLD-COGARCH process. Then the following results are hold.
\begin{itemize}
\item[(a)] There exists a random $(q\times q)-$matrix $J_{s,t}$ and a random vector $\mathbf{K}_{s,t}$ such that
\begin{align}\label{equ3.1}
\mathbf{Y}_{t}=J_{s,t}\mathbf{Y}_{s}+\mathbf{K}_{s,t},\qquad 0\leq s\leq t.
\end{align}
\item[(b)] $\big(J_{s,t},\mathbf{K}_{s,t}\big)$ are periodic in both indices with the same period $\tau$. So, the distribution of $\big(J_{s,t}, \mathbf{K}_{s,t}\big)$ is invariant under time changes of $s,t$ to $s+\tau,t+\tau$.
\item[(c)] $\big(J_{s,t},\mathbf{K}_{s,t}\big)$ and $\big(J_{r,u}, \mathbf{K}_{r,u}\big)$ are independent for $0\leq s\leq t\leq r\leq u$. $\mathbf{Y}_{s}$ is also independent of $\big(J_{s,t},\mathbf{K}_{s,t}\big)$.
\end{itemize}
\end{theorem}
Proof: see Appendix, P2.

In the next theorem, we consider some mild conditions to the state process that converges in distribution to a finite random vector.
\begin{theorem}
Let $(\mathbf{Y}_{t})_{t\geq0}$ be the state process of the SLD-COGARCH process that is satisfied in the followings
\begin{itemize}
\item[(i)] all the eigenvalues of the matrix B are distinct and have strictly negative real parts,
\item[(ii)] there exists some $r\in[1,\infty]$ such that for $t\in[0,\tau]$ and $j=1, \cdots, d$
\begin{align}\label{equ3.2}
\int_{\mathbb{R}}log\big(1+||P^{-1}\mathbf{e}\mathbf{a}^{\prime}P||_{r}z^{2}\big)F_{j}(dz)
<-\frac{\eta\tau}{\Lambda(t+\tau)-\Lambda(t)},
\end{align}
where $P$ is a matrix which causes $P^{-1}BP$ is diagonal.
\end{itemize}
Then, for the fixed $t\in[0,\tau)$, $\mathbf{Y}_{t+m\tau}$ converges in distribution to some vector $\mathbf{U}^{(t)}$, as $m\rightarrow\infty$. Furthermore, $\mathbf{U}^{(t)}$ has a unique distribution that is satisfied in equation
\begin{align}\label{equ3.3}
\mathbf{U}^{(t)}\overset{d}{=}J_{t,t+\tau}\mathbf{U}^{(t)}+\mathbf{K}_{t,t+\tau},
\end{align}
where $\overset{d}{=}$ denotes the equality in
distribution and $\mathbf{U}^{(t)}$ is independent of $\big(J_{t,t+\tau},\mathbf{K}_{t,t+\tau}\big)$.
\end{theorem}
Proof: see Appendix, P3.
\begin{corollary}
Under the conditions (i) and (ii) of Theorem 3.2, the following results are hold.
\begin{itemize}
\item[(a)] If $\mathbf{Y}_{t}\overset{d}{=}\mathbf{U}^{(t)}$ for any $t\in[0, \tau)$, then $\mathbf{Y}_{t}$ and $V_{t}$ defined by (\ref{equ2.6}) are strictly periodically stationary with the period $\tau$. In other words, for any $n\in\mathbb{N}$ and $t_{_{1}}, t_{_{2}}, \cdots, t_{_{n}}\geq0$
\begin{align*}
\big(\mathbf{Y}_{t_{_{1}}}, \mathbf{Y}_{t_{_{2}}}, \cdots, \mathbf{Y}_{t_{_{n}}}\big)\overset{d}{=}
\big(\mathbf{Y}_{t_{_{1}}+\tau}, \mathbf{Y}_{t_{_{2}}+\tau}, \cdots, \mathbf{Y}_{t_{_{n}}+\tau}\big)
\end{align*}
and
\begin{align*}
\big(V_{t_{_{1}}}, V_{t_{_{2}}}, \cdots, V_{t_{_{n}}}\big)\overset{d}{=}\big(V_{t_{_{1}}+\tau}, V_{t_{_{2}}+\tau}, \cdots,
V_{t_{_{n}}+\tau}\big).
\end{align*}
\item[(b)] Increments with the constant length of the SLD-COGARCH process $(G_{t})_{t\geq0}$ is a PC process. In other words, $G_{t}^{(l)}:=\int_{t}^{t+l}\sqrt{V_{u}}dS_{u}$ for any $t, h\geq0$ and fixed $l>0$ satisfies
\begin{align*}
E\big(G_{t}^{(l)}\big)=E\big(G_{t+\tau}^{(l)}\big)
\end{align*}
and
\begin{align*}
cov\big(G_{t}^{(l)},G_{t+h}^{(l)}\big)=cov\big(G_{t+\tau}^{(l)},G_{t+h+\tau}^{(l)}\big).
\end{align*}
\end{itemize}
\end{corollary}
Proof: see Appendix, P4.

One of the striking features of high-frequency financial data is that the returns have negligible correlation while the squared returns are significantly
correlated. These data typically show a PC structure in their squared logarithm returns (known as squared log return).
The next corollary shows that under some conditions, the increments with constant length of the SLD-COGARCH
which were defined in Corollary 3.3 (b) can capture these features.
\begin{corollary}
Let the conditions of Theorem 3.2 hold and the driving process $(S_{t})_{t\geq0}$ defined by (2.2) has zero mean. Then increments with constant length $G_{t}^{(l)}$ for $t\geq0$ and $h\geq l>0$ satisfies
\begin{align*}
&(a)\quad E\big(G_{t}^{(l)}\big)=0, \qquad cov\big(G_{t}^{(l)},G_{t+h}^{(l)}\big)=0,\\ \\
&(b)\quad E\Big((G_{t}^{(l)})^{2}\Big)=E\Big((G_{t+\tau}^{(l)})^{2}\Big),\qquad
cov\Big((G_{t}^{(l)})^{2},(G_{t+h}^{(l)})^{2}\Big)=cov\Big((G_{t+\tau}^{(l)})^{2},(G_{t+h+\tau}^{(l)})^{2}\Big).
\end{align*}
\end{corollary}
Proof: see Appendix, P5.

The following theorem represents the necessary and sufficient conditions for non-negativity of the volatility process $V_{t}$ for any $t\geq0$.
\begin{theorem}
Let $(\mathbf{Y}_{t})_{t\geq0}$ be the state process of the SLD-COGARCH process with parameters $B$, $\mathbf{a}$ and $\alpha_{0}>0$. Suppose that $\gamma\geq-\alpha_{0}$ be a real constant that satisfies in
\begin{align}\label{equ3.4}
\mathbf{a}^{\prime}e^{Bt}\mathbf{e}\geq0\quad \forall t\geq0,
\end{align}
and
\begin{align}\label{equ3.5}
\mathbf{a}^{\prime}e^{Bt}\mathbf{Y}_{0}\geq\gamma \quad a.s.\quad \forall t\geq0.
\end{align}
Then, with probability one, $V_{t}\geq\alpha_{0}+\gamma\geq0$ for $t\geq0$.\\
Conversely, if either (\ref{equ3.5}) fails or (\ref{equ3.5}) holds with $\gamma>-\alpha_{0}$ and (\ref{equ3.4}) fails, then there exists a SLCP process $(S_{t})_{t\geq0}$ and $t\geq0$ such that $P(V_{t} < 0)>0$.
\end{theorem}
Proof: see Appendix, P6.
\section{Simulation}
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In this section, we verify the theoretical results concerning the PC structure of the increments of the SLD-COGARCH process $(G_{t})_{t\geq0}$ is defined in (\ref{equ2.9}) through the simulation. For this, we first simulate the driving semi-L\'evy process $(S_{t})_{t\geq0}$ with period $\tau$ introduced by Definition 2.2. Second, we present the discretized version of the state process $(\mathbf{Y}_{t})_{t\geq0}$ defined by (\ref{equ2.7}) at the time of jump points. We generate the volatility process $(V_{t})_{t\geq0}$ defined by (\ref{equ2.6}) and the process $(G_{t})_{t\geq0}$ at the time of jump points. Then, we evaluate a sampled process from the simulated values of the process $(G_{t})_{t\geq0}$. Finally, we apply some tests to demonstrate the PC structure of the increments for the sampled process.

For simulating the driving process $(S_{t})_{t\geq0}$ with the underlying semi-L\'evy Poisson process $\big(N(t)\big)_{t\geq0}$, we consider $\Upsilon_{_{j}}$ as the time of $j^{th}$ jump $\big(N(t)\big)_{t\geq0}$ and $\Upsilon_{_{0}}=0$. First, by Definition 2.1, we generate the arrival times $\Upsilon_{_{1}}, \Upsilon_{_{2}}, \cdots$ of the process $\big(N(t)\big)_{t\geq0}$ by the following algorithm.
\begin{enumerate}
\item
Consider the positive value $\tau$ as the length of the period of the process $(N_{t})_{t\geq0}$ and some integer $m$ as the number of period intervals for the simulation.
\item
Choose the positive integer $d$ as the number of elements of partition in each period interval.
\item
Consider positive real numbers $l_{1}, \cdots, l_{d}$ such that $\tau=\sum_{j=1}^{d}l_{j}$ and a partition of first period interval $(0,\tau]$ by $A_{1}, A_{2}, \cdots, A_{d}$ where $A_{j}=(s_{_{j-1}},s_{_{j}}]$, $s_{_{0}}=0$ and $s_{_{j}}=\sum_{i=1}^{j}l_{i}$, $j=1, 2, \cdots, d$. Elements of partition of $k^{th}$ period interval $\big((k-1)\tau,k\tau\big]$ are $A_{1+(k-1)d}, \cdots, A_{kd}$ where $|A_{j+(k-1)d}|=|A_{j}|=l_{j}$ for $k=2, \cdots, m$ and $j=1, \cdots, d$.
\item
Let the positive real numbers $\lambda_{1}, \cdots, \lambda_{d}$ be occurrence rates corresponding to the increments of $(N(t))_{t\geq0}$ on $A_{1}, \cdots, A_{d}$. It is also followed from Definition 2.1 that the process $(N(t))_{t\geq0}$ has the occurrence rate $\lambda_{j}$ on $A_{j+(k-1)d}$ for $k=2, \cdots, m$ and $j=1, \cdots, d$.
\item
Generate an independent sequence of Poisson random variables $P_{i}$ with parameter $\lambda_{i}l_{i}$ on $A_{i}$ for $i=1, \cdots, md$ as $p_{1}, \cdots, p_{md}$.
\item
Generate $p_{i}$ independent samples from uniform distribution $U(s_{_{i-1}},s_{_{i}}]$ for $i=1, \cdots,$ $md$. Then, sort these samples and denote these ordered samples by $\Upsilon_{_{1}}, \cdots, \Upsilon_{_{md}}$.
\item
Generate the successive jump size $Z_{n}$ independently and with distribution $F_{j}$ if corresponding arrival time belongs to $\mathfrak{D}_{j}=\bigcup_{k=0}^{\infty}A_{j+kd}$  for $ j=1, \cdots, d$ and evaluate $S_{t}$ by (\ref{equ2.2}).
\end{enumerate}

