text:
\section{Introduction}

McKean-Vlasov stochastic differential equations (MVSDEs in short) are a kind of special stochastic differential equations whose coefficients depend on probability distributions of their solutions. They were first initiated by Henry P. McKean \cite{mckean} in 1966, and then were gradually studied by a lot of researchers. At present, there have been many results about MVSDEs, such as the well-posedness of the solutions in \cite{d, dq1}, the stability of strong solutions in \cite{dq2}, the well-posedness of the mild solutions and their Euler-Maruyama approximation in infinite dimension Hilbert spaces in \cite{q}, and the particle approximations method in \cite{boss}.

As the research of MVSDEs develops, the fields of their application are becoming larger and larger. This leads to some new problems. Estimation of unknown parameters in MVSDEs is one of these problems. Now, there are many results about parameter estimation of stochastic differential equations. Let us mention some works. Liptser and Shiryayev \cite{lipster} considered the maximum likelihood estimation of It$\hat{\rm o}$ diffusions under continuous observations, while Yoshida \cite{y} estimated these diffusion processes with the maximum likelihood estimation based on discrete diffusions. In \cite{bjpn}, Bishwal obtained the exponential bound of the large deviation rate for the maximum likelihood estimator of the drift coefficients.
Other methods of parameter estimation like martingale function estimators, nonparametric methods can be found in \cite{s,ay}.

However, because of the distributions in the drift coefficients and diffusion coefficients, the previous methods and results may not well be applied to MVSDEs. In \cite{r}, Ren and Wu proposed the least squares estimators for a class of path-dependent MVSDEs. Wen et al. \cite{w} discussed the maximum likelihood estimators on MVSDEs with the following form assuming that $\vartheta\in{\mathbb R}$ is known and $\sigma = 1$,
 \begin{eqnarray*}
X_{t}=X_{0}+\int_{0}^{t} \int_{{\mathbb R}}b\left(\theta, X_{s}, y\right)\mu_s(\mathrm{d} y) \mathrm{d} t+\int_{0}^{t} \int_{{\mathbb R}} \sigma\left(\vartheta, X_{s}, y\right) \mu_s(\mathrm{d} y) \mathrm{d} W_{s}, \quad X_{0}=x_0\in{\mathbb R},
\end{eqnarray*}
where $\theta$ is a unknown parameter and $\mu_t$ is the probability distribution of $X_t$.

In this paper, we focus on the following MVSDE in a more general form
\begin{equation}
\mathrm{d} X_t = b(\theta, X_{t\land\cdot}, \mu_t) \mathrm{d} t + \sigma( X_{t\land\cdot}, \mu_t)\mathrm{d} W_t, \quad X_0 =\xi,
\label{11}
\end{equation}
where $\xi$ is a random vector. We not only construct a maximum likelihood estimator $\theta_T$ for $\theta$ but also prove the consistency of $\theta_T$. And then we discretize Eq.(\ref{11}) and also obtain the numerical simulation of $\theta_T$.

The rest of the paper is organized as follows. In Section \ref{exiuni}, we prove the existence and uniqueness of strong solutions for Eq.(\ref{11}) under non-Lipschitz conditions. The maximum likelihood estimators are constructed in Section \ref{mle}. In Section \ref{numsim}, a numerical equation of Eq.(\ref{11}) is given by interacting particles and the Euler-Maruyama method, and then the error between the MVSDE and its approximation is calculated, followed by giving a maximum likelihood estimator of the numerical equation. Finally, in Section \ref{exam}, we apply the method to a specific equation as an example, and explain our results.

The following convention will be used throughout the paper: $C$ with or without indices will denote different positive constants whose values may change from one place to another.

\section{The existence and uniqueness of path-dependent MVSDEs}\label{exiuni}

In the section, we prove the existence and uniqueness of the solutions for Eq.(\ref{11}).

Fix $T>0$. Let $C^d_T$ be the collection of all the continuous functions from $[0,T]$ to ${\mathbb R}^d$. And then we equip it with the compact uniform convergence topology. Let ${\mathcal B}_T^d$ be the $\sigma$-field generated by the topology. For $w\in C^d_T$, set
$$
\|w\|_{T}:=\sup_{0 \leqslant t \leqslant T}|w(t)|.
$$

Let ${\mathscr B}({\mathbb R}^d)$ be the Borel $\sigma$-field on ${\mathbb R}^d$. Let ${\mathcal P}_2({\mathbb R}^d)$ denote the space of probability measures on ${\mathscr B}({\mathbb R}^d)$ with finite second moments. That is, if $\mu\in{\mathcal P}_2({\mathbb R}^d)$, then
$$
\|\mu\|_{\lambda^2}^2:=\int_{{\mathbb R}^d}(1+|x|)^2\mu(\mathrm{d} x)<\infty.
$$
And the distance of $\mu, \nu\in{\mathcal P}_2({\mathbb R}^d)$ is defined as
$$
{\mathbb W}^{2}_{2}(\mu,\nu):=\inf_{\pi\in{\mathscr C}(\mu_1, \mu_2)} \int_{{\mathbb R}^d \times {\mathbb R}^d} |x-y|^2 \pi(\mathrm{d} x, \mathrm{d} y),
$$
where ${\mathscr C}(\mu_1, \mu_2)$ denotes the set of  all the probability measures whose marginal distributions are $\mu_1, \mu_2$, respectively. Thus, $({\mathcal P}_2({\mathbb R}^d),{\mathbb W}_{2})$ is a Polish space.

Let $(\Omega,{\mathscr F},\{{\mathscr F}_t\}_{0 \leqslant t \leqslant T},{\mathbb P})$ be a complete filtered probability space and $\{W_{t},t\geq0\}$ be a $m$-dimensional standard Brownian motion. Consider the following path-dependent MVSDE on ${\mathbb R}^d$:
\begin{eqnarray}\left\{\begin{array}{ll}
X_t=\xi+\int_0^tb(\theta, X_{s\land\cdot},\mu_s)\mathrm{d} s+\int_0^t\sigma(X_{s\land\cdot},\mu_s)\mathrm{d} W_s,\\
\mu_s= $the probability  distribution of\quad$X_s,
\end{array}
\label{eq1}
\right.
\end{eqnarray}
where $\xi$ is a ${\mathscr F}_0$-measurable random vector, $\theta\in\Theta\subset{\mathbb R}^k$ is a unknown parameter, $b: \Theta \times C^d_T \times {\mathcal P}_2({\mathbb R}^d) \mapsto {\mathbb R}^d$, $\sigma: C^d_T \times {\mathcal P}_2({\mathbb R}^d)\mapsto{\mathbb R}^{d\times m}$ are Borel measurable. We assume:
\begin{enumerate}[($\bf{H}_1$)]
\item There exists a nonnegative constant $K_1$ such that for any $w, v\in C_T^d$, $\mu, \nu\in{\mathcal P}_2({\mathbb R}^d)$

(i)
$$
|b(\theta, w,\mu) - b(\theta, v,\nu) |^2+\|\sigma(w,\mu) - \sigma(v,\nu) \|^2 \leqslant K_1\left(\kappa_1(\|w-v\|_T^2) + \kappa_2\({\mathbb W}_2^2(\mu, \nu)\) \right),
$$
where $\|\cdot\|$ denotes the Hilbert-Schmidt norm of a matrix, and $\kappa_i(x), i=1, 2$ are two positive, strictly increasing, continuous concave function and satisfy $\kappa_i(0)=0$,  $\int_{0^+}\frac{1}{\kappa_1(x)+\kappa_2(x)}dx=\infty$;

(ii)
$$
\vert b(\theta, w,\mu)\vert^2+\| \sigma(w,\mu) \|^2 \leqslant K_1\left(1 + \| w\|_T^2 + \|\mu \|_{\lambda^2}^2 \right).
$$
\end{enumerate}

\begin{theorem}\label{wellpose}
Suppose that ($\bf{H}_1$) holds and ${\mathbb E}|\xi|^2<\infty$. Then Eq.(\ref{eq1}) has a unique strong solution $X$ and
$$
{\mathbb E}\left(\sup_{0 \leqslant t \leqslant T}|X(t)|^2\right)<\infty.
$$
\end{theorem}
\begin{proof}
First of all, set
\begin{eqnarray}\left\{\begin{array}{ll}
X_t^{(0)}= \xi,\quad t\in[0,T],\\
X_t^{(n+1)}=\xi+\int_0^tb(\theta, X_{s\land\cdot}^{(n)},\mu_s^{(n)})\mathrm{d} s+\int_0^t\sigma(X_{s\land\cdot}^{(n)},\mu_s^{(n)})\mathrm{d} W_s, \quad n\in{\mathbb N}\cup\{0\},
\end{array}
\label{iteration}
\right.
\end{eqnarray}
where $\mu_s^{(n)}$ is the probability  distribution of $X_s^{(n)}$. We make use of Eq.(\ref{iteration}) to prove the well-posedness of Eq.(\ref{eq1}).

{\bf Step 1.} We prove that the definition of Eq.(\ref{iteration}) is reasonable.

For $n=0$, ${\mathbb E}\left(\sup\limits_{0 \leqslant t \leqslant T}|X_t^{(0)}|^2\right)={\mathbb E}|\xi|^2<\infty$. Assume that for $n\in{\mathbb N}$,
$$
{\mathbb E}\left(\sup\limits_{0 \leqslant t \leqslant T}|X_t^{(n)}|^2\right)<\infty.
$$
And then by the H$\ddot{\rm o}$lder inequality, the Burkholder-Davis-Gundy inequality  and ($\bf{H}_1$), we get that
\begin{eqnarray}
&&{\mathbb E}\left(\sup_{0 \leqslant t \leqslant T}\vert X_t^{(n+1)} \vert ^2\right)\nonumber\\
 &\leqslant& 3{\mathbb E}|\xi|^2 +3{\mathbb E}\left(\sup_{0 \leqslant t \leqslant T}\left|\int_0^tb(\theta, X_{s\land\cdot}^{(n)},\mu_s^{(n)}) \mathrm{d} s \right|^2\right) + 3{\mathbb E} \left(\sup_{0 \leqslant t \leqslant T}\left|\int_0^t\sigma(X_{s\land\cdot}^{(n)},\mu_s^{(n)})\mathrm{d} W_s\right|^2 \right)\nonumber\\
&\leqslant &3{\mathbb E}|\xi|^2 +3T {\mathbb E}\int_0^T \left|b(\theta,X_{s\land\cdot}^{(n)},\mu_s^{(n)})\right|^2 \mathrm{d} s  + 3C{\mathbb E}\int_0^T\|\sigma( X_{s\land\cdot}^{(n)},\mu_s^{(n)})\|^2\mathrm{d} s  \nonumber\\
&\leqslant &3{\mathbb E}|\xi|^2 +3(T+C)K_1{\mathbb E}\int_0^T\left(1 + \| X_{s\land\cdot}^{(n)}\|_T^2 + \| \mu_s^{(n)}\|_{\lambda^2}^2\right)\mathrm{d} s\nonumber\\
&\leqslant &3{\mathbb E}|\xi|^2 +9(T+C)K_1T\left(1 +{\mathbb E}\left(\sup\limits_{0 \leqslant t \leqslant T}|X_t^{(n)}|^2\right)\right),
\label{linest}
\end{eqnarray}
where the last inequality is based on the fact that $\|\mu_s^{(n)}\|^2_{\lambda^2}\leqslant{\mathbb E}(1+|X_s^{(n)}|)^2\leqslant 2{\mathbb E}(1+|X_s^{(n)}|^2)$. From induction on $n$, it follows that
\begin{eqnarray*}
{\mathbb E}\left(\sup\limits_{0 \leqslant t \leqslant T}|X_t^{(n)}|^2\right)<\infty, \quad n\in{\mathbb N}\cup\{0\}.
\end{eqnarray*}

{\bf Step 2.} We prove the existence of the solutions to Eq.(\ref{eq1}).

