text:
\section{Introduction}\label{intro}
The dynamic programming principle (DPP) is a well-known device in studying stochastic optimal control problems. For a standard control problem with finite
horizon, it states that the value function for the control problem starting at time $s \in [0,T]$ from a position $X_{s} = x$ is given by the formula
\begin{equation}\label{eqn:dppjump1}
   V(s,x) = \sup_{u \in \mathcal{A}_{s}}\mathbb{E}\Big(\int_{s}^{\tau}f(r,X_{r},u_{r})dr + V(\tau,X_{\tau})\Big),
\end{equation}
where $\tau$ is some stopping time, $u$ is an admissible control process, $\mathcal{A}_{s}$ is a given admissible control set at time $s$, and $X$ is a
controlled state process.  All terms will be defined in a more precise way later. \\

There are many ways to prove the DPP. When the underlying probability space is fixed in advance, we say a stochastic control problem is under a strong
formulation. In this case, one may use the theory of piecewise constant controls to construct appropriate supermartingales and show that the DPP holds
through properties of supermartingales (see \cite{Kry80} for the diffusion case, and \cite{Ish04} for the jump case). Alternatively, we can prove the DPP
by partitioning the state space, provided the value function satisfies certain regularity conditions or using its semicontinuous envelope (see for example
\cite{BouT11}). When a control problem is defined in the weak sense, that is the underlying probability space is taken to be part of the control, we can
also apply this approach (see for example \cite{AzePW14,YonZ99}). Moreover, recently by interpreting controls in the weak sense, \cite{ElkT13} proved the
DPP by using a probabilistic approach.  \\

To prove the DPP, in most cases, the state process is required to have finite second moments (see for example \cite{AzePW14,BouT11,Ish04,Kry80,YonZ99}). A
stochastic control problem is often formulated with the state process assumed to follow a certain stochastic differential equation (SDE). For SDE driven by
Brownian noise, with appropriate assumptions on the coefficients of SDE, it is well known that the existence of finite second moments is assured. However,
this does not hold in general case when the SDE is driven by a more general L\'{e}vy type noise. For example, let us consider the following (controlled)
state process:
\begin{eqnarray*}
  \left\{\begin{array}{ll} dX_{t} = b(t,X_{t-},u_{t})dt + \sigma(t,X_{t-},u_{t})dW_{t}
                                    + \displaystyle\int_{0 < |\eta| < 1}\gamma(t,X_{t-},u_{t},\eta)\tilde{N}(dt,d\eta) \\
                           \ \ \ \ \ \ \ \ \ + \displaystyle\int_{|\eta| \geq 1}\gamma(t,X_{t-},u_{t},\eta)N(dt,d\eta)                \\
                            \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \        \\
                           X_{s} = x \in \mathbb{R}^{d},  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
                                                             \ \ \ \ \ \ \ \ \ \ \ \ 0 \leq s \leq t \leq T, \end{array}\right.
\end{eqnarray*}
where $W$ is a Brownian motion, $N$ is a Poisson random measure, and $\tilde{N}$ is the associated compensated Poisson random measure. In this case, we need
further assumption on the measure $\nu$, for example
\begin{eqnarray}\label{eqn:aspm}
  \displaystyle\int_{|\eta| \geq 1}|\eta|^{p}\nu(d\eta)dt  &  <  &  \infty, \ \textnormal{for some} \ p \geq 2,
\end{eqnarray}
to assure that there exists a finite second moment for the state process. This would restrict us to only a subclass of L\'{e}vy type noises. However, in order
to study state processes with heavy tailed distributions, one needs to relax the moments assumption. \\

Z\u{a}linescu extended the proof of DPP to stable processes, which requires (\ref{eqn:aspm}) to hold for a certain $p > 0$. He proved the DPP in the context of
a combined control and optimal stopping problem in which a $C^{2}$-approximation of the state process is introduced. In contrast, the recent work of \cite{ElkT13}
formulates the stochastic control problems in terms of controlled martingale problems. Their proof assumes that (\ref{eqn:aspm}) holds for $p = 1$. In this work,
our main contribution is to extend the proof of DPP (under the strong formulation) by relaxing (\ref{eqn:aspm}) in which no finite moments assumption are imposed.
To this end, we use an approximation of state process which is commonly used in construction of solution of SDEs (see for example Theorem 6.2.9 on p374 in \cite{App09}
or pp354-355 in \cite{Kun04}). The idea behind this is to define a new state process by cutting off the jumps if they are too 'large'. Since 'large' jumps cause the
failure of the existence of moments, by cutting of the 'large' jumps we retain the nice property of existence of moments. In contrast to the $C^{2}$-approximation
which considered by Z\u{a}linescu, the approximation which we consider simplifies the proof of the DPP. \\

The paper is organized in the following way. In section 2, we formulate our control problem and state the DPP. In section 3, we present an approximation of the
state process as well as some auxiliary results. Finally, we prove the DPP in section 4.

\section{Problem Formulation}
We will work on the Wiener-Poisson space. Let us recall the construction of such a space given in \cite{BouT11,IshK06}. To this end,
we first recall the definitions of Wiener and Poisson spaces. Fix a $T > 0$. Let $\Omega_{W} = C([0,T];\mathbb{R}^{d})$, and for
$\omega_{1} \in \Omega_{W}$, set $W_{t}(\omega_{1}) := \omega_{1}(t)$. Define $\mathbb{F}^{W} := (\mathcal{F}^{W}_{t})_{t \geq 0}$
as the smallest filtration such that $W_{s}$ is measurable with respect to $\mathcal{F}^{W}_{t}$ for all $s \in [0,t]$. On
$(\Omega_{W},\mathcal{F}^{W})$, let $\mathbb{P}_{W}$ be the probability measure such that $W$ is the $m$-dimensional standard Brownian
motion, where $\mathcal{F}^{W} = \mathcal{F}^{W}_{T}$. Then, we obtain the Wiener space $(\Omega_{W},\mathcal{F}^{W},\mathbb{P}_{W})$.
Let $\mathbb{R}^{q}_{0} =  \mathbb{R}^{q}\backslash\{0\}$, and $\Omega_{N}$ be the set of integer-valued measures on $[0,T] \times
\mathbb{R}^{q}_{0}$. For $\omega_{2} \in \Omega_{N}$, set $N(\omega_{2},I \times A) := \omega_{2}(I \times A)$, where
$I \in \mathcal{B}([0,t])$, and $A \in \mathcal{B}(\mathbb{R}^{q}_{0})$. Define $\mathbb{F}^{N} := (\mathcal{F}^{N}_{t})_{t \geq 0}$
as the smallest filtration such that $N(\cdot, I \times A)$ is measurable with respect to $\mathcal{F}^{N}_{t}$ for all $I \in
\mathcal{B}([0,t])$ and $A \in \mathcal{B}(\mathbb{R}^{q}_{0})$. On $(\Omega_{N}, \mathcal{F}^{N})$, let $\mathbb{P}_{N}$ be the
probability measure such that $N$ is the Poisson random measure with intensity $\nu$, where $\mathcal{F}^{N} = \mathcal{F}^{N}_{T}$,
and $\nu$ is the L\'{e}vy measure, i.e., it satisfies
\begin{eqnarray}
  \displaystyle\int_{\mathbb{R}^{q}_{0}} (|\eta|^{2} \wedge 1) \nu(d\eta) < \infty. \nonumber
\end{eqnarray}
Then, we obtain the Poisson space $(\Omega_{N},\mathcal{F}^{N},\mathbb{P}_{N})$. Now, consider the product space $\Omega = \Omega_{W} \times
\Omega_{N}$. For $\omega = (\omega_{1},\omega_{2}) \in \Omega$, set $W_{t}(\omega) := W_{t}(\omega_{1})$, and $N(\omega,I \times A) :=
N(\omega_{2},I \times A)$.  Let $\mathbb{P} := \mathbb{P}_{W} \otimes \mathbb{P}_{N}$ be the probability measure on $(\Omega,\mathcal{F})$,
where $\mathcal{F}$ is the completion of $\mathcal{F}^{W} \otimes \mathcal{F}^{N}$. This then yields the Wiener-Poisson space
$(\Omega,\mathcal{F},\mathbb{P})$. Without of loss generality, we may assume that this space is complete. On this space, we may associate
a filtration $(\mathcal{F}_{t})_{t \geq 0}$ which is the right-continuous completed revision of the filtration
$(\mathcal{F}^{W}_{t} \otimes \mathcal{F}^{N}_{t})_{t \geq 0}$.  \\

Let $\mathcal{F}^{W,s}_{t}$ be the smallest $\sigma$-algebra such that $W_{r} - W_{s}$ is measurable with respect to $\mathcal{F}^{W,s}_{t}$
for all $r \in [s, t \vee s]$, and $\mathcal{F}^{N,s}_{t}$ be the smallest $\sigma$-algebra such that $N(\cdot,I_{2} \times A) -
N(\cdot, I_{1} \times A)$ is measurable with respect to $\mathcal{F}^{N,s}_{t}$ for all $I_{1}, I_{2} \in \mathcal{B}([s,t \vee s])$, $A \in
\mathcal{B}(\mathbb{R}^{q}_{0})$, where $I_{1} \subset I_{2}$. We define a commonly used filtration $(\mathcal{F}^{s}_{t})_{t \geq s}$ which
is the right-continuous completed revision of $(\mathcal{F}^{W,s}_{t} \otimes \mathcal{F}^{N,s}_{t})_{t \geq s}$ (see for example \cite{BouT11}
for this filtration). For the sake of notations, from now on, we write $N(\omega, (0,t] \times A)$ as $N(t,A)$.  \\

Next, we consider the following control problem. Fix $s \in [0,T)$, the controlled state process $(X_{t})_{t \geq s}$ is assumed to follow
the SDE:
\begin{eqnarray}\label{eqn:dynsdelarge}
    \left\{\begin{array}{ll} dX_{t} = b(t,X_{t-},u_{t})dt + \sigma(t,X_{t-},u_{t})dW_{t}
                                                      + \displaystyle\int_{0 < |\eta| < 1}\gamma(t,X_{t-},u_{t},\eta)\tilde{N}(dt,d\eta) \\
                                    \ \ \ \ \ \ \ \ \ + \displaystyle\int_{|\eta| \geq 1}\gamma(t,X_{t-},u_{t},\eta)N(dt,d\eta)                                  \\
                            \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \                      \\
                           X_{s} = x \in \mathbb{R}^{d},  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
                                                             \ \ \ \ \ \ \ \ \  0 \leq s \leq t \leq T, \end{array}\right.
\end{eqnarray}
where $X_{t-}$ is the left limit of $X_{t}$, and $u: [0,T] \times\Omega \rightarrow \mathbb{R}^{\ell}$ is a predictable process which acts as a control.
Moreover, $b: [0,T] \times \mathbb{R}^{d} \times \mathbb{R}^{\ell} \rightarrow  \mathbb{R}^{d}$ is a continuous function, $\sigma: [0,T] \times\mathbb{R}^{d}
\times\mathbb{R}^{\ell} \rightarrow \mathbb{R}^{d \times m}$ is a continuous function, $\gamma: [0,T]\times\mathbb{R}^{d}\times\mathbb{R}^{\ell}\times
\mathbb{R}^{q}_{0} \rightarrow \mathbb{R}^{d}$ is a Borel measurable function, and $\gamma$ is continuous in $(t,x,u)$ for every $\eta \in \mathbb{R}^{q}_{0}$.
In addition, $\tilde{N}$ is the compensated Poisson random measure associate to $N$, i.e. $\tilde{N}(dt,d\eta) = N(dt,d\eta) - \nu(d\eta)dt$. \\

Fix a compact set $A \subset \mathbb{R}^{\ell}$. The set of admissible controls $(u_{t})_{t \in [0,T]}$ is denoted by $\mathcal{A}_{s}$, where
\begin{eqnarray*}
   \mathcal{A}_{s}   & = & \Big\{u: [0,T] \times \Omega \rightarrow A \ \textnormal{is predictable with respect to}  \ (\mathcal{F}^{s}_{t})_{t \geq 0}\Big\}.
\end{eqnarray*}