Now, by the following steps, we simulate the SLD-COGARH($p,q$) process introduced by Definition 2.3.
\begin{enumerate}
\item
Set integer valued $p$ and $q$ such that $q\geq p\geq1.$
\item
Choose the real parameters $\beta_{1}, \cdots, \beta_{q}$ and $\alpha_{1}, \cdots, \alpha_{p}$ and $\alpha_{0}>0$ such that $\alpha_{p}\neq0$, $\beta_{q}\neq0$, $\alpha_{p+1}=\cdots=\alpha_{q}=0$. Furthermore, the eigenvalues of the matrix $B$, $\eta_{1},\cdots,\eta_{q}$, are distinct and have strictly negative real parts and conditions (\ref{equ3.2}), (\ref{equ3.4}) and (\ref{equ3.5}) are satisfied.\\
To verify the condition (3.2), define the matrix $P$ as follows
\begin{align*}
P = \begin{pmatrix}
1 & \cdots & 1\\
\eta_{1} & \cdots & \eta_{q}\\
\vdots & \cdots & \vdots \\
\eta_{1}^{q-1} & \cdots & \eta_{q}^{q-1}\\
\end{pmatrix}.
\end{align*}
\item
After generating the arrival times $\Upsilon_{_{n}}, n=1, \cdots, md$, by the above algorithm, generate the state process $(\mathbf{Y}_{\Upsilon_{_{n}}})_{n\in\mathbb{N}}$ in (\ref{Th3.1(5.5)}) with an initial value $\mathbf{Y}_{\Upsilon_{0}}$
\begin{align*}
\mathbf{Y}_{\Upsilon_{_{n}}}&=(I+Z_{n}^{2}\mathbf{e}\mathbf{a}^{\prime})e^{B(\Upsilon_{_{n}}-\Upsilon_{_{n-1}})}
\mathbf{Y}_{\Upsilon_{_{n-1}}}+\alpha_{0}Z_{n}^{2}\mathbf{e}\\
&=e^{B\big(\Upsilon_{_{n}}-\Upsilon_{_{n-1}}\big)}\mathbf{Y}_{\Upsilon_{_{n-1}}}+\mathbf{e}\Big(\alpha_{0}+
\mathbf{a}^{\prime}e^{B\big(\Upsilon_{_{n}}-\Upsilon_{_{n-1}}\big)}\mathbf{Y}_{\Upsilon_{_{n-1}}}\Big)Z^{2}_{n}.
\end{align*}
\item
Similar to (\ref{Th3.1(5.2)}) $\mathbf{Y}_{t}$ for $t\in[\Upsilon_{_{n-1}},\Upsilon_{_{n}}^{-}]$ can be written as $\mathbf{Y}_{t}=e^{B(t-\Upsilon_{n-1})}\mathbf{Y}_{\Upsilon_{n-1}}$.
So, by (\ref{equ2.6}), generate the process $(V_{\Upsilon_{n}})_{n\in\mathbb{N}^{0}}$ with
\begin{align}\nonumber
V_{\Upsilon_{_{n}}}&=\alpha_{0}+\mathbf{a}^{\prime}\mathbf{Y}_{\Upsilon_{_{n}}^{-}}\\ \label{equ4.2}
&=\alpha_{0}+\mathbf{a}^{\prime}e^{B(\Upsilon_{_{n}}-\Upsilon_{_{n-1}})}\mathbf{Y}_{\Upsilon_{_{n-1}}}.
\end{align}
\item
Since the driving process $(S_{t})_{t\geq0}$ has one jump at time $\Upsilon_{n}$ over $[\Upsilon_{n-1},\Upsilon_{n}]$, it follows from (\ref{equ2.9}) that
\begin{align}\nonumber
G_{\Upsilon_{n}}-G_{\Upsilon_{n-1}}=\int_{0}^{\Upsilon_{n}}\sqrt{V_{u}}dS_{u}-\int_{0}^{\Upsilon_{n-1}}\sqrt{V_{u}}dS_{u}
&=\int_{\Upsilon_{n-1}}^{\Upsilon_{n}}\sqrt{V_{u}}dS_{u}\\ \label{equ4.3}
&=\sqrt{V_{\Upsilon_{n}}}Z_{n}.
\end{align}
Then generate the process $(G_{\Upsilon_{n}})_{n\in\mathbb{N}^{0}}$ by (\ref{equ4.3}) such that $G_{\Upsilon_{0}}=G_{0}=0$.
\item
Finally, using the values of $V_{\Upsilon_{n}}$ and $G_{\Upsilon_{n}}$ provided by the previous steps, evaluate the sampled processes $(V_{il})_{i\in\mathbb{N}^{0}}$ and $(G_{il})_{i\in\mathbb{N}^{0}}$ for $l=\tau/\varrho$ where $\varrho$ is some integer by the following steps:
\begin{itemize}
\item
For $il\in[\Upsilon_{_{n-1}},\Upsilon_{_{n}}), n\in\mathbb{N}$, it follows from (\ref{Th3.1(5.2)}) that $\mathbf{Y}_{il}=e^{B(il-\Upsilon_{_{n-1}})}\mathbf{Y}_{\Upsilon_{_{n-1}}}$. So, using (\ref{equ2.6}), we have
\begin{align*}
V_{il}&=\alpha_{0}+\mathbf{a}^{\prime}\mathbf{Y}_{il^{-}}\\
&=\alpha_{0}+\mathbf{a}^{\prime}e^{B(il-\Upsilon_{n-1})}\mathbf{Y}_{\Upsilon_{n-1}}.
\end{align*}
Note that if $il=\Upsilon_{_{n-1}}$, by (\ref{equ4.2})
\begin{align*}
V_{il}=V_{\Upsilon_{_{n-1}}}=\alpha_{0}+\mathbf{a}^{\prime}e^{B(\Upsilon_{_{n-1}}-\Upsilon_{_{n-2}})}
\mathbf{Y}_{\Upsilon_{_{n-2}}}.
\end{align*}
\item
Using the fact that the driving process $(S_{t})_{t\geq0}$ has no jump over $[\Upsilon_{_{n-1}},\Upsilon_{_{n}})$, for $n\in\mathbb{N}$, it follows from (\ref{equ2.9}) for $il\in[\Upsilon_{_{n-1}},\Upsilon_{_{n}})$
\begin{align}\label{equ4.4}
G_{il}-G_{\Upsilon_{_{n-1}}}=\int_{\Upsilon_{_{n-1}}}^{il}\sqrt{V_{u}}dS_{u}=0.
\end{align}
Thus, $G_{il}=G_{\Upsilon_{n-1}}$. If $il=\Upsilon_{_{n-1}}$, then it follows from (\ref{equ4.3}) that
\begin{align}\label{equ4.5}
G_{il}=G_{\Upsilon_{_{n-1}}}=G_{\Upsilon_{_{n-2}}}+\sqrt{V_{\Upsilon_{_{n-1}}}}Z_{n-1}.
\end{align}
\end{itemize}
\end{enumerate}

To detect the PC structure of a process, it is shown that the proposed sample spectral coherence statistic can
be used to test whether a discrete-time process is PC \cite{h1, h2}.
In this method, for $n$ samples $X_{0}, X_{1}, \cdots, X_{n-1}$, and a fixed $M$, the following sample spectral coherence statistic is plotted
\begin{align*}
|\hat{\gamma}(P, Q ,M)|^{2}=\frac{|\sum_{m=0}^{M-1}d_{X}(\zeta_{_{P+m}})\overline{d_{X}
(\zeta_{_{Q+m})}}|^{2}}{\sum_{m=0}^{M-1}|d_{X}(\zeta_{_{P+m}})|^{2}\sum_{m=0}^{M-1}|d_{X}(\zeta_{_{Q+m}})|^{2}},
\end{align*}
where $P, Q=0, \cdots, n-1$, $d_{X}(\zeta_{_{P}})=\sum_{k=0}^{n-1}X_{k}e^{ik\zeta_{_{P}}}$ and $\zeta_{_{P}}=\frac{2\pi P}{n}$ for $P=0,\cdots, n-1$.\\
The perception of this sample spectral coherence is aided by plotting the coherence values only at points where the following $\alpha-$threshold is exceeded \cite[p.310]{h2}
\begin{align*}
x_{\alpha}=1-e^{log(\alpha)/(M-1)}.
\end{align*}
For a PC time series with period $\varrho$, it is expected that the sample spectral coherence statistic has a significant value on pairs $(P,Q)$ for $|P-Q|=kn/\varrho, k=0, \cdots, \varrho-1$. The plot of the support of the sample spectral coherence statistic can be used to identify the type of model and analysis of the time series $X_{0}, X_{1}, \cdots, X_{n-1}$, see \cite{d2}:
\begin{itemize}
\item
If only the main diagonal appears, then the time series $X_{i}$ is a stationary time series.
\item
If there are some significant values of statistic and they seem to lie along the parallel equally spaced diagonal lines, then
the time series $X_{i}$ is PC with period $\varrho=n/L$, where $L$ is the line spacing.
\item
If there are some significant values of statistic but they occur in some non-regular places, then
the time series $X_{i}$ is a non-stationary time series.
\end{itemize}
In the following, we provide an example to investigate the process.
\begin{example}
Let $(S_{t})_{t\geq0}$ be a SLCP process with period $\tau=6.5$. Furthermore, the lengths of the
successive subintervals of each period interval are 0.5, 2.5, 3, 0.5 where corresponding occurrence rates
of the semi-L\'evy Poisson process on these subintervals are assumed as 4, 10, 5, 30.
Moreover, the distribution of jump sizes on these subintervals are N(2, 4), N(1.5, 2.5), N(2.5, 1.5) and N(1.75, 3),
where $N(\mu, \sigma^{2})$ denotes a Normal distribution with mean $\mu$ and variance $\sigma^{2}$.
In this example, we consider SLD-COGARCH(1,3) process with parameters $\alpha_{0}=10^{-6}$, $\alpha_{1}=0.005$, $\beta_{1}=2.1$,
$\beta_{2}=6$ and $\beta_{3}=0.6$. Thus, the matrix $B$ is
\begin{align*}
B = \begin{pmatrix}
0 & 1 & 0 \\
0 & 0 & 1 \\
-0.6 & -6 & -2.1\\
\end{pmatrix}
\end{align*}
and conditions (\ref{equ3.2}), (\ref{equ3.4}) and (\ref{equ3.5}) are satisfied. For such SLD-COGARCH process, we simulate
$G_{\Upsilon_{n}}$
for the duration of $m=30$ period intervals by the specified parameters, $\mathbf{Y}_{0}=( 0.37\times10^{-3}, 0.05\times10^{-3},
0.19\times10^{-3})^{\prime}$ and $G_{0}=0$ by the suggested simulation algorithm.
Then, by the step 6 of the last algorithm, we get equally space samples with length $l=0.25$. So, we have 780 discretized
samples of this 30 period intervals.
\end{example}
\begin{figure}[H]
\centering
\includegraphics[scale=0.35]{1.png}
\caption{\scriptsize{Top: the increments ($G_{il}^{(l)}=G_{(i+1)l}-G_{il}$) of the sampled
process $(G_{il})_{i=0,\cdots,779}$; bottom left: the sample autocorrelation plot of $(G_{il}^{(l)})_{i=0,\cdots,779}$;
bottom right: the significant values of the sample spectral coherence statistic of
$(G_{il}^{(l)})_{i=0,\cdots,779}$ with $\alpha=0.05$ and $M=240$.}}
\end{figure}
In Figure 1, we see the differenced series $(G_{il}^{(l)})_{i=0, 1, \cdots, 779}$ (top) and its sample autocorrelation (bottom left). The sample spectral coherence statistic of this series for a specified collection of pairs $(P,Q)$ and $M=240$ that exceed the threshold corresponding to $\alpha=0.05$ which is presented at the bottom right. If $X_{i}:=G_{il}^{(l)}$, then the parallel lines for the sample spectral coherence confirm the series $(X_{i})_{i=0, 1, \cdots, 779}$ is PC with period $\varrho=780/30=26$. This implies that, for $i, j=0, 1, \cdots, 779$,
\begin{align}\label{equ4.6}
E\big(G_{il}^{(l)}\big)=E(X_{i})=E(X_{i+\varrho})=E\big(G_{il+\varrho l}^{(l)}\big)
\end{align}
and
\begin{align}\label{equ4.7}
cov\big(G_{il}^{(l)},G_{jl}^{(l)}\big)=cov(X_{i},X_{j})=cov(X_{i+\varrho},X_{j+\varrho})=cov\big(G_{il+\varrho l}^{(l)},G_{jl+\varrho
l}^{(l)}\big).
\end{align}

So, it follows from (\ref{equ4.6}) and (\ref{equ4.7}) that the series $(G_{t}^{(l)})_{t=0, l, \cdots, 779l}$ has a PC structure with period $\tau=\varrho l=6.5$.

\section{Real data analysis}
\setcounter{equation}{0}
One of the striking features of financial time series is that the log returns have negligible correlation while its square log returns are significantly correlated \cite{b5}. Many of these time series are collected at a high-frequency and typically show a PC structure in their squared log returns \cite{r2}. In this section, we evaluate the results of Corollary 3.4 and show that the SLD-COGARCH process can capture the periodic structure of high-frequency financial data. For this, we use the 15-minute log returns of the Dow Jones Industrial Average (DJIA) index.

This data set is recorded between 9:35 to 16:00 from September 4th, 2015 to August 14th, 2018. There was a total of 735 trading days not including the weekends and holidays with 26 15-minute observations per day, resulting in the total of n = 19110 15-minute observations. Figure 2, shows the DJIA index for the specified times.
\begin{figure}[H]
\centering
\includegraphics[width=10.5cm,height=7.5cm]{2.png}
\caption{\scriptsize{DJIA 15-minute data from September 4th, 2015 to August 14th, 2018.}}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[scale=0.34]{3.png}
\caption{\scriptsize{Top: DJIA log returns; middle: their squares; bottom left: the sample
autocorrelation plot of the squared log returns and bottom right: the significant values of the sample spectral coherence statistic of the squared log
returns.}}
\end{figure}
In Figure 3, we show the 15-minute log returns of DJIA (top), their squares (middle) and sample autocorrelation function (ACF) of the squared log returns (bottom left). It is clear from ACF that 15-minute squared log returns have a seasonal structure with period 26. To show more precisely the PC structure of these data, we sample last 100 trading days. These data were recorded from March 22th, 2018 to August 14th, 2018 which contain 2600 observations. Then, we present the sample spectral coherence statistic of their squared log returns for a specified collection of pairs $(P,Q)$ and $M=100$. It is shown that it exceeds the threshold corresponding to $\alpha=0.05$ in Figure 3 (bottom right). Therefore, sample spectral coherence statistic clearly shows that the squared log returns have a PC structure with period $\varrho=2600/100=26$.

Currently we show that how well SLD-COGARCH process fits the 15-minute log returns of DJIA index. For this, we first simulate the driving process $(S_{t})_{t\geq0}$ which is specified in Example 4.1 in the case that $\delta=0$ and the distribution of jump sizes are assumed to be N(0,4), N(0,2.5), N(0,1.5) and N(0,3). Second, we simulate the SLD-COGARCH(1,3) process for the duration of $m=735$ period intervals with the initial value $\mathbf{Y}_{0}=( 0.37\times10^{-3}, 0.05\times10^{-3}, 0.19\times10^{-3})^{\prime}$ and the parameters $\alpha_{0}=0.8\times10^{-6}$, $\alpha_{1}=0.0275$, $\beta_{1}=2.1$, $\beta_{2}=6$ and $\beta_{3}=0.6$ by the suggested simulation algorithm. Then, by the step 6 of the last algorithm, we get 19110 equally space samples by length $l=0.25$.
\begin{figure}[H]
\centering
\includegraphics[scale=0.34]{4.png}
\caption{\scriptsize{Top: the increments $(G_{il}^{(l)}=G_{(i+1)l}-G_{il})$ of the sampled process $(G_{il})_{i=0,\cdots,19109}$; middle: their squares; bottom left: the sample autocorrelation plot of the squared increments; bottom right: the significant values of the sample spectral coherence statistic of the squared increments with $\alpha=0.05$ and $M=550$.}}
\end{figure}
Figure 4 shows the increments of the sampled process $(G_{il})_{i=0,\cdots,19109}$, (top) and their squares (middle) and ACF of the squared increments (bottom left). We select the last 2600 simulated data as sample and present the sample spectral coherence statistic of their squared increments for a specified collection of pairs $(P,Q)$ and $M=550$. It is shown that it exceeds the threshold corresponding to $\alpha=0.05$ in Figure 4 (bottom right). The ACF and sample spectral coherence statistic demonstrate that the squared increments have a PC structure with period $\varrho=2600/100=26$. Therefore, these graphs show that SLD-COGARCH process can be consistent with 15-minute log returns of DJIA index.