By the same deduction to that of (\ref{linest}), it holds that for $m,n\in{\mathbb N}$
\begin{eqnarray}
 &&{\mathbb E}\left(\sup_{0 \leqslant t \leqslant T}\vert X_t^{(n+1)} - X_t^{(m+1)} \vert ^2 \right)\nonumber\\
 &\leqslant& 2T{\mathbb E} \int_0^T \vert b(\theta, X_{s\land\cdot}^{(n)}, \mu_s^{(n)}) -b(\theta, X_{s\land\cdot}^{(m)}, \mu_s^{(m)})\vert^2 \mathrm{d} s\nonumber\\
 &&+ 2C{\mathbb E}\int_0^T \|\sigma(X_{s\land\cdot}^{(n)}, \mu_s^{(n)}) -\sigma( X_{s\land\cdot}^{(m)}, \mu_s^{(m)} )\|^2 \mathrm{d} s \nonumber\\
&\leqslant& 2(T+C)K_1{\mathbb E} \int_0^T\bigg(\kappa_1(\| X_{s\land\cdot}^{(n)} - X_{s\land\cdot}^{(m)} \|_T^2)\nonumber\\
&&\quad +\kappa_2({\mathbb W}_2^2(\mu_s^{(n)},\mu_s^{(m)}))\bigg)\mathrm{d} s\nonumber\\
&\leqslant& 2(T+C)K_1\int_0^T\bigg[\kappa_1\left({\mathbb E}\left(\sup\limits_{0 \leqslant u\leqslant s}|X_u^{(n)}-X_u^{(m)}|^2\right)\right)\nonumber\\
&&+\kappa_2\left({\mathbb E}\left(\sup\limits_{0 \leqslant u\leqslant s}|X_u^{(n)}-X_u^{(m)}|^2\right)\right)\bigg]\mathrm{d} s,
\label{distesti}
\end{eqnarray}
where the last step is based on the Jensen inequality and the fact that
$$
{\mathbb W}_2^2(\mu_s^{(n)},\mu_s^{(m)}) \leqslant {\mathbb E}|X_s^{(n)}-X_s^{(m)}|^2 \leqslant {\mathbb E}\left(\sup\limits_{0 \leqslant u \leqslant s}|X_u^{(n)}-X_u^{(m)}|^2\right).
$$
Set
$$
g(t):=\lim\limits_{n,m\rightarrow\infty}{\mathbb E}\left(\sup\limits_{0 \leqslant u\leqslant t}|X_u^{(n)}-X_u^{(m)}|^2\right),
$$
and then (\ref{distesti}) admits us to have that
$$
g(T)\leqslant 2(T+C)K_1\int_0^T\(\kappa_1(g(s))+\kappa_2(g(s))\)\mathrm{d} s.
$$
Thus, by \cite[Lemma 3.6]{dq1}, one can get $g(T)=0$. That is, $\{X^{(n)}\}$ is a Cauchy sequence in the space $L^2(\Omega,{\mathscr F},{\mathbb P}, C_T^d)$. From this, we know that there exists a $X\in L^2(\Omega,{\mathscr F},{\mathbb P}, C_T^d)$ such that
\begin{eqnarray}
\lim\limits_{n\to\infty}{\mathbb E}\left(\sup_{0 \leqslant t \leqslant T}\vert X_t^{(n)} - X_t\vert ^2\right)=0.
\label{limit01}
\end{eqnarray}
Note that
$$
\sup_{0 \leqslant t \leqslant T}{\mathbb W}_2^2(\mu_t^{(n)},\mu_t) \leqslant \sup_{0 \leqslant t \leqslant T}{\mathbb E}|X_t^{(n)}-X_t|^2 \leqslant {\mathbb E}\left(\sup\limits_{0 \leqslant t \leqslant T}|X_t^{(n)}-X_t|^2\right).
$$
So, we conclude that
\begin{eqnarray}
\lim\limits_{n\to\infty}\sup_{0 \leqslant t \leqslant T}{\mathbb W}_2^2(\mu_t^{(n)},\mu_t)=0.
\label{limit02}
\end{eqnarray}
And then (\ref{limit01})-(\ref{limit02}) imply that for $\forall t \in [0,T]$,
\begin{eqnarray*}
&&\int_0^tb(\theta, X_{s\land\cdot}^{(n)}, \mu_s^{(n)}) \mathrm{d} s  \to \int_0^t b(\theta, X_{s\land\cdot}, \mu_s) \mathrm{d} s ,\quad a.s.,\\
&&\int_0^t \sigma(X_{s\land\cdot}^{(n)}, \mu_s^{(n)})\mathrm{d} W_s  \to \int_0^t\sigma( X_{s\land\cdot}, \mu_s) \mathrm{d} W_s ~\mbox{in}~ L^2(\Omega, {\mathscr F}_t,{\mathbb P}).
\end{eqnarray*}
Therefore, taking the limit on two hand sides of Eq.(\ref{iteration}) as $n \to \infty$, we have that
$$
X_t=\xi+\int_0^tb(\theta, X_{s\land\cdot},\mu_s)\mathrm{d} s+\int_0^t\sigma(X_{s\land\cdot},\mu_s)\mathrm{d} W_s,
$$
that is, $X$ is a solution of Eq.(\ref{eq1}).

{\bf Step 3.} We prove the uniqueness of the solutions to Eq.(\ref{eq1}).

Suppose that $X$ and $\hat{X}$ are two solutions to Eq.(\ref{eq1}). And then by the similar calculation to that of (\ref{distesti}), it holds that
\begin{eqnarray*}
 {\mathbb E}\left(\sup_{0 \leqslant t \leqslant T}\vert X_t - \hat X_t \vert ^2\right)
&\leqslant& 2T{\mathbb E} \int_0^T\vert b(\theta, X_{s\land\cdot}, \mu_s) -b(\theta, \hat{X}_{s\land\cdot}, \hat \mu_s)\vert^2 \mathrm{d} s   \\
&&+ \ 2C{\mathbb E}\int_0^T \|\sigma(X_{s\land\cdot}, \mu_s) -\sigma( \hat{X}_{s\land\cdot}, \hat \mu_s))\|^2 \mathrm{d} s \\
&\leqslant& 2(T+C)K_1{\mathbb E}\int_0^T\(\kappa_1(\| X_{s\land\cdot} - \hat{X}_{s\land\cdot}\|_T^2) + \kappa_2({\mathbb W}_2^2(\mu_s,\hat \mu_s))\) \mathrm{d} s\\
&\leqslant&2(T+C)K_1\int_0^T \bigg(\kappa_1\left({\mathbb E}\left(\sup_{0 \leqslant u \leqslant s}\vert X_u - \hat X_u \vert ^2\right)\right)\\
&&+\kappa_2\left({\mathbb E}\left(\sup_{0 \leqslant u \leqslant s}\vert X_u - \hat X_u \vert ^2\right)\right)\bigg)\mathrm{d} s,
\end{eqnarray*}
which together with \cite[Lemma 3.6]{dq1} yields that
$$
{\mathbb E}\left(\sup_{0 \leqslant t \leqslant T}\vert X_t - \hat X_t \vert ^2\right)=0.
$$
That is, $X_t=\hat X_t$ for all $t\in[0,T]$ and almost all $\omega$. The proof is complete.
\end{proof}

\section{The maximum likelihood estimation of path-dependent MVSDEs}\label{mle}

In the section, we assume ($\bf{H}_1$) and $d=m=k=1$. And then Eq.(\ref{eq1}) has a unique solution $X^{\theta}$. We construct a maximum likelihood estimator of $\theta$ and prove its properties. $C_T:=C_T^1$.

Assume:
\begin{enumerate}[($\bf{H}_2$)]
\item For any $w\in C_T, \mu\in{\mathcal P}_2({\mathbb R})$, $\sigma(w,\mu)\neq 0$ and
$$
\left| \frac{b(\theta,w,\mu)}{\sigma(w,\mu)}\right|\leqslant K_2,
$$
where $K_2\geqslant 0$ is a constant.
\end{enumerate}

Let $\theta_0$ be the true value of $\theta$. Let ${\mathbb P}_{\theta}^{T}, {\mathbb P}_{\theta_0}^{T}$ be the distributions of $(X^\theta_t)_{t\in[0,T]}$ and $(X^{\theta_0}_t)_{t\in[0,T]}$, respectively. Thus, under ($\bf{H}_2$), it follows from \cite[Theorem 7.19, P. 294]{lipster} that ${\mathbb P}_{\theta}^{T} \ll {\mathbb P}_{\theta_{0}}^{T}$. Define a maximum likelihood function of $\theta$ as
\begin{eqnarray*}
L_{T}(\theta) &:=&\frac{\mathrm{d} {{\mathbb P}_{\theta}^{T}}}{\mathrm{d} {\mathbb P}^{T}_{\theta_{0}}} \\
&=&\exp\left\{{\int_{0}^{T} \frac{1}{\sigma^2(X_{t\land\cdot}^{\theta_{0}}, \mu^{\theta_{0}}_{t}) }\(b(\theta, X_{t\land\cdot}^{\theta_{0}}, \mu^{\theta_{0}}_t) - b(\theta_0, X_{t\land\cdot}^{\theta_{0}}, \mu^{\theta_{0}}_t) \)\mathrm{d} X_{t}^{\theta_0}}\right.\\
&&\qquad\qquad -\left.{\frac{1}{2}\int_{0}^{T} \frac{1}{\sigma^2(X_{t\land\cdot}^{\theta_{0}}, \mu^{\theta_{0}}_{t}) }\(b^2(\theta, X_{t\land\cdot}^{\theta_{0}}, \mu^{\theta_{0}}_t) - b^2(\theta_0, X_{t\land\cdot}^{\theta_{0}}, \mu^{\theta_{0}}_t) \)\mathrm{d} t}\right\}\\
&=&\exp\left\{{\int_{0}^{T} \frac{1}{\sigma(X_{t\land\cdot}^{\theta_{0}}, \mu^{\theta_{0}}_{t}) }\(b(\theta, X_{t\land\cdot}^{\theta_{0}}, \mu^{\theta_{0}}_t) - b(\theta_0, X_{t\land\cdot}^{\theta_{0}}, \mu^{\theta_{0}}_t) \)\mathrm{d} W_{t} }\right.\\
&&\qquad\qquad-\left.{\frac{1}{2}\int_{0}^{T} \frac{1}{\sigma^2(X_{t\land\cdot}^{\theta_{0}}, \mu^{\theta_{0}}_{t}) }\(b(\theta, X_{t\land\cdot}^{\theta_{0}}, \mu^{\theta_{0}}_t) - b(\theta_0, X_{t\land\cdot}^{\theta_{0}}, \mu^{\theta_{0}}_t) \)^2\mathrm{d} t}\right\},
\end{eqnarray*}
where $\mu^\theta_t, \mu^{\theta_0}_t$ are the distributions of $X^\theta_t, X^{\theta_0}_t$, respectively. So, the maximum likelihood estimator of $\theta$ is given by
\begin{eqnarray*}
\theta_T := \arg\max_{\theta \in \Theta}{L_{T}(\theta)}.
\end{eqnarray*}

Next, we study some properties of the maximum likelihood estimator $\theta_T$. To do this, we assume more:
\begin{enumerate}[($\bf{H}_3$)]
\item For any $w\in C_T, \mu\in{\mathcal P}_2({\mathbb R})$, $b(\theta, w, \mu)$ is one-to-one and continuous in $\theta$.
\end{enumerate}