In the rest of the paper, we shall make the following assumption. \\

\hypertarget{assump1}{\textbf{Assumption 1}.}  There exist constants $C>0$ and $C_{M} > 0$ such that for all $t \in [0,T]$, $u \in A$, $x_{1}, x_{2} \in
                                               \mathbb{R}^{d}$, and $0 < |\eta| < M$, we have
                                               \begin{eqnarray}
                                                  \big|\sigma(t,x_{1},u) - \sigma(t,x_{2},u)\big| + \big|b(t,x_{1},u) - b(t,x_{2},u)\big|
                                                                                &  \leq  &  C|x_{1}-x_{2}|, \nonumber \\
                                                  \big|\gamma(t,x_{1},u,\eta)-\gamma(t,x_{2},u,\eta)\big|  &  \leq  &  C_{M}|\eta||x_{1}-x_{2}|, \nonumber \\
                                                  \big|\gamma(t, x,\eta,u)\big|                            &  \leq  &  C_{M}|\eta|(1+|x|). \nonumber
                                               \end{eqnarray}
Here and after, we will use $C$ to denote a generic constant and may differ from one line to the other. Subscripts may be added to $C$ to emphasize dependence
on particular parameters. \\

It is well known that under \hyperlink{assump1}{Assumption 1} and the compactness of $A$, there exists a constant $C>0$ such that for all $t \in [s,T]$, $u \in A$,
and $x \in \mathbb{R}^{n}$, the coefficients $\sigma$ and $b$ satisfy
\begin{eqnarray}
  \big|\sigma(t,x,u)\big| + \big|b(t,x,u)\big| &  \leq  & C(1+|x|).  \nonumber
\end{eqnarray}
Moreover, there exists a unique c\`{a}dl\`{a}g and adapted solution of SDE (\ref{eqn:dynsdelarge}). To emphasize dependence on initial conditions and the control,
we may write $X_{t}$ as $X^{u,s,x}_{t}$. \\

The revenue functional for a given $u \in \mathcal{A}_{s}$ is defined as
\begin{equation}\label{eqn:genrf}
   V^{u}(s,x) =  \mathbb{E}\Big(\int_{s}^{T}f(t,X^{u,s,x}_{t},u_{t})dt + h(X^{u,s,x}_{T}) \Big),
\end{equation}
where $f: \mathbb{R} \times \mathbb{R}^{d} \times \mathbb{R}^{\ell} \rightarrow \mathbb{R}$ and $h: \mathbb{R}^{d} \rightarrow \mathbb{R}$ are continuous bounded
functions. We will say that
\begin{equation}\label{eqn:valuefunc}
  V(s,x) = \sup_{u \in \mathcal{A}_{s}} V^{u}(s,x)
\end{equation}
is the value function. If there exists a maximizer $u^{\ast}(s) : = u^{\ast} \in \mathcal{A}_{s}$, then
\begin{eqnarray*}
  V(s,x)  &   =  & V^{u^{\ast}}(s,x).
\end{eqnarray*}

For $s \in [0,T]$, let $\mathcal{T}_{[s,T]}$ be the set of stopping times in $[s,T]$ adapted to $(\mathcal{F}^{s}_{t})_{t \geq s}$. The DPP is then stated in the
following Theorem.
\begin{theorem}\label{thm:bellmanjump2mr}
  \textbf{(Dynamic Programming Principle)}: For every $\tau \in \mathcal{T}_{[s,T]}$ and all $x \in \mathbb{R}^{d}$,
  \begin{equation}\label{eqn:dpjump}
    V(s,x) = \sup_{u \in \mathcal{A}_{s}}\mathbb{E}\Big(\int_{s}^{\tau}f(r,X^{u,s,x}_{r},u_{r})dr + V(\tau,X^{u,s,x}_{\tau}) \Big).
  \end{equation}
\end{theorem}

In order to prove the DPP, we need some preparations.

\section{Auxiliary Results}
In this subsection, we present an approximation of the state process. Let $\tau_{0} = s$, and for $k = 1, 2, ...$, let $\tau_{k}$ be the arrival
time of $k$th jump of a compound Poisson process $(L_{t})_{t \geq 0}$ after $\tau_{0}$, where
\begin{eqnarray}
   L_{t} = \int_{|\eta| \geq 1} \eta N(t,d\eta). \nonumber
\end{eqnarray}
Then, it is easy to verify that the following lemma holds.
\begin{lemma}\label{lemma:pwconvergence}
   For $M \geq 1$, let $\tau_{M}$ be a stopping time such that
   \begin{eqnarray}
      \tau_{M} = \inf\{t > s: \Delta L_{t} \in E_{M}\},    \nonumber
   \end{eqnarray}
   where $\Delta L_{t} = L_{t} - L_{t-}$, and $E_{M} = \{\eta\in\mathbb{R}^{q}_{0}: |\eta| \geq M\}$. As $M \rightarrow \infty$, we have
   $1_{\{\tau_{M} \leq T\}} \rightarrow 0$, $\mathbb{P}$-a.s. In particular, we have $1_{\{\tau_{M} \leq \tau\}} \rightarrow 0$ $\mathbb{P}$-a.s.
   for every $\tau \in \mathcal{T}_{[s,T]}$.
\end{lemma}

Set $\zeta^{M}_{0} = x$, and for $k = 1, 2, ... $, define
\begin{eqnarray}
  \zeta^{M}_{k} &   =   &    X^{u,\tau_{k-1},\zeta^{M}_{k-1}}_{\tau_{k}}1_{\{ |\Delta L_{\tau_{k}}| < M \}}
                           + X^{u,\tau_{k-1},\zeta^{M}_{k-1}}_{\tau_{k}-}1_{\{ |\Delta L_{\tau_{k}}| \geq M \}},  \nonumber
\end{eqnarray}
and
\begin{equation}\label{eqn:approxistate}
  X^{M}_{t} = \sum^{\infty}_{k=0}X^{u,\tau_{k},\zeta_{k}^{M}}_{t}1_{[\tau_{k},\tau_{k+1})}(t)1_{[s,T]}(t).
\end{equation}
By construction of solution, we see that $(X^{M}_{t})_{t \geq s}$ satisfies the following SDE:
\begin{eqnarray}\label{eqn:dynsdemjumps}
  \left\{\begin{array}{ll} dX^{M}_{t} =    b(t,X^{M}_{t-},u_{t})dt + \sigma(t,X^{M}_{t-},u_{t})dW_{t}
                                         + \displaystyle\int_{0 < |\eta| < 1}\gamma(t,X^{M}_{t-},u_{t},\eta)\tilde{N}(dt,d\eta)  \\
                                         \ \ \ \ \ \ \ \ \  + \displaystyle\int_{1 \leq |\eta| < M}\gamma(t,X^{M}_{t-},u_{t},\eta)N(dt,d\eta)   \\
                            \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \                      \\
                            X^{M}_{s} = x,  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
                                            \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0 \leq s \leq t \leq T. \end{array}\right.
\end{eqnarray}
Again, to emphasize dependence on initial conditions and the control, we may write $X^{M}_{t}$ as $X^{u,s,x,M}_{t}$. \\

Following a standard argument, for example similar as in \cite{Kun04} (see pp340-341 in \cite{Kun04}), we can obtain the estimates below.
\begin{lemma}\label{lemma:estaproxi}
   For every $M \geq 1$, and all $p \geq 2$, there exists a $C_{T,p,M}  > 0$ such that
   \begin{enumerate}
      \item $\displaystyle\mathbb{E}\Big(\sup_{t \in [s,T]}\big|X^{u,s,x,M}_{t}\big|^{p}\Big)
             \leq C_{T,p,M}\Big(1+|x|^{p}\Big)$,
      \item $\displaystyle\mathbb{E}\Big(\sup_{t \in [s,T]}\big|X^{u,s,x,M}_{t}-X^{u,\hat{s},\hat{x},M}_{t}\big|^{p}\Big) \leq C_{T,p,M}
             \Big(|x - \hat{x}|^{p}+ \big(1 +|x|^{p}\big)|s-\hat{s}|\Big)$.
   \end{enumerate}
\end{lemma}

\begin{remark}
  We may extend $X^{u,\hat{s},\hat{x},M}$ by setting $X^{u,\hat{s},\hat{x},M}_{t} = \hat{x}$ for all $t \in [s,\hat{s}]$ (see p175 in \cite{Zal11}).
\end{remark}

For the sequence of state processes $(X^{M}_{t})_{t \geq s}$, we define their corresponding revenual functionals $V^{u,M}$ by
\begin{eqnarray}
   V^{u,M}(s,x) &   =   &  \mathbb{E}\Big(\int_{s}^{T}f(t,X^{u,s,x, M}_{t},u_{t})dt + h(X^{u,s,x, M}_{T})\Big). \nonumber
\end{eqnarray}
The value functions $V^{M}$ is given by
\begin{eqnarray}\label{eqn:VMvaluefunction}
   V^{M}(s,x) &   =   & \sup_{u \in \mathcal{A}_{s}} V^{u,M}(s,x).
\end{eqnarray}

Next, we obtain the following lemma.
\begin{lemma}\label{lemma:convalM}
  For every $(s,x) \in [0, T] \times \mathbb{R}^{d}$, as $M \rightarrow \infty$, $V^{M}(s,x) \rightarrow V(s,x)$.
\end{lemma}
\begin{proof}
  Since $1_{\{\tau_{M} > T\}} X^{u,s,x,M}_{t} =  1_{\{\tau_{M} > T\}} X^{u,s,x}_{t}$ ($\mathbb{P}$-a.s.) for every $t \in [s, T]$, and $f$ and $h$ are bounded,
  we see that for every $u \in \mathcal{A}_{s}$, we have
  \begin{eqnarray}
    V^{u,M}(s,x)   &  \leq  &    \mathbb{E}\Bigg(\int_{s}^{T}f(t,X^{u,s,x}_{t},u_{t})1_{\{\tau_{M} > T\}}dt
                               + 1_{\{\tau_{M} > T\}}h(X^{u,s,x}_{T}) + C_{T}1_{\{\tau_{M} \leq T\}}\Bigg). \nonumber
  \end{eqnarray}

  As $M \rightarrow \infty$, thanks again to boundedness of $f$ and $h$, we can apply the Dominated Convergence Theorem. Thus, together with
  the continuity of $f$ and $h$, and \hyperref[lemma:pwconvergence]{Lemma \ref*{lemma:pwconvergence}}, we obtain
  \begin{eqnarray}\label{eqn:convalM1}
    \lim_{M \rightarrow \infty}V^{u,M}(s,x) &   =   &  \mathbb{E}\Big(\int_{s}^{T}f(t,X^{u,s,x}_{t},u_{t})dt + h(X^{u,s,x}_{T})\Big) \nonumber \\
                                            &   =   & V^{u}(s,x).
  \end{eqnarray}
  This then yields
  \begin{eqnarray}
    \liminf_{M \rightarrow \infty}V^{M}(s,x)  \geq   \lim_{M \rightarrow \infty}V^{u,M}(s,x) = V^{u}(s,x). \nonumber
  \end{eqnarray}
  Taking supremum over $\mathcal{A}_{s}$, we find
  \begin{eqnarray}\label{eqn:convalM2}
    \liminf_{M \rightarrow \infty}V^{M}(s,x) &   \geq   &  V(s,x).
  \end{eqnarray}
  To show the converse inequality, we observe from (\ref{eqn:convalM1}) that for every $u \in \mathcal{A}_{s}$, $(s,x) \in [0,T]\times\mathbb{R}^{d}$,
  and all $\delta > 0$, there exists an $M_{s,x,u,\delta}$ such that for all $M > M_{s,x,u,\delta}$ we have
  \begin{eqnarray}\label{eqn:valfunest}
     \big|V^{u,M}(s,x) - V^{u}(s,x)\big| & \leq  &  \delta.
  \end{eqnarray}
  Using (\ref{eqn:valfunest}), we see that for every $M > M_{s,x,u,\delta}$, there exists an $\epsilon$-optimal control $u^{\epsilon,M}$ such that
  \begin{eqnarray}
    V^{M}(s,x) &   \leq   &  V^{u^{\epsilon,M},M}(s,x) + \epsilon   \nonumber \\
               &   \leq   &  V^{u^{\epsilon,M}}(s,x) + \delta  + \epsilon  \nonumber \\
               &   \leq   &  V(s,x)  + \delta  + \epsilon.  \nonumber
  \end{eqnarray}
  By first letting $M \rightarrow \infty$, we obtain
  \begin{eqnarray}\label{eqn:convalM3}
    \limsup_{M \rightarrow \infty}V^{M}(s,x) &   \leq   &  V(s,x) + \delta +\epsilon.
  \end{eqnarray}
  Since $\delta$ and $\epsilon$ are arbitrary, this yields the converse inequality. The proof is completed.
\end{proof}