\section{Appendix}
\setcounter{equation}{0}
\textbf{P1: Proof of Lemma 2.1}\\
For any $t\geq0$ there exist $r=1, \cdots, d$ such that $t\in A_{r+md}=(s_{_{r-1+md}}, s_{_{r+md}}]$, where $m=[\frac{t}{\tau}]$.
So, it follows from Definition 2.1 and Definition 2.2 that
\begin{align*}
E\big(e^{iuS_{t}}\big)&=E\big(e^{iu\big(\delta t+\sum_{n=1}^{N(t)}Z_{n}\big)}\big)
=e^{iu\delta t}E\big(e^{iu\sum_{n=1}^{N(t)}Z_{n}}\big)\\
&=e^{iu\delta t}E\Big(e^{iu\sum_{n=1}^{N(s_{_{1}})}Z_{n}^{(1)}}\times
e^{iu\sum_{n=N(s_{_{1}})+1}^{N(s_{_{2}})}Z_{n}^{(2)}}\times\cdots\times
e^{iu\sum_{n=N(s_{_{r-1+md}})+1}^{N(t)}Z_{n}^{(r)}}\Big)\\
&=e^{iu\delta t}E\Big(\prod_{j=1}^{d}\prod_{k=0}^{m-1}
e^{iu\sum_{n=N(s_{_{j-1+kd}})+1}^{N(s_{_{j+kd}})}Z_{n}^{(j)}}
\prod_{j=1}^{r-1}e^{iu\sum_{n=N(s_{_{j-1+md}})+1}^{N(s_{_{j+md}})}Z_{n}^{(j)}}\\
&\qquad\qquad\times e^{iu\sum_{n=N(s_{_{r-1+md}})+1}^{N(t)}Z_{n}^{(r)}}\Big).
\end{align*}
Since, by Definition 2.2, the jump sizes $Z_{n}^{(j)}, n\in\mathbb{N},$ are independent for $j=1,\cdots,d$, we have that
\begin{align}\nonumber
E\big(e^{iuS_{t}}\big)&=e^{iu\delta t}\prod_{j=1}^{d}\prod_{k=0}^{m-1}
E\big(e^{iu\sum_{n=N(s_{_{j-1+kd}})+1}^{N(s_{_{j+kd}})}Z_{n}^{(j)}}\big)
\prod_{j=1}^{r-1}E\big(e^{iu\sum_{n=N(s_{_{j-1+md}})+1}^{N(s_{_{j+md}})}Z_{n}^{(j)}}\big)\\ \label{Lem2.1(5.1)}
&\quad\times E\big(e^{iu\sum_{n=N(s_{_{r-1+md}})+1}^{N(t)}Z_{n}^{(r)}}\big).
\end{align}
By the fact that the jump sizes $Z_{n}^{(j)}, n\in\mathbb{N},$ in each partition $A_{j+kd}=(s_{_{j-1+kd}},s_{_{j+kd}}],
k\in\mathbb{N}^{0},$
are i.i.d. random variables with distribution $F_{j}$, it follows from Definition 2.1 and conditional expected value that
\begin{align}\nonumber
&E\big(e^{iu\sum_{n=N(s_{_{j-1+kd}})+1}^{N(s_{_{j+kd}})}Z_{n}^{(j)}}\big)=E\Big(E\big(e^{iu\sum_{n=1}^{N(s_{_{j+kd}})-
N(s_{_{j-1+kd}})}Z_{n}^{(j)}}\vert N(s_{_{j+kd}})-N(s_{_{j-1+kd}})=N\big)\Big)\\ \nonumber
&\qquad\qquad=\sum_{N=0}^{\infty}E\big(e^{iu\sum_{n=1}^{N}Z_{n}^{(j)}}\big)P\big(N(s_{_{j+kd}})-N(s_{_{j-1+kd}})=N\big)\\
\nonumber
&\qquad\qquad=\sum_{N=0}^{\infty}\big(E(e^{iuZ_{n}^{(j)}})\big)^{N}\frac{\big(\Lambda(s_{_{j+kd}})-\Lambda(s_{_{j-1+kd}})\big)^{N}
e^{-\big(\Lambda(s_{_{j+kd}})-\Lambda(s_{_{j-1+kd}})\big)}}{N!}\\ \nonumber
&\qquad\qquad=e^{-(\Lambda(s_{_{j+kd}})-\Lambda(s_{_{j-1+kd}}))}\sum_{N=0}^{\infty}\frac{\Big(\int_{\mathbb{R}}e^{iuz}
\big(\Lambda(s_{_{j+kd}})-\Lambda(s_{_{j-1+kd}})\big)F_{j}(dz)
\Big)^{N}}{N!}\\ \nonumber
&\qquad\qquad=e^{-(\Lambda(s_{_{j+kd}})-\Lambda(s_{_{j-1+kd}}))}e^{\int_{\mathbb{R}}e^{iuz}
(\Lambda(s_{_{j+kd}})-\Lambda(s_{_{j-1+kd}}))F_{j}(dz)}\\ \label{Lem2.1(5.2)}
&\qquad\qquad=e^{\int_{\mathbb{R}}(e^{iuz}-1)\lambda_{j}|A_{j}|F_{j}(dz)},
\end{align}
where $|A_{j}|=|A_{j+kd}|=s_{_{j}}-s_{_{j-1}}$. The last equality follows from (\ref{equ2.1}) and the fact that the semi-L\`evy Poisson
process
$N(t)$ has the occurrence rate $\lambda_{j}$
on $A_{j+kd}$ for $k\in\mathbb{N}^{0}$. An argument similar to that used in (\ref{Lem2.1(5.2)}) shows that
\begin{align}\label{Lem2.1(5.3)}
E\big(e^{iu\sum_{n=N(s_{_{r-1+md}})+1}^{N(t)}Z_{n}^{(r)}}\big)=e^{\int_{\mathbb{R}}(e^{iuz}-1)\lambda_{r}(t-s_{_{r-1+md}})F_{r}
(dz)}
\end{align}
By replacing from (\ref{Lem2.1(5.2)}) and (\ref{Lem2.1(5.3)}) in (\ref{Lem2.1(5.1)}), it reads as
\begin{align*}
E\big(e^{iuS_{t}}\big)&=e^{iu\delta t}\prod_{j=1}^{d}e^{\int_{\mathbb{R}}(e^{iuz}-1)m\lambda_{j}|A_{j}|F_{j}(dz)}
\prod_{j=1}^{r-1}e^{\int_{\mathbb{R}}(e^{iuz}-1)\lambda_{j}|A_{j}|F_{j}(dz)}\\
&\quad\times e^{\int_{\mathbb{R}}(e^{iuz}-1)\lambda_{r}(t-s_{_{r-1+md}})F_{r}(dz)}\\
&=e^{\Big(iu\big(\delta t+\int_{|z|\leq1}z\nu_{t}(dz)\big)+\int_{\mathbb{R}}(e^{iuz}-1-iuzI_{|z|\leq1})\nu_{t}(dz)\Big)},
\end{align*}
where
\begin{align*}
\nu_{t}(dz)=\sum_{j=1}^{r-1}(m+1)\lambda_{j}|A_{j}|F_{j}(dz)+\sum_{j=r}^{d}m\lambda_{j}|A_{j}|F_{j}(dz)+
\lambda_{r}(t-s_{_{r-1+md}})F_{r}(dz).
\end{align*}
\\
\textbf{P2: Proof of Theorem 3.1}
\vspace{2mm}

\textbf{Proof ($a$):} Let $(S_{t})_{t\geq0}$ be the driving process introduced by Definition 2.2, $\Upsilon_{_{n}}$ be the time of $n^{th}$ jump and $\Upsilon_{_{0}}=0$. Following the method of Brockwell et al. \cite{b5}, and as $(S_{t})_{t\geq0}$  has no jump over $[\Upsilon_{_{n}},\Upsilon_{_{n+1}})$, it follows by (\ref{equ2.5}) that $[S,S]_{t}=0$ for $t\in[\Upsilon_{_{n}},\Upsilon_{_{n+1}})$. So (\ref{equ2.7}) implies that $\mathbf{Y}_{t}$ satisfies $d\mathbf{Y}_{t}=B\mathbf{Y}_{t}dt$ for $t\in[\Upsilon_{_{n}},\Upsilon_{_{n+1}})$.  Hence
\begin{align*}
e^{-Bt}d\mathbf{Y}_{t}=Be^{-Bt}\mathbf{Y}_{t}dt
\end{align*}
and
\begin{align}\label{Th3.1(5.1)}
d\big(e^{-Bt}\mathbf{Y}_{t}\big)=0.
\end{align}
Integrating from $\Upsilon_{_{n}}$ to $t\in[\Upsilon_{_{n}},\Upsilon_{_{n+1}})$, we have that
\begin{align}\label{Th3.1(5.2)}
\mathbf{Y}_{t}=e^{B(t-\Upsilon_{_{n}})}\mathbf{Y}_{\Upsilon_{_{n}}}.
\end{align}
By (\ref{equ2.7}) and (\ref{equ2.5}), we have that $\mathbf{Y}_{t}$ has a
jump of size $\mathbf{e}(\alpha_{0}+\mathbf{a}^{\prime}\mathbf{Y}_{\Upsilon_{_{n}}^{-}})Z_{_{n}}^{2}$
at $\Upsilon_{_{n}}$. So, $\mathbf{Y}_{t}$ at $\Upsilon_{_{n}}$ can be written as
\begin{align}\nonumber
\mathbf{Y}_{\Upsilon_{_{n}}}&=\mathbf{Y}_{\Upsilon_{_{n}}^{-}}+\mathbf{e}
(\alpha_{0}+\mathbf{a}^{\prime}\mathbf{Y}_{\Upsilon_{_{n}}^{-}})Z_{_{n}}^{2}\\ \label{Th3.1(5.3)}
&=(I+Z_{_{n}}^{2}\mathbf{e}\mathbf{a}^{\prime})\mathbf{Y}_{\Upsilon_{_{n}}^{-}}+
\alpha_{0}Z_{_{n}}^{2}\mathbf{e}.
\end{align}
Integrating of (\ref{Th3.1(5.1)}) this time from $\Upsilon_{_{n-1}}$ to $\Upsilon_{_{n}}^{-}$, we have that
\begin{align}\label{Th3.1(5.4)}
\mathbf{Y}_{\Upsilon_{_{n}}^{-}}=e^{B(\Upsilon_{_{n}}-\Upsilon_{_{n-1}})}\mathbf{Y}_{\Upsilon_{_{n-1}}}.
\end{align}
Therefore
\begin{align}\label{Th3.1(5.5)}
\mathbf{Y}_{\Upsilon_{_{n}}}=(I+Z_{_{n}}^{2}\mathbf{e}\mathbf{a}^{\prime})
e^{B(\Upsilon_{_{n}}-\Upsilon_{_{n-1}})}\mathbf{Y}_{\Upsilon_{_{n-1}}}+\alpha_{0}Z_{_{n}}^{2}\mathbf{e}.
\end{align}
By assuming $t\in[\Upsilon_{_{N(t)}},\Upsilon_{_{N(t)+1}})$, (\ref{Th3.1(5.2)}) reads as
\begin{align}\label{Th3.1(5.6)}
\mathbf{Y}_{t}=e^{B(t-\Upsilon_{_{N(t)}})}\mathbf{Y}_{\Upsilon_{_{N(t)}}}
\end{align}
and (\ref{Th3.1(5.5)}) as
\begin{align}\label{Th3.1(5.7)}
\mathbf{Y}_{\Upsilon_{_{N(t)}}}=(I+Z_{_{N(t)}}^{2}\mathbf{e}\mathbf{a}^{\prime})
e^{B(\Upsilon_{_{N(t)}}-\Upsilon_{_{N(t)-1}})}\mathbf{Y}_{\Upsilon_{_{N(t)-1}}}
+\alpha_{0}Z_{_{N(t)}}^{2}\mathbf{e}.
\end{align}
By successive replacement from (\ref{Th3.1(5.7)}) in (\ref{Th3.1(5.6)}) we have that
\begin{align}\nonumber
\mathbf{Y}_{t}&=e^{B\big(t-\Upsilon_{_{N(t)}}\big)}\Big[\big(I+Z_{_{N(t)}}^{2}\mathbf{e}\mathbf{a}^{\prime}\big)
e^{B\big(\Upsilon_{_{N(t)}}-\Upsilon_{_{N(t)-1}}\big)}\times\\ \nonumber
&\qquad\qquad\qquad\cdots\times\big(I+Z_{_{N(s)+2}}^{2}\mathbf{e}\mathbf{a}^{\prime}\big)
e^{B\big(\Upsilon_{_{N(s)+2}}-\Upsilon_{_{N(s)+1}}\big)}\Big]\mathbf{Y}_{\Upsilon_{_{N(s)+1}}}\\ \nonumber
&+e^{B\big(t-\Upsilon_{_{N(t)}}\big)}\Big[(\alpha_{0}Z_{_{N(t)}}^{2}\mathbf{e})+\sum_{i=0}^{N(t)-N(s)-3}
\big(I+Z_{_{N(t)}}^{2}\mathbf{e}\mathbf{a}^{\prime}\big)
e^{B\big(\Upsilon_{_{N(t)}}-\Upsilon_{_{N(t)-1}}\big)}\times\\ \label{Th3.1(5.8)}
&\qquad\qquad\qquad\cdots\times\big(I+Z_{_{N(t)-i}}^{2}\mathbf{e}\mathbf{a}^{\prime}\big)
e^{B\big(\Upsilon_{_{N(t)-i}}-\Upsilon_{_{N(t)-i-1}}\big)}(\alpha_{0}Z_{_{N(t)-i-1}}^{2}\mathbf{e})\Big].
\end{align}
Similar to (\ref{Th3.1(5.3)}) $\mathbf{Y}_{t}$ at $\Upsilon_{_{N(s)+1}}$ can be written as
\begin{align}\label{Th3.1(5.9)}
\mathbf{Y}_{\Upsilon_{_{N(s)+1}}}=(I+Z_{_{N(s)+1}}^{2}\mathbf{e}\mathbf{a}^{\prime})
\mathbf{Y}_{\Upsilon_{_{N(s)+1}}^{^{-}}}+\alpha_{0}Z_{_{N(s)+1}}^{2}\mathbf{e}.
\end{align}
By integrating of (\ref{Th3.1(5.1)}) from $s$ to $\Upsilon_{_{N(s)+1}}$, we have that
\begin{align*}
\mathbf{Y}_{\Upsilon_{_{N(s)+1}}^{^{-}}}=e^{B(\Upsilon_{_{N(s)+1}}-s)}\mathbf{Y}_{s}.
\end{align*}
Replacing this in (\ref{Th3.1(5.9)}) implies that
\begin{align}\label{Th3.1(5.10)}
\mathbf{Y}_{\Upsilon_{_{N(s)+1}}}=(I+Z_{_{N(s)+1}}^{2}\mathbf{e}\mathbf{a}^{\prime})
e^{B(\Upsilon_{_{N(s)+1}}-s)}\mathbf{Y}_{s}+\alpha_{0}Z_{_{N(s)+1}}^{2}\mathbf{e}.
\end{align}
By replacing from (\ref{Th3.1(5.10)}) in (\ref{Th3.1(5.8)}), it reads as
\begin{align}\nonumber
\mathbf{Y}_{t}&=e^{B\big(t-\Upsilon_{_{N(t)}}\big)}\Big[\big(I+Z_{_{N(t)}}^{2}\mathbf{e}\mathbf{a}^{\prime}\big)
e^{B\big(\Upsilon_{_{N(t)}}-\Upsilon_{_{N(t)-1}}\big)}\times\\ \nonumber
&\quad\qquad\qquad\cdots\times
\big(I+Z_{_{N(s)+2}}^{2}\mathbf{e}\mathbf{a}^{\prime}\big)
e^{B\big(\Upsilon_{_{N(s)+2}}-\Upsilon_{_{N(s)+1}}\big)}\big(I+Z_{_{N(s)+1}}^{2}\mathbf{e}\mathbf{a}^{\prime}\big)
e^{B\big(\Upsilon_{_{N(s)+1}}-s\big)}\Big]\mathbf{Y}_{s}\\ \nonumber
&+e^{B\big(t-\Upsilon_{_{N(t)}}\big)}\Big[(\alpha_{0}Z_{_{N(t)}}^{2}\mathbf{e})+\sum_{i=0}^{N(t)-N(s)-2}
\big(I+Z_{_{N(t)}}^{2}\mathbf{e}\mathbf{a}^{\prime}\big)
e^{B\big(\Upsilon_{_{N(t)}}-\Upsilon_{_{N(t)-1}}\big)}\times\\ \nonumber
&\quad\qquad\qquad\cdots\times\big(I+Z_{_{N(t)-i}}^{2}\mathbf{e}\mathbf{a}^{\prime}\big)
e^{B\big(\Upsilon_{_{N(t)-i}}-\Upsilon_{_{N(t)-i-1}}\big)}(\alpha_{0}Z_{_{N(t)-i-1}}^{2}\mathbf{e})\Big]\\
\label{Th3.1(5.11)}
&=:J_{s,t}\mathbf{Y}_{s}+\mathbf{K}_{s,t}.
\end{align}