\begin{theorem}\label{et2}(The strong consistency)\\
Under the assumptions $(\bf{H}_1)$-$(\bf{H}_3)$, it holds that
$$
\theta_T\stackrel{a.s.}{\longrightarrow}\theta_0,  \quad T\to \infty.
$$
\end{theorem}
\begin{proof}
Set
$$
l_{T}(\theta) := \log L_T(\theta) =\displaystyle\log\frac{\mathrm{d} {{\mathbb P}_{\theta}^{T}}}{\mathrm{d} {\mathbb P}_{\theta_{0}}^{T}}.
$$
And then it holds that for $\delta > 0$,
\begin{eqnarray*}
l_{T}(\theta_0+\delta) - l_T(\theta_0) &=&\log\frac{\mathrm{d} {{\mathbb P}_{\theta_0+\delta}^{T}}}{\mathrm{d} {\mathbb P}_{\theta_{0}}^{T}} \\
&=&\int_{0}^{T} \frac{1}{\sigma(X_{t\land\cdot}^{\theta_{0}}, \mu^{\theta_{0}}_{t}) }\(b(\theta_0+\delta, X_{t\land\cdot}^{\theta_{0}}, \mu^{\theta_{0}}_t) - b(\theta_0, X_{t\land\cdot}^{\theta_{0}}, \mu^{\theta_{0}}_t) \)\mathrm{d} W_{t} \\
&&-\frac{1}{2}\int_{0}^{T} \frac{1}{\sigma^2(X_{t\land\cdot}^{\theta_{0}}, \mu^{\theta_{0}}_{t}) }\(b(\theta_0+\delta, X_{t\land\cdot}^{\theta_{0}}, \mu^{\theta_{0}}_t) - b(\theta_0, X_{t\land\cdot}^{\theta_{0}}, \mu^{\theta_{0}}_t) \)^2\mathrm{d} t\\
&=:&{\int_{0}^{T} \Gamma_t^{\theta_0} \mathrm{d} W_{t} }-\frac{1}{2}\int_{0}^{T} (\Gamma_t^{\theta_0})^2\mathrm{d} t,
\end{eqnarray*}
where
$$
\Gamma_t^{\theta_0}  := \frac{1}{\sigma(X_{t\land\cdot}^{\theta_{0}}, \mu^{\theta_{0}}_{t}) }\(b(\theta_0+\delta, X_{t\land\cdot}^{\theta_{0}}, \mu^{\theta_{0}}_t) - b(\theta_0, X_{t\land\cdot}^{\theta_{0}}, \mu^{\theta_{0}}_t) \).
$$
Note that
$$
\left[\int_{0}^{\cdot} \Gamma_t^{\theta_0} \mathrm{d} W_{t}\right]_T=\int_{0}^{T}\left|\Gamma_t^{\theta_0}\right|^2\mathrm{d} t,
$$
where $[\cdot]$ stands for the quadratic variation of $\cdot$. Thus, by the time change, we know that
$$
\tilde{W}_t:=\int_{0}^{A_t} \Gamma_s^{\theta_0} \mathrm{d} W_{s}
$$
is a $({\mathscr F}_{A_t})_{t\geqslant 0}$-adapted Brownian motion, where $A_t$ is the inverse function of $\int_{0}^{t}\left|\Gamma_s^{\theta_0}\right|^2\mathrm{d} s$. So,
\begin{eqnarray}
\frac{l_T(\theta_0+\delta)-l_T(\theta_0)}{\int_{0}^{T} (\Gamma_t^{\theta_0})^2\mathrm{d} t}
=\frac{\int_{0}^{T} \Gamma_t^{\theta_0}\mathrm{d} W_t }{\int_{0}^{T} (\Gamma_t^{\theta_0})^2\mathrm{d} t}-\frac{1}{2}=\frac{\tilde{W}_{A_T^{-1}}}{A_T^{-1}}-\frac{1}{2} \stackrel{a.s.}{\longrightarrow}-\frac{1}{2}, \quad T\rightarrow\infty,
\label{limit1}
\end{eqnarray}
where the last step is based on the strong law of large numbers for Brownian motions. By the same deduction to that of (\ref{limit1}), one can get that
\begin{eqnarray}
\frac{l_T(\theta_0-\delta)-l_T(\theta_0)}{\int_{0}^{T} (\Gamma_t^{\theta_0})^2\mathrm{d} t}\stackrel{a.s.}{\longrightarrow}-\frac{1}{2}, \quad T\rightarrow\infty.
\label{limit2}
\end{eqnarray}
Combining (\ref{limit1}) with (\ref{limit2}), we obtain that
\begin{eqnarray}
\frac{l_T(\theta_0\pm\delta)-l_T(\theta_0)}{\int_{0}^{T} (\Gamma_t^{\theta_0})^2\mathrm{d} t}\stackrel{a.s.}{\longrightarrow}-\frac{1}{2}, \quad T\rightarrow\infty.
\label{limit}
\end{eqnarray}

Next, we observe (\ref{limit}). It follows from (\ref{limit}) that for $\delta$ and $\theta_0$, there exists some $t_0>0$ such that
\begin{equation}
l_{T}(\theta_{0}\pm\delta) <l_{T}(\theta_{0}), \quad T \geqslant t_0, \quad a.s..
\label{26}
\end{equation}
Besides, by ($\bf{H}_3$), we know that $l_T(\theta)$ is continuous on $[\theta_0 - \delta, \theta_0 + \delta]$. So, there exists a $\theta^*\in[\theta_0 - \delta, \theta_0 + \delta]$ such that $l_T(\theta^*)$ is the maximum value of $l_T(\theta)$ on $[\theta_0 - \delta, \theta_0 + \delta]$. That is, $\theta_T=\theta^*$ for $\Theta=[\theta_0 - \delta, \theta_0 + \delta]$. Based on (\ref{26}), it holds that $\theta_T \neq \theta_0 \pm \delta $ for $T\geqslant t_0$. Thus, $\theta_T \to \theta_0$ as $T\to\infty$. The proof is over.
\end{proof}

\section{The numerical simulation of path-dependent MVSDEs}\label{numsim}

In the section, we introduce the numerical simulation of Eq.(\ref{eq1}) under ($\bf{H}_1$) and estimate the error between the solution of Eq.(\ref{eq1}) and that of the numerical equation under Lipschitz conditions.

First of all, for $N\in{\mathbb N}$ consider these following MVSDEs
\begin{eqnarray}\left\{\begin{array}{ll}
\mathrm{d} X_{t}^{i,N}= b\left(\theta,  X_{t\land\cdot}^{i,N}, \mu_t^N\right) \mathrm{d} t +\sigma\left( X_{t\land\cdot}^{i,N}, \mu_t^N \right) \mathrm{d} W_t^i ,\\
X_{0}^{i,N}=\xi, \quad i = 1, 2,\ldots, N,
\end{array}
\label{31}
\right.
\end{eqnarray}
where $\mu_t^N:=\dfrac{1}{N}\sum\limits_{j=1}^{N}\delta_{X_t^{j,N}}$, $\delta_{X_t^{j,N}}$ is the Dirac measure at $X_t^{j,N}$, and $W_t^i, i = 1, 2,\ldots, N$ are $N$ mutually independent $m$-dimensional standard Brownian motions. By Theorem \ref{wellpose}, under ($\bf{H}_1$) we know that Eq.(\ref{31}) has a unique solution $X_{t}^{i,N}$. And then we construct  the following numerical simulation equation: for $M\in{\mathbb N}$
\begin{eqnarray}\left\{\begin{array}{ll}
Y_{0}^{i}=\xi,\\
Y_t^{i}=Y_{t_k}^{i}+b\left(\theta,  Y_{t_k\land\cdot}^{i}, \mu_{t_k}^M\right)(t-t_k) +\sigma\left( Y_{t_k\land\cdot}^{i}, \mu_{t_k}^M\right)(W^i_t-W^i_{t_k}), \quad t\in[t_k, t_{k+1}],
\end{array}
\label{34}
\right.
\end{eqnarray}
where $t_k:=k\frac{T}{M}$, $\mu_{t_k}^M :=\dfrac{1}{N}\sum\limits_{j=1}^{N}\delta_{Y_{t_k}^j} $ for $k= 0,...,M-1$. In order to estimate the error between the solution of Eq.(\ref{34}) and the solution of Eq.(\ref{eq1}), we also introduce the following MVSDE:
\begin{eqnarray}
X_t^i = \xi+\int_0^t b(\theta, X_{s\land\cdot}^i, \mu^i_s) \mathrm{d} s + \int_0^t\sigma( X_{s\land\cdot}^i, \mu^i_s)\mathrm{d} W_s^i,
\label{33}
\end{eqnarray}
where $\mu^i_s$ is the distribution of $X_s^i$. Note that the solution of Eq.(\ref{33}) has the same distribution to that of the solution for Eq.(\ref{eq1}). Therefore, we compute the distance between $X_t^i$ and $Y_t^{i}$ to estimate the error between $X_t$ and $Y_t^{i}$. To do this, we need stronger assumptions than ($\bf{H}_1$). Assume:
\begin{enumerate}[($\bf{H}^{\prime}_1$)]
\item There exists a nonnegative constant $K'_1$ such that for any $w, v\in C_T^d$, $\mu, \nu\in{\mathcal P}_2({\mathbb R}^d)$

(i)
$$
|b(\theta, w,\mu) - b(\theta, v,\nu) |^2+\|\sigma(w,\mu) - \sigma(v,\nu) \|^2 \leqslant K'_1\left( \|w-v\|^2 + {\mathbb W}_2^2(\mu, \nu) \right),
$$

(ii)
$$
\vert b(\theta, w,\mu)\vert^2+\| \sigma(w,\mu) \|^2 \leqslant K'_1\left(1 + \| w\|^2 + \|\mu \|_{\lambda^2}^2 \right).
$$
\end{enumerate}

\begin{theorem}
Suppose that ($\bf{H}'_1$) holds and ${\mathbb E}|\xi|^p<\infty$ for $p>4$. Then it follows that
\begin{eqnarray}
\sup_{1 \leqslant i \leqslant N} {\mathbb E}\left[ \sup_{0 \leqslant t \leqslant T}\vert X_t^i - Y_t^i \vert ^2\right] \leqslant C\Gamma_N+C \frac{T}{M}\left(\frac{T}{M}+C\right),
\label{errest}
\end{eqnarray}
where the constant $C>0$ is independent of $N, M$ and \begin{eqnarray*}
\Gamma_N:=\left\{\begin{array}{ll}
N^{-1/2}, \qquad\qquad d<4,\\
N^{-1/2}\log N, ~\quad d=4,\\
N^{-1/d}, \qquad\qquad d>4.
\end{array}
\right.
\end{eqnarray*}
\end{theorem}
\begin{proof}
Note that
\begin{eqnarray}
\sup_{1 \leqslant i \leqslant N} {\mathbb E}\left[ \sup_{0 \leqslant t \leqslant T}\vert X_t^i - Y_t^i \vert ^2\right] &\leqslant&2\sup_{1 \leqslant i \leqslant N} {\mathbb E}\left[ \sup_{0 \leqslant t \leqslant T}\vert X_t^i - X_t^{i,N} \vert ^2\right] \nonumber\\
&&+2\sup_{1 \leqslant i \leqslant N} {\mathbb E}\left[ \sup_{0 \leqslant t \leqslant T}\vert X_t^{i,N} - Y_t^i \vert ^2\right]\nonumber\\
&=:& I_1+I_2.
\label{35}
\end{eqnarray}
For $I_1$, it follows from the same deduction as that of (\ref{linest}) that for $\forall i=1,\ldots, N,$
\begin{eqnarray*}
{\mathbb E}\left(\sup_{0 \leqslant t\leqslant T} \vert X_t^i - X_t^{i,N} \vert ^2\right)&\leqslant& 2{\mathbb E}\sup_{0 \leqslant t\cdot \leqslant T}t\int_0^t\left|b(\theta, X_{s\land\cdot}^i, \mu^i_s) - b(\theta, X_{s\land\cdot}^{i,N}, \mu_s^N)\right|^2 \mathrm{d} s \\
&&+2C{\mathbb E}\int_0^T \left\|\sigma(X_{s\land\cdot}^i, \mu^i_s) -\sigma( X_{s\land\cdot}^{i,N}, \mu_s^N)\right\|^2 \mathrm{d} s \\
&\leqslant& 2(T+C)K'_1{\mathbb E} \int_0^T\(\|X_{s\land\cdot}^i -X_{s\land\cdot}^{i,N}\|_T^2+{\mathbb W}_2^2(\mu^i_s,\mu_s^N)\) \mathrm{d} s \\
&\leqslant& 2(T+C)K'_1\int_0^T{\mathbb E}\left(\sup_{0 \leqslant r \leqslant s}\vert X_r^i - X_r^{i,N} \vert ^2\right)\mathrm{d} s \\
&&+2(T+C)K'_1\int_0^T{\mathbb E}\left({\mathbb W}_2^2(\mu^i_s,\mu_s^N)\right) \mathrm{d} s.
\end{eqnarray*}
Gronwall's inequality admits us to obtain that
\begin{eqnarray*}
{\mathbb E}\left(\sup_{0 \leqslant t\leqslant T} \vert X_t^i - X_t^{i,N} \vert ^2\right)&\leqslant&2(T+C)K'_1\int_0^T  {\mathbb E} {\mathbb W}_2^2(\mu^i_s,\mu_s^N) \mathrm{d} s \cdot \exp\{2(T+C)K'_1T\} \\
&\leqslant&C\Gamma_N,
\end{eqnarray*}
where the last inequality is based on \cite[Theorem 5.8, P. 362]{cf}, and furthermore
\begin{eqnarray}
I_1\leqslant C\Gamma_N.
\label{i1}
\end{eqnarray}