Next, we present two results which we borrowed from \cite{Zal11} (modified version of Lemma 2.3 in \cite{Zal11}). Since the author does not provide a proof,
we prove it here in our context. \\

Now, under the assumption that $f$ and $h$ are continuous, we know that the functions $f$ and $h$ admit a joint modulus of continuity (see Lemma 2.3 in
\hbox{\cite{Zal11}}):
\begin{eqnarray}
   \rho(\alpha, \beta) & = &  \sup_{\substack{t \in [0,T], u \in A, \\ x, \hat{x} \in \overline{B(0,\beta)}, |x-\hat{x}| \leq \alpha}}
                           \Big(|f(t,x,u) - f(t,\hat{x},u)| + |h(x) - h(\hat{x})|\Big), \nonumber
\end{eqnarray}
such that $\displaystyle\lim_{\beta \rightarrow \infty}\lim_{\alpha \rightarrow 0}\rho(\alpha, \beta) = 0$. Thus, we have the first result below.
\begin{props}\label{prop:vfljumpgen}
   There exists constants $C > 0$ and $C_{T,p,M} > 0$, such that for every $u \in \mathcal{A}_{s}$, $(s,x), (\hat{s},\hat{x}) \in [0,T] \times \mathbb{R}^{d}$, and
   all $p \geq 2$, $\alpha > 0$, $\beta > 0$,
   \begin{eqnarray}
                       \big|V^{u,M}(s,x) - V^{u,M}(\hat{s},\hat{x})\big|
       &   \leq   &    C_{T}\rho(\alpha,\beta) + C_{T,p,M}\frac{|x - \hat{x}|^{p}+(1+|\hat{x}|^{p})|s-\hat{s}|}{\alpha^p}  \nonumber \\
       &          &  + C_{T,p,M}\frac{(1+|x|^{p}+|\hat{x}|^{p})}{\beta^{p}}, \nonumber
   \end{eqnarray}
\end{props}

\begin{proof}
   For $u \in \mathcal{A}_{s}$, and $(s,x), (\hat{s},\hat{x}) \in [0,T] \times \mathbb{R}^{d}$, we see that
   \begin{eqnarray}\label{eqn:unifcon0}
       &          &    \big|V^{u,M}(s,x) - V^{u,M}(\hat{s},\hat{x})\big|   \nonumber  \\
       &   \leq   &    \big|V^{u,M}(s,x) - V^{u,M}(s,\hat{x})\big|
                     + \big|V^{u,M}(s,\hat{x}) - V^{u,M}(\hat{s},\hat{x})\big| \nonumber \\
       &    =     &  (I_{1}) + (I_{2}).
   \end{eqnarray}
   The first term in (\ref{eqn:unifcon0}) yields
   \begin{eqnarray}
     (I_{1}) &   =     &     \big|V^{u,M}(s,x) - V^{u,M}(s,\hat{x})\big|  \nonumber \\
             &  \leq   &     \mathbb{E}\Bigg(\int_{s}^{T}\big|f(t,X^{u,s,x, M}_{t},u_{t}) - f(t,X^{u,s,\hat{x}, M}_{t},u_{t})\big|dt
                          +  \big|h(X^{u,s,x, M}_{T}) -  h(X^{u,s,\hat{x}, M}_{T})\big|\Bigg) \nonumber \\
             &    =    &     (I_{1,1}) + (I_{1,2}). \nonumber
   \end{eqnarray}
   The second term in (\ref{eqn:unifcon0}) can be estimated as
   \begin{eqnarray}
     (I_{2}) &    =    &     \big|V^{u,M}(s,\hat{x}) - V^{u,M}(\hat{s},\hat{x})\big|  \nonumber \\
             &  \leq   &     \mathbb{E}\Bigg(\int_{s}^{\hat{s}}\big|f(t,X^{u,s,\hat{x}, M}_{t},u_{t})
                           - f(t,X^{u,\hat{s},\hat{x}, M}_{t},u_{t})\big|dt
                           +  \big|h(X^{u,s,\hat{x}, M}_{T}) -  h(X^{u,\hat{s},\hat{x}, M}_{T})\big|\Bigg) \nonumber \\
             &    =    &     (I_{1,3}) + (I_{1,4}). \nonumber
   \end{eqnarray}
   Each of $(I_{1,1}) - (I_{1,4})$ can be estimated by using the bounds of $f$ and $h$, the Markov inequality, and
   \hyperref[lemma:estaproxi]{Lemma \ref*{lemma:estaproxi}}. For example, for $(I)$ we have
   \begin{eqnarray}
     (I_{1,1}) &   =   &   \mathbb{E}\Bigg(\int_{s}^{T}\big|f(t,X^{u,s,x, M}_{t},u_{t}) - f(t,X^{u,s,\hat{x}, M}_{t},u_{t})\big|dt\Bigg)  \nonumber \\
               &  \leq &    C_{T}\mathbb{P}\Bigg(\sup_{t \in [s,T]}\big|X^{u,s,x, M}_{t} - X^{u,s,\hat{x}, M}_{t}\big|^{p} > \alpha^p\Bigg)
                          + C_{T}\rho(\alpha,\beta) \nonumber
   \end{eqnarray}
   \begin{eqnarray}
              &       &  + C_{T}\mathbb{P}\Bigg(\sup_{t \in [s,T]}\big|X^{u,s,x, M}_{t}\big|^{p} \geq \beta^{p} \Bigg)
                         + C_{T}\mathbb{P}\Bigg(\sup_{t \in [s,T]}\big|X^{u,s,\hat{x}, M}_{t}\big|^{p} \geq \beta^{p} \Bigg) \nonumber \\
              &  \leq &    C_{T}\rho(\alpha,\beta) + C_{T,p,M}\frac{|x - \hat{x}|^{p}}{\alpha^p}
                         + C_{T,p,M}\frac{(1+|x|^{p}+|\hat{x}|^{p})}{\beta^{p}}. \nonumber
   \end{eqnarray}
   In a similar way, we obtain
   \begin{eqnarray}
       (I_{1,2})  &  \leq   &   C_{T}\rho(\alpha,\beta) + C_{T,p,M}\frac{|x - \hat{x}|^{p}}{\alpha^p}
                              + C_{T,p,M}\frac{(1+|x|^{p}+|\hat{x}|^{p})}{\beta^{p}},  \nonumber \\
       (I_{1,3})  &  \leq   &   C_{T}\rho(\alpha,\beta) + C_{T,p,M}\frac{(1+|\hat{x}|^{p})|s-\hat{s}|}{\alpha^p}
                              + C_{T,p,M}\frac{(1+|\hat{x}|^{p})}{\beta^{p}}, \nonumber \\
       (I_{1,4})  &  \leq   &   C_{T}\rho(\alpha,\beta) + C_{T,p,M}\frac{(1+|\hat{x}|^{p})|s-\hat{s}|}{\alpha^p}
                              + C_{T,p,M}\frac{(1+|\hat{x}|^{p})}{\beta^{p}}. \nonumber
   \end{eqnarray}
   Combing $(I)-(IV)$, we complete the proof. \\
\end{proof}

Since
\begin{eqnarray}\label{eqn:lipvalinitialgen}
              |V^{M}(s,x) - V^{M}(s,\hat{x})|
  &    =   &  \Big|\sup_{u \in \mathcal{A}_{s}} V^{u,M}(s,x) - \sup_{u \in \mathcal{A}_{s}} V^{u,M}(s,\hat{x})\Big| \nonumber \\
  &  \leq  &  \sup_{u \in \mathcal{A}_{s}} \Big|V^{u,M}(s,x) - V^{u,M}(s,\hat{x})\Big|, \nonumber
\end{eqnarray}
the following corollary is a direct consequence of \hyperref[prop:vfljumpgen]{Proposition \ref*{prop:vfljumpgen}}.
\begin{corollary}\label{corollary:unifvaluefunM}
    For all $p \geq 2$, there exists constants $C_{T} > 0$ and $C_{T,p,M} > 0$ such that for $\alpha > 0$, $\beta > 0$, and $(s,x), (\hat{s},\hat{x}) \in [0,T] \times
   \mathbb{R}^{d}$, we have
   \begin{eqnarray}
       |V^{M}(s,x) - V^{M}(\hat{s},\hat{x})|   &  \leq  &    C_{T}\rho(\alpha,\beta)
                                                           + C_{T,p,M}\frac{|x - \hat{x}|^{p}+(1+|\hat{x}|^{p})|s-\hat{s}|}{\alpha^p} \nonumber \\
                                               &        &  + C_{T,p,M}\frac{(1+|x|^{p}+|\hat{x}|^{p})}{\beta^{p}}. \nonumber
   \end{eqnarray}
\end{corollary}

In order to prove the DPP, the Markov characterization of the state process (see for example, Lemma 3.2 in \cite{Zal11}) plays an important role.
The next lemma states the controlled Markovian property for jump processes.

\begin{lemma}\label{lemma:eqn:markovstate}
The following two assertions hold.
 \begin{enumerate}
    \item For almost every $\omega \in \Omega$, all $\tau \in \mathcal{T}_{[s,T]}$, and $u \in \mathcal{A}_{s}$, there exists a control $\hat{u}^{\omega}
    \in \mathcal{A}_{\tau}$ such that
    \begin{eqnarray}
             \mathbb{E}\Big(\int_{\tau}^{T}f(r,X^{u,s,x,M}_{r},u_{r})dr + h(X^{u,s,x,M}_{T})\big|\mathcal{F}_{\tau}\Big)(\omega)
           = V^{\hat{u}^{\omega},M}(\tau(\omega),X^{u,s,x,M}_{\tau}(\omega)).  \nonumber
    \end{eqnarray}
    \item For every $t \in [s,T]$, and all $\tau \in \mathcal{T}_{[s,t]}$, there exists a control $\hat{u} \in \mathcal{A}_{s}$, where
          \begin{eqnarray}
            \hat{u}_{r} := u_{r}1_{\{r \in [s,\tau]\}} + \tilde{u}_{r}1_{\{r \in (\tau,T]\}}, \nonumber
          \end{eqnarray}
          and $\tilde{u} \in \mathcal{A}_{t}$, such that
          \begin{eqnarray}
              \mathbb{E}\Big(\int_{\tau}^{T}f(r,X^{\hat{u},s,x,M}_{r},u_{r})dr + h(X^{\hat{u},s,x,M}_{T})\big|\mathcal{F}_{\tau}\Big)(\omega)
            = V^{\tilde{u},M}(\tau(\omega),X^{u,s,x,M}_{\tau}(\omega)) \ \ \mathbb{P}\textnormal{-a.s.}  \nonumber
          \end{eqnarray}
 \end{enumerate}
\end{lemma}
\begin{proof}
   The proof follows from Remark 3.10 and the proof of Proposition 5.4 in \cite{BouT11}. \\
\end{proof}

\section{The Proof of DPP}
We now proceed to the prove of DPP. We will follow \cite{BouT11} and \cite{Zal11}.