\textbf{Proof (b):} We show that the distribution of $J_{s,t}$ and $\mathbf{K}_{s,t}$ are periodic in both indices $s$ and $t$, with the same period $\tau$. Following the fact that $J_{s,t}$ and $\mathbf{K}_{s,t}$ which are defined by (\ref{Th3.1(5.11)}) depend on $s$ and $t$ through
\begin{itemize}
\item
the jump sizes $Z_{\cdot}$'s which by Definition 2.1 are independent and periodically identically distributed random variables with distribution $F_{j}, j=1, \cdots, d$,
\item
$\Upsilon_{_{N(t)}}-\Upsilon_{_{N(t)-1}}, \cdots, \Upsilon_{_{N(s)+2}}-\Upsilon_{_{N(s)+1}}$ which are inter-arrival times of $N(t)$ which by Definition 2.1 has periodically stationary increment with period $\tau$, So are periodic with period $\tau$. Also $N(t)-N(s)$ are periodic by the same reason,
\item
$t-\Upsilon_{_{N(t)}}$ is the current life at time $t$ which is periodic with period $\tau$. Also $J_{s,t}$ is a function of $\Upsilon_{_{N(s)+1}}-s$ which excess life and is periodic with period $\tau$,
\end{itemize}
it follows that
\begin{align*}
J_{s,t}\overset{d}{=}J_{s+\tau,t+\tau} \qquad \textnormal{and}\qquad \mathbf{K}_{s,t}\overset{d}{=}\mathbf{K}_{s+\tau,t+\tau}.
\end{align*}
So, $(J_{s,t},\mathbf{K}_{s,t})$ are periodic in both indices $s$ and $t$, with the same period $\tau$.

\vspace{5mm}
\textbf{Proof (c):} By (\ref{Th3.1(5.11)}) it is clear that
$(J_{s,t},\mathbf{K}_{s,t})$ and $(J_{r,u},\mathbf{K}_{r,u})$ are constructed from segments of the driving process $(S_{t})_{t\geq0}$ in the intervals $(s,t]$ and $(r,u]$, respectively. Thus $(J_{s,t},\mathbf{K}_{s,t})$ and $(J_{r,u},\mathbf{K}_{r,u})$ are independent for $0\leq s\leq t\leq r\leq u$.\\
Now we show that $\mathbf{Y}_{s}$ is independent of $\big(J_{s,t},\mathbf{K}_{s,t}\big)$. By (\ref{Th3.1(5.11)}) we have that
\begin{align*}
\mathbf{Y}_{s}=J_{0,s}\mathbf{Y}_{0}+\mathbf{K}_{0,s}.
\end{align*}
As $(J_{0,s},\mathbf{K}_{0,s})$ and $(J_{s,t},\mathbf{K}_{s,t})$ are independent and that $\mathbf{Y}_{0}$ is independent of the driving process $(S_{t})_{t\geq0}$, assumed by (\ref{equ2.7}), it follows that $\mathbf{Y}_{s}$ and $(J_{s,t},\mathbf{K}_{s,t})$ are independent.
\vspace{6mm}
\\
\textbf{P3: Proof of Theorem 3.2}\\
For fixed $t\in[0,\tau)$ and for $m\in\mathbb{N}$, it follows from (\ref{Th3.1(5.11)}) that
\begin{align}\label{Th3.2(5.12)}
\mathbf{Y}_{t+m\tau}=J_{t+(m-1)\tau,t+m\tau}\mathbf{Y}_{t+(m-1)\tau}+\mathbf{K}_{t+(m-1)\tau,t+m\tau}.
\end{align}
By successive replacement from (\ref{Th3.2(5.12)}) we obtain
\begin{align}\nonumber
\mathbf{Y}_{t+m\tau}&=\Big(J_{t+(m-1)\tau,t+m\tau}\cdots J_{t,t+\tau}\Big)\mathbf{Y}_{t}
+\Big[\mathbf{K}_{t+(m-1)\tau,t+m\tau}\\ \label{Th3.2(5.13)}
&\quad +J_{t+(m-1)\tau,t+m\tau}\mathbf{K}_{t+(m-2)\tau,t+(m-1)\tau}+\cdots+ J_{t+(m-1)\tau,t+m\tau}\cdots J_{t+\tau,t+2\tau}\mathbf{K}_{t,t+\tau}\Big].
\end{align}
Since, by the part ($b$) of Theorem 3.1, $(J_{t,t+\tau}, \mathbf{K}_{t,t+\tau}), (J_{t+\tau,t+2\tau}, \mathbf{K}_{t+\tau,t+2\tau}),\cdots$ are i.i.d., it follows from (\ref{Th3.2(5.13)}) that
\begin{align}\nonumber
\mathbf{Y}_{t+m\tau}&\overset{d}{=}\Big(J_{t,t+\tau}\cdots J_{t+(m-1)\tau,t+m\tau}\Big)\mathbf{Y}_{t}
+\Big[\mathbf{K}_{t,t+\tau}+J_{t,t+\tau}\mathbf{K}_{t+\tau,t+2\tau}\\ \nonumber
&\hspace{5cm}+\cdots+ J_{t,t+\tau}\cdots J_{t+(m-2)\tau,t+(m-1)\tau}\mathbf{K}_{t+(m-1)\tau,t+m\tau}\Big]\\ \nonumber
&=\Big(\prod_{i=1}^{m}J_{t+(i-1)\tau, t+i\tau}\Big)\mathbf{Y}_{t}
+\Big[\mathbf{K}_{t,t+\tau}+\sum_{i=1}^{m-1}J_{t,t+\tau}\cdots J_{t+(i-1)\tau,t+i\tau}
\mathbf{K}_{t+i\tau,t+(i+1)\tau}\Big]\\ \label{Th3.2(5.14)}
&=:H_{m}^{(t)}\mathbf{Y}_{t}+\mathbf{U}_{m}^{(t)}.
\end{align}
The following method of Brockwell et al. \cite{b5}, for the proof of this theorem we use the general theory of multivariate random recurrence equations, see Bougerol and Picard \cite{b3}. For this, we first show that there exist
some $r\in[1,\infty]$ and $(q\times q)-$matrix $P$ such that
\begin{align*}
E\Big(log||J_{t,t+\tau}||_{P,r}\Big)<0 \qquad \textnormal{and}\qquad E\Big(log^{+}||\mathbf{K}_{t,t+\tau}||
_{P,r}\Big)<\infty,
\end{align*}
where $log^{+}(x)=log(max\lbrace1,x\rbrace)$. Then, it follows from the strong law of large numbers that the Lyapunov exponent of the i.i.d. sequence $(J_{t+(n-1)\tau,t+n\tau})_{n\in\mathbb{N}}$ is strictly negative
\begin{align*}
\gamma&=\lim_{n\rightarrow\infty}\frac{1}{n}log||J_{t,t+\tau}\cdots J_{t+(n-1)\tau,t+n\tau}||_{P,r}\\
&\leq \lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}log||J_{t+(i-1)\tau,t+i\tau}||_{P,r}<0
\end{align*}
Therefore, as shown by Bougerol and Picard \cite{b3}, the sum $\mathbf{U}_{m}^{(t)}$ for all $t\in[0,\tau)$ is converges almost surely (a.s.).