For $I_2$, by the similar deduction to that of (\ref{linest}), it holds that
\begin{eqnarray*}
{\mathbb E}\left[ \sup_{0 \leqslant t \leqslant T}\vert X_t^{i,N} - Y_t^i \vert ^2\right]
&\leqslant&2T\int_0^T{\mathbb E}\left|b(\theta,X_{s\land\cdot}^{i,N},\mu_s^N)-b(\theta,Y^i_{\eta(s)\land\cdot},\mu_{\eta(s)}^M)\right|^2\mathrm{d} s\\
&&+2C\int_0^T{\mathbb E}\left|\sigma(X_{s\land\cdot}^{i,N},\mu_s^N)-\sigma(Y^i_{\eta(s)\land\cdot},\mu_{\eta(s)}^M)\right|^2\mathrm{d} s\\
&\leqslant&(2T+2C)K'_1\int_0^T{\mathbb E}\left(\|X_{s\land\cdot}^{i,N}-Y^i_{\eta(s)\land\cdot}\|_T^2+{\mathbb W}_2^2(\mu_s^N,\mu_{\eta(s)}^M)\right)\mathrm{d} s\\
&\leqslant&(2T+2C)K'_1\int_0^T{\mathbb E}\(2\|X_{s\land\cdot}^{i,N}-Y_{s\land\cdot}^i\|_T^2+2\|Y_{s\land\cdot}^i-Y^i_{\eta(s)\land\cdot}\|_T^2\\
&&+2{\mathbb W}_2^2(\mu_s^N,\mu^M_s)+2{\mathbb W}_2^2(\mu^M_s,\mu_{\eta(s)}^M)\)\mathrm{d} s\\
&\leqslant&8(T+C)K'_1\int_0^T{\mathbb E}\(\sup\limits_{0\leqslant r\leqslant s}|X_r^{i,N}-Y_r^i|^2\)\mathrm{d} s\\&&+8(T+C)K'_1T\sup\limits_k{\mathbb E}\left(\sup\limits_{t_k\leqslant r\leqslant t_{k+1}}|Y^i_r-Y^i_{t_k}|^2\right),
\end{eqnarray*}
where $\eta(s)=t_k, s\in[t_k, t_{k+1}]$ and the following fact is used:
$$
{\mathbb E}{\mathbb W}_2^2(\mu^N_s,\mu_s^M)\leqslant{\mathbb E}\left(\dfrac{1}{N}\sum\limits_{j=1}^{N}|X_s^{j,N}-Y_s^j|^2\right)={\mathbb E}|X_s^{i,N}-Y_s^i|^2.
$$
The Gronwall inequality admits us to obtain that
\begin{eqnarray}
{\mathbb E}\left[ \sup_{0 \leqslant t \leqslant T}\vert X_t^{i,N} - Y_t^i \vert ^2\right]\leqslant 8(T+C)K'_1T\sup\limits_k{\mathbb E}\left(\sup\limits_{t_k\leqslant r\leqslant t_{k+1}}|Y^i_r-Y^i_{t_k}|^2\right)e^{8(T+C)K'_1T}.
\label{mides}
\end{eqnarray}

In the following, we estimate ${\mathbb E}\left(\sup\limits_{t_k\leqslant r\leqslant t_{k+1}}|Y^i_r-Y^i_{t_k}|^2\right)$. By (\ref{34}), it holds that
\begin{eqnarray}
{\mathbb E}\left(\sup\limits_{t_k\leqslant r\leqslant t_{k+1}}|Y^i_r-Y^i_{t_k}|^2\right)&\leqslant&2{\mathbb E}\left(\sup\limits_{t_k\leqslant r\leqslant t_{k+1}}\left|\int_{t_k}^r b\left(\theta,  Y_{t_k\land\cdot}^{i}, \mu_{t_k}^M\right)\mathrm{d} u \right|^2\right)\nonumber\\
 &&+2{\mathbb E}\left(\sup\limits_{t_k\leqslant r\leqslant t_{k+1}}\left|\int_{t_k}^r\sigma\left( Y_{t_k\land\cdot}^{i}, \mu_{t_k}^M\right)\mathrm{d} W^i_u\right|^2\right)\nonumber\\
&\leqslant&2\frac{T^2}{M^2}{\mathbb E}|b\left(\theta,  Y_{t_k\land\cdot}^{i}, \mu_{t_k}^M\right)|^2
+2C\int_{t_k}^{t_{k+1}}{\mathbb E}\|\sigma\left( Y_{t_k\land\cdot}^{i}, \mu_{t_k}^M\right)\|^2\mathrm{d} u\nonumber\\
&\leqslant&2\frac{T^2}{M^2}{\mathbb E}|b\left(\theta,  Y_{t_k\land\cdot}^{i}, \mu_{t_k}^M\right)|^2+2C\frac{T}{M}{\mathbb E}\|\sigma\left( Y_{t_k\land\cdot}^{i}, \mu_{t_k}^M\right)\|^2\nonumber\\
&\leqslant&2\frac{T}{M}\left(\frac{T}{M}+C\right){\mathbb E}\left(1+\|Y_{t_k\land\cdot}^{i}\|_T^2+\|\mu_{t_k}^M\|^2_{\lambda^2}\right)\nonumber\\
&\leqslant&2\frac{T}{M}\left(\frac{T}{M}+C\right)\left(1+{\mathbb E}\left(\sup\limits_{0\leqslant r\leqslant t_k}|Y^i_r|^2\right)+{\mathbb E}\(1+|Y^i_{t_k}|\)^2\right)\nonumber\\
&\leqslant&6\frac{T}{M}\left(\frac{T}{M}+C\right)\left(1+{\mathbb E}\left(\sup\limits_{0\leqslant r\leqslant t_k}|Y^i_r|^2\right)\right),
\label{ttkes}
\end{eqnarray}
where in the last second inequality we use the fact that
$$
{\mathbb E}\|\mu_{t_k}^M\|^2_{\lambda^2}=\frac{1}{N}\sum\limits_{j=1}^{N}{\mathbb E}\int_{{\mathbb R}^d}(1+|x|)^2\delta_{Y_{t_k}^j} (\mathrm{d} x)=\frac{1}{N}\sum\limits_{j=1}^{N}{\mathbb E}(1+|Y_{t_k}^j|)^2={\mathbb E}\(1+|Y^i_{t_k}|\)^2.
$$
Besides, from the similar deduction to that of (\ref{linest}), it follows that
\begin{eqnarray*}
&&{\mathbb E}\left(\sup\limits_{0\leqslant t\leqslant T}|Y^i_t|^2\right)\\
&\leqslant&3{\mathbb E}|\xi|^2+3T{\mathbb E}\int_0^T\left|b(\theta,Y^i_{\eta(s)\land\cdot},\mu_{\eta(s)}^M)\right|^2ds+3{\mathbb E}\int_0^T\left\|\sigma(Y^i_{\eta(s)\land\cdot},\mu_{\eta(s)}^M)\right\|^2ds\nonumber\\
&\leqslant&3{\mathbb E}|\xi|^2+3(T+1){\mathbb E}\int_0^TK'_1\left(1+\|Y^i_{\eta(s)\land\cdot}\|_T^2+\|\mu_{\eta(s)}^M\|^2_{\lambda^2}\right)\mathrm{d} s\nonumber\\
&\leqslant&3{\mathbb E}|\xi|^2+3(T+1)\int_0^TK'_1\left(1+{\mathbb E}\left(\sup\limits_{0\leqslant u\leqslant s}|Y^i_u|^2\right)+2{\mathbb E}(1+|Y^i_{\eta(s)}|^2)\right)\mathrm{d} s\nonumber\\
&\leqslant&3{\mathbb E}|\xi|^2+9(T+1)TK'_1+9(T+1)K'_1\int_0^T{\mathbb E}\left(\sup\limits_{0\leqslant u\leqslant s}|Y^i_u|^2\right)\mathrm{d} s.
\end{eqnarray*}
The Gronwall inequality admits us to obtain that
\begin{eqnarray}
{\mathbb E}\left(\sup\limits_{0\leqslant t\leqslant T}|Y^i_t|^2\right)\leqslant C.
\label{2semo}
\end{eqnarray}
Combing (\ref{mides})-(\ref{2semo}), we have that
\begin{eqnarray*}
{\mathbb E}\left[ \sup_{0 \leqslant t \leqslant T}\vert X_t^{i,N} - Y_t^i \vert ^2\right]\leqslant C\frac{T}{M}\left(\frac{T}{M}+C\right),
\end{eqnarray*}
and furthermore
\begin{eqnarray}
I_2\leqslant C\frac{T}{M}\left(\frac{T}{M}+C\right).
\label{i2es}
\end{eqnarray}

Finally, from (\ref{35}) (\ref{i1}) (\ref{i2es}), it follows that
\begin{eqnarray*}
\sup_{1 \leqslant i \leqslant N} {\mathbb E}\left[ \sup_{0 \leqslant t \leqslant T}\vert X_t^i - Y_t^i \vert ^2\right] \leqslant C \Gamma_N+C \frac{T}{M}\left(\frac{T}{M}+C\right).
\end{eqnarray*}
The proof is complete.
\end{proof}

\bigskip

Next, we construct a maximum likelihood estimator of the parameter $\theta$. Let $d=m=k=1$. Assume that ($\bf{H}'_1$)-($\bf{H}_2$) hold. And then define the maximum likelihood function
\begin{eqnarray*}
L^M_{T}(\theta)
&:=&\exp\left\{{\int_{0}^{T} \frac{1}{\sigma(Y_{\eta(t)\land\cdot}^i, \mu_{\eta(t)}^M) }\left(b(\theta, Y_{\eta(t)\land\cdot}^i, \mu_{\eta(t)}^M) - b(\theta_0, Y_{\eta(t)\land\cdot}^i, \mu_{\eta(t)}^M) \right)\mathrm{d} W^i_{t} }\right.\nonumber\\
&&\qquad -\left.{\frac{1}{2}\int_{0}^{T} \frac{1}{\sigma^2(Y_{\eta(t)\land\cdot}^i, \mu_{\eta(t)}^M) }\left(b(\theta, Y_{\eta(t)\land\cdot}^i, \mu_{\eta(t)}^M) - b(\theta_0, Y_{\eta(t)\land\cdot}^i, \mu_{\eta(t)}^M) \right)^2\mathrm{d} t}\right\}.
\end{eqnarray*}
Thus, the maximum likelihood estimator of the parameter $\theta$ is given by
\begin{eqnarray}
\theta_T^M := \arg\max_{\theta \in \Theta}{L^M_{T}(\theta)}.\label{52}
\end{eqnarray}

\section{An example}\label{exam}

In the section, we present an example to explain our results.

Consider the following MVSDE on ${\mathbb R}$:
\begin{eqnarray}
\mathrm{d} X_t = (\theta X_t + \beta {\mathbb E}[X_t]) \mathrm{d} t + \sigma\mathrm{d} W_t, \quad X_0 = x_0\in{\mathbb R},
\label{61}
\end{eqnarray}
where $\theta \in \Theta$ is a unknown parameter and $\beta, \sigma$ are nonzero constants.
Using the numerical simulation method in Section \ref{numsim}, we have the following numerical equation for Eq.(\ref{61})
\begin{eqnarray}\left\{\begin{array}{l}
Y_{0}^{i}=x_{0},\\
{Y_t}^{i}=Y_{t_k}^{i}+\left(\theta Y_{t_{k}}^{i}+\beta \frac{1}{N}\sum_{j=1}^{N} Y_{t_{k}}^{j}\right)(t-t_k) +\sigma(W^i_t-W_{t_k}^i) , \quad t\in[t_k, t_{k+1}].
\end{array}
\right.
\label{numsimeq}
\end{eqnarray}
In terms of the number $N$ of particles and the step size $M$, we draw Figure 1 and Figure 2. That is, we take $N=160, M=16$ in Figure 1, and $N=2560, M=256$ in Figure 2. Comparing Figure 1 with Figure 2, one can find that the numerical solution has higher frequency and smaller amplitude when the number of particles and the step size are larger.

\begin{figure}[ht]
\centering
\includegraphics[scale=0.4]{16-2.jpg}
\caption{Comparison of approximate solution and true solution, taking $\beta=\sigma=1, N=160, M=16$.}
\label{t16}
\end{figure}

\begin{figure}[ht]
\centering
\includegraphics[scale=0.4]{256-2.jpg}
\caption{Comparison of approximate solution and true solution, taking $\beta=\sigma=1, N=2560, M=256$.}
\label{t256}
\end{figure}