\begin{proof}
   We start from the easy direction. For $\tau \in \mathcal{T}_{[s,T]}$, $u \in \mathcal{A}_{s}$, and by the first assertion of
   \hyperref[lemma:eqn:markovstate]{Lemma \ref*{lemma:eqn:markovstate}},  we see that for $M \geq 1$ there exists a control $\hat{u} \in
   \mathcal{A}_{\tau}$ such that
   \begin{eqnarray}
      &         &   \mathbb{E}\Big(\int_{s}^{T}f(t, X^{u,s,x,M}_{t},u_{t})dt + h(X^{u,s,x,M}_{T})\Big)       \nonumber \\
      &    =    &   \mathbb{E}\Big(\int_{s}^{\tau}f(t,X^{u,s,x,M}_{t},u_{t})dt + V^{\hat{u},M}(\tau,X^{u,s,x,M}_{\tau})\Big) \nonumber \\
      &   \leq  &   \mathbb{E}\Big(\int_{s}^{\tau}f(t,X^{u,s,x}_{t},u_{t})1_{\{\tau_{M} > \tau\}}dt + C_{T}1_{\{\tau_{M} \leq \tau\}}
                   + V^{M}(\tau,X^{u,s,x}_{\tau})1_{\{\tau_{M} > \tau\}}\Big). \nonumber
   \end{eqnarray}
   In the last line, we have used the fact that $1_{\{\tau_{M} > \tau\}} X^{u,s,x,M}_{t} =  1_{\{\tau_{M} > \tau\}} X^{u,s,x}_{t}$
   ($\mathbb{P}$-a.s.) for every $t \in [s, \tau]$. As $M \rightarrow \infty$, by boundedness of $f$, and $h$, we can apply the Dominated
   Convergence Theorem. Thus, together with \hyperref[lemma:pwconvergence]{Lemma \ref*{lemma:pwconvergence}} and
   \hyperref[lemma:convalM]{Lemma \ref*{lemma:convalM}}, we have
   \begin{eqnarray}
                    \mathbb{E}\Big(\int_{s}^{T}f(t, X^{u,s,x}_{t},u_{t})dt + h(X^{u,s,x}_{T})\Big)
      &   \leq  &   \mathbb{E}\Big(\int_{s}^{\tau}f(t,X^{u,s,x}_{t},u_{t})dt + V(\tau,X^{u,s,x}_{\tau})\Big). \nonumber
   \end{eqnarray}
   Taking supremum over $\mathcal{A}_{s}$, we obtain
   \begin{eqnarray}\label{eqn:valfunMdpp1}
        V(s,x)   &  \leq   &   \sup_{u \in \mathcal{A}_{s}}\mathbb{E}\Big(\int_{s}^{\tau}f(t,X^{u,s,x}_{t},u_{t})dt + V(\tau,X^{u,s,x}_{\tau})\Big).
   \end{eqnarray}

   To show the converse, fix $\epsilon \in (0, 1)$ and choose $\alpha < \epsilon$. Next, choose $\beta > (\frac{1}{\epsilon})^{\frac{1}{p}}$ such that
   $\rho(\alpha,\beta) < \epsilon$. For a fixed $p \geq 2$, let us take a Borel partition $\{B_{j}\}_{j \geq 1}$ of $\mathbb{R}^{d}$ such that
   \begin{eqnarray}\label{eqn:assumpxdpps}
      \displaystyle\sup_{x_{j}, \hat{x}_{j} \in B_{j}}|x_{j}-\hat{x}_{j}|^{p} \leq \alpha^{p}\epsilon.
   \end{eqnarray}
   For $M \geq 1$, $t \in [s,T]$ and $x \in \mathbb{R}^{d}$, we know that there exists an $\epsilon$-optimal control $\tilde{u}^{\epsilon,M} \in \mathcal{A}_{t}$
   such that
   \begin{eqnarray}\label{eqn:VMep}
      V^{M}(t,x) \leq V^{M,\tilde{u}^{\epsilon,M}}(t,x) + \epsilon.
   \end{eqnarray}
   By \hyperref[corollary:unifvaluefunM]{Corollary \ref*{corollary:unifvaluefunM}}, (\ref{eqn:assumpxdpps})-(\ref{eqn:VMep}), and
   \hyperref[prop:vfljumpgen]{Proposition \ref*{prop:vfljumpgen}}, we see that for every $x_{j}, \hat{x}_{j} \in B_{j}$, there exists
   an $\epsilon$-optimal control $\tilde{u}^{j,\epsilon,M} \in \mathcal{A}_{t}$ such that
   \begin{eqnarray}\label{eqn:3epsilon}
      V^{M}(t,x_{j})   &    \leq   &   V^{M}(t,\hat{x}_{j}) + \epsilon C_{T,p,M}(1+|x_{j}|^{p}) \nonumber \\
                       &    \leq   &   V^{M,\tilde{u}^{j,\epsilon,M}}(t,x_{j}) + \epsilon C_{T,p,M}(1+|x_{j}|^{p}) + \epsilon \nonumber \\
                       &    \leq   &   V^{M,\tilde{u}^{j,\epsilon,M}}(t,x_{j}) + \epsilon C_{T,p,M}(1+|x_{j}|^{p}).
   \end{eqnarray}
   For $u \in \mathcal{A}_{s}$, we take a sequence of controls
   \begin{eqnarray}
      \hat{u}^{j,\epsilon,M}_{r} = \left\{\begin{array}{ll}  u_{r},                          &   \textnormal{if} \ r \in [s,t],     \\
                                                             \tilde{u}^{j,\epsilon,M}_{r},   &   \textnormal{if} \ r \in (t,T] \ \textnormal{and}
                                                                                \ X^{u,s,x,M}_{t} \in B_{j},  \end{array}\right. \nonumber
   \end{eqnarray}
   where $\tilde{u}^{j,\epsilon,M} \in \mathcal{A}_{t}$. It is easy to see that $\hat{u}^{j,\epsilon,M} \in \mathcal{A}_{s}$ which is a consequence
   of the measurability of $X^{u,s,x,M}_{t}$ and the fact that $\mathcal{F}^{t}_{r} \subset \mathcal{F}^{s}_{r}$ for all $s \leq t \leq r$. By
   uniqueness of solution, the second assertion of \hyperref[lemma:eqn:markovstate]{Lemma \ref*{lemma:eqn:markovstate}} and (\ref{eqn:3epsilon}) we
   then obtain
   \begin{eqnarray}
                    V^{M}(s,x)
      &  \geq  &    \mathbb{E}\Bigg(\int^{T}_{s}f(r,X^{\hat{u}^{j,\epsilon,M},s, x, M}_{r},u_{r})dr + h(X^{\hat{u}^{j,\epsilon,M},s, x, M}_{T}) \Bigg) \nonumber \\
      &   =    &   \mathbb{E}\Bigg(\int^{t}_{s}f(r,X^{u,s, x, M}_{r},u_{r})dr\Bigg)
                  + \sum_{j \geq 1}\mathbb{E}\Bigg(\mathbb{E}\Big(\int^{T}_{t}f(r,X^{\tilde{u}^{j,\epsilon,M}, t, X^{u,s,x,M}_{t}}_{r},\tilde{u}^{j,\epsilon,M}_{r})dr \nonumber
   \end{eqnarray}
   \begin{eqnarray}
      &        &  + h(X^{\tilde{u}^{\epsilon,j,M}, t, X^{u,s,x,M}_{t}}_{T}) \big| \mathcal{F}_{t} \Big)1_{\{X^{u,s,x,M}_{t} \in B_{j}\}}\Bigg) \nonumber \\
      &   =    &   \mathbb{E}\Bigg(\int^{t}_{s}f(r,X^{u,s, x, M}_{r},u_{r})dr\Bigg)
                  + \sum_{j \geq 1}\mathbb{E}\Big(V^{u^{\epsilon,j,M},M}(t,X^{u,s,x,M}_{t})1_{\{X^{u,s,x,M}_{t} \in B_{j}\}}\Big) \nonumber  \\
      &  \geq  &    \mathbb{E}\Bigg(\int^{t}_{s}f(r,X^{u,s, x, M}_{r},u_{r})dr\Bigg) + \sum_{j \geq 1}\mathbb{E}\Bigg(\Big(V^{M}(t,X^{u,s,x,M}_{t}) \nonumber \\
      &        &   - \epsilon C_{T,p,M}(1+|X^{u,s,x,M}_{t}|^{p})\Big)1_{\{X^{u,s,x,M}_{t} \in B_{j}\}}\Bigg)  \nonumber \\
      &   =    &    \mathbb{E}\Bigg(\int^{t}_{s}f(r,X^{u,s, x, M}_{r},u_{r})dr + V^{M}(t,X^{u,s,x,M}_{t})\Bigg)
                   - \epsilon C_{T,p,M}\mathbb{E}\Big(1+|X^{u,s,x,M}_{t}|^{p}\Big).  \nonumber
    \end{eqnarray}
   Since $\epsilon$ is arbitrary,  we then have
   \begin{eqnarray}
        V^{M}(s,x)    &  \geq  &  \mathbb{E}\Big(\int^{t}_{s}f(r,X^{u,s, x, M}_{r},u_{r})dr + V^{M}(t,X^{u,s,x,M}_{t})\Big). \nonumber
   \end{eqnarray}
   Let $\mathcal{G}(t) :=  \int_{s}^{t}f(X^{u,s,x}_{r})dr + V^{M}(t,X^{u,s,x}_{t}$). For every $t_{1}, t_{2} \in [s,T)$, and

   \begin{eqnarray}
        \hat{u}_{r} = \left\{\begin{array}{ll}  u_{r},             &   \textnormal{if} \ r \in [s,t_{1}],     \\
                                                \tilde{u}_{r},     &   \textnormal{if} \ r \in (t_{1},T],  \end{array}\right. \nonumber
   \end{eqnarray}
   where $u \in \mathcal{A}_{s}$ and $\tilde{u} \in \mathcal{A}_{t_{1}}$, we have, by uniqueness of solution and the second assertion of
   \hyperref[lemma:eqn:markovstate]{Lemma \ref*{lemma:eqn:markovstate}},
   \begin{eqnarray}
                     \mathbb{E}(\mathcal{G}(t_{2})|\mathcal{F}_{t_{1}})
       &   =   &   \int_{s}^{t_{1}}f(t,X^{u,s,x,M}_{t},u_{t})dt  \nonumber \\
       &       &  + \mathbb{E}\Bigg(\int_{t_{1}}^{t_{2}}f(t,X^{u,s,X^{u,s,x,M}_{t_{1}},u_{t}}_{t},\tilde{u}_{t})dt
                  + V^{M}(t_{2},X^{u,s,X^{u,s,x,M}_{t_{1}}}_{t_{2}})\big|\mathcal{F}_{t_{1}}\Bigg) \nonumber   \\
       &  \leq &   \int_{s}^{t_{1}}f(t,X^{u,s,x,M}_{t},u_{t})dt + V^{M}(t_{1},X^{u,s,x,M}_{t_{1}})   \nonumber \\
       &   =   &   \mathcal{G}(t_{1}). \nonumber
   \end{eqnarray}
   Thus, $\mathcal{G}$ is a supermartingale, and by Doob's Optional Sampling Theorem we know that, for every stopping time $\tau \in \mathcal{T}_{[s,T]}$
   and $u \in \mathcal{A}_{s}$, we have
   \begin{eqnarray}\label{eqn:valfunMdpp2}
     V^{M}(s,x)  &  \geq  & \mathbb{E}\Big(\int_{s}^{\tau}f(t,X^{u,s,x,M}_{t},u_{t})dt + V^{M}(\tau,X^{u,s,x,M}_{\tau})\Big).
   \end{eqnarray}
   Without loss of generality, we assume that $f, h > 0$ for all $(s,x) \in [s,T]\times\mathbb{R}^{d}$. Then, (\ref{eqn:valfunMdpp2}) implies
   \begin{eqnarray}
     V^{M}(s,x) &  \geq  &  \mathbb{E}\Big(\int_{s}^{\tau}f(t,X^{u,s,x}_{t},u_{t})1_{\{\tau_{M} > \tau\}}dt
                           + V^{M}(\tau,X^{u,s,x}_{\tau})1_{\{\tau_{M} > \tau\}} \Big). \nonumber
   \end{eqnarray}
   Here, we use the fact that $1_{\{\tau_{M} > \tau\}} X^{u,s,x,M}_{t} =  1_{\{\tau_{M} > \tau\}} X^{u,s,x}_{t}$ ($\mathbb{P}$-a.s.) for every
   $t \in [s, \tau]$. As $M \rightarrow \infty$, thanks to the boundedness of $f$ and $h$, the Dominated Convergence Theorem can be applied.
   Together with \hyperref[lemma:pwconvergence]{Lemma \ref*{lemma:pwconvergence}} and \hyperref[lemma:convalM]{Lemma \ref*{lemma:convalM}},
   the above yields
   \begin{eqnarray}
      V(s,x) &  \geq  &  \mathbb{E}\Big(\int_{s}^{\tau}f(t,X^{u,s,x}_{t},u_{t})dt + V(\tau,X^{u,s,x}_{\tau})\Big). \nonumber
   \end{eqnarray}
   Taking supremum over $\mathcal{A}_{s}$, and combining with (\ref{eqn:valfunMdpp1}) we obtain the desired result.
\end{proof}

redpajama_set_name: RedPajamaArXiv


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text:
Q: VBA reading XML performance issues I have already looked here for many answers, but none of them seems to cover my problem. In fact, many of the suggestions given in other answers are already implemented in my code.
I have a huge xml file that I must loop through, and the speed is really low, about 1 second per line written (or per node). This speed is also kept when I get a small sample, like 82 lines (with about 15 columns each). This would mean 82 main nodes with 15 child nodes each.
My code is as follows:
Dim wsBase As Worksheet
Dim linEscrita As Long

Dim resp As MSXML2.DOMDocument60
Dim lista As IXMLDOMNodeList
Dim nodeAtual As IXMLDOMNode
Dim childNode As IXMLDOMNode

Dim charIni As Long
Dim charAt As Long
Dim colAtual As Long

Application.ScreenUpdating = False

Set wsBase = Worksheets("Name of worksheet")

Set resp = New DOMDocument60
resp.LoadXML (FunctionForGettingXMLfromWebService)

linEscrita = 2
'name from Node I must find and get child nodes
Set lista = resp.SelectNodes("//node1/node2")

For Each nodeAtual In lista
    colAtual = 1
    If (nodeAtual.HasChildNodes) Then
        For Each childNode In nodeAtual.ChildNodes
            wsBase.Cells(linEscrita, colAtual) = childNode.text
            colAtual = colAtual + 1
        Next childNode
    End If
    linEscrita = linEscrita + 1
Next nodeAtual

Application.ScreenUpdating = True

This code is even really similar to a code found here on an answer, and on that answer the code has results much faster than mine. This is not pc based problem, as I have tried in other computers as well. Has anyone gone through something similar or have any idea what might be the problem? The XML goes fine and fast when looped with PHP, so not a XML problem as well.