Since the eigenvalues of the matrix $B$, $\eta_{1}, \cdots, \eta_{q}$, are distinct, we define the matrix $P$ as follows
\begin{align}\label{Th3.2(5.15)}
P:= \begin{pmatrix}
1 & \cdots & 1\\
\eta_{1} & \cdots & \eta_{q}\\
\vdots & \cdots & \vdots \\
\eta_{1}^{q-1} & \cdots & \eta_{q}^{q-1}
\end{pmatrix}.
\end{align}
So, the matrix $C:=P^{-1}BP$ is the diagonal matrix with entries $\eta_{1}, \cdots, \eta_{q}$ on the diagonal. Using the fact that $P$ is a invertible matrix, it follows from the exponential matrix definition that, for $t\geq0$,
\begin{align*}
e^{Ct}=e^{P^{-1}BtP}&=I+P^{-1}BtP+\frac{(P^{-1}BtP)^{2}}{2!}+\cdots\\
&=P^{-1}\big[I+Bt+\frac{(Bt)^{2}}{2!}+\cdots\big]P=P^{-1}e^{Bt}P.
\end{align*}
Hence, $P^{-1}e^{Bt}P=e^{Ct}$ is the diagonal matrix with entries $e^{\eta_{1}t}, \cdots, e^{\eta_{q}t}$ on the diagonal. Since the eigenvalues of the matrix B have strictly negative real parts, it follows from the Definition 2.5 that, for $r\in[1,\infty]$,
\begin{align}\label{Th3.2(5.16)}
||e^{Bt}||_{P,r}=||Pe^{Ct}P^{-1}||_{P,r}=||P^{-1}Pe^{Ct}P^{-1}P||_{r}=||e^{Ct}||_{r}\leq e^{\eta t}.
\end{align}
By denoting
\begin{align}\nonumber
C_{i}&:=(I+Z_{i}^{2}\mathbf{e}\mathbf{a}^{\prime})e^{B(\Upsilon_{_{i}}-\Upsilon_{_{i-1}})},\\ \label{Th3.2(5.17)}
\mathbf{D}_{i}&:=\alpha_{0}Z_{i}^{2}\mathbf{e},\qquad i\in\mathbb{N},
\end{align}
in (\ref{Th3.1(5.11)}) we have that
\begin{align*}
J_{t,t+\tau}&=e^{B(t+\tau-\Upsilon_{_{N(t+\tau)}})}C_{_{N(t+\tau)}}\cdots C_{_{N(t)+2}}
(I+Z_{_{N(t)+1}}^{2}\mathbf{e}\mathbf{a}^{\prime})e^{B(\Upsilon_{_{N(t)+1}}-t)}\\
&=e^{B(t+\tau-\Upsilon_{_{N(t+\tau)}})}C_{_{N(t+\tau)}}\cdots C_{_{N(t)+2}}C_{_{N(t)+1}}e^{B(\Upsilon_{_{N(t)}}-t)}\\
&=e^{B(t+\tau-\Upsilon_{_{N(t+\tau)}})}\Big[\prod_{i=0}^{N(t+\tau)-N(t)-1}C_{_{N(t+\tau)-i}}\Big]e^{B(\Upsilon_{_{N(t)}}-t)}.
\end{align*}
So, by (\ref{Th3.2(5.16)}) we have that
\begin{align}\label{Th3.2(5.18)}
||J_{t,t+\tau}||_{P,r}\leq e^{\eta(t+\tau-\Upsilon_{_{N(t+\tau)}})}\Big[\prod_{i=0}^{N(t+\tau)-N(t)-1}
||C_{_{N(t+\tau)-i}}||_{P,r}\Big]e^{\eta(\Upsilon_{_{N(t)}}-t)}.
\end{align}
As (\ref{Th3.2(5.16)}) and (\ref{Th3.2(5.17)}) leads to
\begin{align}\nonumber
||C_{i}||_{P,r}&\leq (1+Z_{i}^{2}||\mathbf{e}\mathbf{a}^{\prime}||_{P,r})e^{\eta(\Upsilon_{i}-\Upsilon_{i-1})}\\\label{Th3.2(5.19)}
||\mathbf{D}_{i}||_{P,r}&=\alpha_{0}Z_{i}^{2}||\mathbf{e}||_{P,r},
\end{align}
so, (\ref{Th3.2(5.18)}) reads as
\begin{align*}
||J_{t,t+\tau}||_{P,r}&\leq e^{\eta(t+\tau-\Upsilon_{_{N(t+\tau)}})}\Big[\prod_{i=0}^{N(t+\tau)-N(t)-1}
(1+Z_{_{N(t+\tau)-i}}^{2}
||\mathbf{e}\mathbf{a}^{\prime}||_{P,r})e^{\eta(\Upsilon_{_{N(t+\tau)-i}}-\Upsilon_{_{N(t+\tau)-i-1})}}\Big]\\
&\quad\times e^{\eta(\Upsilon_{_{N(t)}}-t)}.
\end{align*}
Therefore
\begin{align}\label{Th3.2(5.20)}
log||J_{t,t+\tau}||_{P,r}\leq \eta\tau+\sum_{i=0}^{N(t+\tau)-N(t)-1}log(1+Z_{_{N(t+\tau)-i}}^{2}||\mathbf{e}\mathbf{a}^{\prime}||_{P,r}).
\end{align}
So, it follows from (\ref{equ3.2}) and (\ref{Th3.2(5.20)}) that
\begin{align*}
&E\Big(log||J_{t,t+\tau}||_{P,r}\Big)\leq \eta\tau+\Big[\sum_{n,m=0}^{\infty}\sum_{i=0}^{n-1}E\Big(log\big(1+Z_{m-i}^{2}
||\mathbf{e}\mathbf{a}^{\prime}||_{P,r}\big)\Big)\\
&\hspace{5cm}\times P\big(N(t+\tau)-N(t)=n, N(t+\tau)=m\big)\Big]\\
&\qquad<\eta\tau+\sum_{n,m=0}^{\infty}n\times\frac{-\eta\tau}{\Lambda(t+\tau)-\Lambda(t)}P\big(N(t+\tau)-N(t)=n, N(t+\tau)=m\big)\\
&\qquad=\eta\tau+\big(\Lambda(t+\tau)-\Lambda(t)\big)\frac{-\eta\tau}{\Lambda(t+\tau)-\Lambda(t)}=0
\end{align*}
To show that $E\big(log^{+}||\mathbf{K}_{t,t+\tau}||_{P,r}\big)<\infty$, observe from (\ref{Th3.1(5.11)}) and (\ref{Th3.2(5.17)}) that
\begin{align*}
\mathbf{K}_{t,t+\tau}=e^{B(t+\tau-\Upsilon_{_{N(t+\tau)}})}\Big[\mathbf{D}_{_{N(t+\tau)}}+\sum_{i=0}^{N(t+\tau)-N(t)-2}
C_{_{N(t+\tau)}}\cdots C_{_{N(t+\tau)-i}}\mathbf{D}_{_{N(t+\tau)-i-1}}\Big].
\end{align*}
So, by (\ref{Th3.2(5.16)}) we have that
\begin{align}\nonumber
||\mathbf{K}_{t,t+\tau}||_{P,r}\leq e^{\eta(t+\tau-\Upsilon_{_{N(t+\tau)}})}&\Big[||\mathbf{D}_{_{N(t+\tau)}}||_{P,r}+
\Big(\sum_{i=0}^{N(t+\tau)-N(t)-2}||C_{_{N(t+\tau)}}||_{P,r}\cdots||C_{_{N(t+\tau)-i}}||_{p,r}\\\label{Th3.2(5.21)}
&\quad\times||\mathbf{D}_{_{N(t+\tau)-i-1}}||_{P,r}\Big)\Big].
\end{align}
The eigenvalues of the matrix B have strictly negative real parts, say $\mathcal{R}e(\eta_{i})<0$. So, it follows that $\eta:=\max_{i=1,\cdots,q}\mathcal{R}e(\eta_{i})<0$ and for $t\geq0$
\begin{align*}
e^{\eta t}\leq1.
\end{align*}
From this and (\ref{Th3.2(5.19)}) we have that
\begin{align*}
||C_{i}||_{P,r}\leq(1+Z_{i}^{2}||\mathbf{e}\mathbf{a}^{\prime}||_{P,r}).
\end{align*}
Thus, (\ref{Th3.2(5.21)}) reads as
\begin{align*}
||\mathbf{K}_{t,t+\tau}||_{P,r}\leq \alpha_{0}Z_{_{N(t+\tau)}}^{2}||\mathbf{e}||_{P,r}
+&\Big(\sum_{i=0}^{N(t+\tau)- N(t)-2}\big(1+Z_{_{N(t+\tau)}}^{2}||\mathbf{e}\mathbf{a}^{\prime}||_{P,r}\big)\times\\
&\quad\cdots\times\big(1+Z_{_{N(t+\tau)-i}}^{2}||\mathbf{e}\mathbf{a}^{\prime}||_{P,r}\big)
\times\alpha_{0}Z_{_{N(t+\tau)-i-1}}^{2}||\mathbf{e}||_{P,r}\Big).
\end{align*}
Since $E\big(Z_{i}^{2}\big)<\infty$ and $E\big(N(t+\tau)-N(t)\big)=\Lambda(t+\tau)-\Lambda(t)<\infty,$ it follows that $E\big(||\mathbf{K}_{t,t+\tau}||_{P,r}\big)<\infty$. So, by Jensen's inequality, we have that
\begin{align*}
E\Big(log^{+}||\mathbf{K}_{t,t+\tau}||_{P,r}\Big)=E\Big(log\big(max\lbrace1,||\mathbf{K}_{t,t+\tau}||_{P,r}\rbrace\big)\Big)<\infty.
\end{align*}
Now, since $E\Big(log||J_{t,t+\tau}||_{P,r}\Big)<0$ and $E\Big(log^{+}||\mathbf{K}_{t,t+\tau}||_{P,r}\Big)<\infty$, it follows from the general theory of random recurrence equations (see \cite{b3}) that $H_{m}^{(t)}$, in (\ref{Th3.2(5.14)}), converges a.s. to 0 as $m\rightarrow\infty$ and that $\mathbf{U}_{m}^{(t)}$ converges a.s. absolutely to some random vector $\mathbf{U}^{(t)}$ as $m\rightarrow\infty$, for fixed $t\in[0,\tau)$. Using the fact that the process $(\mathbf{Y}_{t})_{t\geq0}$ has c\`adl\`ag paths, it follows that $||\mathbf{Y}_{t}||_{P,r}$ is a.s. finite for fixed $t\in[0,\tau)$. So
\begin{align*}
\lim_{m\rightarrow\infty}||H_{m}^{(t)}\mathbf{Y}_{t}||_{P,r}=0\qquad\textnormal{a.s.,}
\end{align*}
and it follows the $\mathbf{Y}_{t+m\tau}$ converges in distribution to $\mathbf{U}^{(t)}$ as $m\rightarrow\infty$, for fixed $t\in[0,\tau)$.\\
It remains to show that $\mathbf{U}^{(t)}$ satisfies (\ref{equ3.3}) for fixed $t\in[0,\tau)$. It follows from (\ref{Th3.2(5.12)}) and the part (c) of Theorem 3.1 that $\mathbf{Y}_{t+(m-1)\tau}$ is independent of $(J_{t+(m-1)\tau,t+m\tau}$, $\mathbf{K}_{t+(m-1)\tau,t+m\tau})$. So, by the fact that $(J_{t+(m-1)\tau,t+m\tau},\mathbf{K}_{t+(m-1)\tau,t+m\tau})$ and $(J_{t,t+\tau},\mathbf{K}_{t,t+\tau})$ are i.i.d., for any $m\geq2$, and that $\mathbf{Y}_{t+m\tau}$ converges in distribution to $\mathbf{U}^{(t)}$ as $m\rightarrow\infty$, we have that
\begin{align}\label{Th3.2(5.22)}
\big(J_{t+(m-1)\tau,t+m\tau},\mathbf{K}_{t+(m-1)\tau,t+m\tau},\mathbf{Y}_{t+(m-1)\tau}\big)\overset{d}{\longrightarrow}
\big(J_{t,t+\tau},\mathbf{K}_{t,t+\tau},\mathbf{U}^{(t)}\big)\quad \textnormal{as}\quad m\rightarrow\infty,
\end{align}
where $\mathbf{U}^{(t)}$ is independent of $(J_{t,t+\tau},\mathbf{K}_{t,t+\tau})$ and $\overset{d}{\longrightarrow}$ denotes the convergence in distribution. Hence, by (\ref{Th3.2(5.12)}) and (\ref{Th3.2(5.22)}), it follows (\ref{equ3.3}).
\vspace{7mm}
\\
\textbf{P4: Proof of Corollary 3.3}
\vspace{2mm}