According to (\ref{errest}) in Section \ref{numsim}, we calculate the errors between the solutions of Eq.(\ref{numsimeq}) and the solution of Eq.(\ref{61})  and list them in Table 1. From Table 1, one can find that the error decreases when the number of particles and the step size increase.
\begin{table}[ht]
    \centering
    \begin{center}
    \fontsize{12}{12}\selectfont
    \caption{The errors between the numerical solution and the original solution when N, M take different values.}
    \setlength{\tabcolsep}{4mm
     \begin{tabular}{ cccccc }
   \toprule
\diagbox [width=1.8cm, trim=l] {$N$}{$M$} & 16 & 32 & 64 & 128 & 256 \\
   \midrule
		 160&0.0753     &0.0389    &0.0182   &0.0093    &0.0040\\
    		 320&0.0686    &0.0354    &0.0176    &0.0086    &0.0048\\
   		 640&0.0656    &0.0337    &0.0167    &0.0077    &0.0037\\
   		1280&0.0670    &0.0331     &0.0158    &0.0077    &0.0034\\
    		2560&0.0672    &0.0323    &0.0157    &0.0073    &0.0032\\
    \bottomrule
\end{tabular}
\label{error}
\end{center}
\end{table}

Finally, by the formula (\ref{52}) in Section \ref{numsim}, we get the maximum likelihood estimator $\theta^M_T$ as follows:
\begin{eqnarray}
\theta^M_T = \frac{\sum\limits_{k=0}^{M-1} Y_{t_{k}}^{i}(Y_{t_{k+1}}^{i}-Y_{t_k}^{i})-\sum\limits_{k=0}^{M-1} \beta Y_{t_{k}}^{i}·\frac{1}{N}\sum\limits_{j=1}^{N} Y_{t_{k}}^{j}\frac{T}{M}}{\sum\limits_{k=0}^{M-1} (Y_{t_{k}}^{i})^2\frac{T}{M}}.\label{63}
\end{eqnarray}
In terms of $T$, the values of $\theta^M_T$ present in Table 2, which indicates that the value of $\theta_T$ is closer to the true value $\theta_0 = -0.5$ when the time $T$ becomes larger.
\begin{table}[ht]
    \centering
    \fontsize{12}{12}\selectfont
    \caption{The maximum likelihood estimator $\theta^M_T$ with $\beta=\sigma=1, N=2560, M=256$.}
    \setlength{\tabcolsep}{4mm
    \begin{tabular}{ cccccc }
    \toprule
   		$T$ & 1 & 2 & 5 & 8 & 10 \\
   \midrule
	    $\theta^M_T$ &-1.0510  &-0.7420   &-0.5107   &-0.5009   &-0.4999\\
    \bottomrule
\end{tabular}
\label{esti}
\end{table}

\bigskip

\textbf{Acknowledgements:}

The authors are very grateful to Professor Xicheng Zhang for valuable discussions. The second author also thanks Professor Renming Song for providing her an excellent environment to work in the University of Illinois at Urbana-Champaign.

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I Am Confused By The Various Parts of Medicare.
Medicare consists of four parts: Part A covers hospital insurance, Part B covers Medical insurance, Part C is offered through insurance companies and offers an alternative to Medicare Supplement policies, and Part D is offered through insurance companies to provide prescription drug coverage.
For most people, there is no charge for Part A, but you must pay for Part B. You are also responsible for the premiums whether you elect Part C Medicare Advantage or a Medicare Supplement. You are also responsible for the premiums and copayments for Part D prescription drug coverage.
It is important to know that if yo do not elect certain parts of Medicare when you are first eligible, you will be subject to a penalty for late enrollment. The following link provides a good overview, which benefits you should apply for, when and where to apply, and which parts have penalties for late enrollment: http://www.medicare.gov/Publications/Pubs/pdf/11038.pdf.
Although the age to receive full Social Security retirement benefits is gradually being raised, individuals turning 65 should still enroll for Medicare. You are eligible for Medicare benefits the first of the month in which you turn 65 even if your birthday isn't until the end of the month. You can enroll in Medicare the three months before your birthday, the month of your birthday, and three months after. If you wish to have Medicare benefits when you turn 65, you should apply in the three months before your birth month.
I'm Covered by Medicare – Do I Need Additional Coverage?
Medicare doesn't cover everything and if you need a service not covered by Medicare, you will need to pay for it yourself unless you have other health insurance that will cover it. Even if Medicare Parts A and B cover a service, you are responsible for your deductible, coinsurance, and copayment. Under Medicare there is no limit to your maximum out-of-pocket. For this reason, most people insure themselves to cover the deductible and coinsurance. Your choices to supplement Medicare are a Medicare Supplement or Medicare Advantage plan.
Medicare also does not cover prescription drugs. To get Rx coverage, you must join a plan offered by an insurance company approved by Medicare. There are two options to get prescription drug coverage: through a Part D plan offered by insurance companies or through a Medicare Advantage (Part C) plan which is also offered by insurance companies. Regardless of which option you choose, you must be enrolled for Medicare Part A and B in order to be eligible for Part C or D. If you don't elect prescription drug coverage when you are first eligible, you'll likely pay a late enrollment penalty.
What if I am Still Working and Covered by my Employer's Group Health Benefits?
You should first check with your employer's plan administrator. You may be able to choose either your group plan or Medicare Parts A and B with a Medicare Supplement and Part D or with a Medicare Advantage Plan. If you remain covered under your employer's group plan, you can delay enrolling in and paying for Part B until you need it.
Are the Medicare Supplement and Part C Medicare Advantage Plans All the Same?
No. The Medicare Supplement plans are standardized with the same plan name and benefits available in all states except Massachusetts, Minnesota, or Wisconsin. Thus a Plan F will have the same benefits regardless of which company you purchase from. The differences will be in the premium they charge and their underwriting requirements.
Part C Medicare Advantage plans can vary greatly. They are approved on a county-wide basis and may not be available in all counties. Medicare approves these plans annually prior to Open Enrollment. With a Medicare Advantage plan, the premium and benefits will vary by company and county. Some plans will offer additional benefits such as vision, hearing aids, and health clubs memberships among others.
What is the Difference Between the Different Part D Prescription Drug Plans?
Each plan has their own list of drugs that they cover, commonly known as their formulary list. Depending on which medications you take, your drug may be on one company's formulary list, but not on another. That is why it is important to run your medications through the calculator on the Medicare website to be informed about which plan is most advantageous for you (the REGIT staff will be happy to help you with this). Each company's plan design can also vary in the amount of the deductible (if any) and the copay or coinsurance, as well as the premium.
What is the Open Enrollment Period I've Read About?
Each year between October 15 and December 7 you have an opportunity to enroll in or change Part C Medicare Advantage or Part D Prescription drug plans. Each year you should review your personal situation to see if moving to another plan will be beneficial to you. This is especially important if your medications have changed. When you enroll in a Part C Medicare Advantage or Part D prescription drug plan, you generally are committed to that plan for one year (subject to limited exceptions). Medicare Supplement plans are not subject to open enrollment periods and you can change at any time you choose, although you may be subject to underwriting.
Although some people think that Medicare and Medicaid are the same, they are actually very different programs. Medicare is generally for people over 65, although some disabled people under 65 qualify for Medicare. Medicaid is a state-run program that provides medical and hospital coverage for people with a low income and little or no resources. Because Medicaid is run by the state, each state has their own rules about who is eligible and what is covered by Medicaid. You can call your state Medicaid office for more information.


text:
Q: How to use multilines in textbox I have a calculator and want to have exactly 3 lines in my textbox. First line will be number A, second operator and third will be number 2. Now I am using just one line for my application. Can somebody show me how to do that? And I also need to know how CLEAR ONLY first line for exaple and how TO ADD NEW TEXT for exaple to line second without changing other lines.

A: I would use 3 textboxes on top of each other to do this. In this way you can easily clear the particular line you want to clear.
Of course, the exact way you do this depends on whether you are using silverlight, WPF, Windows Forms, or Asp.net.
But for Silverlight or WPF you'd do something like this:
<StackPanel>
    <TextBox x:Name="Number1" BorderThickness="0" />
    <TextBox x:Name="Operator" BorderThickness="0" />
    <TextBox x:Name="Number2" BorderThickness="0" />
</StackPanel>

Setting BorderThickness to 0 effectively removes the borders from the textboxes. You can also set FontSize to a bigger font, and set HorizontalAlinment="Right" for a typical calculator style.
You'd then need an event handler (I'm making assumptions about how you want this to work here), that would respond to the enter key, so that the focus moves to the next textbox.

A: Out of the box you can have a multiline textbox by setting the Multiline to true, but not a text box with three lines .If you want a textbox with three lines I guess you need to develop a custom control yourself.

A: This doesn't sound like a very user-friendly interface. You're trying to reinvent the wheel in the world of GUI, but it's never a good idea to do so. Why don't you just use three text boxes, one for each input?
On the technical side, in order to create a multi-line text box, you have to set its MultiLine property to true. And then you can split the text into lines using Environment.NewLine:
textBox.Text = "1" + Environment.NewLine + "+" + Environment.NewLine + "3";

A: Why don't you use an Multi Line Textbox (Textarea)? In your application keep an array of strings for individual lines. When displaying concatenate the array and set the text on text area.

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I recorded myself doing dumbbell military presses recently, and as always, I was surprised. My wrists were slightly bent, which I never would have noticed otherwise and could cause issues.
When I saw the recording, I realized I needed a cue to keep my wrists neutral. "Knuckles up" does the trick, and I filed it away to be used in my next workout. This cue will be one more technique in my invisible toolbox that helps me push harder, safely. Call it lifting mindfully or whatever you like, but it can be the difference between getting results with strength training and getting frustrated and giving up.
I believe you can tell who is lifting mindfully and who is not. Every time I walk into the gym, I see people flinging weights around, losing tension between reps, using momentum, and rushing through sets.
People who are lifting with purpose – feeling every contraction and focusing on the muscles working – are hard to spot. But they stand out from the crowd because (a) they've achieved some impressive physique goals, and (b) they're lifting weights differently from everyone else.
Tensing my entire body during every lift. When you let your muscles go slack between reps, you're missing out on one of the most important mechanisms of building muscle and strength – constant tension.
Letting the dumbbells "float" in my hands. This helps me feel my front and lateral deltoids working, prevents my hands from fatiguing as fast, and lets me push harder. In other words, I'm actively feeling the muscles working instead of powering my way through.
Lifting on auto-pilot is a two-headed monster. If you know how to do it well, it benefits you. If you get lazy, it'll take over and waste your time. It's obvious who's using it well and who is not.
When you lift on auto-pilot in a good way, you're so familiar with proper technique that you don't have to focus on it too much. You know what an exercise should feel like and which muscles it should activate so that you can push hard safely.
When you lift on auto-pilot in a not-so-good way, you're just going through the motions.
Let's zoom in on the gal doing bicep curls a few feet away. She's letting her arms fall slack at her sides between every rep, losing muscular tension in her biceps. Her weights even swing a little at the bottom.
First of all, losing tension is a terrible habit that cheats you out of gains. Second, your weights should be under control at all times – no swinging or moving them outside their narrow path of motion.
Of course, you could say that these mistakes are form issues. But to me, it's all about mindfulness: Are your movements careful and methodical or careless and rushed?
I know – it's more difficult to be mindful. You're in a hurry, and does it really matter that much if you feel the muscles working?
Look at people who lift mindfully versus those who don't, and you'll have your answer.
Back to letting the dumbbells "float" in your hands during military presses. This technique helps you feel the muscles that should really be working while preventing excessive fatigue in your hands.
It's easy to clinch the weights and try to force them overhead with your hands. Anyone can do this. But how strong and sculpted will your deltoids really get?
When I say that feeling which muscles are working is important, I'm not throwing around touchy-feely yoga room talk. It's hugely important and something that most people don't do.
That gal (or guy) doing the bicep curls? If she focused on the eccentric motion – lowering under control and maintaining tension – she would feel her biceps a heck of a lot more. A nice pause and squeeze at the top and her biceps would feel a lot more fatigued. In other words, her biceps would be getting bigger and stronger.
Studies have even suggested that self-talk and visualization between sets can increase strength and performance gains. Mindfulness is only now gaining the spotlight, but bodybuilders have known this for years.
Mindfulness is the difference between getting results and just wasting your time, getting frustrated, and eventually quitting.
I'd say it's worth it to go the extra mile, wouldn't you?
YESSSSSSSSSSSSSSSSSSSSS !!!!! I call it mind muscle link but same thing! It is so crucial not only for progress but not injuring oneself. If you let the mind wander while lifting & not paying attention to form & feel, not good! I am sooooooooooooooooo unapproachable in the gym & proud of it!!!
Great post Suzanne & you are looking fantastic!!!!
This, yes! The mind-muscle link needs to gain the same importance as in other sports. Weightlifting has become so mainstream and will only continue to grow. As we talked about elsewhere (I think), injuries will become epic if that link isn't learned. (Plus, well, GAINZ ;)).
I like the concept of mindful exercise. Because of injuries my workouts are almost completely bodyweight, and that's a great way to be mindful, especially if you've been lifting for a while. There are lots of creative bodyweight exercises, which almost force you to be conscious of your movements. You can probably get the same effect with weights by trying different exercises. But for me the bodyweight ones have completely changed my mindset about working out.
I like your perspective, Kris and agree with you. Bodyweight training is tough. I like it though, because as you say, it requires strong mindfulness. I'm so happy you've found something you love!
Nothing wrong with a little "touchy-feely talk". 🙂 Weights have always been a form of mindfulness for me and I coach my clients to think of their gym time that way. They need to leave the world outside, focus on the the muscles they are working and breathe.
You are one epic woman!!!
I won't forget the "Knuckles up" as I'm the same, I'm very weak in my wrists and find that my wrists aren't always neutral!
My partner Ben is always on me about keeping tension throughout the exercise. I used to be :that girl" who would relax after each rep, I've seen a bigger difference i my "gains" when i do things properly, i think sometimes its just a lack of knowledge.
Well it sounds like you've come a long way Sam. Nice job!
Great article reinforcing the mind and muscle!
Musculate, now that's a new one for me ;).
I do think that recording yourself doing certain exercises that you have been doing awhile or even just starting or having someone watch you can be a good tool as you may not realize you are not using correct form. It is important to be mindful of every move. Great information, thanks for sharing!