A: Array method
Looping through a range via VBA is always time consuming. Therefore the following code example - as close as possible to your original post *) - writes the wanted XML contents into a predefined 2-dim array (section [3]and [4]) and writes it back to sheet via one single code line (section [5]).
Furthermore it's assumed in this very basic XPath model that all 15 subnodes follow the same strict order; empty lines will be omitted.
*) You could speed this up by direct reference to the childnodes as proposed by @Absinthe.
Code example as close as possible to your OP
Sub WriteXMLContents2Sheet()
  Const MAXCOLUMNS& = 15                          ' << change to columns limit in xml
  Dim linEscrita As Long: linEscrita = 2          ' << start row

  Application.ScreenUpdating = False
  Dim wsBase     As Worksheet
  Set wsBase = ThisWorkbook.Worksheets("Name of worksheet")
'[1] load xml
  Dim resp       As MSXML2.DOMDocument60
  Dim lista      As IXMLDOMNodeList
  Dim nodeAtual  As IXMLDOMNode
  Dim childNode  As IXMLDOMNode
  Set resp = New DOMDocument60
  resp.LoadXML (FunctionForGettingXMLfromWebService)
'[2] set nodelist to memory (XPath possibly could be refined :-)
  Set lista = resp.SelectNodes("//node1/node2")
'[3] dimension temporary variant (1-based) 2-dim array to hold contents
  ReDim tmp(1 To lista.Length, 1 To MAXCOLUMNS)
' [4] loop thru nodelist
  Dim c&, r&                                            ' declare row/column counters
  r = 1
  For Each nodeAtual In lista
    c = 1
    If nodeAtual.HasChildNodes Then
        For Each childNode In nodeAtual.ChildNodes
            tmp(r, c) = childNode.Text
            c = c + 1                                   ' increment column counter
        Next childNode
        r = r + 1                                       ' << move row counter into If condition to avoid empty lines
    End If
  Next nodeAtual
'[5] write array content to sheet
  wsBase.Range("A" & linEscrita).Resize(UBound(tmp), UBound(tmp, 2)) = tmp

  Application.ScreenUpdating = True
End Sub

redpajama_set_name: RedPajamaStackExchange


We've all been under the weather around here. Sometime last week a horrible bug descended on us and on my husband in particular. I have never seen him as sick as he was last week, hacking away and running a high fever. I myself nursed a body-wracking cough, which continues to persist to this day. Despite our best efforts, soon the germs reached our little ones and they too fell under this weird bug. Suffice to say we were all feeling more than a bit glum.
But we rallied, the less sick taking care of the more sick. Or, more accurately, mama taking care of everyone else. Although admittedly, I was (thankfully!) truly the "less sick". Little C did do her part though, my little Florence Nightingale, holding a cold compress to her papa's burning forehead, and checking his temperature periodically by laying her little hand on his cheek or leg declaring with authority, "It's still hot papa!" Our nanny (bless her!) made a pot of soothing arroz caldo, our local chicken and rice soup/porridge, which we gulped at gratefully.
Before all this hit, I had been talking about making pancakes with Little C. Pancakes are her favorite "special" breakfast food (her favorite all-around breakfast food being egg). I had been experimenting with several recipes in the hopes of not having to succumb to pancake mix…or at least only having to succumb to it in extreme emergency cases (like on those mornings that mama seriously has to sleep in because she was up all night watching Nigella Lawson videos). I chanced upon this recipe and quickly bookmarked it because: 1. We almost always have Greek yogurt in the house because I am hopelessly addicted to it, and 2. It looked criminally easy.
Well, when we were all down for the count Little C decides to ask for a pancake breakfast, complete with the goo-goo eyes she has been working hard to perfect, which, as evidenced by the pancakes pictured here, mama cannot resist.
– In a small bowl, stir the yogurt to make it smooth and creamy. Add the egg to the yogurt and stir to thoroughly combine.
– In a separate bowl, whisk together the flour and the baking soda.
– Add the wet ingredients into the dry and mix until combined, but do not over-mix. The batter will be alarmingly thick…that's ok.
– Heat a griddle pan or skillet on medium high heat. Add a pat of butter and when it has melted add a scoop of batter to the pan.
– When the pancake starts to bubble a bit on the surface, and the underside is golden brown, flip it and cook until both sides are golden. Remove to a pan and enjoy warm.
I am not exaggerating when I say you can drag your cough-ridden self out of bed at 6:30 am and prepare these pancakes half asleep. That is how easy they are to make. No special equipment, no fancy steps, no whipping of egg whites. I literally felt, through the flu-smog that enveloped my brain, like I was clumsily plopping things into bowls, and then suddenly…golden pancakes before me! They are magical I tell you.
I am also happy to report that these were a hit with the little ones! I made them medium sized and I was able to get six pancakes out of this…which was perfect: one for each kid and four for the photo you see here. By the time I finished taking pictures the little ones were ready for their second serving.
I made the pancakes before watching Everyday Belle's video so I didn't know she actually used peach yogurt and if using plain I should have added my own sweetener. But I've kept the recipe the same here because, despite the lack of sweetener, the pancakes came out absolutely fine. Just drizzle with maple syrup or honey (or any of your preferred pancake toppings) and that will be sweetening enough.
Another small random tip I thought I'd share: I always sieve my baking soda through a small fine mesh strainer. Yup, even if it's just a teaspoonful. Baking soda tends to lump when stored and there is nothing worse that biting into an un-dissolved lump of baking soda in your baked goods. Just typing this gives me the heebie jeebies.
Right now we are on the road to recovery (again, thankfully!). C is back to work. Little C is back to her old tricks. And Little H will be well enough to go back to school on Friday. Meanwhile, I am still nursing a lingering cough, but doing much better as well. Nothing a few mugs of warm lemon water, and a stack of pancakes, can't cure.
We love quick and easy recipes, especially ones that provide for a delicious breakfast. Greek yogurt is such a versatile ingredient, and perfect for use in so many dishes. Our favorites include Fage, Stonyfield, and Nancy's! What are yours?
I think that's breakfast sorted tomorrow – I have been working my rear end off for the last 2 weeks on a book update, and been neglecting the hubby. He'd TOTALLY jump for joy at the prospect of these! Thanks Joey, and all of you feel better soon!!
Hope everybody gets better. Will try your recipe because my recipe calls for egg whites beaten into soft peaks. Pancakes are great with bacon. Nice one from Bacchus in Commerce Center.
We tried making this recipe for breakfast and were very disappointed. My husband and I are good cooks so we don't believe our skill set was to blame. The pancakes were so dark before they were cooked throughly and they were so greasy in the mouth that we threw them out. The batter is like glue. Perhaps you left something out when you blogged it.
Sorry for the negative comment but this recipe failed.
Hi Elfie, thanks for leaving an honest comment. I'm sorry to hear you had a bad experience with this recipe…I did is as published on Everyday Belle and it turned out really good, just as it is picture here. I double checked my ingredients with the original recipe after reading your comments and everything checks. The batter is alarmingly thick, almost like a dough, but when I cooked it it turned out fine. Wish I could be more help!


A chemical peel is a technique used to improve and smooth the texture of the. Facial skin is mostly treated, and scarring can be improved. Chemical peels are intended to remove the outermost layers of the skin. To accomplish this task, the chosen peel solution induces a controlled injury to the skin. Resulting wound healing processes begin to regenerate new tissues.
The dead skin eventually peels off. The regenerated skin is usually smoother and less wrinkled than the old skin. Some types of chemical peels can be purchased and administered without a medical license, however people are advised to seek professional help from a Medical Trained Provider. Post treatment follow up is important to enhance the the healing of the skin.
Chemical peels can be used to address uneven skin tone, pigmented and textural acne scarring, acne, clogged pores, dull skin, fine lines and brown spots. Peel away skin blemishes and fine wrinkles.
Exfoliation is the skin from a chemical peel involves the removal of the oldest dead skin cells on the skin's outermost surface. Exfoliation is involved in the process of all facials, during microdermabrasion or chemical peels. Exfoliation can be achieved through mechanical or chemical means.


15th century, Angkor had been abandoned.
Likewise, the magnificence of Cambodia is now marred by unexploded landmines and dangerous roads, largely abandoned by tourists until quite recently. Khmer Rouge Communist guerrillas took control of Cambodia in 1975 with Prime Minister Pol Pot at the helm. Pol Pot manufactured a unique ideology based on elements of Maoist thought and Medieval quasi-mysticism, rooted in the history of the Angkor state. 'Year Zero' was established in 1975, under which Cambodia was to be converted into a pure Communist state centered on basic agricultural production. Currency was abolished, intellectuals purged, churches and temples destroyed and thousands of urban dwellers driven into the countryside for 're-education' and primitive labor. The outcome was a regime of horrific brutality, which was responsible for another of the 20th century's genocides – it is estimated that one third of the population died during the four years of Khmer Rouge rule.
But tourists are now beginning to return to Cambodia. Pol Pot died of natural causes in 1998 and his death seemed to symbolise a collective 'moving on'. Following the overthrow of the Khmers by the Vietnamese army, Phnom Penh, a ghost city under the Khmer Rouge, was re-populated by 1982. Although Cambodia drifted in and out of penury and semi-chaos during the 1980s, 1998 was also the year that Cambodia would finally comply with the international community and was reinstated as a member of ASEAN. Cambodians have also regained pride in their country, which is as beautiful as it ever was: for the traveller who seeks it, there is sprawling jungle, verdant fields, snaking rivers and golden beaches. Cambodia remains afflicted by poverty and authoritarian regimes that hide behind the veneer of democratic practice but for those who dare believe in it, there is also hope.
Cambodia shares borders in the north with Laos and Thailand, in the east with Vietnam and in the southwest with the Gulf of Thailand. The landscape comprises tropical rainforest and fertile cultivated land traversed by many rivers. In the northeast area rise highlands. The capital is located at the junction of the Mekong and Tonle Sap rivers. The latter flows from a large inland lake, also called Tonle Sap, situated in the center of the country. There are numerous offshore islands along the southwest coast.
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Add : 66 Hang Than St, Hanoi, Vietnam.
Add : 127 Ban Co road - Dict 3 -HCMC - VN.