\textbf{Proof ($a$):} By Theorem 3.2, $\mathbf{U}^{(t)}$ is the long run behavior of $\mathbf{Y}_{t+m\tau}$ for fixed $t\in[0,\tau)$ and large $m$, so that $\mathbf{U}^{(t)}\overset{d}{=}J_{t,t+\tau}\mathbf{U}^{(t)}+\mathbf{K}_{t,t+\tau}$ and $\mathbf{U}^{(t)}$ is independent of $(J_{t,t+\tau},\mathbf{K}_{t,t+\tau})$. Since, by the part (b) of Theorem 3.1, $(J_{t,t+\tau}, \mathbf{K}_{t,t+\tau})$ and $(J_{t+(k-1)\tau,t+k\tau},\mathbf{K}_{t+(k-1)\tau,t+k\tau})$, for $k\geq2$, are i.i.d., we have that
\begin{align}\nonumber
\mathbf{U}^{(t)}&\overset{d}{=}J_{t,t+\tau}\mathbf{U}^{(t)}+\mathbf{K}_{t,t+\tau}\\ \label{Coro3.3(5.23)}
&\overset{d}{=}J_{t+(k-1)\tau,t+k\tau}\mathbf{U}^{(t)}+\mathbf{K}_{t+(k-1)\tau,t+k\tau}.
\end{align}
It also follows from the parts ($a$) and (c) of Theorem 3.1 that, for fixed $t\in[0,\tau)$ and $k\in\mathbb{N}$,
\begin{align}\label{Coro3.3(5.24)}
\mathbf{Y}_{t+k\tau}=J_{t+(k-1)\tau,t+k\tau}\mathbf{Y}_{t+(k-1)\tau}+\mathbf{K}_{t+(k-1)\tau,t+k\tau}
\end{align}
and that $\mathbf{Y}_{t+(k-1)\tau}$ is independent of $(J_{t+(k-1)\tau,t+k\tau},\mathbf{K}_{t+(k-1)\tau,t+k\tau})$. Thus if $\mathbf{Y}_{t}\overset{d}{=}\mathbf{U}^{(t)}$ for $t\in[0,\tau)$, then (\ref{Coro3.3(5.23)}) and (\ref{Coro3.3(5.24)}) imply that
$\mathbf{Y}_{t+\tau}\overset{d}{=}\mathbf{U}^{(t)}$. Further, it follows by induction on $k\in\mathbb{N}$ that
\begin{align}\label{Coro3.3(5.25)}
\mathbf{Y}_{t+k\tau}\overset{d}{=}\mathbf{U}^{(t)}.
\end{align}
We show that the process $(\mathbf{Y}_{t})_{t\geq0}$ is strictly periodically stationary with period $\tau$. If $n=1$, then
for any $t_{_{1}}\geq0$ there exist $t\in[0,\tau)$ and $k\in\mathbb{N}^{0}$ such that $t_{_{1}}=t+k\tau$. So, by (\ref{Coro3.3(5.25)})
we have that
\begin{align*}
\mathbf{Y}_{t_{_{1}}}\overset{d}{=}\mathbf{Y}_{t_{_{1}}+\tau}.
\end{align*}
Now, let $n\geq2$ and $0\leq t_{_{1}}\leq t_{_{2}}\leq\cdots\leq t_{_{n}}$. Since, by the part ($a$) of Theorem 3.1,
\begin{align}\label{Coro3.3(5.26)}
\mathbf{Y}_{t_{_{n}}}=J_{t_{_{n-1}},t_{_{n}}}\mathbf{Y}_{t_{_{n-1}}}+\mathbf{K}_{t_{_{n-1}},t_{_{n}}},
\end{align}
it follows that $(\mathbf{Y}_{t})_{t\geq0}$ is a Markov process. So, by the fact that $(J_{t_{_{n-1}},t_{_{n}}},\mathbf{K}_{t_{_{n-1}},t_{_{n}}})\overset{d}{=}(J_{t_{_{n-1}}+\tau,t_{_{n}}+\tau}, \mathbf{K}_{t_{_{n-1}}+\tau,t_{_{n}}+\tau})$, we have that
\begin{align}\nonumber
\big(\mathbf{Y}_{t_{_{n}}}|\mathbf{Y}_{t_{_{n-1}}}=\mathbf{y}_{n-1},\cdots, \mathbf{Y}_{t_{_{1}}}=\mathbf{y}_{1}\big)
&=J_{t_{_{n-1}},t_{_{n}}}\mathbf{y}_{n-1}+\mathbf{K}_{t_{_{n-1}},t_{_{n}}}\\ \nonumber
&\overset{d}{=}J_{t_{_{n-1}}+\tau,t_{_{n}}+\tau}\mathbf{y}_{n-1}+\mathbf{K}_{t_{_{n-1}}+\tau,t_{_{n}}+\tau}\\ \label{Coro3.3(5.26.2)}
&=\big(\mathbf{Y}_{t_{_{n}}+\tau}|\mathbf{Y}_{t_{_{n-1}}+\tau}=\mathbf{y}_{n-1},\cdots,
\mathbf{Y}_{t_{_{1}}+\tau}=\mathbf{y}_{1}\big).
\end{align}
Therefore, an induction argument for $n=2, 3, \cdots$ and (\ref{Coro3.3(5.26.2)}) imply
\begin{align*}
\big(\mathbf{Y}_{t_{_{1}}}, \mathbf{Y}_{t_{_{2}}}, \cdots, \mathbf{Y}_{t_{_{n}}}\big)\overset{d}
{=}\big(\mathbf{Y}_{t_{_{1}}+\tau}, \mathbf{Y}_{t_{_{2}}+\tau}, \cdots, \mathbf{Y}_{t_{_{n}}+\tau}\big).
\end{align*}
Hence, by (\ref{equ2.6}) it follows that
\begin{align}\label{Coro3.3(5.27)}
\big(V_{t_{_{1}}}, V_{t_{_{2}}}, \cdots, V_{t_{_{n}}}\big)\overset{d}
{=}\big(V_{t_{_{1}}+\tau}, V_{t_{_{2}}+\tau}, \cdots, V_{t_{_{n}}+\tau}\big).
\end{align}

\textbf{Proof (b):} By (\ref{equ2.6}) and (\ref{Th3.1(5.11)}) we have that
\begin{align}\nonumber
V_{u}&=\alpha_{0}+\mathbf{a}^{\prime}\mathbf{Y}_{u^{-}}\\ \label{Coro3.3(5.28)}
&=\alpha_{0}+\mathbf{a}^{\prime}J_{0,u^{-}}\mathbf{Y}_{0}+\mathbf{a}^{\prime}\mathbf{K}_{0,u^{-}}\quad \forall u\geq0.
\end{align}
As $(J_{0,u^{-}},\mathbf{K}_{0,u^{-}})$ are constructed from segments of the driving process $S_{t}$ in the interval $(0,u^{-}]$, see Theorem 3.1 (c), and that $\mathbf{Y}_{0}$ is independent of $(S_{t})_{t\geq0}$, assumed by (\ref{equ2.7}), it follows that $V_{u}$ and $dS_{u}=S_{u+du}-S_{u}$ are independent. So, (\ref{Coro3.3(5.27)}) and semi-L\'evy structure of $S_{t}$ imply that $V_{u}$ and $dS_{u}=S_{u+du}-S_{u}$ both are periodic with the same period $\tau$, say
\begin{align}\label{Coro3.3(5.29)}
\sqrt{V_{u}}dS_{u}\overset{d}{=}\sqrt{V_{u+\tau}}dS_{u+\tau}.
\end{align}
Thus it follows from (\ref{Coro3.3(5.29)}) that
\begin{align}\nonumber
E\big(G_{t}^{(l)}\big)&=E\Big(\int_{t}^{t+l}\sqrt{V_{u}}dS_{u}\Big)\\ \label{Coro3.3(5.30)}
&=E\Big(\int_{t}^{t+l}\sqrt{V_{u+\tau}}dS_{u+\tau}\Big)=E\big(G_{t+\tau}^{(l)}\big)
\end{align}
and
\begin{align}\nonumber
E\big(G_{t}^{(l)}G_{t+h}^{(l)}\big)&=
E\Big(\int_{t}^{t+l}\sqrt{V_{u}}dS_{u}\int_{t+h}^{t+h+l}\sqrt{V_{u}}dS_{u}\Big)\\ \label{Coro3.3(5.31)}
&=E\Big(\int_{t}^{t+l}\sqrt{V_{u+\tau}}dS_{u+\tau}\int_{t+h}^{t+h+l}\sqrt{V_{u+\tau}}dS_{u+\tau}\Big)
=E\big(G_{t+\tau}^{(l)}G_{t+h+\tau}^{(l)}\big).
\end{align}
Hence, from (\ref{Coro3.3(5.30)}) and (\ref{Coro3.3(5.31)}) we have that
\begin{align*}
cov\big(G_{t}^{(l)},G_{t+h}^{(l)}\big)=cov\big(G_{t+\tau}^{(l)},G_{t+h+\tau}^{(l)}\big).
\end{align*}
\\
\textbf{P5: Proof of Corollary 3.4}
\vspace{2mm}

\textbf{Proof ($a$):} By the part (b) of Corollary 3.3, it follows that $V_{u}$ and $dS_{u}=S_{u+du}-S_{u}$ are independent for any $u\geq0$. So, $E(dS_{u})=E(S_{u+du})-E(S_{u})=0$ and
\begin{align*}\nonumber
E\big(G_{t}^{(l)}\big)&=E\Big(\int_{t}^{t+l}\sqrt{V_{u}}dS_{u}\Big)\\
&=\int_{t}^{t+l}E(\sqrt{V_{u}})E(dS_{u})=0.
\end{align*}
Since, by Definition 2.2, the jump sizes $Z_{n}$ have finite second order moment it is easy to see that $E(S_{t})^{2}<\infty$. So, it follows from the proposition 3.17 of Cont and Tankov \cite{c} that $S_{t}$ is a martingale. Thus, from It\^o isometry for square integrable martingales as integrators (e.g. Rogers and Williams \cite{r1}, IV 27) follows
\begin{align*}\nonumber
E\big(G_{t}^{(l)}G_{t+h}^{(l)}\big)=E\Big(\int_{0}^{t+h+l}V_{u}I_{(t,t+l]}(u)I_{(t+h,t+h+l]}(u)d[S,S]_{u}\Big)=0
\end{align*}
for $h\geq l$. Hence, $cov\big(G_{t}^{(l)},G_{t+h}^{(l)}\big)=0$.
\vspace{3mm}

\textbf{Proof (b):} By (\ref{Coro3.3(5.29)}) we have that
\begin{align}\nonumber
E\Big((G_{t}^{(l)})^{2}\Big)&=E\Big(\int_{t}^{t+l}\sqrt{V_{u}}dS_{u}\Big)^{2}\\ \label{Coro3.4(5.32)}
&=E\Big(\int_{t}^{t+l}\sqrt{V_{u+\tau}}dS_{u+\tau}\Big)^{2}=E\Big((G_{t+\tau}^{(l)})^{2}\Big)
\end{align}
and
\begin{align}\nonumber
E\Big((G_{t}^{(l)})^{2}(G_{t+h}^{(l)})^{2}\Big)&=
E\Big(\big(\int_{t}^{t+l}\sqrt{V_{u}}dS_{u}\big)^{2}\big(\int_{t+h}^{t+h+l}\sqrt{V_{u}}dS_{u}\big)^{2}\Big)\\ \nonumber
&=E\Big(\big(\int_{t}^{t+l}\sqrt{V_{u+\tau}}dS_{u+\tau}\big)^{2}\big(\int_{t+h}^{t+h+l}\sqrt{V_{u+\tau}}dS_{u+\tau}\big)^{2}\Big)\\
\label{Coro3.4(5.33)}
&=E\Big((G_{t+\tau}^{(l)})^{2}(G_{t+h+\tau}^{(l)})^{2}\Big).
\end{align}
Hence, from (\ref{Coro3.4(5.32)}) and (\ref{Coro3.4(5.33)}) it follows that
\begin{align*}
cov\Big((G_{t}^{(l)})^{2},(G_{t+h}^{(l)})^{2}\Big)=cov\Big((G_{t+\tau}^{(l)})^{2},(G_{t+h+\tau}^{(l)})^{2}\Big).
\end{align*}
\\
\textbf{P6: Proof of Theorem 3.5}\\
Suppose that (\ref{equ3.4}) and (\ref{equ3.5}) both hold. The following proof of Theorem 3.1, for any $t\geq0$ it follows that $t\in[\Upsilon_{_{N(t)}},\Upsilon_{_{N(t)+1}})$. So, we have by (\ref{Th3.1(5.6)}) that
\begin{align}\label{Th3.5(5.34)}
\mathbf{Y}_{t}=e^{B(t-\Upsilon_{_{N(t)}})}\mathbf{Y}_{\Upsilon_{_{N(t)}}}.
\end{align}
By (\ref{equ2.6}) and replacing from $N(t)$ in $n$, (\ref{Th3.1(5.3)}) reads as
\begin{align}\label{Th3.5(5.35)}
\mathbf{Y}_{\Upsilon_{_{N(t)}}}=\mathbf{Y}_{\Upsilon^{-}_{_{N(t)}}}+\mathbf{e}V_{\Upsilon_{_{N(t)}}}Z_{_{N(t)}}^{2}
\end{align}
and (\ref{Th3.1(5.4)}) as
\begin{align}\label{Th3.5(5.36)}
\mathbf{Y}_{\Upsilon^{-}_{_{N(t)}}}=e^{B(\Upsilon_{_{N(t)}}-\Upsilon_{_{N(t)-1}})}\mathbf{Y}_{\Upsilon_{_{N(t)-1}}}.
\end{align}
Replacing (\ref{Th3.5(5.36)}) in (\ref{Th3.5(5.35)}) implies that
\begin{align}\label{Th3.5(5.37)}
\mathbf{Y}_{\Upsilon_{_{N(t)}}}=e^{B(\Upsilon_{_{N(t)}}-\Upsilon_{_{N(t)-1}})}\mathbf{Y}_{\Upsilon_{_{N(t)-1}}}
+\mathbf{e}V_{\Upsilon_{_{N(t)}}}Z_{_{N(t)}}^{2}.
\end{align}
So, it follows from (\ref{Th3.5(5.34)}) and (\ref{Th3.5(5.37)}) that
\begin{align}\nonumber
\mathbf{Y}_{t}&=e^{B(t-\Upsilon_{_{N(t)}})}\Big[e^{B(\Upsilon_{_{N(t)}}-\Upsilon_{_{N(t)-1}})}\mathbf{Y}_{\Upsilon_{_{N(t)-1}}}
+\mathbf{e}V_{\Upsilon_{_{N(t)}}}Z_{_{N(t)}}^{2}\Big]\\ \label{Th3.5(5.38)}
&=e^{B(t-\Upsilon_{_{N(t)-1}})}\mathbf{Y}_{\Upsilon_{_{N(t)-1}}}
+e^{B(t-\Upsilon_{_{N(t)}})}\mathbf{e}V_{\Upsilon_{_{N(t)}}}Z_{_{N(t)}}^{2}.
\end{align}
By successive replacement from (\ref{Th3.5(5.37)}) in (\ref{Th3.5(5.38)}) we have that
\begin{align*}
\mathbf{Y}_{t}=e^{Bt}\mathbf{Y}_{0}+\sum_{i=1}^{N(t)}e^{B(t-\Upsilon_{_{i}})}\mathbf{e}V_{\Upsilon_{_{i}}}Z_{_{i}}^{2},
\qquad t\geq0.
\end{align*}
So, from this and (\ref{equ3.5}) follows that
\begin{align}\label{Th3.5(5.39)}
\mathbf{a}^{\prime}\mathbf{Y}_{t}&=\mathbf{a}^{\prime} e^{Bt}\mathbf{Y}_{0}
+\sum_{i=1}^{N(t)}\mathbf{a}^{\prime}e^{B(t-\Upsilon_{_{i}})}\mathbf{e}V_{\Upsilon_{_{i}}}Z_{_{i}}^{2}\\ \label{Th3.5(5.40)}
&\geq\gamma+\sum_{i=1}^{N(t)}\mathbf{a}^{\prime}e^{B(t-\Upsilon_{_{i}})}\mathbf{e}V_{\Upsilon_{_{i}}}Z_{_{i}}^{2}.
\end{align}
By a similar method in Brockwell et al. \cite{b5}, we find by (\ref{equ2.6}) and (\ref{Th3.5(5.40)}) that $V_{t}=\alpha_{0}+\mathbf{a}^{\prime}\mathbf{Y}_{t^{-}}\geq\alpha_{0}+\gamma$ for $t\in[0,\Upsilon_{_{1}}]$. It also follows from (\ref{Th3.5(5.40)}), (\ref{equ3.5}) and (\ref{equ3.4}) that
\begin{align*}
V_{\Upsilon^{+}_{_{1}}}=\alpha_{0}+\mathbf{a}^{\prime}\mathbf{Y}_{\Upsilon_{_{1}}}
&=\alpha_{0}+\mathbf{a}^{\prime}e^{B\Upsilon_{_{1}}}\mathbf{Y}_{0}+
\mathbf{a}^{\prime}\mathbf{e}V_{\Upsilon_{_{1}}}Z_{_{1}}^{2}\\
&\geq\alpha_{0}+\gamma.
\end{align*}
So, by induction one can easily verify $V_{t}\geq\alpha_{0}+\gamma\geq0$ for any $t\geq0$.
\vspace{2mm}