I've been fishing w. Capt. White for over 15 years. He has top notch gear, works hard and is very professional. He introduced me to flats fishing in the Hamptons for striped bass, something I thought to be utterly impossible. No matter how good the fishing is, we always have a great day on the water. I recommend Merritt to all my friends and customers and they all tell me they had a terrific time. It's nice to know I have a guide and friend for life!
In 2004 I turned to Hell's Bay Boatworks as my sole supplier of both the Whip Ray Pro 17.8′ flats skiff and the Cape Sable 21′ center console. Hell's Bay is not one of those large scale boat builders that pumps thousands of hulls out of their door per year. Instead their objective has been about uncompromising quality and attention to detail. Hell's Bay Boatworks is not an outsourcer! They are a one stop shop, a complete boatworks, involved in customization, tooling, design and rigging. With the finest raw materials at their disposal, Hell's Bay utilizes state of the art vacuum bagged ATC core-cell construction resulting in lightweight, versatile hulls of incredible strength and stiffness.
Hell's Bay addressed my needs perfectly when it came to fishing the Fall striped bass migration off of Montauk Point, New York. There are many days it seems when only an aircraft carrier will do when it comes to fishing the surrounding waters of Montauk. The Hell's Bay Cape Sable 21′ outshines all other 21′ center consoles in its class at every level! Utilizing the same state of the art construction techniques as its flats skiff brethren, the Cape Sable 21′ offers up a bombproof hull capable of handling the punishing seas of Montauk. Not only is the Cape Sable 21′ a solid, reliable workhorse, anyone who has witnessed this boat in action can attest to its inherent beauty with some of the finest fishing appointments available only on boats twice its size!!
As far as a flats skiff goes, the Hell's Bay Whip Ray Pro 17.8′ was an obvious choice. Any fisherman who has spent as much time on the water as I have knows that those idyllic sea state conditions that we all dream of are more often the exception than the norm. Whether its crossing the Boca Grande Channel in Key West or navigating the cantankerous waters of Gardiner's Bay, Long Island, the Whip Ray Pro 17.8′ delivers! This is just one side of the equation, however. Not only does the Whip Ray Pro 17.8′ offer a dry, comfortable ride in the most severe conditions, it poles into the wind and up tide with little effort, all the while negotiating the shallowest of waters. You say this can't be true! Hell's Bay makes the impossible, possible!
I have had the good fortune to fish with Captain White for many years in the bays of Eastern Long Island and at Montauk. Merritt is a superb guide and a true gentleman. He knows his home waters very well, and he is willing to go to any lengths to locate fish. Whether we experience an epic bite or have to work for the fish, I always have a great time fishing with Merritt.
As with my boats, my fishing tackle is of the highest quality and "bombproof construction." Rest assured that the reel drags on my equipment are silky smooth, the knots are 100% and the hooks are razor sharp! The only thing you have to be concerned about are the fish in front of you! Catch 'em up!


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News/Media/Blog/News/Cordoba Corporation President & CEO George L. Pla Adds "Author" to his Resume with the Launch of the Book "Power Shift"
Cordoba Corporation President & CEO George L. Pla Adds "Author" to his Resume with the Launch of the Book "Power Shift"
August 10, 2018	| News
Cordoba Corporation is pleased to announce that President & CEO George L. Pla has written a book, Power Shift, How Latinos in California Transformed Politics in America, which has been published by Berkeley Public Policy Press. Power Shift was written after intense research and extensive oral interviews conducted for almost a decade. The book was co-authored by David R. Ayón, a Senior Strategist and Advisor to Latino Decisions, and Senior Fellow at the Center for the Study of Los Angeles.
In celebration of the release of Power Shift, Los Angeles Mayor Eric Garcetti and Los Angeles City Council Member Gil Cedillo (CD 1) are hosting a Power Shift Book Launch and Signing on Friday, August 10, 2018 in the Los Angeles City Hall Council Chambers from 4:00 – 6:00 p.m. Messrs. Pla and Ayón will be in attendance, along with important community leaders featured in the book.
Power Shift recounts the origins and development of 10 Los Angeles Latino leaders who transformed politics and government, forging a progressive political tradition in the process. Challenging disadvantage and discrimination, these key figures integrated the Los Angeles City Council as well as school and county boards; established Latino collective political action in the state legislature and in Congress; restructured the US Census and how California's electoral districts are drawn; and played major roles in reforming the US labor movement and US-Mexico relations.
The authors' objective in writing Power Shift was to chronicle how a political movement in Los Angeles had a profound and transformative impact in both California as well as the nation. At its heart, Power Shift is a story of the American Dream. While the perspective focuses on Latino politics, it is an American story and echoes the experience of other ethnic groups such as Irish-Americans and Italian-Americans as they rose to prominence in earlier generations of American politics. The book also discusses how this social movement spearheaded critical changes in public health, law enforcement, education, housing, immigration, transportation, and environmental policies.
Power Shift will serve as a textbook for high schools, colleges and universities to supplement current teachings in such subjects as American History, Social Studies, Civics and Political Science. The book will be taught at Loyola Marymount University and the University of California, Berkeley beginning this fall. Book signing events are being planned at the California State Capitol, Long Beach, Washington, D.C., Boston, New York, and at colleges and universities including East Los Angeles College, USC, UCLA, Berkeley, Stanford, Cal State Los Angeles, Harvard, and NYU, among others.
"We believe the book will inspire and educate students about this unique political movement that is rooted in Los Angeles," stated George L. Pla. "I am humbled by the support we have received from scholars, educators, and political leaders, who have provided comments in support of the book, including Former Secretary of Defense Leon Panetta, who graciously penned the Foreword for Power Shift."
Power Shift is available for presale on the Amazon books website at the following link: https://www.amazon.com/Power-Shift-California-Transformed-Politics/dp/0877724563
Contact: Communications Department, Cordoba Corporation
Phone: (213) 895-0224 Email: communications@cordobacorp.com
About Cordoba Corporation: Founded in 1983, Cordoba Corporation is statewide full-service engineering, construction management and program management firm specializing in the delivery of infrastructure projects in transportation, education, water and energy sectors. Cordoba Corporation has offices in Sacramento, San Francisco, Chatsworth, Los Angeles, Santa Ana and San Diego, and has been recognized by Engineering News-Record as a Top 50 Program Management Firm and a Top 100 Construction Management-for-Fee Firm in the nation, and a Top 100 Design Firm in California.

redpajama_set_name: RedPajamaCommonCrawl


Great recommendation PBM. Small menu of 3 starters and mains which told me they're serious about the food, plus 5 deserts (this is Paris after all). Less than €100 with wine. Walked there and back through beautiful St Germaine. Date night heaven!


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Fees? Insurance? Appointments? Cancellation Policy?
Initial Chiropractic fees range between $60-$160 depending on the severity and complexity of the condition. Most PPO and Medicare insurance plans are accepted for chiropractic care. However, if your insurance company doesn't' cover chiropractic, you have a HMO, or you don't have insurance coverage at all, we offer a range of affordable cash payment plans that correlate to your current condition and health goals. Payment options are check, cash, Visa/Master card/ Discover/AM EX, Care Credit, and Flexible Spending Accounts. In any case, Dr. Noey will work with you on getting you the care you need at a cost you can afford.
Colon Hydrotherapy is the safe, efficient way to remove built-up toxins and waste from the large intestine. Our office uses the FDA approved LIBBE open system, twice purified water and sterile, disposable tubes.
Please call us at 815-733-5282 to schedule your appointment, we will do our best to get you in as soon as possible. We understand that occasionally same day appointments are necessary. We will make every effort to accommodate you as quickly as possible.
24 hours notice is required for all cancellations, we make every effort to contact all patients the day before their appointment. However, should an emergency arise and you need to reschedule please leave a message on our voice mail system by 8AM (on the day of the appointment). Not doing so will result in a $35 fee for same day cancellations.


Something I like to do at the start of each new year is take stock of the unread books still sitting on my bookshelves.
I have just the one shelf for unread books, and the reason being is that it helps me keep my unread books to a minimum - I do not put unread books in my actual book collection until I've read them. Although I have just the one shelf, I will say that I make good use of all that space on the shelf - I pretty much play tetris with my unread books!
This year, I'm going into 2017 with a total of 34 books on my unread shelf. In 2016, one of my goals was to end the year with just 15 books on my unread shelf, and whilst clearly that goal hasn't been achieved, I'm very happy with where I am with my unread shelf - I think it is manageable, with a variety of books to suit pretty much any mood of mine. I recently shared about how my perspective on books and reading has shifted over the past year or so, and in doing so, I see myself ending 2017 with a lot less than 34 books on my unread shelf.
In previous years whilst completing this inventory task I've left out books sitting on my Kindle, however this year I'm adding them in. They were left out before as I read so few ebooks in comparison to physical books, however I'm keen to make more use of my Kindle this year and there are a reasonable number of unread titles on there.
If you're keen on decreasing your own stack of unread books this year, then I'd highly recommend taking stock of the titles you have. One of the reasons I do this task routinely is to have a clear knowledge of exactly what I own, and from there being able to focus on a plan of action. In years prior, I have also decluttered my shelves at the same time, however I felt no need to do that this year.
Wishing you all a wonderful year of reading!
I'm always going back and forward with ebooks - I go through phases! Thanks!
I have a bit more than 34 unread books in my house. I took stock partway through last year and scared myself a bit. lol I just did a big purge to tidy up my shelves a bit and now I guess I should take stock again!
Haha! It's good to have a purge of books every now and then. I hope you're able to enjoy a bunch of your unread books this year.
This is a great idea, I should really do it for my Kindle as I'm sure I have amazing titles and have forgotten I downloaded. I highly recommend Behind Closed Doors from your Kindle. I'm currently reading Small Great Things, very hard read but good.
You should - you may discover a hidden gem on your Kindle! I actually purchased Behind Closed Doors after reading your review of it - I'm hoping to get round to it soon.
I also keep my read and unread books separated. I currently have 42 unread books, and I'm doing the Mount TBR challenge to help me get through them. By the end of the year, I want to read at least 36 of them.
I just can't put them all together as I know many people do - I'd probably forget which ones I had and hadn't read!
I think 42 is a manageable amount of unread books, and I'm sure you'll achieve the goal of reading 36 this year. Good luck!
Once again, your unread shelf looks soooo manageable! I know you will do so well reading from your unread shelf this year. I'm going to work on it too, only difference is I am reading from my unread shelves as in many, many shelves -- haha!
Haha! I think you're going to do great with getting the books on your unread shelves down this year, Christine! It seems like you genuinely want to tackle them and that definitely helps in working through them.
I only wish my unread books were this many!! Lots of good titles for you to look forward to this year.
Haha! I've got a great number of titles I just want to jump right into.. But one at a time eh!
I'll be sure to share my thoughts here when I do get round to reading Philippa Gregory, which I hope will be very shortly!
I hope you reach your goal of ending the year with fewer TBR books than you started with. I will be happy if I can get the number down below 700 before the end of this year.
Good luck with lowering your TBR also!