text:
Thursday, 18 Jun 2009
Order of Business.
The Tánaiste
It is proposed to take No. a11, motion re ministerial rota for parliamentary questions, to be decided without debate; No. 24, the Broadcasting Bill 2008 [Seanad] — Report Stage (resumed) and Final Stages, which shall, if not previously concluded, be brought to a conclusion at 1.30 p.m. by one question, which shall be put from the Chair and which shall, in relation to amendments, include only those set down or accepted by the Minister for Communications, Energy and Natural Resources; No 1, the Companies (Amendment) Bill 2009 [Seanad] — Second Stage, which shall, if not previously concluded, be brought to a conclusion at 5.30 p.m; Question Time shall be taken at 5.30 p.m. for 75 minutes and in the event of a Private Notice Question being allowed, it shall be taken after 45 minutes; and the order shall not resume thereafter. It is proposed, notwithstanding anything in Standing Orders, that the Dáil shall sit later than 4.45 p.m. and business shall be interrupted not later than 6.45 p.m.
There are five proposals to put to the House. Is the proposal that the Dáil should sit later than 4.45 p.m. agreed? Agreed. Is the proposal for dealing with No. a11, motion re ministerial rota for parliamentary questions agreed?
Deputy Richard Bruton
On that issue, I should like to ask whether this is designed to give the Department of Education and Science more time to explain how after decades of saying literacy was its priority, we find that a third of children in disadvantaged schools have severe reading problems. Despite that, we have no literacy strategy in the Department and it does not even include literacy as an output to be achieved.
This is a simple procedural motion regarding a ministerial rota. The Deputy, I am sure, will raise that important issue in a different way.
I do not believe it would be possible. The Department of Education and Science is one of the Departments whose Question Time is being moved in this motion and I believe it is significant that this is happening on a day when this appalling report comes to our attention.
Is the proposal for dealing with the rota arrangements agreed? Agreed.
Is the proposal for dealing with No. 24, Broadcasting Bill 2008, Report and Final Stages, agreed?
Deputy Joan Burton
On that, we have had five guillotines this week. I want to lay down a marker as regards the seven proposed guillotines in next week's business, in particular in relation to the Financial Services (Miscellaneous Provisions) Bill, for which the Government has provided at the most three hours for the entire Bill, including Second Stage, Committee and Report Stages. This includes the right of the Minister for Finance to extend by ministerial order the bank guarantee and there will be no time to debate or examine this properly in the House. There is no Green Paper or White Paper, no financial information accompanying the Bill which——
——is going to hang like a millstone around the necks of future generations and give the Minister power over something like €60 billion of an increase in the national debt.
We are dealing with the Report and Final Stages of the Broadcasting Bill.
Is the Government now determined to effectively rule by guillotine, which is the same as ruling by decree? We are changing now into a Dáil which is ruled by decree and in this case, legislation which is of historic importance and which will double the country's national debt, will be subject to a guillotine and a debate in a total of three hours. That is not good enough in terms of ordinary taxpayers' interests.
We are concerned here with the Broadcasting Bill, Deputy. Is the Deputy opposing arrangements for dealing with the Broadcasting Bill?
(Interruptions).
We are opposing the principle of guillotine and rule by decree by Fianna Fáil.
Deputy Caoimhghín Ó Caoláin
With regard to the Broadcasting Bill, it has now become almost a matter of course that guillotines are part of the Order Paper proposal. While it has traditionally been the penchant of Governments to employ guillotines in the closing weeks before a summer or Christmas recess, it is now a matter of course. It must be objected to. It is not required. As we demonstrated yesterday, and made reference to the previous week's work, it cannot be accepted. The Government should accept that Bills should be debated and the sections of Bills, be it Report and Final Stages or Second Stage, as the next proposition on No. 1 will highlight again. It is unacceptable and should not be proceeded with.
Deputy Finian McGrath
It is all "Scary Carey's" fault. He is at it again.
While the House can tolerate the occasional use of a guillotine, we have reached a stage where even matters of major importance are not presented with the necessary sort of information to underpin them and there is not the sort of debate required. The guillotine is undermining proper legislation. In the United States today a White Paper on the regulation of the financial sector is being published. The US Legislature will have a long time to consider the options.
Here it is likely that the very meagre proposals on financial regulation will come in with no White Paper, will be debated in a curtailed debate and will be rammed through with a guillotine. We must draw a sharp contrast between the legislative approach of other responsible governments and the one this Government is inclined to pursue far too frequently.
To be in order, this relates to the Broadcasting Bill, which is to provide for new governance for the broadcasting industry. There have been extensive discussions on this and I cannot understand why there would be objections to it.
Deputy Aengus Ó Snodaigh
The objection is to the guillotine, not the Bill.
A considerable amount of legislation has yet to be finalised here and perhaps if we give ourselves more time, in the context of the Order of Business, we could get more work done and have more debate.
That is a very simplistic answer. The Tánaiste should take her job seriously.
Question, "That the proposal for dealing with No. 24 be agreed to", put and declared carried.
Is the proposal for dealing with No. 1, Second Stage of the Companies (Amendment) Bill, agreed to?
Deputy Emmet Stagg
It is not agreed to. This is the same issue again. The Executive is using the guillotine as a routine management tool to manage the legislative programme, thereby bypassing this legislative assembly whose constitutional role it is to examine and pass legislation. It cannot fulfil that role if legislation is routinely guillotined before it is debated fully. There was no need for the particular guillotine we have just dealt with because the Bill is practically completed. A guillotine was applied yesterday to a Bill and there was no need for it; the Bill would have been passed readily without the guillotine.
The guillotine is being used as a routine tool and we object to that. I ask the Government to apply the guillotine only if it is absolutely required, in an emergency. That is when it is intended to be used. There were periods of this Dáil during which we debated everything and anything because no legislation was coming forward from the Government. If it organised its programme properly it would not have this bottleneck at the end of every session.
The Companies (Amendment) Bill 2009 is a particularly important piece of legislation.
Of course it is; that is why it should be debated.
It is a short Bill. This is only Second Stage.
It is very far-reaching and important.
It is a technical Bill which the Minister of State will take on my behalf. There was a considerable amount of debate in the Seanad on this. Deputy Stagg's colleague, the Chairman of the relevant committee, is very anxious to facilitate an in-depth discussion on Committee Stage on the basis that we are all very anxious to have this legislation enacted before the House rises. Committee Stage, which will be facilitated by the Chairman and members of the committee, would be a more appropriate opportunity to tease out some of the issues. However we refer now to Second Stage.
Question, "That the proposal for dealing with No. 1 be agreed to," put and declared carried.
Are the arrangements for Question Time agreed to? Agreed.
In the past couple of days we saw a report that the details of 75,000 individual personal bank accounts were stolen in a computer held by Bord Gáis. These were not encrypted and it will be two weeks before the company will have contacted the individuals concerned. There are legal weaknesses here. Will the Government consider enacting Deputy Coveney's Private Members' legislation published last year on this subject?
Is there promised legislation in this area?
Deputy Simon Coveney
There is.
Is there a Government promise of legislation?
That remains to be seen. It has been promised it in the past.
Deputy Paul Kehoe
The Government has made many promises, but it might not have legislation.
No Government legislation is promised. It is up to the Deputy to order his Private Members' time.
The Government has indicated it is supportive of legislative measures in this area. In light of the risk to 75,000 people that has been exposed, the Tánaiste should be examining this seriously. Fine Gael anticipated the problem well over a year ago.
We cannot deal with the matter now.
Perhaps Deputy Coveney will return to this issue. Is the Government preparing legislation on a bank levy to accompany the National Asset Management Agency Bill? We are informed that any deficit will be made up by way of a bank levy. Is that legislation being prepared?
No, the legislation is on NAMA.
I wish the Taoiseach well today in his negotiations in Brussels aimed at obtaining clarification on a number of issues that were of concern to Irish people during the Lisbon treaty referendum. What is the situation on the guarantees of workers' rights, which was one of the matters of particular concern? This is a matter at EU level but it is also a matter for domestic legislation. In the review of Towards 2016, which was concluded last September, a number of initiatives were agreed between the Government parties and trade unions on workers' rights legislation.
Two Bills have been on the list of promised legislation for some time, the employment agency regulation Bill and the industrial relations (amendment) Bill. The employment agency regulation Bill was on the list to be published before summer 2008. What has happened to it? When will we see these pieces of legislation? Does the Tánaiste accept that in the context of another Lisbon treaty referendum they would be very helpful to voters, who are deeply concerned about the implications for workers' rights in the context of EU legislation and frameworks?
It is over a year since agreement was reached on the agency workers directives. There were marches to this House and the Government parties stepped out to reassure workers and their representatives that this would be legislated for, and we have heard nothing about it. I have another issue to raise.
Three pieces of legislation were agreed with the social partners, which my colleague, the Minister of State, brought forward. One is the Employment Law Compliance Bill, which has been read a Second Time and will be facilitated on Committee Stage by the committee.
The industrial relations Bill is with the social partners for consideration and once that has been completed, which I hope will be next week, we will be in a position to publish the Bill. There are two agency Bills. The agency regulation Bill is ready for publication. As has been indicated and agreed when the Minister of State, Deputy Kelleher, negotiated the agency workers directive on behalf of the Government, it will take at least two years for its enactment on the basis that very careful consideration, discussion and deliberation need to take place with the unions on the matter.
The Tánaiste may be aware of the unprecedented spate of murders on the fringes of the west and north sides of Dublin resulting in a significant loss of human life. People are being gunned down in ways that are reminiscent of Al Capone's days in Chicago. The public living in these areas, including in my constituency feel helpless.
Does the Deputy have a relevant question on legislation?
Will the Government provide time for a debate on the spree of lawlessness, killing and crime——
I am sure that is a matter that can be discussed with the Whips.
—— that is blighting entire areas of our major cities and towns, linked to the drugs trade?
That would be an appropriate matter for the Adjournment or the Whips could discuss having a debate.
Deputy Pádraic McCormack
Could we have some zero tolerance, a Leas Cheann Comhairle?
Deputy Ciarán Lynch
When will the criminal justice (money laundering) Bill to give effect to an EU directive on the matter come before the House? The Housing (Miscellaneous Provisions) Bill passed Committee Stage last week and Report and Final Stages will be taken next week. I am informed that more than 30 amendments have been tabled and four hours' debating time has been allocated to it next week. Given that the Bill has been before the House since 2006 and given that on Committee Stage last week entire sections were inserted into the Bill——
We can deal with the arrangements when it is properly before the House.
This is a significant Bill. The major section introduced last week relates to the tenant purchase of local authority flats.
We can deal with the arrangements when it is properly before the House next week.
Opposition spokespersons are entitled to be informed and to be given time to digest this type of information before a Bill, which has been around the House since 2006, gets moved into almost a crisis-driven situation. Given the Bill's complexities, the Tánaiste should consider two matters. Can the Report Stage of the Bill be postponed by a week so that Opposition spokespersons can consider the Government amendments and submit their own? Can more time be given to Report Stage so that the complexities of the Bill can be discussed at length?
The Bill is being guillotined.
Before proper proposals are put to the House on that matter, I am sure the Tánaiste will give the Deputy's intervention consideration.
The money laundering Bill will be taken this session. The Minister has made a memorandum available to the Opposition spokespersons on the complexity of this issue. I am sure we can facilitate some discussions if there are particular issues between the Minister and the officials.
Deputy Seymour Crawford
I was involved in a meeting with Co-operation Ireland regarding the withdrawal of the €3 million capital funding from the Middletown Centre for Autism, which was part of the Good Friday Agreement. When can we have discussion on North-South co-operation in the Dáil? Could we at least have a discussion on the amendment Bill to the George Mitchell Scholarship Fund Act 1998 to allow us to discuss that?
When will that legislation be taken and is a debate promised?
There is no date for the legislation. The debate is a matter for the Whips.
Given that patients are finding great difficulty in getting service on the front line in hospitals — yesterday I highlighted the situation——
The Deputy should ask a question that is relevant.
It is very relevant to sick people who cannot even get their breakfast.