To prove the converse, suppose first that (\ref{equ3.5}) fails. Then, it follows from (\ref{Th3.5(5.39)}) that, for  $t\in[0,\Upsilon_{_{1}}]$,
\begin{align}\label{Th3.5(5.41)}
V_{t}=\alpha_{0}+\mathbf{a}^{\prime}\mathbf{Y}_{t^{-}}<\alpha_{0}+\gamma.
\end{align}
Since $P(\Upsilon_{_{1}}>0)>0$, it follows from (\ref{Th3.5(5.41)}) and $\gamma=-\alpha_{0}$ that $P(V_{t}<0)>0$ for $t\in[0,\Upsilon_{_{1}}]$. Now assume that (\ref{equ3.5}) holds with $\gamma>-\alpha_{0}$, but (\ref{equ3.4}) fails. Suppose that the support of the jumps distribution $Z_{i}$ is unbounded. Let $(t_{1},t_{2})\subset(0,\infty)$ be an interval such that $\mathbf{a}^{\prime}e^{Bt}\mathbf{e}<c$ for any $t\in(t_{1},t_{2})$ and for some $c<0$. By (\ref{equ2.6}), (\ref{Th3.5(5.39)}) and (\ref{equ3.5}) we have that
\begin{align}\nonumber
V_{\Upsilon_{_{1}}}=\alpha_{0}+\mathbf{a}^{\prime}\mathbf{Y}_{\Upsilon_{_{1}}^{-}}&=\alpha_{0}
+\mathbf{a}^{\prime}e^{B\Upsilon_{_{1}}}\mathbf{Y}_{0}\\
\label{Th3.5(5.42)}
&\geq\alpha_{0}+\gamma.
\end{align}
It follows from (\ref{Th3.5(5.42)}) and assumption $\gamma>-\alpha_{0}$, that $P(V_{\Upsilon_{_{1}}}\geq\alpha_{0}+\gamma>0)=1$. So, one can easily verify that the set
\begin{align*}\nonumber
A:=\lbrace \Upsilon_{_{1}}<t_{3}<\Upsilon_{_{2}}, t_{3}-\Upsilon_{_{1}}\in(t_{1}, t_{2}), V_{\Upsilon_{_{1}}}>0\rbrace
\end{align*}
has positive probability for $t_{3}>t_{2}$. On $A$, by (\ref{Th3.5(5.39)}) we have that
\begin{align}\label{Th3.5(5.43)}
V_{t_{3}}=\alpha_{0}+\mathbf{a}^{\prime}e^{Bt_{3}}\mathbf{Y}_{0}+\mathbf{a}^{\prime}e^{B(t_{3}-\Upsilon_{_{1}})}\mathbf{e}V_{\Upsilon_{_{1}}}Z_{1}^{2}.
\end{align}
Since $\mathbf{a}^{\prime}e^{B(t_{3}-\Upsilon_{_{1}})}\mathbf{e}<c<0$, by choosing $Z_{1}$ large enough it follows from (\ref{Th3.5(5.43)}) that $P(V_{t_{3}}<0)>0$.

redpajama_set_name: RedPajamaArXiv


With a slogan of 'you break it, we fix it', the bike shop team solves the perennial problem of basic bike repairs as a service to students, staff and faculty. The team also carries a small line of replacement parts and partners with The Cottages at Berry to offer bike rentals to campus guests. Partnerships with local area bike shops are in place where customers are referred for purchases and more complicated repairs.
Viking Cycle Works will now be offering a maintenance plan where students can pay a one-time fee and receive services for a full year. Get a year of complete bike maintenance for $150, two years for $250, or four years for $350.
We now offer daily bike rentals starting at $10. Use this service to explore campus with friends, spend time with family, or temporarily use a bike while we are repairing yours.
Viking Cycle Works provides bicycle storage during the summer break. When you store your bike with us you can have the peace of mind that it will be protected and ready to go when you get back in the fall. You won't have to worry about it being collected and impounded while you are away! Also, your bike will be stored in a covered area ensuring that it will not be damaged by rain. We also offer tune-ups for your bike for $25.
Monday 3 p.m. - 5 p.m.
Tuesday 1 p.m. - 5 p.m.


If you have a plumbing problem, it could negatively affect the home and all the stuff you have in it. These tips below can help you remedy your plumbing problems.
To deal with a frozen pipe, turn on the faucet closest to the pipe. This gives the water a place to exit as the pipe defrosts. This reduces pressure to keep the pipe from bursting, which could prevent a big mess in your home.
Your septic tank should be pumped out at least once every five years if you want it to stay in optimum working order. Having your septic pumped will keep sediments from settling to the bottom of the septic tank and causing the tank to malfunction or backup. Hiring a professional to pump out your septic tank is costly, but it is far preferable to the damages that could be incurred if the tank becomes backed-up or inoperative.
TIP! You should make sure that the holes are not stopped up on the overflow. Sinks use overflow holes if they start to overfill, and this isn't usually a problem, but it can be.
Toilet discs and odor removers should never be used in your toilet. Either kind of toilet tablet might break down obnoxious smells in your toilet, but they might also break down the rubber parts, resulting in malfunction.
TIP! Make sure to clean out the dryers lint trap. This could prevent you from a tons of trouble, and it can also prevent fires.
You can save a lot of money in your energy bills by putting in efficient shower heads. Most of the hot water we use is in the shower. Install energy saving faucets throughout your home and you can recapture your cost within the year.
TIP! Prevention is necessary when you want to keep plumbing bills down. Clogs are one of the main reasons people have to call a plumber.
To limit toilet problems, don't use it as a trash can. Don't flush anything that is unlikely to dissolve and wind up clogging pipes, such as diapers, tissues, paper towels and cotton balls. Also, be conservative in your use of toilet paper.
TIP! Some people say that when you use the garbage disposal you don't have to run the water. Many people make the mistake of thinking that a disposal can handle anything as long as they have water running when they turn it on.
If you take care of your plumbing, you will have lower plumbing bills. One of the major repairs is clogs. Drains can be clogged by hair. You can stop this by using a screen or drain cover that is designed to keep hair out of your drain and pipes. It's a lot easier to take hair out of a screen than to pull it from a pipe.
TIP! Try not to use drain cleaners too often, it at all. The chemicals in these type of cleaners are corrosive, and they could damage your pipes by using them too much.
You can take the stress and expense out of major plumbing problems by addressing issues while they are still small. Asking around and doing a bit of homework may get you some positive results. Hopefully this article has provided you with some useful advice for the next time you face another plumbing problem.
Most people want to know about click for source, but do not always know how to go about it on there own. This material will give you some great info about click for source. It is up to you to apply the ideas you have reviewed.


If you, a loved one or a friend served in military uniform from World War II to present, you are eligible to participate for a FREE trip to Washington D.C. to see the memorials built to honor the sacrifices you made and to the many whom were lost protecting our right to freedom.
ONLINE ELECTRONIC APPLICATIONS NOW AVAILABLE!!!
Veteran applications are taking in the order they are submitted, the order of war/conflict they served and medical situations such as being terminal. Example: A Vietnam Veteran or later could submit an application and they would be entered into our database, however as Korean, Cold War or World War II applications continue to come in, they will be placed ahead based on their war/conflict time frame. Veterans of Vietnam and later should consider applying now and getting into our database, however it could be several years before your turn arrives. We ask for your patience and understanding.
Honor Flight Northeast Indiana provides four (4) flights each year. Two (2) in the spring and two (2) in the fall. We take applications all year long and add our veterans to our waiting list database.
If you or your veteran should have health condition that deteriorates and would require a doctors determination of terminal or similar status, please contact our offices immediately to update your status.
IF YOU PREFER A PRINTABLE PAPER APPLICATION TO BE US MAILED TO US, THOSE LINKS ARE DOWN BELOW.
We have veterans as well as family members of veterans wanting to verify service dates. These are the official service dates that we use as published by the D.O.D. and other government sources.
DOWNLOAD PAPER VETERAN APPLICATION - This is a PDF version that can be printed, manually filled out and mailed or scanned to us.
Guardians are individuals who accompany a Veteran on their trip to Washington D.C. for the day. Typically a guardian is a family member (not a spouse) or other loved one or friend. Here you will find the application to be a guardian. If you are a veteran that does not have a guardian, we are able to provide you with a guardian. PLEASE NOTE: At this time we do not require guardian only applications. If you have a veteran to whom you will be traveling with that meets the eligibility dates, then use this application and make sure to reference the veteran whom you will be escorting.
THIS IS NOT A "Volunteer to Help" APPLICATION.
DOWNLOAD PAPER GUARDIAN APPLICATION - This is a PDF version that can be printed, manually filled out and mailed or scanned to us.


BEWARE OF DAVID CHERNOFF AND HIS SALES ASSOCIATE JOHN AND OTHERS; This company is taking thousands of dollars from individuals promising them lead generation in the areas of mortgages and insurance. The company not only violates SANS do not call, but they constently over promise, take money and deliver 10% of the time. The owner will not see to it that you are satsified in any circumstance. Buyer beware. Big time con men.
Magic Jack fails to deliver!


Anyone who has ever tried to handle a legal matter on his own has probably made the acquaintance of a court clerk. These clerks are responsible for the administrative work involved in running municipal, county, state and federal court systems. Whether you want to file a legal complaint or just pay a fine, you'll most likely turn your paperwork or money over to a clerk.
The responsibilities of this position can vary greatly depending on the courts served, the clerk's level of experience, and the region in which he or she works. You may begin in an entry level position and work your way up from there. With education and experience, court clerks can advance to higher positions of responsibility.
Deputy clerk: Most court clerks begin as deputy clerks, also known as assistant court clerks. They perform a variety of administrative duties, including preparing and processing legal documents, correspondence, motions and orders, and providing customer service to the public, judicial officers, attorneys, and staff.
Court clerk: Deputy clerks can advance to the position of court clerks. Court clerk responsibilities are similar to those of deputy clerks, but they involve higher levels of responsibility and compensation.
Chief court clerk: Chief court clerks, also known as chief deputy clerks or chief clerks, are the highest level in the court clerk system. In some jurisdictions, the chief court clerk is an executive level position. Chief court clerks are responsible for all administrative and operational elements of the Clerk's Office. They often supervise other staff. This is effectively a management position.
At a minimum, court clerks possess a high school diploma or its equivalent. Experience counts for a lot on the upward-mobility ladder, but you may be able to at least get your foot in the door with no further education beyond high school. Some court systems require at least two years of college, and many jurisdictions prefer a bachelor's degree. A background in business or public administration, political science, criminal justice, law or a related field is helpful.
Salaries for court clerks vary according to jurisdiction, court, and position. Clerks working for the federal court system generally earn the highest salaries. Clerks who are just starting out with only a high school degree tend to earn the least. The median salary as of February 2019 was approximately $39,499, which means just as many clerks earn more than this as those who earn less.
Court clerks generally work in an office setting and may be required to sit or stand for long periods of time as they perform filing, copying, and administrative work. Clerks often must bend or stoop to retrieve files and lift boxes, files and other materials weighing up to 30 pounds or more.
Court clerks usually work a five-day, 40-hour week. Their hours may vary according to federal or state laws, jurisdictional rules and the orders of judges or others with the power to regulate hours of work. Most holidays are paid days off.
Clerks cannot dispense legal advice, although they're often asked to do so. This can be frustrating for the citizens they serve because they can explain how to file a document, but they can't detail the legal ramifications of doing so. This can be a fine line that is difficult to maintain. An entry-level clerk typically has much more contact with the public than a chief court clerk, who rarely or ever has reason to interact with the average litigant. The customer service aspect of this job can be particularly trying for those without experience in dealing with the worst human nature has to offer.
People don't go to courthouses because everything is great in their lives — they have problems, they want answers, and they may become angry and even abusive when they're told that only a lawyer can give them those answers. A thick skin is required.
Court Clerks may belong to professional associations, such as the Federal Court Clerks Association or the National Association for Court Management.