Diesel supply bedevils investments in Africa's internet: solar can help.
"Such nighttime frets arise from the fact that telecoms towers – which much of Africa relies upon to enable internet access – need diesel-powered generators to keep humming. And the supply of diesel in sub-Saharan Africa is prey to theft, political upheavals and economic volatility.
As a result, the business model of "towercos" such as Kulshrestha's, as well as the operation of some 170,000 telecoms towers and the provision of wireless internet for communications, banking and commerce in large swathes of the continent all hinge on the vicissitudes of the market in the fuel.
"Towercos", which have attracted high profile investors from the West such as Goldman Sachs and George Soros, operate by buying portfolios of towers from mobile phone operators and then lease back space for their transmission equipment, reducing costs for the operators while creating economies of scale by opening up the towers to sharing with rival operators.
The problem is that in much of sub-Saharan Africa (aside from South Africa), the electricity supply from the grid is unreliable, necessitating the deployment of the diesel generators either as back up or, in poorly-connected rural areas, the prime source of power.
The fuel is far more expensive than mains electricity and is the largest single overhead in running towers, accounting for up to 75 per cent of costs where a generator runs for 16 hours a day. In Nigeria, for example, power sourced from generators costs around $0.35 for a kilowatt-hour – four to five times the price of mains electricity.
"The price spikes I worry less about, it's more the supply," says Kevin Koch, chief operating officer of Helios Towers Africa, which has 7,800 towers in and is backed by the funds of George Soros, Madeleine Albright and Jacob Rothschild.
In Ghana, which has a comparatively high 72 per cent electricity grid access, according to World Bank figures, and up to 21 hours a day of supply, this is less of an issue for towercos.
By contrast, the less-than-half of Nigerians who have mains access often only get a handful of hours of daily supply. Yet the alternative of obtaining diesel from a licensed vendor can be difficult, compounded by frequent shortages, limited refining capacity and and strikes by workers.
Elsewhere, the vastness of the Democratic Republic of Congo – a land mass four times the size of France– means that while a litre of diesel costs around $1.40 in the capital, Kinshasha, in the remote interior it sells for as much as $3 to $4, says Koch.
These factors have led some tower owners to hedge against fuel price increases with oil derivatives such as forward contracts, according to one advisor. More commonly, leasing agreements contain clauses that share fuel price increases with the tenants – the mobile operators. And many companies contract with several bulk suppliers to preclude shortfalls.
"We hold between 800 to 1000 litres on a site so even if there's a two-week shortage it will be OK," adds Koch.
Yet as tower sites are unmanned most of the time this makes them a prime target for fuel pilfering.
"Some operators will admit they are losing a quarter of their diesel to theft," says Terry Rhodes, co-founder of Eaton, which operates 1,500 towers across Ghana, Kenya, Uganda and South Africa. "This isn't just a guy with a jerry can – it's organised crime. When you get deliveries you have to make sure it's the right amount and that it hasn't been diluted en route".
Competition is spurring towercos to drive down costs by limiting diesel use. Many have systems that remotely monitor the level of fuel in tanks and how long generators run for. Increasingly popular are 'hybrid' power set-ups, comprising long-lasting deep cycle batteries that charge either from the grid or a diesel generator.
"Solar panels are an option for more spacious rural sites," says Bob Hurley of Eltek, a Norwegian Nasdaq-listed company that has developed more than 500 power systems for towers in Africa at a cost of around $30,000 each.
In the long run this makes commercial sense. A study by the mobile industry body GMSA claimed that adoption of green power in Ghana, Cameroon, Senegal and Nigeria could realise annual savings of up to $200m and reduce diesel consumption by more than 70 per cent.
….Perhaps the biggest driver towards greater energy efficiency is rising consumer demand. Mobile subscriptions in sub-Saharan countries have grown to 475m from 90m in seven years, but fewer than one in three Africans has a mobile phone, according to GMSA.
As 3G is rolled out, some industry figures believe the number of towers needs to double.
This opportunity for investment is attracting international investors such as Goldman Sachs, which participated in a $490m capital raising earlier this year deal by Nigeria-based IHS, the continent's biggest independent towerco with 8,250 towers.


You understand that content is what attracts prospects to your website. What if you could make your marketing even more effective? Incorporating video content into your content strategy can add significant lift to your marketing results.
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We believe that people deserve to hear your story. There's more to your business than just what are your products and services. People need to know why you do what you do. They want to connect with you and build a relationship with you and there's only so much that written content and photos can do to connect on an emotional level. This is why video is so powerful.
Product demo videos or e-commerce videos showcase your products and present viewers with a clear idea and details exactly how your product works.These videos are especially helpful to show how your products move and function. Video provides an ideal environment for displaying your product's functionalities.
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Video content marketing videos are a perfect complement business blogging. Video blogging or video content marketing videos help attract leads to your website, build relationships with your viewers, educate your ideal prospects and is an extremely effective sales tool.
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Nahum 1-2 God is jealous, and the LORD revengeth; the LORD revengeth, and is furious; the LORD will take vengeance on his adversaries, and he reserveth wrath for his enemies. 3 The LORD is slow to anger, and great in power, and will not at all acquit the wicked: the LORD hath his way in the whirlwind and in the storm, and the clouds are the dust of his feet.
All emergency response orders have been issued and are being carried out at this moment with a high degree of competence and skill.
Members of the congregation in Florida have been evacuated and are in route to a safer destination.
I would like to thank the Apostles over the state of Florida for being efficient and effective during this crisis as well as the Chief Priests from the state of Florida. Apostle Qatazapahla for his dedication and hard work with the Chief Priests and brothers staffed to work with him.
I would also like to thank the Elect Lady, the Chief of Staff, for her manly stomach in managing the various orders I have delivered to her, with the upmost professionalism.
Apostle Yatazar and the Ministry of Health and the Nation Services for being ready and on point and jumping right into action to make life easier for the brothers, sisters, and their children.
Always and forever thanking the Almighty Christ for bestowing upon us his wisdom and his knowledge to stabilize these evil times we are living in.
May God continue to bless the nation of Israel forever.
Matthew 25-35 For I was an hungred, and ye gave me meat: I was thirsty, and ye gave me drink: I was a stranger, and ye took me in: 36 Naked, and ye clothed me: I was sick, and ye visited me: I was in prison, and ye came unto me. 37 Then shall the righteous answer him, saying, Lord, when saw we thee an hungred, and fed thee? or thirsty, and gave thee drink? 38 When saw we thee a stranger, and took thee in? or naked, and clothed thee? 39 Or when saw we thee sick, or in prison, and came unto thee? 40 And the King shall answer and say unto them, Verily I say unto you, Inasmuch as ye have done it unto one of the least of these my brethren, ye have done it unto me.
Shalom to the congregation of the Nation of Israel in the Caribbean Islands that was affected by the storm and the members in Puerto Rico as well.
Shalom to all the congregation in the state of Florida, waiting to see if Irma will hit with deadly force. I am monitoring the situation. I have ordered the Apostles of the state of Florida to keep me updated on the situation on the ground and to keep in close contact with the Chief Priests for continuous reports.
I have put the Chief of Staff on notice to be prepared to issue the emergency response orders if needed, if necessary to relieve the members of the nation that may need our assistance.
Based on the developments in the next hours we will issue the necessary response orders.
Shalom and may God bless the Nation of Israel forever.
Shalom Nation of Israel, to all twelve tribes, peace and blessings forever more to the whole holy family.
Thus saith the LORD GOD (ISAIAH 43:10) "Ye are my witnesses, saith the LORD, and my servants whom I have chosen: that ye may know and believe me, and understand that I am he: before me there was no GOD formed, neither shall there be after me."
You have seen, and are seeing, what the ALMIGHTY is doing in the world. Floods, hurricanes, tornadoes .. DO not doubt, GOD ALMIGHTY is making himself known in the world and among us. Among us because he is letting us know these things is going to happen before they happen. (AMOS 3:6,7) "Shall a trumpet be blown in the city, and the people not be afraid? shall there be evil in a city, and the Lord hath not done it? 7- Surely the Lord GOD will do nothing, but he revealeth his secret unto his servants the prophets."
This is what GOD is doing, so let not your hearts be troubled because there is so much more to come, starting with a major EARTHQUAKE right here in AMERICA!!!!! GOD ALMIGHTY has begun to do these things because the Bible tells us HE IS directly responsible for these sudden calamitous events because of the unbelief. (YHWH) is showing his word to be true. (ST JOHN 16:13) "Howbeit when the Spirit of truth is come, he will guide you into all truth: for he shall not speak of himself; But whatsoever he shall hear, that shall he speak: and he will SHEW YOU THINGS TO COME."
Also the nation of Israel is the PLACE TO BE this Sabbath evening, which is Friday night (6-27-08), IN NEW YORK CITY HEADQUARTERS, THE TEMPLE, for there will I be giving a CLASS not to be missed, which is THE COMPLETE BREAKDOWN OF THE SECOND COMING OF JESUS CHRIST! ALL ARE WELCOME. Come early because I will be starting early seeing I have alot to go through.
racial unrest and civil injustice.
We all see that concerning events have reached your city.
have peace of mind knowing the end of all things.
The Killing of the Black Man.
I will be posting 30min clips on my website.
they know that a prophet have been among them.
whom be glory for ever and ever. Amen."
everyone that felt its tremors, and make them stop and think what's going on?
preserved by their faith in God and in their firm belief in this truth.
ALL PROJECTS IN MOVING PROGRESS.
1SG Zayan: "The God Sent Comforter said this is the year of judgment!!!!!! Did you know this past weekend the city of New York's 911 calls were so extremely high that the last time there were anywhere near as high was the last blackout in 2003.... A half million people were without power in the New York, New Jersey, and Conn. area....There were reported deaths from falling trees and even drownings from the judgment of the Almighty God... Praise the Lord for keeping us from the evil and preserving us and for sending us the God Sent Comforter ACHP Tazadaqyah to shew us all these things before the happen!!!!!"
"And he shall redeem Israel from all his iniquities."


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Q: Irreducibility in $\mathbb{Z}[\sqrt{10}]$. and modular arithmetic I am trying to show that 17 is irreducible in $\mathbb{Z}[\sqrt{10}]$. So, I suppose $17=ab$, Then, I consider the norm, $N$ and thus obtain:
$289=N(a)N(b)=(a^2-10b^2)(c^2-10d^2)$
Hence it suffices to consider the following two cases:
(1) $(a^2-10b^2)=17$ and $(c^2-10d^2)=17$
(2)$(a^2-10b^2)=-17$ and $(c^2-10d^2)=-17$
Now , $a^2\equiv 7 \pmod{10}$ is a consequence of the above case. Now, I am not sure what to do with the first case.
Please do not use reciprocity

A: The first case implies $a^2\equiv 2\mod 5$. However, the only squares modulo $5$ are $0,1$ and $4$. So this case can't happen.
The second case implies $a\equiv -2\equiv 3\mod 5$, and  the above remark shows it can't happen either.

A: In either case, we have $a^2-10b^2\equiv 0\mod 17$. If $b\equiv 0$, then $a\equiv 0$, so we $17^2$ divides $a^2-10b^2 = 17$, which is nonsense.
Thus we can divide by $b$, and letting $x=\frac{a}{b}$, we get $x^2\equiv 10\mod 17$. Thus by definition of the Jacobi symbol we must have $\left(\frac{10}{17}\right)=1$. But one can compute that actually we have $\left(\frac{10}{17}\right)=-1$, so this is a contradiction.