It might be appropriate for the Adjournment or for a motion proper.
When will the nurses and midwives Bill be published?
Early next year.
Deputy Pat Breen
This relates to a forthcoming Bill. A very serious situation is emerging in the aviation sector given the news over the weekend that services are being withdrawn from Shannon Airport and Ryanair's announcement yesterday of a reduction of 36 flights a week and one fewer aircraft. I am pleased to see the Ministers for Transport, and Arts, Sport and Tourism present. Will the Government call Michael O'Leary's bluff and abandon the €10 travel tax——
Does the Deputy have a question that is relevant?
——which is having considerable effect on tourism——
The Deputy knows this is not the appropriate way of raising that important issue.
——in the entire country and particularly in the mid-west area?
I call Deputy Ó Caoláin.
It is a very serious situation.
I am sure it is and that is why it should be dealt with in a serious fashion by way of a substantive motion or perhaps an Adjournment matter.
Perhaps the Tánaiste would allow some time to discuss the matter, which is having an adverse effect on tourism.
It relates to forthcoming legislation.
The Deputy knows there are proper ways of raising it. He has had a good run. It is out of order.
It relates to the air navigation and transport Bill, which is due before the House shortly and the €10 tax.
When is the air navigation Bill due?
Is it the Aviation (Preclearance) Bill?
Yes, it would be part of that. When is that Bill coming before the House?
The House will come to order for Deputy Ó Caoláin.
Can the Tánaiste confirm that as has been reported, the Minister for the Environment, Heritage and Local Government, Deputy Gormley, intends introducing an amendment of the Constitution Bill to facilitate a second referendum on the Lisbon treaty and that Bill is to be addressed in the House in the coming week? Will it at least be published next week? What is the timeframe for the legislation? Does the Government intend to press it through before the summer recess? Will the Bill facilitate putting to the electorate the exact same Lisbon treaty on which we have already passed judgment? I understand the text will not be altered in any way either by so-called legal guarantees or by declarations.
We have debated the report of the Commission to Inquire into Child Abuse and the House passed a unanimous motion, which was very welcome. Everyone is to be commended on the approach to that matter. On the publication of the report of the Dublin Archdiocese Commission of Investigation into clerical sexual abuse of children in the Dublin archdiocese, will that report also be debated on the Floor of the House?
There are two matters for the Tánaiste.
Given the precedent already established——
Go raibh maith agat, a Theachta.
—— and the import of the subject, will the Government facilitate address of that?
I am trying to facilitate a number of Deputies.
I have one last item to ask about, with your patience a Leas Cheann Comhairle. However, on the matter of the report into clerical sexual abuse in the Dublin archdiocese, will the Government facilitate address of that report here?
The Deputy has asked the question.
My final question is about an internal HSE report leaked last week that exposed the totally inadequate system of child protection in the State. It refers specifically to the infrastructure in terms of social workers, emergency accommodation and out of hours services——
That is not in order on the Order of Business.
——that are clearly failing the vulnerable and endangered children in Irish society.
Deputy, please.
Will the Government introduce the child care Bill in order that the House can address in a real and earnest way the continuing threat to children in Irish society today and tomorrow?
The Deputy has asked about two Bills and a debate on the report into clerical sexual abuse in the Dublin archdiocese when it is received.
The Government met and approved the wording of the Twenty-ninth Amendment of the Constitution Bill. As the Deputy knows we would like to get the matter initiated within the House as quickly as possible. Of course that is on the basis of a satisfactory outcome of the deliberations today and tomorrow. It is the intention that this legislation would go before the House and be complete before we rise.
The Government has not received the report into clerical sexual abuse in the Dublin archdiocese, but it will consider it and make a decision afterwards. The child care Bill will be taken in this session.
Deputy Denis Naughten
Until the end of last year the Government planned to publish the education (Ireland) Bill to establish a body to register the provision of education services including the teaching of English to foreign students. The Bill was to be published in 2009 but the Minister for Education and Science decided to shelve it. The Europol report last summer stated that educational establishments have gained importance in facilitating illegal immigration into the country and Members heard yesterday that language schools in Dublin are now breeding grounds for prostitution. I understand the Minister has an alternative Bill, No. 60, the Qualifications, (Education and Training) Bill. When will that be published? In light of the comments yesterday will it be expedited to deal with what is now major exploitation using a loophole in our immigration law affecting the language schools in Dublin?
The Qualifications (Education and Training) Bill is due next year but it does not deal with that issue.
Why will it not deal with that issue?
If I remember correctly it is not to deal with that issue. The Qualifications (Education and Training) Bill will be dealt with next year.
There is no provision to deal with that issue. That is appalling.
The Tánaiste is in sparkling form today.
Deputy Jimmy Deenihan
Will the Tánaiste confirm a report inThe Irish Times yesterday that the Government has drafted legislation to be enacted in advance of the next Lisbon referendum ensuring that participation in European Defence Agency operations is subject to a Cabinet decision and Dáil approval? When will this legislation come before the Dáil?
The Tánaiste has already answered that question.
She has not. This is a different issue.
This is about the European Defence Agency.
This is totally different, it concerns the European Defence Agency. A Cabinet report was obviously leaked toThe Irish Times stating that in future our participation in the European Defence Agency would have to be subject to Dáil and Government approval. Has that been drafted? Is that accurate?
That legislation is being prepared.
When will it be introduced?
I cannot say when the Parliamentary Draftsman will complete it.
It is a major issue for the referendum. Will it be introduced before the referendum? Will it be discussed in the House?
The legislation is being drafted. The intention is to finalise the Lisbon treaty legislation before the House rises. The other Bill is being drafted but I cannot give a commitment on when it will be available.
Will it be before or after the referendum?
What is written in the newspaper is not always right.
Who leaked the report?
I am indicating what is happening to that legislation.
It is very important legislation.
It is a terrible reflection onThe Irish Times.
The legislation is being drafted and we will bring it before the House as soon as possible.
Deputy Bernard Allen
I thought Deputy Deenihan was raising an issue about the Kerry defence last Saturday.
Last February, the Committee on Public Accounts, PAC, issued an interim report on the irregularities, to put it mildly, in FÁS. We issued several key recommendations and sent them to the Tánaiste but to my knowledge no action has been taken on any of those recommendations. According to some media outlet, the Tánaiste is preparing legislation on the reconstitution of the FÁS board and the reduction of numbers on the board. When can we expect to see that legislation and when will there be action on some of the other recommendations? The public, the taxpayers, were aghast at some of the disclosures and at the fact that no action has been taken yet.
Is legislation promised in that area?
How many will be sacked?
I want to raise another issue. PAC also wrote to the Minister for Finance to ask that the roles of the Comptroller and Auditor General and PAC be taken into consideration in the drafting of the National Assets Management Agency, NAMA, legislation to protect the taxpayers' interest in this potentially massive cost to them.
Can I have an answer to both questions, and in respect of the second will the Government accept the recommendations of PAC that it and the Comptroller and Auditor General would have a key role in monitoring the activities of NAMA?
I indicated to the Deputy privately last week that I intend to introduce a new legislative framework for FÁS. Many of the PAC recommendations have been enacted and dealt with. We have to finalise consideration of further reports.
The second question is a matter for the Minister for Finance to consider. I will indicate to him that the Deputy raised it today. I cannot give a commitment on that issue.
The Tánaiste said that many of the recommendations have been enacted. Which ones have been enacted?
That is a matter for a good parliamentary question.
I wish to return to a question Deputy Bruton asked.
I hope it is a different question.
It is. It arises from the sloppy treatment of sensitive personal data by Bord Gáis and the Health Service Executive, HSE, in the past two days which has resulted in bank account details of 75,000 people and the personal details of nine families being compromised.
Last October Fine Gael published a data protection disclosure Bill.
The Deputy must ask a question.
I am coming to it. The Leas-Cheann Comhairle has allowed other people to have a preamble.
In response to that Bill the Minister for Justice, Equality and Law Reform set up a data protection review group and promised it would result in a legislative response from Government to cover data disclosure——
The Tánaiste to reply on legislation in relation to——
The Leas-Cheann Comhairle should let me finish. It deals with data protection, disclosure and, potentially, encryption obligations, the absence of which has caused the latest data protection breach. A legislative response was promised and I want to know where it is now.
The promised legislation will be finalised fairly soon. It is expected in this session, the Communications (Retention of Data) Bill is based on an EU directive.
That is a separate issue.
On the other issue, the legislation has not been listed yet but I will get a further update from the Minister.
The Deputy will be communicated with then.
Deputy Thomas P. Broughan
When will we see the Road Traffic Bill? Is there any chance we will have it before the end of the session?
Is the Tánaiste aware that Quinn Direct is apparently discriminating against citizens in certain areas of this city and of Cork, Limerick and other cities in respect of car insurance?
The Deputy knows that is not in order on the Order of Business.
Would the Tánaiste consider introducing legislation?
This is not Question Time. The only thing the Deputy may ask about is promised legislation.
Will the Tánaiste promise legislation in respect of this outrageous discrimination against people in major urban areas?
The Deputy may ask only about legislation previously announced and promised by Government.
I have another question, in support of Deputy Breen. The Tánaiste is the Minister for Enterprise, Trade and Employment with by far the worst record in respect of jobs in the history of the State. Given that we are now losing another 600 jobs in aviation and related areas, would the Tánaiste consider asking the Minister for Finance to urgently——
That is not in order, as I indicated to Deputy Breen and I must be consistent.
I want to talk about the forthcoming emergency financial statement in late September or early October to remove the travel tax which is devastating jobs.
Deputy Noel Dempsey
The Deputy is speaking for Ryanair.
That is not in order. The Tánaiste may reply to the first question.
I want to know if she can do something about this problem.
The Deputy is supporting Ryanair. Michael O'Leary would be proud of him.
The Minister hopes to bring forward the Road Traffic Bill in this session if possible.
Deputy Kathleen Lynch
He is the only one who stuck with Shannon. I do not know why the Minister is talking about him.
Deputy Noel J. Coonan
An industrial manufacturing company in Tipperary, Eltex, sought the assistance of the Tánaiste's Department, through the IDA, to protect jobs.
That is clearly not in order.
Deputy Noel Coonan
Has the Tánaiste any plans to introduce legislation to protect jobs in the manufacturing industry before she goes on holidays?
That would be appropriate to a parliamentary question.
The IDA's response to the manufacturing company was that the stabilisation fund was being reduced by the Department, was over-restrictive and was unworkable. Has the Tánaiste plans to introduce emergency legislation before the recess, so that small manufacturing companies in this country will not be closed when we return?
I call Deputy Kathleen Lynch.
It is not good enough for the IDA to state that——
The Deputy has had a good innings.
——the fund was unworkable and too restrictive. Has the Tánaiste plans——
Deputy Coonan, please.
I am asking whether she plans for this. I am sure you do not want to see the country closing down.
I ask you to resume your seat.
Three Deputies before have been out of order, so I am terribly worried about myself at this stage.
I have every confidence in you, Deputy.
On the basis that they were all men, I am sure there will be a bit of latitude. I have two issues to raise.
Both in order, no doubt.
I hope so. Are the computers in the Houses of the Oireachtas encrypted?
That would be a parliamentary question.
Okay. Under the industrial relations (protection of employment) (amendment) Bill, which is promised legislation, there are 100 jobs in Cork that have not been lost yet in Corden Pharmachem. With intervention, these jobs could be saved. Downstream, there are 600 jobs involved, so I ask the Tánaiste in her capacity as Minister for Enterprise, Trade and Employment to intervene as quickly as possible.
The Deputy knows that is not in order.
The question about the legislation is in order.
The matter is with the social partners. We hope to have their views back quickly and to publish it this session.
Deputy Tom Sheahan
She will not go to Cork to save jobs, but she went to America to try to save Dell.
Deputy Jan O'Sullivan
There is a Bill promised on entitlement to health care. I raise this as some of the sickest children in the country have had their treatment delayed because of a 3% cutback in the children's hospital in Crumlin. Many of us have raised this in the past week to deaf ears from the Government. I know that entitlement does not automatically bring treatment, but will the Government not do something to provide the funding for these very sick children?
That Bill will not be published until next year.
Will the Government do something soon?
Requests to move Adjournment of Dáil under Standing Order 32.
Ministerial Rota for Parliamentary Questions : Motion.