text:
Photogravure of Westward Ho
51 cm (20")
87 cm (34 1/4")
Royal North Devon Golf Club 1893 by Mayall & Co. Ltd.
A great image of the members of the 'Royal North Devon Golf Club, Westward Ho!' from 1893. The figures are all photographs superimposed onto a painted background of the course. In the margin is printed 'Permanent Autotype, Produced by Mayall & Co. Ltd. 73 Piccadilly'. The image appears very much photographic but is more of a photomechanical print rather than a photograph.
John Jabez Edwin Paisley Mayall (1813 - 1901) was an English photographer who in 1860 took the first carte-de-visite photographs of Queen Victoria.
taken from Royal North Devon website:-
Royal North Devon at Westward Ho! can rightly claim to be the cradle of English Golf. Founded in 1864, it is the oldest golf course in the country and is regarded as the St Andrews of the South. The golf course is as tough as any of the UK's more famous links layouts and has recently been placed in Golf World 's 'Top 100 Courses in the World' that a golfer 'must play'.
Not only is Royal North Devon a great course, it is also a place of huge historical importance. Inside the warmly welcoming clubhouse, the golf museum, honours boards and the clubs top competition trophies are worth a visit on their own and help to tell the story of the game and this remarkable club's unique place in it.
Above all, this is great, natural golf that is largely unchanged from 100 years ago when Harold Hilton et al pronounced Westward Ho! As England's No 1 competition golf course. RND is pure, raw exhilaration and ultimately the experience of playing a fast running links in a stiff breeze still tests the very best in the game.
Many of the great players of the last 100 years were regular competitors at RND but the greatest of them all is without doubt John Henry Taylor. Born in the village of Northam overlooking the course, he started his long association with the Club as a caddie boy and soon became an accomplished player. As every golf lover will know he went on to form one third of the great triumvirate, winning five Open Championships. Alongside Harry Vardon and James Braid, JH Taylor dominated the game for 30 years. The club honoured Taylor with a presidency in 1957 and his portrait and some of the clubs which he used to win his Open titles are proudly displayed in the clubhouse.
A look at the Club's honours boards, dating back to 1864, is like a 'Who's Who' of golfing greats. Listing many of the games founding fathers such as Horace Hutchinson, Harold Hilton, Herbert Fowler, Capt Molesworth, etc. Although golf has changed hugely in the last century and continues to develop and grow, a great links will always stand the test of time. Royal North Devon is just that, a great links and as far removed as possible from the manicured and manufactured championship courses we see every day on television.
Good image but with foxing
Champions, Hookers & Dew Sweepers No.47
Enquire about Photogravure of Westward Ho
Dealer Reference: 26635
Original Watercolour, Playing The 18th Green, Wentworth West Course By Arthur Weaver
Humorous Golf Print
Henry Callender ESQ, Golf Mezzotint

redpajama_set_name: RedPajamaCommonCrawl


text:
Shakkyo
Revision as of 23:43, 18 January 2014 by LordAmeth (talk | contribs) (Created page with "*''Japanese'': 石橋 ''(shakkyou)'' ''Shakkyô'' (lit. "stone bridge") is a fifth-category (''kiri-nô'') Noh play featuring a lion dance (''shishimai''). Numerous o...")
Japanese: 石橋 (shakkyou)
Shakkyô (lit. "stone bridge") is a fifth-category (kiri-nô) Noh play featuring a lion dance (shishimai). Numerous other plays, in kabuki and other forms, take after this play, and are known as belonging to the sub-genre, or category, of Shakkyô-mono ("Shakkyô pieces"). Most often, today, only the second half of the play, that is, the lion dance itself, is performed.
A monk by the name of Jakushô (also known as Ôe no Sadamoto) comes upon a stone bridge at Mt. Shôryôzen (in modern-day Shanxi province) during his travels in China. He meets a small boy, who says that the land on the other side is the Buddhist Pure Land belonging to the bodhisattva Monju (Skt: Manjusri), but that only those who have spent many years in ascetic training in preparation for crossing the bridge have been able to do so safely.
One or two lions (depending on the performance)[1] appear and come across the bridge to where Jakushô is, dancing and playing among peony flowers. When their dance is finished, they cross the bridge again, and return to the Pure Land.
Shakkyô is a fifth-category play, meaning it is best performed as the closing act of a longer performance. Short, energetic, and auspicious, it corresponds to the kyû of the jo-ha-kyû pacing of a Noh program, when preceded by other plays.
Because of the special skills associated with performing the lion dance, Shakkyô is considered a hiraki-mono, the first piece performed by an actor after achieving a particular rank or level of experience, and emblematic of his talent and skill.
While in most schools, the young boy and the lion are both played by a shite actor, in the Hôshô school, the boy is played by a tsure actor, and the shite only appears in the second half. The Hôshô school also omits the kyôgen interlude.
"Shakkyô (Stone Bridge)," the-NOH.com, 2014.
↑ When two lions dance, the dance is known as renjishi. A kabuki play by that name is likely the most popular, or most often performed, lion dance piece in the kabuki repertoire.
Retrieved from "http://samurai-archives.com/w/index.php?title=Shakkyo&oldid=26436"
Poetry and Theater


The following credit cards are accepted: AX,DC,EC,JC,XS.
Time zone GMT +1. Check in time: 2:00 PM. Check out time: 12:00 PM.
Go right into Lichtenberg Street.
Go left at Strahlauer Allee Muhlenstr.
Take the first left - the hotel on right hand side.
Rate disclaimer: Indicative rates for search purposes only; check for specific rates when making a reservation.


Dimmable AR111 (G23 Base) Lower Energy Saving Eco halogen light bulbs / lamps.
AR111 (G53) light bulbs / lamps are a high performance reflector type lamp and provide high contrast, low glare lighting and are also known as Halospot 111 (By Osram) and Masterline 111 (By Philips).
This type of lighting is often used in the retail industry such as in restaurants, shops, hotels and galleries. Get a great deal today on our low energy saving AR111 light bulbs.


text:
Free Admission for Veterans and Military during Veterans Day Weekend
Special Activities Will Celebrate America's First Veterans and the Marine Corps' Birthday
PHILADELPHIA, OCT. 10, 2017 — More than 200,000 people fought for American independence during the Revolutionary War. During Veterans Day Weekend, the Museum of the American Revolution will celebrate America's first veterans as well as those that continue to serve our country today.
From Friday, Nov. 10 – Sunday, Nov. 12, 2017, the Museum will offer free admission to veterans and active/retired members of the military (with ID), as well as special programs and activities related to early American soldiers. A printed gallery guide will highlight the contributions of those who served in the Revolutionary War. Each visitor to the Museum will receive a card with the name of a Revolutionary veteran – follow their journey through the war and then see an actual photograph at the conclusion of the exhibits.
Throughout the weekend, visitors can write notes thanking the soldiers of today for their service. Visitors also can donate to the American Legion Auxiliary and receive a red crepe paper poppy that was handmade by veterans as part of their therapeutic rehabilitation. Proceeds will assist disabled and hospitalized veterans.
On Friday, Nov. 10, the Museum will celebrate the birthday of the Marine Corps, which formed just down the street from the Museum at Tun Tavern in 1775. Walking tours to the location where Tun Tavern used to stand (corner of Chestnut and Front Streets) will depart from the Museum's outdoor plaza at 11 a.m. and 2 p.m. In the Museum's exhibits, visitors can board the Museum's privateer sloop to learn about the ships upon which early Marines served.
On Saturday, Nov. 11 at 2 p.m., Director of Curatorial Affairs and Chief Historian Dr. Philip Mead will present a special talk on veteran and memoirist Joseph Plumb Martin in the Museum's Liberty Hall. Plumb Martin wrote one of the best accounts of the Revolutionary War from a private soldier's perspective almost 50 years after the war ended. The 45-minute talk will be followed by a question-and-answer session with the audience. Visitors also will have the opportunity to engage with a costumed Joseph Plumb Martin historical interpreter.
Throughout the weekend, visitors can experience programs and storytelling based on Revolutionary War pension narratives in which veterans and widows wrote about their wartime experiences, their families, and their postwar lives. These spontaneous 10-minute pop-up talks will take place in the galleries, and will focus on Joseph Plumb Martin and another young Continental Army soldier, Israel Trask, whose lives are portrayed in the galleries' life-size recreated scenes.
Visitors can also explore the stories behind the rare photographs of people who lived through the Revolutionary War that conclude the Museum's exhibits. Photography did not exist during the War, but many members of the Revolutionary generation survived into the age of photography. In fact, the last known Revolutionary War veterans died shortly after the Civil War. Historians have identified more than 150 images of them, and about half of these are on display at the Museum. These haunting photographs offer an extraordinary opportunity to see the faces of the Revolutionary Generation.
The Museum of the American Revolution explores the dynamic story of the American Revolution using its rich collection of Revolutionary-era weapons, personal items, letters, diaries, and works of art. Immersive galleries, theater experiences, and recreated historical environments bring to life the events, people, and ideals of our nation's founding and engage people in the history and continuing relevance of the American Revolution. Located just steps away from Independence Hall, Carpenters' Hall, and Franklin Court, the Museum serves as a portal to the region's many Revolutionary sites, sparking interest, providing context, and encouraging exploration. The Museum is a private, non-profit, and non-partisan organization. For more information, visit www.AmRevMuseum.org or call 877.740.1776.


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Watch Tim and Eric's Billion Dollar Movie (2012) Online
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Don Juan DeMarco (1994)
John Arnold DeMarco is a man who believes he is Don Juan, the greatest lover in the world. Clad in a cape and mask, DeMarco undergoes psychiatric treatment with Dr….
Knockin' on Heaven's Door (1997)
Two young men, Martin and Rudi, both suffering from terminal cancer, get to know each other in a hospital room. They drown their desperation in tequila and decide to take…
Genre: Action, Comedy, Crime, Drama
The Upside (2017)
Phillip is a wealthy quadriplegic who needs a caretaker to help him with his day-to-day routine in his New York penthouse. He decides to hire Dell, a struggling parolee who's…
Boys vs. Girls (2019)
Written and directed by Windsor's own Mike Stasko, Boys vs. Girls is loosely based on his experiences at a summer camp during the 90s. When camps around the country were…
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Genre: Adventure, Animation, Comedy, Family, Fantasy, Romance
What's Your Number? (2011)
Ally Darling is realizing she's a little lost in life. Her latest romance has just fizzled out, and she's just been fired from her marketing job. Then she reads an…
Arachnophobia (1990)
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Genre: Comedy, Fantasy, Horror, Thriller
Good Boys (2019)
A group of young boys on the cusp of becoming teenagers embark on an epic quest to fix their broken drone before their parents get home.
American Ultra (2015)
Mike is an unmotivated stoner whose small-town life with his live-in girlfriend, Phoebe, is suddenly turned upside down. Unbeknownst to him, Mike is actually a highly trained, lethal sleeper agent….
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Liar Liar (1997)
Fletcher Reede is a fast-talking attorney and habitual liar. When his son Max blows out the candles on his fifth birthday he has just one wish – that his dad…
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Becca, an anxious, non-vampire teen is torn between two boys. Before she can choose, Becca must get around her controlling father, who embarrasses Becca by treating her like a child….
Trailer: Tim and Eric's Billion Dollar Movie (2012)


The Union Finance Minister Shri P. Chidambaram will be leaving tomorrow for a five day visit to Australia.
The Finance Minister Shri Chidambaram will be leading a high level Indian delegation to attend the two day G-20 Finance Ministers' and Central Bank Governors' Meeting scheduled to be held in Sydney, Australia on 22nd and 23rd February, 2014.
Beside the Finance Minister, the Indian delegation includes Dr. Raghuram G. Rajan, Governor of Reserve Bank of India (RBI), Dr. Arvind Mayaram, Secretary, Department of Economic Affairs and other senior officials of Ministry of Finance among others.
During the G-20 Meet, the Finance Minister Shri Chidambaram will be discussing issues relating to Global Economy including Key Challenges facing the Global Economy, issues relating to Investment and Infrastructure, Growth Challenge Including Building a Common Vision for Growth Strategies, issues relating to IMF Reforms, Financial Regulation and tax matters among others.
During his stay in Sydney, Australia, the Finance Minister Shri Chidambaram will also hold bilateral meetings with his counterparts from various G-20 member countries.
During the two day G-20 Finance Ministers' and Central Bank Governors' Meeting, the Finance Minister will be participating in a Session to endorse the package of Financial Reforms to be completed by the Brisbane Summit.
There will also be another Session to endorse a draft Global Standard on Automatic Exchange of Information and G-20 Tax Agenda for 2014 and signing off any outstanding communiqué issues on tax among others.
Besides above, the Finance Minister Shri Chidambaram will also be addressing the Investors' Meet in Sydney after the conclusion of G-20 Finance Ministers' and Central Bank Governors' Meeting on 23rd February, 2014.
The Finance Minister Shri Chidambaram will also meet the Premier and representatives of New South Wales Government during his stay in Australia.
The Finance Minister will later also hold meetings with Major Institutional Investors' and hold Round Table Meeting with major Businesses during his stay in Sydney, Australia.
The Finance Minister will leave for India on 25th February, 2014.


College Park MD: University of Maryland, 1935, 1936, 1937.
Three volumes. First and only editions. Quartos. Textured leatherette with embossed terrapin. Each has a bit of soiling, else about fine. The yearbooks for Charlie Keller's varsity baseball career at Maryland. Keller starred for Maryland in his sophomore (1935) and junior year (1936), and the presence of major league scouts is noted in the baseball section which also pictures Keller, and enumerates his accomplishments. The 1937 yearbook, Keller's senior year, mentions that Keller left the school for the Newark Bears, just before the college season started. Keller played on the great Bear teams of 1937 and 1938 before being brought up to star for the Yankee from 1939-1951, with time out to serve during the war. He was a five-time all-star.