"Jorge Agustín Nicolás Ruiz de Santayana y Borrás", known as "George Santayana", was a philosopher, essayist, poet, and novelist. Santayana was raised and educated in the United States and identified himself as an American people/American, although he always kept a valid Spanish passport. He wrote in English and is generally considered an American man of letters. At the age of forty-eight, Santayana left his position at Harvard and returned to Europe permanently, never to return to the United States. His last wish was to be buried in the Spanish pantheon in Rome.
Santayana is known for famous sayings, such as "Those who cannot remember the past are condemned to repeat it", or "Only the dead have seen the end of war." Santayana, like many philosophers from the late nineteenth century, was a naturalist (that is, he denied the existence of supernatural beings, like gods and ghosts), but he found profound meaning in literary writings and in religious ideas and texts (which he regarded as fundamentally akin to literature). Santayana was a broad ranging cultural critic whose observations spanned many disciplines. He said that he stood in philosophy "exactly where [he stood] in daily life."
If you enjoy these quotes, be sure to check out other famous philosophers! More George Santayana on Wikipedia.


text:
Q: Converting special HTML characters back into their original strings I'm building a small parser that scrapes web pages and logs the data on them. One of the things to log is the post title of forums. I'm using a XML parser to look through the DOM and get this information, and I'm storing it like this:
// Strip out the post's title
$title = $page->find('a[rel=bookmark]', 0);
$title = htmlspecialchars_decode(html_entity_decode(trim($title->plaintext)));

This works for the most part, but some posts have certain special HTML character codes like – which is dash (-). How would I go about converting these special character codes back into their original strings?
Thanks.

A: Use html_entity_decode. Here's a quick example.
$string = "hyphenated&#8211words";

$new = html_entity_decode($string);

echo $new;

You should see...
hyphenated–words

A: Documentation is your friend:
html_entity_decode(trim($title->plaintext), ENT_XHTML, YOUR_ENCODING);
                                            ^^^^^^^^^^^^^^^^^^^^^^^^

A: This might help:
<?php
 function clean_up($str){
 $str = stripslashes($str);
 $str = strtr($str, get_html_translation_table(HTML_ENTITIES));
 $str = str_replace( array("\x82", "\x84", "\x85", "\x91", "\x92", "\x93", "\x94", "\x95", "\x96",  "\x97"), array("‚", "„", "…", "‘", "’", "“", "”", "•", "–", "—"),$str);
return $str;
}
?>


The index is designed to measure the performance of 30 top stocks by dividend yield in the Asia/Pacific region, which includes Australia, Hong Kong, Japan, New Zealand and Singapore.
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The following table presents a proprietary ETFdb rating for ETFs tracking the Dow Jones Asia/Pacific Select Dividend 30 Index.


text:
BOI launches new campaign to Woo foreign investors
In the past 50 years, Thailand has had to face some challenges, such as natural disasters for instance, and deal with their negative impact on businesses. The capacity to recover such setbacks is a priority for the global economy and international trade.
Marion Boccon-Gibod
With the support the ministry of Industry, BOI announced on March 2nd in Bangkok the launching of its new campaign to regain investor's confidence. The Japanese community is the main target of this new campaign, as Japanese investors have suffered the most during the recent flooding.
The campaign will start in Japan as the Prime Minister, Yingluck Shinawatra, is on state visit. "Unbeatable Thailand, Unparalleled opportunities" is meant to answer Thailand's new priority to restore its image as a preferred trade and investment destination and to strengthen investor confidence in the country's economy.
"Unbeatable Thailand, Unparalleled opportunities" is meant to answer Thailand's new priority to restore its image as a preferred trade and investment destination
The Japan Chamber of Commerce (JCC), which represents 1,370 Japanese firms operating in Thailand, has been taking questions from its members on whether the floods will hit Thailand again this year. This reflects the concerns of not only Japanese but all foreign investors in Thailand. About 70 % of Japanese plants damaged by the severe inundation last year now are able to resume their operations, while 20 per cent are not yet ready.
The Japan International Co-operation Agency (JICA) is cooperating with the Thai government in designing the flood-prevention system.
"Japanese and foreign investors here in Thailand export 50 per cent or more. Toyota, for example, exports pickup trucks to 113 countries, and it plans to increase the export volume of passenger cars as well. We believe that other products would do the same,"
said Tanada, who is president of Toyota Motor Thailand.
A New Campaign for a better Future Thailand's board of Investments (BOI)
In the past 50 years, Thailand has had to face some challenges, such as natural disasters for instance, and deal with their negative impact on businesses. The capacity to recover such setbacks is a priority for the global economy and international trade. The Thai government wishes to announce the new measures and policies that will be taken in order to help investors and business operators to recover in such cases, and therefore lower the impact of any future challenges.
The main argument of the BOI in this campaign is Thailand's ability to recover and overcome any issue, coming back stronger when facing an economical obstacle. The BOI is in fact pointing out Thai Fighting Spirit, saying it is "embedded in the DNA of all Thai people".
The fact that business, industrial and service sectors are recovering from last year's flood at a rapid pace is an argument the government wants to show international partners, in order to keep the growth rate of investments in the country. "Unbeatable Thailand" illustrates that spirit of fighting and risen above difficulties thanks to the country competitiveness and the unique strategy location, strong by its infrastructure and its skilled and thriving economy.
BOI to demonstrate Thailand's potential as an investment destination.
"Unparalleled Opportunities" refers to the variety of the different business opportunities in the country, from the agriculture to the automobile industry, as well as the electronic or electrical sectors. In fact, Thailand could become to this day the production base for the new ASEAN global market which would account for more than 600 million people.
Thailand is starting with Japan as they are confident in the trust of their Japanese investors, which have massively confirmed their participation to the campaign-related events. A good collaboration as well as a convincing "Thai Fighting spirit" should ensure the international investors' confidence in Thailand and protect the global economy, as well as Thailand's position in Asia. After Japan, the Thai government should try to take the campaign to other neighbors, starting with China and South Korea, their strong target.
Many events are already planned and the campaign should hopefully be a success in restoring Thailand's international image as one of the main economic powers in Asia.
In 2007, Japan and Thailand signed on Japan-Thailand Economic Partnership Agreement (JTEPA) and it came into effect. JTEPA is expected to enhance trade, investment and cooperation between two countries. The total trade volume between Thailand and Japan as well as Japan's direct investment in Thailand had increased after JTEPA came into force.
Related Topics:Board of Investment
China's economic slowdown may impact Thailand's export sector
India emerges as cheaper alternative to rice from exporters like Thailand
Foreign Investors' Confidence in Thailand Still High Despite Covid-19 Impact, BOI Survey Shows
Thailand's BOI Applications Show a 15% decline in value
Thailand's Board of Investment (BOI) approves 35.7 billion baht worth of investment projects
Investment37 mins ago


Would like to learn to use command StatusBarGetText and import to excel. However it doesnt seem to work. I could be missing something big here. Tried other software but result is the same.
Is this the correct way or is the other way?
StatusBarGetText "Retrieves the text from a standard status bar control." Excel 2003 does not use a "standard" status bar control. AU3_Spy.exe shows statusbar text as blank instead of "Ready" which indicates it is not a standard control.
What are you actually trying to read out of Excel? Perhaps it could be done in a different manner, possibly using a COM object.
I wrote a function that gets the text from Excel 2007's status bar.
Please confirm if it works on other versions of Excel. Or use AccViewer to work out and share modified versions of the function that work on other versions of Excel.
Doesn't seem to work on EXCEL 2003. Returns NULL for hCtl. Apparently NETUIHWND1 is undefined in EXCEL2003, needs to be defined or I'm missing something else. Don't see where NETUIHWND1 is defined in either Acc.ahk or AccViewer.ahk.
msgbox, %hWnd% ;Gets reasonable hWnd and ahk_Class is XLMAIN so this is OK.
Try AutoHotkey's Window Spy to identify the control name for the status bar, and try focusing the status bar with AccViewer, to find the information needed in order to modify the Acc_Get line.
It doesn't surprise me if the control name is different, for example, DirectUIHWND controls in Explorer folder windows, weren't in Windows XP, but were in Windows 7 (and probably Windows Vista).
The numbers where it says 'Path' and the numbers in the treeview hierarchy, might provide some clues.
The control name is EXCEL41 according to AccViewer. However, when I changed that the vText returned is still NULL. So there might be something else different (besides EXCEL41) needed in the Acc_Get command.hCtl is same as reported by AccViewer so that isn't the problem.
Probably you need to change "4.1" to something else, at which point the function will probably work.
Path is 4 for if that helps.
Last edited by dmatch on 22 Apr 2017, 13:23, edited 1 time in total.
I can't be sure that it will work, however if you ever see the status bar text, in the AccViewer window e.g. in the Name or Value field, that would suggest it is possible to retrieve the text.
[EDIT:] I believe AccViewer uses 1-based indexes, so the numbers are either 1 or greater.
Last edited by jeeswg on 22 Apr 2017, 14:06, edited 1 time in total.
jeeswg wrote: I can't be sure that it will work, however if you ever see the status bar text, in the AccViewer window e.g. in the Name or Value field, that would suggest it is possible to retrieve the text.
Therein lies the possible SNAFU. Both name and value are blank in AccViewer for Path 4 Client (Status bar EXCEL41).
Btw it may be that there is a subelement of 4 Client, from which the text can be retrieved. Otherwise, when I had Excel 2003, I used to use ImageSearch to check for Ready/Edit/Enter.
No subelement for Path 4 Client.
Please confirm if it works on other versions of Excel.
Don't really have need right now.
Yeah if there's a programmatic way to get that status bar text, I would like to know, even though I don't use Excel 2003 any more, just because ... you should never let yourself be defeated by the software.
2 simple functions relating to highlighting, tested on Excel 2007. I generally do everything via macros, but the 3 functions on this page are the few exceptions.
Loop, Parse, vList, % ","
Ut Oh! Hijack in progress.


text:
Reading: The enchanting beauty of Sachal Sarmast's shrine
PhotoNews > Entertainment > The enchanting beauty of Sachal Sarmast's shrine
The enchanting beauty of Sachal Sarmast's shrine
Abdul Wahab, fondly known by his disciples as Sachal Sarmast, (intoxicated man of God and truth) was born in 1739 in a village called Daraza in Khairpur. He spread the message of divine love through his poetry, which embodied the principles of mysticism and according to him was 'divine' inspiration.
He was fond of 'sama' (music)and loved to meditate and pontificate on the wonders of spirituality He wrote his poetry in seven languages, including Sindhi, Seraiki and Persian, and earned titles from his devotees like Haft-i-Zuban Shair (the poet of seven languages) and Sartaj-us-Shuara. There are nine compilations of his Persian poetry, notably the Diwan-i-Ashkar and Dard Nama, which consist of mainly philosophical verses. However, it is his Sindhi and Seraiki work that is celebrated the most.
He was an ascetic and lived to the age of 90 years. He passed away on Ramazan 1829 AD and his belongings were buried alongside the graves of his ancestors. Later his mausoleum was built by Mir Rustam Khan Talpur, the ruler of Khairpur.
Each year in Ramazan, thousands of people gather at his shrine in Daraza for his Urs.
TAGGED: Abdul Wahab, Dard Nama, Diwan-i-Ashkar, Haft-i-Zuban Shair, Mir Rustam Khan Talpur, Sachal Sarmast's shrine, Sartaj-us-Shuara, shrine in Daraza


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Today, U.S. Secretary of Commerce Wilbur Ross announced that the Department's Economic Development Administration (EDA) is awarding a $2 million grant to Jackson County, North Carolina, to help build a new workforce training facility. According to grantee estimates, the project is expected to create 236 jobs, retain 127 jobs, and generate $1.25 million in private investment.
This investment will construct a workforce training facility called the Health Sciences Occupation Building on the main campus of Southwestern Community College. The facility will provide a home for programs leading to healthcare employment opportunities in the area and will also house a clinic for students to gain hands-on experience. This project was made possible by the regional planning efforts led by the Southwestern Commission. EDA funds the Southwestern Commission to bring together the public and private sectors to create an economic development roadmap to strengthen the regional economy, support private capital investment, and create jobs.
For more information, contact John Atwood at 202-482-4085 or jatwood@eda.gov.


text:
Q: Нужна помощь по регулярным выражениям php Пока не очень хорошо понимаю регулярное выражение php.
Прошу подсказать, как обрезать спереди ноли
примеры чисел которые есть:
000010
000210
000112
004010
Мне нужно отрезать ноли перед тем числом больше 0
а вот пример чисел которые я хочу получить:
10
210
112
4010

A: Можно использовать функцию preg_replace. В качестве параметров передаём регулярное выражение, чем заменить найденные в выражении символы и строку в которой делаем замену.
preg_replace('/^0+/', "", "00010")

^ - ищем в начале строки
0+ - 1 или более символов

A: Еще проще
$str = ltrim($str, '0');


Should I let him get away with this?
Mercedes Sprinter 311CDi brand new engine price?
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