redpajama_set_name: RedPajamaCommonCrawl


text:
Written Confirmations: Time is running out - new EU Progress Report
As of 2 January 2013, all APIs imported in the EU have to be manufactured in accordance with GMP which is at least equivalent with EU-GMP. From 2 July 2013 on, APIs from Third Countries will be allowed for importation into the European Union only if one of the following options is fulfilled:
Option 1: the consignment is accompanied by a 'Written Confirmation' by the authority of the third country, certifying that the plant manufacturing active substances operates in compliance with EU-GMP, or with equivalent rules; or
Option 2: the third country has been listed by the European Commission as a country with an equivalent system of supervision and inspection as in the EU; or
Option 3: on exception and where necessary to ensure the availability of medicinal products, the need for the written confirmation can be waived by a Member State if a Member State has inspected the specific plant.
However at the moment there is still a lot of uncertainty in Europe what the future will bring. There are currently severe doubts that all APIs imported into the EU will comply with these requirements.
Here is an overview of the current status:
Option 1: Challenges with India and China
The EU Commission (COM) has published a progress report giving an overview on 20 major countries and their status of preparation for adhering to the new rules. According to this report, India and China seem to have to do more to comply with the EU requirements. Both countries currently have almost 1.000 API manufacturing sites supplying to the EU. For both, the EU confirms "good progress" but for both more work is needed - "in particular by industry stakeholders". It's similar with countries like Russia, Argentina and Malaysia.
Another thing to consider is the fact that the Chinese State Food and Drug Administration SFDA has already informed the Commission that it would not issue 'written confirmations' for manufacturing sites which are not under SFDA's supervision. These are about 30 sites. EMA is already co-ordinating the inspections of these sites (option 3).
South-Korea, South-Africa and the Ukraine have already started issuing written confirmations. According to the statement, the "situation is under control". Turkey, Mexico, Brazil, Taiwan and Canada will issue written confirmations.
Option 2: Only one country listed yet
To get on the list the EU Commission evaluates the equivalence of control standards in the respective country. So far, only Switzerland is on this list. The U.S. FDA has also filed a request for being evaluated. The authority does not mention in a statement whether they will also issue 'written confirmations' in the case they will not be listed until 2 July. However, it is very likely that the FDA's request will be answered positively. An on-site audit visit by the EU is scheduled for mid-May 2013. Japan also wants to get on the list. An on-site audit visit is planned for mid-April 2013.
The equivalence assessment for Australia is almost concluded and they seem to be the next country on the list.
Brazil has applied for listing. However, the respective documentation has not been received yet. As soon as COM receives the information, COM will start the 'equivalence assessment'.
Israel and also Singapore's request have been rejected. Both countries will start issuing written confirmations.
Option 3: Who will do all these inspections?
In December 2012, the HMA (Head of Medicines Agencies) expressed their concerns about the implementation of the new requirements in a statement which can be found on the HMA webpage. HMA representatives say: "This situation might endanger public health within the EU due to possible shortages of medicinal products for the European market and consequently become a serious public health risk." According to the HMA information 300 sites might have to be inspected. The HMA's protocol shows that the resources available in the EU countries would enable the performance of only 25 inspections by the end of June 2013.
It is impossible to predict whether the current challenges will lead to supply shortage from 2 July on. As a precautionary measure, many pharmaceutical manufacturers are currently increasing their stocks of APIs from non-EU countries.
Conference Recommendations
Related GMP News
Brexit Agreement - what now?
Avoiding Non-Compliance in Packaging Areas
EMA Publishes New Q&A Guide on Brexit
Brexit: Phased Process for implementing Medicines Regulation in Northern Ireland
Does GMP apply to Development?
WHO Draft Document on GMP for IMPs


CHICAGO (Reuters) - Come January, most U.S. retirees will get some welcome news when they check their bank accounts - the biggest inflation adjustment to Social Security benefits in eight years.
The federal government on Thursday announced a 2.8 percent Social Security cost-of-living adjustment (COLA) for 2019; seniors will see the raise in their January benefit payment. That is the largest increase since 2012, when the COLA was 3.6 percent.
COLAs are determined by an automatic formula tied to the Consumer Price Index for Urban Wage Earners and Clerical Workers (CPI-W). With inflation running very flat since the recession of 2009-2010, COLAs have been anemic in some recent years - there was no COLA at all in 2015, and it was three-tenths of 1 percent in 2016. The COLA awarded for 2018 was a more generous 2 percent.
But for many retirees, that increase was blunted by the impact of the little-understood hold-harmless rule, which prohibits the dollar amount of Part B premium increases from exceeding the dollar amount of the COLA for roughly 70 percent of beneficiaries. The rule ensures that net Social Security benefits do not fall when the dollar amount of the Part B increase is greater than the dollar amount of the COLA increase. Last year, however, due to some quirky dynamics of how the COLA and Part B premium interact, the COLA was wiped out for many seniors. [nL1N1NQ0ZA] (reut.rs/2iuZM3J).
This year should be different for most beneficiaries. For example, assuming the standard Part B premium winds up at $135.50 in 2019, the 2.8 percent COLA will translate to a $40.50 monthly net raise (after the Medicare premium adjustment) for a beneficiary receiving $1,500 this year.
The math will be less straightforward for beneficiaries who currently pay less than the standard Part B premium - again due to the hold-harmless rule, which kept their premiums down in recent years. That group includes roughly 25 percent of all Medicare beneficiaries, according to research by the Senior Citizens League - in most cases, lower-income seniors. Most of these retirees will see their premium jump up to the standard 2019 level, consuming a larger portion of the COLA.
"That's where we will see the biggest bite taken out of the COLA next year," said Mary Johnson, a Social Security policy analyst for the League.
Despite the overall good COLA news, rising healthcare costs continue to pose a long-term threat to net Social Security benefits. For example, the Medicare trustees project that the Part B premium will start rising at a faster pace beginning in 2020, rising anywhere from 5.6 to 10 percent annually through 2026.
Overall healthcare inflation is projected to rise 4.22 percent over the coming 20 years, according to a report released last week by research firm HealthView Services. That is down from HealthView's 2017 projection of 5.47 percent, due mainly to moderation of projected prescription drug costs. HealthView cited ongoing shifts by consumers from brand names to less expensive generics, and the earlier-than-expected closing of the doughnut-hole gap in plan coverage.
Still, HealthView calculates that healthcare expenses will consume about half (48 percent) of lifetime Social Security benefits for a healthy 66-year-old couple retiring this year. And the squeeze will get worse in the years ahead due to healthcare cost inflation. HealthView calculates that a 55-year-old healthy couple will need 57 percent of their benefits to cover future retirement healthcare costs, and a 45-year-old couple will spend 63 percent.
"That's very worrisome, since so many seniors rely on Social Security for most of their income," said Ron Mastrogiovanni, HealthView's chief executive officer.
One way that seniors can control cost is to take advantage of the annual fall Medicare enrollment season, which begins on Oct. 15 and runs through Dec. 7. [nL2N1WJ111] (reut.rs/2yj24dB) This is the time of year when you can make changes to your basic coverage and prescription drug insurance. Prescription drug premiums, in particular, are volatile and can jump dramatically from year to year.


text:
package example.service;

import example.repo.Customer1337Repository;

import org.springframework.stereotype.Service;

@Service
public class Customer1337Service {
	public Customer1337Service(Customer1337Repository repo) {}
}

redpajama_set_name: RedPajamaGithub


This is not an official copy of the Philippine Electrical Code. This is a Design Guide according to the provisions of the Philippine Electrical Code. It also provides interpretation and annotations of the clauses of the Philippine Electrical Code. Any information, interpretation and/or annotation presented herein are based on personal opinion of the authors and does not represent the official interpretation of the Philippine Electrical Code.
This page was last modified on 14 March 2016, at 17:56.
This page has been accessed 100,140 times.


text:
Performance Art in the 80's? A drift towards music?
Cláudia Madeira
Instituto de Filosofia da NOVA (IFILNOVA)
The creation of an art market in Portugal in the 1980s had an impact on the field of Portuguese performance art. In the 1960s and 1970s, some of the precursors of performance art in Portugal no longer followed this option and devoted themselves to other professional fields or a more object-based art, such as painting, sculpture etc. – and even talked of a kind of "death of performance art". Yet, at the same time, new agents and groups emerged that still had a connection with performance art, including Homeostética and Happy the Faith. Some of them moved in the direction of alternative music with a strong satirical base. This article will seek to examine the specificities and the intergenerational points of contact, continuity and difference among these groups in relation not only to their predecessors but also to the positions held by some of their members in the field of contemporary art and to the "social places" that performance art occupies today in the national and international scene.
Book Proceedings International Conference, Keep it Simple, Make it Fast!
Paula Guerra, Tânia Moreira
Performance art and alternative music
Fingerprint Dive into the research topics of 'Performance Art in the 80's? A drift towards music?'. Together they form a unique fingerprint.
Performance Art Arts & Humanities
Music Arts & Humanities
Portugal Arts & Humanities
Art Arts & Humanities
Art Market Arts & Humanities
Faith Arts & Humanities
Precursor Arts & Humanities
Specificity Arts & Humanities
Madeira, C. (2016). Performance Art in the 80's? A drift towards music? In P. Guerra, & T. Moreira (Eds.), Book Proceedings International Conference, Keep it Simple, Make it Fast! (Vol. 2, pp. 109-115). Universidade do Porto.
Madeira, Cláudia. / Performance Art in the 80's? A drift towards music?. Book Proceedings International Conference, Keep it Simple, Make it Fast!. editor / Paula Guerra ; Tânia Moreira. Vol. 2 Porto : Universidade do Porto, 2016. pp. 109-115
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booktitle = "Book Proceedings International Conference, Keep it Simple, Make it Fast!",
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Madeira, C 2016, Performance Art in the 80's? A drift towards music? in P Guerra & T Moreira (eds), Book Proceedings International Conference, Keep it Simple, Make it Fast!. vol. 2, Universidade do Porto, Porto, pp. 109-115.
Performance Art in the 80's? A drift towards music? / Madeira, Cláudia.
Book Proceedings International Conference, Keep it Simple, Make it Fast!. ed. / Paula Guerra; Tânia Moreira. Vol. 2 Porto : Universidade do Porto, 2016. p. 109-115.
T1 - Performance Art in the 80's?
T2 - A drift towards music?
AU - Madeira, Cláudia
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N2 - The creation of an art market in Portugal in the 1980s had an impact on the field of Portuguese performance art. In the 1960s and 1970s, some of the precursors of performance art in Portugal no longer followed this option and devoted themselves to other professional fields or a more object-based art, such as painting, sculpture etc. – and even talked of a kind of "death of performance art". Yet, at the same time, new agents and groups emerged that still had a connection with performance art, including Homeostética and Happy the Faith. Some of them moved in the direction of alternative music with a strong satirical base. This article will seek to examine the specificities and the intergenerational points of contact, continuity and difference among these groups in relation not only to their predecessors but also to the positions held by some of their members in the field of contemporary art and to the "social places" that performance art occupies today in the national and international scene.
AB - The creation of an art market in Portugal in the 1980s had an impact on the field of Portuguese performance art. In the 1960s and 1970s, some of the precursors of performance art in Portugal no longer followed this option and devoted themselves to other professional fields or a more object-based art, such as painting, sculpture etc. – and even talked of a kind of "death of performance art". Yet, at the same time, new agents and groups emerged that still had a connection with performance art, including Homeostética and Happy the Faith. Some of them moved in the direction of alternative music with a strong satirical base. This article will seek to examine the specificities and the intergenerational points of contact, continuity and difference among these groups in relation not only to their predecessors but also to the positions held by some of their members in the field of contemporary art and to the "social places" that performance art occupies today in the national and international scene.
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Madeira C. Performance Art in the 80's? A drift towards music? In Guerra P, Moreira T, editors, Book Proceedings International Conference, Keep it Simple, Make it Fast!. Vol. 2. Porto: Universidade do Porto. 2016. p. 109-115


text:
Claudio Gómez Castro, (born 6 April 1980 in San Felipe) is a Chilean lawyer and university professor serving, , as a member of the Constitutional Convention, the body in charge of drafting a proposal for a new Chilean constitution.

Background
Gómez earned his law degree from Central University of Chile in 2007, writing his thesis on changes in Chilean labor law. He also holds a master's degree in Democratic Society, State and Law from the University of the Basque Country in Spain. In 2010, he joined the law faculty at Universidad del Aconcagua in San Felipe as professor of political and municipal law.

Public career
Since 2016, Gómez has worked with environmental movements in the Aconcagua region, as well as efforts to for the provinces of Los Andes, San Felipe de Aconcagua, and Petorca to work together.

In 2021, he stood for election in the 6th District of Valparaíso for the Apruebo Dignidad party, earning 8,171 votes and a seat in the convention. During the first three months of the convention, he was a member of the Regulation Commission and is currently part of the State Form and Decentralization Commission.

References

External links

 https://www.instagram.com/claudioconstituyente/

Living people
1980 births
21st-century Chilean politicians
Central University of Chile alumni
University of the Basque Country alumni
Members of the Chilean Constitutional Convention

redpajama_set_name: RedPajamaWikipedia